So we will actually do go through a design of a 2-section input filter, and a particular form of damping is selected right here. Obviously you can have any one of these damping strategies employed in individual sections. Or we can even mix and match different damping strategies in different sections of the filter. In this case here we'll go for the r l parallel damping in both sections 1 and 2. It is the same example that we did before with the single section. So the requirements are the same as before, we would like to have 80 dB of attenuation at 250 kHz. What do you thing 250 kHz actually is actually coming from? Why 250 kHz? That's switching frequency of the converter, so this is our f sub s and we want to have a certain amount of attenuation of the switching frequency ripple. And of course harmonics of the switching frequency ripple are going to be attenuated even further. The Zn and Zd, that's what we were looking for right here, those are converters Zn and Zd. They're shown right here. And when we designed the single section filter, remember that we are looking for the corner frequency of the filter summary in this range right here. We want to stay away from the resonance of the converter itself and we move that corner frequency of the single section filter to the left of that resonance. So this is where the single section, this was the corner frequency of the single section design. And you see how that makes sense, the further down you move the corner frequency of the filter, the higher the attenuation and higher frequency is going to be. And so if you need to meet the 80 dB attenuation with the single section you have to move that corner frequency low enough. So you want to avoid this region right here and have this frequency low enough so that you meet the attenuation requirement. What do you think will happen with a 2-section filter? We still want to avoid this resonance right here. Where do you think the corner frequencies of these sections, 1 and 2, are going to reside? You could, of course, put them right here. That would be fine in terms of staying away from the resonance. So we stagger them, right, so we actually stagger them right here. So I put these two frequencies away from each other, we do want to stagger them. Or I can put them right here, stagger them. But I'm still thinking of them being on the left hand side of this resonance that we have right here. Why do you think this is generally not a good place, it's not perhaps necessary for the 2-section filter. The primary reason for going for multiple sections is to reduce the size of the components which means individual section corner frequencies are going to be higher. We can work with higher individual section corner frequencies and still meet the attenuation requirements because we now have higher order filter, we have two sections. The actual position, as you will see as we go through the numerical example, is going to be somewhere around here. We will be placing the corner frequencies of the additional sections to the right of the resonance of the converter. And that's really enabled by having a higher order filter in place with two sections available for achieving the same attenuation as we did with the single section earlier. So that's the main benefit of a 2-section design. We can have a higher corner frequency and smaller component values per section. We actually need to meet this requirement of 80 dB attenuation. The way we think about that is that we say this total attenuation of 80 dB can now be split into two parts. We can, for example, go for achieving 80 dB of attenuation by having each of the two sections have 40 dB of attenuation. That's one possibility in this design. Why is this not a good choice? This would result in ff1 and ff2 being the same. And the same corner frequencies of the two filters will make it difficult to meet impedance criteria when adding section 2 to section 1. Just as we tried to avoid the resonance of the converter when we add a single section of the filter, here we want to avoid the resonance of section 1 when we add section 2. Now of course, the other extreme choice would be, put everything on section 1 and nothing on section 2. And that gives us also 80 dB of attenuation. That's not going to be a great choice because that means you take out section 2 and you're back into section 1. So the choice that we have in the multiple section design, the first choice, is really how to split this attenuation between the two sections. And yet again this is not a formula, we are going to give a suggestion here, for example we will say that the first section is going to be 45 dB plus the second section of 35 dB and that's going to total. But this is not the only choice. So this is a particular choice that we can follow. This choice is not unique, you can try out different values. In fact, I'm sure you can actually expand this into optimizing the overall filter design by treating the split between the attenuation as an optimization parameter. And you can formulate the optimization goal in terms of total volume of components, and so on. But you can see a reasonable engineering choice is here made by really splitting these by let's say about 10 dB. So we have good separation between the corner frequencies of the two sections. So we'll take that as an example. So section 1 should be designed for attenuation of 45 dB and in this version of the damping network we have around section 1, we have the degradation of the high frequency attenuation given by this factor 1 + 1 over n. It's a way to make an engineering choice of what n is going to be, in this case here, I'm going to choose n = 0.5, 0.5 value means that we are reducing the attenuation by 1 over 0.5, that's by 3. That 3 comes out to be 9.5 dB of reduced attenuation of the filter response at high frequencies. So Lf, Cf should be selected to achieve attenuation of 45dB plus however much we've lost because of the damping, 9.5 dB, so that is going to require attenuation of 45.5 dB from the Lf part of that filter. The filter itself, the Lf, Cf. So what is the high frequency attenuation of this filter? The high frequency attenuation is simply going to be equal to fs over ff1 squared. fs is the frequency that we are interested in. That's the switching frequency of the conductor. The ff1 is 1 over 2 pi square root of Lf Cf. How do we get this, well at high frequencies it's the second order filter, it's minus 40 dB per decade. Attenuation goes down with the square of the frequency and the attenuation achieved is going to be just the ratio of the frequency we are looking at to the corner frequency ff1 of the filter. And we want that to be 54.5 dB, that means the ratio of fs over ff1 of, so based on this we calpulate ff1 is 10.8 kHz. All right, so at that point we have the corner frequency of the filter and what we can do next is to choose magnitude of Z output 1 Mm to be sufficiently smaller than either Zn or Zd. And at that point here we would probably need to look at Zn and Zd again, and look at the interaction possibly of that section 1 filter with Zn or Zd. I think on the next slide they have Zn and Zd, that's it, here we have Zn and Zd, notice that we are now on the right-hand side of that converter resonance. And we are looking for the introduction at this point right here, this is going to be our fm1 and this is going to be our magnitude Z over 1mm. For the optimized design of the filter, we are going to have the peak value of the output impedance of that filter when it's standing alone should be sufficiently lower than either Zn or Zd. In this case here, it's really just Zn that matters right here. Although here you might want to take a look at this point right here as well. The choice that we make right here, again, is somewhat an engineering decision. So we put this to be 3 ohms. So remember this was 12 ohms right here, so, we have the ratio of about 3 over 12, 1 over 4, in meeting that impedance criteria. Choose that to be equal to 3 ohms, again an engineering choice, not necessarily unique, you may have chosen a somewhat higher or lower value. And once you do that then you have the cookbook step by step procedure on finding the optimum damping. All right so that's the section 1 design. Now we are going to go to section 2 design and the section 2 design requires 35 dB of attenuation at the same switching frequency, so yet again we have to make a choice of n, we will again pick n = 0.5. For n = 0.5, we need another 9.5 dB of attenuation. So Lf and Cf for section 2 need to meet attenuation of 35 + 9.5, that's 44.5 dB. And so yet again you're going to go by, remember that for ff1 we had 10.8 kHz, so you see this staggered tuning we have here spreading the position of the corner frequencies for the two sections of the filter to try to make it easier to make the impedance criteria for the second section. Next we need to select a Z output 2 mm which we call Za, and it's going to be mm value. So, we select this value here to meet impedance criteria with respect to section 1. That means with respect to Zn1 and Zd1 and so we need to find out what Zn1 and Zd1, plot them and then overlay the output impedance of section two to that. Here's the Zd1 and Zn1 overlayed. You see them here. This is Zd1 right here, this is Zn1 right here. And you see this dip in Zd1 occurs at the corner frequency of the first section. So right here you see this point right here, being this fm1 frequency of the first section. So, now Za is the output impedance of the additional section. And you see that if we were to position the corner frequencies to be on top of each other we would have much harder time meeting that impedance criteria right here because we dip with Zd1 at that point. So fortunately we've done this well and so we've moved the current frequencies of the second section up, and so we have a chance of meeting that criteria right here, although it's not quite convincing as we had before. So if you look at this part right here, the choice that we have made for this Za mm looks a little bit questionable, right? So we have, this is in a 1 ohm. This is 0 dB ohms. And when you look up right here you see that the distance between that value and what we have in Zd1 is just maybe a couple of dB. It's not convincing difference between these three impedances. And so at that point here you may look down the phase and see what we have as a phase difference between Za and Zd. So we have maybe about 45 degrees or so and then you consult that impedance of the chart that shows the contours of constant values of this part of the correction factor. And if we have 45 degrees and we have here maybe a few dBs, going to be right here and right here, we see that we are going to have some impact on that correction factor. Right, so we are not completely decoupling the two sections of the filter, they do have some interaction. And we will typically employ simulation to figure out whether exactly we have the nice response of the overall output impedance of the filter with the additional section edit. So we'll go with that, so we go back here and we say all right, we chose the za mm = 1 ohm and then from that point it's just calculation through the formulas given you have the ability to calculate all components in section 2. Then ultimately once you're done with this, since there is quite a few points where they made these engineering decisions, they're not coming from formulas, right? They're making from your view of what you think is adequate. Ultimately we will resort to checking the overall filter by simulation. So let's get to the final result right here. When you look at the overlay of Zd and Zn, now with the output impedance of the filter including both sections. We see that we are meeting the original impedance criteria quite well. So this is the solid line right here is with cascaded sections 1 and 2. This is the curve that we have right here, and we have a lot of space between that and Zn or Zd in the converter. So the overall filter with both sections included meets the original impedance criteria very well. Put that into a simulation tool and find out what the transfer function of the filter actually is. By the way, what is the transfer function we are looking at? This is our filter, and this is the output side of the filter, this is the input side of the filter. What is the setup for actually finding the transfer function of the filter. There are two different setups that are going to give exactly the same results. One is the setup where you have the input voltage right here. You may say this is Vg hat, and this is the Vo hat, And H is equal to Vo hat over Vg hat. That's one setup. The other setup which is really what this filter should be doing, is the setup where we have H box right here. We have the output Io hat, this is presumably the current that we are trying to attenuate. And we are looking through the current on the input side, that's Ig hat. And so it turns out by reciprocity these two transfer functions are exactly the same, so whether you look at it from the voltage input to output or for the input current versus the output current you get exactly the same result. Either one of these two are equivalent ways to looking at the transfer function of the filter. So you look at the transfer function of the filter, and we are looking for the attenuation of 80 dB at 250 kHz, this is f sub s of 250 kHz. And we verify that we have the gain of the filter of -80 dB. That's att = 80 dB. It's attenuation with respect to the low frequency where we have 0 dB gain. Let's just quickly review before we wrap up for today, the component value. So, look at the component values in the single section filter. So we have 330 microfarads, 470 microfarads. Pretty large DC blocking can pass through right here. If you look at the component values in the two-section filter you see much much lower values. That's the benefit of multiple section filter. Same attenuation with much smaller values of components, now of course there are more components. Right, so what stops us from adding another stage right here, nothing. It is entirely possible to extend this procedure to a third section and then you're going to be talking about even higher individual corner frequencies for those sections, which means even smaller individual components. But the number of components goes up and so at some point you decide what is a good engineering choice in that trade off.
