Determine \(g_m\) and \(r_d\) from the specification sheets.
\(\boxed{g_m = g_{fs} = y_{fs}}\)
\(\boxed{r_d = \frac{1}{g_{os}} = \frac{1}{y_{os}}}\)
If \(g_{fs}\) or \(y_{fs}\) is not available, determine \(g_m\) using the values of \(V_{GS}\) and \(I_D\) from the DC biasing arrangement.
\(\boxed{g_m = 2k\left(V_{GS} - V_{GS(Th)}\right)}\)
\(\boxed{k = \frac{I_{D(on)}}{\left( V_{GS(on)} - V_{GS(Th)} \right) ^ 2}}\)
Once the levels of \(g_m\) and \(r_d\) are determined, the AC equivalent model can be substituted between the appropriate terminals. Set all capacitors and DC sources to short-circuit equivalent.
Input impedance \(Z_i\)
\(\displaystyle V_i = V_{gs}\)
Applying Kirchhoff’s voltage law from input to output.
\(\displaystyle -V_i + I_i R_F + V_o = 0\)
\(\displaystyle V_o = V_i - I_i R_F\)
Applying Kirchhoff’s current law at node \(D\).
\(\displaystyle I_i = g_m V_{gs} + \frac{V_o}{r_d \parallel R_D}\)
\(\displaystyle I_i = g_m V_i + \frac{V_i - I_i R_F}{r_d \parallel R_D}\)
\(\displaystyle I_i = g_m V_i + \frac{V_i}{r_d \parallel R_D} - \frac{I_i R_F}{r_d \parallel R_D}\)
\(\displaystyle I_i + \frac{I_i R_F}{r_d \parallel R_D} = g_m V_i + \frac{V_i}{r_d \parallel R_D}\)
\(\displaystyle I_i\left(1 + \frac{R_F}{r_d \parallel R_D}\right) = V_i\left(g_m + \frac{1}{r_d \parallel R_D}\right)\)
\(\displaystyle Z_i = \frac{V_i}{I_i} = \frac{\displaystyle 1 + \frac{R_F}{r_d \parallel R_D}}{\displaystyle g_m + \frac{1}{r_d \parallel R_D}}\)
\(\boxed{Z_i = \frac{R_F + \left(r_d \parallel R_D\right)}{1 + g_m\left(r_d \parallel R_D\right)}}\)
Output impedance \(Z_o\) (set \(V_i = 0\ V\))
\(\boxed{Z_o = R_F \parallel r_d \parallel R_D}\)
Voltage gain \(A_v\)
\(\displaystyle V_i = V_{gs}\)
Applying Kirchhoff’s current law at node \(D\).
\(\displaystyle I_i = g_m V_{gs} + \frac{V_o}{r_d \parallel R_D}\)
\(\displaystyle \frac{V_i - V_o}{R_F} = g_m V_i + \frac{V_o}{r_d \parallel R_D}\)
\(\displaystyle \frac{V_i}{R_F} - \frac{V_o}{R_F} = g_m V_i + \frac{V_o}{r_d \parallel R_D}\)
\(\displaystyle \frac{V_i}{R_F} - g_m V_i = \frac{V_o}{R_F} + \frac{V_o}{r_d \parallel R_D}\)
\(\displaystyle V_i\left(\frac{1}{R_F} - g_m\right) = V_o\left(\frac{1}{R_F} + \frac{1}{r_d \parallel R_D}\right)\)
\(\displaystyle A_v = \frac{V_o}{V_i} = \frac{\displaystyle \frac{1}{R_F} - g_m}{\displaystyle \frac{1}{R_F} + \frac{1}{r_d \parallel R_D}} = \frac{\displaystyle \frac{1}{R_F} - g_m}{\displaystyle \frac{1}{R_F} + \frac{1}{r_d} + \frac{1}{R_D}}\)
\(\boxed{A_v = \left(\frac{1}{R_F} - g_m\right)\left(R_F \parallel r_d \parallel R_D\right)}\)
Since \(\displaystyle g_m \gg \frac{1}{R_F}\)
\(\boxed{A_v = -g_m\left(R_F \parallel r_d \parallel R_D\right)}\)
The negative sign for \(A_v\) reveals that \(V_o\) and \(V_i\) are out of phase by \(180^\circ\).