 Welcome to the eighth lecture in the course Engineering Electromagnetics. As you can see on the overhead projector, the topics for discussion today are velocities of wave propagation and the second topic is transmission line charts. Coming up the first topic that is the velocities of wave propagation, we introduced last time the concept of phase velocity and we obtained an expression for phase velocity as vp equal to omega by beta with omega and beta having their standard meanings. For a signal with a single frequency sinusoid, the interpretation of vp is straight forward. It represents the velocity with which an observer must travel to keep pace with the constant phase point. An observer who wishes to remain on a constant phase point must travel with the velocity vp that is omega by beta. However, when we consider signals with arbitrary shapes, one immediately realizes that there could be problems. What kind of problems? Of course, a signal with an arbitrary shape can be decomposed using the Fourier analysis into its component sinusoidal frequency components. It would be consisting of number of frequency components and each of these frequency components could in principle have a different value of the attenuation constant and different value of the phase velocity. The simplest situation would be that alpha, the attenuation constant which in general is a function of omega is constant for all frequency components. That is what we mean by constant here and the phase velocity which also in general is a function of frequency or omega is constant. If these two simplifying conditions are satisfied, then no matter how long the transmission line, no matter how much distance the signal travels on the transmission line, the signal shape will be faithfully reconstructed at the other end. So, these two conditions we associate with the faithful reconstruction. Those who are interested in music would also consider it as a final system. The input signal is faithfully reconstructed at the output. Now, these considerations are not new. This kind of considerations we have been making in the context of filters, amplifiers, etcetera also. And we say that the amplitude response should be constant with frequency and instead of velocity we have been saying that the phase response should be linear with frequency and we will see shortly how a constant phase velocity translates into a linear phase response. So, these are not new considerations. Anyway, if these conditions are not satisfied that is either of these, the attenuation is not constant as a function of frequency or the phase velocity is not constant with frequency. What is going to happen? One can see fairly easily that the proportion in which the various components started with that is going to be disturbed at the output end depending on the distance travelled on the transmission line. And there may appear relative time delays between each of these components. In general one would say that if these conditions are not satisfied the signal is going to be distorted. That is the original shape is not going to be faithfully reproduced. The system will lose fidelity and as I said this distortion we have been concerned with in many other systems. And we will be concerned with this in transmission lines and wave guides as well. When the attenuation is not constant or the amplitude is not constant with frequency we have been calling it the amplitude distortion. And similarly when the second condition is not satisfied we have been calling it the phase distortion. The second condition is given a special name in this context and it is called dispersion. Dispersion now one can define is the phenomena in which the various frequency components travel at different phase velocities. An immediate result of dispersion would be distortion. But as one can see dispersion is not the only cause of distortion. It is one of the two causes of distortion. When did we first come across this phenomenon named dispersion. This was introduced by Newton through a simple experiment in which he passed white light through a prism. And this is the incident white light. And upon refraction the white light decomposed into its various components. The colors of the spectrum and what was the reason for this decomposition. One could consider the law satisfied by the or obeyed by the phenomenon of refraction Snell's law. Let us say these are media 1 and 2 with corresponding permittivities as epsilon 1 and epsilon 2. Let the angle of incidence be theta 1 and the angle of refraction let us say for one of these be theta 2. And then we write Snell's law saying that sin theta 1 upon sin theta 2 is equal to the ratio of the phase velocities in the two media. Therefore, it becomes v p 1 by v p 2 and since the phase velocity is related to the permittivity and permeability. And we are dealing with non magnetic media here. So, mu is mu naught and therefore, this ratio turns out to be square root of epsilon 2 by epsilon 1. And because v p 2 or by the same token epsilon 2 are not constant with frequency. Therefore, the angle of refraction is different for different frequency components thereby causing a separation of incident white light into its various constituent colors. One kind of distortion one could say the original signal is not faithfully reproduced here. Now, while this may appear interesting to us at the moment in many other situations this kind of distortion is going to be only troublesome and is not going to be acceptable. In general the phase velocity would not be constant with frequency and as we shall see the attenuation constant is also not going to be constant with frequency. And therefore, in general some amount of distortion caused by these two factors is going to be present. Having said this it is very interesting to consider some condition that one could mathematically derive or mathematically verify where these two conditions are satisfied by an appropriate choice of the transmission line parameters. In which case even in the presence of attenuation one could perhaps have no distortion all right. And that condition fortunately exists it is called the distortion less condition or it is also called the heavy side condition after the scientist who first proposed it. What is it that we require for no distortion? We require that these conditions should be satisfied that is alpha and v p should be constant with frequency. Now, this distortion less condition or the heavy side condition states that if R by L is equal to G by C you can recognize R L G C these are the distributed parameters of the transmission line resistance per unit length inductance per unit length etcetera. So, if they satisfy such a relationship then alpha and v p turn out constant with frequency. It would of course, be interesting to start from this requirement and then end up with this condition, but here we go the other way round assuming that such a condition holds good we show that this condition or these two conditions are satisfied. What would be the starting point? The starting point would be the expression for the propagation constant gamma. What is the expression for the propagation constant? For the general transmission line it is R plus j omega L into G plus j omega C whole square. Now, we take out the factors j omega L and j omega C common that will give us j omega square root of L C here and then we will be left with 1 plus R by j omega L into 1 plus G by j omega C whole square root. However, if this condition is satisfied then the two terms within the brackets are identical and therefore, it can be rewritten as j omega square root of L C into we can retain either of these terms say into 1 plus R by j omega L. One can multiply through and one can see that this is going to be R by square root of L by C plus j omega square root of L C. So, that alpha which in general is a function of frequency or in general can be a function of frequency is simply R by z naught and what is beta? Beta is omega square root of L C. So, that what is the corresponding phase velocity v p omega by beta and we find that v p is 1 by square root of L C and this is without any approximation this kind of relations for phase shift constant and the phase velocity we obtained for lossless transmission lines or for low loss transmission lines, but under the low loss approximation these were approximations. Now, when this condition is satisfied these are exact expressions and even in the presence of losses and we cannot restricting the losses to be small. This is the expression for the phase shift constant and this is the expression for the phase velocity and we see that in particular alpha is independent of frequency and the phase velocity is also independent of frequency and therefore, on such a transmission line whatever the original signal shape it will be faithfully reproduced without distortion. One can also make out the phase behavior being linear being associated with no distortion the phase shift constant is linearly changing with frequency. In amplifiers and filters we are we stop here that is the amplitude response should be constant with frequency and the phase response should be linear with frequency. Since we are specifically talking about wave propagation and the time delay effects we normally say that the phase velocity should be constant with frequency. Now, this observation has a very important application. Now, it should be clear that the effect of distortion will be more and more pronounced as the length of the transmission line keeps on increasing and what would be the longest transmission lines that you can imagine. You know we lay trans oceanic cables for the purpose of telecommunication which would be thousands of kilometers long and in general the attenuation and the phase velocity will not be constant with frequency and they will be considerable amount of distortion. But if we make sure that this distortion continuous condition r by l equal to g by c is satisfied then one will have done away with the distortion in this idealized consideration. We will have a little more to say about this later on. What happens on such long transmission lines is that the inductance falls short of satisfying this condition. There is a certain resistance even when we use good conducting materials for constructing the cables and there is a certain conductance associated with the dielectric material and a certain capacitance associated with the geometry of the transmission line. So, for those parameters it is the inductance l which comes out to be less compared to the value required for satisfying this condition and therefore, one uses a regularly placed or periodically placed loading coils to make up for this short fall in the value of the inductance and to ensure that r by l becomes equal to g by c. In the beginning it appeared an excellent idea to lay trans oceanic cables and to connect continents for the purpose of telecommunication, but the quality was not satisfactory and therefore, this was a great improvement in the quality of telecommunication. Now, we have digressed a little to go into the ideas the concepts of distortion and we have seen how dispersion contributes to distortion and how distortion can be eliminated in principle on transmission lines, but we were starting we started with the concept of the velocities of wave propagation. So, when we do have a group of frequencies constituting the signal then what will be the velocity we will associate with that group of frequencies that consideration still remains. In general they will travel at different phase velocities so what will be the velocity that we will associate with such a group of frequencies that is what we return to now and for this purpose one talks about what is called the group velocity. The concept of group velocity helps us in identifying a single velocity value for a group of frequencies. However, this group of frequencies must not have frequencies which are widely separated as long as the frequency components within this group are close by we can associate a unique velocity of propagation with this group of frequencies. How that can be done is shown in the following. Let us say that we have frequency components which are omega naught minus delta omega and omega naught plus delta omega. We considered two frequency components in the group that we have taken for illustration and the frequencies are only slightly separated as a percentage of the center frequency is a very small percentage. Now, we further assume that these frequency components travel with different phase velocities phase shift constants and let the phase shift constants be beta naught minus delta beta and beta naught plus delta beta. If one had a signal with frequency omega naught traveling with a phase shift constant beta naught how would it be represented on the transmission line as a signal varying with time and as a signal varying with the distance along the transmission line and the expression we have been using is sin of omega naught t minus beta naught z. We have seen that this kind of expressions satisfy the wave equation for voltage or current on the transmission line and they represent propagating waves waves propagating in the positive z direction considering that particular direction of propagation for illustration. Accordingly these two components would also travel as follows it will be sin of omega naught minus delta omega t minus beta naught minus delta beta z. This is how the first component would look like as a wave traveling on the transmission line. Similarly, the second component would be written as sin of omega naught plus delta omega t minus beta naught plus delta beta. Now, we recombine terms in a particular manner so that it reads as sin of omega naught t minus beta naught z minus delta omega t minus delta beta z. That is how the first component is rewritten the argument is rearranged and the second component is rearranged in a similar manner to read as omega naught t minus beta naught z plus delta omega t minus delta beta z. Now, these are two sin terms with arguments reading as a minus b and a plus b and therefore, these can be easily combined to give us twice sin of omega naught t minus beta naught z and then cosine of delta omega t minus delta beta z. Mathematically, this is how far we reach this of course, is the high frequency term and compared to this the second term is a low frequency term because we have considered that delta omega is small. Now, which of these terms contains the effect of the frequency spread in the signal it is the second term which contains the effect of the frequency spread in the signal and the spread in the phase shift constant and that as you can make out is acting as a low frequency envelope of the entire signal. Now, since this is the second it is the second term which is containing the effect of the frequency spread if one now sits at a constant phase point on this low frequency envelope and the velocity with which one has to travel for that purpose then that can be associated as the velocity of this group of frequencies. So, on the envelope if we try to keep delta omega t minus delta beta z as constant similar to what we did for arriving at the phase velocity for a single frequency sinusoid and try to find out the velocity with which an observer will have to travel to maintain this condition we will have d z by d t equal to delta omega by delta beta or in the limit when delta omega and delta beta become very small we will be able to write d z by d t as d omega by d beta and since this velocity is related to the characteristics of the group in some sense this is called the group velocity and given the symbol v g all right. Now, in general on transmission lines the phase velocity remains more or less constant with frequency and then the concept of group velocity is not required to be invoked since all frequency components are travelling with more or less the same velocity and that becomes the velocity of a group of frequencies as well. But on wave guides typically the phase velocity is not constant with frequency and one has to talk about the group velocity when we require the concept of group velocity or when the phase velocity is not constant with frequency these things can be seen very easily on what is called an omega beta diagram which is very simply a plot between the phase shift constant beta and the radiant frequency omega and as I just said on transmission lines the omega beta diagram is quite straight forward since they could be waves travelling in two different directions on a transmission line we have two branches in the omega beta diagram with equal slopes and these are more or less straight lines and one can see that whether one calculates the phase velocity at a particular frequency by omega by beta or one calculates the group velocity by d omega by d beta it will come out to be the same that is a very strong advantage with transmission lines, but they have their own limitations. On wave guides on the other hand the omega beta diagram looks like this this is the typical omega beta diagram on the kind of wave guides rectangular wave guides that we are dealing with in the laboratory and one can see that the phase shift is not constant with frequency and there is going to be a different value of the group velocity at different points. So, this is just an aid to the consideration of the various velocities if you have any questions at this point of time we can take those otherwise we move on to the next topic. Having said these things about the velocities of propagation we now go on to consider the second topic for today and that is the transmission line charts. You have done some calculations regarding the input impedance at different points on a transmission line and the calculations are fairly complex. It is true that with the modern calculators these calculations can be handled quite conveniently, but there was a time when such calculators were not available and also the calculators do not provide any physical insight into the problem what is really happening. Now these two problems can be resolved can be eased out by using what are called transmission line charts and now we shall try to see how one such transmission line chart can be evolved. We start with the familiar expressions for voltage and current on a transmission line. So, that V is V naught plus e to the power minus j beta z plus V naught minus e to the power plus j beta z and the voltage at some distance z equal to minus l from the load is going to be V naught plus e to the power j beta l plus V naught minus e to the power minus j beta l which can be written as V naught plus e to the power j beta l into 1 plus rho e to the power minus e to the power minus 2 j beta l where rho is V naught minus upon V naught plus. We could generalize the expression for rho or the interpretation for rho the reflection coefficient by writing this as V minus upon V plus at l equal to 0 or at the load impedance. This is how we have been defining the reflection coefficient. Similarly, we can write the expression for the current at the same location at which we have written the expression for the voltage and that is going to be 1 by z naught V naught plus e to the power j beta l minus V naught minus e to the power minus j beta l and therefore, can be written as V naught plus by z naught e to the power j beta l into 1 minus rho e to the power minus 2 j beta l z naught here is the characteristic impedance of the transmission line. So, that the input impedance at this location z equal to minus l is the ratio of V and i at this location and this is turns out to be z naught times 1 plus rho e to the power minus 2 j beta l upon 1 minus rho e to the power minus 2 j beta l quite straight forward. Next we introduce some new symbols. So, that we can refer to this expression in a compact manner. We say that let small z be the ratio of the input impedance and the characteristic impedance. There is a standard terminology used for this. Let small z be the normalized input impedance. Also, we let a new symbol rho l be introduced which is equal to 0 rho times e to the power minus 2 j beta l. What is the relationship between rho l and rho whereas, rho represented the reflection coefficient at the load at l equal to 0 that is where the load is connected. This will represent the reflection coefficient at some other point on the transmission line. So, this one could say it is the reflection coefficient that is V minus upon V plus at z equal to minus l taking into account the phase shifts of the incident wave and the reflected wave. In terms of these two new symbols we have z equal to 1 plus rho l upon 1 minus rho l where the interpretation of rho l and z has already been mentioned. Correspondingly, one could also have rho l equal to z minus 1 upon z plus 1 and without much difficulty one can correlate this or see the correspondence of this new expression with the expression for the reflection coefficient that we have been using for some time now. It is just that we are using normalized impedances here and the reflection coefficient at some distance from the load. Now, these are both complex quantities. So, they can be written as r plus g x the real part being the normalized resistance and the imaginary part being the normalized reactance. Similarly, rho l could also be written as a real part and an imaginary part and mathematically one could consider either of these two expressions as a transformation from one complex plane to the other or as a mapping of one complex variable on to the other. One can consider some specific or typical values to see what this mapping looks like. What is this expression transformation amounting to? We consider the complex z plane on one hand and represent it as follows and for passive components, passive circuits r greater than 0 greater than or equal to 0 will be the region of this complex plane which will be applicable. On the other hand the complex rho l plane can be represented in terms of the real and the imaginary axis u and v. Now, one can consider some typical values. The first point that we consider is let us say a and let the value of a b r equal to 1 and x equal to 0. It would be plotted somewhere here depending on the scale. This is point a. Can you recognize this particular value of impedance? This is the normalized resistance equal to 1 and the reactance is 0. So, this is equal to the characteristic impedance of the transmission line. What kind of reflection coefficient do we expect for this? It should be 0 which is what you would get if you substitute this value of z real and imaginary parts in this expression and we get the corresponding point a here as u equal to 0 and v equal to 0. This is point a. Next one could consider the point at infinity which will correspond to an open circuit. One can see that this will map on to a point which is u equal to 1 and v equal to 0 that is some point here. One can also consider the short circuit point c with r equal to 0 and x equal to 0 and without much difficulty one makes out that it will map on to a point where u is minus 1 and v is 0. One can consider other points also. For example, a point d with r equal to 0 and x equal to 1 that is a point here. So, z is equal to j for this point and the corresponding point on the complex real plane is u equal to 0 and v equal to 1. This is point d and in symmetry with this if we consider a point e with x equal to minus 1. We will see that it maps at u equal to 0 and v equal to minus 1 that is a point. Now, in fact we have considered a fairly large range of points on the r greater than or equal to 0 complex z plane. You could consider some other point also and you would find that all points map on to or within a circle of radius rho l equal to 1. That is within this circle lie all the points corresponding to the r greater than or equal to 0 portion of the complex z plane which should have been expected because the magnitude of rho is less than or equal to 1. So, this is nothing new. Now, this mapping or this transformation becomes the basis of a very popular transmission line chart which is what we shall take up next time. So, in today's lecture we have discussed two topics. One is the velocities of the wave propagation. We introduce the concepts of phase velocity and group velocity and we have seen their role in distortion. We also introduce the phenomenon of dispersion and then we went on to consider the basis of a particular type of transmission line chart. Thank you.