Middlebrook's extra element theorem applies to linear time invariant systems. Such as the system that's shown right here with an input, and an output. And, that system is assumed to have a port where certain impedance is connected. So that impedance connected to a port of that linear circuit is considered the extra element and the extra element here then gives us ability to write the transfer function from input to output in a form that is based on the transfer function when the extra element, or this additional impedance is not present in the circuit. It's either open or short. And in terms of the impedance is seen into the port, where the impedance is actually connected. So that's the nature of the extra element theorem. We have again, a general setup. The input shown here is a voltage, the output here is shown as a voltage but in general, input can be current and output can be current or any combination of current or voltage inputs and outputs. Let's see what it meant by this impedances seen into the port where the element is connected. So, here I have that diagram of the linear circuit that we start from in general and we look at two different situations for finding gull impedance. Seeing into the port where the element is connected. Those two impedances are called ZD and ZN. Let's see first the setup for ZD. So ZD is impedance seen into the port, where the element is connected, when all other independent sources are set to zero. So in this set up right here, you have a test source, I. You set all other independent inputs to 0. They're set to 0 which means that in case you have the input voltages as an input, you have a short. Right. So short circuit really is the way of saying that input is set to zero. Then you find the response ZD is equal to voltage seen across the test source or the current produced by that test source. And that's again under the condition that VN=0. And inside the box, there are no other independent sources. And if there are, they are going to be set to 0, current sources, open-circuited, voltage sources, short-circuited. So when you look at this particular definition of ZD, this is a traditional way to define impedance seen into the port. Examples of which you can find when we look into the output impedance. So output impedance of a converter is output voltage over the current that we inject into the output node. With everything else set to zero. D hat is zero, VG hat to zero, all other independent sources are set to zero and you find out that the response to the test current are placed at the output of the converter in terms of the voltages you see or cause the output of the converter, it divide the voltage by current and you get really the output impedence. That output impedance is really ZD. ZD is just generally saying this port does not have to be the output, it can be any port you like, and in particularly, this is the port where we have this extra element connected to. So, nothing special about ZD, ZD is a very traditional introductory circuit notion of an impedance seen at a certain port of a linear circuit. Now, ZN is the one that's actually more interesting. So ZN has exactly the same setup of placing the test source at the port where the impedance was connected. But at the same time, we have V in present. So together VN and IS are now two independent sources present in the circuit. No other independent source is here. And you find ZN has the ratio of V, the voltage seen across that test source. And the current I under the condition that the output signal is null. So V out is null and that's, as you can see here, two sources acting together to null something. It is a case of null double injection. So now we have these two impedances defined. And we are in good position to actually give the Extra Element Theorem results. So the Extra Element Theorem main results are shown right here. There are two possible cases. In one you can consider the extra element to be open. Okay the other one is really considered the extra element to be short. So in this circuit right here, if you just open the extra element you will get the output voltage over input voltage that gives a transfer function as contained right here. So this is when the Z extra element is open circuited. G of S right here is refered to, I'm going to call that G old. So this part right here is called G old. G old in a sense before the impedance is inserted unto the circuit. So without impedance, when you say without impedance. Now in this case here it means open circuited impedance, Z goes to infinity. We have that G old in front. Okay. That's the front end part of the system transfer function with impedance present. So this is what we're interested in. This is G with impedance is equal to G old, meaning the transfer function without the impedance open circuited times what it's called a correction factor that is expressed in terms of the value of Z. Of course it's going to depend on the value that impedance and it depends on these two impedances seen at the port where Z is connected, ZN and ZD And this is how the correction factor looks like. So you have 1 + Zn over Z over 1+ Zd over Z. It is going to be very easy to remember the correction factor here for the case where the extra element is considered open in the old transfer function. It's easy to remember how to set up these ratios right here because you want to realize that if Z does go to infinity, indeed, if there's no Z, right? It's open circuited. This term goes away, this term goes away and you go back to the old transfer function without the impedance, specifically with impedance open circuited. Okay. Now, the other version of the theorem is when the old transfer function is found under the condition that the extra impedance is short circuited. So we can choose either one of these two and in some cases, one will be more favorable to us than the other. But if we start off with the impedance short circuited, we have now G old in this form right here. That's Z to zero. So, this is the G old for the case where the impedance is short circuited, and we have a different version of the correction factor right now. Which has a form of 1+Z over Zn over 1+Z over Zd, okay. So those are two main expression that you're going to use over and over again. In addition to that, as in the feedback theorem, we will recognize that there is a reciprocity relationship in this case as well and that the reciprocity relationship is given at the bottom. The two old transfer functions, the one with Z open to the one with Z short, the ratio of those is equal to ZD over ZN which in itself is quite interesting And it can be used in some cases where you know with three of these you can find the fourth one just from the reciprocity relationship. So that's the main result.
