Voltage-transfer characteristic

The voltage-transfer characteristic (VTC) of a resistor-loaded MOSFET stage is the plot of output voltage $V_{DS}$ against input voltage $V_{GS}$. Connect the drain to $V_{DD}$ through a resistor $R_D$, sweep $V_{GS}$, and watch $V_{DS}$. The curve has three distinct regions, one for each MOSFET region of operation, and reading it shows at a glance where the stage amplifies and where it clips.

Output $V_{DS}$ vs input $V_{GS}$: cut-off, saturation (amplifying), triode.

The three regions

• Cut-off ($V_{GS}<V_t$): the MOSFET is off (see MOSFET cut-off region), no current, no drop across $R_D$, so the output sits flat at the top:

$V_{DS}=V_{DD}$

• Saturation ($V_{GS}>V_t$, $V_{DS}>V_{OV}$): the device is a $V_{GS}$-controlled current source (MOSFET saturation region). With $I_D=\frac{1}{2}{k}_n(V_{GS}-V_t{)}^2$ flowing through $R_D$:

$V_{DS}=V_{DD}-I_D{R}_D=V_{DD}-\frac{1}{2}{k}_n{R}_D(V_{GS}-V_t{)}^2$

The output drops quadratically with the input. This is the steep middle of the curve — the active region where amplification happens.

• Triode ($V_{DS}<V_{OV}$): the device is on hard (MOSFET triode region), $V_{DS}$ is small, and further increases in $V_{GS}$ produce only tiny additional drops. The curve flattens out near $0\text{V}$ — the output is “saturated low.”

Here $V_t$ is the Threshold voltage, $V_{OV}=V_{GS}-V_t$ the Overdrive voltage, and $k_n$ the device MOSFET transconductance parameter.

The slope is the gain

For a small input wiggle around a bias point $Q$ in the saturation region, the output wiggle is the input times the slope of the VTC at $Q$. Differentiate the saturation expression:

$\frac{d{V}_{DS}}{d{V}_{GS}}=-k_n{R}_D(V_{GS}-V_t)=-(k_n{V}_{OV})R_D=-g_m{R}_D$

using $g_m=k_n{V}_{OV}$ (the MOSFET transconductance). So the small-signal gain $A_v=-g_m{R}_D$ is literally the slope of the VTC at the Operating point — exactly the Common-source amplifier result, now seen as the local steepness of this curve.

This is also why bias-point placement matters: $Q$ must sit on the steep saturation portion with enough room on both sides for the signal to swing without sliding into the flat cut-off plateau (clipped at $V_{DD}$) or the flat triode plateau (clipped near 0). For maximum symmetric swing, $Q$ goes roughly at the midpoint of the saturation segment.