Let's complicate the problem...
For large signal, mosfet works in saturatation region:
- Ids1 = 0.5*Kn*W1/L1*(Vgs-Vth)^2
- Ids2 = 0.5*Kn*W2/L2*(Vgs-Vth)^2
Hence,
- Ids1/Ids2 = (W1/L1) / (W2/L2) = Ratio1/Ratio2
Now turn to small signal analysis, gm of a mosfet is defined by "gm=ids/vgs = delta_Ids/delta_Vgs"
- gm1 = 2*Ids1/(Vgs-Vth)
- gm2 = 2*Ids2/(Vgs-Vth)
The small current "iin" injected into "X" is fisrt converted to "vin" by gm1, then converted to "iout" by gm2, hence
- iout/in = gm2/gm1 = Ids2/Ids1 = Ratio2/Ratio1
All discussion above assume ”gds“ is much smaller than "gm“ , the interesting frequency of the signal "iin" is much smaller than "gm/(2*pi*cp)"
# cp is mainly contributed by the gate capacitance of M1 and M2
# gds is defined by "gds = ids/vds"
# the function of M3 is used to decrease the effective gds of M1 observed at "P", then the current source behaves more like a ideal one