MOSFET triode region

The triode region (also called the linear or ohmic region) is the MOSFET operating mode where a channel is formed end-to-end and the device behaves like a gate-controlled resistor. The conditions are

$V_{GS}>V_t\text{and}{V}_{DS}<V_{OV}$

i.e. the gate is above Threshold voltage so a channel exists, but the drain-to-source voltage hasn’t reached the Overdrive voltage $V_{OV}=V_{GS}-V_t$, so the channel is still continuous from source to drain (not yet pinched off).

The drain-current equation

In triode the drain current is

$i_D=k_n^{\prime }\frac{W}{L}\left[V_{OV}{V}_{DS}-\frac{V_{DS}^2}{2}\right]$

where $k_n^{\prime }={\mu }_n{C}_{ox}$ is the process MOSFET transconductance parameter, $W/L$ the channel width-to-length ratio, $V_{OV}=V_{GS}-V_t$ the overdrive, and $V_{DS}$ the drain-source voltage. The combination $k_n=k_n^{\prime }(W/L)$ is the device transconductance parameter, so this is also $i_D=k_n[V_{OV}{V}_{DS}-V_{DS}^2/2]$.

Read the equation. For a fixed $V_{GS}$ (fixed $V_{OV}$), $i_D$ rises with $V_{DS}$, but the $-V_{DS}^2/2$ term bends the curve over: the current grows quickly at first, then flattens as $V_{DS}$ approaches $V_{OV}$. At $V_{DS}=V_{OV}$ the slope $\partial i_D/\partial V_{DS}$ goes to zero and the curve hands off smoothly to the flat saturation characteristic. That continuity is why setting $V_{DS}=V_{OV}$ here yields the MOSFET square-law.

Channel fully formed; for small $V_{DS}$ the device is a gate-controlled linear resistor.

The voltage-controlled-resistor limit

For small $V_{DS}$ the quadratic $V_{DS}^2/2$ term is negligible next to $V_{OV}{V}_{DS}$, so

$i_D\approx k_n^{\prime }\frac{W}{L}{V}_{OV}{V}_{DS}$

This is Ohm’s law: $i_D$ is proportional to $V_{DS}$, with a channel resistance

$r_{DS}=\frac{V_{DS}}{i_D}\approx \frac{1}{k_n^{\prime }(W/L)V_{OV}}=\frac{1}{{\mu }_n{C}_{ox}(W/L)V_{OV}}$

That resistance is tunable by the gate. Raising $V_{GS}$ raises $V_{OV}$ and lowers $r_{DS}$. The MOSFET in triode is a voltage-controlled resistor.

What it is used for

The triode region is the on-state of digital logic (a transistor pulling an output hard toward a rail is a small on-resistance to that rail), and it is what makes MOSFET analog switches and transmission gates work: close the switch by driving the gate hard so $r_{DS}$ is small, open it by dropping $V_{GS}$ below $V_t$ into cut-off. For amplification you don’t want triode, you want saturation, where $i_D$ is set by $V_{GS}$ and is nearly independent of $V_{DS}$.