We want to really go for actually putting everything together. We'll do that for a numerical example. It is given right here. The values are given here for the input voltage, output voltage, the two capacitance values, two inductance values and the load. That is the model of the SEPIC that we've solved using the extra element theorem. One thing to note is that in a practical circuit, you would often see capacitance C_1 being relatively smaller than the capacitance C_2. C_2 is at the output, C_2 is going to be really sized for a ripple considerations and perhaps transient responses. Whereas C_1 is internal, we don't really care that much about how much ripple we may have across C_1. It is often observed that in cases like SEPIC or core chuck, you'd have this energy transfer cap inside being relatively small value. You still need to size it properly so that it is capable of carrying the RMS current, which is significant in this case here, but beyond that, usually we prefer to use a relatively smaller value for C_1. The results that you obtain from the extra element theorem, this first level of extra element theorem was looking into the extra element impedance. I remind you the first level, we first got rid of this C_1 as an open. Z itself, 1 over sC_1. That is shown right here as a straight line. The magnitude response to that impedance is just 1 over Omega C_1, and you see that as minus 20 dB per decade straight line in the magnitude response. Now, these are the Z_N and Z_D that we just obtained. Remember we actually did the extra element even to find Z_D, but once we have Z_D now in place, we see that it actually has this inductive behavior at low frequencies, followed by a pair of zeros and a pair of poles. On the other hand, Z_N has the same low-frequency inductive behavior, but then it's followed by right half-plane zero and right half-plane pole. When you plot the magnitude responses of Z_D and Z_N at low frequencies, you see this part right here, they actually match. They have the same values at low frequencies and follow Omega L_1 plus L_2 response. At high frequencies, what is the nature of the impedance is Z_N and Z_D. Right here, what can we say about this ultimate asymptotes of Z_N and Z_D. They're still going to be inductive. Are there going to be the same? Not necessarily. How do we find out what they are, one of the inductive behavior is at very high frequencies, while you take these terms here and neglect one compared to those, and you look at the Z_N. S's are going to cancel out and you will have Omega Z over Omega Z_N. In terms of the N impedance, Omega L_1 plus L_2 times Omega Z over Omega Z_N. In terms of the Z_D behavior, you do exactly the same thing, and you're going to have the behavior that looks like Omega L_1 plus L_2 times Omega naught squared over Omega Z_D squared. Those are again inductive straight lines that are increasing up at plus 20 dB per decade, but they have different values. They're not necessarily the same. In between, you have different things going on. There is right half-plane zero and pole are relatively close to each other. If you look carefully at the impedance of the Z_N, you won't see much there. It looks like there is a maybe a pole and a zero right here. Calculate the numerical value and see where they are. They don't really make much of a difference, it would be hard to tell. This is just not a straight line everywhere. On the other hand, Z_D does have this behavior right here. What is this? This is a pair of resonant zeros followed by a pair of resonant pulse. They're relatively close to each other, but they have this bump up and down in the magnitude response of Z_D. These are overlays of these impedances, Z, Z_N and Z_D. Why do we actually put this overlay together? Because the correction factor has the form of one plus Z_N over Z over one plus Z_D over Z. This is the correction factor that we are interested in. At low frequencies, correction factor is going to be approximately one. We can say right away that at low frequencies G_vd, the transfer function of the complete circuit is going to follow that transfer function of the buck boost like equivalent circuit model. How about high-frequency behavior? High-frequency Z is much smaller than Z_N or Z_D. What is the approximate expression for that correction factor going to be, Z_N over Z_D. What is Z_N over Z_D? However, both are inductive and so the ratio of the two is going to be a constant value. At low frequencies we have no impact on that buck boost transfer function. At high frequencies, the impact is reduced to just a scale factor, just the ratio of Z_N over Z_D, which is a constant. The interesting stuff happens here where these impedances are comparable in value. What is going to happen when we have this range of frequencies where the magnitude responses of Z's are similar. That's where we expect these interactions to happen, and going a little bit ahead, we can actually see how that is going to look like. This is now plotting the old transfer function and the complete solution for G_vd with the correction factor. This is the old, and the old looks like a familiar two pole with high Q-factor followed by right half-plane zero so that G_vd-bb is going to have this familiar look that we see for the buck-boost type converter. The solid line here shows the magnitude and phase responses for the G_vd that's going to include this sharp spike right here, and is going to include a substantial phase lag in addition to what we see in the buck-boost response, look at the phase response for a second. What is the final phase asymptote in the buck-boost case, where's this going? Because we have two poles and right half-plane zero. What happens in the SEPIC? It's going to be minus 630 degrees. That's going to be a lot of phase lag. In addition to what we start with, two poles and one right half-plane zero. We'll have additional two poles and additional two right half-plane zeros. You have really a combined effect of another 360 degrees of phase like. When you look at these responses right here, you can see why we said at the beginning that SEPIC in this form as given, is difficult to control because we have this high spiky resonance in the magnitude response, and the huge phase lag. It is very difficult to close the control loop anywhere except for some low frequencies well before this.
