We previously discussed some circuit manipulations. To ex, expose the relationships between the different converters and to derive some new ones. There's another way to do this. We can actually derive all possible converters that have a given set of constraints. Such as having, say, two switching intervals and one inductor. I'm not going to derive all the possible converters. It turns out that there are an infinite number of them. And they do some pretty interesting and diverse things. But I do want to briefly summarize some of the basic results. so, what's shown here is I think the most rudimentary class of converters, where we have one inductor that is switched between the input source VG and the output load V in two different ways. One way during interval one, and another way during interval two. Now, there's only a limited number of possible ways to connect in, an inductor between two voltages and pick one for the first interval and one for the second. And we can write down all the possible ways to do that and see what converters occur. And the answer is right here, there are eight converters that are interesting. in addition to a few degenerate or redundant cases that don't add anything new. And we can group these converters, actually, into three different groups. the first group are converters that I would call dc-dc. Namely that if you have a positive input voltage then the output voltage is always of the same polarity regardless of the duty cycle. there's also two converters where when you vary the duty cycle from zero to one, the output voltage changes polarity in such a way that these are useful as dc to ac type inverters. And then the the last two are the inversion of the inverters. So, they're more suitable for ac to dc type applications or rectifiers. So here they are. the first two are the buck and the boost, which we already know about. the next two, we've also discussed. We have the buck-boost and the noninverting buck-boost. so these are the four converters in this class. they're all you can make with a single inductor and two intervals that are dc/dc type converters. Okay? Converters five and six are ones that you might say are suitable as inverters. And we've already discussed the H bridge in the last lecture. This is converter number five. Converter number six is a pretty strange one. It was invented, actually, the transformer isolated version of this was invented at the Watkins-Johnson company. And we call this the Watkins-Johnson converter. It has this conversion ratio 2d minus 1 divided by d. It looks like this. but it passes through 0 at d of a half and it can make either positive or negative output voltages, although the m of d function is a non-linear one. and then the last two in this class are the inverse of numbers five and six. So converter number seven is what you get if you swap the source and the load of the H bridge and this one is sometimes called the current-fed bridge. So its conversion ratio is 1 over 2D minus 1. And at d of a half, the ideal conversion ratio goes to infinity. So here's a plot of its conversion ratio. This is a pretty strange conversion ratio. But, in fact, if the input is a sinusoid, and you want the output to be dc, this is the conversion ratio that you need to, to perform this rectification function. So, when the sinusoid at the input passes through zero, and the output voltage is non-zero, then the ratio, or the conversion ratio, goes to infinity. But in fact, we're not making infinite output voltage, we're just holding up the output voltage when the input is going through zero. So, if you vary the duty cycle sinusoidally about a half, then you can convert an ac input voltage to a dc output. And then the last one is what we get from inversion of source and load with the Watkins-Johnson converter and its conversion ratio is the inverse, which also goes to infinity at d of a half. So those are all the converters that are possible if you only have one inductor and you have two switching intervals. we can think of other classes of converters, and the next most logical class might be the class of two inductor type converters. So we have two inductors and we switch them in one way between the source and load for the first interval, and a different way for the second interval. There are many more than eight members of this class of converters and they can do some pretty interesting different things. here's just a couple of them. The Cuk converter, we've previously mentioned derived this by cascade of a boost followed by a buck converter. it's an inverting buck-boost type converter. The SEPIC is another converter in this class. the transformer isolated version of the SEPIC was invented at Bell Labs in the 1970's. And they called it a single ended primary inductance converter, whose initials are SEPIC. this turns out to be a noninverting buck-boost converter. And it's interesting, actually finds quite a few applications. It's interesting, for one, because it can, its transistor has its source connected to ground. And you realize the switch is with transistor and diode like this. so it's easy to drive this transistor with a low side driver and get this noninverting buck-boost type characteristic. Here's another converter, is the inverse of the SEPIC. It's also a noninverting buck-boost type converter. some people call this the zeta converter. then there's a lot of other very different converters. So some of them have what we call quadratic conversion ratios that are functions of the duty cycle, squared. Here's one of those that was in an earlier homework assignment that has a conversion ratio equal to D squared. and it's interesting that you can get such a conversion ratio in a converter that has three diodes and only one transistor. We can get quadratic conversion ratios that have other functions also like buck-boost squared or other types of conversion ratios by some of the members of this class. So there are some very interesting and exotic conversion ratios that are possible. In general you know, if you have an arbitrary number of inductors and switches, we can build there are an infinite number of converters that are possible. And in this universe of converters, there are many different and unusual functions of duty cycle and some of these can can have practical applications.