what happens when the gate of a mosfet is connected to the drain

Question

anonymous · Answer

This configuration of MOSFET is named as "DIODE CONNECTED MOSFET". In this configuration mosfet operates in saturation region always. For mosfet to operate in saturation region following condition should meet 
$$Vds>=Vgs-Vth$$
where Vds is Drain to source potential difference, Vgs is gate to source potential difference and Vth is threshold voltage of mosfet. This condition is true for both mosfets NMOS and PMOS. Just for PMOS Vgs becomes Vsg and Vds becomes Vsd. 
Since Vgs=Vds thus Vds will alwasys be more than Vgs-Vth , so this clearly show that mosfet in saturation 
Now we see that MOSFET is ON and it is operation is saturation region, apply the current equation of MOSFET in this region 
$$Id= W/2L*\mu*Cox*(Vgs -Vth)^{2}= W/2L*\mu*Cox*(Vds -Vth)^{2}$$
plot Id vs. Vgs  or Id vs. Vds. You will get a diode like characteristics.

anonymous · Answer

But there is a difference between diode connected mosfet and actual diode characteristics.  Basically the difference between an exponential (in case of diode) and a square-law function (in case of mosfet). basically this circuit ( with some modifications ) is used almost in all CMOS analog circuits. Like op-amp, bandgap, comparators etc.

anonymous · Answer

It becomes constant current source (if vds is fixed) as the MOSFET becomes saturated for condition Vgs > Vth and VDS>= Vgs - Vth

anonymous · Answer

I do agree with vkp.

anonymous · Answer

Can we make a active load with it?