As shown here, this will cause the current to change by some amount delta

IDS1. One stands for case number one we were

considering and we will consider two more cases.

So, the cause is delta VGS and the effect is delta IDS1.

We divide the effect by the cause. And we call the result a

transconductance. G sub M is the symbol for the

transconductance. More precisely it is the small signal

gate transconductance. The reason we use an approximately equals

sign is that later we will refine these definitions by allowing delta v to go to

zero. so this will become a derivative.

If instead of varying VGS we vary VBS like this, then the corresponding change

in the drain source current will be delta IED S2.

Notice that I vary the voltage from body to source by the amount delta VBS.

You can then divide again the effect by the cause, and get another type of trans

conductance called the body trans conductance.

It tells you how much the drain source current varies when you vary the body

voltage. And again, we have an approximate sign

because later on we will refine this one. Finally I'm going to leave VGS and VSP at

their DC value. And vary instead VDS by a small amount

delta VDS. The drain source carbon will change by an

amount delta IDS 3. Dividing we get something we call the

small signal source drain conductors. So now we have three conductance

parameters. They all have the mentions of

conductance, that's current over voltage. One is the gate transconductance, one is

the body transconductance and one is the source drain conductance.

All of these parameters are small signal parameters but in order not to make the

names too long, I will just say transconductance, gate transconductance,

body transconductance. Source drain conductance.

And it will be understood that we're only talking about very small changes of

voltages and currents and their ratios, which give rise to small signal

conductance parameters. Take any model of the ones we have

described. We can write the drain source current.

As a function of the source, excuse me, of the gate source, body source and drain

source voltages. If the model is an old region model that

involves the surface potential. We can express the surface potential as

an approximate function of the external terminal voltages and again we can assume

that, in principle, we can express the current in this form.

Now, the definitions we gave before, refined, become like this.

The trans conductance is the partial derivative of IDS with respect to VGS.

Assuming that VBS and VDS are held constant.

This corresponds to the experiments I showed you before, which actually were

close to how measurements are done. And you just let the delta v's and delta

i's go to zero. So, then by definition, you get the

partial derivatives. This is the gate trans conductors, the

corresponding body trans conductors is the partial derivative of YDS with

respect to VBS, assuming VGS and VDS are held constant.

And finally, this source drain conductance is the partial derivative of

IDS with respect to VDS, assuming VGS and VBS are held constant.

As you can see, each of these definitions involve changing only one quantity,

assuming the other two are held constant. But what happens if all three, of these

terminal voltages are changing? First of all, if delta VGS is changing,

then I can take the rate of change of the current, with respect to VGS, multiply by

delta VGS, and get the corresponding change in the drain source current.

But if VBS is changing also, then to that I have to add the contribution because of

that change. It is a partial derivative of IDS with

respect to VBS times delta VBS. This gives me the contribution of the

change in VBS to the change in the current.

And then if VDS is changing as well, then I take the rate of change of I with

respect to VDS, multiply by delta VDS, this is the contribution of the change of

the drain source voltage. To the current.

The sum of this, assuming these delta Vs are very small, the sum of these three

gives you the total change in the current.

Again, the delta Vs and delta i's are supposed to be small signals, very small

changes, that is why you can write a relation for the current that is a linear

combination of the changes in the corresponding voltages.

Now if you replace these partial derivatives by the corresponding

definitions here, you get this relation. Delta IDS is GM delta VGS plus GMB delta

VBS plus GSD, delta VDS. I repeat this here.