Chapter 12, Problem 12.69P

To determine

To derive: The expression for the loop gain.

Answer to Problem 12.69P

The expression for the loop gain is $\frac{\left(g_{m1}{g}_{m2}{R}_{C2}\left(\frac{1}{R_{E3}}+\frac{1}{R_F}\right)\left(\frac{1}{r_{\pi 3}}+g_{m3}\right)\right)\left(R_{C1}\left|\right|r_{\pi 2}\right)}{\left[\begin{array}{l}\left[\frac{1}{r_{\pi }+g_{m3}}+\left(\frac{1}{R_{E1}}+\frac{1}{R_F}\right)\right]R_F\left(\frac{1}{E_3}+\frac{1}{R_F}\right)\\ \left[\left(\frac{1}{R_{C2}}+\frac{1}{r_{\pi 3}}\right)R_{C2}\left(\frac{1}{R_{E3}}+\frac{1}{R_F}\right)\right]+\left(\frac{1}{r_{\pi }}+g_{m3}\right)-\\ \left[\frac{1}{R_F}\left(-1+\left[\left(\frac{1}{R_{C2}}+\frac{1}{r_{\pi 3}}\right)R_{C2}\left(\frac{1}{R_{E3}}+\frac{1}{R_F}\right)\right]+\left(\frac{1}{r_{\pi }}+g_{m3}\right)\right)\right]\end{array}\right]}$ .

Explanation of Solution

Given:

The given circuit is shown in Figure 1

  

Figure 1

Calculation:

The small signal equivalent diagram for the Figure is shown in Figure 2

  

Figure 2

Apply KCL at node $V_{e1}$ .

  $\begin{array}{l}\frac{V_{\pi 1}}{r_{\pi 1}}+g_{m1}{V}_{\pi 1}=\frac{V_{e1}}{R_{E1}}+\frac{V_{e1}-V_O}{R_F}\\ V_{\pi 1}\left[\frac{1}{r_{\pi 1}}+g_{m1}\right]=V_{e1}\left[\frac{1}{R_{E1}}+\frac{1}{R_F}\right]-\frac{V_O}{R_F}\end{array}$ ....................(1)

Apply KCL at node $V_r$ .

  $\begin{array}{l}{g}_{m1}{V}_{\pi 1}+\frac{V_r}{R_{C1}\left|\right|r_{\pi 2}}=0\\ V_{\pi 1}=-\left(g_{m1}{V}_{\pi 1}\right)\left(R_{C1}\left|\right|r_{\pi 2}\right)\end{array}$ ....................(2)

The expression for the voltage $V_{C2}$ is given by,

  $V_{C2}=V_{\pi 3}+V_O$

Apply KCL at node $V_{C2}$ .

  $g_{m2}{V}_t+\frac{V_{C2}}{R_{C2}}+\frac{V_{\pi 3}}{r_{\pi 3}}=0$

Substitute $V_{\pi 3}+V_O$ and $V_{C2}$ in the above equation.

  $\begin{array}{l}{g}_{m2}{V}_t+\frac{V_{\pi 3}+V_O}{R_{C2}}+\frac{V_{\pi 3}}{r_{\pi 3}}=0\\ g_{m2}{V}_t+V_{\pi 3}\left(\frac{1}{R_{C2}}+\frac{1}{r_{\pi 3}}\right)+\frac{V_O}{R_{C2}}=0\end{array}$ ....................(3)

Apply KCL at node $V_O$ .

  $\begin{array}{l}\frac{V_{\pi 3}}{r_{\pi 3}}+g_{m3}{V}_{\pi 3}=\frac{V_O}{R_{E3}}+\frac{V_O-V_{e1}}{R_F}\\ V_{\pi 3}\left(\frac{1}{r_{\pi 3}}+g_{m3}\right)+\frac{V_{e1}}{R_{E3}}=V_O\left(\frac{1}{R_{E3}}+\frac{1}{R_F}\right)\end{array}$

Substitute $-V_{\pi 1}$ for $V_{e1}$ in the above equation.

  $\begin{array}{l}{V}_{\pi 3}\left(\frac{1}{r_{\pi 3}}+g_{m3}\right)-\frac{-V_{\pi 1}}{R_{E3}}=V_O\left(\frac{1}{R_{E3}}+\frac{1}{R_F}\right)\\ V_O=\frac{V_{\pi 3}\left(\frac{1}{r_{\pi 3}}+g_{m3}\right)-\frac{V_{\pi 1}}{R_{E3}}}{\left(\frac{1}{R_{E3}}+\frac{1}{R_F}\right)}\end{array}$

Substitute $\frac{V_{\pi 3}\left(\frac{1}{r_{\pi 3}}+g_{m3}\right)-\frac{V_{\pi 1}}{R_{E3}}}{\left(\frac{1}{R_{E3}}+\frac{1}{R_F}\right)}$ for $V_O$ and $-V_{\pi 1}$ for $V_{e1}$ in equation (1).

  $\begin{array}{l}{V}_{\pi 1}\left[\frac{1}{r_{\pi 1}}+g_{m1}\right]=-V_{\pi 1}\left[\frac{1}{R_{E1}}+\frac{1}{R_F}\right]-\frac{\left(\frac{V_{\pi 3}\left(\frac{1}{r_{\pi 3}}+g_{m3}\right)-\frac{V_{\pi 1}}{R_{E3}}}{\left(\frac{1}{R_{E3}}+\frac{1}{R_F}\right)}\right)}{R_F}\\ V_{\pi 1}\left[\frac{1}{r_{\pi 1}}+g_{m1}\right]+V_{\pi 1}\left[\frac{1}{R_{E1}}+\frac{1}{R_F}\right]=\frac{-V_{\pi 3}\left(\frac{1}{r_{\pi 3}}+g_{m3}\right)+\frac{V_{\pi 1}}{R_{E3}}}{R_F\left(\frac{1}{R_{E3}}+\frac{1}{R_F}\right)}\end{array}$ ....................(4)

Substitute $\frac{V_{\pi 3}\left(\frac{1}{r_{\pi 3}}+g_{m3}\right)-\frac{V_{\pi 1}}{R_{E3}}}{\left(\frac{1}{R_{E3}}+\frac{1}{R_F}\right)}$ for $V_O$ in equation (3).

  $\begin{array}{l}{g}_{m2}{V}_t+V_{\pi 3}\left(\frac{1}{R_{C2}}+\frac{1}{r_{\pi 3}}\right)+\frac{\frac{V_{\pi 3}\left(\frac{1}{r_{\pi 3}}+g_{m3}\right)-\frac{V_{\pi 1}}{R_{E3}}}{\left(\frac{1}{R_{E3}}+\frac{1}{R_F}\right)}}{R_{C2}}=0\\ V_{\pi 3}=\frac{\frac{V_{\pi 1}}{R_F}-g_{m2}{V}_t{R}_{C2}\left(\frac{1}{R_{E3}}+\frac{1}{R_F}\right)}{\left[\left(\frac{1}{R_{C2}}+\frac{1}{r_{\pi 3}}\right)R_{C2}\left(\frac{1}{R_{E3}}+\frac{1}{R_F}\right)+g_{m3}\right]}\end{array}$

Substitute $\frac{\frac{V_{\pi 1}}{R_F}-g_{m2}{V}_t{R}_{C2}\left(\frac{1}{R_{E3}}+\frac{1}{R_F}\right)}{\left[\left(\frac{1}{R_{C2}}+\frac{1}{r_{\pi 3}}\right)R_{C2}\left(\frac{1}{R_{E3}}+\frac{1}{R_F}\right)+g_{m3}\right]}$ for $V_{\pi 3}$ in the above equation.

  $V_{\pi 1}\left[\frac{1}{r_{\pi 1}}+g_{m1}\right]+V_{\pi 1}\left[\frac{1}{R_{E1}}+\frac{1}{R_F}\right]=\frac{-\left(\frac{\frac{V_{\pi 1}}{R_F}-g_{m2}{V}_t{R}_{C2}\left(\frac{1}{R_{E3}}+\frac{1}{R_F}\right)}{\left[\left(\frac{1}{R_{C2}}+\frac{1}{r_{\pi 3}}\right)R_{C2}\left(\frac{1}{R_{E3}}+\frac{1}{R_F}\right)+g_{m3}\right]}\right)\left(\frac{1}{r_{\pi 3}}+g_{m3}\right)+\frac{V_{\pi 1}}{R_{E3}}}{R_F\left(\frac{1}{R_{E3}}+\frac{1}{R_F}\right)}$

  $V_{\pi 1}\left[\begin{array}{l}\frac{1}{r_{\pi 1}}+g_{m1}\\ +\frac{1}{R_{E1}}+\frac{1}{R_F}\end{array}\right]\left[R_F\left(\frac{1}{E_3}+\frac{1}{R_F}\right)\right]=\frac{-\frac{V_{\pi 1}}{R_F}+g_{m2}{V}_t{R}_{C2}\left(\frac{1}{R_{E3}}+\frac{1}{R_F}\right)\left(\frac{1}{r_{\pi 3}}+g_{m3}\right)}{\left[\begin{array}{l}\left(\frac{1}{R_{C2}}+\frac{1}{r_{\pi 3}}\right)R_{C2}\\ \left(\frac{1}{R_{E3}}+\frac{1}{R_F}\right)+g_{m3}\end{array}\right]+\left(\frac{1}{r_{\pi }+g_{m3}}\right)}+\frac{V_{\pi 1}}{R_{E3}}$

  $V_{\pi 1}=\frac{V_t\left(g_{m2}{R}_{C2}\left(\frac{1}{R_{E3}}+\frac{1}{R_F}\right)\left(\frac{1}{r_{\pi 3}}+g_{m3}\right)\right)}{\left[\begin{array}{l}\left[\frac{1}{r_{\pi }+g_{m3}}+\left(\frac{1}{R_{E1}}+\frac{1}{R_F}\right)\right]R_F\left(\frac{1}{E_3}+\frac{1}{R_F}\right)\\ \left[\left(\frac{1}{R_{C2}}+\frac{1}{r_{\pi 3}}\right)R_{C2}\left(\frac{1}{R_{E3}}+\frac{1}{R_F}\right)\right]+\left(\frac{1}{r_{\pi }}+g_{m3}\right)-\\ \left[\frac{1}{R_F}\left(\left[-1+\left[\left(\frac{1}{R_{C2}}+\frac{1}{r_{\pi 3}}\right)R_{C2}\left(\frac{1}{R_{E3}}+\frac{1}{R_F}\right)\right]+\left(\frac{1}{r_{\pi }}+g_{m3}\right)\right]\right)\right]\end{array}\right]}$.

Substitute $\frac{V_t\left(g_{m2}{R}_{C2}\left(\frac{1}{R_{E3}}+\frac{1}{R_F}\right)\left(\frac{1}{r_{\pi 3}}+g_{m3}\right)\right)}{\left[\begin{array}{l}\left[\frac{1}{r_{\pi }+g_{m3}}+\left(\frac{1}{R_{E1}}+\frac{1}{R_F}\right)\right]R_F\left(\frac{1}{E_3}+\frac{1}{R_F}\right)\\ \left[\left(\frac{1}{R_{C2}}+\frac{1}{r_{\pi 3}}\right)R_{C2}\left(\frac{1}{R_{E3}}+\frac{1}{R_F}\right)\right]+\left(\frac{1}{r_{\pi }}+g_{m3}\right)-\\ \frac{1}{R_F}\left(\left[-1+\left[\left(\frac{1}{R_{C2}}+\frac{1}{r_{\pi 3}}\right)R_{C2}\left(\frac{1}{R_{E3}}+\frac{1}{R_F}\right)\right]+\left(\frac{1}{r_{\pi }}+g_{m3}\right)\right]\right)\end{array}\right]}$ for $V_{\pi 1}$ in equation (2).

  $V_r=-g_{m1}\frac{V_t\left(g_{m2}{R}_{C2}\left(\frac{1}{R_{E3}}+\frac{1}{R_F}\right)\left(\frac{1}{r_{\pi 3}}+g_{m3}\right)\right)\left(R_{C1}\left|\right|r_{\pi 2}\right)}{\left[\begin{array}{l}\left[\frac{1}{r_{\pi }+g_{m3}}+\left(\frac{1}{R_{E1}}+\frac{1}{R_F}\right)\right]R_F\left(\frac{1}{E_3}+\frac{1}{R_F}\right)\\ \left[\left(\frac{1}{R_{C2}}+\frac{1}{r_{\pi 3}}\right)R_{C2}\left(\frac{1}{R_{E3}}+\frac{1}{R_F}\right)\right]+\left(\frac{1}{r_{\pi }}+g_{m3}\right)-\\ \left[\frac{1}{R_F}\left(\left[-1+\left[\left(\frac{1}{R_{C2}}+\frac{1}{r_{\pi 3}}\right)R_{C2}\left(\frac{1}{R_{E3}}+\frac{1}{R_F}\right)\right]+\left(\frac{1}{r_{\pi }}+g_{m3}\right)\right]\right)\right]\end{array}\right]}$

  $-\frac{V_r}{V_t}=\frac{\left(g_{m1}{g}_{m2}{R}_{C2}\left(\frac{1}{R_{E3}}+\frac{1}{R_F}\right)\left(\frac{1}{r_{\pi 3}}+g_{m3}\right)\right)\left(R_{C1}\left|\right|r_{\pi 2}\right)}{\left[\begin{array}{l}\left[\frac{1}{r_{\pi }+g_{m3}}+\left(\frac{1}{R_{E1}}+\frac{1}{R_F}\right)\right]R_F\left(\frac{1}{E_3}+\frac{1}{R_F}\right)\\ \left[\left(\frac{1}{R_{C2}}+\frac{1}{r_{\pi 3}}\right)R_{C2}\left(\frac{1}{R_{E3}}+\frac{1}{R_F}\right)\right]+\left(\frac{1}{r_{\pi }}+g_{m3}\right)-\\ \left[\frac{1}{R_F}\left(-1+\left[\left(\frac{1}{R_{C2}}+\frac{1}{r_{\pi 3}}\right)R_{C2}\left(\frac{1}{R_{E3}}+\frac{1}{R_F}\right)\right]+\left(\frac{1}{r_{\pi }}+g_{m3}\right)\right)\right]\end{array}\right]}$

Conclusion:

Therefore, the expression for the loop gain is $\frac{\left(g_{m1}{g}_{m2}{R}_{C2}\left(\frac{1}{R_{E3}}+\frac{1}{R_F}\right)\left(\frac{1}{r_{\pi 3}}+g_{m3}\right)\right)\left(R_{C1}\left|\right|r_{\pi 2}\right)}{\left[\begin{array}{l}\left[\frac{1}{r_{\pi }+g_{m3}}+\left(\frac{1}{R_{E1}}+\frac{1}{R_F}\right)\right]R_F\left(\frac{1}{E_3}+\frac{1}{R_F}\right)\\ \left[\left(\frac{1}{R_{C2}}+\frac{1}{r_{\pi 3}}\right)R_{C2}\left(\frac{1}{R_{E3}}+\frac{1}{R_F}\right)\right]+\left(\frac{1}{r_{\pi }}+g_{m3}\right)-\\ \left[\frac{1}{R_F}\left(-1+\left[\left(\frac{1}{R_{C2}}+\frac{1}{r_{\pi 3}}\right)R_{C2}\left(\frac{1}{R_{E3}}+\frac{1}{R_F}\right)\right]+\left(\frac{1}{r_{\pi }}+g_{m3}\right)\right)\right]\end{array}\right]}$ .