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| | <!-- Head End --> | | <!-- Head End --> |
| − | <body><div id="MathJax_Message" style="display: none;"></div> | + | <!-- Content Begin --> |
| − | <h1 id="introduction">Introduction</h1>
| + | <table width="960" border=0 cellspacing=0 cellpadding=10> |
| − | <p>Our pathogen detection approach relies on Cas13a digesting RNA. A change in fluorescence is the most convenient way to take trace of such enzyme kinetics. This is impractical for in-field applications because commercial fluorescence detectors are expensive and inconveniently large. A straight forward approach for making our pathogen detection fit for in-field applications is a cheap and handy fluorescence detector. Other iGEM teams already constructed fluorescence detectors. We therefore rated detectors of other teams in a cost vs sensitivity diagram. Nevertheless non of them had a high enough sensitivity or the ability to measured fluorescence quantitatively. We therefore constructed a detector matching our requirements. Our detector is paper-based and can detect fluorescein concentration down to 200 nM. The detector is able measure to automatically measure fluorescence in units of equivalent fluorescein concentrations. The detector is very compact and costs less than 15 EUR. We were able to measure a time trace of Cas13a digesting RNaseAlert with our detector. For comparison we also measured a positive control containing RNase A and a negative control containing only RNaseAlert. The data are displayed in image [img:k3]. The time traces show an enzymatic reaction taking place on filter paper. This proves that our detector is sensitive enough and meets our requirements. However the detector is not limited to the detection of our specific application. It can be used for the detection of any fluorescence signal in biological or chemical systems. We think that our detector can benefit other iGEM teams and also research groups that want to make a fluorescence based detection fit for in-field applications.</p>
| + | <tr> |
| − | <div class="figure"> | + | <td width=160></td> |
| − | <img src="./testSave_files/timetrace.png" alt="Time trace of Cas13a digesting RNaseAlert. Positive control containing RNaseA and RNaseAlert. Negative control containing only RNaseAlert. Shaded areas represent uncertainties for data points with the corresponding color">
| + | <td width=160></td> |
| − | <p class="caption">Time trace of Cas13a digesting RNaseAlert. Positive control containing RNaseA and RNaseAlert. Negative control containing only RNaseAlert. Shaded areas represent uncertainties for data points with the corresponding color<span data-label="img:k3"></span></p>
| + | <td width=160></td> |
| − | </div>
| + | <td width=160></td> |
| − | <h1 id="principle-design">Principle design</h1>
| + | <td width=160></td> |
| − | <p>Light from a blue LED is filtered by the blue filter foil. The light from the LED excites fluorophores on a filter paper. The excitation light is blocked by an orange filter foil. The fluorescence light from the excited fluoroscopes passes through the orange filter foil and illuminates and light dependent resistor (LDR). The LDR changes its resistance corresponding to the degree of illumination with fluorescence light. An Arduino nano measures the resistance via a voltage divider and calculates the fluorophore concentration.</p>
| + | <td width=160></td> |
| − | <h1 id="components">Components</h1>
| + | |
| − | <h2 id="micro-controller">Micro controller</h2>
| + | |
| − | <p>We used an Arduino nano for automatized data collection. This micro controller has analogue pins that can measure voltages from 0 to 5 V. The output is an integer from 0 to 1023 corresponding to the voltage at the analogue pin. The micro controller is connected via an USB port with a computer where the data can be processed further.</p>
| + | |
| − | <h2 id="light-dependent-resistor-ldr">Light dependent resistor (LDR)</h2>
| + | |
| − | <p>For the detection of fluorescence light we used an light depending resistor (LDR). An LDR is a resistor which decreases its resistance <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-1-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-1" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2" class="mjx-mrow"><span id="MJXc-Node-3" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-4" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-5" class="mjx-texatom" style=""><span id="MJXc-Node-6" class="mjx-mrow"><span id="MJXc-Node-7" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">L</span></span><span id="MJXc-Node-8" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">D</span></span><span id="MJXc-Node-9" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-1">R_{LDR}</script></span> with increasing light intensity <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-2-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-10" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-11" class="mjx-mrow"><span id="MJXc-Node-12" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em; padding-right: 0.064em;">I</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi></math></span></span><script type="math/tex" id="MathJax-Element-2">I</script></span>. The dependence of the resistance <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-3-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-13" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-14" class="mjx-mrow"><span id="MJXc-Node-15" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-16" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-17" class="mjx-texatom" style=""><span id="MJXc-Node-18" class="mjx-mrow"><span id="MJXc-Node-19" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">L</span></span><span id="MJXc-Node-20" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">D</span></span><span id="MJXc-Node-21" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-3">R_{LDR}</script></span> on the light intensity <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-4-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-22" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-23" class="mjx-mrow"><span id="MJXc-Node-24" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em; padding-right: 0.064em;">I</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi></math></span></span><script type="math/tex" id="MathJax-Element-4">I</script></span> is <span class="math display"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-5-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub><mo>&#x221D;</mo><msup><mi>I</mi><mrow class="MJX-TeXAtom-ORD"><mo>&#x2212;</mo><mi>&#x03B3;</mi></mrow></msup><mo>,</mo></math>" role="presentation" style="font-size: 101%; text-align: center; position: relative;"><span id="MJXc-Node-25" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-26" class="mjx-mrow"><span id="MJXc-Node-27" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-28" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-29" class="mjx-texatom" style=""><span id="MJXc-Node-30" class="mjx-mrow"><span id="MJXc-Node-31" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">L</span></span><span id="MJXc-Node-32" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">D</span></span><span id="MJXc-Node-33" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span></span></span></span><span id="MJXc-Node-34" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.127em; padding-bottom: 0.374em;">∝</span></span><span id="MJXc-Node-35" class="mjx-msubsup MJXc-space3"><span class="mjx-base" style="margin-right: -0.064em;"><span id="MJXc-Node-36" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em; padding-right: 0.064em;">I</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.584em; padding-left: 0.188em; padding-right: 0.071em;"><span id="MJXc-Node-37" class="mjx-texatom" style=""><span id="MJXc-Node-38" class="mjx-mrow"><span id="MJXc-Node-39" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.313em; padding-bottom: 0.436em;">−</span></span><span id="MJXc-Node-40" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.189em; padding-bottom: 0.498em; padding-right: 0.025em;">γ</span></span></span></span></span></span><span id="MJXc-Node-41" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.183em; padding-bottom: 0.56em;">,</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub><mo>∝</mo><msup><mi>I</mi><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mi>γ</mi></mrow></msup><mo>,</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-5">\label{eq:k1}
| + | |
| − | R_{LDR} \propto I^{-\gamma},</script></span> where <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-6-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03B3;</mi></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-42" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-43" class="mjx-mrow"><span id="MJXc-Node-44" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.189em; padding-bottom: 0.498em; padding-right: 0.025em;">γ</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>γ</mi></math></span></span><script type="math/tex" id="MathJax-Element-6">\gamma</script></span> is a parameter depending on the type of resistor being used and can even differ for LDRs with the same type designation.</p>
| + | |
| − | <p>Equation [eq:k1] is motivated from the equation <span class="math display"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-7-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>&#x03B3;</mi><mo>=</mo><mfrac><mrow><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">l</mi><mi mathvariant="normal">g</mi></mrow><mrow><mo>(</mo><mfrac><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mn>10</mn></mrow></msub><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mn>100</mn></mrow></msub></mfrac><mo>)</mo></mrow></mrow><mrow><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">l</mi><mi mathvariant="normal">g</mi></mrow><mrow><mo>(</mo><mfrac><mrow><mn>100</mn><mtext>&#xA0;</mtext><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">L</mi><mi mathvariant="normal">x</mi></mrow></mrow><mrow><mn>10</mn><mtext>&#xA0;</mtext><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">L</mi><mi mathvariant="normal">x</mi></mrow></mrow></mfrac><mo>)</mo></mrow></mrow></mfrac><mo>,</mo></math>" role="presentation" style="font-size: 101%; text-align: center; position: relative;"><span id="MJXc-Node-45" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-46" class="mjx-mrow"><span id="MJXc-Node-47" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.189em; padding-bottom: 0.498em; padding-right: 0.025em;">γ</span></span><span id="MJXc-Node-48" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.065em; padding-bottom: 0.313em;">=</span></span><span id="MJXc-Node-49" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 4.773em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 4.773em; top: -2.352em;"><span id="MJXc-Node-50" class="mjx-mrow"><span id="MJXc-Node-51" class="mjx-texatom"><span id="MJXc-Node-52" class="mjx-mrow"><span id="MJXc-Node-53" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.436em; padding-bottom: 0.313em;">l</span></span><span id="MJXc-Node-54" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.189em; padding-bottom: 0.56em;">g</span></span></span></span><span id="MJXc-Node-55" class="mjx-mrow MJXc-space1"><span id="MJXc-Node-56" class="mjx-mo"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 0.931em; padding-bottom: 0.931em;">(</span></span><span id="MJXc-Node-57" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 1.453em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 2.055em; top: -1.506em;"><span id="MJXc-Node-58" class="mjx-msubsup" style=""><span class="mjx-base"><span id="MJXc-Node-59" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-60" class="mjx-texatom" style=""><span id="MJXc-Node-61" class="mjx-mrow"><span id="MJXc-Node-62" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.374em;">10</span></span></span></span></span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 2.055em; bottom: -0.799em;"><span id="MJXc-Node-63" class="mjx-msubsup" style=""><span class="mjx-base"><span id="MJXc-Node-64" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-65" class="mjx-texatom" style=""><span id="MJXc-Node-66" class="mjx-mrow"><span id="MJXc-Node-67" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.374em;">100</span></span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.289em; width: 1.453em;"></span></span><span class="mjx-vsize" style="height: 1.63em; vertical-align: -0.565em;"></span></span><span id="MJXc-Node-68" class="mjx-mo"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 0.931em; padding-bottom: 0.931em;">)</span></span></span></span></span><span class="mjx-denominator" style="width: 4.773em; bottom: -1.852em;"><span id="MJXc-Node-69" class="mjx-mrow"><span id="MJXc-Node-70" class="mjx-texatom"><span id="MJXc-Node-71" class="mjx-mrow"><span id="MJXc-Node-72" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.436em; padding-bottom: 0.313em;">l</span></span><span id="MJXc-Node-73" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.189em; padding-bottom: 0.56em;">g</span></span></span></span><span id="MJXc-Node-74" class="mjx-mrow MJXc-space1"><span id="MJXc-Node-75" class="mjx-mo"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 0.931em; padding-bottom: 0.931em;">(</span></span><span id="MJXc-Node-76" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 2.194em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 3.103em; top: -1.37em;"><span id="MJXc-Node-77" class="mjx-mrow" style=""><span id="MJXc-Node-78" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.374em;">100</span></span><span id="MJXc-Node-79" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.306em; padding-bottom: 0.313em;"> </span></span><span id="MJXc-Node-80" class="mjx-texatom"><span id="MJXc-Node-81" class="mjx-mrow"><span id="MJXc-Node-82" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.313em;">L</span></span><span id="MJXc-Node-83" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.127em; padding-bottom: 0.313em;">x</span></span></span></span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 3.103em; bottom: -0.663em;"><span id="MJXc-Node-84" class="mjx-mrow" style=""><span id="MJXc-Node-85" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.374em;">10</span></span><span id="MJXc-Node-86" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.306em; padding-bottom: 0.313em;"> </span></span><span id="MJXc-Node-87" class="mjx-texatom"><span id="MJXc-Node-88" class="mjx-mrow"><span id="MJXc-Node-89" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.313em;">L</span></span><span id="MJXc-Node-90" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.127em; padding-bottom: 0.313em;">x</span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.289em; width: 2.194em;"></span></span><span class="mjx-vsize" style="height: 1.437em; vertical-align: -0.469em;"></span></span><span id="MJXc-Node-91" class="mjx-mo"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 0.931em; padding-bottom: 0.931em;">)</span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.289em; width: 4.773em;"></span></span><span class="mjx-vsize" style="height: 4.205em; vertical-align: -1.852em;"></span></span><span id="MJXc-Node-92" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.183em; padding-bottom: 0.56em;">,</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>γ</mi><mo>=</mo><mfrac><mrow><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">l</mi><mi mathvariant="normal">g</mi></mrow><mrow><mo>(</mo><mfrac><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mn>10</mn></mrow></msub><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mn>100</mn></mrow></msub></mfrac><mo>)</mo></mrow></mrow><mrow><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">l</mi><mi mathvariant="normal">g</mi></mrow><mrow><mo>(</mo><mfrac><mrow><mn>100</mn><mtext> </mtext><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">L</mi><mi mathvariant="normal">x</mi></mrow></mrow><mrow><mn>10</mn><mtext> </mtext><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">L</mi><mi mathvariant="normal">x</mi></mrow></mrow></mfrac><mo>)</mo></mrow></mrow></mfrac><mo>,</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-7">\label{eq:k2}
| + | |
| − | \gamma = \frac{\mathrm{lg}\left(\frac{R_{10}}{R_{100}}\right)}{\mathrm{lg}\left(\frac{100~\mathrm{Lx}}{10~\mathrm{Lx}}\right)},</script></span> which is given in the data sheet of the LDR. The dominator is the decadic logarithm of the fraction of two light intensities of 100 Lx and 10 Lx. <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-8-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mn>10</mn></mrow></msub></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-93" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-94" class="mjx-mrow"><span id="MJXc-Node-95" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-96" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-97" class="mjx-texatom" style=""><span id="MJXc-Node-98" class="mjx-mrow"><span id="MJXc-Node-99" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.374em;">10</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mn>10</mn></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-8">R_{10}</script></span> and <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-9-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mn>100</mn></mrow></msub></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-100" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-101" class="mjx-mrow"><span id="MJXc-Node-102" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-103" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-104" class="mjx-texatom" style=""><span id="MJXc-Node-105" class="mjx-mrow"><span id="MJXc-Node-106" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.374em;">100</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mn>100</mn></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-9">R_{100}</script></span> are the corresponding resistances at these light intensities. The used resistor with the type designation GL5516 NT00183 has a parameter <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-10-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03B3;</mi></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-107" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-108" class="mjx-mrow"><span id="MJXc-Node-109" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.189em; padding-bottom: 0.498em; padding-right: 0.025em;">γ</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>γ</mi></math></span></span><script type="math/tex" id="MathJax-Element-10">\gamma</script></span> of 0.8.</p>
| + | |
| − | <p>The response of a LDR depends on the wavelength <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-11-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03BB;</mi></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-110" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-111" class="mjx-mrow"><span id="MJXc-Node-112" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.498em; padding-bottom: 0.313em;">λ</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi></math></span></span><script type="math/tex" id="MathJax-Element-11">\lambda</script></span> of the incoming light. The data sheet provides information on the relative response normalized to the maximal response. The relative response is maximum for a wavelength of 540 nm and is therefore appropriate for detection of green fluorophores.</p> | + | |
| − | <h2 id="sec:k1">Circuit for resistance measurements</h2>
| + | |
| − | <p>A voltage divider is the simplest way to measure resistance. Image shows the circuit of a voltage divider. Applying Kirchhoff’s laws we get <span class="math display"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-12-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mfrac><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>r</mi><mi>e</mi><mi>f</mi></mrow></msub></mfrac><mo>=</mo><mfrac><msub><mi>U</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub><msub><mi>U</mi><mrow class="MJX-TeXAtom-ORD"><mi>r</mi><mi>e</mi><mi>f</mi></mrow></msub></mfrac></math>" role="presentation" style="font-size: 101%; text-align: center; position: relative;"><span id="MJXc-Node-113" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-114" class="mjx-mrow"><span id="MJXc-Node-115" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 2.598em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 2.598em; top: -1.394em;"><span id="MJXc-Node-116" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-117" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-118" class="mjx-texatom" style=""><span id="MJXc-Node-119" class="mjx-mrow"><span id="MJXc-Node-120" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">L</span></span><span id="MJXc-Node-121" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">D</span></span><span id="MJXc-Node-122" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span></span></span></span></span><span class="mjx-denominator" style="width: 2.598em; bottom: -1.037em;"><span id="MJXc-Node-123" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-124" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.23em; padding-right: 0.071em;"><span id="MJXc-Node-125" class="mjx-texatom" style=""><span id="MJXc-Node-126" class="mjx-mrow"><span id="MJXc-Node-127" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">r</span></span><span id="MJXc-Node-128" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">e</span></span><span id="MJXc-Node-129" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.498em; padding-bottom: 0.498em; padding-right: 0.06em;">f</span></span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.289em; width: 2.598em;"></span></span><span class="mjx-vsize" style="height: 2.431em; vertical-align: -1.037em;"></span></span><span id="MJXc-Node-130" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.065em; padding-bottom: 0.313em;">=</span></span><span id="MJXc-Node-131" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 2.522em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 2.522em; top: -1.394em;"><span id="MJXc-Node-132" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-133" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-134" class="mjx-texatom" style=""><span id="MJXc-Node-135" class="mjx-mrow"><span id="MJXc-Node-136" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">L</span></span><span id="MJXc-Node-137" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">D</span></span><span id="MJXc-Node-138" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span></span></span></span></span><span class="mjx-denominator" style="width: 2.522em; bottom: -1.037em;"><span id="MJXc-Node-139" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-140" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.23em; padding-right: 0.071em;"><span id="MJXc-Node-141" class="mjx-texatom" style=""><span id="MJXc-Node-142" class="mjx-mrow"><span id="MJXc-Node-143" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">r</span></span><span id="MJXc-Node-144" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">e</span></span><span id="MJXc-Node-145" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.498em; padding-bottom: 0.498em; padding-right: 0.06em;">f</span></span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.289em; width: 2.522em;"></span></span><span class="mjx-vsize" style="height: 2.431em; vertical-align: -1.037em;"></span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mfrac><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>r</mi><mi>e</mi><mi>f</mi></mrow></msub></mfrac><mo>=</mo><mfrac><msub><mi>U</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub><msub><mi>U</mi><mrow class="MJX-TeXAtom-ORD"><mi>r</mi><mi>e</mi><mi>f</mi></mrow></msub></mfrac></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-12">\label{eq:k20}
| + | |
| − | \frac{R_{LDR}}{R_{ref}}=\frac{U_{LDR}}{U_{ref}}</script></span> and <span class="math display"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-13-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>U</mi><mn>0</mn></msub><mo>=</mo><msub><mi>U</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub><mo>+</mo><msub><mi>U</mi><mrow class="MJX-TeXAtom-ORD"><mi>r</mi><mi>e</mi><mi>f</mi></mrow></msub><mo>.</mo></math>" role="presentation" style="font-size: 101%; text-align: center; position: relative;"><span id="MJXc-Node-146" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-147" class="mjx-mrow"><span id="MJXc-Node-148" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-149" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-150" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.374em;">0</span></span></span></span><span id="MJXc-Node-151" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.065em; padding-bottom: 0.313em;">=</span></span><span id="MJXc-Node-152" class="mjx-msubsup MJXc-space3"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-153" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-154" class="mjx-texatom" style=""><span id="MJXc-Node-155" class="mjx-mrow"><span id="MJXc-Node-156" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">L</span></span><span id="MJXc-Node-157" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">D</span></span><span id="MJXc-Node-158" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span></span></span></span><span id="MJXc-Node-159" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.313em; padding-bottom: 0.436em;">+</span></span><span id="MJXc-Node-160" class="mjx-msubsup MJXc-space2"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-161" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.23em; padding-right: 0.071em;"><span id="MJXc-Node-162" class="mjx-texatom" style=""><span id="MJXc-Node-163" class="mjx-mrow"><span id="MJXc-Node-164" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">r</span></span><span id="MJXc-Node-165" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">e</span></span><span id="MJXc-Node-166" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.498em; padding-bottom: 0.498em; padding-right: 0.06em;">f</span></span></span></span></span></span><span id="MJXc-Node-167" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.183em; padding-bottom: 0.313em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>U</mi><mn>0</mn></msub><mo>=</mo><msub><mi>U</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub><mo>+</mo><msub><mi>U</mi><mrow class="MJX-TeXAtom-ORD"><mi>r</mi><mi>e</mi><mi>f</mi></mrow></msub><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-13">\label{eq:k21}
| + | |
| − | U_0= U_{LDR}+U_{ref}.</script></span> <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-14-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-168" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-169" class="mjx-mrow"><span id="MJXc-Node-170" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-171" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-172" class="mjx-texatom" style=""><span id="MJXc-Node-173" class="mjx-mrow"><span id="MJXc-Node-174" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">L</span></span><span id="MJXc-Node-175" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">D</span></span><span id="MJXc-Node-176" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-14">R_{LDR}</script></span> and <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-15-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-177" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-178" class="mjx-mrow"><span id="MJXc-Node-179" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-180" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-181" class="mjx-texatom" style=""><span id="MJXc-Node-182" class="mjx-mrow"><span id="MJXc-Node-183" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">L</span></span><span id="MJXc-Node-184" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">D</span></span><span id="MJXc-Node-185" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-15">U_{LDR}</script></span> is the resistance and voltage drop at the LDR. <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-16-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>r</mi><mi>e</mi><mi>f</mi></mrow></msub></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-186" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-187" class="mjx-mrow"><span id="MJXc-Node-188" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-189" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.23em; padding-right: 0.071em;"><span id="MJXc-Node-190" class="mjx-texatom" style=""><span id="MJXc-Node-191" class="mjx-mrow"><span id="MJXc-Node-192" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">r</span></span><span id="MJXc-Node-193" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">e</span></span><span id="MJXc-Node-194" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.498em; padding-bottom: 0.498em; padding-right: 0.06em;">f</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>r</mi><mi>e</mi><mi>f</mi></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-16">R_{ref}</script></span> and <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-17-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mrow class="MJX-TeXAtom-ORD"><mi>r</mi><mi>e</mi><mi>f</mi></mrow></msub></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-195" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-196" class="mjx-mrow"><span id="MJXc-Node-197" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-198" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.23em; padding-right: 0.071em;"><span id="MJXc-Node-199" class="mjx-texatom" style=""><span id="MJXc-Node-200" class="mjx-mrow"><span id="MJXc-Node-201" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">r</span></span><span id="MJXc-Node-202" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">e</span></span><span id="MJXc-Node-203" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.498em; padding-bottom: 0.498em; padding-right: 0.06em;">f</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mrow class="MJX-TeXAtom-ORD"><mi>r</mi><mi>e</mi><mi>f</mi></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-17">U_{ref}</script></span> is the resistance and voltage of a reference Resistor. <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-18-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-204" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-205" class="mjx-mrow"><span id="MJXc-Node-206" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-207" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-208" class="mjx-texatom" style=""><span id="MJXc-Node-209" class="mjx-mrow"><span id="MJXc-Node-210" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.374em;">0</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-18">U_{0}</script></span> is the supply voltage which we choose to be <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-19-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">V</mi></mrow></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-211" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-212" class="mjx-mrow"><span id="MJXc-Node-213" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.374em;">5</span></span><span id="MJXc-Node-214" class="mjx-texatom"><span id="MJXc-Node-215" class="mjx-mrow"><span id="MJXc-Node-216" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.374em;">V</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">V</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-19">5\mathrm{V}</script></span>. This gives <span class="math display"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-20-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub><mo>=</mo><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>r</mi><mi>e</mi><mi>f</mi></mrow></msub><mo>&#x22C5;</mo><mfrac><mn>1</mn><mrow><mfrac><msub><mi>U</mi><mn>0</mn></msub><msub><mi>U</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></mfrac><mo>&#x2212;</mo><mn>1</mn></mrow></mfrac><mo>,</mo></math>" role="presentation" style="font-size: 101%; text-align: center; position: relative;"><span id="MJXc-Node-217" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-218" class="mjx-mrow"><span id="MJXc-Node-219" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-220" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-221" class="mjx-texatom" style=""><span id="MJXc-Node-222" class="mjx-mrow"><span id="MJXc-Node-223" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">L</span></span><span id="MJXc-Node-224" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">D</span></span><span id="MJXc-Node-225" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span></span></span></span><span id="MJXc-Node-226" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.065em; padding-bottom: 0.313em;">=</span></span><span id="MJXc-Node-227" class="mjx-msubsup MJXc-space3"><span class="mjx-base"><span id="MJXc-Node-228" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.23em; padding-right: 0.071em;"><span id="MJXc-Node-229" class="mjx-texatom" style=""><span id="MJXc-Node-230" class="mjx-mrow"><span id="MJXc-Node-231" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">r</span></span><span id="MJXc-Node-232" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">e</span></span><span id="MJXc-Node-233" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.498em; padding-bottom: 0.498em; padding-right: 0.06em;">f</span></span></span></span></span></span><span id="MJXc-Node-234" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.003em; padding-bottom: 0.313em;">⋅</span></span><span id="MJXc-Node-235" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 3.946em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 3.946em; top: -1.368em;"><span id="MJXc-Node-236" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.313em;">1</span></span></span><span class="mjx-denominator" style="width: 3.946em; bottom: -1.633em;"><span id="MJXc-Node-237" class="mjx-mrow"><span id="MJXc-Node-238" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 1.783em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 2.522em; top: -1.506em;"><span id="MJXc-Node-239" class="mjx-msubsup" style=""><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-240" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-241" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.374em;">0</span></span></span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 2.522em; bottom: -0.798em;"><span id="MJXc-Node-242" class="mjx-msubsup" style=""><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-243" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-244" class="mjx-texatom" style=""><span id="MJXc-Node-245" class="mjx-mrow"><span id="MJXc-Node-246" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">L</span></span><span id="MJXc-Node-247" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">D</span></span><span id="MJXc-Node-248" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.289em; width: 1.783em;"></span></span><span class="mjx-vsize" style="height: 1.629em; vertical-align: -0.564em;"></span></span><span id="MJXc-Node-249" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.313em; padding-bottom: 0.436em;">−</span></span><span id="MJXc-Node-250" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.313em;">1</span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.289em; width: 3.946em;"></span></span><span class="mjx-vsize" style="height: 3em; vertical-align: -1.633em;"></span></span><span id="MJXc-Node-251" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.183em; padding-bottom: 0.56em;">,</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub><mo>=</mo><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>r</mi><mi>e</mi><mi>f</mi></mrow></msub><mo>⋅</mo><mfrac><mn>1</mn><mrow><mfrac><msub><mi>U</mi><mn>0</mn></msub><msub><mi>U</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></mfrac><mo>−</mo><mn>1</mn></mrow></mfrac><mo>,</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-20">\label{eq:k22}
| + | |
| − | R_{LDR}=R_{ref}\cdot\frac{1}{\frac{U_0}{U_{LDR}}-1},</script></span> an equation to calculate the resistance <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-21-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-252" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-253" class="mjx-mrow"><span id="MJXc-Node-254" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-255" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-256" class="mjx-texatom" style=""><span id="MJXc-Node-257" class="mjx-mrow"><span id="MJXc-Node-258" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">L</span></span><span id="MJXc-Node-259" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">D</span></span><span id="MJXc-Node-260" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-21">R_{LDR}</script></span> of the LDR from the voltage drop <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-22-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-261" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-262" class="mjx-mrow"><span id="MJXc-Node-263" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-264" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-265" class="mjx-texatom" style=""><span id="MJXc-Node-266" class="mjx-mrow"><span id="MJXc-Node-267" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">L</span></span><span id="MJXc-Node-268" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">D</span></span><span id="MJXc-Node-269" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-22">U_{LDR}</script></span> at the LDR. <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-23-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-270" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-271" class="mjx-mrow"><span id="MJXc-Node-272" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-273" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-274" class="mjx-texatom" style=""><span id="MJXc-Node-275" class="mjx-mrow"><span id="MJXc-Node-276" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">L</span></span><span id="MJXc-Node-277" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">D</span></span><span id="MJXc-Node-278" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-23">U_{LDR}</script></span> can be measured with the micro controller.</p>
| + | |
| − | <p>We need to find an equation to choose an optimal resistor <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-24-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>r</mi><mi>e</mi><mi>f</mi></mrow></msub></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-279" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-280" class="mjx-mrow"><span id="MJXc-Node-281" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-282" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.23em; padding-right: 0.071em;"><span id="MJXc-Node-283" class="mjx-texatom" style=""><span id="MJXc-Node-284" class="mjx-mrow"><span id="MJXc-Node-285" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">r</span></span><span id="MJXc-Node-286" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">e</span></span><span id="MJXc-Node-287" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.498em; padding-bottom: 0.498em; padding-right: 0.06em;">f</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>r</mi><mi>e</mi><mi>f</mi></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-24">R_{ref}</script></span>. We want an maximal change of <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-25-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-288" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-289" class="mjx-mrow"><span id="MJXc-Node-290" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-291" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-292" class="mjx-texatom" style=""><span id="MJXc-Node-293" class="mjx-mrow"><span id="MJXc-Node-294" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">L</span></span><span id="MJXc-Node-295" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">D</span></span><span id="MJXc-Node-296" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-25">U_{LDR}</script></span> for a certain detection range of <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-26-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-297" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-298" class="mjx-mrow"><span id="MJXc-Node-299" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-300" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-301" class="mjx-texatom" style=""><span id="MJXc-Node-302" class="mjx-mrow"><span id="MJXc-Node-303" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">L</span></span><span id="MJXc-Node-304" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">D</span></span><span id="MJXc-Node-305" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-26">R_{LDR}</script></span>. Therefore equation [eq:k22] is solved for <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-27-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-306" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-307" class="mjx-mrow"><span id="MJXc-Node-308" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-309" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-310" class="mjx-texatom" style=""><span id="MJXc-Node-311" class="mjx-mrow"><span id="MJXc-Node-312" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">L</span></span><span id="MJXc-Node-313" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">D</span></span><span id="MJXc-Node-314" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-27">U_{LDR}</script></span> giving <span class="math display"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-28-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>U</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub><mo>=</mo><mfrac><msub><mi>U</mi><mn>0</mn></msub><mrow><mn>1</mn><mo>+</mo><mfrac><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>r</mi><mi>e</mi><mi>f</mi></mrow></msub><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></mfrac></mrow></mfrac><mo>.</mo></math>" role="presentation" style="font-size: 101%; text-align: center; position: relative;"><span id="MJXc-Node-315" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-316" class="mjx-mrow"><span id="MJXc-Node-317" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-318" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-319" class="mjx-texatom" style=""><span id="MJXc-Node-320" class="mjx-mrow"><span id="MJXc-Node-321" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">L</span></span><span id="MJXc-Node-322" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">D</span></span><span id="MJXc-Node-323" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span></span></span></span><span id="MJXc-Node-324" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.065em; padding-bottom: 0.313em;">=</span></span><span id="MJXc-Node-325" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 4em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 4em; top: -1.395em;"><span id="MJXc-Node-326" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-327" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-328" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.374em;">0</span></span></span></span></span><span class="mjx-denominator" style="width: 4em; bottom: -1.733em;"><span id="MJXc-Node-329" class="mjx-mrow"><span id="MJXc-Node-330" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.313em;">1</span></span><span id="MJXc-Node-331" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.313em; padding-bottom: 0.436em;">+</span></span><span id="MJXc-Node-332" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 1.837em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 2.598em; top: -1.648em;"><span id="MJXc-Node-333" class="mjx-msubsup" style=""><span class="mjx-base"><span id="MJXc-Node-334" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.23em; padding-right: 0.071em;"><span id="MJXc-Node-335" class="mjx-texatom" style=""><span id="MJXc-Node-336" class="mjx-mrow"><span id="MJXc-Node-337" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">r</span></span><span id="MJXc-Node-338" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">e</span></span><span id="MJXc-Node-339" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.498em; padding-bottom: 0.498em; padding-right: 0.06em;">f</span></span></span></span></span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 2.598em; bottom: -0.798em;"><span id="MJXc-Node-340" class="mjx-msubsup" style=""><span class="mjx-base"><span id="MJXc-Node-341" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-342" class="mjx-texatom" style=""><span id="MJXc-Node-343" class="mjx-mrow"><span id="MJXc-Node-344" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">L</span></span><span id="MJXc-Node-345" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">D</span></span><span id="MJXc-Node-346" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.289em; width: 1.837em;"></span></span><span class="mjx-vsize" style="height: 1.73em; vertical-align: -0.564em;"></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.289em; width: 4em;"></span></span><span class="mjx-vsize" style="height: 3.128em; vertical-align: -1.733em;"></span></span><span id="MJXc-Node-347" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.183em; padding-bottom: 0.313em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>U</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub><mo>=</mo><mfrac><msub><mi>U</mi><mn>0</mn></msub><mrow><mn>1</mn><mo>+</mo><mfrac><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>r</mi><mi>e</mi><mi>f</mi></mrow></msub><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></mfrac></mrow></mfrac><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-28">\label{eq:k30}
| + | |
| − | U_{LDR} = \frac{U_0}{1+\frac{R_{ref}}{R_{LDR}}}.</script></span> The change<span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-29-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">&#x0394;</mi><msub><mi>U</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-348" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-349" class="mjx-mrow"><span id="MJXc-Node-350" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.436em; padding-bottom: 0.313em;">Δ</span></span><span id="MJXc-Node-351" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-352" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-353" class="mjx-texatom" style=""><span id="MJXc-Node-354" class="mjx-mrow"><span id="MJXc-Node-355" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">L</span></span><span id="MJXc-Node-356" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">D</span></span><span id="MJXc-Node-357" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">Δ</mi><msub><mi>U</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-29">\Delta U_{LDR}</script></span> of <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-30-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-358" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-359" class="mjx-mrow"><span id="MJXc-Node-360" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-361" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-362" class="mjx-texatom" style=""><span id="MJXc-Node-363" class="mjx-mrow"><span id="MJXc-Node-364" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">L</span></span><span id="MJXc-Node-365" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">D</span></span><span id="MJXc-Node-366" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-30">U_{LDR}</script></span> between a maximum value <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-31-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-367" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-368" class="mjx-mrow"><span id="MJXc-Node-369" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-370" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-371" class="mjx-texatom" style=""><span id="MJXc-Node-372" class="mjx-mrow"><span id="MJXc-Node-373" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">m</span></span><span id="MJXc-Node-374" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.189em; padding-bottom: 0.313em;">a</span></span><span id="MJXc-Node-375" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">x</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-31">R_{max}</script></span> and a minimal value <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-32-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>m</mi><mi>i</mi><mi>n</mi></mrow></msub></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-376" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-377" class="mjx-mrow"><span id="MJXc-Node-378" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-379" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-380" class="mjx-texatom" style=""><span id="MJXc-Node-381" class="mjx-mrow"><span id="MJXc-Node-382" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">m</span></span><span id="MJXc-Node-383" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">i</span></span><span id="MJXc-Node-384" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">n</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>m</mi><mi>i</mi><mi>n</mi></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-32">R_{min}</script></span> of <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-33-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-385" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-386" class="mjx-mrow"><span id="MJXc-Node-387" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-388" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-389" class="mjx-texatom" style=""><span id="MJXc-Node-390" class="mjx-mrow"><span id="MJXc-Node-391" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">L</span></span><span id="MJXc-Node-392" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">D</span></span><span id="MJXc-Node-393" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-33">R_{LDR}</script></span> is <span class="math display"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-34-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi mathvariant="normal">&#x0394;</mi><msub><mi>U</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>d</mi><mi>R</mi></mrow></msub><mo>=</mo><msub><mi>U</mi><mn>0</mn></msub><mo>&#x22C5;</mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mrow><mn>1</mn><mo>+</mo><mfrac><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>r</mi><mi>e</mi><mi>f</mi></mrow></msub><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub></mfrac></mrow></mfrac><mo>&#x2212;</mo><mfrac><mn>1</mn><mrow><mn>1</mn><mo>+</mo><mfrac><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>r</mi><mi>e</mi><mi>f</mi></mrow></msub><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>m</mi><mi>i</mi><mi>n</mi></mrow></msub></mfrac></mrow></mfrac></mrow><mo>)</mo></mrow><mo>,</mo></math>" role="presentation" style="font-size: 101%; text-align: center; position: relative;"><span id="MJXc-Node-394" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-395" class="mjx-mrow"><span id="MJXc-Node-396" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.436em; padding-bottom: 0.313em;">Δ</span></span><span id="MJXc-Node-397" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-398" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.219em; padding-right: 0.071em;"><span id="MJXc-Node-399" class="mjx-texatom" style=""><span id="MJXc-Node-400" class="mjx-mrow"><span id="MJXc-Node-401" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">L</span></span><span id="MJXc-Node-402" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.498em; padding-bottom: 0.313em; padding-right: 0.003em;">d</span></span><span id="MJXc-Node-403" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span></span></span></span><span id="MJXc-Node-404" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.065em; padding-bottom: 0.313em;">=</span></span><span id="MJXc-Node-405" class="mjx-msubsup MJXc-space3"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-406" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-407" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.374em;">0</span></span></span></span><span id="MJXc-Node-408" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.003em; padding-bottom: 0.313em;">⋅</span></span><span id="MJXc-Node-409" class="mjx-mrow MJXc-space2"><span id="MJXc-Node-410" class="mjx-mo" style="vertical-align: -1.059em;"><span class="mjx-delim-v"><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.931em; padding-bottom: 0.931em;">⎛</span><span class="mjx-char MJXc-TeX-size4-R" style="line-height: 0.338em; margin-bottom: -0.1em; margin-top: 0px;">⎜</span><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.931em; padding-bottom: 0.931em;">⎝</span></span></span><span id="MJXc-Node-411" class="mjx-mrow"><span id="MJXc-Node-412" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 3.855em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 3.855em; top: -1.368em;"><span id="MJXc-Node-413" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.313em;">1</span></span></span><span class="mjx-denominator" style="width: 3.855em; bottom: -1.728em;"><span id="MJXc-Node-414" class="mjx-mrow"><span id="MJXc-Node-415" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.313em;">1</span></span><span id="MJXc-Node-416" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.313em; padding-bottom: 0.436em;">+</span></span><span id="MJXc-Node-417" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 1.693em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 2.394em; top: -1.648em;"><span id="MJXc-Node-418" class="mjx-msubsup" style=""><span class="mjx-base"><span id="MJXc-Node-419" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.23em; padding-right: 0.071em;"><span id="MJXc-Node-420" class="mjx-texatom" style=""><span id="MJXc-Node-421" class="mjx-mrow"><span id="MJXc-Node-422" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">r</span></span><span id="MJXc-Node-423" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">e</span></span><span id="MJXc-Node-424" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.498em; padding-bottom: 0.498em; padding-right: 0.06em;">f</span></span></span></span></span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 2.394em; bottom: -0.791em;"><span id="MJXc-Node-425" class="mjx-msubsup" style=""><span class="mjx-base"><span id="MJXc-Node-426" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-427" class="mjx-texatom" style=""><span id="MJXc-Node-428" class="mjx-mrow"><span id="MJXc-Node-429" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">m</span></span><span id="MJXc-Node-430" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.189em; padding-bottom: 0.313em;">a</span></span><span id="MJXc-Node-431" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">x</span></span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.289em; width: 1.693em;"></span></span><span class="mjx-vsize" style="height: 1.725em; vertical-align: -0.559em;"></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.289em; width: 3.855em;"></span></span><span class="mjx-vsize" style="height: 3.096em; vertical-align: -1.728em;"></span></span><span id="MJXc-Node-432" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.313em; padding-bottom: 0.436em;">−</span></span><span id="MJXc-Node-433" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 3.777em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 3.777em; top: -1.368em;"><span id="MJXc-Node-434" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.313em;">1</span></span></span><span class="mjx-denominator" style="width: 3.777em; bottom: -1.728em;"><span id="MJXc-Node-435" class="mjx-mrow"><span id="MJXc-Node-436" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.313em;">1</span></span><span id="MJXc-Node-437" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.313em; padding-bottom: 0.436em;">+</span></span><span id="MJXc-Node-438" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 1.615em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 2.283em; top: -1.648em;"><span id="MJXc-Node-439" class="mjx-msubsup" style=""><span class="mjx-base"><span id="MJXc-Node-440" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.23em; padding-right: 0.071em;"><span id="MJXc-Node-441" class="mjx-texatom" style=""><span id="MJXc-Node-442" class="mjx-mrow"><span id="MJXc-Node-443" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">r</span></span><span id="MJXc-Node-444" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">e</span></span><span id="MJXc-Node-445" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.498em; padding-bottom: 0.498em; padding-right: 0.06em;">f</span></span></span></span></span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 2.283em; bottom: -0.791em;"><span id="MJXc-Node-446" class="mjx-msubsup" style=""><span class="mjx-base"><span id="MJXc-Node-447" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-448" class="mjx-texatom" style=""><span id="MJXc-Node-449" class="mjx-mrow"><span id="MJXc-Node-450" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">m</span></span><span id="MJXc-Node-451" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">i</span></span><span id="MJXc-Node-452" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">n</span></span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.289em; width: 1.615em;"></span></span><span class="mjx-vsize" style="height: 1.725em; vertical-align: -0.559em;"></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.289em; width: 3.777em;"></span></span><span class="mjx-vsize" style="height: 3.096em; vertical-align: -1.728em;"></span></span></span><span id="MJXc-Node-453" class="mjx-mo" style="vertical-align: -1.059em;"><span class="mjx-delim-v"><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.931em; padding-bottom: 0.931em;">⎞</span><span class="mjx-char MJXc-TeX-size4-R" style="line-height: 0.338em; margin-bottom: -0.1em; margin-top: 0px;">⎟</span><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.931em; padding-bottom: 0.931em;">⎠</span></span></span></span><span id="MJXc-Node-454" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.183em; padding-bottom: 0.56em;">,</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi mathvariant="normal">Δ</mi><msub><mi>U</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>d</mi><mi>R</mi></mrow></msub><mo>=</mo><msub><mi>U</mi><mn>0</mn></msub><mo>⋅</mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mrow><mn>1</mn><mo>+</mo><mfrac><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>r</mi><mi>e</mi><mi>f</mi></mrow></msub><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub></mfrac></mrow></mfrac><mo>−</mo><mfrac><mn>1</mn><mrow><mn>1</mn><mo>+</mo><mfrac><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>r</mi><mi>e</mi><mi>f</mi></mrow></msub><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>m</mi><mi>i</mi><mi>n</mi></mrow></msub></mfrac></mrow></mfrac></mrow><mo>)</mo></mrow><mo>,</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-34">\label{eq:k31}
| + | |
| − | \Delta U_{LdR}=U_0 \cdot \left(\frac{1}{1+\frac{R_{ref}}{R_{max}}}-\frac{1}{1+\frac{R_{ref}}{R_{min}}} \right),</script></span> which has a maximum for <span class="math display"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-35-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>r</mi><mi>e</mi><mi>f</mi></mrow></msub><mo>=</mo><msqrt><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub><mo>&#x22C5;</mo><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>m</mi><mi>i</mi><mi>n</mi></mrow></msub></msqrt><mo>.</mo></math>" role="presentation" style="font-size: 101%; text-align: center; position: relative;"><span id="MJXc-Node-455" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-456" class="mjx-mrow"><span id="MJXc-Node-457" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-458" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.23em; padding-right: 0.071em;"><span id="MJXc-Node-459" class="mjx-texatom" style=""><span id="MJXc-Node-460" class="mjx-mrow"><span id="MJXc-Node-461" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">r</span></span><span id="MJXc-Node-462" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">e</span></span><span id="MJXc-Node-463" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.498em; padding-bottom: 0.498em; padding-right: 0.06em;">f</span></span></span></span></span></span><span id="MJXc-Node-464" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.065em; padding-bottom: 0.313em;">=</span></span><span id="MJXc-Node-465" class="mjx-msqrt MJXc-space3"><span class="mjx-box" style="padding-top: 0.045em;"><span class="mjx-surd"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.622em; padding-bottom: 0.622em;">√</span></span><span class="mjx-box" style="padding-top: 0.098em; border-top: 1.2px solid;"><span id="MJXc-Node-466" class="mjx-mrow"><span id="MJXc-Node-467" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-468" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-469" class="mjx-texatom" style=""><span id="MJXc-Node-470" class="mjx-mrow"><span id="MJXc-Node-471" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">m</span></span><span id="MJXc-Node-472" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.189em; padding-bottom: 0.313em;">a</span></span><span id="MJXc-Node-473" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">x</span></span></span></span></span></span><span id="MJXc-Node-474" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.003em; padding-bottom: 0.313em;">⋅</span></span><span id="MJXc-Node-475" class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span id="MJXc-Node-476" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-477" class="mjx-texatom" style=""><span id="MJXc-Node-478" class="mjx-mrow"><span id="MJXc-Node-479" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">m</span></span><span id="MJXc-Node-480" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">i</span></span><span id="MJXc-Node-481" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">n</span></span></span></span></span></span></span></span></span></span><span id="MJXc-Node-482" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.183em; padding-bottom: 0.313em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>r</mi><mi>e</mi><mi>f</mi></mrow></msub><mo>=</mo><msqrt><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub><mo>⋅</mo><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>m</mi><mi>i</mi><mi>n</mi></mrow></msub></msqrt><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-35">\label{eq:k23}
| + | |
| − | R_{ref}= \sqrt{R_{max}\cdot R_{min}}.</script></span> We expect a <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-36-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-483" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-484" class="mjx-mrow"><span id="MJXc-Node-485" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-486" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-487" class="mjx-texatom" style=""><span id="MJXc-Node-488" class="mjx-mrow"><span id="MJXc-Node-489" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">m</span></span><span id="MJXc-Node-490" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.189em; padding-bottom: 0.313em;">a</span></span><span id="MJXc-Node-491" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">x</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-36">R_{max}</script></span> of approximately <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-37-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mtext>&#xA0;</mtext><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">M</mi><mi mathvariant="normal">&#x03A9;</mi></mrow></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-492" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-493" class="mjx-mrow"><span id="MJXc-Node-494" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.313em;">2</span></span><span id="MJXc-Node-495" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.306em; padding-bottom: 0.313em;"> </span></span><span id="MJXc-Node-496" class="mjx-texatom"><span id="MJXc-Node-497" class="mjx-mrow"><span id="MJXc-Node-498" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.313em;">M</span></span><span id="MJXc-Node-499" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.436em; padding-bottom: 0.313em;">Ω</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mtext> </mtext><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">M</mi><mi mathvariant="normal">Ω</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-37">2~\mathrm{M\Omega}</script></span> and a <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-38-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>m</mi><mi>i</mi><mi>n</mi></mrow></msub></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-500" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-501" class="mjx-mrow"><span id="MJXc-Node-502" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-503" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-504" class="mjx-texatom" style=""><span id="MJXc-Node-505" class="mjx-mrow"><span id="MJXc-Node-506" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">m</span></span><span id="MJXc-Node-507" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">i</span></span><span id="MJXc-Node-508" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">n</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>m</mi><mi>i</mi><mi>n</mi></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-38">R_{min}</script></span> of approximately <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-39-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mtext>&#xA0;</mtext><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">M</mi><mi mathvariant="normal">&#x03A9;</mi></mrow></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-509" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-510" class="mjx-mrow"><span id="MJXc-Node-511" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.313em;">1</span></span><span id="MJXc-Node-512" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.306em; padding-bottom: 0.313em;"> </span></span><span id="MJXc-Node-513" class="mjx-texatom"><span id="MJXc-Node-514" class="mjx-mrow"><span id="MJXc-Node-515" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.313em;">M</span></span><span id="MJXc-Node-516" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.436em; padding-bottom: 0.313em;">Ω</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mtext> </mtext><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">M</mi><mi mathvariant="normal">Ω</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-39">1~\mathrm{M\Omega}</script></span>. By using equation [eq:k23] as a guideline we choose <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-40-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>r</mi><mi>e</mi><mi>f</mi></mrow></msub></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-517" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-518" class="mjx-mrow"><span id="MJXc-Node-519" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-520" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.23em; padding-right: 0.071em;"><span id="MJXc-Node-521" class="mjx-texatom" style=""><span id="MJXc-Node-522" class="mjx-mrow"><span id="MJXc-Node-523" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">r</span></span><span id="MJXc-Node-524" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">e</span></span><span id="MJXc-Node-525" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.498em; padding-bottom: 0.498em; padding-right: 0.06em;">f</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>r</mi><mi>e</mi><mi>f</mi></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-40">R_{ref}</script></span> to be <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-41-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1.5</mn><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">M</mi><mi mathvariant="normal">&#x03A9;</mi></mrow></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-526" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-527" class="mjx-mrow"><span id="MJXc-Node-528" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.374em;">1.5</span></span><span id="MJXc-Node-529" class="mjx-texatom"><span id="MJXc-Node-530" class="mjx-mrow"><span id="MJXc-Node-531" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.313em;">M</span></span><span id="MJXc-Node-532" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.436em; padding-bottom: 0.313em;">Ω</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1.5</mn><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">M</mi><mi mathvariant="normal">Ω</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-41">1.5\mathrm{M\Omega}</script></span></p>
| + | |
| − | <h2 id="light-emitting-diode-led">Light emitting diode (LED)</h2>
| + | |
| − | <p>To detect green fluorophores we choose a blue LED with peak emission at 470 nm. For optimum performance we choose the brightest LED we could find. The LED used has a luminous intensity of 12 cd, a maximum current of 20 mA and a forward voltage of 3.2 V</p>
| + | |
| − | <h2 id="npn-transistor">NPN transistor</h2>
| + | |
| − | <p>To provide a stable illumination, it is crucial to supply the LED with a constant voltage. We therefore control it via a NPN transistor with type designation BC635. Its base-emitter-on voltage is 2 V.</p>
| + | |
| − | <h2 id="control-circuit-for-the-led">Control circuit for the LED</h2>
| + | |
| − | <p>The digital output pin is connected via a voltage divider to the base of the transistor. The voltage divider consist of the resistor <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-42-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mn>1</mn></msub></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-533" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-534" class="mjx-mrow"><span id="MJXc-Node-535" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-536" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-537" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.313em;">1</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mn>1</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-42">R_1</script></span> with a resistance of <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-43-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mn>1</mn><mtext>&#xA0;</mtext><mi mathvariant="normal">k</mi><mi mathvariant="normal">&#x03A9;</mi></mrow></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-538" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-539" class="mjx-mrow"><span id="MJXc-Node-540" class="mjx-texatom"><span id="MJXc-Node-541" class="mjx-mrow"><span id="MJXc-Node-542" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.313em;">1</span></span><span id="MJXc-Node-543" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.306em; padding-bottom: 0.313em;"> </span></span><span id="MJXc-Node-544" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.436em; padding-bottom: 0.313em;">k</span></span><span id="MJXc-Node-545" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.436em; padding-bottom: 0.313em;">Ω</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mn>1</mn><mtext> </mtext><mi mathvariant="normal">k</mi><mi mathvariant="normal">Ω</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-43">\mathrm{1~k\Omega}</script></span> and the resistor <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-44-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mn>2</mn></msub></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-546" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-547" class="mjx-mrow"><span id="MJXc-Node-548" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-549" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-550" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.313em;">2</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mn>2</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-44">R_2</script></span> with a resistance of <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-45-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mn>9.1</mn><mtext>&#xA0;</mtext><mi mathvariant="normal">k</mi><mi mathvariant="normal">&#x03A9;</mi></mrow></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-551" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-552" class="mjx-mrow"><span id="MJXc-Node-553" class="mjx-texatom"><span id="MJXc-Node-554" class="mjx-mrow"><span id="MJXc-Node-555" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.374em;">9.1</span></span><span id="MJXc-Node-556" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.306em; padding-bottom: 0.313em;"> </span></span><span id="MJXc-Node-557" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.436em; padding-bottom: 0.313em;">k</span></span><span id="MJXc-Node-558" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.436em; padding-bottom: 0.313em;">Ω</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mn>9.1</mn><mtext> </mtext><mi mathvariant="normal">k</mi><mi mathvariant="normal">Ω</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-45">\mathrm{9.1~k\Omega}</script></span>. When the output pin is set to 5 V a voltage of 4.5 V is present at the base of the transistor. This is above the base-emitter-on voltage and the LED is turned on. The resistor <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-46-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mn>3</mn></msub></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-559" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-560" class="mjx-mrow"><span id="MJXc-Node-561" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-562" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-563" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.374em;">3</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mn>3</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-46">R_3</script></span> with resistance <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-47-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>39</mn><mrow class="MJX-TeXAtom-ORD"><mtext>&#xA0;</mtext><mi mathvariant="normal">&#x03A9;</mi></mrow></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-564" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-565" class="mjx-mrow"><span id="MJXc-Node-566" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.374em;">39</span></span><span id="MJXc-Node-567" class="mjx-texatom"><span id="MJXc-Node-568" class="mjx-mrow"><span id="MJXc-Node-569" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.306em; padding-bottom: 0.313em;"> </span></span><span id="MJXc-Node-570" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.436em; padding-bottom: 0.313em;">Ω</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>39</mn><mrow class="MJX-TeXAtom-ORD"><mtext> </mtext><mi mathvariant="normal">Ω</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-47">39\mathrm{~\Omega}</script></span> was chosen empirically to limit the LEDs working current and to power it at maximum brightness.</p>
| + | |
| − | <h2 id="light-filter-foils">Light filter foils</h2>
| + | |
| − | <p>For filtering the emission and excitation light we used filter foils from LEE filters. The producer provides transmission spectra for every filter foil. For the excitation filter we choose the filter colour “TOKYO BLUE” and for the emission filter we choose the filter colour “RUST”. We took care that the transmission spectra of the filters have no overlap. We measured the transmissions spectra of the foils our self. The data are shown in image [img:k2]. We also measured the transmission spectra of two blue filter foils combined with one orange filter foil. For this combination there is nearly no transmission up to a wavelength of 700 nm. This combination is sufficient for blocking the excitation light of the blue LED from reaching the LDR.</p>
| + | |
| − | <div class="figure">
| + | |
| − | <img src="./testSave_files/spectra.png" alt="emission spectra of light filter foils">
| + | |
| − | <p class="caption">emission spectra of light filter foils<span data-label="img:k2"></span></p>
| + | |
| − | </div>
| + | |
| − | <h2 id="filter-paper">Filter paper</h2>
| + | |
| − | <p>We choose glass fiber filter paper from Whatman Laboratory Products with type designation “934-AH” to detect fluorescence on. Because normal cellulose filter paper is auto fluorescent and causes a high background signal.</p>
| + | |
| − | <h2 id="d-printed-parts">3D printed parts</h2>
| + | |
| − | <p>We intended to have the LED,the fluorescence sample and the LDR in direct proximity. This ensures a maximum use excitation light from the LED and emission light from the fluorophores. We therefore choose a sandwich-like design for our sample holder that can be placed into a slot were a detection system snaps in keeps the sample in position.</p>
| + | |
| − | <p>In detail, two blue filter foils are stacked and glued with tape in front of two excitation windows of the upper halve of the sandwich. One orange filter foil is glued in front of the two detection windows on the lower half of the sandwich. The two detection and emission windows enable us to measure a blank sample and an actual sample with the exact same set-up. Care needs to be taken that no tape is covering the detection or excitation windows because tape is usually auto fluorescent and causes a high background signal. A piece of filter paper is placed between the two halves of the sandwich. The upper and lower part of this sandwich and pressed together with magnets and hold the filter paper in position. The magnets ensure an user-friendly exchange of filter papers. The sandwich can now be inserted into the slot of the detection device. The LED and the LDR are mounted onto beams at opposite sites of the detection device. The sandwich snaps in when the LED and the LDR are at the right position under the excitation and over detection window. Four magnets apply an additional force to the beams and press the LED and the LDR close together. The design ensures that the distance between filter paper, LED and LDR is only limited by the thickness of the filter foils.</p>
| + | |
| − | <h2 id="list-of-materials-and-cost-calculation">List of materials and cost calculation</h2>
| + | |
| − | <table>
| + | |
| − | <caption>List of materials</caption>
| + | |
| − | <thead>
| + | |
| − | <tr class="header">
| + | |
| − | <th align="left">used item</th>
| + | |
| − | <th align="right">price of components per detector in EUR</th>
| + | |
| − | </tr>
| + | |
| − | </thead>
| + | |
| − | <tbody>
| + | |
| − | <tr class="odd">
| + | |
| − | <td align="left">10 Magnets</td>
| + | |
| − | <td align="right">4,8</td> | + | |
| − | </tr>
| + | |
| − | <tr class="even">
| + | |
| − | <td align="left">8 Resistors</td> | + | |
| − | <td align="right">0,48</td> | + | |
| − | </tr>
| + | |
| − | <tr class="odd">
| + | |
| − | <td align="left">2 NPN Transistors</td> | + | |
| − | <td align="right">0,12</td> | + | |
| − | </tr>
| + | |
| − | <tr class="even">
| + | |
| − | <td align="left">Green LED</td>
| + | |
| − | <td align="right">0,1</td>
| + | |
| − | </tr>
| + | |
| − | <tr class="odd">
| + | |
| − | <td align="left">Blue LED</td>
| + | |
| − | <td align="right">0,1</td>
| + | |
| − | </tr>
| + | |
| − | <tr class="even">
| + | |
| − | <td align="left">Geekcreit ATmega328P Arduino Kompatibel Nano</td>
| + | |
| − | <td align="right">2,51</td>
| + | |
| | </tr> | | </tr> |
| − | <tr class="odd"> | + | <tr id="hardwareFrontPage"> |
| − | <td align="left">LDR</td>
| + | <td colspan=2> |
| − | <td align="right">0,15</td>
| + | <table width=320> |
| − | </tr>
| + | |
| − | <tr class="even">
| + | |
| − | <td align="left">USB cable</td>
| + | |
| − | <td align="right">1,5</td>
| + | |
| − | </tr>
| + | |
| − | <tr class="odd">
| + | |
| − | <td align="left">Hole raster board</td>
| + | |
| − | <td align="right">0,42</td>
| + | |
| − | </tr>
| + | |
| − | <tr class="even">
| + | |
| − | <td align="left">ca 100 g PLA</td>
| + | |
| − | <td align="right">3</td>
| + | |
| − | </tr>
| + | |
| − | <tr class="odd">
| + | |
| − | <td align="left">M3 screws</td>
| + | |
| − | <td align="right">0,01</td>
| + | |
| − | </tr>
| + | |
| − | <tr class="even">
| + | |
| − | <td align="left">M3 nut</td>
| + | |
| − | <td align="right">0,01</td>
| + | |
| − | </tr>
| + | |
| − | <tr class="odd">
| + | |
| − | <td align="left">Filter foils</td>
| + | |
| − | <td align="right">0,1</td>
| + | |
| − | </tr>
| + | |
| − | <tr class="even">
| + | |
| − | <td align="left">Total cost</td>
| + | |
| − | <td align="right">13,3</td>
| + | |
| − | </tr>
| + | |
| − | </tbody>
| + | |
| − | </table>
| + | |
| − | <p>[tab:k20]</p>
| + | |
| − | <p>We keept an eye on using only low cost and easy available items for the construction of our detector.</p>
| + | |
| − | <h1 id="calibration">Calibration</h1>
| + | |
| − | <h2 id="derivation-of-a-calibration-function">Derivation of a calibration function</h2>
| + | |
| − | <p>We want to find an equation that relates fluorophore concentration <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-48-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-571" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-572" class="mjx-mrow"><span id="MJXc-Node-573" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">c</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math></span></span><script type="math/tex" id="MathJax-Element-48">c</script></span> and the corresponding resistance <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-49-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-574" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-575" class="mjx-mrow"><span id="MJXc-Node-576" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math></span></span><script type="math/tex" id="MathJax-Element-49">R</script></span> of the LDR.</p>
| + | |
| − | <p>The light intensity I at the LDR during a fluorescence measurement is a sum of signal intensity <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-50-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>I</mi><mi>s</mi></msub></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-577" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-578" class="mjx-mrow"><span id="MJXc-Node-579" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.064em;"><span id="MJXc-Node-580" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em; padding-right: 0.064em;">I</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-581" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">s</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>I</mi><mi>s</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-50">I_s</script></span> and background intensity <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-51-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>I</mi><mi>b</mi></msub></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-582" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-583" class="mjx-mrow"><span id="MJXc-Node-584" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.064em;"><span id="MJXc-Node-585" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em; padding-right: 0.064em;">I</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.219em; padding-right: 0.071em;"><span id="MJXc-Node-586" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.498em; padding-bottom: 0.313em;">b</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>I</mi><mi>b</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-51">I_b</script></span>: <span class="math display"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-52-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>I</mi><mo>=</mo><msub><mi>I</mi><mi>s</mi></msub><mo>+</mo><msub><mi>I</mi><mi>b</mi></msub></math>" role="presentation" style="font-size: 101%; text-align: center; position: relative;"><span id="MJXc-Node-587" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-588" class="mjx-mrow"><span id="MJXc-Node-589" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em; padding-right: 0.064em;">I</span></span><span id="MJXc-Node-590" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.065em; padding-bottom: 0.313em;">=</span></span><span id="MJXc-Node-591" class="mjx-msubsup MJXc-space3"><span class="mjx-base" style="margin-right: -0.064em;"><span id="MJXc-Node-592" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em; padding-right: 0.064em;">I</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-593" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">s</span></span></span></span><span id="MJXc-Node-594" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.313em; padding-bottom: 0.436em;">+</span></span><span id="MJXc-Node-595" class="mjx-msubsup MJXc-space2"><span class="mjx-base" style="margin-right: -0.064em;"><span id="MJXc-Node-596" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em; padding-right: 0.064em;">I</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.219em; padding-right: 0.071em;"><span id="MJXc-Node-597" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.498em; padding-bottom: 0.313em;">b</span></span></span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>I</mi><mo>=</mo><msub><mi>I</mi><mi>s</mi></msub><mo>+</mo><msub><mi>I</mi><mi>b</mi></msub></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-52">\label{eq:lk1}
| + | |
| − | I = I_s + I_b</script></span></p>
| + | |
| − | <p>We assumed that <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-53-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>I</mi><mi>b</mi></msub></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-598" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-599" class="mjx-mrow"><span id="MJXc-Node-600" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.064em;"><span id="MJXc-Node-601" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em; padding-right: 0.064em;">I</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.219em; padding-right: 0.071em;"><span id="MJXc-Node-602" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.498em; padding-bottom: 0.313em;">b</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>I</mi><mi>b</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-53">I_b</script></span> is the intensity for a water sample. Importantly, wet samples give different background signal than dry ones, suspectedly due to different light scattering in the filter paper. With equation [eq:k1] this gives an equation for the resistance <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-54-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mi>b</mi></msub></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-603" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-604" class="mjx-mrow"><span id="MJXc-Node-605" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-606" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.219em; padding-right: 0.071em;"><span id="MJXc-Node-607" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.498em; padding-bottom: 0.313em;">b</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mi>b</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-54">R_b</script></span> of a water sample and for the resistance <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-55-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-608" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-609" class="mjx-mrow"><span id="MJXc-Node-610" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-611" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-612" class="mjx-texatom" style=""><span id="MJXc-Node-613" class="mjx-mrow"><span id="MJXc-Node-614" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">L</span></span><span id="MJXc-Node-615" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">D</span></span><span id="MJXc-Node-616" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-55">R_{LDR}</script></span> for a fluorescence sample: <span class="math display"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-56-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>b</mi></mrow></msub><mo>&#x221D;</mo></mtd><mtd><msubsup><mi>I</mi><mrow class="MJX-TeXAtom-ORD"><mi>b</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mo>&#x2212;</mo><mi>&#x03B3;</mi></mrow></msubsup></mtd></mtr><mtr><mtd><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub><mo>&#x221D;</mo></mtd><mtd><msup><mrow><mo>(</mo><mrow><msub><mi>I</mi><mrow class="MJX-TeXAtom-ORD"><mi>b</mi></mrow></msub><mo>+</mo><msub><mi>I</mi><mrow class="MJX-TeXAtom-ORD"><mi>s</mi></mrow></msub></mrow><mo>)</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo>&#x2212;</mo><mi>&#x03B3;</mi></mrow></msup></mtd></mtr></mtable></math>" role="presentation" style="font-size: 101%; text-align: center; position: relative;"><span id="MJXc-Node-617" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-618" class="mjx-mrow"><span id="MJXc-Node-619" class="mjx-mtable" style="vertical-align: -1.161em; padding: 0px 0.167em;"><span class="mjx-table"><span id="MJXc-Node-620" class="mjx-mtr" style="height: 1.464em;"><span id="MJXc-Node-621" class="mjx-mtd" style="padding: 0px; text-align: right; width: 3.454em;"><span id="MJXc-Node-622" class="mjx-mrow" style="margin-top: -0.046em;"><span id="MJXc-Node-623" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-624" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.219em; padding-right: 0.071em;"><span id="MJXc-Node-625" class="mjx-texatom" style=""><span id="MJXc-Node-626" class="mjx-mrow"><span id="MJXc-Node-627" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.498em; padding-bottom: 0.313em;">b</span></span></span></span></span></span><span id="MJXc-Node-628" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.127em; padding-bottom: 0.374em;">∝</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-629" class="mjx-mtd" style="padding: 0px; text-align: left; width: 4.556em;"><span id="MJXc-Node-630" class="mjx-mrow" style="margin-top: -0.046em;"><span id="MJXc-Node-631" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.064em;"><span id="MJXc-Node-632" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em; padding-right: 0.064em;">I</span></span></span><span class="mjx-stack" style="vertical-align: -0.335em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.255em; padding-left: 0.188em; padding-right: 0.071em;"><span id="MJXc-Node-636" class="mjx-texatom" style=""><span id="MJXc-Node-637" class="mjx-mrow"><span id="MJXc-Node-638" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.313em; padding-bottom: 0.436em;">−</span></span><span id="MJXc-Node-639" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.189em; padding-bottom: 0.498em; padding-right: 0.025em;">γ</span></span></span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-633" class="mjx-texatom" style=""><span id="MJXc-Node-634" class="mjx-mrow"><span id="MJXc-Node-635" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.498em; padding-bottom: 0.313em;">b</span></span></span></span></span></span></span><span class="mjx-strut"></span></span></span></span><span id="MJXc-Node-640" class="mjx-mtr" style="height: 1.357em;"><span id="MJXc-Node-641" class="mjx-mtd" style="padding: 0.15em 0px 0px; text-align: right;"><span id="MJXc-Node-642" class="mjx-mrow" style="margin-top: -0.068em;"><span id="MJXc-Node-643" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-644" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-645" class="mjx-texatom" style=""><span id="MJXc-Node-646" class="mjx-mrow"><span id="MJXc-Node-647" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">L</span></span><span id="MJXc-Node-648" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">D</span></span><span id="MJXc-Node-649" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span></span></span></span><span id="MJXc-Node-650" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.127em; padding-bottom: 0.374em;">∝</span></span><span class="mjx-strut"></span></span></span><span id="MJXc-Node-651" class="mjx-mtd" style="padding: 0.15em 0px 0px; text-align: left;"><span id="MJXc-Node-652" class="mjx-mrow" style="margin-top: -0.068em;"><span id="MJXc-Node-653" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-654" class="mjx-mrow"><span id="MJXc-Node-655" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.436em; padding-bottom: 0.622em;">(</span></span><span id="MJXc-Node-656" class="mjx-mrow"><span id="MJXc-Node-657" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.064em;"><span id="MJXc-Node-658" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em; padding-right: 0.064em;">I</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.219em; padding-right: 0.071em;"><span id="MJXc-Node-659" class="mjx-texatom" style=""><span id="MJXc-Node-660" class="mjx-mrow"><span id="MJXc-Node-661" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.498em; padding-bottom: 0.313em;">b</span></span></span></span></span></span><span id="MJXc-Node-662" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.313em; padding-bottom: 0.436em;">+</span></span><span id="MJXc-Node-663" class="mjx-msubsup MJXc-space2"><span class="mjx-base" style="margin-right: -0.064em;"><span id="MJXc-Node-664" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em; padding-right: 0.064em;">I</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-665" class="mjx-texatom" style=""><span id="MJXc-Node-666" class="mjx-mrow"><span id="MJXc-Node-667" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">s</span></span></span></span></span></span></span><span id="MJXc-Node-668" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.436em; padding-bottom: 0.622em;">)</span></span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.71em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-669" class="mjx-texatom" style=""><span id="MJXc-Node-670" class="mjx-mrow"><span id="MJXc-Node-671" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.313em; padding-bottom: 0.436em;">−</span></span><span id="MJXc-Node-672" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.189em; padding-bottom: 0.498em; padding-right: 0.025em;">γ</span></span></span></span></span></span><span class="mjx-strut"></span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>b</mi></mrow></msub><mo>∝</mo></mtd><mtd><msubsup><mi>I</mi><mrow class="MJX-TeXAtom-ORD"><mi>b</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mi>γ</mi></mrow></msubsup></mtd></mtr><mtr><mtd><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub><mo>∝</mo></mtd><mtd><msup><mrow><mo>(</mo><mrow><msub><mi>I</mi><mrow class="MJX-TeXAtom-ORD"><mi>b</mi></mrow></msub><mo>+</mo><msub><mi>I</mi><mrow class="MJX-TeXAtom-ORD"><mi>s</mi></mrow></msub></mrow><mo>)</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mi>γ</mi></mrow></msup></mtd></mtr></mtable></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-56">\begin{aligned}
| + | |
| − | \label{eq:k11}
| + | |
| − | R_{b} \propto & I_{b}^{-\gamma}\\
| + | |
| − | \label{eq:k12}
| + | |
| − | R_{LDR} \propto & \left( I_{b}+ I_{s} \right)^{-\gamma}\end{aligned}</script></span> Equation [eq:k12] is normalizes by equation [eq:k11]: <span class="math display"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-57-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mfrac><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>b</mi></mrow></msub></mfrac><mo>=</mo><msup><mrow><mo>(</mo><mfrac><mrow><msub><mi>I</mi><mrow class="MJX-TeXAtom-ORD"><mi>b</mi></mrow></msub><mo>+</mo><msub><mi>I</mi><mrow class="MJX-TeXAtom-ORD"><mi>s</mi></mrow></msub></mrow><msub><mi>I</mi><mrow class="MJX-TeXAtom-ORD"><mi>b</mi></mrow></msub></mfrac><mo>)</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo>&#x2212;</mo><mi>&#x03B3;</mi></mrow></msup><mo>=</mo><msup><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mfrac><msub><mi>I</mi><mrow class="MJX-TeXAtom-ORD"><mi>s</mi></mrow></msub><msub><mi>I</mi><mrow class="MJX-TeXAtom-ORD"><mi>b</mi></mrow></msub></mfrac></mrow><mo>)</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo>&#x2212;</mo><mi>&#x03B3;</mi></mrow></msup></math>" role="presentation" style="font-size: 101%; text-align: center; position: relative;"><span id="MJXc-Node-673" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-674" class="mjx-mrow"><span id="MJXc-Node-675" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 2.598em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 2.598em; top: -1.394em;"><span id="MJXc-Node-676" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-677" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-678" class="mjx-texatom" style=""><span id="MJXc-Node-679" class="mjx-mrow"><span id="MJXc-Node-680" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">L</span></span><span id="MJXc-Node-681" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">D</span></span><span id="MJXc-Node-682" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span></span></span></span></span><span class="mjx-denominator" style="width: 2.598em; bottom: -0.892em;"><span id="MJXc-Node-683" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-684" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.219em; padding-right: 0.071em;"><span id="MJXc-Node-685" class="mjx-texatom" style=""><span id="MJXc-Node-686" class="mjx-mrow"><span id="MJXc-Node-687" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.498em; padding-bottom: 0.313em;">b</span></span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.289em; width: 2.598em;"></span></span><span class="mjx-vsize" style="height: 2.286em; vertical-align: -0.892em;"></span></span><span id="MJXc-Node-688" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.065em; padding-bottom: 0.313em;">=</span></span><span id="MJXc-Node-689" class="mjx-msubsup MJXc-space3"><span class="mjx-base"><span id="MJXc-Node-690" class="mjx-mrow"><span id="MJXc-Node-691" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.241em; padding-bottom: 1.241em;">(</span></span><span id="MJXc-Node-692" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 3.008em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 3.008em; top: -1.392em;"><span id="MJXc-Node-693" class="mjx-mrow"><span id="MJXc-Node-694" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.064em;"><span id="MJXc-Node-695" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em; padding-right: 0.064em;">I</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.219em; padding-right: 0.071em;"><span id="MJXc-Node-696" class="mjx-texatom" style=""><span id="MJXc-Node-697" class="mjx-mrow"><span id="MJXc-Node-698" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.498em; padding-bottom: 0.313em;">b</span></span></span></span></span></span><span id="MJXc-Node-699" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.313em; padding-bottom: 0.436em;">+</span></span><span id="MJXc-Node-700" class="mjx-msubsup MJXc-space2"><span class="mjx-base" style="margin-right: -0.064em;"><span id="MJXc-Node-701" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em; padding-right: 0.064em;">I</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-702" class="mjx-texatom" style=""><span id="MJXc-Node-703" class="mjx-mrow"><span id="MJXc-Node-704" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">s</span></span></span></span></span></span></span></span><span class="mjx-denominator" style="width: 3.008em; bottom: -0.892em;"><span id="MJXc-Node-705" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.064em;"><span id="MJXc-Node-706" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em; padding-right: 0.064em;">I</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.219em; padding-right: 0.071em;"><span id="MJXc-Node-707" class="mjx-texatom" style=""><span id="MJXc-Node-708" class="mjx-mrow"><span id="MJXc-Node-709" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.498em; padding-bottom: 0.313em;">b</span></span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.289em; width: 3.008em;"></span></span><span class="mjx-vsize" style="height: 2.283em; vertical-align: -0.892em;"></span></span><span id="MJXc-Node-710" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.241em; padding-bottom: 1.241em;">)</span></span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 1.7em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-711" class="mjx-texatom" style=""><span id="MJXc-Node-712" class="mjx-mrow"><span id="MJXc-Node-713" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.313em; padding-bottom: 0.436em;">−</span></span><span id="MJXc-Node-714" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.189em; padding-bottom: 0.498em; padding-right: 0.025em;">γ</span></span></span></span></span></span><span id="MJXc-Node-715" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.065em; padding-bottom: 0.313em;">=</span></span><span id="MJXc-Node-716" class="mjx-msubsup MJXc-space3"><span class="mjx-base"><span id="MJXc-Node-717" class="mjx-mrow"><span id="MJXc-Node-718" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.241em; padding-bottom: 1.241em;">(</span></span><span id="MJXc-Node-719" class="mjx-mrow"><span id="MJXc-Node-720" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.313em;">1</span></span><span id="MJXc-Node-721" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.313em; padding-bottom: 0.436em;">+</span></span><span id="MJXc-Node-722" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 1.007em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.007em; top: -1.386em;"><span id="MJXc-Node-723" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.064em;"><span id="MJXc-Node-724" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em; padding-right: 0.064em;">I</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-725" class="mjx-texatom" style=""><span id="MJXc-Node-726" class="mjx-mrow"><span id="MJXc-Node-727" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">s</span></span></span></span></span></span></span><span class="mjx-denominator" style="width: 1.007em; bottom: -0.892em;"><span id="MJXc-Node-728" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.064em;"><span id="MJXc-Node-729" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em; padding-right: 0.064em;">I</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.219em; padding-right: 0.071em;"><span id="MJXc-Node-730" class="mjx-texatom" style=""><span id="MJXc-Node-731" class="mjx-mrow"><span id="MJXc-Node-732" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.498em; padding-bottom: 0.313em;">b</span></span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.289em; width: 1.007em;"></span></span><span class="mjx-vsize" style="height: 2.278em; vertical-align: -0.892em;"></span></span></span><span id="MJXc-Node-733" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.241em; padding-bottom: 1.241em;">)</span></span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 1.7em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-734" class="mjx-texatom" style=""><span id="MJXc-Node-735" class="mjx-mrow"><span id="MJXc-Node-736" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.313em; padding-bottom: 0.436em;">−</span></span><span id="MJXc-Node-737" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.189em; padding-bottom: 0.498em; padding-right: 0.025em;">γ</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mfrac><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>b</mi></mrow></msub></mfrac><mo>=</mo><msup><mrow><mo>(</mo><mfrac><mrow><msub><mi>I</mi><mrow class="MJX-TeXAtom-ORD"><mi>b</mi></mrow></msub><mo>+</mo><msub><mi>I</mi><mrow class="MJX-TeXAtom-ORD"><mi>s</mi></mrow></msub></mrow><msub><mi>I</mi><mrow class="MJX-TeXAtom-ORD"><mi>b</mi></mrow></msub></mfrac><mo>)</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mi>γ</mi></mrow></msup><mo>=</mo><msup><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mfrac><msub><mi>I</mi><mrow class="MJX-TeXAtom-ORD"><mi>s</mi></mrow></msub><msub><mi>I</mi><mrow class="MJX-TeXAtom-ORD"><mi>b</mi></mrow></msub></mfrac></mrow><mo>)</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mi>γ</mi></mrow></msup></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-57">\label{eq:k13}
| + | |
| − | \frac{R_{LDR}}{R_{b}}=\left(\frac{I_{b}+ I_{s}}{ I_{b}}\right)^{-\gamma}=\left(1+\frac{I_{s}}{I_{b}}\right)^{-\gamma}</script></span></p>
| + | |
| − | <p>We assumed that the intensity <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-58-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>I</mi><mrow class="MJX-TeXAtom-ORD"><mi>s</mi></mrow></msub></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-738" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-739" class="mjx-mrow"><span id="MJXc-Node-740" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.064em;"><span id="MJXc-Node-741" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em; padding-right: 0.064em;">I</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-742" class="mjx-texatom" style=""><span id="MJXc-Node-743" class="mjx-mrow"><span id="MJXc-Node-744" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">s</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>I</mi><mrow class="MJX-TeXAtom-ORD"><mi>s</mi></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-58">I_{s}</script></span> of fluorescence light depends linearly on the concentration <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-59-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-745" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-746" class="mjx-mrow"><span id="MJXc-Node-747" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">c</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math></span></span><script type="math/tex" id="MathJax-Element-59">c</script></span> of fluorophores and the light intensity <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-60-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>I</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-748" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-749" class="mjx-mrow"><span id="MJXc-Node-750" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.064em;"><span id="MJXc-Node-751" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em; padding-right: 0.064em;">I</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-752" class="mjx-texatom" style=""><span id="MJXc-Node-753" class="mjx-mrow"><span id="MJXc-Node-754" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.374em;">0</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>I</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-60">I_{0}</script></span> produced by the LED: <span class="math display"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-61-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>I</mi><mrow class="MJX-TeXAtom-ORD"><mi>s</mi></mrow></msub><mo>&#x221D;</mo><msub><mi>I</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>&#x22C5;</mo><mi>c</mi></math>" role="presentation" style="font-size: 101%; text-align: center; position: relative;"><span id="MJXc-Node-755" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-756" class="mjx-mrow"><span id="MJXc-Node-757" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.064em;"><span id="MJXc-Node-758" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em; padding-right: 0.064em;">I</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-759" class="mjx-texatom" style=""><span id="MJXc-Node-760" class="mjx-mrow"><span id="MJXc-Node-761" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">s</span></span></span></span></span></span><span id="MJXc-Node-762" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.127em; padding-bottom: 0.374em;">∝</span></span><span id="MJXc-Node-763" class="mjx-msubsup MJXc-space3"><span class="mjx-base" style="margin-right: -0.064em;"><span id="MJXc-Node-764" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em; padding-right: 0.064em;">I</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-765" class="mjx-texatom" style=""><span id="MJXc-Node-766" class="mjx-mrow"><span id="MJXc-Node-767" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.374em;">0</span></span></span></span></span></span><span id="MJXc-Node-768" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.003em; padding-bottom: 0.313em;">⋅</span></span><span id="MJXc-Node-769" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">c</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>I</mi><mrow class="MJX-TeXAtom-ORD"><mi>s</mi></mrow></msub><mo>∝</mo><msub><mi>I</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub><mo>⋅</mo><mi>c</mi></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-61">\label{eq:k15}
| + | |
| − | I_{s}\propto I_{0}\cdot c</script></span> And we assumed that the background intensity <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-62-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>I</mi><mrow class="MJX-TeXAtom-ORD"><mi>b</mi></mrow></msub></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-770" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-771" class="mjx-mrow"><span id="MJXc-Node-772" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.064em;"><span id="MJXc-Node-773" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em; padding-right: 0.064em;">I</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.219em; padding-right: 0.071em;"><span id="MJXc-Node-774" class="mjx-texatom" style=""><span id="MJXc-Node-775" class="mjx-mrow"><span id="MJXc-Node-776" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.498em; padding-bottom: 0.313em;">b</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>I</mi><mrow class="MJX-TeXAtom-ORD"><mi>b</mi></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-62">I_{b}</script></span> depends also linearly on <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-63-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>I</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-777" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-778" class="mjx-mrow"><span id="MJXc-Node-779" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.064em;"><span id="MJXc-Node-780" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em; padding-right: 0.064em;">I</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-781" class="mjx-texatom" style=""><span id="MJXc-Node-782" class="mjx-mrow"><span id="MJXc-Node-783" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.374em;">0</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>I</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-63">I_{0}</script></span>. <span class="math display"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-64-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>I</mi><mrow class="MJX-TeXAtom-ORD"><mi>b</mi></mrow></msub><mo>&#x221D;</mo><msub><mi>I</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub></math>" role="presentation" style="font-size: 101%; text-align: center; position: relative;"><span id="MJXc-Node-784" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-785" class="mjx-mrow"><span id="MJXc-Node-786" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.064em;"><span id="MJXc-Node-787" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em; padding-right: 0.064em;">I</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.219em; padding-right: 0.071em;"><span id="MJXc-Node-788" class="mjx-texatom" style=""><span id="MJXc-Node-789" class="mjx-mrow"><span id="MJXc-Node-790" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.498em; padding-bottom: 0.313em;">b</span></span></span></span></span></span><span id="MJXc-Node-791" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.127em; padding-bottom: 0.374em;">∝</span></span><span id="MJXc-Node-792" class="mjx-msubsup MJXc-space3"><span class="mjx-base" style="margin-right: -0.064em;"><span id="MJXc-Node-793" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em; padding-right: 0.064em;">I</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-794" class="mjx-texatom" style=""><span id="MJXc-Node-795" class="mjx-mrow"><span id="MJXc-Node-796" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.374em;">0</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>I</mi><mrow class="MJX-TeXAtom-ORD"><mi>b</mi></mrow></msub><mo>∝</mo><msub><mi>I</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-64">\label{eq:k16}
| + | |
| − | I_{b} \propto I_{0}</script></span></p>
| + | |
| − | <p>The fraction <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-65-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msub><mi>I</mi><mrow class="MJX-TeXAtom-ORD"><mi>s</mi></mrow></msub><msub><mi>I</mi><mrow class="MJX-TeXAtom-ORD"><mi>b</mi></mrow></msub></mfrac></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-797" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-798" class="mjx-mrow"><span id="MJXc-Node-799" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.712em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 1.007em; top: -1.498em;"><span id="MJXc-Node-800" class="mjx-msubsup" style=""><span class="mjx-base" style="margin-right: -0.064em;"><span id="MJXc-Node-801" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em; padding-right: 0.064em;">I</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-802" class="mjx-texatom" style=""><span id="MJXc-Node-803" class="mjx-mrow"><span id="MJXc-Node-804" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">s</span></span></span></span></span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 1.007em; bottom: -0.796em;"><span id="MJXc-Node-805" class="mjx-msubsup" style=""><span class="mjx-base" style="margin-right: -0.064em;"><span id="MJXc-Node-806" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em; padding-right: 0.064em;">I</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.219em; padding-right: 0.071em;"><span id="MJXc-Node-807" class="mjx-texatom" style=""><span id="MJXc-Node-808" class="mjx-mrow"><span id="MJXc-Node-809" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.498em; padding-bottom: 0.313em;">b</span></span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.289em; width: 0.712em;"></span></span><span class="mjx-vsize" style="height: 1.622em; vertical-align: -0.563em;"></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msub><mi>I</mi><mrow class="MJX-TeXAtom-ORD"><mi>s</mi></mrow></msub><msub><mi>I</mi><mrow class="MJX-TeXAtom-ORD"><mi>b</mi></mrow></msub></mfrac></math></span></span><script type="math/tex" id="MathJax-Element-65">\frac{I_{s}}{I_{b}}</script></span> does not depend on the intensity <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-66-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>I</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-810" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-811" class="mjx-mrow"><span id="MJXc-Node-812" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.064em;"><span id="MJXc-Node-813" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em; padding-right: 0.064em;">I</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-814" class="mjx-texatom" style=""><span id="MJXc-Node-815" class="mjx-mrow"><span id="MJXc-Node-816" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.374em;">0</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>I</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-66">I_{0}</script></span> of the LED and depends linearly on the concentration <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-67-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-817" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-818" class="mjx-mrow"><span id="MJXc-Node-819" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">c</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math></span></span><script type="math/tex" id="MathJax-Element-67">c</script></span>. <span class="math display"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-68-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtext>&#xA0;</mtext><mfrac><msub><mi>I</mi><mrow class="MJX-TeXAtom-ORD"><mi>s</mi></mrow></msub><msub><mi>I</mi><mrow class="MJX-TeXAtom-ORD"><mi>b</mi></mrow></msub></mfrac><mo>=</mo><mi>k</mi><mo>&#x22C5;</mo><mi>c</mi></math>" role="presentation" style="font-size: 101%; text-align: center; position: relative;"><span id="MJXc-Node-820" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-821" class="mjx-mrow"><span id="MJXc-Node-822" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.306em; padding-bottom: 0.313em;"> </span></span><span id="MJXc-Node-823" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 1.007em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.007em; top: -1.386em;"><span id="MJXc-Node-824" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.064em;"><span id="MJXc-Node-825" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em; padding-right: 0.064em;">I</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-826" class="mjx-texatom" style=""><span id="MJXc-Node-827" class="mjx-mrow"><span id="MJXc-Node-828" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">s</span></span></span></span></span></span></span><span class="mjx-denominator" style="width: 1.007em; bottom: -0.892em;"><span id="MJXc-Node-829" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.064em;"><span id="MJXc-Node-830" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em; padding-right: 0.064em;">I</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.219em; padding-right: 0.071em;"><span id="MJXc-Node-831" class="mjx-texatom" style=""><span id="MJXc-Node-832" class="mjx-mrow"><span id="MJXc-Node-833" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.498em; padding-bottom: 0.313em;">b</span></span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.289em; width: 1.007em;"></span></span><span class="mjx-vsize" style="height: 2.278em; vertical-align: -0.892em;"></span></span><span id="MJXc-Node-834" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.065em; padding-bottom: 0.313em;">=</span></span><span id="MJXc-Node-835" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.498em; padding-bottom: 0.313em;">k</span></span><span id="MJXc-Node-836" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.003em; padding-bottom: 0.313em;">⋅</span></span><span id="MJXc-Node-837" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">c</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtext> </mtext><mfrac><msub><mi>I</mi><mrow class="MJX-TeXAtom-ORD"><mi>s</mi></mrow></msub><msub><mi>I</mi><mrow class="MJX-TeXAtom-ORD"><mi>b</mi></mrow></msub></mfrac><mo>=</mo><mi>k</mi><mo>⋅</mo><mi>c</mi></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-68">~\label{eq:k17}
| + | |
| − | \frac{I_{s}}{I_{b}} = k\cdot c</script></span></p>
| + | |
| − | <p><span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-69-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-838" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-839" class="mjx-mrow"><span id="MJXc-Node-840" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.498em; padding-bottom: 0.313em;">k</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math></span></span><script type="math/tex" id="MathJax-Element-69">k</script></span> is a constant that depends on the transmission spectra of the filter foils, the spectra of the fluorophore, the spectra of the LED and light scattering effects of the filter paper. <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-70-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-841" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-842" class="mjx-mrow"><span id="MJXc-Node-843" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.498em; padding-bottom: 0.313em;">k</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math></span></span><script type="math/tex" id="MathJax-Element-70">k</script></span> can be assumed to be constant for one specific measurement set up. Equation [eq:k17] inserted into equation [eq:k13] gives the final equation relating fluorophore concentration <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-71-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-844" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-845" class="mjx-mrow"><span id="MJXc-Node-846" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">c</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math></span></span><script type="math/tex" id="MathJax-Element-71">c</script></span> and the resistance <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-72-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-847" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-848" class="mjx-mrow"><span id="MJXc-Node-849" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-850" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-851" class="mjx-texatom" style=""><span id="MJXc-Node-852" class="mjx-mrow"><span id="MJXc-Node-853" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">L</span></span><span id="MJXc-Node-854" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">D</span></span><span id="MJXc-Node-855" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-72">R_{LDR}</script></span> of the LDR. <span class="math display"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-73-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mfrac><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>b</mi></mrow></msub></mfrac><mo>=</mo><msup><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mtext>&#xA0;</mtext><mi>k</mi><mo>&#x22C5;</mo><mi>c</mi></mrow><mo>)</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo>&#x2212;</mo><mi>&#x03B3;</mi></mrow></msup></math>" role="presentation" style="font-size: 101%; text-align: center; position: relative;"><span id="MJXc-Node-856" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-857" class="mjx-mrow"><span id="MJXc-Node-858" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 2.598em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 2.598em; top: -1.394em;"><span id="MJXc-Node-859" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-860" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-861" class="mjx-texatom" style=""><span id="MJXc-Node-862" class="mjx-mrow"><span id="MJXc-Node-863" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">L</span></span><span id="MJXc-Node-864" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">D</span></span><span id="MJXc-Node-865" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span></span></span></span></span><span class="mjx-denominator" style="width: 2.598em; bottom: -0.892em;"><span id="MJXc-Node-866" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-867" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.219em; padding-right: 0.071em;"><span id="MJXc-Node-868" class="mjx-texatom" style=""><span id="MJXc-Node-869" class="mjx-mrow"><span id="MJXc-Node-870" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.498em; padding-bottom: 0.313em;">b</span></span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.289em; width: 2.598em;"></span></span><span class="mjx-vsize" style="height: 2.286em; vertical-align: -0.892em;"></span></span><span id="MJXc-Node-871" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.065em; padding-bottom: 0.313em;">=</span></span><span id="MJXc-Node-872" class="mjx-msubsup MJXc-space3"><span class="mjx-base"><span id="MJXc-Node-873" class="mjx-mrow"><span id="MJXc-Node-874" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.436em; padding-bottom: 0.622em;">(</span></span><span id="MJXc-Node-875" class="mjx-mrow"><span id="MJXc-Node-876" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.313em;">1</span></span><span id="MJXc-Node-877" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.313em; padding-bottom: 0.436em;">+</span></span><span id="MJXc-Node-878" class="mjx-mtext MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.306em; padding-bottom: 0.313em;"> </span></span><span id="MJXc-Node-879" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.498em; padding-bottom: 0.313em;">k</span></span><span id="MJXc-Node-880" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.003em; padding-bottom: 0.313em;">⋅</span></span><span id="MJXc-Node-881" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">c</span></span></span><span id="MJXc-Node-882" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.436em; padding-bottom: 0.622em;">)</span></span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.71em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-883" class="mjx-texatom" style=""><span id="MJXc-Node-884" class="mjx-mrow"><span id="MJXc-Node-885" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.313em; padding-bottom: 0.436em;">−</span></span><span id="MJXc-Node-886" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.189em; padding-bottom: 0.498em; padding-right: 0.025em;">γ</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mfrac><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>b</mi></mrow></msub></mfrac><mo>=</mo><msup><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mtext> </mtext><mi>k</mi><mo>⋅</mo><mi>c</mi></mrow><mo>)</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mi>γ</mi></mrow></msup></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-73">\label{eq:k18}
| + | |
| − | \frac{R_{LDR}}{R_{b}}=\left(1+\ k\cdot c\right)^{-\gamma}</script></span></p>
| + | |
| − | <h2 id="determination-of-the-calibration-parameter-k">Determination of the calibration parameter k</h2>
| + | |
| − | <p>We want to find a value for <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-74-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-887" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-888" class="mjx-mrow"><span id="MJXc-Node-889" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.498em; padding-bottom: 0.313em;">k</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math></span></span><script type="math/tex" id="MathJax-Element-74">k</script></span> from equation [eq:k18]. To determine that <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-75-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-890" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-891" class="mjx-mrow"><span id="MJXc-Node-892" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.498em; padding-bottom: 0.313em;">k</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math></span></span><script type="math/tex" id="MathJax-Element-75">k</script></span> does not depend on the intensity <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-76-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>I</mi><mn>0</mn></msub></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-893" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-894" class="mjx-mrow"><span id="MJXc-Node-895" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.064em;"><span id="MJXc-Node-896" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em; padding-right: 0.064em;">I</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-897" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.374em;">0</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>I</mi><mn>0</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-76">I_0</script></span> we made two measurement series. We changed the resistance <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-77-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mn>3</mn></msub></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-898" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-899" class="mjx-mrow"><span id="MJXc-Node-900" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-901" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-902" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.374em;">3</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mn>3</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-77">R_3</script></span> in series to the LED to change the light Intensity <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-78-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>I</mi><mn>0</mn></msub></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-903" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-904" class="mjx-mrow"><span id="MJXc-Node-905" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.064em;"><span id="MJXc-Node-906" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em; padding-right: 0.064em;">I</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-907" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.374em;">0</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>I</mi><mn>0</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-78">I_0</script></span>. We used a <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-79-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>39</mn><mtext>&#xA0;</mtext><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">&#x03A9;</mi></mrow></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-908" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-909" class="mjx-mrow"><span id="MJXc-Node-910" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.374em;">39</span></span><span id="MJXc-Node-911" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.306em; padding-bottom: 0.313em;"> </span></span><span id="MJXc-Node-912" class="mjx-texatom"><span id="MJXc-Node-913" class="mjx-mrow"><span id="MJXc-Node-914" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.436em; padding-bottom: 0.313em;">Ω</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>39</mn><mtext> </mtext><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">Ω</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-79">39~\mathrm{\Omega}</script></span> resistor and a <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-80-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>60</mn><mtext>&#xA0;</mtext><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">&#x03A9;</mi></mrow></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-915" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-916" class="mjx-mrow"><span id="MJXc-Node-917" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.374em;">60</span></span><span id="MJXc-Node-918" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.306em; padding-bottom: 0.313em;"> </span></span><span id="MJXc-Node-919" class="mjx-texatom"><span id="MJXc-Node-920" class="mjx-mrow"><span id="MJXc-Node-921" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.436em; padding-bottom: 0.313em;">Ω</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>60</mn><mtext> </mtext><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">Ω</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-80">60~\mathrm{\Omega}</script></span> resistor.</p>
| + | |
| − | <p>We prepared a dilution series of fluorescein by 10-fold dilution from <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-81-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>100</mn><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">n</mi><mi mathvariant="normal">M</mi></mrow></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-922" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-923" class="mjx-mrow"><span id="MJXc-Node-924" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.374em;">100</span></span><span id="MJXc-Node-925" class="mjx-texatom"><span id="MJXc-Node-926" class="mjx-mrow"><span id="MJXc-Node-927" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.127em; padding-bottom: 0.313em;">n</span></span><span id="MJXc-Node-928" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.313em;">M</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>100</mn><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">n</mi><mi mathvariant="normal">M</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-81">100 \mathrm{nM}</script></span> to <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-82-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">m</mi><mi mathvariant="normal">M</mi></mrow></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-929" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-930" class="mjx-mrow"><span id="MJXc-Node-931" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.313em;">1</span></span><span id="MJXc-Node-932" class="mjx-texatom"><span id="MJXc-Node-933" class="mjx-mrow"><span id="MJXc-Node-934" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.127em; padding-bottom: 0.313em;">m</span></span><span id="MJXc-Node-935" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.313em;">M</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">m</mi><mi mathvariant="normal">M</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-82">1\mathrm{mM}</script></span>. For each measurement we pipetted 30 <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-83-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi>&#x03BC;</mi><mi mathvariant="normal">l</mi></mrow></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-936" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-937" class="mjx-mrow"><span id="MJXc-Node-938" class="mjx-texatom"><span id="MJXc-Node-939" class="mjx-mrow"><span id="MJXc-Node-940" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.498em;">μ</span></span><span id="MJXc-Node-941" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.436em; padding-bottom: 0.313em;">l</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi>μ</mi><mi mathvariant="normal">l</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-83">\mathrm{\mu l}</script></span> of sample on a fresh filter paper, placed it in the detector and turned on the LED. We measured the resistance <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-84-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-942" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-943" class="mjx-mrow"><span id="MJXc-Node-944" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-945" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-946" class="mjx-texatom" style=""><span id="MJXc-Node-947" class="mjx-mrow"><span id="MJXc-Node-948" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">L</span></span><span id="MJXc-Node-949" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">D</span></span><span id="MJXc-Node-950" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-84">R_{LDR}</script></span> directly with a multimeter. For the set-up with the <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-85-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>39</mn><mtext>&#xA0;</mtext><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">&#x03A9;</mi></mrow></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-951" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-952" class="mjx-mrow"><span id="MJXc-Node-953" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.374em;">39</span></span><span id="MJXc-Node-954" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.306em; padding-bottom: 0.313em;"> </span></span><span id="MJXc-Node-955" class="mjx-texatom"><span id="MJXc-Node-956" class="mjx-mrow"><span id="MJXc-Node-957" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.436em; padding-bottom: 0.313em;">Ω</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>39</mn><mtext> </mtext><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">Ω</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-85">39~\mathrm{\Omega}</script></span> resistor we additionally measured a <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-86-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>200</mn><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">n</mi><mi mathvariant="normal">M</mi></mrow></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-958" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-959" class="mjx-mrow"><span id="MJXc-Node-960" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.374em;">200</span></span><span id="MJXc-Node-961" class="mjx-texatom"><span id="MJXc-Node-962" class="mjx-mrow"><span id="MJXc-Node-963" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.127em; padding-bottom: 0.313em;">n</span></span><span id="MJXc-Node-964" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.313em;">M</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>200</mn><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">n</mi><mi mathvariant="normal">M</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-86">200 \mathrm{nM}</script></span> sample because this is the expected final working condition of the detector. After each measurement the detector was cleaned gently with ethanol. The first and last measurement of each series was conducted with plain water to determine <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-87-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mi>b</mi></msub></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-965" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-966" class="mjx-mrow"><span id="MJXc-Node-967" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-968" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.219em; padding-right: 0.071em;"><span id="MJXc-Node-969" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.498em; padding-bottom: 0.313em;">b</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mi>b</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-87">R_b</script></span> and to confirm the absence of contaminations. The measured resistances can be seen in table [tab:k1]. We normalized every resistance of one measurement series with the resistance <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-88-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mi>b</mi></msub></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-970" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-971" class="mjx-mrow"><span id="MJXc-Node-972" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-973" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.219em; padding-right: 0.071em;"><span id="MJXc-Node-974" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.498em; padding-bottom: 0.313em;">b</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mi>b</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-88">R_b</script></span> for plain water. A plot of the normalized resistances is shown in image [gra:k1]. We fitted the data with equation [eq:k18] to determine a value for k . For the fit we used an value of 0.8 for <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-89-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03B3;</mi></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-975" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-976" class="mjx-mrow"><span id="MJXc-Node-977" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.189em; padding-bottom: 0.498em; padding-right: 0.025em;">γ</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>γ</mi></math></span></span><script type="math/tex" id="MathJax-Element-89">\gamma</script></span>. The fitted value for k with absolute uncertainty is <span class="math display"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-90-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>k</mi><mo>=</mo><mo stretchy="false">(</mo><mn>3.758</mn><mo>&#x00B1;</mo><mn>0.133</mn><mo stretchy="false">)</mo><mfrac><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mi>&#x03BC;</mi><mi mathvariant="normal">M</mi></mrow></mfrac></math>" role="presentation" style="font-size: 101%; text-align: center; position: relative;"><span id="MJXc-Node-978" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-979" class="mjx-mrow"><span id="MJXc-Node-980" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.498em; padding-bottom: 0.313em;">k</span></span><span id="MJXc-Node-981" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.065em; padding-bottom: 0.313em;">=</span></span><span id="MJXc-Node-982" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.436em; padding-bottom: 0.622em;">(</span></span><span id="MJXc-Node-983" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.374em;">3.758</span></span><span id="MJXc-Node-984" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.313em;">±</span></span><span id="MJXc-Node-985" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.374em;">0.133</span></span><span id="MJXc-Node-986" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.436em; padding-bottom: 0.622em;">)</span></span><span id="MJXc-Node-987" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 1.72em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.72em; top: -1.368em;"><span id="MJXc-Node-988" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.313em;">1</span></span></span><span class="mjx-denominator" style="width: 1.72em; bottom: -0.952em;"><span id="MJXc-Node-989" class="mjx-texatom"><span id="MJXc-Node-990" class="mjx-mrow"><span id="MJXc-Node-991" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.498em;">μ</span></span><span id="MJXc-Node-992" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.313em;">M</span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.289em; width: 1.72em;"></span></span><span class="mjx-vsize" style="height: 2.32em; vertical-align: -0.952em;"></span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>k</mi><mo>=</mo><mo stretchy="false">(</mo><mn>3.758</mn><mo>±</mo><mn>0.133</mn><mo stretchy="false">)</mo><mfrac><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mi>μ</mi><mi mathvariant="normal">M</mi></mrow></mfrac></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-90">k= (3.758 \pm 0.133)\frac{1}{\mathrm{\mu M}}</script></span> and the relative uncertainty <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-91-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>&#x03C3;</mi><mi>k</mi></msub></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-993" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-994" class="mjx-mrow"><span id="MJXc-Node-995" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.001em;"><span id="MJXc-Node-996" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.189em; padding-bottom: 0.313em; padding-right: 0.001em;">σ</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.219em; padding-right: 0.071em;"><span id="MJXc-Node-997" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.498em; padding-bottom: 0.313em;">k</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mi>k</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-91">\sigma_k</script></span> is therefore <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-92-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0.035</mn></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-998" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-999" class="mjx-mrow"><span id="MJXc-Node-1000" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.374em;">0.035</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0.035</mn></math></span></span><script type="math/tex" id="MathJax-Element-92">0.035</script></span></p>
| + | |
| − | <div class="figure">
| + | |
| − | <img src="./testSave_files/final.png" alt="normalized resistances \frac{R_{LDR}}{R_b} vs fluorescein concentration c and corresponding fit function">
| + | |
| − | <p class="caption">normalized resistances <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-93-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub><msub><mi>R</mi><mi>b</mi></msub></mfrac></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-1001" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1002" class="mjx-mrow"><span id="MJXc-Node-1003" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 1.837em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 2.598em; top: -1.505em;"><span id="MJXc-Node-1004" class="mjx-msubsup" style=""><span class="mjx-base"><span id="MJXc-Node-1005" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1006" class="mjx-texatom" style=""><span id="MJXc-Node-1007" class="mjx-mrow"><span id="MJXc-Node-1008" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">L</span></span><span id="MJXc-Node-1009" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">D</span></span><span id="MJXc-Node-1010" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span></span></span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 2.598em; bottom: -0.796em;"><span id="MJXc-Node-1011" class="mjx-msubsup" style=""><span class="mjx-base"><span id="MJXc-Node-1012" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.219em; padding-right: 0.071em;"><span id="MJXc-Node-1013" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.498em; padding-bottom: 0.313em;">b</span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.289em; width: 1.837em;"></span></span><span class="mjx-vsize" style="height: 1.627em; vertical-align: -0.563em;"></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub><msub><mi>R</mi><mi>b</mi></msub></mfrac></math></span></span><script type="math/tex" id="MathJax-Element-93">\frac{R_{LDR}}{R_b}</script></span> vs fluorescein concentration c and corresponding fit function<span data-label="gra:k1"></span></p>
| + | |
| − | </div>
| + | |
| − | <table>
| + | |
| − | <caption>Calibration measurement</caption>
| + | |
| − | <thead>
| + | |
| − | <tr class="header">
| + | |
| − | <th align="left">Concentration in <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-94-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi>&#x03BC;</mi><mi mathvariant="normal">M</mi></mrow></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-1014" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1015" class="mjx-mrow"><span id="MJXc-Node-1016" class="mjx-texatom"><span id="MJXc-Node-1017" class="mjx-mrow"><span id="MJXc-Node-1018" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.498em;">μ</span></span><span id="MJXc-Node-1019" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.313em;">M</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi>μ</mi><mi mathvariant="normal">M</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-94">\mathrm{\mu M}</script></span></th>
| + | |
| − | <th align="right">Plain water</th>
| + | |
| − | <th align="right">0.1</th>
| + | |
| − | <th align="right">0.2</th>
| + | |
| − | <th align="right">1</th>
| + | |
| − | <th align="right">10</th>
| + | |
| − | <th align="right">100</th>
| + | |
| − | <th align="right">1000</th>
| + | |
| − | </tr>
| + | |
| − | </thead>
| + | |
| − | <tbody>
| + | |
| − | <tr class="odd">
| + | |
| − | <td align="left"><span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-95-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-1020" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1021" class="mjx-mrow"><span id="MJXc-Node-1022" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1023" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1024" class="mjx-texatom" style=""><span id="MJXc-Node-1025" class="mjx-mrow"><span id="MJXc-Node-1026" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">L</span></span><span id="MJXc-Node-1027" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">D</span></span><span id="MJXc-Node-1028" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-95">R_{LDR}</script></span> in <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-96-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">k</mi><mi mathvariant="normal">&#x03A9;</mi></mrow></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-1029" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1030" class="mjx-mrow"><span id="MJXc-Node-1031" class="mjx-texatom"><span id="MJXc-Node-1032" class="mjx-mrow"><span id="MJXc-Node-1033" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.436em; padding-bottom: 0.313em;">k</span></span><span id="MJXc-Node-1034" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.436em; padding-bottom: 0.313em;">Ω</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">k</mi><mi mathvariant="normal">Ω</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-96">\mathrm{k\Omega}</script></span> for bright LED</td> | + | |
| − | <td align="right">1200</td>
| + | |
| − | <td align="right">960</td>
| + | |
| − | <td align="right">748</td>
| + | |
| − | <td align="right">340</td>
| + | |
| − | <td align="right">58.0</td>
| + | |
| − | <td align="right">10.2</td>
| + | |
| − | <td align="right">2.4</td> | + | |
| − | </tr>
| + | |
| − | <tr class="even">
| + | |
| − | <td align="left"><span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-97-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-1035" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1036" class="mjx-mrow"><span id="MJXc-Node-1037" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1038" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1039" class="mjx-texatom" style=""><span id="MJXc-Node-1040" class="mjx-mrow"><span id="MJXc-Node-1041" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">L</span></span><span id="MJXc-Node-1042" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">D</span></span><span id="MJXc-Node-1043" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-97">R_{LDR}</script></span> in <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-98-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">k</mi><mi mathvariant="normal">&#x03A9;</mi></mrow></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-1044" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1045" class="mjx-mrow"><span id="MJXc-Node-1046" class="mjx-texatom"><span id="MJXc-Node-1047" class="mjx-mrow"><span id="MJXc-Node-1048" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.436em; padding-bottom: 0.313em;">k</span></span><span id="MJXc-Node-1049" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.436em; padding-bottom: 0.313em;">Ω</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">k</mi><mi mathvariant="normal">Ω</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-98">\mathrm{k\Omega}</script></span> for dimmed LED</td>
| + | |
| − | <td align="right">2050</td>
| + | |
| − | <td align="right">1620</td>
| + | |
| − | <td align="right"></td>
| + | |
| − | <td align="right">529</td>
| + | |
| − | <td align="right">80.6</td>
| + | |
| − | <td align="right">12.9</td>
| + | |
| − | <td align="right">3.1</td>
| + | |
| − | </tr>
| + | |
| − | </tbody>
| + | |
| − | </table>
| + | |
| − | <p>[tab:k1]</p>
| + | |
| − | <h1 id="read-out">Read out</h1>
| + | |
| − | <h2 id="data-collecting-and-processing-with-the-micro-controller">Data collecting and processing with the micro controller</h2>
| + | |
| − | <p>We want to verify if a reaction with a fluorescence product is taking place on a filter paper. Therefore we need to measure the resistance <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-99-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-1050" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1051" class="mjx-mrow"><span id="MJXc-Node-1052" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1053" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1054" class="mjx-texatom" style=""><span id="MJXc-Node-1055" class="mjx-mrow"><span id="MJXc-Node-1056" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">L</span></span><span id="MJXc-Node-1057" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">D</span></span><span id="MJXc-Node-1058" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-99">R_{LDR}</script></span> of the LDR over a certain time certain period. We choose to measure a data point every 5 min over a time period of 1 h to take trace of the reaction kinesics. Before such a measurement series a blank measurement with plain water is performed to determine the resistance <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-100-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mi>b</mi></msub></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-1059" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1060" class="mjx-mrow"><span id="MJXc-Node-1061" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1062" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.219em; padding-right: 0.071em;"><span id="MJXc-Node-1063" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.498em; padding-bottom: 0.313em;">b</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mi>b</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-100">R_b</script></span>.</p>
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| − | <p>As a first step the exact value of the supply voltage <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-101-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mn>0</mn></msub></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-1064" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1065" class="mjx-mrow"><span id="MJXc-Node-1066" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-1067" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1068" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.374em;">0</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mn>0</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-101">U_0</script></span> needs to be measured. The supply voltage is connected via an additional voltage divider with an analogue pin. The voltage divider consists of a <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-102-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>100</mn><mtext>&#xA0;</mtext><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">&#x03A9;</mi></mrow></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-1069" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1070" class="mjx-mrow"><span id="MJXc-Node-1071" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.374em;">100</span></span><span id="MJXc-Node-1072" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.306em; padding-bottom: 0.313em;"> </span></span><span id="MJXc-Node-1073" class="mjx-texatom"><span id="MJXc-Node-1074" class="mjx-mrow"><span id="MJXc-Node-1075" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.436em; padding-bottom: 0.313em;">Ω</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>100</mn><mtext> </mtext><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">Ω</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-102">100~\mathrm{\Omega}</script></span> resistor and a <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-103-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>910</mn><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">k</mi><mi mathvariant="normal">&#x03A9;</mi></mrow></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-1076" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1077" class="mjx-mrow"><span id="MJXc-Node-1078" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.374em;">910</span></span><span id="MJXc-Node-1079" class="mjx-texatom"><span id="MJXc-Node-1080" class="mjx-mrow"><span id="MJXc-Node-1081" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.436em; padding-bottom: 0.313em;">k</span></span><span id="MJXc-Node-1082" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.436em; padding-bottom: 0.313em;">Ω</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>910</mn><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">k</mi><mi mathvariant="normal">Ω</mi></mrow></math></span></span><script type="math/tex" id="MathJax-Element-103">910\mathrm{k\Omega}</script></span> resistor. The analogue pin measures still a correct value for the supply voltage <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-104-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mn>0</mn></msub></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-1083" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1084" class="mjx-mrow"><span id="MJXc-Node-1085" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-1086" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1087" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.374em;">0</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mn>0</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-104">U_0</script></span> because the first resistor is negligible small compared to the second resistor. We did not connect the analogue pin directly with the power supply to prevent the micro controller from damage in case of a short circuit or a peak voltage caused by an other component of the overall device. The micro controller measures the supply voltage 50 times with a delay time of 50 ms between measurements. It calculates the average of <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-105-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mn>0</mn></msub></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-1088" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1089" class="mjx-mrow"><span id="MJXc-Node-1090" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-1091" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1092" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.374em;">0</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mn>0</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-105">U_0</script></span> and the empirical standard deviation <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-106-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>&#x03C3;</mi><mrow class="MJX-TeXAtom-ORD"><msub><mi>U</mi><mn>0</mn></msub></mrow></msub></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-1093" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1094" class="mjx-mrow"><span id="MJXc-Node-1095" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.001em;"><span id="MJXc-Node-1096" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.189em; padding-bottom: 0.313em; padding-right: 0.001em;">σ</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1097" class="mjx-texatom" style=""><span id="MJXc-Node-1098" class="mjx-mrow"><span id="MJXc-Node-1099" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-1100" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1101" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.374em;">0</span></span></span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow class="MJX-TeXAtom-ORD"><msub><mi>U</mi><mn>0</mn></msub></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-106">\sigma_{U_0}</script></span> in percent, which is used as measurement uncertainty for further calculations.</p>
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| − | <p>To initiate a resistance measurement of the LDR the LED needs to be turned on by setting a digital output pin to 5 V. An additional green LED on the device is turned on as well to indicate that a measurement is taking place. After a waiting time of 30 s the actual measurement starts. This waiting time was determined empirically and is required because of the slow response of the LDR. <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-107-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-1102" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1103" class="mjx-mrow"><span id="MJXc-Node-1104" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-1105" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1106" class="mjx-texatom" style=""><span id="MJXc-Node-1107" class="mjx-mrow"><span id="MJXc-Node-1108" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">L</span></span><span id="MJXc-Node-1109" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">D</span></span><span id="MJXc-Node-1110" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-107">U_{LDR}</script></span> is measured as <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-108-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mn>0</mn></msub></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-1111" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1112" class="mjx-mrow"><span id="MJXc-Node-1113" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-1114" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1115" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.374em;">0</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mn>0</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-108">U_0</script></span>, 50 times with a delay time of 50ms between measurements. The average of <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-109-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-1116" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1117" class="mjx-mrow"><span id="MJXc-Node-1118" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-1119" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1120" class="mjx-texatom" style=""><span id="MJXc-Node-1121" class="mjx-mrow"><span id="MJXc-Node-1122" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">L</span></span><span id="MJXc-Node-1123" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">D</span></span><span id="MJXc-Node-1124" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-109">U_{LDR}</script></span> and the relative empirical standard deviation <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-110-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>&#x03C3;</mi><mrow class="MJX-TeXAtom-ORD"><msub><mi>U</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></mrow></msub></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-1125" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1126" class="mjx-mrow"><span id="MJXc-Node-1127" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.001em;"><span id="MJXc-Node-1128" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.189em; padding-bottom: 0.313em; padding-right: 0.001em;">σ</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1129" class="mjx-texatom" style=""><span id="MJXc-Node-1130" class="mjx-mrow"><span id="MJXc-Node-1131" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-1132" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1133" class="mjx-texatom" style=""><span id="MJXc-Node-1134" class="mjx-mrow"><span id="MJXc-Node-1135" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">L</span></span><span id="MJXc-Node-1136" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">D</span></span><span id="MJXc-Node-1137" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span></span></span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow class="MJX-TeXAtom-ORD"><msub><mi>U</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-110">\sigma_{U_{LDR}}</script></span> are calculated. Equation [eq:k22] is used to calculate <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-111-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-1138" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1139" class="mjx-mrow"><span id="MJXc-Node-1140" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1141" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1142" class="mjx-texatom" style=""><span id="MJXc-Node-1143" class="mjx-mrow"><span id="MJXc-Node-1144" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">L</span></span><span id="MJXc-Node-1145" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">D</span></span><span id="MJXc-Node-1146" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-111">R_{LDR}</script></span> from the average of <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-112-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-1147" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1148" class="mjx-mrow"><span id="MJXc-Node-1149" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-1150" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1151" class="mjx-texatom" style=""><span id="MJXc-Node-1152" class="mjx-mrow"><span id="MJXc-Node-1153" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">L</span></span><span id="MJXc-Node-1154" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">D</span></span><span id="MJXc-Node-1155" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-112">U_{LDR}</script></span>. We derived an equation for the propagation of the relative systematic and the relative statistic uncertainty of <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-113-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mn>0</mn></msub></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-1156" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1157" class="mjx-mrow"><span id="MJXc-Node-1158" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-1159" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1160" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.374em;">0</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mn>0</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-113">U_0</script></span> and <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-114-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-1161" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1162" class="mjx-mrow"><span id="MJXc-Node-1163" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-1164" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1165" class="mjx-texatom" style=""><span id="MJXc-Node-1166" class="mjx-mrow"><span id="MJXc-Node-1167" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">L</span></span><span id="MJXc-Node-1168" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">D</span></span><span id="MJXc-Node-1169" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-114">U_{LDR}</script></span>. For the relative statistic uncertainty <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-115-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>&#x03C3;</mi><mrow class="MJX-TeXAtom-ORD"><mi>s</mi><mi>t</mi><mi>a</mi><mi>t</mi></mrow></msub></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-1170" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1171" class="mjx-mrow"><span id="MJXc-Node-1172" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.001em;"><span id="MJXc-Node-1173" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.189em; padding-bottom: 0.313em; padding-right: 0.001em;">σ</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1174" class="mjx-texatom" style=""><span id="MJXc-Node-1175" class="mjx-mrow"><span id="MJXc-Node-1176" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">s</span></span><span id="MJXc-Node-1177" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.374em; padding-bottom: 0.313em;">t</span></span><span id="MJXc-Node-1178" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.189em; padding-bottom: 0.313em;">a</span></span><span id="MJXc-Node-1179" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.374em; padding-bottom: 0.313em;">t</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow class="MJX-TeXAtom-ORD"><mi>s</mi><mi>t</mi><mi>a</mi><mi>t</mi></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-115">\sigma_{stat}</script></span> of <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-116-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-1180" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1181" class="mjx-mrow"><span id="MJXc-Node-1182" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1183" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1184" class="mjx-texatom" style=""><span id="MJXc-Node-1185" class="mjx-mrow"><span id="MJXc-Node-1186" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">L</span></span><span id="MJXc-Node-1187" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">D</span></span><span id="MJXc-Node-1188" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-116">R_{LDR}</script></span> we get <span class="math display"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-117-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>&#x03C3;</mi><mrow class="MJX-TeXAtom-ORD"><mi>s</mi><mi>t</mi><mi>a</mi><mi>t</mi></mrow></msub><mo>=</mo><mfrac><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>r</mi><mi>e</mi><mi>f</mi></mrow></msub></mfrac><mo>&#x22C5;</mo><mfrac><msub><mi>U</mi><mn>0</mn></msub><msub><mi>U</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></mfrac><mo>&#x22C5;</mo><msqrt><msubsup><mi>&#x03C3;</mi><mrow class="MJX-TeXAtom-ORD"><msub><mi>U</mi><mn>0</mn></msub></mrow><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>&#x03C3;</mi><mrow class="MJX-TeXAtom-ORD"><msub><mi>U</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></mrow><mn>2</mn></msubsup></msqrt><mo>.</mo></math>" role="presentation" style="font-size: 101%; text-align: center; position: relative;"><span id="MJXc-Node-1189" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1190" class="mjx-mrow"><span id="MJXc-Node-1191" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.001em;"><span id="MJXc-Node-1192" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.189em; padding-bottom: 0.313em; padding-right: 0.001em;">σ</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1193" class="mjx-texatom" style=""><span id="MJXc-Node-1194" class="mjx-mrow"><span id="MJXc-Node-1195" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">s</span></span><span id="MJXc-Node-1196" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.374em; padding-bottom: 0.313em;">t</span></span><span id="MJXc-Node-1197" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.189em; padding-bottom: 0.313em;">a</span></span><span id="MJXc-Node-1198" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.374em; padding-bottom: 0.313em;">t</span></span></span></span></span></span><span id="MJXc-Node-1199" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.065em; padding-bottom: 0.313em;">=</span></span><span id="MJXc-Node-1200" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 2.598em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 2.598em; top: -1.394em;"><span id="MJXc-Node-1201" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1202" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1203" class="mjx-texatom" style=""><span id="MJXc-Node-1204" class="mjx-mrow"><span id="MJXc-Node-1205" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">L</span></span><span id="MJXc-Node-1206" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">D</span></span><span id="MJXc-Node-1207" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span></span></span></span></span><span class="mjx-denominator" style="width: 2.598em; bottom: -1.037em;"><span id="MJXc-Node-1208" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1209" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.23em; padding-right: 0.071em;"><span id="MJXc-Node-1210" class="mjx-texatom" style=""><span id="MJXc-Node-1211" class="mjx-mrow"><span id="MJXc-Node-1212" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">r</span></span><span id="MJXc-Node-1213" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">e</span></span><span id="MJXc-Node-1214" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.498em; padding-bottom: 0.498em; padding-right: 0.06em;">f</span></span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.289em; width: 2.598em;"></span></span><span class="mjx-vsize" style="height: 2.431em; vertical-align: -1.037em;"></span></span><span id="MJXc-Node-1215" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.003em; padding-bottom: 0.313em;">⋅</span></span><span id="MJXc-Node-1216" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 2.522em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 2.522em; top: -1.395em;"><span id="MJXc-Node-1217" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-1218" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1219" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.374em;">0</span></span></span></span></span><span class="mjx-denominator" style="width: 2.522em; bottom: -0.894em;"><span id="MJXc-Node-1220" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-1221" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1222" class="mjx-texatom" style=""><span id="MJXc-Node-1223" class="mjx-mrow"><span id="MJXc-Node-1224" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">L</span></span><span id="MJXc-Node-1225" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">D</span></span><span id="MJXc-Node-1226" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.289em; width: 2.522em;"></span></span><span class="mjx-vsize" style="height: 2.289em; vertical-align: -0.894em;"></span></span><span id="MJXc-Node-1227" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.003em; padding-bottom: 0.313em;">⋅</span></span><span id="MJXc-Node-1228" class="mjx-msqrt MJXc-space2"><span class="mjx-box" style="padding-top: 0.045em;"><span class="mjx-surd"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 0.931em; padding-bottom: 0.931em;">√</span></span><span class="mjx-box" style="padding-top: 0.182em; border-top: 1.2px solid;"><span id="MJXc-Node-1229" class="mjx-mrow"><span id="MJXc-Node-1230" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.001em;"><span id="MJXc-Node-1231" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.189em; padding-bottom: 0.313em; padding-right: 0.001em;">σ</span></span></span><span class="mjx-stack" style="vertical-align: -0.327em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.255em; padding-left: 0.073em; padding-right: 0.071em;"><span id="MJXc-Node-1237" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.313em;">2</span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-1232" class="mjx-texatom" style=""><span id="MJXc-Node-1233" class="mjx-mrow"><span id="MJXc-Node-1234" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-1235" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1236" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.374em;">0</span></span></span></span></span></span></span></span></span><span id="MJXc-Node-1238" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.313em; padding-bottom: 0.436em;">+</span></span><span id="MJXc-Node-1239" class="mjx-msubsup MJXc-space2"><span class="mjx-base" style="margin-right: -0.001em;"><span id="MJXc-Node-1240" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.189em; padding-bottom: 0.313em; padding-right: 0.001em;">σ</span></span></span><span class="mjx-stack" style="vertical-align: -0.327em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.255em; padding-left: 0.073em; padding-right: 0.071em;"><span id="MJXc-Node-1250" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.313em;">2</span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-1241" class="mjx-texatom" style=""><span id="MJXc-Node-1242" class="mjx-mrow"><span id="MJXc-Node-1243" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-1244" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1245" class="mjx-texatom" style=""><span id="MJXc-Node-1246" class="mjx-mrow"><span id="MJXc-Node-1247" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">L</span></span><span id="MJXc-Node-1248" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">D</span></span><span id="MJXc-Node-1249" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span id="MJXc-Node-1251" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.183em; padding-bottom: 0.313em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>σ</mi><mrow class="MJX-TeXAtom-ORD"><mi>s</mi><mi>t</mi><mi>a</mi><mi>t</mi></mrow></msub><mo>=</mo><mfrac><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>r</mi><mi>e</mi><mi>f</mi></mrow></msub></mfrac><mo>⋅</mo><mfrac><msub><mi>U</mi><mn>0</mn></msub><msub><mi>U</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></mfrac><mo>⋅</mo><msqrt><msubsup><mi>σ</mi><mrow class="MJX-TeXAtom-ORD"><msub><mi>U</mi><mn>0</mn></msub></mrow><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>σ</mi><mrow class="MJX-TeXAtom-ORD"><msub><mi>U</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></mrow><mn>2</mn></msubsup></msqrt><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-117">\sigma_{stat}=\frac{R_{LDR}}{R_{ref}}\cdot \frac{U_0}{U_{LDR}}\cdot \sqrt{\sigma_{U_0}^2+\sigma_{U_{LDR}}^2}.</script></span> We used an value of 1 digit for the absolute systematic uncertainty for a voltage measurement. The relative systematic uncertainty is <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-118-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mi>U</mi></mfrac></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-1252" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1253" class="mjx-mrow"><span id="MJXc-Node-1254" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.684em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.967em; top: -1.331em;"><span id="MJXc-Node-1255" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.313em;">1</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.967em; bottom: -0.663em;"><span id="MJXc-Node-1256" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em; padding-right: 0.084em;">U</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.289em; width: 0.684em;"></span></span><span class="mjx-vsize" style="height: 1.41em; vertical-align: -0.469em;"></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mi>U</mi></mfrac></math></span></span><script type="math/tex" id="MathJax-Element-118">\frac{1}{U}</script></span> for a measured voltage U. For the relative systematic uncertainty <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-119-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>&#x03C3;</mi><mrow class="MJX-TeXAtom-ORD"><mi>s</mi><mi>y</mi><mi>s</mi></mrow></msub></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-1257" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1258" class="mjx-mrow"><span id="MJXc-Node-1259" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.001em;"><span id="MJXc-Node-1260" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.189em; padding-bottom: 0.313em; padding-right: 0.001em;">σ</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1261" class="mjx-texatom" style=""><span id="MJXc-Node-1262" class="mjx-mrow"><span id="MJXc-Node-1263" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">s</span></span><span id="MJXc-Node-1264" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.498em; padding-right: 0.006em;">y</span></span><span id="MJXc-Node-1265" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">s</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow class="MJX-TeXAtom-ORD"><mi>s</mi><mi>y</mi><mi>s</mi></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-119">\sigma_{sys}</script></span> of <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-120-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-1266" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1267" class="mjx-mrow"><span id="MJXc-Node-1268" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1269" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1270" class="mjx-texatom" style=""><span id="MJXc-Node-1271" class="mjx-mrow"><span id="MJXc-Node-1272" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">L</span></span><span id="MJXc-Node-1273" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">D</span></span><span id="MJXc-Node-1274" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-120">R_{LDR}</script></span> we get therefore <span class="math display"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-121-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>&#x03C3;</mi><mrow class="MJX-TeXAtom-ORD"><mi>s</mi><mi>y</mi><mi>s</mi></mrow></msub><mo>=</mo><mfrac><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>r</mi><mi>e</mi><mi>f</mi></mrow></msub></mfrac><mo>&#x22C5;</mo><mfrac><msub><mi>U</mi><mn>0</mn></msub><msub><mi>U</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></mfrac><mo>&#x22C5;</mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><msub><mi>U</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></mfrac><mo>+</mo><mfrac><mn>1</mn><msub><mi>U</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub></mfrac></mrow><mo>)</mo></mrow><mo>.</mo></math>" role="presentation" style="font-size: 101%; text-align: center; position: relative;"><span id="MJXc-Node-1275" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1276" class="mjx-mrow"><span id="MJXc-Node-1277" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.001em;"><span id="MJXc-Node-1278" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.189em; padding-bottom: 0.313em; padding-right: 0.001em;">σ</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1279" class="mjx-texatom" style=""><span id="MJXc-Node-1280" class="mjx-mrow"><span id="MJXc-Node-1281" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">s</span></span><span id="MJXc-Node-1282" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.498em; padding-right: 0.006em;">y</span></span><span id="MJXc-Node-1283" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">s</span></span></span></span></span></span><span id="MJXc-Node-1284" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.065em; padding-bottom: 0.313em;">=</span></span><span id="MJXc-Node-1285" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 2.598em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 2.598em; top: -1.394em;"><span id="MJXc-Node-1286" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1287" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1288" class="mjx-texatom" style=""><span id="MJXc-Node-1289" class="mjx-mrow"><span id="MJXc-Node-1290" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">L</span></span><span id="MJXc-Node-1291" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">D</span></span><span id="MJXc-Node-1292" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span></span></span></span></span><span class="mjx-denominator" style="width: 2.598em; bottom: -1.037em;"><span id="MJXc-Node-1293" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1294" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.23em; padding-right: 0.071em;"><span id="MJXc-Node-1295" class="mjx-texatom" style=""><span id="MJXc-Node-1296" class="mjx-mrow"><span id="MJXc-Node-1297" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">r</span></span><span id="MJXc-Node-1298" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">e</span></span><span id="MJXc-Node-1299" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.498em; padding-bottom: 0.498em; padding-right: 0.06em;">f</span></span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.289em; width: 2.598em;"></span></span><span class="mjx-vsize" style="height: 2.431em; vertical-align: -1.037em;"></span></span><span id="MJXc-Node-1300" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.003em; padding-bottom: 0.313em;">⋅</span></span><span id="MJXc-Node-1301" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 2.522em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 2.522em; top: -1.395em;"><span id="MJXc-Node-1302" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-1303" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1304" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.374em;">0</span></span></span></span></span><span class="mjx-denominator" style="width: 2.522em; bottom: -0.894em;"><span id="MJXc-Node-1305" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-1306" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1307" class="mjx-texatom" style=""><span id="MJXc-Node-1308" class="mjx-mrow"><span id="MJXc-Node-1309" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">L</span></span><span id="MJXc-Node-1310" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">D</span></span><span id="MJXc-Node-1311" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.289em; width: 2.522em;"></span></span><span class="mjx-vsize" style="height: 2.289em; vertical-align: -0.894em;"></span></span><span id="MJXc-Node-1312" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.003em; padding-bottom: 0.313em;">⋅</span></span><span id="MJXc-Node-1313" class="mjx-mrow MJXc-space2"><span id="MJXc-Node-1314" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.241em; padding-bottom: 1.241em;">(</span></span><span id="MJXc-Node-1315" class="mjx-mrow"><span id="MJXc-Node-1316" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 2.522em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 2.522em; top: -1.368em;"><span id="MJXc-Node-1317" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.313em;">1</span></span></span><span class="mjx-denominator" style="width: 2.522em; bottom: -0.894em;"><span id="MJXc-Node-1318" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-1319" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1320" class="mjx-texatom" style=""><span id="MJXc-Node-1321" class="mjx-mrow"><span id="MJXc-Node-1322" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">L</span></span><span id="MJXc-Node-1323" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">D</span></span><span id="MJXc-Node-1324" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.289em; width: 2.522em;"></span></span><span class="mjx-vsize" style="height: 2.261em; vertical-align: -0.894em;"></span></span><span id="MJXc-Node-1325" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.313em; padding-bottom: 0.436em;">+</span></span><span id="MJXc-Node-1326" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 1.272em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.272em; top: -1.368em;"><span id="MJXc-Node-1327" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.313em;">1</span></span></span><span class="mjx-denominator" style="width: 1.272em; bottom: -0.895em;"><span id="MJXc-Node-1328" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-1329" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1330" class="mjx-texatom" style=""><span id="MJXc-Node-1331" class="mjx-mrow"><span id="MJXc-Node-1332" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.374em;">0</span></span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.289em; width: 1.272em;"></span></span><span class="mjx-vsize" style="height: 2.262em; vertical-align: -0.895em;"></span></span></span><span id="MJXc-Node-1333" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.241em; padding-bottom: 1.241em;">)</span></span></span><span id="MJXc-Node-1334" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.183em; padding-bottom: 0.313em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>σ</mi><mrow class="MJX-TeXAtom-ORD"><mi>s</mi><mi>y</mi><mi>s</mi></mrow></msub><mo>=</mo><mfrac><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>r</mi><mi>e</mi><mi>f</mi></mrow></msub></mfrac><mo>⋅</mo><mfrac><msub><mi>U</mi><mn>0</mn></msub><msub><mi>U</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></mfrac><mo>⋅</mo><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><msub><mi>U</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></mfrac><mo>+</mo><mfrac><mn>1</mn><msub><mi>U</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub></mfrac></mrow><mo>)</mo></mrow><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-121">\sigma_{sys}=\frac{R_{LDR}}{R_{ref}}\cdot \frac{U_0}{U_{LDR}}\cdot \left(\frac{1}{U_{LDR}}+\frac{1}{U_{0}}\right).</script></span> The equation for the total uncertainty <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-122-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>&#x03C3;</mi><mrow class="MJX-TeXAtom-ORD"><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></mrow></msub></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-1335" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1336" class="mjx-mrow"><span id="MJXc-Node-1337" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.001em;"><span id="MJXc-Node-1338" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.189em; padding-bottom: 0.313em; padding-right: 0.001em;">σ</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1339" class="mjx-texatom" style=""><span id="MJXc-Node-1340" class="mjx-mrow"><span id="MJXc-Node-1341" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1342" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1343" class="mjx-texatom" style=""><span id="MJXc-Node-1344" class="mjx-mrow"><span id="MJXc-Node-1345" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">L</span></span><span id="MJXc-Node-1346" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">D</span></span><span id="MJXc-Node-1347" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span></span></span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mrow class="MJX-TeXAtom-ORD"><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-122">\sigma_{R_{LDR}}</script></span> is then <span class="math display"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-123-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>&#x03C3;</mi><mrow class="MJX-TeXAtom-ORD"><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></mrow></msub><mo>=</mo><msub><mi>&#x03C3;</mi><mrow class="MJX-TeXAtom-ORD"><mi>s</mi><mi>y</mi><mi>s</mi></mrow></msub><mo>+</mo><msub><mi>&#x03C3;</mi><mrow class="MJX-TeXAtom-ORD"><mi>s</mi><mi>t</mi><mi>a</mi><mi>t</mi></mrow></msub><mo>=</mo><mfrac><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>r</mi><mi>e</mi><mi>f</mi></mrow></msub></mfrac><mo>&#x22C5;</mo><mfrac><msub><mi>U</mi><mn>0</mn></msub><msub><mi>U</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></mfrac><mo>&#x22C5;</mo><mrow><mo>(</mo><mrow><msqrt><msubsup><mi>&#x03C3;</mi><mrow class="MJX-TeXAtom-ORD"><msub><mi>U</mi><mn>0</mn></msub></mrow><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>&#x03C3;</mi><mrow class="MJX-TeXAtom-ORD"><msub><mi>U</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></mrow><mn>2</mn></msubsup></msqrt><mo>+</mo><mfrac><mn>1</mn><msub><mi>U</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></mfrac><mo>+</mo><mfrac><mn>1</mn><msub><mi>U</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub></mfrac></mrow><mo>)</mo></mrow><mo>.</mo></math>" role="presentation" style="font-size: 101%; text-align: center; position: relative;"><span id="MJXc-Node-1348" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1349" class="mjx-mrow"><span id="MJXc-Node-1350" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.001em;"><span id="MJXc-Node-1351" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.189em; padding-bottom: 0.313em; padding-right: 0.001em;">σ</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1352" class="mjx-texatom" style=""><span id="MJXc-Node-1353" class="mjx-mrow"><span id="MJXc-Node-1354" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1355" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1356" class="mjx-texatom" style=""><span id="MJXc-Node-1357" class="mjx-mrow"><span id="MJXc-Node-1358" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">L</span></span><span id="MJXc-Node-1359" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">D</span></span><span id="MJXc-Node-1360" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span></span></span></span></span></span></span></span><span id="MJXc-Node-1361" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.065em; padding-bottom: 0.313em;">=</span></span><span id="MJXc-Node-1362" class="mjx-msubsup MJXc-space3"><span class="mjx-base" style="margin-right: -0.001em;"><span id="MJXc-Node-1363" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.189em; padding-bottom: 0.313em; padding-right: 0.001em;">σ</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1364" class="mjx-texatom" style=""><span id="MJXc-Node-1365" class="mjx-mrow"><span id="MJXc-Node-1366" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">s</span></span><span id="MJXc-Node-1367" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.498em; padding-right: 0.006em;">y</span></span><span id="MJXc-Node-1368" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">s</span></span></span></span></span></span><span id="MJXc-Node-1369" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.313em; padding-bottom: 0.436em;">+</span></span><span id="MJXc-Node-1370" class="mjx-msubsup MJXc-space2"><span class="mjx-base" style="margin-right: -0.001em;"><span id="MJXc-Node-1371" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.189em; padding-bottom: 0.313em; padding-right: 0.001em;">σ</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1372" class="mjx-texatom" style=""><span id="MJXc-Node-1373" class="mjx-mrow"><span id="MJXc-Node-1374" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">s</span></span><span id="MJXc-Node-1375" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.374em; padding-bottom: 0.313em;">t</span></span><span id="MJXc-Node-1376" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.189em; padding-bottom: 0.313em;">a</span></span><span id="MJXc-Node-1377" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.374em; padding-bottom: 0.313em;">t</span></span></span></span></span></span><span id="MJXc-Node-1378" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.065em; padding-bottom: 0.313em;">=</span></span><span id="MJXc-Node-1379" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 2.598em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 2.598em; top: -1.394em;"><span id="MJXc-Node-1380" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1381" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1382" class="mjx-texatom" style=""><span id="MJXc-Node-1383" class="mjx-mrow"><span id="MJXc-Node-1384" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">L</span></span><span id="MJXc-Node-1385" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">D</span></span><span id="MJXc-Node-1386" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span></span></span></span></span><span class="mjx-denominator" style="width: 2.598em; bottom: -1.037em;"><span id="MJXc-Node-1387" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1388" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.23em; padding-right: 0.071em;"><span id="MJXc-Node-1389" class="mjx-texatom" style=""><span id="MJXc-Node-1390" class="mjx-mrow"><span id="MJXc-Node-1391" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">r</span></span><span id="MJXc-Node-1392" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">e</span></span><span id="MJXc-Node-1393" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.498em; padding-bottom: 0.498em; padding-right: 0.06em;">f</span></span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.289em; width: 2.598em;"></span></span><span class="mjx-vsize" style="height: 2.431em; vertical-align: -1.037em;"></span></span><span id="MJXc-Node-1394" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.003em; padding-bottom: 0.313em;">⋅</span></span><span id="MJXc-Node-1395" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 2.522em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 2.522em; top: -1.395em;"><span id="MJXc-Node-1396" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-1397" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1398" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.374em;">0</span></span></span></span></span><span class="mjx-denominator" style="width: 2.522em; bottom: -0.894em;"><span id="MJXc-Node-1399" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-1400" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1401" class="mjx-texatom" style=""><span id="MJXc-Node-1402" class="mjx-mrow"><span id="MJXc-Node-1403" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">L</span></span><span id="MJXc-Node-1404" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">D</span></span><span id="MJXc-Node-1405" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.289em; width: 2.522em;"></span></span><span class="mjx-vsize" style="height: 2.289em; vertical-align: -0.894em;"></span></span><span id="MJXc-Node-1406" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.003em; padding-bottom: 0.313em;">⋅</span></span><span id="MJXc-Node-1407" class="mjx-mrow MJXc-space2"><span id="MJXc-Node-1408" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.241em; padding-bottom: 1.241em;">(</span></span><span id="MJXc-Node-1409" class="mjx-mrow"><span id="MJXc-Node-1410" class="mjx-msqrt"><span class="mjx-box" style="padding-top: 0.045em;"><span class="mjx-surd"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 0.931em; padding-bottom: 0.931em;">√</span></span><span class="mjx-box" style="padding-top: 0.182em; border-top: 1.2px solid;"><span id="MJXc-Node-1411" class="mjx-mrow"><span id="MJXc-Node-1412" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.001em;"><span id="MJXc-Node-1413" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.189em; padding-bottom: 0.313em; padding-right: 0.001em;">σ</span></span></span><span class="mjx-stack" style="vertical-align: -0.327em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.255em; padding-left: 0.073em; padding-right: 0.071em;"><span id="MJXc-Node-1419" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.313em;">2</span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-1414" class="mjx-texatom" style=""><span id="MJXc-Node-1415" class="mjx-mrow"><span id="MJXc-Node-1416" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-1417" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1418" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.374em;">0</span></span></span></span></span></span></span></span></span><span id="MJXc-Node-1420" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.313em; padding-bottom: 0.436em;">+</span></span><span id="MJXc-Node-1421" class="mjx-msubsup MJXc-space2"><span class="mjx-base" style="margin-right: -0.001em;"><span id="MJXc-Node-1422" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.189em; padding-bottom: 0.313em; padding-right: 0.001em;">σ</span></span></span><span class="mjx-stack" style="vertical-align: -0.327em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.255em; padding-left: 0.073em; padding-right: 0.071em;"><span id="MJXc-Node-1432" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.313em;">2</span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-1423" class="mjx-texatom" style=""><span id="MJXc-Node-1424" class="mjx-mrow"><span id="MJXc-Node-1425" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-1426" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1427" class="mjx-texatom" style=""><span id="MJXc-Node-1428" class="mjx-mrow"><span id="MJXc-Node-1429" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">L</span></span><span id="MJXc-Node-1430" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">D</span></span><span id="MJXc-Node-1431" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span id="MJXc-Node-1433" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.313em; padding-bottom: 0.436em;">+</span></span><span id="MJXc-Node-1434" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 2.522em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 2.522em; top: -1.368em;"><span id="MJXc-Node-1435" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.313em;">1</span></span></span><span class="mjx-denominator" style="width: 2.522em; bottom: -0.894em;"><span id="MJXc-Node-1436" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-1437" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1438" class="mjx-texatom" style=""><span id="MJXc-Node-1439" class="mjx-mrow"><span id="MJXc-Node-1440" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">L</span></span><span id="MJXc-Node-1441" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">D</span></span><span id="MJXc-Node-1442" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.289em; width: 2.522em;"></span></span><span class="mjx-vsize" style="height: 2.261em; vertical-align: -0.894em;"></span></span><span id="MJXc-Node-1443" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.313em; padding-bottom: 0.436em;">+</span></span><span id="MJXc-Node-1444" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 1.272em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.272em; top: -1.368em;"><span id="MJXc-Node-1445" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.313em;">1</span></span></span><span class="mjx-denominator" style="width: 1.272em; bottom: -0.895em;"><span id="MJXc-Node-1446" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.084em;"><span id="MJXc-Node-1447" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em; padding-right: 0.084em;">U</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1448" class="mjx-texatom" style=""><span id="MJXc-Node-1449" class="mjx-mrow"><span id="MJXc-Node-1450" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.374em;">0</span></span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.289em; width: 1.272em;"></span></span><span class="mjx-vsize" style="height: 2.262em; vertical-align: -0.895em;"></span></span></span><span id="MJXc-Node-1451" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.241em; padding-bottom: 1.241em;">)</span></span></span><span id="MJXc-Node-1452" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.183em; padding-bottom: 0.313em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>σ</mi><mrow class="MJX-TeXAtom-ORD"><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></mrow></msub><mo>=</mo><msub><mi>σ</mi><mrow class="MJX-TeXAtom-ORD"><mi>s</mi><mi>y</mi><mi>s</mi></mrow></msub><mo>+</mo><msub><mi>σ</mi><mrow class="MJX-TeXAtom-ORD"><mi>s</mi><mi>t</mi><mi>a</mi><mi>t</mi></mrow></msub><mo>=</mo><mfrac><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>r</mi><mi>e</mi><mi>f</mi></mrow></msub></mfrac><mo>⋅</mo><mfrac><msub><mi>U</mi><mn>0</mn></msub><msub><mi>U</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></mfrac><mo>⋅</mo><mrow><mo>(</mo><mrow><msqrt><msubsup><mi>σ</mi><mrow class="MJX-TeXAtom-ORD"><msub><mi>U</mi><mn>0</mn></msub></mrow><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>σ</mi><mrow class="MJX-TeXAtom-ORD"><msub><mi>U</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></mrow><mn>2</mn></msubsup></msqrt><mo>+</mo><mfrac><mn>1</mn><msub><mi>U</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></mfrac><mo>+</mo><mfrac><mn>1</mn><msub><mi>U</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub></mfrac></mrow><mo>)</mo></mrow><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-123">\sigma_{R_{LDR}}=\sigma_{sys}+\sigma_{stat} = \frac{R_{LDR}}{R_{ref}}\cdot \frac{U_0}{U_{LDR}}\cdot \left(\sqrt{\sigma_{U_0}^2+\sigma_{U_{LDR}}^2}+\frac{1}{U_{LDR}}+\frac{1}{U_{0}}\right).</script></span></p>
| + | |
| − | <p><span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-124-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-1453" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1454" class="mjx-mrow"><span id="MJXc-Node-1455" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1456" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1457" class="mjx-texatom" style=""><span id="MJXc-Node-1458" class="mjx-mrow"><span id="MJXc-Node-1459" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">L</span></span><span id="MJXc-Node-1460" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">D</span></span><span id="MJXc-Node-1461" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-124">R_{LDR}</script></span>, <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-125-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mi>b</mi></msub></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-1462" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1463" class="mjx-mrow"><span id="MJXc-Node-1464" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1465" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.219em; padding-right: 0.071em;"><span id="MJXc-Node-1466" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.498em; padding-bottom: 0.313em;">b</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mi>b</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-125">R_b</script></span> and their uncertainties are read by the computer and saved for further analysis in a text file.</p>
| + | |
| − | <h2 id="data-analysis-and-final-result">Data analysis and final result</h2>
| + | |
| − | <p>With equation [eq:k13] and the fitted value for k the measured resistances can be translated into fluorescein concentrations <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-126-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-1467" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1468" class="mjx-mrow"><span id="MJXc-Node-1469" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">c</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math></span></span><script type="math/tex" id="MathJax-Element-126">c</script></span>. Equation [eq:k13] solved for <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-127-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-1470" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1471" class="mjx-mrow"><span id="MJXc-Node-1472" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">c</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math></span></span><script type="math/tex" id="MathJax-Element-127">c</script></span> is</p>
| + | |
| − | <p><span class="math display"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-128-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>c</mi><mo>=</mo><mfrac><mrow><msup><mrow><mo>(</mo><mfrac><msub><mi>R</mi><mi>b</mi></msub><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></mfrac><mo>)</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mfrac><mn>1</mn><mi>&#x03B3;</mi></mfrac></mrow></msup><mo>&#x2212;</mo><mn>1</mn></mrow><mi>k</mi></mfrac><mo>.</mo></math>" role="presentation" style="font-size: 101%; text-align: center; position: relative;"><span id="MJXc-Node-1473" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1474" class="mjx-mrow"><span id="MJXc-Node-1475" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">c</span></span><span id="MJXc-Node-1476" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.065em; padding-bottom: 0.313em;">=</span></span><span id="MJXc-Node-1477" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 5.77em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 5.77em; top: -2.756em;"><span id="MJXc-Node-1478" class="mjx-mrow"><span id="MJXc-Node-1479" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1480" class="mjx-mrow"><span id="MJXc-Node-1481" class="mjx-mo"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 0.931em; padding-bottom: 0.931em;">(</span></span><span id="MJXc-Node-1482" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 1.837em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 2.598em; top: -1.503em;"><span id="MJXc-Node-1483" class="mjx-msubsup" style=""><span class="mjx-base"><span id="MJXc-Node-1484" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.219em; padding-right: 0.071em;"><span id="MJXc-Node-1485" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.498em; padding-bottom: 0.313em;">b</span></span></span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 2.598em; bottom: -0.798em;"><span id="MJXc-Node-1486" class="mjx-msubsup" style=""><span class="mjx-base"><span id="MJXc-Node-1487" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1488" class="mjx-texatom" style=""><span id="MJXc-Node-1489" class="mjx-mrow"><span id="MJXc-Node-1490" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">L</span></span><span id="MJXc-Node-1491" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">D</span></span><span id="MJXc-Node-1492" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.289em; width: 1.837em;"></span></span><span class="mjx-vsize" style="height: 1.627em; vertical-align: -0.564em;"></span></span><span id="MJXc-Node-1493" class="mjx-mo"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 0.931em; padding-bottom: 0.931em;">)</span></span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 1.276em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-1494" class="mjx-texatom" style=""><span id="MJXc-Node-1495" class="mjx-mrow"><span id="MJXc-Node-1496" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.525em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.743em; top: -1.354em;"><span id="MJXc-Node-1497" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.313em;">1</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.743em; bottom: -0.729em;"><span id="MJXc-Node-1498" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.189em; padding-bottom: 0.498em; padding-right: 0.025em;">γ</span></span></span><span class="mjx-line" style="border-bottom: 1px solid; top: -0.292em; width: 0.525em;"></span></span><span class="mjx-vsize" style="height: 1.472em; vertical-align: -0.515em;"></span></span></span></span></span></span><span id="MJXc-Node-1499" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.313em; padding-bottom: 0.436em;">−</span></span><span id="MJXc-Node-1500" class="mjx-mn MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.313em;">1</span></span></span></span><span class="mjx-denominator" style="width: 5.77em; bottom: -0.758em;"><span id="MJXc-Node-1501" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.498em; padding-bottom: 0.313em;">k</span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.289em; width: 5.77em;"></span></span><span class="mjx-vsize" style="height: 3.515em; vertical-align: -0.758em;"></span></span><span id="MJXc-Node-1502" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.183em; padding-bottom: 0.313em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>c</mi><mo>=</mo><mfrac><mrow><msup><mrow><mo>(</mo><mfrac><msub><mi>R</mi><mi>b</mi></msub><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></mfrac><mo>)</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mfrac><mn>1</mn><mi>γ</mi></mfrac></mrow></msup><mo>−</mo><mn>1</mn></mrow><mi>k</mi></mfrac><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-128">\label{eq:k41}
| + | |
| − | c= \frac{\left( \frac{R_b}{R_{LDR}} \right)^{\frac{1}{\gamma}}-1}{k}.</script></span></p>
| + | |
| − | <p>The equation for the absolute uncertainty <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-129-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>&#x03C3;</mi><mi>c</mi></msub></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-1503" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1504" class="mjx-mrow"><span id="MJXc-Node-1505" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.001em;"><span id="MJXc-Node-1506" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.189em; padding-bottom: 0.313em; padding-right: 0.001em;">σ</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1507" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">c</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>σ</mi><mi>c</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-129">\sigma_c</script></span> of the fluorescein concentration <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-130-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-1508" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1509" class="mjx-mrow"><span id="MJXc-Node-1510" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">c</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math></span></span><script type="math/tex" id="MathJax-Element-130">c</script></span> is <span class="math display"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="mjx-chtml MJXc-display" style="text-align: center;"><span id="MathJax-Element-131-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>&#x03C3;</mi><mi>c</mi></msub><mo>=</mo><msqrt><msup><mrow><mo>(</mo><mfrac><msub><mi>R</mi><mi>b</mi></msub><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></mfrac><mo>)</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mfrac><mn>2</mn><mi>&#x03B3;</mi></mfrac></mrow></msup><mo>&#x22C5;</mo><mfrac><mn>1</mn><msup><mi>k</mi><mn>2</mn></msup></mfrac><mo>&#x22C5;</mo><mrow><mo>(</mo><mrow><msubsup><mi>&#x03C3;</mi><mrow class="MJX-TeXAtom-ORD"><msub><mi>R</mi><mi>b</mi></msub></mrow><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>&#x03C3;</mi><mrow class="MJX-TeXAtom-ORD"><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></mrow><mn>2</mn></msubsup></mrow><mo>)</mo></mrow><mo>+</mo><msup><mi>c</mi><mn>2</mn></msup><msubsup><mi>&#x03C3;</mi><mi>k</mi><mn>2</mn></msubsup></msqrt><mo>.</mo></math>" role="presentation" style="font-size: 101%; text-align: center; position: relative;"><span id="MJXc-Node-1511" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1512" class="mjx-mrow"><span id="MJXc-Node-1513" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.001em;"><span id="MJXc-Node-1514" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.189em; padding-bottom: 0.313em; padding-right: 0.001em;">σ</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1515" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">c</span></span></span></span><span id="MJXc-Node-1516" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.065em; padding-bottom: 0.313em;">=</span></span><span id="MJXc-Node-1517" class="mjx-msqrt MJXc-space3"><span class="mjx-box" style="padding-top: 0.045em;"><span class="mjx-surd"><span class="mjx-delim-v"><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.374em; padding-bottom: 0.313em;"></span><span class="mjx-char MJXc-TeX-size4-R" style="line-height: 0.589em; margin-bottom: -0.106em; margin-top: 0.006em;">
| + | |
| − | </span><span class="mjx-char MJXc-TeX-size4-R" style="padding-top: 0.684em; padding-bottom: 1.179em;">⎷</span></span></span><span class="mjx-box" style="padding-top: 0.297em; border-top: 1.2px solid;"><span id="MJXc-Node-1518" class="mjx-mrow"><span id="MJXc-Node-1519" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1520" class="mjx-mrow"><span id="MJXc-Node-1521" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.241em; padding-bottom: 1.241em;">(</span></span><span id="MJXc-Node-1522" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 2.598em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 2.598em; top: -1.392em;"><span id="MJXc-Node-1523" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1524" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.219em; padding-right: 0.071em;"><span id="MJXc-Node-1525" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.498em; padding-bottom: 0.313em;">b</span></span></span></span></span><span class="mjx-denominator" style="width: 2.598em; bottom: -0.894em;"><span id="MJXc-Node-1526" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1527" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1528" class="mjx-texatom" style=""><span id="MJXc-Node-1529" class="mjx-mrow"><span id="MJXc-Node-1530" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">L</span></span><span id="MJXc-Node-1531" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">D</span></span><span id="MJXc-Node-1532" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.289em; width: 2.598em;"></span></span><span class="mjx-vsize" style="height: 2.286em; vertical-align: -0.894em;"></span></span><span id="MJXc-Node-1533" class="mjx-mo"><span class="mjx-char MJXc-TeX-size3-R" style="padding-top: 1.241em; padding-bottom: 1.241em;">)</span></span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 1.7em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-1534" class="mjx-texatom" style=""><span id="MJXc-Node-1535" class="mjx-mrow"><span id="MJXc-Node-1536" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.525em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.743em; top: -1.354em;"><span id="MJXc-Node-1537" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.313em;">2</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.743em; bottom: -0.729em;"><span id="MJXc-Node-1538" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.189em; padding-bottom: 0.498em; padding-right: 0.025em;">γ</span></span></span><span class="mjx-line" style="border-bottom: 1px solid; top: -0.292em; width: 0.525em;"></span></span><span class="mjx-vsize" style="height: 1.472em; vertical-align: -0.515em;"></span></span></span></span></span></span><span id="MJXc-Node-1539" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.003em; padding-bottom: 0.313em;">⋅</span></span><span id="MJXc-Node-1540" class="mjx-mfrac MJXc-space2"><span class="mjx-box MJXc-stacked" style="width: 1.11em; padding: 0px 0.12em;"><span class="mjx-numerator" style="width: 1.11em; top: -1.368em;"><span id="MJXc-Node-1541" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.313em;">1</span></span></span><span class="mjx-denominator" style="width: 1.11em; bottom: -0.817em;"><span id="MJXc-Node-1542" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1543" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.498em; padding-bottom: 0.313em;">k</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.409em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-1544" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.313em;">2</span></span></span></span></span><span class="mjx-line" style="border-bottom: 1.3px solid; top: -0.289em; width: 1.11em;"></span></span><span class="mjx-vsize" style="height: 2.184em; vertical-align: -0.817em;"></span></span><span id="MJXc-Node-1545" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.003em; padding-bottom: 0.313em;">⋅</span></span><span id="MJXc-Node-1546" class="mjx-mrow MJXc-space2"><span id="MJXc-Node-1547" class="mjx-mo"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 0.931em; padding-bottom: 0.931em;">(</span></span><span id="MJXc-Node-1548" class="mjx-mrow"><span id="MJXc-Node-1549" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.001em;"><span id="MJXc-Node-1550" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.189em; padding-bottom: 0.313em; padding-right: 0.001em;">σ</span></span></span><span class="mjx-stack" style="vertical-align: -0.327em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.255em; padding-left: 0.073em; padding-right: 0.071em;"><span id="MJXc-Node-1556" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.313em;">2</span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-1551" class="mjx-texatom" style=""><span id="MJXc-Node-1552" class="mjx-mrow"><span id="MJXc-Node-1553" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1554" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.219em; padding-right: 0.071em;"><span id="MJXc-Node-1555" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.498em; padding-bottom: 0.313em;">b</span></span></span></span></span></span></span></span></span><span id="MJXc-Node-1557" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.313em; padding-bottom: 0.436em;">+</span></span><span id="MJXc-Node-1558" class="mjx-msubsup MJXc-space2"><span class="mjx-base" style="margin-right: -0.001em;"><span id="MJXc-Node-1559" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.189em; padding-bottom: 0.313em; padding-right: 0.001em;">σ</span></span></span><span class="mjx-stack" style="vertical-align: -0.327em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.255em; padding-left: 0.073em; padding-right: 0.071em;"><span id="MJXc-Node-1569" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.313em;">2</span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-1560" class="mjx-texatom" style=""><span id="MJXc-Node-1561" class="mjx-mrow"><span id="MJXc-Node-1562" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-1563" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.212em; padding-right: 0.071em;"><span id="MJXc-Node-1564" class="mjx-texatom" style=""><span id="MJXc-Node-1565" class="mjx-mrow"><span id="MJXc-Node-1566" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">L</span></span><span id="MJXc-Node-1567" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.251em;">D</span></span><span id="MJXc-Node-1568" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.436em; padding-bottom: 0.313em;">R</span></span></span></span></span></span></span></span></span></span></span></span><span id="MJXc-Node-1570" class="mjx-mo"><span class="mjx-char MJXc-TeX-size2-R" style="padding-top: 0.931em; padding-bottom: 0.931em;">)</span></span></span><span id="MJXc-Node-1571" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.313em; padding-bottom: 0.436em;">+</span></span><span id="MJXc-Node-1572" class="mjx-msubsup MJXc-space2"><span class="mjx-base"><span id="MJXc-Node-1573" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">c</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.584em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-1574" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.313em;">2</span></span></span></span><span id="MJXc-Node-1575" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.001em;"><span id="MJXc-Node-1576" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.189em; padding-bottom: 0.313em; padding-right: 0.001em;">σ</span></span></span><span class="mjx-stack" style="vertical-align: -0.335em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.255em; padding-left: 0.073em; padding-right: 0.071em;"><span id="MJXc-Node-1578" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.374em; padding-bottom: 0.313em;">2</span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-1577" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.498em; padding-bottom: 0.313em;">k</span></span></span></span></span></span></span></span></span><span id="MJXc-Node-1579" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.183em; padding-bottom: 0.313em;">.</span></span></span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>σ</mi><mi>c</mi></msub><mo>=</mo><msqrt><msup><mrow><mo>(</mo><mfrac><msub><mi>R</mi><mi>b</mi></msub><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></mfrac><mo>)</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mfrac><mn>2</mn><mi>γ</mi></mfrac></mrow></msup><mo>⋅</mo><mfrac><mn>1</mn><msup><mi>k</mi><mn>2</mn></msup></mfrac><mo>⋅</mo><mrow><mo>(</mo><mrow><msubsup><mi>σ</mi><mrow class="MJX-TeXAtom-ORD"><msub><mi>R</mi><mi>b</mi></msub></mrow><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>σ</mi><mrow class="MJX-TeXAtom-ORD"><msub><mi>R</mi><mrow class="MJX-TeXAtom-ORD"><mi>L</mi><mi>D</mi><mi>R</mi></mrow></msub></mrow><mn>2</mn></msubsup></mrow><mo>)</mo></mrow><mo>+</mo><msup><mi>c</mi><mn>2</mn></msup><msubsup><mi>σ</mi><mi>k</mi><mn>2</mn></msubsup></msqrt><mo>.</mo></math></span></span></span><script type="math/tex; mode=display" id="MathJax-Element-131">\label{eq:k42}
| + | |
| − | \sigma_c = \sqrt{\left(\frac{R_b}{R_{LDR}}\right)^{\frac{2}{\gamma}}\cdot\frac{1}{k^2}\cdot \left(\sigma_{R_b}^2+\sigma_{R_{LDR}}^2\right)+c^2\sigma_k^2}.</script></span> We are now enabled to measure fluorescence in units of equivalent fluorescein concentrations <span class="math inline"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span id="MathJax-Element-132-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math>" role="presentation" style="font-size: 101%; position: relative;"><span id="MJXc-Node-1580" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-1581" class="mjx-mrow"><span id="MJXc-Node-1582" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.251em; padding-bottom: 0.313em;">c</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math></span></span><script type="math/tex" id="MathJax-Element-132">c</script></span>.</p>
| + | |
| − | <p>As a first proof of principle we measured a kinetic curve of Cas13a digesting RNaseAlert. For a negative control we measured a kinetic curve for Cas13a without target RNA. and for a positive control we digested RNaseAlert with RNase A. We analysed the data with equation [eq:k41] and [eq:k42]. Image [img:k4] shows fluorescence in equivalent fluorescein concentrations vs time. The time traces show an enzymatic reaction taking place on filter paper. This proves that our detector is sensitive enough for our application.</p>
| + | |
| − | <div class="figure">
| + | |
| − | <img src="./testSave_files/timetrace.png" alt="Time trace of Cas13a digesting RNaseAlert. Positive control containing RNaseA and RNaseAlert. Negativ control containing only RNaseAlert. Shaded areas represent uncertainties for data points with the corresponding color">
| + | |
| − | <p class="caption">Time trace of Cas13a digesting RNaseAlert. Positive control containing RNaseA and RNaseAlert. Negativ control containing only RNaseAlert. Shaded areas represent uncertainties for data points with the corresponding color<span data-label="img:k4"></span></p>
| + | |
| − | </div>
| + | |
| | | | |
| − |
| |
| − | </body>
| |
| − |
| |
| − | <tr><td colspan=6 align=center valign=center>
| |
| − | <h3>CascAID</h3>
| |
| − | <p>
| |
| − | Our project, which we named Cas13a controlled assay for infectious diseases (CascAID), features the recently identified CRISPR/Cas effector Cas13a<sup><a class="myLink" href="#ref_10">10</a></sup>. Unlike other proteins in the familiy, Cas13a has the unique ability to bind and cleave specific RNA targets rather than DNA ones. Moreover, after cleaving its target, Cas13a is able to unspecifically cleave RNA molecules. By using this collateral activity from Cas13a, our system is capable of detecting virtually any RNA target. This is done by changing the crRNA in the protein, that is a short RNA sequence that determines what is recognized as target.</p>
| |
| − | </td>
| |
| − | </tr>
| |
| | <tr> | | <tr> |
| − | <td colspan=6 align=center valign=center> | + | <td> |
| − | <img src="https://static.igem.org/mediawiki/2017/0/04/T--Munich--Description_Cas13a_Mechanism.svg" alt="Diagram for Cas13a's function"> | + | <a href="/Team:Munich/Hardware/QuakeValve"><img class="picture1" src="https://static.igem.org/mediawiki/2017/0/ 08/T--Munich-- Overlay. png"></ a> |
| − | <p>Cas13a binds specific target RNA depending on the crRNA sequence. After activation, Cas13a cleaves RNA indiscriminately.</p>
| + | |
| | </td> | | </td> |
| − | | + | <td> |
| − | </tr>
| + | |
| − | | + | |
| − | <tr><td align=center valign=center colspan=4>
| + | |
| − | <p>
| + | |
| − | We wanted to start our project by showing that Cas13a's collateral activity could be used to detect the presence of specific RNA. For this, we used the RNAse alert system, as done in a recent publication<sup><a class="myLink" href="#ref_11">11</a></sup>, to detect RNA digestion. In this assay, the presence of RNAse-like activity is detected by an increase in green fluorescence. Our experiments yielded a convincing proof-of-principle which we went on to model. Moreover, CascAID can be used to detect a wide spectrum of pathogens, as our experiments with gram-positive and viral targets suggested. As we wanted to make CascAID available for everyone, we focused on building an inexpensive fluorescence detector to measure the presence of the target. Our detector “Lightbringer” was designed to be able to detect the fluorescence produced by the fluorescein in the Rnase alert system<sup><a class="myLink" href="#ref_12">12</a></sup>, but we theorize that changing the filters allows detection of other fluorophores. In addition, we experimented with freeze-drying on paper to make CascAID durable and easily portable.
| + | |
| − | </p>
| + | |
| | </td> | | </td> |
| − | <td align=center valign=center colspan=2> | + | </tr> |
| − | <img src="https://static.igem.org/mediawiki/2017/7/7f/T--Munich--Description_Cas13a_Readout_Comparision.svg">
| + | </table> |
| − | <p>Cas13a can be used to detect specific RNA sequences</p>
| + | |
| − | </td>
| + | |
| − | </tr>
| + | |
| | | | |
| − | <tr class="lastRow"> | + | <a href="/Team:Munich/Hardware/SampleProcessing"><img id="picture3" src="https://static.igem.org/mediawiki/2017/0/08/T--Munich--Overlay.png"></a> |
| − | <td align=center valign=center colspan=2> | + | <a href="/Team:Munich/Hardware/Paperstrip"><img id="picture2" src="https://static.igem.org/mediawiki/2017/ 0/ 08/T--Munich-- Overlay. png"></a> |
| − | <a href=" http:// www. uni-muenchen. de/ studium/ lehre_at_lmu/ index. html"><img src="https://static.igem.org/mediawiki/2017/ 9/ 9a/T--Munich-- Logo_LehreLMU. gif" width="200"></a > | + | |
| − | <p>Picture of the Thermocycler</p>
| + | |
| | </td> | | </td> |
| − | <td align=center valign=center colspan=4> | + | <td colspan=4> |
| − | <p> | + | <a href="/Team:Munich/Hardware/Detector"><img id="picture4" src="https://static.igem.org/mediawiki/2017/0/08/T--Munich--Overlay.png"></a> |
| − | For RNA extraction from the samples we tested three methods: extraction with silica beads, extraction with silica membrane and heat lysis. We custom-built an affordable thermocycler for signal amplification by RT-PCR to improve the detection limit. We explored recombinase polymerase amplification (RPA), an isothermal amplification procedure, to use over more conventional PCR methods as its simplicity makes it the more attractive option.
| + | |
| − | </p>
| + | |
| − | </td>
| + | |
| − | </tr> | + | |
| − | | + | |
| − | <tr><td colspan=6 align=center valign=center>
| + | |
| − | <h3>Colorimetric read-outs</h3>
| + | |
| − | <p>
| + | |
| − | To couple CascAID with an easy read-out method we explored three colorimetric read-outs:
| + | |
| − | </p> | + | |
| | </td> | | </td> |
| | </tr> | | </tr> |
| | | | |
| − | <tr><td colspan=2 align=center valign=center> | + | <tr> |
| − | <p> | + | <td colspan=6> |
| − | <b>AeBlue</b>: The RNA strand in a specially designed RNA/DNA dimer is cut by Cas13a's collateral
| + | <div class="captionPicture"> |
| − | activity. After digestion, the interaction between the two strands is too weak to hold the dimer and it
| + | <p> |
| − | decays. We can then use the DNA-strand as template to translate the chromoprotein <a href="http://parts.igem.org/Part:BBa_K864401">aeBlue</a>.
| + | Complete overview of all modular hardware parts in our pathogen detection system. Shown are counterclockwise and starting in in the upper left corner: The Quake valve that controls fluid flow, our sample processing device, the paper strip where a reaction mix is stored and the readout reaction takes place and finally our low-cost fluorescence detector "Lightbringer" that performs the readout measurement. Images are clickable and linked to the corresponding wiki subsection. |
| − | </p>
| + | |
| − | </td>
| + | |
| − | <td colspan=4 align=center valign=center>
| + | |
| − | <img src="https://static.igem.org/mediawiki/2017/9/90/T--Munich--Description_aeBlue.svg">
| + | |
| − | <p>Diagram of aeBlue</p>
| + | |
| − | </td>
| + | |
| − | </tr>
| + | |
| | | | |
| − | <tr><td colspan=2 align=center valign=center>
| |
| − | <p>
| |
| − | <b>Intein-Extein</b>: By binding TEV-protease with a RNA-linker we can use Cas13a's collateral activity
| |
| − | to regulate the protease's diffusion and use it to cleave a TEV tag separating the intein regions of a
| |
| − | modified chromophore. After the first cleavage, the intein segment excises itself<sup><a class="myLink" href="#13">13</a></sup>, bringing together the
| |
| − | halves of the chromophore. Only then is the chromophore functional and produces the colorimetric
| |
| − | read-out.
| |
| | </p> | | </p> |
| − | </td> | + | </div></td> |
| − | <td colspan=4 align=center valign=center>
| + | </tr> |
| − | <a href="http://www.uni-muenchen.de/studium/lehre_at_lmu/index.html"><img src="https://static.igem.org/mediawiki/2017/9/9a/T--Munich--Logo_LehreLMU.gif" width="200"></a>
| + | |
| − | <p>Diagram of Intein-Extein</p>
| + | |
| − | </td> | + | |
| − | </tr> | + | |
| | | | |
| − | <tr class="lastRow"><td colspan=2 align=center valign=center> | + | <tr><td colspan=6 align=left valign=center> |
| − | <p> | + | <font size=7 color=#51a7f9><b style="color: #51a7f9">Hardware</b></font> |
| − | <b>Gold nanoparticles</b>: Gold nanoparticles coated with short DNA sequences are held closely | + | |
| − | together by a complementary linker RNA, which makes the solution intense blue<sup><a class="myLink" href="#14">14</a></sup>. Activated Cas13a cuts
| + | |
| − | the linker RNA, causing the nanoparticles to diffuse away from each other. This increase in distance
| + | |
| − | causes a color change to intense red.
| + | |
| − | </p>
| + | |
| | </td> | | </td> |
| − | <td colspan=4 align=center valign=center> | + | </tr> |
| − | <img src="https://static.igem.org/mediawiki/2017/b/b3/T--Munich--Description_Goldnanoparticles.svg"> | + | <tr> |
| − | <p>Gold nanoparticles</p> | + | <td colspan = 6 align=center valign=center> |
| − | </td> | + | <p class="introduction"> |
| − | </tr> | + | |
| | + | The liberation of diagnostic tests from expensive lab infrastructure requires innovative ways of sample processing and measuring. We therefore developed a set of portable hardware tools with the goal of providing an automated sample-to-answer solution. The heart of our system is <a class="myLink" href="https://2017.igem.org/Team:Munich/Hardware/Detector">‘Lightbringer’</a>, our fluorescence detector, which is capable of measuring kinetics of biological or chemical reactions on <a class="myLink" href= "https://2017.igem.org/Team:Munich/Hardware/Paperstrip" >paper.</a> Built from 3D–printed parts and standard electronic components, it can be assembled for less than 15$, while offering a sensitivity competitive to commercial fluorescence readers. Additionally, tackling the challenge of sample pre-processing in field, we developed a portable <a class="myLink" href="https://2017.igem.org/Team:Munich/Hardware/SampleProcessing"> fluidic system</a>, featuring a temperature control unit for lysis and isothermal PCR. Conceiving a platform independent of lab infrastructure, we demonstrate the feasibility of <a class="myLink" href="https://2017.igem.org/Team:Munich/Hardware/QuakeValve"> controlling fluid flow</a> with bike tires and air balloons. All hardware components are designed and documented with the aim of enabling the community to reproduce and extend our set of tools. |
| | + | </p> |
| | | | |
| − | <tr><td colspan=6 align=center valign=center> | + | <div class="captionPicture"> |
| − | <h3>Software</h3>
| + | <img width=960 src="https://static.igem.org/mediawiki/2017/2/26/Schema_final_lowres.png"> |
| − | <p> | + | <p> |
| − | To help facilitate the design of crRNA, the sequences that give CascAID its specificity, we developed a
| + | |
| − | software tool that checks crRNA for unwanted secondary structures. This gives valuable insight on
| + | |
| − | whether the sequence is suited to use with Cas13a or whether some modifications are needed.
| + | |
| − | Together with Team Delft's software tool which designs the corresponding crRNA based on the target,
| + | |
| − | we collaborated to develop a powerful tool that suggests crRNA sequences and checks their usability
| + | |
| | </p> | | </p> |
| − | </td> | + | </div> |
| − | </tr>
| + | |
| | | | |
| − | <tr><td colspan=6 align=center valign=center>
| |
| − | <h3>References</h3>
| |
| − | <p>
| |
| − | <ol style="text-align: left">
| |
| − | <li id="ref_1">Cohen, Limor, and David R. Walt. "Single-Molecule Arrays for Protein and Nucleic Acid Analysis." Annual Review of Analytical Chemistry 0 (2017).</li>
| |
| − | <li id="ref_2">Nakano, Michihiko, et al. "Single-molecule PCR using water-in-oil emulsion." Journal of biotechnology 102.2 (2003): 117-124.</li>
| |
| − | <li id="ref_3">Taniguchi, Yuichi, et al. "Quantifying E. coli proteome and transcriptome with single-molecule sensitivity in single cells." science 329.5991 (2010): 533-538.</li>
| |
| − | <li id="ref_4">Rissin, David M., et al. "Single-molecule enzyme-linked immunosorbent assay detects serum proteins at subfemtomolar concentrations." Nature biotechnology 28.6 (2010): 595-599.</li>
| |
| − | <li id="ref_5">Pardee, Keith, et al. "Rapid, low-cost detection of Zika virus using programmable biomolecular components." Cell 165.5 (2016): 1255-1266.</li>
| |
| − | <li id="ref_6">Slomovic, Shimyn, Keith Pardee, and James J. Collins. "Synthetic biology devices for in vitro and in vivo diagnostics." Proceedings of the National Academy of Sciences 112.47 (2015): 14429-14435.</li>
| |
| − | <li id="ref_7">Tang, Ruihua, et al. "A fully disposable and integrated paper-based device for nucleic acid extraction, amplification and detection." Lab on a Chip 17.7 (2017): 1270-1279.</li>
| |
| − | <li id="ref_8">Vashist, Sandeep Kumar, et al. "Emerging technologies for next-generation point-of-care testing." Trends in biotechnology 33.11 (2015): 692-705.</li>
| |
| − | <li id="ref_9">Gubala, Vladimir, et al. "Point of care diagnostics: status and future." Analytical chemistry 84.2 (2011): 487-515.</li>
| |
| − | <li id="ref_10">Abudayyeh, Omar O., et al. "C2c2 is a single-component programmable RNA-guided RNA-targeting CRISPR effector." Science 353.6299 (2016): aaf5573.</li>
| |
| − | <li id="ref_11">Gootenberg, Jonathan S., et al. "Nucleic acid detection with CRISPR-Cas13a/C2c2." Science (2017): eaam9321.</li>
| |
| − | <li id="ref_12">https://www.idtdna.com/pages/docs/technical-reports/in_vitro_nuclease_detectionD325FDB69855.pdf (retrieved: 13.10.17)</li>
| |
| − | <li id="ref_13"> Anraku, Yasuhiro, Ryuta Mizutani, and Yoshinori Satow. "Protein splicing: its discovery and structural insight into novel chemical mechanisms." IUBMB life 57.8 (2005): 563-574.</li>
| |
| − | <li id="ref_14">Link, Stephan, and Mostafa A. El-Sayed. "Size and temperature dependence of the plasmon absorption of colloidal gold nanoparticles." The Journal of Physical Chemistry B 103.21 (1999): 4212-4217.</li>
| |
| − | </ol>
| |
| − | </p>
| |
| | </td> | | </td> |
| | </tr> | | </tr> |
| − |
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