We will now derive a very general model called a y-Parameter Model that places no restrictions on the frequency of operation. I have to warn you that the development is rather long and we'll have to skip a lot of steps. You'll find the detailed steps in the book. I will show you highlights of that development and their results in this video. We will now consider the transistor in a very general situation. Not necessarily in quasi static operation. Where the four terminal voltages with respect to ground, contain small signal voltages that can be varying very fast. Possibly fast, or possibly slow, but we will not impose any restriction on their rate of variation. We will consider only the small signal equivalent circuit for this situation. This is the small signal equivalent of the transistor, which we have yet to derive, and these are the small-signal voltages shown at the terminal voltages. The DC bias quantities are replaced by short-circuits, because this circuit only represents small signal operation. The corresponding currents are also small-signal currents. Now, we will assume that the voltages vary sinusoidally, for example, Vg, the gate small signal voltage, is supposed to have magnitude M and a phase phi, and it is varying at radian frequency omega, so it's M cos omega t plus phi. Now, in circuit analysis, you know that we very often handle this situation by defining complex phasors that retain the magnitude and phase correspondingly M is the amplitude of the time function, phi is the phase of the time function and omega is the same for all quantities in the circuit and it is understood. So if instead of voltages and currents you use voltage phasors and current phasors, you end up with the corresponding circuit that looks like this. So for example, Vg is this complex number, Ig is another complex number that represents the magnitude and phase of the corresponding small signal gate current, and so on for all of the other quantities. So we will be using a phasor representation. But for brevity, instead of saying the voltage, the gate voltage phasor and the gate current phasor, we will be talking about the gate voltage and the gate current, and the word phasor will be understood. First we will define certain so-called y parameters. In the circuit we vary the drain voltage V sub d. The corresponding phaser is capital V sub d, and we observe the current phaser I sub d. The ratio of the two is defined as Ydd, assuming that the other voltages, the other small signal voltages are zero. This is why the terminals gate, source, and body are shown grounded. Similarly, if we still look at the drain current, but now we apply a gate voltage we can define Ydg as the ratio of Id to Vg, with the other voltages being zero. We continue in this fashion, still looking at the drain current, but now we're varying the body voltage. We define as Ydb the ratio of Id to Vb with the other voltages being zero. And finally, we do the same by varying Vs, and looking at Id, the ratio of the two is Yds, with the other voltages being zero. So now, if we have a general situation where there are four voltages, all known zero, and you want to find the combined effect on the drain current. Because we're talking about very small signals, we can assume linearity around the operating point and use super position. So we can say that the total drain current would be the drain current that would have been predicted by only looking at the drain voltage variation, with the other voltage being zero. Plus the corresponding drain current due to a variation in Vg with the other voltage being zero and so on. So if for each of them you use the definitions that we showed in the previous four slides you end up with this equation. So this gives you the total drain current. More precisely, drain current phaser, as a function of the corresponding voltage phasers at each of the terminals with respect to ground. Now, if you have the other currents, so if you want to look at also Ig, Is and Ib, you get similar equations. You define in general Ykl is Ik over Vl. For the drain current, we have this equation, which we have already derived, for the gate current, you have this equation, for the body current, this one and for the first current, this one. Now, notice that, to be consistent with circuit theory, all of the y-parameters are defined as I over V with no minus sign here. Now, this is different from what we did for some of the capacitances where the minus sign was introduced, and we had to explain why. So, because now we do not have the minus sign, keep in mind that this new formulation may end up in some algebraic sign difference, which would be apparent later on. And we should not be surprised when we see it. Now, if you do the same development as we did for the capacitances, we can find relations between the y-parameters. And this is one relation, this is the second and the third and the fourth. The development of these equations follows step by step what we did for the capacitances, so it would not be repeated here. Now we can transform the current equations above, which use terminal voltages with respect to ground to a set of equations using terminal to terminal voltages. The details are shown in the book. Now we keep three independent equations. The Drain, Gate, and Body equations. The source current equation can be found from these three, because the sum of all four currents entering the device is zero. So these will be the equations we'll be working with. And now we take those equations, and we transform them, by using the y-parameter relations I showed you a couple of slides ago. And also using transformations of voltages provided by Kirchhoff's Voltage Law, which we have discussed in the previous video. And we finally end up with this set of equations. The development is entirely analogous to the equations we derived for the complete quasistatic model. And it will not be repeated here, but you'll find more details in the book. So these equations now have certain parameters Ym, which is defined as Ydg minus Ygd. In other words, it represents the non-reciprocal behavior between drain and gate. Ymb is Ydb minus Ybd, and Ymx is defined as Ybg minus Ygb. We have a total of nine independent y-parameters, and if you look at these equations, you can see that they are represented by this equivalent circuit. It may take some time for you to make the correspondence, but if you take any one of these currents, let's say you take Id and you write the equation for it. So this current is the sum of this current, plus this, plus this, plus this, plus this, you'll find this first equation here. And similarly, you can do it for the others. Notice that there are these minor signs that appear in the admittance of the corresponding elements here, and here, and so on. These are the result of not having using the minor sign in the definitions. Please do not let this bother you, this is just a matter of definition. The fact that there is a minus sign in front of Ygs doesn't mean that this is a negative quantity. In fact, this is a complex quantity, in general. You don't say if it's positive or negative. So this will become more clear in a minute. Now, notice that we derived this equivalent circuit without making any assumption as to how fast the terminal voltages were varied. In other words, here we do not make the assumption of quasistatic operation. So this model is completely general. It is valid for all speeds, for any four terminal transistor, and as a matter of fact, for any four timing terminal device, we did not even say that this was MYS transistor in all this development. We just use the circuit theoretic approach to this. So this is a very general model. And as it turns out, it contains all other models we have developed as special cases. For example, let's compare our complete quasistatic model, which we derived in the previous video, to the model we just derived. Between drain and source, there is a two terminal admittance called minus Ysd. This box here contains for the special case of quasistatic operation, contains a Gsd and a Csd in parallel. So the parallel combination of these two is what we call minus Ysd here. This here, minus Ygd in quasistatic operation becomes just like a past tense CGD. Okay? And so on. This one, Ym, is the total control current that describe the variation of the drain current when the voltage varies at the gate. And here we can see that there are two terms in it. This one, and this one. Of course, Y being in the phasor, being a ratio of two phasors is in the frequency domain, and this is in the time domain. But you know from your circuit analysis knowledge that the derivative corresponds to multiplication by J omega. So basically this is in the frequency domain, J omega Vgs times Cm. So now if you take this current minus this current, because they are in opposite directions, it corresponds to this source, Ym Vgs. Similar for the other elements. So there is a general model that is valid for any speed and in the case of quasistatic operation it reduces to the complete quasistatic model, if everything is done correct. So we have shown in this video a way to derive a completely general wide parameter model. In principle valid at arbitrarily high frequencies. We will use it as part of a derivation of a non-quasistatic model, which will be sketched in the next video.