In the last lecture, we derived the small signal model of a peak current mode controller in a block diagram form. And we also discussed how that model can be combined with the known equivalent circuit model for converter of interest. This is an example for the boost converter. When you look at the structure of what we have here, which is a complete small signal model for a CPM controlled boost converter, you notice some difficulties: the model includes feed-forward action and feedback action, because of the nature of the peak current mode controller. The input and output voltage determine the slopes of the inductor current and in turn show up as gains in the equivalent circuit model in a block diagram form for the peak current mode controller. So solving this complete equivalent circuit model is not easy or it does not appear to be easy because there are multiple dependencies and feedforward and feedback paths. After viewing this lecture, you will be able to derive transfer functions of CPM controlled converters, starting with an equivalent circuit model of this type. We will prefer to present this model in a block diagram from. That gives us a simple approach to deriving desired transfer functions. Let's look at the system block diagram. So on the left hand side, you have the CPM controller model in a block diagram form already. The input to that model is the control current ic hat. On the right hand side, instead of the equivalent circuit model for the converter we present the converter transfer functions in a block diagram form. For example, the output voltage v hat can be shown as Gvd, control to output transfer function from d hat to v hat, plus Gvg, line to output transfer function times vg hat. Similiarly, iL hat can be written in the form of Gid times d hat plus Gig times vg hat where Gid and Gig are the transfer functions from duty cycle or vg hat to the inductor current. All these four transfer functions can be readily derived for any of the duty cycle controlled converters based on known small-signal models. Now, we can solve the block diagram for desired transfer functions. What would be the desired transfer functions? Well, first of all control to output, ic hat to v hat. Let's start with the derivation. So we will take first that vg hat is equal to 0 and this term goes away, this term goes away. We have that iL hat is equal to Gid times d hat. Now in the CPM controller model, we can say that ic hat minus Gid times d hat minus Fv times v hat, this is Fv v hat all multiplied by F sub m gives us the duty cycle d hat. So we're simply tracing the signals through the block diagram to establish relationships between the quantities of interest. This can be immediately solved for d hat. We have d hat x (1 + Fm Gid) on the left hand side equal to ic hat minus Fv v hat times Fm on the right-hand side. And so, we have d hat = Fm ic hat minus Fv v hat over one plus Fm Gid. Next, we can notice that v hat is equal to Gvd times d hat. And this relationship here, together with this one, can be solved for v hat over ic hat, which is the control to output transfer function for the CPM controlled converter. So here are the results summarized. So we can solve for d hat in general, including vg hat perturbation, substitute that in the expression for v hat. Here is the result. That can it be then solved for v hat in terms of the ic hat and vg hat. The expressions obtained from the block diagram model are really the control to output transfer function from the control current ic hat to the output voltage, and the line to output transfer function Gvg with a dash cpm to remind us that this result here holds for the converter that includes peak current mode control. These expressions are not simple, they don't look very nice but nevertheless can be worked out in any particular example and that's what we'll do next. They give us valuable insight into where dynamics, the poles and zeroes of CPM controlled converters really come from. So here is an example, we will derive transfer functions for the buck converter. We start by deriving the Gvd, Gvg, Gid, and Gig. These four transfer functions that enter in a block diagram form, and a block diagram version of the complete circuit model. So by analysis, that was done earlier in the specialization, we find these transfer functions. We realized that they share the same denominator, denominator that has the form of 1 + s L / R, R being the load resistance, + s squared LC. And the numerators depend on which transfer function we are actually looking at. Similarly, the gains in front depend on the particular transfer function. Now that we have these four transfer functions for the duty cycle controlled converter, we can combine them in a block diagram form and determine the control-to-output transfer function for the CPM controlled buck converter. Here's the general expression that we have derived on the earlier slide. And when you work out through that expression now taking into account the particular values for Gvd and Gid for the buck converter. The end result in this form, can be finally represented in the form that is shown at the bottom of this derivation right here. That final form shows that the control-to-output transfer function in the CPM control buck converter is a two-pole response with a low frequency gain and a pair of poles at a center frequency fc and a Q factor of Qc. So the form of the control-to-output transfer function for the CPM controlled buck converter is the same as the form for the control to output transfer function of the duty cycle controlled converter, Gvd. But you will see in a moment, that the Q factor in a peak current mode controlled converter tends to be much smaller, and so this pair of poles is going to split into two poles, a low frequency pole and a high frequency pole. The complete results for a buck converter are shown on this page, and we will now discuss details in the lecture, but you can use this page as a reference. Let's just see what is shown here. On the left-hand top side, we have the simple model result. This is the case where we assume the inductor current is directly controlled by the control input, ic. The right hand side, shows the converter transfer functions with a duty cycle control. Those transfer functions are used in a block diagram model to derive transfer functions for the peak current mode controlled converter. And those transfer functions are shown in the bottom part of this reference slide. The control-to-output transfer function has a pair of poles with a low frequency gain. There is an expression for the locational of the pair of poles, there's an expression for the low frequency gain and the Q factor and similarly, we have a complete expression for the line to output transfer function with the low frequency gain shown right here. And the denominator is exactly the same as in the control to output transfer function. Let's discuss this result, so here's the expression for the Q factor of the peak current mode controlled buck converter. You will notice that the front end or the front part of that Q factor expression is the Q factor for the pair of poles in the duty cycle controlled converter. The second term is what modifies that Q factor, and that second term depends on the gain in the peak current mode controller and the factor F sub v, the gain with respect to the output voltage. When you evaluate this expression right here, in most practical cases you will find that the Q factor of the pair of poles is substantially reduced compared to the Q factor that we have in the duty cycle controlled converter. So, if the artificial ramp is not too large, and that's the particular case where Fm is relatively large, then the poles in the peak current mode controlled converter become real and well separated. Given that the poles are well separated, we can employ the low-Q approximation. The low-Q approximation tells us that the low-frequency pole can be obtained simply by taking the center frequency omega c and multiplying that by the value of the Q factor. The expression for the low frequency pole is shown right here, and for the large value of F sub m and small value of F sub v, that expression can further be approximated as being just 1 over RC which is identical to the prediction of the simple CPM model we had seen before. As a reference, we have a page here with complete results for the boost CPM control converter. And again we have results for the simple model, the transfer function for the duty cycle controlled converter, and control to output and line to output transfer functions for the peak current mode controlled boost converter, and a similar reference page is available for the buck boost converter. Again, you will notice that the expressions are not simple, do not look particularly nice but nevertheless, very useful in the analysis of transfer functions of CPM controlled converters. In conclusion, more accurate CPM small-signal model can be combined, with known transfer functions of duty-cycle controlled PWM converters in a block diagram form. There is an approach to work through the difficulties of feedforward and feedback paths that we observe in the complete equivalent circuit models for the current mode controlled converters. The block diagram can be solved for all CPM transfer functions of interest. We have found that the CPM control reduces the Q factor of the converter poles. The converter poles are split into a dominant low frequency pole, which is approximately the same as the pole that we had found earlier using the simple approximation, and the high frequency pole, so there is a prediction of high frequency dynamics available in this more accurate small signal model. It is important to know that the right half plane zero remains present in the control to output transfer functions of peak current mode controlled boost and buck-boost converters. In fact, the location of that right half-plane zero is the same as the location of the right half-plane zero in the duty cycle controlled converter given the same operating point. A large value of the artificial ramp results in higher value of the Q-factor of the pair of poles in the peak current mode controlled converter and the high frequency pole moves to lower frequency. And that coincides with the example we have seen earlier where by simulation we have shown that we have additional phase lag associated with CPM controlled converter when the artificial ramp slope is too large. But in general, it is easier to design a voltage loop compensator around the CPM controlled converter than around the duty cycle controlled converter because the frequency responses are dominated by a relatively low frequency pole. So the transfer functions look like single pole transfer functions for wide ranges of frequencies.