MOSFET transfer characteristic

The MOSFET transfer characteristic is the plot of drain current $i_D$ versus gate-source voltage $V_{GS}$ at a fixed $V_{DS}$ (with the device in saturation). It is the curve that says how the input voltage controls the output current — exactly the relationship an amplifier exploits.

Shape of the curve

Two pieces:

• Below threshold ($V_{GS}<V_t$): no channel, so $i_D=0$. The curve sits flat on the axis. This is the MOSFET cut-off region.
• Above threshold ($V_{GS}>V_t$): the square-law takes over,

$i_D=\frac{1}{2}{k}_n(V_{GS}-V_t{)}^2$

so the current rises as the square of the overdrive $V_{OV}=V_{GS}-V_t$ (see Overdrive voltage). The curve leaves the axis at $V_{GS}=V_t$ with zero slope and then steepens — a parabola shifted right by $V_t$. Here $k_n$ is the device MOSFET transconductance parameter and $V_t$ the Threshold voltage.

$i_D$ vs $V_{GS}$: zero below $V_t$, then rises as $(V_{GS}-V_t{)}^2$.

Why this curve is the whole game

The transfer characteristic is the big-signal control law of the device, drawn as a graph. Two things come from it:

1. The small-signal model. Pick a bias point on the curve and zoom in. Over a small enough input swing the parabola looks like a straight line; its slope at that point is the MOSFET transconductance $g_m=\partial i_D/\partial V_{GS}{|}_Q=k_n{V}_{OV}$. Linearising the transfer characteristic is exactly how the MOSFET small-signal model is derived (see Linearisation around an operating point). Bias where the curve is steep and you get more gain — but the same steepness is what makes the device nonlinear over large swings.
2. The full I–V picture. The transfer characteristic is only half the story. The other half is the family of $i_D$-vs-$v_{DS}$ output curves at different $V_{GS}$. Together they describe the entire large-signal behaviour: the output family shows the triode/saturation split bounded by $v_{DS}=v_{OV}$, and the transfer characteristic shows how the saturated current depends on the input.

Left: $i_D$–$v_{DS}$ family bounded by $v_{DS}=v_{OV}$; right: square-law transfer characteristic.