 Welcome to the seventh lecture in the course engineering electromagnetics. In the last lecture we considered the behavior of voltage and current on a general transmission line general in the sense that it incorporated losses. And we saw the effect of these losses on the various aspects related to the transmission line the formation of standing waves the input impedance etcetera. Today what we will consider is the use of transmission line sections as circuit elements. The first question that will come to mind perhaps is why or in what situation do we need to use transmission line sections as circuit elements. The reason for this is the following what happens is that conventional circuit elements do not perform as expected or do not behave as expected at high frequencies. The reasons for this misbehavior can be traced back to time delay and the high frequency effects. But what kind of misbehavior do we have in mind here we are quite familiar with an inductance or an element which is supposed to behave as an inductor showing a lot of resistance. In addition at high frequencies what may be quite significant is the inter turn capacitance in the windings that are made to obtain an inductor. Similarly an element which is expected to behave as a capacitor may have associated with it what is called a considerable amount of lead inductance. And of course wire bound resistor may have considerable inductance associated with it. Now it is not that these effects are completely absent at low frequencies. The problem is that the values of these elements that are required to be used at high frequencies are themselves quite small. And therefore these small effects become quite significant as a fraction of the actual element value that is required to be used. And then the frequency is high therefore the associated reactance with even small capacitance or inductance can have a considerable effect. And it is because of these reasons that the conventional forms of the circuit elements do not behave as expected at high frequencies. There are solutions to this problem one could use what are called thin film circuit elements. But what we talk here today is the use of transmission line sections as circuit elements which also is a very convenient way of solving some of these problems. And the problem is of considerable significance in the frequency range which is a few 100 megahertz. Hundreds of megahertz that is the kind of frequency range where this kind of problems come up. The basis for the use of transmission line sections as circuit elements is of course the input impedance seen at the terminals of a transmission line with some termination at the other end. The expression is quite familiar to you by now the input impedance is z naught times z l plus j z naught tan beta l upon z naught plus j z l tan beta l. Under what conditions or under what situation is this the expression for the input impedance either lossless situation or low loss approximation. This is the input impedance at a distance l from the load impedance z l. Now let us consider the behavior of a short circuited transmission line. A short circuited transmission line would be one where z l is equal to 0. And therefore, z n is simply j z naught tan beta l. It is completely imaginary it is no real part and varies as the tan function varies. And therefore, in a straight forward manner one can make out that depending on the value of the argument beta l the overall input impedance could have an inductive behavior or could have a capacitive behavior. And therefore, appropriate lengths of sections of short circuited transmission lines could be utilized as inductors or capacitors. One can put it down more definitively here beta is 2 pi by lambda. And the behavior is inductive if beta l is less than or equal to pi by 2. Let us say less than pi by 2. Of course, this is the situation when the argument lies in the first quadrant. Similar behavior will be obtained when the argument lies in the third quadrant. For the sake of simplicity we are not listing all mathematically possible values of beta l. So, for beta l less than pi by 2 the behavior is inductive and the corresponding length is less than lambda by 4. Short circuited transmission line section with length which is less than lambda by 4 will show an input impedance behavior similar to that of an inductive. The actual inductance value that one will be realizing at the input will depend upon the length and the wavelength or the product beta l. And since tan function can have any value from 0 to infinity any inductance value can be realized. Similarly, the behavior will be capacitive if beta l lies between pi by 2 and pi or l lies between lambda by 4 and lambda by 2. There will be other ranges of beta l or l where again similar kind of behavior will be shown. Similarly, an open circuited transmission line can also be utilized for the same purpose. Here, z l is infinite and therefore, z n has an expression which is minus j z naught cot beta l which again has similar characteristics as the tan function and therefore, an open circuited section of a transmission line could also be utilized as an inductor or a capacitor. The ranges of the inductive behavior or capacitive behavior can be easily identified it is going to be inductive. If beta l lies in the second quadrant that is pi by 2 is less than beta l is less than pi or correspondingly l lies between lambda by 4 and lambda by 2. Similarly, it is going to be capacitive if beta l is less than pi by 2 or l is less than lambda by 4. And this way you see that we can use sections of transmission lines effectively either as inductors or capacitors. These are going to be low loss transmission lines therefore, associated resistance is going to be quite small and they will behave as expected as designed. They will not be corrupted by lead inductance or stray capacitance effects. If we consider a particular length of short circuited transmission line let us say a quarter wave short circuited transmission line lambda by 4 short circuited transmission line. Then based on this expression for the input impedance which is derived from the general expression for the input impedance pertaining to low loss approximation or loss less approximation. We see that for this situation where l is lambda by 4 and beta l is exactly pi by 2 and infinite reactance is indicated. But in practice things remain finite and if you want to find out the exact value of the input impedance then we need to go back to the exact expression for the input impedance incorporating the small amount of loss that is invariably present. Let us work it out. We have z i equal to z naught tan hyperbolic gamma l. This is the general expression for transmission lines. For the short circuited transmission line that is z l is 0 and this we write out as z naught then sin hyperbolic gamma l. So, that it is alpha l plus j beta l upon cos hyperbolic gamma l which is again alpha l plus j beta l which can be expanded quite like sin and cosine of a sum of two angles. With a bit of care one can see that this expands into z naught into sin hyperbolic alpha l times cos beta l plus cos hyperbolic alpha l and then j sin beta l divided by cos hyperbolic alpha l cos beta l plus j sin hyperbolic alpha l sin beta l. Now, substituting the condition that it is a quarter wave short circuited transmission line that is beta l is pi by 2. This simplifies considerably and we get z i as z naught times cos hyperbolic alpha l upon sin hyperbolic alpha l. One can simplify this further recognizing that alpha l is small in practice. We use low loss transmission lines for practical purposes and within that approximation using the small argument approximation for cos hyperbolic and for sin hyperbolic. This simplifies to z naught into 1 upon alpha l. What is the expression for alpha for low loss transmission lines? That was derived earlier the expression for alpha was half r by z naught plus g z naught and we said that in practice the second term is much smaller than the first term and therefore, this is simply r by 2 z naught. And therefore, the input impedance for the case we are considering is simply 2 z naught squared by r l which is the case not only for beta l equal to pi by 2. Now, we could generalize which is the case for beta l equal to all odd multiples of pi by 2 2 n plus 1 into pi by 2 or alternatively when l is an odd multiple of quarter wave lengths which is dot infinite. It is still high a high value because for a well constructed transmission line r will be quite small using good conductors. So, it is still a high value, but you see that it does not go to infinity in the practical case. Same behavior will be shown by half wave or lambda by 2 open circuited transmission line sections. Once again starting with the exact expression for the input impedance and then substituting z l equal to infinity in that expression. One will find that z i in this case is once again z naught into cot hyperbolic alpha l there is no approximation here which can be approximated to 2 z naught squared by r l an expression similar to that we have got for the short circuited quarter wave section. It is alpha l quite similar to this you will start with z naught cot hyperbolic gamma l and then proceed in this manner and substitute the condition that beta l is now an even multiple of pi by 2 and l is a multiple of lambda by 2 and then you will get this. If you have any other questions you please let me know. You could, but this will be a very high value and there are other alternative means of using circuit elements as registers. For example, thin film resistors which are quite convenient. Next we go on to a very special length of transmission line that is the quarter wave length and we will find that it has very interesting properties. We call it a quarter wave transformer what we will do is we will use the general expression for the input impedance z i is z naught times z l plus j z naught tan beta l upon z naught plus j z l tan beta l under the low loss approximation and now we say that we are using a quarter wave length of the transmission line or odd multiples of quarter wave length as we have seen. So, that beta l can be written as 2 n plus 1 into pi by 2 or l is an odd multiple of quarter wave length no. So, far we have considered the short circuited sections or open circuited sections. Now, we are not considering any particular restriction on z l, but we are considering that the length is restricted to an odd multiple of quarter wave length. So, this case has not been considered earlier. For this one can see easily that z i is going to be z naught squared upon z l there are two things that one can notice from this the way the input impedance seen at the end of a quarter wave section of a transmission line has come out what are these two things first is the input impedance z i is inversely proportional to z l there is an inverse relationship between z l and z i z i is inversely proportional to z l which is very interesting. If z l is 0 if it is a short circuit at the input end of the quarter wave section it will be seen as an open circuit and vice versa instead of taking these extreme values one could say that if z l is high z i will be low and if z l is low z i will be high. So, this reciprocal relationship between the input impedance and the load impedance holds good for a quarter wave transformer. Hence it is called a transformer it is transforms the load impedance in this manner that is one thing the second thing which makes a quarter wave section of a transmission line quite useful is the following let us make the following consideration. Let us say that we have got a transmission line with characteristic impedance z 1 which for some reason for some practical application needs to be connected to a different transmission line let us say like this with z naught equal to z 2 different in the sense that it is characteristic impedance is different. In practice it could be a load impedance which is different from the characteristic impedance of this transmission line or it could be simply another transmission line with a different characteristic impedance. The application that we are going to illustrate is not affected by this. Now let us say that it is connected through a transmission line in this manner let this section be of length lambda by 4 at the design frequency and we pose the question that can we choose a value of the characteristic impedance of this intermediate section. So, that the input impedance seen here at this point z i it is equal to z 1 that is by interposing such a section between a transmission line of characteristic impedance z 1 and a transmission line of characteristic impedance z 2 can be match this transmission line essentially that is what it would be if the input impedance seen here is equal to the characteristic impedance of the first transmission line. Now since we have said that this is a quarter wave section it will have the properties of the quarter wave section that we have just put down and we will have z 1 equal to or z i which is required to be equal to z 1 for matching it should be equal to z naught squared by z 2 and therefore, if z naught is chosen such that it is equal to the geometric mean of the impedances on either side then this quarter wave section will have matched this item to this item of a quarter wave section. As I said earlier while for illustration we have said that this is a transmission line of a characteristic impedance z 2 it could be just a more difference whether it is equal to z 1 that we can form is that they can be connected through a quarter wave section with a characteristic impedance which is the geometric mean. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . it will not remain perfectly matched. This thing can be seen easily on the overhead predictor where we consider the application of the part wave section of a transmission line to match a lone impedance Z n to a transmission line with characteristic impedance Z 1. And as we have said if we select the characteristic impedance of this part wave section as the duet treat of Z l n Z 1, the transmission line will be matched at the design frequency. Or the base point is indicated here as beta l equal to pi by 2 for l equal to lambda. What we have shown here is the application of a part wave section of a transmission line to match a load impedance Z l to a transmission line of characteristic impedance Z 1. And as we stated earlier if we select the characteristic impedance of this part wave section as the duet treat of these two impedances then the line will be matched at the design frequency. Now here we are showing a plot of the reflected coefficient that one sees at this point as a function of beta l. Beta l can be considered to be the electrical length of this section beta is 2 pi by lambda. And therefore, theta is the electrical length of this section at the design frequency or the design wave length the electrical length is exactly pi by 2. And at this point the load is perfectly matched the reflection coefficient is if as we change the operating frequency for example, if we increase it then the electrical length is going to increase. If we reduce the frequency the electrical length is going to reduce. In particular when the electrical length is brought down to 0 meaning thereby that this part wave section is absent. Then we have a reflection coefficient which is just as if Z l is directly connected to Z 1. And the same thing obtains at when the electrical length is pi and therefore, this is a periodic variation of the reflection coefficient that one is going to see. Therefore, we see that water wave transformer used for matching purposes is going to have a restrictive bandwidth the bandwidth will depend upon the maximum reflection coefficient that one is going to be able to follow. This rho m value depends upon the applications typical values that are acceptable would be 0.1 for 0.2. Depending on that there will be a certain bandwidth certain frequency range within which the reflection coefficient will be below this maximum permissible value beyond which the reflection coefficient is going to exceed this permissible value. And therefore, we say that that is the bandwidth of the wave transformer. The question is whether how we relate this to bandwidth what we are saying is that at the design frequency or at the design frequency the match is perfect. As we deviate from this on either side either because the physical length is different from lambda by 4 or the operating frequency is changing the reflection coefficient is going to increase rather steeply. Up to a certain point where the reflection coefficient is below the maximum permissible value you could still operate this and consider that the load is matched to the transmission line. And therefore, you get this upper frequency limit and lower frequency limit. And the separation between these two becomes the bandwidth over which this quadrupole section will keep this load more or less matched within the specified limits of the reflection coefficient. Normally this bandwidth is quite small it is just a few percent bandwidth few percent of the center frequency. There are ways and means available for improving us. For example, one could use multiple section water wave transporters, but we are not going to going into that here. Excuse me sir. Yes. Is this variation sinusoidal? Electrified sinusoidal. Electrified sinusoidal. Electrified sinusoidal. I think so yes sinusoidal rectified sinusoidal. One can calculate the reflection coefficient instead of using as a fixed electrical length beta equal to pi by 2. Keep this general and then estimate the reflection coefficient. So, we will get you find out the input impedance general expression for the input impedance here and calculate the reflection coefficient. That is how this curve has been plotted. Excuse me sir. Yes please. This is the junction we have in mind. Why should it be there? Because the input impedance that is seen by this section at the design frequency is equal to z 1. One can go into the details of this alright and see how the reflection at this junction is cancelled by the reflection at this junction. Overall effect is that at this junction there is no net reflection. So, the x axis it is the length of the extended portion. After that z 1 junction that length is for you. Yes the electrical length that we are plotting here is the electrical length of the quiet wave section alright. So, I suppose we can now go on to the next topic and that relates to the velocity of propagation. This is a natural question what is the velocity of wave propagation on a transmission line. One term that we have been using is the phase velocity designated as v p and we have said that it is equal to omega by beta. How we identified this expression omega by beta as the velocity was as follows. For example, for sinusoidally time varying signals we have e to the power j omega t minus beta z which is manipulated to read as e to the power minus j beta and then z minus omega by beta which when compared with a function of argument z minus v t tanges as that v is equal to omega by beta. But a better interpretation of this velocity is obtained if we consider omega t minus beta z alright and say that alright we want to stay on a constant phase point omega t minus beta z represents the phase of the wave as a function of z and as a function of tang. Let us say that we want to stay on a constant phase point then with what velocity an observer or v will have to track that one can find out by finding out the derivative of this with respect to time that is p by d t of omega t minus beta z equal to 0 and one can see that the d z by d t that we get is equal to omega by beta and since it is like that from the consideration of staying on a constant phase point we call this the phase velocity stop here today and we will make further considerations in the next lecture.