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.
\(\boxed{g_m = \frac{2I_{DSS}}{\left|V_P\right|}\left(1 - \frac{V_{GS}}{V_P}\right) = \frac{2I_{DSS}}{\left|V_P\right|} \sqrt{\frac{I_D}{I_{DSS}}}}\)
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 \(Z_i\)
\(\displaystyle V_{gs} = -V^\prime\)
\(\displaystyle V_{R_D} = I^\prime R_D\)
Applying Kirchoff’s current law at node \(a\).
\(\displaystyle I^\prime + g_m V_{gs} = I_{r_d}\)
\(\displaystyle I^\prime - g_m V^\prime = \frac{V^\prime - V_{R_D}}{r_d}\)
\(\displaystyle I^\prime - g_m V^\prime = \frac{V^\prime}{r_d} - \frac{I^\prime R_D}{r_d}\)
\(\displaystyle I^\prime + \frac{I^\prime R_D}{r_d} = \frac{V^\prime}{r_d} + g_m V^\prime\)
\(\displaystyle I^\prime \left(1 + \frac{R_D}{r_d}\right) = V^\prime \left(\frac{1}{r_d} + g_m\right)\)
\(\displaystyle Z_i^\prime = \frac{V^\prime}{I^\prime} = \frac{\displaystyle 1 + \frac{R_D}{r_d}}{\displaystyle \frac{1}{r_d} + g_m} = \frac{r_d + R_D}{1 + g_m r_d}\)
\(\boxed{Z_i = R_S \parallel Z_i^\prime = R_S \parallel \left(\frac{r_d + R_D}{1 + g_m r_d}\right)}\)
Output impedance \(Z_o\) (set \(V_i = 0\ V\))
\(\boxed{Z_o = r_d \parallel R_D}\)
Voltage gain \(A_v\)
\(\displaystyle V_o = -I_D R_D\)
\(\displaystyle V_{gs} = -V_i\)
\(\displaystyle I_D = -\frac{V_o}{R_D}\)
Applying Kirchoff’s current law at node \(b\).
\(\displaystyle I_D = g_m V_{gs} + I_{r_d}\)
\(\displaystyle -\frac{V_o}{R_D} = -g_m V_i + \frac{V_o - V_i}{r_d}\)
\(\displaystyle -\frac{V_o}{R_D} = -g_m V_i + \frac{V_o}{r_d} - \frac{V_i}{r_d}\)
\(\displaystyle -\frac{V_o}{R_D} - \frac{V_o}{r_d} = -g_m V_i - \frac{V_i}{r_d}\)
\(\displaystyle -V_o \left(\frac{1}{R_D} + \frac{1}{r_d}\right) = -V_i \left(g_m + \frac{1}{r_d}\right)\)
\(\boxed{A_v = \frac{V_o}{V_i} = \frac{\displaystyle g_m + \frac{1}{r_d}}{\displaystyle \frac{1}{R_D} + \frac{1}{r_d}} = \frac{1 + g_m r_d}{\displaystyle 1 + \frac{r_d}{R_D}}}\)
The fact that \(A_v\) is a positive number will result in an in-phase relationship between \(V_o\) and \(V_i\) for the common-gate configuration.