Now we're going to start talking about the Techniques of Design-Oriented Analysis, in particular, The Feedback Theorem. So it is a material that I think you will find quite interesting and useful in the design process. It has a name associated with it. The feedback theorem is attributed and it's really contributed by Professor Middlebrook at Caltech. It is a general approach to analyzing feedback circuits and systems, as long as those feedback circuits and systems are linear and time-invariant. When we say general, it is a method that is capable of performing that analysis of feedback circuits in the cases that do not fall into the standard control theory block diagram forum. In real circuits, it is very often difficult to actually put a circuit into a block diagram form. Sometimes we can do it, sometimes we can't. The feedback theorem allows us to actually do very quickly and relatively easily Analysis of feedback circuits that don't necessarily form an easily identifiable blog diagram. We can also allow the possibilities of having arbitrary input and output impedances, and we can also allow for the possibility of having the signal flow being bidirectional, as really is the case in real circuits. The derivation of the feedback theorem is based on the linearity assumption about the system, and it's going to apply techniques of superposition and something that you may have not seen, it's called the null double injection. I said at the beginning, the feedback theorem applies more generally to cases where we are not able to necessarily easily identify a block diagram. But when you do the derivation and you try to do that in a general sense, you'd have no choice but to resort to a block-diagram as representations. So that's a little bit of a contradiction right here. So notice that the starting point that we have in the discussion on the feedback theorem is in fact a block-diagram, and we can identify a single feedback loop in that block diagram. Here's an input signal, here is an output signal. In the input to output signal, we have what we call a forward path, and forward path would be generally processing the error between the input and the sensed value of the output signal, whatever that may be, and that processed error signal is then ultimately passed to the output. In that block diagram, which I said is really an attempt to represent this in a general manner, we also have importantly a point that if we're going to call an injection point. So we'll be looking at this block diagram right here as a generic form of a feedback circuit we would like to solve. What do we actually want to solve? When you have a feedback circuit of this type right here, what is the primary transfer function of interest? What is the ultimately the end goal of any analysis of that system? Output over input. That's what we ultimately want. We want to find out a way to find what is the output over the input, and that goal is what we are going to keep in mind as we go through the feedback theorem. Couple of important points about this injection point: So we say there is a an injection point that we can refer to as being ideal. Ideal injection point includes two aspects. One is that the signal u_y, the signal right here at this point right here, if you wish, to the left of the injection point, is directly proportional to the error signal. So it is directly proportional to the error signal and it's not directly affected by the injection itself. In the sense of measurement of this gain, you have seen this particular type of setup when we use the injection signal to enable measurement of the loop-gain. In those sections 9.6.1 and 9.6.2, we noted that that injection signal can be a voltage injection signal or a current injection signal. Indeed that is the case, that's going to be the case in the application of the feedback theorem as well. The way we put that in a more general form is to call this u sub z. So u sub z is in a genetic sense and injection signal could be a voltage type or a current type. Second aspect to the ideality of the injection point is that there are no alternative signal paths through the forward portion of the feedback loop, other than what is shown right here. More specifically, if you set u_x signal equal to zero, then none of the error signal is going to reach the output. So those are two characteristics that we associate with the ideal injection point. So number 1, u_y is directly proportional to the error signal, and second, there are no alternative paths, there is nothing that is bypassing this right here or going down right here. The forward path is as shown right here and there is no way for the error signal to reach the output other than going through the path that is shown right here that goes through the point where the injection is actually met. So again if you u_x is equal to zero, the output would be unaffected by the error signal. Let's point to the fact that this is the system that has two independent inputs. Those two independent inputs being the input signal that's referred to as u_i, and the injection signal that is referred to as u_z. So those are two independent inputs. The system has three outputs of interest. One is the output signal itself, and the other two outputs of interests are going to be this u_i and u_x. So in a block-diagram form, you think of the feedback system to have is two inputs; the actual input and the injection, and has three outputs of interest. Those three outputs being the output signal itself, and this y and x signals around the injection point. So now we're ready to actually just state the theorem result and then we're going to derive it. So the feedback theorem and main result goes like this. So we said the main purpose is to find the overall closed-loop transfer function. The way the feedback theorem gives that result in the following form. It says, the overall transfer function u naught over u_i, the output signal over the input signal, is equal to this expression G infinity,T over one plus T, plus G naught over over 1 plus T. Great. But what are these individual values inside? So T is the loop gain. So if you have a look at the definition of the loop-gain is the ratio of the y over x signals u_y over u_x, when the input signal is equal to zero. This is definition of the loop gain. In fact it's exactly the same definition that we used earlier when we examined measurement techniques for feedback systems. So again, the loop gain is u_i or u_x when the input signal in general could be voltage or current is equal to zero. What are the other elements in this expression right here? A particularly important element, and the one that is very easy to give an intuitive interpretation to is called G-infinity. So let's look at the G-infinity term right here. So G-infinity of S is defined again as output over input. But not when the input signal is equal to zero, but instead when the u_y signal is zero. You'll be careful to notice this symbol right here, it's called null symbol. We are basically saying this is output over input when the u_y signal is nulled. We'll be careful to distinguish between having an independent signal equal to zero, or dependent signal being nulled, those are two very different situations. So independent signal can be zero or non-zero, a dependent signal can be nulled or not nulled. So nulling the u_y signal is the condition under which we evaluate output signal over the input signal, and that gives us the quantities called G_infinity. The interpretation of that G-infinity is is that it represents the ideal forward gain. Why do we say that? If you go back to the block-diagram right here, and you say, "What happens if we null u_y?" If we null u_y, that implies that the error signal is nulled as well. If the error signal is nulled as well, that means we have perfect regulation, there is no error between the input signal and whatever is the sensed value of the output signal. So nulling u_y by this property of the ideal injection point implies that the error signal is nulled, which implies perfect ideal operation of the feedback system. In fact in many cases, we design the feedback system having this G-infinity, having the ideal response in mind to start with. So G-infinity formally is defined as u_o over u_i when u_y is nulled. Now, there is another interesting G component right here that actually showed shows up right here. It's called G naught. So G naught again is the ratio of U-naught over u_i. So again, output over input, but when you u_x signal is nulled instead. Let's see what that really means in the block-diagram. So we are saying, all right, what is u_o over u_i, when u_x is nulled? How can that be non-zero? We already said in ideal injection point will be such that having u_x equal to zero is going to have no error signal propagating through to the output. So if that's the case, if we null u_x, how could we possibly have anything at the output showing up at all? Why is this not just zero? Why is this G-naught, not zero right away, why do we bother with it at all? Because in real circuits, it is entirely possible to have signal propagation in the opposite direction. That signal propagation can actually give G-naught that's not equal to zero. That's what we can evaluate how exactly large that G-naught is and what it consists of is again formally defined as the ratio of i_o over I input when u_x is nulled. Then there's finally yet another point and it actually doesn't show up here directly in the expression that we are showing right here, it's called the null loop gain. Null loop gain, similar to the actual loop gain, is u_y over u_x, but under the condition that the output is nulled. So you see this actually looks a little bit of strange. We say what is the loop gain when the output is nulled? It does not have so easy intuitive interpretation as the other three quantities do have. But there is an interesting relationship that's called the reciprocity relationship that actually links these four quantities together. So we will find out that T_n over T is equal to G- infinity over G-naught. You will see that in some cases, is it in fact actually easier to find T_n as opposed to T, and we can use the reciprocity relationship to then find the loop gain. So it could be a shortcut in the analysis process if if you wish. So that's the main result, and what follows next is really an attempt to derive where these main result is actually coming from.
