Chapter 19, Problem 88P

Using the y parameters, derive formulas for Z_in, Z_out, A_i, and A_v for the common-emitter transistor circuit.

To determine

Derive the formulae for input impedance, output impedance, current gain, and voltage gain for the common-emitter transistor circuit using y parameters.

Explanation of Solution

Formula used:

Write the expressions for admittance parameters of a two-port network as follows:

$I_{\text{1}}=y_{\text{11}}{V}_1+y_{\text{12}}{V}_{\text{2}}$ (1)

$I_{\text{2}}=y_{\text{21}}{V}_{\text{1}}+y_{\text{22}}{V}_{\text{2}}$ (2)

Refer to Figure 19.56 (a) for schematic circuit of common-emitter amplifier (transistor) and write the formulae for input impedance, output impedance, current gain, and voltage gain for the common-emitter transistor circuit as follows:

Write the expression for voltage gain of a common-emitter transistor as follows:

$A_v=\frac{V_c}{V_b}$ (3)

Here,

$V_c$ is the collector voltage of the transistor, which is the output voltage and

$V_b$ is the base voltage of the transistor.

Write the expression for current gain of a common-emitter transistor as follows:

$A_i=\frac{I_c}{I_b}$ (4)

Here,

$I_c$ is the collector current of the transistor and

$I_b$ is the base current of the transistor.

Write the expression for input impedance of a common-emitter transistor follows:

$Z_{\text{in}}=\frac{V_b}{I_b}$ (5)

Write the expression for output impedance of a common-emitter transistor as follows:

$Z_{\text{out}}=\frac{V_c}{I_c}$ (6)

Calculation:

Draw a two-port circuit to obtain input impedance, current gain, and voltage gain for the common-emitter transistor circuit as shown in Figure 1.

From Figure 1, write the expression for current $I_{\text{2}}$ as follows:

$I_{\text{2}}=-\frac{V_{\text{2}}}{R_L}$

Substitute $\left(-\frac{V_{\text{2}}}{R_L}\right)$ for $I_{\text{2}}$ in Equation (2) as follows:

$\begin{array}{l}{y}_{\text{22}}{V}_{\text{2}}+\frac{V_{\text{2}}}{R_L}=-y_{\text{21}}{V}_{\text{1}}\\ V_{\text{2}}\left(y_{\text{22}}+\frac{1}{R_L}\right)=-y_{\text{21}}{V}_{\text{1}}\end{array}$

$V_{\text{2}}=\frac{-y_{\text{21}}{V}_{\text{1}}}{y_{\text{22}}+\frac{1}{R_L}}$ (7)

Substitute $\left(\frac{-y_{\text{21}}{V}_{\text{1}}}{y_{\text{22}}+\frac{1}{R_L}}\right)$ for $V_{\text{2}}$ in Equation (1) as follows:

$I_{\text{1}}=y_{\text{11}}{V}_1+y_{\text{12}}\left(\frac{-y_{\text{21}}{V}_{\text{1}}}{y_{\text{22}}+\frac{1}{R_L}}\right)$

Consider $\left(\frac{1}{R_L}\right)$ as $Y_L$ and rewrite the expression as follows:

$\begin{array}{c}{I}_{\text{1}}=y_{\text{11}}{V}_1+y_{\text{12}}\left(\frac{-y_{\text{21}}{V}_{\text{1}}}{y_{\text{22}}+Y_L}\right)\\ =\left(y_{\text{11}}-\frac{y_{\text{12}}{y}_{\text{21}}}{y_{\text{22}}+Y_L}\right)V_{\text{1}}\\ =\left(\frac{y_{\text{11}}{Y}_L+y_{\text{11}}{y}_{\text{22}}-y_{\text{12}}{y}_{\text{21}}}{y_{\text{22}}+Y_L}\right)V_{\text{1}}\\ =\left(\frac{y_{\text{11}}{Y}_L+{\Delta }_y}{y_{\text{22}}+Y_L}\right)V_{\text{1}}\left\{\because y_{\text{11}}{y}_{\text{22}}-y_{\text{12}}{y}_{\text{21}}={\Delta }_y\right\}\end{array}$

As the $V_{\text{1}}$ is the input port voltage and $I_{\text{1}}$ is the input port current for the two-port network, consider $V_{\text{1}}$ as base voltage and $I_{\text{1}}$ as base current for the common-emitter transistor circuit. Therefore, rewrite the expression for common-emitter transistor circuit as follows:

$I_b=\left(\frac{y_{\text{11}}{Y}_L+{\Delta }_y}{y_{\text{22}}+Y_L}\right)V_b$

Rearrange the expression as follows:

$\frac{V_b}{I_b}=\frac{y_{\text{22}}+Y_L}{y_{\text{11}}{Y}_L+{\Delta }_y}$

Substitute $\left(\frac{y_{\text{22}}+Y_L}{y_{\text{11}}{Y}_L+{\Delta }_y}\right)$ for $\frac{V_b}{I_b}$ in Equation (5) to obtain the input impedance of the common-emitter transistor circuit.

$Z_{\text{in}}=\frac{y_{\text{22}}+Y_L}{y_{\text{11}}{Y}_L+{\Delta }_y}$ (8)

As the current $I_2$ is the output port current and $V_{\text{2}}$ is the output port voltage for the two-port network, consider $I_2$ as collector current and $V_{\text{2}}$ as collector for the common-emitter transistor circuit. Therefore, rewrite the expression in Equation (2) for common-emitter transistor circuit as follows:

$I_c=y_{\text{21}}{V}_b+y_{\text{22}}{V}_c$

Modify the expression in Equation (7) for common-emitter transistor circuit as follows:

$V_c=\frac{-y_{\text{21}}{V}_b}{y_{\text{22}}+Y_L}$

Substitute $\left(y_{\text{21}}{V}_b+y_{\text{22}}{V}_c\right)$ for $I_c$ in Equation (4) as follows:

$\begin{array}{c}{A}_i=\frac{y_{\text{21}}{V}_b+y_{\text{22}}{V}_c}{I_b}\\ =y_{\text{21}}\left(\frac{V_b}{I_b}\right)+y_{\text{22}}\left(\frac{V_c}{I_b}\right)\end{array}$

Substitute $\left(\frac{-y_{\text{21}}{V}_b}{y_{\text{22}}+Y_L}\right)$ for $V_c$ and from Equation (5), substitute $Z_{\text{in}}$ for $\left(\frac{V_b}{I_b}\right)$ as follows:

$\begin{array}{c}{A}_i=\frac{y_{\text{21}}{V}_b+y_{\text{22}}{V}_c}{I_b}\\ =y_{\text{21}}{Z}_{\text{in}}+y_{\text{22}}\left[\frac{\left(\frac{-y_{\text{21}}{V}_b}{y_{\text{22}}+Y_L}\right)}{I_b}\right]\\ =y_{\text{21}}{Z}_{\text{in}}-\left(\frac{y_{\text{21}}{y}_{\text{22}}}{y_{\text{22}}+Y_L}\right)\left(\frac{V_b}{I_b}\right)\\ =y_{\text{21}}{Z}_{\text{in}}-\left(\frac{y_{\text{21}}{y}_{\text{22}}}{y_{\text{22}}+Y_L}\right)\left(Z_{\text{in}}\right)\left\{\because \frac{V_b}{I_b}=Z_{\text{in}}\right\}\end{array}$

$A_i=Z_{\text{in}}\left(y_{\text{21}}-\frac{y_{\text{21}}{y}_{\text{22}}}{y_{\text{22}}+Y_L}\right)$

From Equation (8), substitute $\left(\frac{y_{\text{22}}+Y_L}{y_{\text{11}}{Y}_L+{\Delta }_y}\right)$ for $Z_{\text{in}}$ to obtain the current gain for the common-emitter transistor circuit.

$\begin{array}{c}{A}_i=\left(\frac{y_{\text{22}}+Y_L}{y_{\text{11}}{Y}_L+{\Delta }_y}\right)\left(y_{\text{21}}-\frac{y_{\text{21}}{y}_{\text{22}}}{y_{\text{22}}+Y_L}\right)\\ =\left(\frac{y_{\text{22}}+Y_L}{y_{\text{11}}{Y}_L+{\Delta }_y}\right)\left(\frac{y_{\text{21}}\left(y_{\text{22}}+Y_L\right)-y_{\text{21}}{y}_{\text{22}}}{y_{\text{22}}+Y_L}\right)\\ =\frac{y_{\text{21}}{Y}_L}{y_{\text{11}}{Y}_L+{\Delta }_y}\end{array}$

Substitute $\left(\frac{-y_{\text{21}}{V}_b}{y_{\text{22}}+Y_L}\right)$ for $V_c$ and in Equation (3) to obtain the voltage gain for common-emitter transistor circuit as follows:

$\begin{array}{c}{A}_v=\frac{\left(\frac{-y_{\text{21}}{V}_b}{y_{\text{22}}+Y_L}\right)}{V_b}\\ =\frac{-y_{\text{21}}}{y_{\text{22}}+Y_L}\end{array}$

Draw a two-port circuit to obtain output impedance for the common-emitter transistor circuit as shown in Figure 2.

From Figure 2, write the expression for output impedance as follows:

$Z_{\text{out}}=\frac{V_2}{I_2}$

From Equation (2), substitute $\left(y_{\text{21}}{V}_{\text{1}}+y_{\text{22}}{V}_{\text{2}}\right)$ for $I_{\text{2}}$ as follows:

$Z_{\text{out}}=\frac{V_2}{y_{\text{21}}{V}_{\text{1}}+y_{\text{22}}{V}_{\text{2}}}$ (9)

From Figure 2, write the expression for $V_{\text{1}}$ as follows:

$V_{\text{1}}=-R_s{I}_1$

Substitute $\left(-R_s{I}_1\right)$ for $V_{\text{1}}$ in Equation (1) as follows:

$\begin{array}{l}{I}_{\text{1}}=y_{\text{11}}\left(-R_s{I}_1\right)+y_{\text{12}}{V}_{\text{2}}\\ I_{\text{1}}\left(1+y_{\text{11}}{R}_s\right)=y_{\text{12}}{V}_{\text{2}}\\ V_{\text{2}}=\frac{I_{\text{1}}\left(1+y_{\text{11}}{R}_s\right)}{y_{\text{12}}}\end{array}$

Substitute $\left(-R_s{I}_1\right)$ for $V_{\text{1}}$ and $\left[\frac{I_{\text{1}}\left(1+y_{\text{11}}{R}_s\right)}{y_{\text{12}}}\right]$ for $V_{\text{2}}$ in Equation (9) as follows:

$\begin{array}{c}{Z}_{\text{out}}=\frac{\frac{I_{\text{1}}\left(1+y_{\text{11}}{R}_s\right)}{y_{\text{12}}}}{y_{\text{21}}\left(-R_s{I}_1\right)+y_{\text{22}}\left[\frac{I_{\text{1}}\left(1+y_{\text{11}}{R}_s\right)}{y_{\text{12}}}\right]}\\ =\frac{I_{\text{1}}\left(1+y_{\text{11}}{R}_s\right)}{-y_{\text{12}}{y}_{\text{21}}{R}_s{I}_1+\left(y_{\text{22}}+y_{\text{11}}{y}_{\text{22}}{R}_s\right)I_{\text{1}}}\\ =\frac{R_s{I}_{\text{1}}\left(y_{\text{11}}+\frac{1}{R_s}\right)}{\left[-y_{\text{12}}{y}_{\text{21}}+\left(\frac{y_{\text{22}}}{R_s}+y_{\text{11}}{y}_{\text{22}}\right)\right]I_{\text{1}}{R}_s}\\ =\frac{y_{\text{11}}+Y_s}{y_{\text{22}}{Y}_s+{\Delta }_y}\left\{\because \frac{1}{R_s}=Y_s\right\}\end{array}$

From the calculations, the formulae for input impedance, output impedance, current gain, and voltage gain for the common-emitter transistor circuit are derived as follows:

$\begin{array}{c}{Z}_{\text{in}}=\frac{y_{\text{22}}+Y_L}{y_{\text{11}}{Y}_L+{\Delta }_y}\\ Z_{\text{out}}=\frac{y_{\text{11}}+Y_s}{y_{\text{22}}{Y}_s+{\Delta }_y}\\ A_i=\frac{y_{\text{21}}{Y}_L}{y_{\text{11}}{Y}_L+{\Delta }_y}\\ A_v=\frac{-y_{\text{21}}}{y_{\text{22}}+Y_L}\end{array}$

Conclusion:

Thus, the formulae for input impedance, output impedance, current gain, and voltage gain for the common-emitter transistor circuit are derived.