Question2. $\mathrm{R}_{\mathrm{D}}=2 \mathrm{k} \Omega, \mathrm{R}_{\mathrm{G}}=2 \mathrm{M} \Omega, \mathrm{R}_{\mathrm{S}}=1 \mathrm{k} \Omega, \mathrm{V}_{\mathrm{DD}}=15 \mathrm{~V}, \mathrm{I}_{\mathrm{DSS}}=8 \mathrm{~mA}, \mathrm{~V}_{\mathrm{P}}=-6 \mathrm{~V}$. (1) Calculate DC bias $\mathrm{V}_{\mathrm{GS}}, \mathrm{I}_{\mathrm{D}}$, and $V_{D S}$. (2) Draw the equivalent circuit for $A C$ analysis, and determine $g_{m}, Z_{i}, Z_{o}$ and $A_{v}$.

Studdy Solution
Now let's calculate the small-signal parameters:
1. Transconductance (gm): $$g_m = \frac{2I_{DSS}}{|V_P|}\left(1 - \frac{V_{GS}}{V_P}\right) = \frac{2 \cdot 8 \text{ mA}}{6 \text{ V}}\left(1 - \frac{-3.43 \text{ V}}{-6 \text{ V}}\right) \approx 2.28 \text{ mS}$$
2. Input impedance (Zi): Zi is simply RG: $$Z_i = R_G = 2 \text{ M}\Omega$$
3. Output impedance (Zo): For a JFET, the output impedance is approximately equal to rd, which can be calculated as:
$$r_d = \frac{1}{\lambda I_D}$$
where λ (channel-length modulation parameter) is typically around 0.01 V^-1 for JFETs.
$$r_d = \frac{1}{0.01 \text{ V}^{-1} \cdot 3.43 \text{ mA}} \approx 29.15 \text{ k}\Omega$$
The output impedance Zo is the parallel combination of rd and RD:
$$Z_o = r_d || R_D = \frac{r_d \cdot R_D}{r_d + R_D} = \frac{29.15 \text{ k}\Omega \cdot 2 \text{ k}\Omega}{29.15 \text{ k}\Omega + 2 \text{ k}\Omega} \approx 1.87 \text{ k}\Omega$$
4. Voltage gain (Av): $$A_v = -g_m Z_o = -(2.28 \text{ mS})(1.87 \text{ k}\Omega) \approx -4.26$$
The final results are:
DC bias values: VGS = -3.43 V ID = 3.43 mA VDS = 4.71 V
AC small-signal parameters: gm = 2.28 mS Zi = 2 MΩ Zo = 1.87 kΩ Av = -4.26