Why is the VGS of a diode-connected MOSFET relatively supply-independent?

[aravind2705]

For example, in the Beta-Multiplier circuit I have attached here, when I sweep the VDD, the VSD of M4 increases at a higher rate than does the VGS of M2.
Also, the gate of M3 follows VDD and keeps VSG constant. Why does this happen, Why is the gate-source voltage of a diode-connected transistor magically supply-independent (in the large-signal domain) ??

Image: https://obrazki.elektroda.pl/84_1325303679.png

 

[dave9000]

You can think a diode connection (gate connected to drain) as a local feedback loop.
Imagine Vgs as the input and Vds (= Id * Rload) as the output: raising Vgs causes Vds to decrease.
When you close the loop (Vgs=Vds) the net effect is to "lock" Vds to Vth (at first order).
This behaviour is probably more evident for BJTs.