In a peak current mode controller, a sent switch current or sent inductor current, is compared to a control signal and the comparison determines the end of this switch on time. So far, we have noted that the control signal and the average inductor current are relatively close to each other and use a simple approximation based on that observation to derive models for the peak current mode control 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 control. The objectives in the denigration of the more accurate average CPM model are to include effects of inductive 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 the 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 average CPM controller model, to establish a relationship between the control input and the average inductor current, I_sub_L average, average voltages from the converter and the duty-cycle d, without making the approximation that the average inductor current is approximately equal to the controlled current. Once we establish this relationship, then we can implement that model as a Spice sub-circuit or we can linearize that model to obtain small-signal averaged controller model for analysis purposes. In the end, we will have a set of design-oreinted analysis and simulation tools to support our design efforts around peak current mode controlled PWM converters. Here are the waveforms in the peak current mode control 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, and 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 controller and find the average inductor current as a function of control input i_sub_c, the duty-cycle d and the slopes m1, m2 and ma. Let's catch 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 a time when the transistor is off. Sketching the average value of the inductor current wave shape 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 sub-interval, d prime Ts, we identify a 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 we 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 relationships for the currents 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 the dTs. So that's the expression we have for i1, and the expression for i2 is obtained in a very similar manner. i2 is equal to 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/2. And so here, we have i2 is equal to ip minus one half and two d prime T s. Based on these relationships here, now the question is, what is actually the average value of the end of the 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 a point where the duty-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 this average value of i_sub_L. Here's the set of waveforms again. We see that the time interval between the points i1 and i2 is equal to exactly Ts/2. The length of the first subinterval here is going to be dts/2 and the length of the second subinterval is going to be d prime Ts/2. So we can write down that i sub L 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 minus i1, over the length of interval Ts/2 times the length of the interval right here, which is dTs/2. Now Ts/2 and Ts/2 go away. 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 times i1 plus di2. Then, of course, d prime is equal to one minus d. Here is the summary of what we have so far. iL average is equal to d prime i1 plus 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 minus ma dTs. And so, i L average which is equal to d prime times i1 plus d times i2 is going to be that same value i_sub_c minus ma dTs. That's common in both. And then we have minus one half and one d, d prime T s from i1 and minus one half and two d, d prime T s from i2. Finally, we can write i L average is equal to ic minus madTs minus one half and one plus m2 d, d prime Ts. Here is that final expression for the average inductor current, which is now found as a function of control input, i_sub_c, duty-cycle d, which shows up here and here, and slope 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 prime Ts interval. We do assume, still, the converter operates in continuous conduction mode. Given that large-signal nonlinear average 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 control 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 equivalents circuit models for CPM control converters, which can then be used in design-oriented hand analysis. We will pursue both paths in the lectures that follow.
