In the previous lecture, we have derived an average switch model for the current mode controlled converter based on the simple approximation that states that the inductor current in the converter is approximately directly set by the control input to the peak current mode modulator. In this lecture, you will learn how to linearize this model and how to derive transfer functions for a peak current-mode controlled converter. The linearization step is exactly the same as usual. For each quantity in the average switch model, let's say average value of v1 we assume that it is represented as a sum of the DC component capital V1 and a small ac component v1 hat. And we do so for all other quantities in the average switch model including the control input ic represented as the value of the dc set point Ic and a small signal perturbation ic hat. If you plug these into the average switch model relationship that states that the product of the input current and the input voltage is equal to ic and the output voltage, we multiply and neglect the terms that contain products of small-signal quantities. The result can be solved for the value of the input port small signal current, i1 hat, in terms of the ic hat, v2 hat, and v1 hat, where the terms that multiply these small signal quantities are parameters of the small signal model that, as you can see, depend on the operating point in the converter. The output port equation is as simple as stating that i2 = ic hat. So i2 hat = ic hat. So here is the model in the circuit form, an equivalent circuit model that is representing the small signal model obtained by linearization. of the averaged switch model for peak current mode control is shown here, i1 hat current depends on ic hat, v2 hat and v1 hat in this manner. And you see these three components are represented by three components on the input port of the small-signal model. The current source, the control current source, it depends on the ic and the operating point. A resistance value, which has a negative term in front, we'll talk about it in just a second, and a controlled current source depends on the v2 hat voltage. On the output side, we have a very simple circuit that then presents the approximation that i2 hat, which is equal to the inductor current in the small-signal sense, equals the small-signal ic hat. Now that input resistance that we see as a negative value has a very simple physical interpretation. What is shown right here is the average value of the input port current as a function of the average value of the input port voltage. Let's suppose the input port voltage goes up. Since the voltage goes up the slope of the inductor current is going to increase. But we assume that the control current input has a fixed value. And so if the slope of the inductor current increases as a function of the input voltage increasing, the current will reach the set value, the control input sooner, and the duty cycle of the pulse is going to shrink. As a result, the average value of the input current will drop down, so increasing the voltage of the input port results in lower input current. That is a feature of the characteristic of the peak current mode controlled converter that has an impact on the dynamics of the converter, design of the input filter, and so on. Now let's look at how the small-signal model can be used to derive converter transfer functions. So for the control to output transfer function which is the output voltage hat right here, here is for the buck example we have this right here. over the control input being ic hat, we simply have to solve the output port circuit for the small signal model and that solution is extremely simple, because we have ic hat current directly feeding the parallel combination of R and C at the output. And so the control to output transfer function is a single pole transfer function. It is obtained simply by taking the parallel combination of the load resistance R and 1 over sC, the impedance of the output filter capacitor. If you look at the line to output transfer function, Gvg, from the input voltage perturbations to the output voltage perturbations, in the model shown right here, we get a beautiful result of zero. So we're basically getting the result here that tells us that any input voltage variations will have no effect on the output voltage in a buck converter. Now that is not exactly true and as we move forward with lectures next week you will see that this is all here while approximately true can actually be improved and we can get a better prediction for the line to output transfer function in a peak current mode controlled buck converter. The same small-signal model can be applied to other converters. And in general, the shape of the model stays always the same, which is why we call it a CPM canonical model based on the simple approximation. So the shape of the model is always the same. There are three elements that represent the input port, three elements that in general represent the output port, and the values of those elements are given in this table here for the basic converters, the Buck, Boost, and the Buck-Boost. You can use this page as a reference for finding transfer functions of the basic converters with peak current-mode control. Now, transfer functions predicted by the simple model, as we have already seen in the case of a Buck example, include, control-to-output transfer function which in general has the following form. It's equal to f2, f2 being the value that multiplies the control-current ic times the power combination of the output resistance in the model, C and R, the load resistance. What those values are, again you can see on the page that includes the table with f2, r2 and so on for the basic converters. For the buck-boost example we have, again at low frequencies, on the right we have a right half plain zero and a single poll response. Notice that right half plain zero in the buck-boost case, and the boost case as well is present in the peak current mode controlled converters. But the response is a first order, there is just a single pole in the denominator of the control to output and other transfer functions. For the line-to-output transfer function, again, we have a general result where you can plug in values from the table for different converters. In particular, for the buck-boost converter, we have an expression that looks like this. The same pole as before, of course, a different low frequency gain in front. Even in the ideal case, the rejection of disturbances in the input voltage is not equal to zero for the buck-boost case. And then finally the output impedance is simply a parallel combination of r2, the load resistance, and the impedance of the output filter capacitor and you can do the derivation for the output impedance for the three basic converters based on the table results for r2. For the buck-boost example we have the output impedance that has the same pole as before and has in front a low frequency part. In summary, the small-signal model based on the simple approximation has a canonical form shown right here. In the model, we see that the inductor current disappears as a state variable so that a transfer function predicted by the simple model are a first order for the basic converters. And those simple single-pole converter transfer functions make it very easy and convenient to design voltage control loops around peak current mode controlled converters. And that's what we will see in the following lectures.