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.
Bypassed \(R_S\)
The resulting configuration is same as appearing in JFET fixed-bias configuration, the resulting equations for \(Z_i\), \(Z_o\), and \(A_v\) will be the same.
Unbypassed \(R_S\)
Input impedance \(Z_i\)
\(\boxed{Z_i = R_G}\)
Output impedance \(Z_o\) (set \(V_i = 0\,V\))
\(\displaystyle V_o = -I_D R_D\)
\(\displaystyle V_{gs} = -\left(I_D + I_o\right) R_S\)
\(\displaystyle V_S = -V_{gs}\)
\(\displaystyle I_D = -\frac{V_o}{R_D}\)
\(\displaystyle I^\prime = \frac{V_o - V_S}{r_d} = \frac{-I_D R_D + V_{gs}}{r_d}\)
\(\displaystyle I^\prime = \frac{-I_D R_D -\left(I_D + I_o\right) R_S}{r_d} = -\frac{I_D\left(R_D + R_S\right)}{r_d} - \frac{I_o R_S}{r_d}\)
Apply Kirchoff’s current law (KCL)
\(\displaystyle I_D + I_o = g_m V_{gs} + I^\prime\)
\(\displaystyle I_D + I_o = -g_m \left(I_D + I_o\right) R_S - \frac{I_D\left(R_D + R_S\right)}{r_d} - \frac{I_o R_S}{r_d}\)
\(\displaystyle I_D + I_o = - g_m R_S I_D - g_m R_S I_o - \frac{I_D\left(R_D + R_S\right)}{r_d} - \frac{I_o R_S}{r_d}\)
\(\displaystyle I_D + g_m R_S I_D + \frac{I_D\left(R_D + R_S\right)}{r_d} = - I_o - g_m R_S I_o - \frac{I_o R_S}{r_d}\)
\(\displaystyle I_D\left(1 + g_m R_S + \frac{R_D + R_S}{r_d}\right) = -I_o\left(1 + g_m R_S + \frac{R_S}{r_d}\right)\)
\(\displaystyle -\frac{V_o}{R_D}\left(1 + g_m R_S + \frac{R_D + R_S}{r_d}\right) = -I_o\left(1 + g_m R_S + \frac{R_S}{r_d}\right)\)
\(\displaystyle \boxed{Z_o = \frac{V_o}{I_o} = \frac{\displaystyle R_D\left(1 + g_m R_S + \frac{R_S}{r_d}\right)}{\displaystyle 1 + g_m R_S + \frac{R_D + R_S}{r_d}}}\)
For \(r_d \geq 10R_D\)
\(\boxed{Z_o \simeq R_D}\)
Voltage gain \(A_v\)
\(\displaystyle V_o = -I_D R_D\)
\(\displaystyle I_D = -\frac{V_o}{R_D}\)
\(\displaystyle V_{R_S} = I_D R_S\)
Apply Kirchoff’s voltage law (KVL) to the input circuit
\(\displaystyle -V_i + V_{gs} + V_{R_S} = 0\)
\(\displaystyle V_{gs} = V_i - I_D R_S\)
Find the current in \(r_d\)
\(\displaystyle I^\prime = \frac{V_o - V_{R_S}}{r_d} = \frac{-I_D R_D - I_D R_S}{r_d} = -\frac{I_D\left(R_D + R_S\right)}{r_d}\)
Apply Kirchoff’s current law (KCL) to the output circuit
\(\displaystyle I_D = g_m V_{gs} + I^\prime\)
\(\displaystyle I_D = g_m\left(V_i - I_D R_S\right) - \frac{I_D\left(R_D + R_S\right)}{r_d}\)
\(\displaystyle I_D = g_m V_i - g_m I_D R_S - \frac{I_D\left(R_D + R_S\right)}{r_d}\)
\(\displaystyle I_D + g_m I_D R_S + \frac{I_D\left(R_D + R_S\right)}{r_d} = g_m V_i\)
\(\displaystyle I_D\left(1 + g_m R_S + \frac{R_D + R_S}{r_d}\right) = g_m V_i\)
\(\displaystyle -\frac{V_o}{R_D}\left(1 + g_m R_S + \frac{R_D + R_S}{r_d}\right) = g_m V_i\)
\(\boxed{A_v = \frac{V_o}{V_i} = -\frac{g_m R_D}{\displaystyle 1 + g_m R_S + \frac{R_D + R_S}{r_d}}}\)
For \(r_d \geq 10\left(R_D + R_S\right)\)
\(\boxed{A_v = \frac{V_o}{V_i} = -\frac{g_m R_D}{\displaystyle 1 + g_m R_S}}\)
The negative sign in the resulting equation for \(A_v\) clearly reveals a phase shift of \(180^\circ\) between input and output voltages.