Here is a handy table that lists the salient features of the transfer functions of the three basic converters. So, if you have an ideal converter or if the losses are not significant, you can use this table to quickly get the transfer functions of the basic converters. So, what I would say here is that in each case, the Gvg or line to output transfer function. Can be written in this form with two poles and a DC gain. Gbd, the control throughout the transfer function, will have the same two poles, it'll have a different DC gain, and it may have a right half plane zero. In the case of the boost and the buck boost converters, we have a right half plane zero and in the buck converter we don't. This table can also be used for isolated converters containing transformers that are based or derived from these buck boost, buck boost converters. So, for example if we have a forward converter we can use the buck transfer functions. And all we need to do though, is include a turns ratio in Gbg, what, which is the transformer turns ratio. And as far as Gbd is concerned, the game is actually written in terms of the output voltage V rather than the input voltage Vg. And so, the effect of the turns ratio cancels out, and you can apply this Gvd transfer function with this, these coefficients directly. If the converter has inductors on the primary side, then you would have to reflect the inductors through the turns ratio. Of the transformer. This is the physical turns ratio in, before applying these, these formulas. Now, we've seen right half plane zeros arise in the buck, the the buck loose converters. And they arise in other converters as well. And what I'd like to do now is briefly discuss where the right half plane zero comes from. And what its nature is. It actually causes some serious problems in the dynamics of the converter that are, make it difficult to control. So, one way to view the right half-plane zero is this. That at low frequency, the one dominates over the s over omega naught term. And so, the transfer function is one, which is positive. On the other hand, at high frequency, where S over omega zero has a magnitude substantially greater than one, then this term dominates and there's a minus sign involved. So, at high frequency, we have a phase reversal from the minus sign. In a block diagram form, one way to do this transfer function, is that we have a low frequency path this way, that is, has a gain of plus 1. Where as that high frequency, this path dominates And it has the S over omega z term, and it has the minus sign right there. So, one interpretation of this from the frequency domain is that initially for high frequencies The transient response goes the wrong direction because of the minus sign, and then later, when the low frequency gain becomes significant, the phase becomes correct, the plus sign and the output will follow the input. So, one way to think about this, then, is suppose that our input signal to our converter in some kind of step. Say we start at a low value and we step up to a larger value. So, that's our input. And we would hope that the the output would follow the input. Perhaps there is some transient response. That has some lag and takes some time to get there but, we would like the output to eventually rise and follow the input but the right hand plane zero response the characteristic is that the output goes the wrong direction. So, the output goes down instead of up. And then eventually after some substantial time, it catches up and goes the right way and follows the input. So, why do the boost converter and the buck boost converter exhibit this kind of behavior? Well it, it comes out of how the switching. Occurs to connect the inductor to the output. In each of these converters, the current that drives the output coming out of the converter is a diode current, and the diode conducts when the transistor is off. In fact we can right the equation for the average value of the diode current. And it is D prime times the inductor current. So, the inductor current flows through the switch to the output during the prime interval. And this is true in both the buck and the buck-boost converters. So, let's consider a trangent where we turn up the duty cycle. Here are some wave forms. So, this is our inductor current, including it's ripple. It's operating here, initially, with D of 0.4, and we're switching so, we get some switching ripple like you see here. The diode current is equal to the inductor current when the diode is on and it's equal to 0 when the diode is off. So, here's the sketch of that. The diode current waveform and its average which is D prime times iL is also illustrated here. Finally, I've illustrated the output voltage as well. So, at this point right here, we turn the duty cycle up from 0.4 to 0.6. And we would expect the boost converter you know, with a larger duty cycle would make a larger output voltage. And so the output voltage should rise. But what happens to the average diode current? It takes time for the inductor current to change and right after we change to B of, equals 0.6, the inductor current hasn't have time to change much, it's a continous wave form. So, the inductor current goes through some trangent, and eventually it builds up to a large value, but initially it hasn't had time to do that. On the other hand when we change the duty cycle from 0.4 to 0.6, at this instant, the duty cycle immediately goes down. So, see what happens to the diode current. The width of the D prime interval is less and the average diode current immediately jumps down. Well, if you drive the output with less current, then the load can discharge the output capacitor and make the output voltage go down, so you'll see the output voltage initially deep. After some time, the inductor current builds up to a big value. In fact, in steady state the inductor current goes like Bg over D prime squared R, so with that square the inductor current will build up to a much larger value. And we see eventually that the average diode current builds up to a value where the average is greater than it was before even though the D prime interval was shorter. And so with this larger current we have more current coming out of the converter that can charge the output capacitor up to a higher volt voltage. And eventually we come into a steady state with a higher output voltage. But initially the voltage goes down instead of up, and so this part of the response is indicative of a right half plane zero. Now, imagine the effectiveness on a feedback loop, we build a control system for our converter that. Is trying to regulate the output voltage to follow some reference. When the feedback loop sees the output voltage go down, or if the alpha voltage is less than the reference, then the feedback loop will turn up the duty cycle. So, what do you suppose will happen? If we have a wide bandwidth or fast feedback loop it will see the voltage sag and it will say oh, the voltage is going down, I need to turn up the duty cycle even know. So, it will turn up the duty cycle again to a larger value which will make the output sag even more and in fact this can be unstable. Where the, the convertible just hit the stops at D equals 1. And the output voltage will go to a low value. It can also cause the feedback loop to oscillate. One way to avoid this is to make our feedback loop slow so that it never even sees the sag. We make the, the feedback loop change the duty cycle only very slowly. And so it may not even notice the output voltage has gone down, or even see anything until we get out of here at a later time where the effect of the right half-plane zero has gone. So, boost converters, [UNKNOWN] boost converters, and other converters that have right half-plane zeroes are notoriously difficult to. control, and in fact, the four of these average models were developed, people didn't understand why. But they did know that boost converters and buck boost converters tended to oscillate, and they were difficult to control. And it was one of the early, I think, victories of averaged converter modeling to be able to predict right half plane zero. And and show why these things were happening.
