Diode-connected MOSFET

A diode-connected MOSFET has its gate shorted to its drain, turning the three-terminal transistor into a two-terminal nonlinear element that behaves much like a diode.

Why it is always saturated when conducting

Tie gate to drain and $V_{GD}=0$, so $V_{DS}=V_{GS}$. The saturation condition is $V_{DS}\ge V_{OV}=V_{GS}-V_t$. Substituting $V_{DS}=V_{GS}$:

$V_{GS}\ge V_{GS}-V_t⟺0\ge -V_t⟺V_t\ge 0$

which is always true (for a conducting n-MOSFET, $V_{GS}>V_t>0$). So a diode-connected MOSFET sits in saturation automatically whenever it conducts. You never have to check the region or worry about it dropping into triode. That’s the appeal: it removes a case from the analysis.

Gate shorted to drain. Always in saturation when it conducts; a two-terminal square-law element.

The two-terminal I–V

Because it is always saturated, its terminal current follows the MOSFET square-law with $V_{GS}=v$ (the single terminal voltage across the device):

$i=\frac{1}{2}{k}_n(v-V_t{)}^2\text{for }v>V_t$

and $i=0$ for $v<V_t$. Here $k_n$ is the device MOSFET transconductance parameter and $V_t$ the Threshold voltage. So the $i$–$v$ curve is zero current until $v$ exceeds $V_t$, then current rising quadratically. Qualitatively similar to a forward-biased Diode (turns on near $0.7\text{V}$, then conducts steeply), but square-law rather than exponential.

What it is used for

It’s the MOSFET counterpart of using a forward-biased diode as a Voltage reference: a soft, predictable voltage drop that depends on the current through it. Pass a known current and it develops a well-defined $V_{GS}$.