The last section of the input filter design topic is the fact that very often, we design input filter that contains multiple sections. So, so far, we've only discussed a single section filter, and L-C network basically. So we can, in principle and in practice, we often do have multiple L-C filter sections to achieve a given high-frequency attenuation with smaller component values. We'll go through an example and show by comparison of component varies how much smaller multiple section filter can be compared to a single section filter. But then the question is, how do we actually design a multiple section filter? And we will talk a bit about a particular approach that extends the discussion that we had so far, it's going to employ the next element and yet again, and it is going to insist on properly damping each section of the filter. I should mention that this is not the only possible approach. You will see a variety of techniques in practice, some of which do not necessarily damp each individual filter section separately. We think that the damping of each section is advantageous, because it prevents any internal resonances that you may have in the filter, exposing components to high stresses in case of disturbances in the input voltage, or converter input currents. And so we think it is a good practice really to damp properly each section of the filter and the design procedure that we can follow right here is based on that idea. So the basic approach is to just add sections to the filter using the same criteria or similar criteria based on the extra element theory. So you say it right, I'm going to design section one first, the one that's connected to the converter itself, and I do that based on the impedance criteria. Then, yeah, that the next section based on not disturbing, not changing the output impedance of the first section. So yet again, another round of application of the extra element here. One important point that you will see, and we'll try to emphasize into discussion here is that we are going to what is called stagger-tune, the filter sections. I'll explain that more as we go o, but stagger-tuning means that we will choose the coordinate frequencies of individual filter sections to not coincide with each other, to be spread out. Okay, so stagger-tuning means that the coordinate frequencies of individual sections are spread out in frequency. We'll explain why that is the case. So, how addition of another section affects the existing filter section? So here is the setup, this is where the converter is. And we are now concerned about the output impedance of the first section, so this is what we will be called, section one. So we have designed section one, so that its output impedance does not affect Zn and Zd. It's not interacting with Zn and Zd of the converter, fine. Then we say, we need to add another filter section. And now we are concerned about how the output impedance of that filter section, which is Za, affects the output impedance Z naught of the existing filter section one. So this is additional section two, just for notation. So modified Z output, this is Z output of the originally designed section one, is going to be the output impedance of section one when the additional section is removed, meaning that Za is equal to 0. That's what we have right here, times the correction factor following the extra element here, whether ZN1 and ZD1. Those are the input impedances of the existing filter under the conditions that either the test here is nulled or that i-test is nulled. So the setup is, where the test of the output impedance Z naught is v-test over i-test, and the two impedance is of interest, ZN1 and ZD1, are the impedances that we see at the input of section one when v-test is nulled, that's the first one right here, that's Zn1. Or when i-test is equal to 0, and that's the Z input of section one. Now, one thing to note here is that the transfer function of interest is Z naught, right? Zn is requiring null link of v-test across the very input to the transfer function. So i-test is the input, and null link v-test, in this case, in this particular case, and only really in this case, actually means is equivalent to shorting the output. So this situation here where we have v-test nulled. And the input source itself is across that nulled point, that's equivalent to having a short across v-test, so this is the same as short. One particular case where null link is equivalent to a short is shown right here. Carries the, again, the summary of what the Zn and Zd1 are. So Zn1 is the case where we're looking at this input impedance right here where the output port of the filter is short circuited, and Zd1 is the input impedance with the output filter open circuited. And so what we want to achieve, the design criteria for section two is that the output impedance of that section two meets the impedance criteria with respect to section one, and that's these two. So we will be looking into Zn1 and Zd1 as the input impedances of the existing filter section one, and use the impedance criteria to design damping for the additional section two. And now you can see how that can be extended to section three, four, five, and how many you have. Obviously, adding more sections is going to result in smaller and smaller component values per section but larger and larger number of components, so yet again, we have an engineering trade off in that process. Typically, would see two or three filter sections using practice