# KVL and KCL in MOSFET small-signal model, how?

I want to find the gain in this MOSFET small-signal model.

KVL in loop 1:

$+V_{in}-V_{gs}-g_mV_{gs}R_s=0 \iff V_{in}=V_{gs}(1+g_mR_s)$

KVL in loop 2:

$+V_{out}-g_mV_{gs}R_s=0 \iff V_{out}=g_mV_{gs}R_s$

Questions:

In the answer $V_{out}$ should have a negative sign, $V_{out}=-g_mR_s \cdot V_{in}$. What have I missed?

The final answer is $V_{out} / V_{in}=-g_mR_d/(1+g_mR_s)$. (*)

How can the current through $R_d$ be $g_mV_{gs}$? Shouln't there instead be a KCL in $V_a$, so the current through $R_d$ is $V_{out}/R_d$?

simulate this circuit – Schematic created using CircuitLab

Update with KCL in $V_a$:

$I_{out}-i_d-I_r=0$

where $i_d=g_mV_{gs}$ and $I_r=V_{out}/R_d$, so

$I_{out}=g_mV_{gs}+V_{out}/R_d$.

I'm stuck here, how can I find equation (*) above?

• Your KVL loop2 is incorrect, Vout is not gmVgsRs, you made Vs = Vout. You should not use a voltage loop as there's a current source (gmVgs) in the loop. Use the current law instead and make sure Rd is in the expression because it is crucial. The overall transfer (Vout/Vin) will have a negative sign and both Rs and Rd must be in the expression. Commented Apr 13, 2017 at 14:30
• Where is Iout going? (Is there a loop there)? Commented Apr 13, 2017 at 15:11

## 2 Answers

KVL in loop 2 is missing the voltage across the current source. Because that's an unknown, instead combine your KCL expression with your KVL loop 1 expression:

$$V_{in} = V_{gs}(1+g_mR_s) \\ I_{out} = g_mV_{gs}+V_{out}/R_d \implies V_{out} = (I_{out} - g_mV_{gs})R_d \\ \therefore \frac{V_{out}}{V_{in}}=\frac{I_{out}R_d - g_mV_{gs}R_d}{V_{gs}(1+g_mR_s)}=\frac{I_{out}R_d/V_{gs} - g_mR_d}{1+g_mR_s}$$

Now if and only if $I_{out} = 0$, the expression reduces to:

$$\frac{V_{out}}{V_{in}}=\frac{-g_mR_d}{1+g_mR_s}$$

which is the result you're after.

Donsert you can find the solution in the following figure