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\begin{document}

%TITLE OF REPORT
\title{Determining $\beta$ for a 2N2222 transistor}

% AUTHORS ... MAKE SURE YOUR NAME GOES FIRST, FOLLOWED BY LAB PARTNERS
\author{Darrell F. Schroeter}
\email[]{schroetd@reed.edu}
\author{Joel S. Franklin}
\author{John M. Essick}
\affiliation{Reed College, Portland, OR 97202}

\date{\today}

\begin{abstract}
% ABSTRACT GOES HERE
In this experiment we measure the constant $\beta \equiv I_c/I_b$ for a 2N2222 transistor.  We show that $\beta$ is in fact constant in the linear regime and equal to approximately $190$.  In addition, we show that the linear regime is realized for base currents of $I_b < 50 \, \mu A$ and that in this regime the voltage hierarchy is satisfied.
\end{abstract}

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\section{Introduction}

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The transistor is the fundamental building block of all modern electronic devices.  A detailed description of transistor operation and circuit analysis can be found in  Horowitz and Hill~\cite{handh}. In a transistor, the current $I_b$ flowing into the base controls the current $I_c$ that flows from the collector to the emitter.  A picture of a transistor, along with its circuit drawing is shown in Figure~\ref{fig:transistor}.

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	\centerline{\includegraphics[scale=0.375]{Figures/TransistorDiagram}}
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	\caption{The diagram shows a schematic of the 2N2222 silicon NPN transistor used in this experiment.  The figure on the left shows the correct labeling of the leads for the transistor.  The figure on the right shows the circuit diagram.}
	% FINALLY WE LABEL THIS FIGURE SO THAT WE CAN REFERENCE IT AS DONE ABOVE
	\label{fig:transistor}
\end{figure}

There are two different regimes of operation for a transistor when it is on.  In the linear regime, the current that flows into the collector is proportional to the current that flows into the base:
% HERE WE INCLUDE AN EQUATION
\begin{equation}
	I_c = \beta \, I_b \, . 
	% THIS EQUATION IS LABELED SO WE CAN REFER TO IT IN THE TEXT
	\label{eq:transistorlinear1}
\end{equation}
The constant $\beta$ is large, typically  $100-200$, and effectively constant over the linear regime~\cite{EssickLecture}.  Because the current that flows into the collector is negligible when the transistor is operating in the linear regime, the current that flows out of the emitter is essentially equal to the current that flows into the collector
\begin{equation}
	I_e \approx I_b \, . 
	\label{eq:transistorlinear2}
\end{equation}
Finally, the transistor maintains a fixed voltage between the base and the collector.  Referred to as $V_{be} \equiv V_b - V_e$, this value is approximately 
\begin{equation}
	V_{be} \approx 0.6 \, V \, .
	\label{eq:transistorlinear3}
\end{equation}
The three equations~\ref{eq:transistorlinear1}, \ref{eq:transistorlinear2}, and \ref{eq:transistorlinear3} may be used to analyze a transistor operating in the linear regime. 

The transistor will not operate as described above if $I_b$ becomes too large.  When this happens the transistor is said to be \emph{saturated} and  $\beta$ is no longer constant.  When the transistor is saturated the voltage difference between the collector and the emitter drops to $V_{ce} \equiv V_c - V_e \approx 0.1$--$0.4 V$  while for a transistor operating in the linear regime this difference is always at least as big as $V_{be}$ so that the voltage hierarchy $V_b > V_c > V_e$ is preserved. The experiment performed here will provide a quantitative measure of how large $I_b$ needs to be to drive the transistor into saturation mode.

% ANOTHER FIGURE
\begin{figure}[h]
	\centerline{\includegraphics[scale=0.25]{Figures/CircuitDiagram}}
	\caption{Circuit used to  measure $\beta$ of the 2N2222 silicon NPN transistor.  The variable resistor $R$ is used to control the base voltage $I_b$.  The collector current $I_c$ is measured with the ammeter inserted between the $+12 \, V$ supply and the $1 \, \mathrm{k}\Omega$ resistor.}
	\label{fig:circuit}
\end{figure}


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\section{Procedure}

The circuit used to determine $\beta$ for the 2N2222 transistor is shown in Figure~\ref{fig:circuit}.  The base current $I_b$ is controlled by the variable resistor $R$.  Applying Ohm's Law to the base leg of the circuit gives 
\begin{equation}
	5 \, V - I_b \, R - I_b \, 4.7 \, \mathrm{k}\Omega = V_b \, . 
	\label{eq:OhmBase}
\end{equation}
Equation~\ref{eq:transistorlinear3} applies to both the linear and saturated regimes and we therefore expect the base-emitter voltage difference of the transistor to be $V_b - V_e \approx 0.6 \, \mathrm{V}$.  Combining this with Equation~\ref{eq:OhmBase} we see that the base current will be given by
\begin{equation}
	I_b = \frac{4.4 \, V}{R + 4.7 \, \mathrm{k}\Omega} \, .
	\label{eq:Ibcalc}
\end{equation}
The experiment is set up so that we will measure the collector current $I_c$ directly.  We can use this to determine the value of $V_c$ by applying Ohm's Law to the collector leg of the circuit:
\begin{equation}
	+12 \, V - I_c \, R = V_c \, . 
	\label{eq:Vccalc}
\end{equation}



\section{Results}

The collector current $I_c$ was measured for a range of resistor values $R$ between $0$ and $1 \, \mathrm{M}\Omega$.  The results are shown in Table~\ref{tab:data}.  For each value of $R$, $I_b$ was calculated using Equation~\ref{eq:Ibcalc} and $V_c$ was calculated using Equation~\ref{eq:Vccalc}.  From the measured value of $I_c$ and the calculated value of $I_b$, $\beta$ was calculated according to Equation~\ref{eq:transistorlinear1}.

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\begin{table}[h]
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	\caption{Experimental data.  For each value of the variable resistor $R$, the collector current $I_c$ was measured, and the collector voltage $V_c$, base current $I_b$ and proportionality constant $\beta$ were calculated according to Equations~\ref{eq:Vccalc}, \ref{eq:Ibcalc}, and \ref{eq:transistorlinear2} respectively. }
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	$R \, \of{\mathrm{k} \Omega}$ & $I_c \, \of{\mathrm{m} A}$ & $V_c \, \of{V}$ & $I_b \, \of{\mathrm{m} A}$ & $\beta$ \\  \hline
	0 & 11.67 & 0.04 &  0.94 &  12.4 \\ 
	47 & 11.54 & 0.17  & 0.09  & 130 \\ 
	100 & 8.33 & 3.3  & 0.04  & 189 \\ 
	470 & 1.90 & 9.78  & 0.01  & 196 \\ 
	1000 & 0.9 & 10.75  & 0.005  & 196 \\ 
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	\label{tab:data}
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Figure~\ref{fig:betavsR} shows a plot of $\beta$ versus the resistance $R$.  For values of $R$ of $100 \, \mathrm{k}\Omega$ and above, the transistor is clearly operating in the linear regime and $\beta$ is approximately constant and equal to $\beta \approx 190$.  From Table~\ref{tab:data} we see that these values of $R$ correspond to base currents of $I_b < 50 \,  \mu A$.  We also see from Table~\ref{tab:data} that for these values of $R$ the voltage hierarchy is satisfied with $V_c > V_b = 0.6 \, V$.

\begin{figure}[b]
	\centerline{\includegraphics[scale=0.8]{Figures/BetavsR}}
	\caption{Plot of $\beta$ as a function of the variable resistance $R$.  For $R\geq100 \, \mathrm{k}\Omega$, $\beta$ is effectively constant and equal to approximately 190.}
	\label{fig:betavsR}
\end{figure}



% THE FINAL SECTION IS DISCUSSION
\section{Discussion}

This experiment has shown that the 2N2222 transistors operate in the linear regime for base currents of $I_b < 50 \, \mu A$ and that for base currents in this range the voltage hierarchy is satisfied.  When operating in the linear regime, it has been verified that the base and collector currents are proportional with $\beta \equiv I_c/I_b \approx 190$.  

The experiment could have been improved by measuring the actual voltage of the $+12 \, V$ and $+5 \, V$ terminals, and by directly measuring the voltage difference $V_{ce}$ between the collector and the emitter. Since our determination of $I_b$ (and therefore $\beta$) relied on knowledge of these values, it would have been preferable to measure them directly.

\vfill


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