Question about MOSFET I-V equation

Most textbooks describe MOSFET I-V characteristic equation with channel length modulation effect as
Id(sat)=K(W/L)(Vgs-Vth)²(1+λVds).

But because the effect of pinch-off(or channel length modulation) begins when Vds>Vsat(Vgs-Vth), I think that it should be
Id(sat)=K(W/L)(Vgs-Vth)²[1+λ(Vds-Vsat)].

And when it comes to continuity between Id(triode) and Id(sat), I think later one is better.

Am I wrong? I'm confused.

Thank you

p.s. Sorry, I'm not good at English.

mosfet equations are usually specified in two equations:
i)in linear region (vds<vsat)(or vgs ...)
ii)in constant current region (vds>vsat)

your equation is in ii). where lamda effect is pronounced.

Hi Rainbee,

I see your point. But, using the equation that you proposed: Id(sat)=K(W/L)(Vgs-Vth)²[1+λ(Vds-Vsat)]
The effect of channel length modulation will be proportional to VDS-VSAT, what it is wrong. This effect is proportional to VDS.

For VDS<VGS-VT ==> Device in linear operation: I = 1/2*un*Cox*W/L[2*(VGS-VT)VDS - VDS^2].
For VDS = or > VGS-VT ==> Device in saturation:Id(sat)=K(W/L)(Vgs-Vth)².
Thus, at this point VDS=VGS-VT, the pinch-off happens, and the channel length modulation happens.... so, if you consider this effect:

Id(sat)=K(W/L)(Vgs-Vth)²(1+λVds).


Is it clear?

Regards

temporary reply for image link

---------- Post added at 12:26 ---------- Previous post was at 12:26 ----------

Thanks for reply.

But when I draw graphs of two equations(triode region and saturation region), they don't have a intersection point (that is expected to be made at Vds=Vgs-Vt).


This was my confusing point...

palmeiras said:

Hi Rainbee,

The effect of channel length modulation will be proportional to VDS-VSAT, what it is wrong. This effect is proportional to VDS.

Click to expand...

Hi,
Well they will never have a meeting point because the equations do not follow a linear trend and hence there would be a mismatch.Only spice models can give the best accurate combined model.

Rainbee said:

temporary reply for image link

---------- Post added at 12:26 ---------- Previous post was at 12:26 ----------

Thanks for reply.

But when I draw graphs of two equations(triode region and saturation region), they don't have a intersection point (that is expected to be made at Vds=Vgs-Vt).


This was my confusing point...

Click to expand...

Hi Rainbee,

This is normal. Remember that these equations only can be used for hand-analysis calculations. The continuity of equations that model the transistor behavior is a very discussed topic.
The real equations used in the simulators are much more complicated. For example, BSIM3 or BSIM4. If you have curiosity, take a look at the models. There are many parameters (and many of them without physical meaning – only fitting purpose). Other models, such as, EKV and ACM, are simpler and all parameters having physical meaning.
Regards,

Obs: one last remark, note that the derivative of ID (@ID/@VDS) = 0, representing the peak of your parabola, occurs at VDS = VGS - VTH: The beginning of saturation operation. Interesting, righ?!