transconductance with channel length modulation

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namel

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hi, while reading design of analog cmos integrated circuits by razavi chapter 2, he has two of gm expressions with
gm=1/2*k*W/L*(vgs-vth)*(1+lambda*vds) and sqrt((2Id*K*W/L)/(1+lambda*vds)). To my understanding, they both should have been same after
expressing vgs-vth in terms of Id but they are not. Anybody have any idea?
Thanks in advance
 

saturation formula is I=K/2*W/L*(Vgs-Vth)^2 and transconductance is gm=partial(I)/partial(Vgs)=K*W/L*(Vgs-Vth)
rearrange saturation formula gives Vgs-Vth=sqrt(2*I*L/(K*W)), inserted gives gm=sqrt(2*I*K*W/L) and finally gm=2*I/(Vg-Vth) (a very handy formula!)
 

yes, I know that, but what I have asked is that what should be the formulas after considering the channel length modulation effect?
according to razavi he has all three of them but they do not match
 

namel said:
gm=1/2*k*W/L*(vgs-vth)*(1+lambda*vds) and sqrt((2Id*K*W/L)/(1+lambda*vds)).
Click to expand...

Right. I think it should mean gm=sqrt((2Id*K*W/L)*(1+lambda*vds)) .
 

thank you. I also think that but wanted to make sure.
 

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