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  1. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  2. % LaTeX template for writing reports for the AA320 series classes (AA320, AA321, AA322).
  3. %
  4. % To build to a pdf:
  5. % pdflatex filename.tex
  6. % bibtex filename
  7. % pdflatex filename.tex
  8. % pdflatex filename.tex
  9. %
  10. % Note that pdflatex requires filename extensions, while bibtex requires you
  11. % DO NOT have the extension. You must re-run pdflatex twice after running bibtex
  12. % to make sure references are correct in the pdf output.
  13. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  14. \documentclass[letterpaper,12pt]{article}
  15.  
  16. \usepackage{amsmath}
  17. \usepackage{amssymb}
  18.  
  19. \usepackage{caption}
  20. \usepackage{gensymb}
  21. \usepackage{subcaption}
  22. \usepackage[onehalfspacing]{setspace}
  23. \usepackage{indentfirst}
  24. \usepackage{array}
  25. \usepackage{graphicx} %this package enables the \includegraphics to create a figure
  26. %\usepackage{subfig}.    %this package enables properly labelled subfigures
  27. \usepackage[square,comma,sort&compress]{natbib}
  28.  
  29.  
  30. % place figures and tables where it seems like they should be
  31. \usepackage{float}
  32. \floatplacement{figure}{H}
  33. \floatplacement{table}{H}
  34.  
  35. % automatically center figures/tables
  36. \makeatletter
  37. \g@addto@macro\@floatboxreset{\centering}
  38. \makeatother
  39.  
  40. \usepackage{tocvsec2}
  41. \usepackage[bookmarksdepth=subsection]{hyperref}
  42. \usepackage{bookmark}
  43. \usepackage[margin=1in]{geometry}
  44. % \usepackage{authblk}
  45. \usepackage{titling}
  46. \setlength{\droptitle}{-1in}
  47.  
  48. \RequirePackage[style]{abstract}
  49. \renewcommand{\abstitlestyle}[1]{}
  50. \renewcommand{\abstracttextfont}{\normalsize}
  51. \setlength{\absleftindent}{0.5in}
  52. \setlength{\absrightindent}{0.5in}
  53. \setlength{\parindent}{0cm}
  54.  
  55. \captionsetup[figure]{labelfont=bf}
  56. \captionsetup[table]{labelfont=bf}
  57. \renewcommand{\figurename}{\textbf{Fig.}}
  58. \renewcommand{\tablename}{\textbf{Table}}
  59.  
  60. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  61. % Change the listed Section to your section
  62. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  63. \pretitle{\begin{center}\fontsize{16pt}{1em}\bfseries\selectfont\vspace{14pt}}
  64.  
  65. \posttitle{\end{center}}
  66. \preauthor{\begin{center}\fontsize{12pt}{1.5em}\selectfont}
  67. \postauthor{\end{center}}
  68. \predate{\begin{center}}
  69. \postdate{\end{center}}
  70.  
  71. % section heading formatting
  72. \renewcommand{\thesection}{\Roman{section}.}
  73.  
  74. \usepackage{sectsty}
  75. \sectionfont{\centering\fontsize{12pt}{1em}\selectfont}
  76. \subsectionfont{\centering\fontsize{11pt}{1em}\selectfont}
  77. \renewcommand{\thesubsection}{\Roman{section}.\Alph{subsection}.}
  78. \subsubsectionfont{\fontsize{11pt}{1em}\selectfont}
  79. \renewcommand{\thesubsubsection}{}
  80.  
  81. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  82. % The title of the lab
  83. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  84. \title{Photodiode, Transimpedance Amplifier, and Analog Integration and Differentiation}
  85.  
  86. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  87. % Your names go here (one should be underlined)
  88. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  89. \author{Atharva Agashe and Tony Tan\\
  90. AA 321 �� Aerospace Laboratory I  Section AD\\William E. Boeing Department of Aeronautics\\University of Washington, Seattle, WA, 98195}
  91.  
  92. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  93. % Due date of the lab
  94. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  95. \date{2018-11-30}
  96.  
  97. \begin{document}
  98. \maketitle
  99. \thispagestyle{empty}
  100.  
  101. \begin{abstract}
  102. \textbf{Abstract}: The main objective of this lab was to understand how photodiodes operate and additionally the operation of integrator and differentiator circuits with different input waves. An oscilloscope, function generator, speaker, photo diode , LED , capacitor and resistors were used to monitor the output voltage and frequency. The results of the experiment were that when a bright light was shone there was an output voltage. The integrator and differentiator circuits were built using resistors, capacitors, op amp, oscilloscope and function generator. Different input voltage waves of different shape were inputted into the circuit and the out voltage was monitored.The results of the experiment were that different input signals produced different output voltages and in addition theses output signals were directly proportional to RC.
  103. \end{abstract}
  104.  
  105. \section*{Nomenclature}
  106.  
  107. \begin{tabular}{>{\itshape}l l}
  108.  
  109. C & Capacitance\\
  110. P_{L} & Power generated by light source\\
  111. R & Resistance\\
  112. R_{F} & Feedback resistance\\
  113. V_{in} & Input voltage\\
  114. V_{out} & Output voltage\\
  115. \Re & Proportionality constant between current and power generated by light\\
  116. \end{tabular}
  117.  
  118. \setlength{\parindent}{.5in}
  119. \section{Introduction}
  120.  
  121. The objective of this lab was to examine photodiodes and their behavior when exposed to light. By converting the current produced by the photodiode into voltage using a transimpedance amplifier, the relationship between light and current generated could be determined. Integrating and differentiating circuits using op-amps were also examined. Using different input signals at varying frequencies their resulting outputs can be observed with the oscilloscope and compared to the original signals. The resulting integrated or differentiated signals were compared with predicted output values based on known relationships between $V_{in}$ and $V_{out}$ to better understand the limitations of using an analog circuit to perform these calculations.
  122.  
  123. \section{Theory}
  124.  
  125. Equation 1 describes the relationship between output voltage and the power of light source in a circuit with a transimpedance amp and photodiode. $\Re$ is the proportionality constant between current and the power of light:\cite{TextBook}
  126.  
  127. \begin{equation}
  128.  V_{out} = R_{F}\Re P_{L}
  129.  \label{photodiode}
  130. \end{equation}
  131.  
  132. In an analog integrator circuit, $V_{in}$ is integrated such that $V_{out}$ is the integral of the input signal with a factor of -$\frac{1}{RC}$. Equation 2 shows this relationship:
  133.  
  134. \begin{equation}
  135.  V_{out} = -\int{\frac{1}{RC}V_{in}dt}
  136.  \label{integrator}
  137. \end{equation}
  138.  
  139. In an analog differentiator circuit, $V_{out}$ is the time derivative of $V_{in}$, scaled by a factor of -RC. Equation 3 describes this relationship:
  140.  
  141. \begin{equation}
  142.  V_{out} = -RC\frac{dV_{in}}{dt}
  143.  \label{differentiator}
  144. \end{equation}
  145.  
  146. \noindent To calculate the uncertainties when measurements are taken digitally Equation \ref{diguncer} is used and when measurements are taken by analog methods Equation \ref{alguncer} is used.
  147. \begin{equation}
  148.    u_{digital}(x) = \dfrac{\alpha}{2\sqrt{3}}
  149.    \label{diguncer}
  150. \end{equation}
  151.  
  152. \noindent The propagation of uncertainties through a calculation is given by Equation \ref{propogation} and is the primary method for comparing the accuracy of a resultant value to its predicted value.\cite{error}
  153.  
  154. \begin{equation}
  155.    \Delta  f = \sqrt{(\dfrac{\partial f}{\partial x}\Delta x)^2+(\dfrac{\partial f}{\partial y}\Delta y)^2+(\dfrac{\partial f}{\partial z}\Delta z)^2 +. . .  }
  156.    \label{propogation}
  157. \end{equation}
  158.  
  159. \section{Experimental Apparatus}
  160.  
  161. The photodiode used in this experiment acted like a sensor that produced current proportional to the power of any light it was exposed to. The LF411 op-amp converted that current into an output voltage which could be measured, this type of op-amp is also known as a transimpedance amplifier. \cite{instrctns}
  162.  
  163. \begin{table}
  164. \caption{Used Equipment}
  165. \label{equipment}
  166. \begin{tabular}{|c|c|c|}
  167.  \hline
  168.  \textbf{Model Number} & \textbf{Name} & \textbf{Notes} \\
  169.  \hline
  170.  Instek GPC-3030D & DC Power Supply & Provided power\\
  171.  \hline
  172.  Keithley 2110 5-1 & Digital Multimeter & Measured voltage \\
  173.  \hline
  174.  & Proto-board & \\
  175.  \hline
  176.  AFG1022 & Function Generator & Generate signals \\
  177.  \hline
  178.  LF411 & Operational Amplifier & Generate voltage proportional to current\\
  179.  \hline
  180.  & Multi-turn Potentiometer & 10 k$\Omega$ \\
  181.  \hline
  182.  & Resistors & 100 k$\Omega$, 30 k$\Omega$, 100 $\Omega$, 1 M$\Omega$ \\
  183.  \hline
  184.  & Photodiode & Light sensor\\
  185.  \hline
  186.  & Speaker &\\
  187.  \hline
  188.  & LED & \\
  189.  \hline
  190. \end{tabular}
  191. \end{table}
  192.  
  193. \section{Procedure}
  194.  
  195. The circuit in Fig 1 consisting of a photodiode and a LF411 transimpedance amplifier was assembled, noting that the long lead on the photodiode is the anode. Shining a light on the photodiode with a phone and rotating the circuit board to maximize the signal, the scope of $V_{out}$ was monitored and recorded on the oscilloscope. The light source was blocked from the photodiode to confirm that the output voltage was generated from the light.
  196.  
  197. \begin{figure}
  198.  \includegraphics[width=0.4\textwidth]{fig1.JPG}
  199.  \caption{Circuit with LF411 op-amp and photodiode}
  200.  \label{Fig1}
  201. \end{figure}
  202.  
  203. A separate circuit with an LED hooked up to a function generator was constructed with the LED being as close to the photodiode as possible. The function generator frequency was set to 500 Hz, with a square wave and amplitude of 4 V. A speaker was also connected to the output of the transimpedance amplifier as shown in Fig 2 below. Illuminating the photodiode with the LED, $V_{out}$ was again monitored and recorded on Ch.1 and $V_{sig}$ on Ch.2 of the oscilloscope. The frequency on the function generator was adjusted to determine the maximum and minimum frequencies that could be heard from the speaker.
  204.  
  205. \begin{figure}
  206.  \includegraphics[width=0.4\textwidth]{fig2.JPG}
  207.  \caption{Circuit with LF411 op-amp, photodiode, LED, and speaker}
  208.  \label{Fig2}
  209. \end{figure}
  210.  
  211. An analog circuit used for integrating input signals was assembled and connected to the function generator. $V_{in}$ and $V_{out}$ were connected to Ch.1 and Ch.2 of the oscilloscope respectively as shown in Fig 3 below, with Ch.1 set to 2 V/div and Ch.2 set to 1 V/div. The generator was again set to a square wave, with a frequency of 500 Hz and amplitude of 4 V peak to peak, and adjusting the generator frequency so that the HI(+) duration is 1 ms. $V_{out}$ was recorded, noting the extreme voltages of the signal. The same measurements were repeated with a triangle wave and sine wave, keeping the generator settings the same. $V_{out}$ of the square wave with a frequency of 1 kHz and HI time of 0.5 ms was also observed.
  212.  
  213. \begin{figure}
  214.  \includegraphics[width=0.4\textwidth]{fig3.JPG}
  215.  \caption{Analog integrator circuit}
  216.  \label{Fig3}
  217. \end{figure}
  218.  
  219. An analog differentiator circuit was then constructed as shown in Fig 4 below, again with the function generator and oscilloscope. The function generator was set to 4 V peak to peak and a frequency of 500 Hz, first with a sine wave. The differentiated peak positive and negative output voltage was recorded and monitored on Ch.2 of the oscilloscope. Keeping the function generator settings the same the same measurements were repeated with a triangle wave. The op-amp was then replaced with a 10 k$\Omega$ potentiometer with resistance set to 10 k$\Omega$ using the DMM. This potentiometer was wired as a rheostat, using only one end terminal and the center tap. $V_{out}$ was observed with a sine wave output from the function generator, adjusting the rheostat such that $V_{out}$ was reduced to half its original value. Resistance of the rheostat was measured and recorded at this point.
  220.  
  221. \begin{figure}
  222.  \includegraphics[width=0.4\textwidth]{fig4.JPG}
  223.  \caption{Analog differentiator circuit}
  224.  \label{Fig4}
  225. \end{figure}
  226.  
  227. \section{Discussion of Results}
  228.  
  229. When a light was shone on the photodiode, $V_{out}$ was measured to be 7.36 V. When light was blocked from the photodiode, $V_{out}$ = 0.171 $\pm$ 0.001 V, confirming that the output voltage was a result of the light from the phone. The maximum frequency that could be heard was 16 $\pm$ 1 kHz and the lowest was 1 $\pm$ 1 kHz. At 1 kHz a square wave was out and a beeping noise was heard on the speaker. The reason was this range of frequencies is because at 16 khz the light is blinking very fast so there is no change in the power of the incoming light from the LED so a very high squeak is heard. For the lower frequency since the power from the light varied over a period of time the sound heard varied and so a beeping sound was heard. Humans also have a limit of hearing an there could have been high frequency  or lower frequency sounds which humans many not be able to hear.
  230.  
  231. A square wave input from the function generator with an amplitude of $\pm$2 V and frequency of 500 Hz through the integrator circuit resulted in a triangular wave output with an amplitude of $\pm$1 V. Since a square wave function is a constant oscillating between its positive and negative value, integrating a constant gives a function of the form f(x) = kx + C, which explains the resulting triangular wave. A plot of the input and output signals from the oscilloscope of this result is shown in Fig 5. The predicted value of the voltage was calculated using equation \ref{integrator} and the value obtained was 8E-3 $\pm$ 1E-3 V. This value is not accurate as the difference between the predicted and the measured value is very large but the value predicted is precise due to the small uncertainty value.  
  232.  
  233. \begin{figure}
  234.  \includegraphics[width=0.4\textwidth]{int_square.jpg}
  235.  \caption{Input sine wave signal (green) and its integrated signal output (red)}
  236.  \label{Fig5}
  237. \end{figure}
  238.  
  239. When a triangular wave input at the same amplitude and frequency as the square wave was generated, the output resembled an oscillating quadratic function with an amplitude $V_{out}$ = 0.5 $\pm$ 0.1 V. Examining a triangular wave such as this, its function is of the form f(x) = $\pm$ a $\pm$ bx, where a and b are constants. Integrating this results in a quadratic function as shown in Fig 6, scaled by a factor of $\frac{1}{RC}$ which is known from Eq 2. The predicted value of the voltage was calculated using equation \ref{integrator} and the value obtained was 0.46 $\pm$ 1E-3 V. This  predicted value is accurate as the difference between the measured and predicted value is small and it is precise due to its low uncertainty.
  240.  
  241. \begin{figure}
  242.  \includegraphics[width=0.4\textwidth]{int_triangle.jpg}
  243.  \caption{Input square wave signal (green) and its integrated signal output (red)}
  244.  \label{Fig6}
  245. \end{figure}
  246.  
  247. The next signal integrated was a sine wave, with an amplitude of $V_{in}$ = 2 V. The resulting integral was a cosine wave with amplitude $V_{out}$ = 0.64 V. The sine wave has the function f(x) = Asin($\omega$t) = 2sin(3141.59t), where $\omega$ = 2$\pi$f. The integral of this function is f'(x) = -0.6366cos(3141.59t) after being scaled by the factor of $\frac{1}{RC}$. The oredicted maximum voltage value is 0.6366 $\pm$ 2.7E-3 V. This value is very accurate and as the difference between the predicted and measured value is low and it is precise due to its small uncertainty.
  248.  
  249. \begin{figure}
  250.  \includegraphics[width=0.4\textwidth]{int_sine.jpg}
  251.  \caption{Input triangle wave signal (green) and its integrated signal output (red)}
  252.  \label{Fig7}
  253. \end{figure}
  254.  
  255. When the frequency of the function generator was set to 1 kHz with a square wave input, the result was that the period of the function was half of what it was at f = 500 Hz.
  256.  
  257. \begin{figure}
  258.  \includegraphics[width=0.35\textwidth]{int_square2.jpg}
  259.  \caption{Input square wave signal (green) at 1 kHz and its integrated signal output (red)}
  260.  \label{Fig8}
  261. \end{figure}
  262.  
  263. For the differntiator circuit  when a input voltage was in the form of a sine wave, the peak to peak output voltage that was measured was 1.4 $\pm$ 0.1 V. The trace of both the input and output voltage is show in in figure \ref{d}a. The output voltage trace was consistent with the input voltage trace as using equation \ref{differentiator} the differentiation of a sine function is a cosine function and that is what is observed on the oscilloscope as at t = 0 the output voltage(CH.2) is at its maximum value which is similar to a cosine function and the input voltage (CH.1) is at its minimum value similar to the sine function. The phase difference measured between the output and input voltages was 95 $\pm$ 1 \degree this value when compared to the actual value between a sine and cosine wave which is 90 $\pm$ 1 \degree. The predicted output voltage that was calculated using equation \ref{differentiator} was 1.36 $\pm$ 1E-3 V. The uncertainty for the measured value was calculated using equation \ref{propogation}. The uncertainty values are small so the output measured is precise and accurate, there is also a small error percentage between the measured and actual values. When the input wave was changed to a triangular wave the output peak to peak voltage was measured to be  960 $\pm$ 1 mV. The phase difference between both the input and output voltages was 92.6 $\pm$ 1 \degree. The trace scopes for the output and input voltages are shown in figure \ref{d}a. The traces are consistent because the derivative of a line a constant and so the output should be a square wave and that is captured on the oscilloscope. The predicted value of the output voltage is calculated using equation\ref{differentiator}, the value predicted was 1 $\pm$ 4.47E-3 V. The uncertainty value was calculated using equation \ref{propogation} and the value predicted is very close to the value measured with a small uncertainty which shows the value is precise and accurate.
  264. \noindent When the rheostat was added and set to approximately 10 k$\Omega$ the measured output voltage was 1.36 $\pm$ 0.01 V which was the input voltage when a 10 k$\Omega$ resistor was used for the previous parts. When the rheostat was adjusted such that the amplitude of the output signal was reduced by half the measured resistance was 4.65 $\pm$ 0.01 k$\Omega$, the scaling factor is 1/2 and this caused the amplitude in the output signal to be reduced by a 1/2 as well. If the frequency was set to 5000 Hz the circuit would not have to be changed because the out signal is independent of the input frequency as this can be using in equation \ref{differentiator}.
  265.  
  266. \begin{figure}
  267.    \centering
  268.    \begin{subfigure}[b]{0.35\textwidth}
  269.        \includegraphics[width=\textwidth]{F0009TEK.png}
  270.        \label{d}
  271.        \caption{}
  272.    \end{subfigure}
  273.    ~
  274.    \begin{subfigure}[b]{0.35\textwidth}
  275.        \includegraphics[width=\textwidth]{F0010TEK.png}
  276.        \label{d}
  277.        \caption{}
  278.    \end{subfigure}
  279.    ~
  280. \caption {(a) Trace scope when input signal is a sine wave. (b) Trace Scope when input signal is a triangular wave. }
  281.    \label{d}
  282.    \end{figure}
  283. \section{Conclusions}
  284.  
  285. In conclusion, for the first part of the lab which was analyzing photodiodes it was found that when a bright light was shone there was an output voltage that was measured , this helped in the understanding of photodiodes and how they operate. The operation of a transimpedance amplifier with a photodiode was understood as well as the maximum an minimum frequency range and how this relates to equation \ref{photodiode}.The integrator and differentiator circuits produced various output signals when various input signals were inputted, and the integrator and differentiator circuits were inversely and directly proporitonal to the constant RC respectively. This lab could have been improved by taking more measurements, using more sensitive equipment and varying power levels of light and observing the output voltage for the transimpedance amplifier.
  286.  
  287.  
  288. \section{References}
  289.  
  290. \bibliographystyle{aiaa}
  291. \begingroup
  292. \renewcommand{\section}[2]{}
  293. \bibliography{ExampleBib}
  294. \endgroup
  295.  
  296.  
  297.  
  298. \end{document}
  299.  
  300. %%% Local Variables:
  301. %%% mode: latex
  302. %%% TeX-master: t
  303. %%% End:
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