In a peak current mode controller, a sensed switch current or sensed inductor current is compared to a control signal, and the comparison determines the end of the switch on time. So far, we have noted that the control signal and the average inductor current are relatively close to each other. And we use the simple approximation based on that observation to derive models for the peak current mode controlled converters. In this lecture, you will learn how to derive a more accurate average model which takes into account the inductor current ripple and the presence of the artificial ramp in the peak current mode controller. So the objectives in the derivation of the more accurate average CPM model are to include effects of inductor current ripple and artificial ramp, to obtain more accurate modeling of all transfer functions of interest, control to output, line to output, input and output impedances, and so on. In particular, the more accurate average CPM model will give us more accurate predictions for high frequency dynamics, and will enable wide bandwidth designs. The approach taken is to develop a large signal averaged CPM controller model to establish a relationship between the control input and the average inductor current, iL average, average voltages for the converter and the duty-cycle d, without making the approximation that the average inductive current is approximately equal to the control current. Once we establish this relationship, we can implement that model as a SPICE sub-circuit or we can linearize that model to obtain small signal average controller model for analysis purposes. In the end, we'll have a set of design-oriented analysis and simulation tools to support our design efforts around peak current mode controlled PWM converters. Here, the waveforms in the peak current mode controlled converter. Control current i sub c, the artificial ramp with a slope of ma, inductor current with a slope of m1, during the time when the switch is on, minus m2 during the time when the switch is off. The goal is to look at the geometry of the waveforms in the peak current mode controllers and find the average inductor current as a functional controlling input i sub c, duty-cycle d and the slopes m1, m2, and ma. Let's scatch the average inductor current waveform. Given that the inductor current itself is a triangular wave shape, with approximately linear segments in both the time when the transistor is on and the time when the transistor is off, sketching the average value of the inductor current waveshape is relatively simple. In the first segment when the switch is on, we identify current i1 right in the middle of that segment. Similarly in the second subinterval, d prime Ts, we identify the current i2 right in the middle of that linear segment, and the line passing through these middle points is the average value of the inductor current. We can actually write simple relationships for the quantities based on the geometry that you see here. The peak value of the inductor current is equal to the control input i sub c minus the slope of the artificial ramp times the length of the interval dTs. Similarly, we can obtain the relationships for the current i1 and i2. Notice that i1 can be written as the peak value of the inductor current minus the slope of the inductor current during dTs over a time interval equal to one-half of dTs. So that's the expression we have for i1, and the expression for i2 is obtained in a very similar manner. i2 = ip minus the slope of the inductor current during d prime Ts interval, times the length of time, which is equal to d prime Ts over 2. And so here, we have i2 = ip- one-half m2 d-prime Ts. Based on these relationships here now, the question is what is actually the average value of the inductor current? To answer that question, we need to think about at what point in time we really care about the average value of the inductor current. And the answer to that question is that we really care about it at the point where the jury cycle is modulated at the edge, the falling edge of the control signal for the switch. So we are really looking for the average value of the inductor current at a point in time where the duty cycle is modulated. Let's find out how to determine the average value of iL. So here is the set of waveforms again. We see that the time interval between the points i1 and i2 is equal to exactly Ts over 2. The length of the first subinterval here is going to be dTs over 2. And the length of the second subinterval is going to be d-prime Ts over 2. So we can write down that iL average is equal to i1 as a starting point right here, plus the slope of the average inductor current, and that slope can be written as i2-i1 over the length of interval Ts over 2, times the length of the interval right here, which is dTs over 2. Now Ts over 2 and Ts over 2 go away, and you can see that the average value of the inductor current can be written as i1 plus d times i2 minus d times i1. And that's equal to d prime i1 + d i2, where of course d prime is equal to 1-d. So here is the summary of what we have so far, iL average is equal to d prime i1 + d times i2, and we have expressions for i1 and i2. Next, we can eliminate i1 and i2, and obtain one convenient expression for the iL average. Let's take i1 and plug here, i2 and plug here. Notice that both i1 and i2 contain exactly the same term in front, ic - ma dTs. And so iL average, which is equal to d prime times i1 plus d times i2 is going to be that same value i sub c - ma dTs that's common in both, and then we have minus one-half m1 d, d prime Ts from i1 and minus one-half m2 d d prime Ts from i2. And so finally, we can write iL average = ic- ma dTs- one-half m1 + m2 d d prime Ts. So here is that final expression for the average inductor current, which is now found as a function of controlling input i sub c, duty-cycle d, which shows up here and here, and slopes m1, m2, and ma. This relationship here does not assume that the inductor current ripple is small nor it assumes that we don't have any artificial ramp employed. In fact, it takes into account the slope of the artificial ramp and it takes into account the slopes of the inductor current in the dTs interval and the d prime Ts interval. We do assume still that a converter operates in continuous conduction mode. Now given that large signal nonlinear averaged CPM model in continuous conduction mode, we can pursue two paths. Based on this expression right here, we can construct a Spice subcircuit. Using that Spice subcircuit we will be able to construct average Spice circuit models of CPM controlled converters, and use those in design verification using Spice DC, transient or AC simulations. The other path is to linearize this expression and obtain more accurate small-signal equivalent circuit models for CPM controlled converters, which can then be used in design-oriented analysis. We will pursue both paths in the lectures that follow.