transconductance with channel length modulation

[namel]

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

 

[analog-designer]

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!)

 

[namel]

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

 

[erikl]

gm=1/2*k*W/L*(vgs-vth)*(1+lambda*vds) and sqrt((2Id*K*W/L)/(1+lambda*vds)).

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

 

[namel]

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