The small$-$signal equivalent circuit is a simplified version of the amplifier circuit that represents the AC behavior of the transistor. Here, the transistor is often represented by its small$-$signal model $($for example, using the hybrid$-$pi or re model$).$

Calculate $\backslash ($ r$\_\backslash$pi $\backslash )$ and the Voltage Gain $\backslash ($ A$\_$V $\backslash )$

$\backslash ($ r$\_\backslash$pi $\backslash )$ is the input resistance of the transistor's base and is given by:

$\backslash [$ r$\_\backslash$pi $=\backslash$frac$\{\backslash$beta$\}\{$g$\_$m$\}\backslash ]$

where $\backslash ($ g$\_$m $\backslash )$ is the transconductance and can be calculated using:

$\backslash [$ g$\_$m $=\backslash$frac$\{$I$\_$C$\}\{$V$\_$T$\}\backslash ]$

$\backslash ($ V$\_$T $\backslash )$ is the thermal voltage, approximately $26$mV at room temperature, and $\backslash ($ I$\_$C $\backslash )$ is the quiescent collector current.

The voltage gain $\backslash ($ A$\_$V $\backslash )$ can be estimated by:

$\backslash [$ A$\_$V $=-$g$\_$m $\backslash$cdot $($R$\_$C $||$ R$\_$L$)\backslash ]$

where $\backslash ($ R$\_$C $\backslash )$ and $\backslash ($ R$\_$L $\backslash )$ are the collector and load resistances, respectively, and $\backslash (||\backslash )$ denotes parallel combination.

Determine the Peak Amplitude of $\backslash ($ V$\_\{$IN$\}\backslash )$ for Maximum Swing

Explanation:

The maximum undistorted swing in the output is limited by the power supply voltage $\backslash ($ V$\_\{$CC$\}\backslash )$ and the quiescent point of the transistor $($the DC operating point$).$ The maximum peak$-$to$-$peak swing can be roughly estimated as:

$\backslash [$ V$\_\{$pp$\}\backslash$approx V$\_\{$CC$\}-$ V$\_\{$CE$($sat$)\}-$ V$\_\{$BE$($on$)\}\backslash ]$

where $\backslash ($ V$\_\{$CE$($sat$)\}\backslash )$ is the saturation voltage of the collector$-$emitter junction, and $\backslash ($ V$\_\{$BE$($on$)\}\backslash )$ is the base$-$emitter on voltage.

For a symmetric swing, we'd want to set the quiescent point $\backslash ($ V$\_\{$CEQ$\}\backslash )$ to be roughly half of $\backslash ($ V$\_\{$CC$\}\backslash ),$ and then we can estimate the maximum input amplitude by considering the gain of the amplifier.

Calculate the Quiescent Point

To set the quiescent point, you typically use the following equations:

$\backslash [$ V$\_\{$CEQ$\}=$ V$\_\{$CC$\}-$ I$\_\{$CQ$\}\backslash$cdot R$\_$C $\backslash ]$

$\backslash [$ I$\_\{$CQ$\}=\backslash$beta $\backslash$cdot I$\_\{$BQ$\}\backslash ]$

$\backslash [$ I$\_\{$BQ$\}=\backslash$frac$\{$V$\_\{$CC$\}-$ V$\_\{$BE$\}\}\{$R$\_1+$ R$\_2+(\backslash$beta $+1)$R$\_$E$\}\backslash ]$

where $\backslash ($ V$\_\{$BE$\}\backslash )$ is typically around $0\mathrm{.}7$V for silicon BJTs$.$

Step $2$

Answer

Based on the given values and calculations, here are the results:

$-$ The quiescent base current $\backslash ($ I$\_\{$BQ$\}\backslash )$ is approximately $76\mathrm{.}48$A$.$

$-$ The quiescent collector current $\backslash ($ I$\_\{$CQ$\}\backslash )$ is approximately $8\mathrm{.}03$ mA$.$

$-$ The quiescent collector$-$emitter voltage $\backslash ($ V$\_\{$CEQ$\}\backslash )$ is approximately $1\mathrm{.}97$ V$.$

With these quiescent point values, we can calculate further parameters:

$-$ The transconductance $\backslash ($ g$\_$m $\backslash )$ is approximately $0\mathrm{.}309$ S $($Siemens$).$

$-$ The input resistance $\backslash ($ r$\_\backslash$pi $\backslash )$ is approximately $340\backslash (\backslash$Omega $\backslash ).$

The voltage gain $\backslash ($ A$\_$V $\backslash )$ of the amplifier is approximately $-154\mathrm{.}43.$ The negative sign indicates an inversion of phase between input and output, which is characteristic of common$-$emitter amplifier configurations.

For the maximum undistorted output swing, which is half the supply voltage for a symmetrical swing, we have $\backslash ($ V$\_\{$max$\backslash$ swing$\}\backslash )$ as approximately $5$ V $($since $\backslash ($ V$\_\{$CC$\}\backslash )$ is $10$ V$).$

Given this voltage gain, the peak amplitude of $\backslash ($ V$\_\{$IN$\}\backslash )$ for the maximum symmetrical swing is approximately $32\mathrm{.}38$ mV