Continuing our review of Bode plots, here are some more standard terms that often come up. First one we look at, is the single zero. This is a polynomial that has a root, that we can view as a root of the numerator of the transfer function. So, this term actually is one over the term for the pole that we discussed in the last lecture. So we can go through the same arguments as for the simple pole to find which term dominates at high or low frequency and derive the asymptotes. But another way to see it is simply to consider what happens to the magnitude and phase of a complex number when we invert it or take one over the number. So suppose we have some complex number, G of j omega. That at some given value of omega, has a magnitude and a phase, which we can actually express in polar form, as some magnitude I'm going to call R, times e to the j theta where theta is the phase. Then what happens to 1 over G of j omega? Well, 1 over G will be 1 over R e to the j theta, and this has a magnitude of 1 over R, and it, it has a phase, phase term we can write e to the minus j theta. So the phase of this term is minus theta, or minus the phase of the original G. So, 1 over the complex number gives us a phase with a minus sign, and for the magnitude, when the magnitude is expressed you know, in decibels using the log. This is, can be written as R to the minus 1 power. So we get minus the number of dB. So 1 over G to, to do that we end up putting a minus sign on both the dB magnitude and on the phase of degrees. So as a result then here are the, the asymptotes for the, the zero. With the magnitude asymptotes and the phase asymptotes, they look just like the, the pole asymptotes but with a minus sign. So at high frequency, the magnitude asymptote has a plus 20 dB per decade slope, rather than minus, and for the phase at high frequency, the phase asymptote is plus 90 degrees rather than minus. And the intermediate frequency asymptote has a slope of plus 45 degrees per decade. But otherwise, all of, we have all of the same rules for constructing the asymptotes as we have for the pole. Here's another term that is important in power converters, we are going to see that it. in some upcoming lectures, that this kind of term comes up in boost and buck-boost converters, as well as some others, and that it causes some major problems and headaches in our control loops. It's called the right half-plane zero. It looks just like the zero, except for this minus sign here. We call it a right half plane zero, because the root occurs where s is a positive real number, or has a positive real part. Because of this minus sign. Now what does this do to our bode plot? First of all what does this do to the magnitude? If we let s equal j omega and take the magnitude or, find the sum of the squares of the real and imaginary parts, we find that the minus sign doesn't affect anything, and we end up with the same expression for the magnitude as in the, the more conventional left half plane or real zero. However, the minus sign does change the phase. When we take the arc tangent of the imaginary over the real part, the minus sign comes into play, and so we got a minus sign here, in the expression for the phase. So, the right half-plane zero has magnitude like any other zero, but its phase is not like a zero, it has another minus sign, and looks like a pole. So here are the asymptotes for the right half plane zero, with magnitude like at any other zero. But phase that looks like the phase of a pole. The next two terms are ones that we don't have to use, but they can be useful in many applications. Give us a, a more physical way to write the transfer functions of some some kinds of systems. And this trick is called frequency inversion. It's well known in the classical filter design field, where we, we invert the frequency axis. So wherever we had a term that was like, say, s over omega naught, we turn it into a term that goes like omega naught over s. Now our logarithmic frequency axis. What this does is it makes the frequency axis operate in the opposite direction. It's like a minus sign on the log. so this is a well known transformation in filter design that can turn a low pass filter for example into a high pass filter. Okay, in the case then of say, this term illustrated here which is an inverted pole. The regular pole will look like this. And what frequency inversion does is it flips the frequency axis around so that our term looks like this. So we have a flat gain of zero dB at high frequency. And then as we decrease in frequency below omega naught, the gain reduces or rolls off. Now, you don't ever have to use these kinds of terms. For example, you could multiply top and bottom here by s over omega naught. And if we do that, we'll get s over omega naught in the numerator, and the denominator will give us s over omega naught plus 1, which we can write like this. So this, this alternate form has no inverted terms and is completely equivalent to the in, the frequency inverted term. But the nice thing about the frequency inversion term is that it gives us this interpretation that we have a high frequency gain of one or zero dB and then we roll off at low frequency. And in combination with other terms, this can be a nice, useful thing. We'll talk about that in the next lecture. And [INAUDIBLE] examples. So here are the asymptotes in summary form for the inverted pole. Likewise, we can invert any of the, our other terms. So for example, an inverted zero is found by applying the same transformation. So if our regular zero has asymptotes in magnitude that look like this. Then frequency inversion will turn them into a high frequency flat asymptote and a low frequency asymptote that increases as we reduce frequency. And again, we can write this in, in algebraically equivalent form like this, that does not use inverted terms. But, we, we are interested in emphasizing what the high frequency asymptote is, this is a good way to write the expression. And here is a summary of the asymptotes for the inverted zero. Okay? So, we've gone through all of the basic real pole and real zero type terms that we're going to use. in the next lecture we're going to put them together and construct the larger transfer functions and their frequency responses.