fet ac all formulaes

Here is the complete, detailed list of formulas for the AC small-signal analysis of FET amplifiers (JFETs, Depletion-MOSFETs, and Enhancement-MOSFETs). It includes the expressions for input impedance ($Z_i$), output impedance ($Z_o$), and voltage gain ($A_v$), both with and without the effect of the internal output resistance ($r_d$).

JFET & Depletion-Type MOSFET Configurations

1. Fixed-Bias Configuration

• Input Impedance ($Z_i$):
  • With and Without $r_d$: $Z_i=R_G$
• Output Impedance ($Z_o$):
  • With $r_d$: $Z_o=R_D||r_d$
  • Without $r_d$: $Z_o\cong R_D$
• Voltage Gain ($A_v$):
  • With $r_d$: $A_v=-g_m(R_D||r_d)$
  • Without $r_d$: $A_v\cong -g_m{R}_D$

2. Self-Bias Configuration (with Bypassed $R_S$) (Note: Because $R_S$ is bypassed by a capacitor, the AC equivalent circuit is identical to the fixed-bias configuration.)

• Input Impedance ($Z_i$):
  • With and Without $r_d$: $Z_i=R_G$
• Output Impedance ($Z_o$):
  • With $r_d$: $Z_o=R_D||r_d$
  • Without $r_d$: $Z_o\cong R_D$
• Voltage Gain ($A_v$):
  • With $r_d$: $A_v=-g_m(R_D||r_d)$
  • Without $r_d$: $A_v\cong -g_m{R}_D$

3. Self-Bias Configuration (with Unbypassed $R_S$)

• Input Impedance ($Z_i$):
  • With and Without $r_d$: $Z_i=R_G$
• Output Impedance ($Z_o$):
  • With $r_d$: $Z_o=\left[\frac{1+g_m{R}_S+\frac{R_S}{r_d}}{1+g_m{R}_S+\frac{R_S}{r_d}+\frac{R_D}{r_d}}\right]R_D$
  • Without $r_d$: $Z_o\cong R_D$
• Voltage Gain ($A_v$):
  • With $r_d$: $A_v=\frac{-g_m{R}_D}{1+g_m{R}_S+\frac{R_D+R_S}{r_d}}$
  • Without $r_d$: $A_v\cong \frac{-g_m{R}_D}{1+g_m{R}_S}$

4. Voltage-Divider Bias Configuration

• Input Impedance ($Z_i$):
  • With and Without $r_d$: $Z_i=R_1||R_2$
• Output Impedance ($Z_o$):
  • With $r_d$: $Z_o=R_D||r_d$
  • Without $r_d$: $Z_o\cong R_D$
• Voltage Gain ($A_v$):
  • With $r_d$: $A_v=-g_m(R_D||r_d)$
  • Without $r_d$: $A_v\cong -g_m{R}_D$

5. Common-Gate Configuration

• Input Impedance ($Z_i$):
  • With $r_d$: $Z_i=R_S||\left[\frac{r_d+R_D}{1+g_m{r}_d}\right]$
  • Without $r_d$: $Z_i\cong R_S||\frac{1}{g_m}$
• Output Impedance ($Z_o$):
  • With $r_d$: $Z_o=R_D||r_d$
  • Without $r_d$: $Z_o\cong R_D$
• Voltage Gain ($A_v$):
  • With $r_d$: $A_v=\frac{g_m{R}_D+\frac{R_D}{r_d}}{1+\frac{R_D}{r_d}}$
  • Without $r_d$: $A_v\cong g_m{R}_D$

6. Source-Follower (Common-Drain) Configuration

• Input Impedance ($Z_i$):
  • With and Without $r_d$: $Z_i=R_G$
• Output Impedance ($Z_o$):
  • With $r_d$: $Z_o=r_d||R_S||\frac{1}{g_m}$
  • Without $r_d$: $Z_o\cong R_S||\frac{1}{g_m}$
• Voltage Gain ($A_v$):
  • With $r_d$: $A_v=\frac{g_m(r_d||R_S)}{1+g_m(r_d||R_S)}$
  • Without $r_d$: $A_v\cong \frac{g_m{R}_S}{1+g_m{R}_S}$

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Enhancement-Type MOSFET Configurations

(Note: For E-MOSFETs, $g_m=2k(V_{GSQ}-V_{GS(Th)})$.)

7. Drain-Feedback Bias Configuration

• Input Impedance ($Z_i$):
  • With $r_d$: $Z_i=\frac{R_F+(r_d||R_D)}{1+g_m(r_d||R_D)}$
  • Without $r_d$: $Z_i\cong \frac{R_F}{1+g_m{R}_D}$
• Output Impedance ($Z_o$):
  • With $r_d$: $Z_o=R_F||r_d||R_D$
  • Without $r_d$: $Z_o\cong R_D$ (assuming $R_F\gg R_D$)
• Voltage Gain ($A_v$):
  • With $r_d$: $A_v=-g_m(R_F||r_d||R_D)$
  • Without $r_d$: $A_v\cong -g_m{R}_D$ (assuming $R_F\gg R_D$)

8. Voltage-Divider Bias Configuration

• Input Impedance ($Z_i$):
  • With and Without $r_d$: $Z_i=R_1||R_2$
• Output Impedance ($Z_o$):
  • With $r_d$: $Z_o=r_d||R_D$
  • Without $r_d$: $Z_o\cong R_D$
• Voltage Gain ($A_v$):
  • With $r_d$: $A_v=-g_m(r_d||R_D)$
  • Without $r_d$: $A_v\cong -g_m{R}_D$