In this we will derive a simple, non quasi, static model. We call it the first order non quasi static model, that allows us to go to higher frequencies than those that were possible with the models we have developed so far. Again, I would like to emphasize that the development of this model is very long. The details are given in the book. In this video I will only show you the highlights of that development, sketch the procedures, but we will not go through the detailed mathematical steps. I will go directly to the results. As a reminder, we're only assuming intrinsic effects until further notice. Let us assume we have a transistor, with the DC bias voltages shown by capital Vs on which we have superimposed small-signal voltages show by lower vs. If we allow the voltages to vary very fast, we can no longer assume quasi-static operation, as we have already discussed. So the voltages and charges in the channel in general will be functions of both position x and time t. To find a non-quasi static small-signal model, we assume that the voltages, the small-signal voltages at the terminals are sinusoidal, we convert them to phasors and we go through a very long analysis, which of course I will have to skip, but you will find it in the book. The analysis involves DC equations for the bias part of the various quantities. It involves the continuity equation which we have given, back when we discussed large signal transient operation. And then the resulting equations can be solved in two ways, using either Bessel functions or using symbolic iterations. These are summarized in the book. Here I will only show you the results of this type of analysis, entirely skipping the algebra. You end up with phasors for the drain current, the gate current and the body current, and these phasors are functions of omega. As an example, the above analysis shows that the phasor for the drain current is of this form. It has a denominator which is a function of omega, the radian frequency. In other words, 2 pi f, and in the numerator, we have a linear combination of the various voltage phasors, the drain source voltage. The gate source voltage and the body source voltage in the equations in front of each of these three voltage phases are denoted here by a A of omega, B of omega, and C of omega. This equation now can be written like this and can be compared to this equation which we derived earlier. You will recall this was the equation we obtained when we discussed the wide parameter model. Now by comparison of these two equations, we can decide what the values of parameters of the wide parameters are. For example, we recognize this by comparison to be Ydd. This to ydg, and this to be ydb, and therefore, we can write that ydd is A / D, ydg is B / D, and ydb is C / D. All of these are functions of omega, and I will show you some examples for them in a minute. So proceeding in this way, we can determine nine independent parameters, which correspond to the nine y parameters in the model we developed in the previous video. This was the model we derived in the previous video, and using the procedure I just very, very briefly sketched, we can find values for all of the y parameters in it. For example, -ygs, this admittance here turns out to be given by this expression, J omega c g s. Remember, c g s was the gate to source capacitance. So J omega c g s is the admittance corresponding to that capacitance and that is modified by what turns out to be a ratio of two polynomials, and each polynomial is in terms of J omega. So we have J omega, J omega squared, J omega cubed and so on. So I'm only showing you the zeroth and the first order term in each of numerator and denominator. The taus here can be found from the above procedure and they depend on omega zero. Omega zero is a quantity we have defined in the past. It is mu V GS minus V T Over alpha times L squared in the strong inversion region. And also, they depend on the operating point, which can be taken into account by introducing that eta parameter, which was the degree of non-saturation. So if you know the operating point, you can find tau1 and tau2 and insert them here. Now, our objective Is to find a way to represent this with a simple equivalent circuit, and this is what I will show you how to do next. So, let's take this example. This is the equation I showed you before. The higher ordered terms are neglected assuming that omega is not very high. It is high enough to be a non quasi static operation, but not too high, and I will be more quantitative as to what this means a little later. For now, we assume that omega is such that we can neglect the high order terms. And as long as omega tau 2 is much less than 1, we can take the numerator and write it like this. This uses the well known property that 1 plus a small quantity is approximately equal to 1 over 1 minus the same quantity. So you replace the numerator by 1/(1- j omega tau 2). You multiply by the denominator, ignore higher order terms, and then you end up with this expression. Now, we have something that is simple enough to represent by a simple equivalent circuit. Look at this, this is an admittance which is a capacity of admittance, divided by 1 + j omega times some kind of a time constant here. This circuit exactly represents this. The simple way to see that is the admittance of this is one over the impedance, and the impedance is Rgs Plus 1 over J omega CGs. So if you work out the algebra, you end up with exactly this, provided the third Gs is given by this expression. So now we have then a way to take a general Y parameter, which was shown as a box in our equivalent circuit, and say that as long as the frequency is not too high. In particular, this is true. It will be approximately given by this, and therefore it can be represented by a serious RC combination. We do that with the other y parameters. There is a lot involved that I am skipping. As I mentioned, I am only sketching the procedure at this point, and finally, the model becomes like this. Here, I have replaced all of the boxes that represented the y parameters by their equivalent circuit. This is the particular equivalent circuit I derived on the previous slide. You have something similar for the drain gate admittance, and similar things for the body-related y parameters. And what is very important is that ym, which was the transadmittance in the general model is now replaced by gm over 1 + j omega tower 1. Where tau 1, you know once you know the transitional parameters and the bias point. The results are in the book. So, we see what used to be the transconductance at high frequencies become something else. We call it a trans submittance and as the frequency goes up the magnitude of the numerator goes up, and the trans submittance goes down. In other words, at very high frequencies the device gives up. It cannot provide a large a tranconductance or transubmittance rather to you. This one again can be found by taking the corresponding equation for YSD, the box we had in here. And going through a similar development as on the previous slide, and then you look at the equation and decide that it can be represented by a serious combination of a resistor and an inductor. Now, notice the following. All of these elements were derived by looking at the math. We went through a mathematical development with the arrived equations then looked at them and say how can we represent these admittances by simple equivalent circuits? And it turned out that by almost, by inspection, looking at this expressions, it turns out that they can be represented by these elements. It's not that we decide that okay I look at this part of the device and it has something that looks like an inductance physically and that's why I put and inductor here. No, this is the result of a mathematical development, again which results in equations that can be represented by equivalent circuits. So this entire equivalent small signal circuit comes out of the math of the situation in the non-quasistatic operation case assuming that the frequency is not too high. Nevertheless, the math should be checked by some physical understanding. So if you look at this small signal equivalent circuit, you should be able to tell what is a resistance doing in series with a gate source capacitance, for example. Physically, why would we even expect it to be there, and for that, let us look at the transistor, again only the intrinsic part. Every voltage has a DC part and a small signal part. In the past, we had seen that between gate and source, there was a capacitance, Cgs. In a sense, this was the capacitance between the gate and the bottom plate that consisted of the inversion layer, and part of that capacitance was Cgs, and another part was Cgd. Now, if you are in non-quasi-static operation, and you vary the source voltage very fast. It takes some time for this variation to be transmitted to the rest of the inversion layer, there is some inertia in the inversion layer, and that inertia is really what is represented by this resistance. So when you vary the source voltage very fast, you will see a variation in the gate current and this gate current will come a little later. There will be some phase shift to it and this phase shift is what results if you have a RC combination over here. Another way to see it is that between the gate and the inversion layer, you have a capacitance. And between the inversion layer and the source, you have some resistance. The resistance of the inversion layer. Now, this doesn't tell you how much the capacitance is, and how much the resistance is, but it does give you a physical feel for both the presence of both capacitance and resistance in this model, which itself came out of the math. What about this inductance? Similar things you can say for the inductance. Let's say you are in non-saturation and you vary the voltage at the drain very fast, it takes some time for this variation to be transmitted and be felt at the source. Between the current at the source and the voltage at the drain there will be some phase lag and this phase lag is what is represented by the serious combination of their resistance and the inductance. And although these two elements came out of the math, it does make physical sense that they came out to be in this way. Finally, when we vary the gate vault that's very fast, the inversal layer charge doesn't have time to immediately adjust to those variations. There is some inertia to each variation, and that inertia is represented by the denominator of the trance admittance. And the faster we vary the gate voltage, the smaller the change that we will see in the drain current because simply the inversal layer cannot follow that fast. That's why as omega goes up, the magnitude of the trance admittance goes down. Let's recall that we derived this model from this model. So I would like to emphasize once again that this model, a simple non-quasi static model Is a special case of this very general model. This one can go up to high frequencies, higher than the complete quasi-static model, but it cannot go to very high frequencies. If you want to go to very frequencies, you cannot make the approximations I made. That I took minus YGS and represented it as a first order circuit, you have to retain more terms. You may end up with a second order circuit, and so on. We will not discuss such high-order models. For most practical purposes, this model is satisfactory, even at high frequencies, assuming of course you augment it by the extrinsic parasitics, which will come later on. In this video, we showed how starting from the Y parameter model and making a series of approximations. We can derive a non quasi static model shown here, and we discussed how the various elements in the model that came out of the mathematical development have some physical meaning to them. In the next video, we will take the various models we have derived and we will compare them, both in terms of their topology and in terms of their frequency range of validity.