# Why gmbs is not zero when B and S are tied together?

I tried to run this simple circuit for DC operating point. The transistor is in saturation with Vth = 800mV.

What makes me confused is about gmbs. It is not zero but also very large compared to gm.

Why gmbs is not zero when B and S are tied together?

Thank you.

The value $g_{mbs}$ is a small-signal parameter. It is a linearization around a given operating point.

It answers the question how much the dependent variable would change if the independent variable is varied. Even though $V_{BS}=0$ we could change this voltage by a small amount $v_{bs}$ and consequently the drain current would change as well. The amount is given by $v_{bs} g_{mbs}$.

Likewise the $g_m$ of a transistor is not zero even if the gate is connected to a fixed voltage source.

Make a transient simulation of the following example and try to understand what's going on. I am sure the value of gmbs will suddenly make sense to you.

Update: The reason why gmbs is not equal to zero lies in the fact that backgate voltage acts through the backgate-effect.

The drain current in saturation is given by the following equation $$I_D = \frac{K'}{2} \frac{W}{L} \left(V_{GS} - V_T(V_{BS})\right)^2$$ where VT is a function of VBS!

Without going into the details of the derivation this finally results in $$g_{mbs} = g_m \frac{\gamma}{2\sqrt{2\Phi - V_{BS}}} = g_m \cdot \eta$$

So, gmbs is proportional to gm!

Even if $V_{BS}$ is zero a small change will alter the threshold voltage an therefore the drain current changes as well.

• Thank you. gm is not zero because at DC bias voltage VGS is not zero. However, in the case gmb, DC bias voltage VBS is also zero. They are not the same. Oct 14 '16 at 13:17
• No, it's the same. Think about it. Oct 14 '16 at 13:25
• See my update to the answer. Oct 14 '16 at 13:40
• Thank you. I will run the circuit tomorrow when I work. However, from the circuit, what should I do? See how gmbs changes with time? Oct 14 '16 at 13:45
• You should see how Id varies for a 1mV sinusoidal signal with zero DC bias. The variation should be related to gmbs. Oct 14 '16 at 13:47

The model for the NMOS which is used in the simulator is a lot more complex than what you will have seen in books etc. The model we humans use have to be more simple, models as used in the simulator (BSIM, MOS Model 9, MOS Model 11, etc) are too complex to work with.

Although you shorted Bulk and Source, you only shorted the outer pins of the NMOS. There will also be series resistances present between the pins and the "real" NMOS model.

I would not pay too much attention to all these model parameters, they sometimes have unexpected values because of the way the values which go into the model are created. This a largely automated process, devices are measured and then the model parameters are tweaked (by a program) such that the model will fit the measurements.

In over 20 years of designing circuits I've never concerned myself with the value of this gmbs so neither should you !

• Thank you. It makes sense. However, for low voltage design I need to pay attention to it. Oct 14 '16 at 13:18
• for low voltage design I need to pay attention to it Why ? In those 20 years all I did was low voltage design. Why is gmsb important ? Oct 14 '16 at 14:55
• "In over 20 years of designing circuits I've never concerned myself with the value of this gmbs so neither should you" : maybe you just aren't designing circuits where this is an issue. Just because YOU didn't need to understand this parameter does not need others cannot exploit it in their designs. Oct 14 '16 at 14:59
• @jbord39 Then show me an example of the usefullness of gmbs in a circuit. There can be up to a hundred or so parameter values coming out of a model for a MOS. Only a few of them are important to circuit designers. Many of the parameters just make the model "work". Switch to a different model and there might be no gmbs or it will have a completely different value. Oct 14 '16 at 15:18
• @FakeMoustache: I do not need to prove that this is a useful concept for the OP to understand; I did not ask the question. I am only pointing out that just because YOU did not find it useful does not mean others may not exploit its behaviour. Perhaps the OP will be the first to create a circuit which uses the gmbs characteristics to their advantage. Oct 14 '16 at 16:02