After viewing this lecture, you will be able to derive and use a simple model for peak current mode control. Here are the waveforms to remind you how peak current mode control works. At the beginning of a switching period, the controller turns on the transistor switch. The transistor current ramps up as an approximately linear function of time. At time DTs, that current reaches the control signal ic and that determines the end of the switching time when the transistor is on, or in turn determines the duty cycle of the switch. When you compare signal ic, and the actual inductor current in the converter, you can see that they're relatively close to each other. In fact, if the ripple of the inductor current is relatively small, the peak value of the inductor current is very close to the average value, and we could say that the control signal directly determines the value of the inductor current iL. That is the basis for the simple model that we will talk about in this lecture. So a simple first order model will be derived now. That simple first order model will be valid for any switching converter that has a CPM controller around it, with sensed voltages and currents affecting directly the duty cycle of the converter, and we can have an outer voltage control loop that sets the control input for the peak current mode controller. So the basic idea behind the first-order approximation, the simplest model we have for the peak current mode control is to simply say that the inductor current on average is directly determined by the control input ic. This simple model neglects switching ripple and as you will see later on, what is called artificial ramp. But it is a very useful, practical model because it gives us a simple physical insight into the operation of the peak current mode controlled converter. It also gives us a simple first order model which can be used in the design process. This model is very accurate when the converter operates well in the continuous conduction mode and when the ripple of the inductor current is relatively small. But in all cases, it does provide a first order insight into the effects of peak current mode control on the converter dynamics and will allow us to design voltage control loop around it. Now, this is a large signal relationship. We simply say the actual value of the average of the inductor current is set by the control input ic. So this is very easy to linearize. In fact, directly we have that the small-signal relationship between the small signal inductor current perturbation and the small signal perturbation in the control input are approximately the same. Based on this simple approximation, we can derive an average switch model for the peak current mode controlled converter. First, notice how the switch network is defined. We take a buck converter as an example, but we could do that in other converters as well. The switch network is defined with the input port v1 and i1 being across the two switches connected in series, whereas the output port is defined as the port across the rectifying switch. Let's scatch the waveforms for v2 and i1. So v2 waveform is going to be equal to the input voltage when the switch is on and is going to be approximately 0 when the switch is off, neglecting now conduction losses on the rectifier. So this is vg and approximately 0 right here. i1 is a pulsating waveform that follows the inductor current during the time when the switch is on, and that is equal to 0 when the switch is off. So here we have i1 = iL, which in turn is equal to what we call current i2, and here the current is equal to 0. Now, averaging of these waveforms can be done very easily. We have the v2 average is simply equal to d times vg average, and similarly, i1 average is equal to d times i2 average. Now as we have done earlier in the average switch modelling, we can eliminate the duty cycle and establish the relationships between the average values of the switch network. So here is the summary of the average terminal waveforms for continuous conduction mode for the switch network of the converter that operates with peak current mode control. That model is based on the simple approximation that the i2 average current is equal to the average value of the control signal ic. So if we take the relationships we have derived and if we eliminate duty cycle, we have the i1 average is equal to d times the ic average. Where ic average is the same as i2 average. And replace d with the expression that we have from the first relationship right here, we have d is equal to v2 average over v1 average. We obtained that i1 average ultimately is equal to the ratio of v2 average over v1 average and ic. Taking the product of v1 and i1 average on the left hand side, you obtain that product which really represents the power absorbed on the input port of the averaged switch network equals the product of the control set current ic average value, and the average value of the output voltage v2, or the port 2 voltage v2. So in the average switch model the output port of the switch network operates as a current source. The current value i2 is directly determined by the control input. As a result, the input port behaves as a dependent power sink, an interesting component. And that power sink is really defined by this relationship right here. Showing that the product of the input current and input voltage in the average sense, equals the power that is set on the output side of the average switch network with the output current being directly determined by the control input ic. So here is the diagram that shows the averaged switch model for the peak current mode controlled buck converter. So notice the input side, the v1 i1 side, represented by the symbol for the power sink. The output side is simply represented by a symbol for a current source because that current is directly set by the control input ic. Notice that current source in the buck converter is directly in series with the inductor. That is basically physically telling us that the inductor current is directly set by the control input, and inductor current will disappear as a state variable from the model. The same approach can be applied to other converters and you can see that the exact same average switch model positioned of course, properly can be applied to the boost converter where the current control can be observed now on the side where the inductor is connected. And again, the inductor current is directly set by the control current ic. And then the output port looks like a power source with respect to the output voltage v. In a buck-boost converter we have similarly averaged switch network represented by the power sink and the current source again being in series with the inductor of the converter. In summary, the simple approximation in the average switch modeling of the peak current mode control is that the inductor current is approximately equal to the control input ic. That leads to the average switch model that represents the switch network in this form as a power sink in combination with a current source. This is a large signal model, and our next task will be to linearize this model and determine an AC equivalent circuit model that can be used to find the converter transfer functions and then use those transfer functions to design a voltage loop compensator around the peak current mode controlled converter.