Our overall objectives are to learn how to use modeling and simulation techniques in the design process. As an example, here is a synchronous buck switching converter used as a regulator, which is capable of precisely regulating the output voltage in the presence of low transient or input voltage transients or in the presence of variations in circuit parameters. We have seen how spice simulations of the switching circuit model can be used to verify transient responses of the voltage regulator. But how about frequency responses? How about finding crossover frequency or phase margin? How about output impedance? Circuit averaging and average switch modeling provide effective ways to address these important issues. Here is the same example with switches replaced by a subcircuit that represents an average switch model for the switch network, whereas the rest of the circuit is essentially the same as before. Using this approach, we can perform transient, DC, and AC simulation of a circuits in both time and frequency domains and obtain very valuable information about its performance. And very valuable ways of verifying its performance in the presence of different sets of parameter values or different operating points. Average switch modeling is based on circuit averaging as opposed to averaging equations. It is a very popular or practical approach that can be applied to a wide range of cases. Including pulse-width modulated converters operating in continuous conduction mode or discontinuous conduction mode. Or current-mode controlled pulse-width modulated converters, as we will see in the rest of the course. A switching converter consists of switches, magnetic, and capacitive components. The first step in circuit averaging is to separate the switch network from the remainder of the converter, and define ports of the switch network. The second step is a step of circuit averaging, averaging switching waveforms of the switch network. It is best to see how this works on an example. Let's take a SEPIC. It's a pulse-width modulated converter that we assume is operating in continuous conduction mode. In this example, we will neglect losses. The main switch Q1 is controlled by a pulsating waveform duty cycle d. So, the first step is to separate the switch network from the remainder of the converter. And you could see here how it is done simply by identifying port one of the switch network as the terminals of the transistor switch and port two is defined by the terminals of the diode rectifier. We then examine the waveforms on the two ports of the switch network. For example, v1 is the voltage across the transistor Q1. It is equal to 0 when the switch is on, and it is equal to the sum of the two capacitor voltages vc1 + vc2 when the switch is off. Remember that we consider no losses in this particular example. For the current i1 through the switch, we see that it's equal to the sum of the two inductor currents when the switch is on, n is equal to 0 when the switch is off. Similarly for the diode, we have that the voltage across the diode when it's off is equal to vc1 + vc2, and is equal to 0 when the diode is conducting. And then finally, the diode current or the port 2i2 current is going to be equal to 0 when the main switch is on and is equal to il1 + il2 when the diode is conducting. So now that we have the waveforms we can perform the averaging of the waveforms. So v1 average and you can see that by inspection, it's very easy. Can be written as d prime times the average vc1 + average vc2. Real small-ripple approximation for the capacitor voltage and currents because we have assumed the converter operates in continuous conduction mode. For the diode voltage, that's the port 2 voltage, v2, we similarly have v2 = d(vc1 average) + (vc2 average). And then for currents we have i1 = d(il1 + il2 average). And finally, i2 = d prime( il1 average + il2 average). Here's a summary of the expressions found for average values of switch network terminal waveforms. It would now be highly desirable to eliminate from these expressions everything other than the port quantities. It is, in fact, possible to do that. Let's see how. Take a look at the two voltage expressions. From these two voltage expressions, we can eliminate, The capacitor voltages, vc1 + vc2. And we can obtain a relationship between the average v1, And the average v2. This relationship holds for the port quantities of the switch network alone, regardless of any other quantities in the rest of the converter. For the currents, we can eliminate, iL1 + iL2 and find that i2 can be written as d prime over d times i1. This is important because now we see relationships between the port quantities average that are independent of capacitor voltages or inductor currents, which really implies that the same model can be used for any two switch PWM converter. So here are the summary of the relationships that we have derived. And the result down here can be recognized in the form of an equivalent circuit model, shown down here, that consists of two control sources. In summary, we can identify a switch network consisting of a transistor port and a diode port in any two switch PWM converter. By averaging the switch waveforms, then we can obtain an average switch model that is valid for any two switch PWM converters operating in continuous production mode, neglecting losses at the moment. We will address losses in one of the upcoming lectures. The average switch model is a large signal, non-linear, but time-invariant model. And so it can be effectively used as a sub-circuit in spice simulations. Or it can be linearized to lead to DC or AC analysis of switching converters. And we will explore both applications of the average switch modeling approach in the coming lectures.