Damping off this particular type of filter, the parallel R and C placed in parallel with the filter capacitor. If the additional DC blocking capacitor is very large in value, then the damping is very simple. At the corner frequency the center frequency of the pair of poles, you will simply have infinite impedance of the parallel combination of L_f and C_f. So if you look at this here and you say, what is the output impedance that we see here, you will see that this part at F Naught goes to infinity, and if C_b is very large, that implies that z knot is going to R_f at the frequency of F_f. F_f is the center frequency of the pair of poles in the filters. So with large C_b, damping is very simple, you choose R_f so that the peak value or the output impedance meets then the design criteria. So we would like to choose R_f to be much smaller than Z_n and R_f to be much smaller than Z_d. This is the large C_b design. We'll revisit the choice of C_b in just a moment. But at this stage right now, we are looking into the case where C_b is just chosen to be relatively large value, it blocks DC voltage, it makes it so that the R_f is not going to disobey significant amount of power. But it's large enough so that AC wise we can consider it a short-circuit, which is why we have this super simple picture and the design approach in the choice of the damping resistance. Now, related to these statements right here that we want to have these inequalities satisfied, one question that comes up is, how much? We choose R_f to be 1 over 1,000 times Z_n, or is 1 over 10 enough? Or maybe that's too conservative as well. So there is a question of how do we actually choose the value of R_f in terms of meeting these criteria. But let's here look at a particular case where R_f is equal to one ohm, that gives us the magnitude of the output impedance and the F_f frequency equal to zero dB ohm, so that's what we have right here. You see that we have, apparently from this diagram right here, met the conditions that the output impedance of the filter is significantly smaller than either Z_n and Z_d. Well, how much smaller really? Depends, of course, on the frequency you're looking at. If you look at this particular frequency right here, you see that at the frequency F_f where we have this peaking in the magnitude response over the output impedance, we are down by about 20 dB. So 20 dB less, which means R_f is about one-tenth of what the Z_n or Z_d are. In fact, if you look at the actual value of Z_d and Z_n, you see 12 ohms right here, and we have one ohm right here. So we're meeting that condition by about 20 dB. But in general, you want to be careful about this evaluation because there may be other frequencies where we have interaction that is not directly affected by the damping. In particular, you see right here that the distance between the Z_d in this case and the output impedance of the filter is in fact smaller in dB compared to what we have at the F_f frequency, the peak of the output impedance. So as a general rule when you engage into a filter design, you want to observe the overlay of the impedance plots over a range of frequencies and be careful about seeing were these introductions are potentially the strongest. So back to the question of, how far should this be? Is this actually good enough or not? Are these distances here sufficiently away from Z_d or Z_n so that we don't disturb RGBD significantly? So the question is really, how much smaller should Z_0 the output impedance of the filter B compared to either Z_N or Z_D. That's when we come back to these general plots that give us quantitatively what is the effect of a certain ratio of impedances on the correction factor. So let's review again what is actually shown right here. So in this particular plot, we have contours of constant one plus Z over Z_D. Remember this is a denominator in the correction factor. So this here is the denominator in the correction factor, you would have exactly the same results for the numerator. So looking at the denominator of the correction factor, you would like that to be close to what value, if you don't want to disturb GVD? One. One. So you would like this to be equal to, ideally one or close to one should be relatively close to one, and see 0 dB would be this contour that would be perfect. It would make no difference whatsoever in the GVD response. Because the GVD, would not be affected at all by the presence of this correction factor. Now, how exactly is that correction factor is going to affect our result is going to depend on how large is the ratio of the output impedance of the filter. So here known as this is really Z_0, Z_0 right here compared to Z_D or similarly Z_0 over Z_N. The same really graphs apply to that ratio as well. The impact of that on the correction factor is going to depend on the magnitude of that ratio and the phase of that ratio. So this chart gives us a general view of what happens. Let's suppose the phase between Z_0 and Z_D is equal to 0. Let's suppose we are right here. How does that actually translate into what we have in this diagram right here? Well, could we say that the phase between Z_N or Z_D and Z output is close to zero. That would be right here. So this point right here, output impedance of the filter looks like just a damping resistor. It's not exactly zero, but it's in the vicinity of zero. So we are looking at that particular point, now on this graph right here, and then looking into what is the impact of the choice of different Z out over Z_D or Z out over Z_N ratios in magnitude on the correction factor. So let's suppose we have chosen 20 dB, 20 dB being Z_0 over Z_D. The difference between these two being about 20 dB. That means the magnitude of Z_0 over Z_D would be 0.1. In fact, we actually have a little bit less than that in the example that we are considering. Why is that observed right here? Well, if you go down here to minus 20 dB right here, and you now look at this intersection right here, you will see that the correction factor has a value that is less than in magnitude of one dB. So the magnitude of 1 plus Z_0 over Z_D is going to be less than one dB. So we have a measure of how much of an impact this interaction is going to produce on the correction factor and thereby on the GVD transfer function that we are interested in. You can look, of course, with respect to the Z_N as well. In this case, it would be exactly the same because Z_N and Z_D are the same, and that frequency f's above that we are looking at. Is this enough or not? Well, one dB is a very small change. It's not going to make much of a difference in the magnitude response. You could in fact be certainly less conservative and go for less stringent meeting of that Z_0 over Z_D. For example, if you choose Z_0 over Z_D in magnitude to be as high as 0.4. That's minus 8 dB. Let's look at that as an example. If you take this one right here, that's a ratio of 0.4. Minus 8 dB would come in right here. If you go this way, you will see that it will have an impact of about 3 dB on the correction factor and as a result about 3 dB on the G_vd transfer function. You see that the explanation here of how small this ratio should be is really not a given single number. I'm not going to now say, "0.1 is always good, don't worry about it," because you may want to look at the phase responses as well and see what the impact on the phase is. It is really understanding the nature of the interaction and being able to use these quantitative plots to evaluate what small or large may be in that comparison between Z_naught and Z_N and Z_D. It is somewhat like a phase margin. What's a good phase margin? 53.5 degrees. We don't say that we understand the relationship between the phase margin and the quality of the response of the closed loop system and we make use of that in the engineering practice. Similarly here, we want to make Z_naught over Z_N and Z_naught over Z_D much smaller than one in magnitude. But how much smaller in one depends on from case to case and you make an engineering decision in that process. The reason it is an engineering decision is that the smaller you want to make this, the larger the components are going to be. You will have larger passive components in the filter, if you wanted to make this be 1 over a 100 or 1 over 1000, you're going to end up having huge components in the filter circuit. The trade-off in the engineering sense is going to be in meeting the design conditions by using relatively small values of components. Good question. We were looking here at this particular point where we think the phase between the impedances about zero degrees. But let's suppose you look at other points. For example, this is an interesting frequency here, f_naught. What is the phase of the, let's say, and the one that we are really looking at is Z_D. Z_N is away from it. Z_D is an interesting point. Z_D deeps here to a minimum value because of the relatively high Q factor of the filter in the converter itself. Then we are certain number of dB's away from the output impedance of the filter. What is approximately the phase difference between these two impedances? It's about 90 degrees because this behaves capacitively. This is approximately resistive. So we have about 90 degrees of phase difference between these two. Now, if you look at the 90 degrees right here or here, it doesn't matter, it's completely symmetric. Whether it's plus or minus 90 degrees, it makes no difference. You will see that again for, Iet's say it is minus 20 dB and minus 90 degrees, it maybe less than this, maybe it's minus 15 dB and 90 degrees, you have close to 0 dB for the correction factor. Here is this particular example. We have the original filter being 330 microhenries and 470 microfarads. Those components are selected to provide sufficient attenuation or the switching ripple. They're also selected so that the center frequency of L_f and C_f, the ff frequency is smaller than f_naught of the converter filter. Then we choose a large value of C_b. Here, it is equal to 10 times C_f. Right now if we just choose a large value for C_b and then simple choice of one ohm for damping. The end result is very small effect on the magnitude and phase responses of G_vd.
