We are now ready to begin small signal modeling. This is the modeling of the behavior of the device under small variations of its terminal voltages. We will explain how those small variations of the terminal voltages cause corresponding small variations in the drain current. We will begin with the so called Small-Signal Conductance Parameters. These describe small signal behavior at very low frequencies. In this video we will concentrate on the drain-to-source current path. We begin by considering a transistor biased with three DC. Voltage sources. VGS zero, VSB zero, and VDS zero. The device is assumed to be in steady state, so it will have a drain current IDS zero. All these quantities are DC quantities. We will now begin varying each of these terminal voltages by a small amount delta V, and discussing what happens to the drain source current, which of course will change. By some amount delta iDS. we assume that these changes in the voltages are effected and then we wait until we reach steady state and then we study the change in the current delta iDS. First, I will vary VGS zero by a small amount delta VGS. 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. This is the change in a drain source current and we also need the change the gate current which you assume to be zero, because we're talking about a DC situation here. And, no change in the body current also. We assume that we have gate and body currents which are zero, and therefore their changes are also zero under DC. Excitation. Now, we're going to develop a, an equivalence circuit, so-called small signal equivalence circuit that describes these three relations. First of all, since the gate current is zero, I will present the corresponding terminal with an open circuit, so that no current can flow. I will do the same for the body current, but for the drain current I need the sum of three individual currents. So, this circuit will have three branches. So, here is the equivalent circuit. The small signal equivalent circuit relates only changes in voltages delta VGS, delta VBS, and delta VDS, to change in the current delta IDS. It does not represent total currents as functions of total voltage, but only small changes of currents as functions of small changes of voltages. So let's take the, this term. GSD delta VDS. Delta VDS is the voltage from drain to source in this equivalent circuit times a conductance GSV, will give you the current through the conductance, which is GSD delta VDS, in other words it gives you this term. We could do it so simply and present this by a resistor because the current in this was proportional to the voltage across this resistor, delta VDS happens to be across this resistor. That's why this simple representation was possible. But if you now take this term delta, GM VGS, this one relates the drain source current to a voltage somewhere else, okay. The drain source current is here, but the voltage delta VGS that controls it, it's somewhere else. It is not across the corresponding device, so I cannot use a resistor. I have to use a controlled current source. So this current source has a value, gm delta VGS, and it is controlled by this c-, voltage delta VGS. I do the same for this term GMB delta VBS, and other controlled current source of this value, controlled by this voltage delta VBS. So now if you write the total current delta IDS from [UNKNOWN] cos current low will be what? GM delta VGS plus the current in GSD which is GSD delta VDS plus the current here which is GMB delta VBS. As you can see, this current is given by exactly this equation. This is an open circuit so it gives you delta IG is equal to zero. This is an open circuit it gives you delta IB equal to zero. Therefore. This small signal equivalent circuit represents this set of equations. This circuit does not exist, it is a fictitious circuit that is equivalent to this mathematical representation of the relations between small signal changes in voltages to small signal changes in currents. This circuit is valid for very low frequencies, because if the frequencies start varying faster, then you're going to be changing the charges rather fast, and this will imply significant charging components that can not be represented by conductances. Then, you need evoke capacitances, and we will do that separately. In a future video. In this video we introduce the concept of small signal parameters. We, specifically we concentrated on small signal conductance parameters and we defined 3 conductance parameters, the gate transconductance, the body transconductance, and the source drain conductance. In the next video we will introduce additional conductance parameters that describe in small-signal operation. The gate and body leakage currents.
