 Welcome to the 26th lecture in the course Engineering Electromagnetics. We have been discussing the parallel plane guide for the last few lectures. Today we consider the following topics, the characteristics of the transverse electric and transverse magnetic waves and then we go on to consider transverse electric and magnetic waves on the parallel plane guide and we consider the concept of wave impedance as it is applicable to various types of wave guides. Coming up the first topic listed here, the characteristics of TE and TM waves, you would recall that we have seen that the fields that the parallel plane guide supports for the TE and the TM modes are quite different from those of the uniform plane waves and we should try to see what are the properties of these waves on the parallel plane guide. For this purpose we consider the various expressions that we have got as a solution of the Maxwell's equations subject to the boundary conditions imposed by the parallel plane guide and you would recall that we have the following expressions. We have h squared equal to omega squared mu of silent plus gamma bar squared where gamma bar is the propagation constant and we have seen that h comes out to have the value m pi by a for various modes TE m 0 or TM m 0 modes. The expression for the propagation constant therefore, can be obtained from here as gamma bar equal to m pi by a whole squared minus omega squared mu of silent whole squared root and from here we make out that the nature of the propagation constant gamma bar changes depending on a certain frequency is exceeded or not and that certain frequency is called the cutoff frequency omega c which is obtained as m pi by a divided by square root of mu of silent and the corresponding frequency f c is 1 by 2 pi m pi by a square root of mu of silent. So, above this cutoff frequency gamma bar will be completely imaginary and below this cutoff frequency gamma bar will be completely real. So, at frequencies greater than this cutoff frequency f greater than f c we are going to have gamma bar which will go to say j beta bar and we will have the expression for beta bar as omega squared mu of silent minus m pi by a whole squared whole squared root. Above the cutoff frequency the wave propagates with the phase shift that we are familiar with associated with the propagating waves without any attenuation, but below this cutoff frequency gamma bar would be completely real and the wave will just attenuate and actually there will be no wave motion alright. This expression for the phase shift constant beta bar can be considered for plotting the omega beta diagram which we had considered earlier the diagram can be plotted right here. Let us say this is the beta bar axis and this is the omega axis and at frequencies below omega c the phase shift constant does not exist and therefore, the plot is going to be of this nature symmetric for positive and negative values of beta bar since the propagation could take place both in the positive z and the negative z directions. We can make out from here that as the frequency increases beta bar will gradually approach beta the phase shift constant for the infinite medium. If we were to plot the omega beta diagram for the wave propagating in an infinite medium with the same characteristics mu and epsilon those would be simply straight lines like this in this manner and we see that as the frequency increases the omega beta diagram for the parallel plane guide for TE or TM modes asymptotically approaches that of the infinite medium. As the frequency decreases a greater effect of the confinement because of the parallel plane guide becomes evident. This frequency is omega c the cut off frequency that we have been mentioning. Once we have got the expression for the phase shift constant various other quantities can be easily put down. For example, we can have the expression for the phase velocity let us call it v p bar. So, as in the case of the uniform plane waves the expression in terms of the phase shift constant is not going to change, but the phase shift constant itself is different now. The expression is omega by beta bar and therefore, it will be omega upon omega squared mu epsilon minus m pi by a whole squared whole square. Now, one can make out in a relatively simple manner that this is going to be greater than the phase in the infinite medium which is the velocity of light in that medium. For that one needs to carry out some very simple processing and it will turn out that we will have an expression where c the velocity of light in that medium will be in the numerator and there will be a quantity in the denominator which is less than 1. And therefore, the phase velocity for the parallel plane guide is going to be greater than the velocity of light in the medium filling the parallel plane wave guide. This we have discussed earlier and this can also be linked to the concept of expressing the wave guide field in terms of plane waves. And since they are propagating at an inclination to the z direction one can clearly understand why the phase velocity is coming out greater than the velocity of light. We have also discussed the concept of group velocity. The velocity that a group of frequencies which are spaced relatively closely together what is the velocity with which that group travels. That is the group velocity we may call this V G bar and we had obtained an expression for this as d omega by d beta and in this case d beta bar. Since we have the expression for beta bar we can calculate the derivative with respect to omega here. We may have d beta bar by d omega as omega mu epsilon by the same expression that is omega squared mu epsilon minus m pi by a whole squared whole squared root or omega mu epsilon by d beta bar. So, that the expression for the group velocity becomes beta bar by omega mu epsilon and by processing similar to what was mentioned for the case of the phase velocity with the help of which we saw that it is going to be greater than the velocity of light. It can be seen that this is going to be less than the velocity of light and therefore, information or energy travel will always be less than we will always travel at a speed less than the speed of light. Since this is equal to omega by beta bar if we consider the product of V P and V G it is going to be simply 1 by 1 by 1 by 1 by 1 by 1 by 1 by 1 mu epsilon or the square of the velocity of light in the medium filling the parallel plane waveguide. This is a fairly general expression it will hold good for all metal waveguides like the parallel plane waveguide or the rectangular waveguide or even circular waveguide that we are going to consider later on. But obviously, it will not hold good when we deal with types of waveguides where the phase velocity is less than the velocity of light. We have seen that that can happen sometimes those types of waveguides will be called slow waveguides. Apart from that wherever we have fast waveguides where the phase velocity is greater than the velocity of light this expression is going to hold good. You can make out the wave length also lambda bar or in many books the symbol lambda G is also used the guide wave length here we have been using these various symbols with an over bar. This is going to be 2 pi by beta bar and therefore, it is going to be 2 pi by omega squared mu epsilon minus m pi by a whole square whole square. We have already considered the field configurations corresponding to the T E and T M modes. Now, we can see the various other characteristics or properties of these waves and we see that these are different from a uniform plane wave that would be supported in the infinite medium having the same mu and epsilon as that of the medium in the parallel plane waveguide. A few things are quite noteworthy and they are very significant. First is the phase velocity or the group velocity as also the guide wave length. These are different for different types of modes I mean modes which have which correspond to different values of m. They are going to have different velocities and different wave lengths and therefore, if we allow a signal to propagate in these various modes there is going to be quite a great amount of confusion. It is obvious that after travelling a certain distance the signal is going to get distorted because the various parts of the signal in various modes are going to travel at different velocities and it is also going to be very difficult to have efficient excitation or collection of power. Because we can design the excitation mechanism or the reception mechanism for a particular mode and all these modes have different field configurations and therefore, it is preferable that the signal propagates in a single mode. Having made these comments we will consider the importance of a single mode of propagation in a little more detail later on in the context of the rectangular waveguide. Any clarifications required so far? With this discussion on the characteristics of the TE and TM waves we now pick up the second topic listed for the day and that is the propagation of transverse electric and magnetic waves on the parallel plane waveguide. This is the parallel plane waveguide that we have been considering and we have got the expressions for the TE modes and the TM modes. These are the field expressions for the TE modes, TEM0 modes in terms of the index M and we said that M can have values 1, 2, 3, etcetera integral values. What happens if M is 0? Then all field components drop to a 0 value which is a trivial case and therefore, we say that M equal to 0 is not allowed. Let us look at the similar field expressions for the TM modes and we consider if it is possible to have M equal to 0. These were the field expressions that we obtained for the TM modes and now we see that if we put M equal to 0 it is not that all field components become 0. In fact, two of the field components become indeterminate and therefore, we say that perhaps it is possible to have a situation where M is equal to 0 and therefore, we go back to the basic Maxwell's equations and consider this possibility. You would recall that we wrote the Maxwell's equations for the region, for the perfect electric region between the parallel plane waveguides. We expanded these Maxwell's equations in the Cartesian coordinates. We considered that the wave is propagating in the positive z direction and since there were no restraints or conditions to be fulfilled along the y direction, we said that del by del y can be equated to 0. With these simplifications, we obtained the following processed form of the Maxwell's equations involving all six field components and the wave equations for the magnetic field and the electric field. Now what happens if both E z and H z are 0? Earlier what we did was we expressed the transverse field components in terms of the longitudinal field components and we said that in general one of these E z or H z should be non-zero. But supposing both are 0, then do we get a meaningful situation? Supposing both E z and H z are 0, then from here we see that an H y could exist but it will have to be uniform with respect to x, constant with respect to x, alright. Now in itself that does not violate anything. If H z is allowed to be 0, then we see that in a similar manner perhaps an E y which is uniform with respect to x could exist, okay. That is what is indicated mathematically. However if we look at the wave guide that we are dealing with, E y is an electric field component which is tendential to the perfectly conducting planes and it must be 0 at least at the conducting planes and therefore an E y which is uniform with respect to x within the guide is ruled out. And therefore we conclude that there may exist an H y but an E y which is uniform with respect to x cannot exist. See we are considering what is going to happen if both E z and H z are 0. And we find that mathematically an H y which is constant with respect to x is indicated and an E y which is constant with respect to x is also indicated. But E y must drop to 0 at the conducting planes being a tendential field component, okay. And therefore something which is uniform within the wave guide but 0 at the conducting planes is not possible. They cannot take place such a sudden change, alright. And therefore from that consideration an E y which is uniform with respect to x is ruled out. It may have a sinusoidal kind of variation as we have seen for some of the modes between the conducting planes. So that it is 0 at the planes and then it gradually changes value within the region in the planes. That is alright. But something which is uniform with respect to x is not possible considering the boundary conditions that E y will eventually have to satisfy. It will have to be 0 for all values of x actually if it is to be uniform and 0 at the conducting planes. So E y becomes 0. So we see that an H y could exist and correspondingly an E x could exist, okay. And a similar situation is indicated from this equation also and the other field components are going to be 0. And therefore we consider the case when both E z and H z are 0 and we make out that we may have field components which are E x and H y. These may exist alright and these will have to be uniform with respect to x constant within the wave kind. What about the propagation constant gamma bar for a wave which has these characteristics? We can see from the corresponding wave equations. We are talking about the case where there is no variation with respect to x. That is what is indicated if both of these are 0. So the x variation becomes 0 and we see that gamma bar is simply the square root of this quantity. And therefore gamma bar becomes j omega square root of mu epsilon and we may write this as j beta bar where beta bar is simply equal to the phase shift constant beta for the infinite medium having the same mu and epsilon as the medium filling the parallel plane wave kind. What are the expressions for these field components? They are fairly straight forward. We will have say H y equal to some constant times E to the power minus gamma bar z. That is all that we will be left with in this case. You would recall that earlier for TE and TM modes we had a part which was varying with respect to x and then the z part was like this. Now even the x part becomes a constant by virtue of this restrain and therefore that part also drops out and there is no variation with respect to y anyway. So there are no variations with respect to x and y. In the entire cross section the field is uniform at a particular z equal to constant plane and in the z direction it propagates. We may write this as c times E to the power minus j minus j minus j beta bar z or c times E to the power minus j beta z where beta is equal to omega times square root of mu epsilon. As far as the other field component which exists in this case is concerned EX that can be obtained from let us say this equation and we are going to have EX equal to beta bar by omega epsilon times c and E to the power minus j beta bar z or beta z since both are the same in this case. And these two field components satisfy any boundary conditions that exist. In fact there are no boundary conditions on these two field components. This is a normal electric field component to the perfectly conducting planes and they are constant everywhere they just propagate in the z direction. And in this case we also see that the expression for beta bar is very simple and there is no corresponding phenomenon of the cutoff. And therefore this is quite like the waves that travel on the transmission line. And these are transverse to each other orthogonal to each other and also orthogonal to the direction of propagation. And hence this becomes the case which is called the transverse electric and magnetic wave supported on the parallel plane guide. If you remember we had mentioned that the parallel plane guide serves as a transition between the transmission line and the wave guides. This was the reason for making that statement. It supports waves which are similar to the waves on the transmission lines with no cutoff. And the fields and the direction of propagation being mutually perpendicular. And it supports types of waves for example TE and TM waves which are similar to those supported on say the rectangular wave guides. And therefore it forms a very convenient transition from the transmission lines to the wave guides. One can consider further the nature of these fields which we have been doing by obtaining the corresponding time varying field expressions. That can be done in a straight forward manner. We have these expressions and we multiply these by e to the power j omega t and take out the real part to obtain the time varying expressions for these fields. And what we get is something very simple. We will have h y equal to c times cosine of h y equal to c times cosine of omega t minus beta bar z or beta z. There is no distinction between the two in this case and if you look at beta bar by omega epsilon you will see that since beta bar has a very simple expression that is omega into square root of mu epsilon. It is simply square root of mu by epsilon which is the intrinsic impedance of the wave guide. Of the medium filling the parallel plane wave guide and therefore we can write here e x equal to eta times c and then cosine of omega t minus beta z. So these are the expressions which will help us in visualizing the fields corresponding to this third type of mode that can be supported on the parallel plane guide. We utilize the template that we have been using earlier and as far as the fields are concerned they are x and y directed. So one way of representing these is to consider the x y plane at some z equal to constant value. So that this is x equal to 0, this is x equal to a and this is x equal to a by 2. Now the representation is really quite straight forward since there are no variations whatsoever along the x direction. There were no variations of the fields along the y direction in the parallel plane guide anyway. Here x direction, x variation in the x direction also drops out and therefore we can have uniformly spaced directed line arrows representing the electric field extending from one plane to the other. And depending on the value of this argument the time instant and the z value that we have selected it is let us say going to be directed upwards. We can choose a value of this argument so that the cosine function is positive and then assuming that c also is positive. This is the way the electric field lines are going to look like. What about the magnetic field lines they will be y directed otherwise their ratio will be similar. They will be uniformly spaced like this in the region between the conducting planes and assuming that the argument and c have same values as for ex this direction is going to be also positive. So that the cross product of the electric field and the magnetic field is in the direction of propagation. We can also consider an alternative representation where the effect of the variation in the value of z can come in. So we may consider the representation in the x z plane also where now these reference planes can be assumed to have let us say the following values it is pi by 2 let us say omega t minus beta z equal to say pi by 2 0 minus pi by 2 minus pi etcetera. And at these planes pi by 2 and minus pi by 2 since the cosine function is going to be 0 the fields will be 0 both fields will be 0. And at these planes we are going to have fairly strong field and it is going to be of this form closely spaced around this plane but as we move away from this plane because of this argument decreasing in value and therefore the spacing will increase as we deviate from this. And there is no other field component coming in therefore we do not shorten the lengths here. They remain straight wherever they exist only the magnitude changes which is indicated by an increased spacing between the planes. And similarly one will have these here also in a similar manner and the only difference that will come is in the arrow direction. If you want to represent the magnetic field also on this we will have lots of arrows or the arrow heads coming out of this plane here where the field is strongest and as we move away from this reference plane the number of these dots can be reduced indicating that the field strength is going down of course the direction does not change. And similarly here the magnetic field direction will also reverse and it can be indicated by arrows which are going down into the plane of display and strongest at this plane and then decreasing in value as we move away from this plane because of the function cosine of omega t minus beta z. And everywhere you see that the cross product e cross h continues to have the direction z. So this is the way the TEM wave fields look like. There is no index m that appears here because there is no variation with respect to the transverse directions. Next we take up the last topic listed for the day that is the concept of wave direction wave impedance. You would recall that we had the concept of characteristic impedance and also the concept of intrinsic impedance in the cases of the transmission line and the uniform plane wave respectively. And the concept was quite useful for example it was not necessary to have a knowledge of both V and I in the case of the transmission line or a knowledge of both E and H in the case of the uniform plane waves to be able to let us say find out the power flow. In the case of the transmission line or let us say the uniform plane waves it was just one direction that we were interested in the direction of wave propagation. In the general case of the wave guides depending on the orientation and the construction of the wave guide the wave could be propagating in any direction. And therefore we need to have a more general definition of wave impedance. It is consistent with the definitions we have been using earlier it is just that we use a little more elaborate notation to make it unambiguous. The wave impedance is also given in terms of the transverse field components the field components that are transverse to the direction of propagation. And we write the following kind of defining relations Z x y plus I will explain the notation in a short while equal to E x by H y. The subscripts indicate the field components that are going to be involved in the evaluation of the wave impedance. And the superscript indicates the third direction and let us say in this case positive z direction. So looking along the positive z direction if we calculate the impedance involving the x component of the electric field and the y component of the magnetic field then it will be given this symbol Z x y plus. They could be we could be interested in finding out impedances looking in other directions as well. And those expressions can be obtained in a simple manner by rotating the subscripts in a cyclic manner. For example, we will have z y z looking in the positive x direction as E y by H z. And similarly z z x looking in the positive y direction will be E z by H x. What happens if we change the order of the subscripts? I mean for example, in a particular situation we may not have E x and H y. We may have E y and H x available. So it is not that there will be no wave impedance in that case. And that will be written as z y x looking in the positive z direction will be minus E y by H x. Which is quite consistent with the definition of the intrinsic impedance we wrote earlier where we said that we see whether these subscripts are in the cyclic order with respect to the direction of propagation or not. We will comment on this choice of sign a little later but we can complete these other things. We will have z z y plus equal to minus of E z by H x and z H y. Thank you. And z x z looking in the positive y direction will be E x by H z with a minus sign. So already you have understood the notation. Now, the significance of this can be understood if you make the following consideration. Let us consider the complex pointing vector. We have p equal to half of E cross H star. And if you consider the power the complex power that is flowing in the z direction we consider the z component of this cross product. So that we have p z equal to half of E x H y star minus E y H x star. Now, let us express E x in the positive y direction. E y in terms of these wave impedances. So that we are going to have half of E x can be expressed in terms of H y and z x y plus. So we have z x y plus H y H y star. And E y can be expressed in terms of H x and z y x plus. And since there is a negative sign here E becomes z y x plus and H x H x star. Now, the products H y H y star and H x H x star are going to be positive. And therefore, if these impedances have positive real parts. Then there will be a net average power flow in the z direction. So, this is how the signs have been chosen that if these impedances have positive real parts they will contribute to average power flow. And that is true of wave impedance looking in any positive direction. If you do not look at the positive z or the positive x direction if you look at the negative direction we once again change the sign. And that will also serve the same purpose that if the real part of the impedances is positive it will contribute to average power flow in the corresponding direction. So, with this definition of these different types of wave impedances we can consider the wave impedances that apply or that we obtain for the various types of waves that the parallel plane guide supports. We have just considered the TEM wave fields. And if we apply this definition we can consider the wave impedances of the wave impedance. We have EX and HY components. And therefore, we should get for the TEM wave a wave impedance which is written as Z x y and looking in the positive z direction how much is it going to be? It will be EX by HY and it is simply beta bar by omega epsilon as and as we made out it is equal to eta. Now, let us look at the other fields that is fields for the TE mode and the TM mode. For the TE modes, TE M0 modes these were the field expressions that we got. And for this what will be the wave impedance? Now we have the y directed E field and the x directed H field. And therefore, what should we write? We will be writing Z y x in the positive z direction. Since it is a wave propagating in the positive z direction let us look at the impedance along that direction and that will be minus E y by H x. And the expressions are all here and we are going to get the value which is omega mu by beta bar that is we can write the expression for beta bar here. It will be omega mu upon omega square mu epsilon minus m pi by a whole squared and the square root of the whole denominator. And we find that as the frequency decreases towards the cut off frequency and exactly at the cut off frequency this is going to be infinite. And as the frequency increases this is going to asymptotically touch the intrinsic impedance. This behavior we have seen for various types of parameters for the TE waves. In a similar manner we can consider the field expressions for the TM modes. These were the fields for the TM M0 modes and we can get the wave impedance corresponding to this TM modes. And going by our definitions that have been written on the board what will we have now? We will have once again Z x y and with the positive superscript since we have E x and H y field components. And they are in phase all other variations are identical except for these amplitude factors. And taking these into account we obtain the expression for the wave impedance here. It is going to be E x by H y and equal to beta bar by omega epsilon as you can make out. In the range where the wave is propagating without attenuation. In the range where gamma bar goes to j beta bar. So, that substitution has been made and then this ratio will come out equal to this. The expression for beta bar can be incorporated here also. We will have omega squared mu epsilon minus m pi by a whole squared whole square root divided by omega epsilon. Two comments here are important. First is that the expressions in terms of beta bar for these wave impedances are quite general. They are independent of the type of waveguide. For example, these will hold good for the rectangular waveguides also. And for the circular waveguides also. In terms of beta bar, beta bar may have a different expression from this. That is a separate matter. But in terms of beta bar these expressions are fairly universal. Secondly what happens to the impedance in the range of frequencies below the cutoff frequencies? You will find that below the cutoff frequency the impedances will become completely imaginary. So, they will have no positive real part and therefore, they will not correspond to any power flow in the direction of propagation. And as we have been mentioning it will just correspond to field decay and there will be no wave motion in the frequency range where the frequency does not exceed the cutoff frequency. So, this concept of wave impedance throws further light on what happens in the different frequency ranges with respect to the cutoff frequencies. Except for the TEM case the wave impedances are not uniform with respect to frequency. And this variation can be plotted as shown here. These are the wave impedances for different types of waves. This is the frequency axis the abscissa and this is the wave impedance. This straight line is the wave impedance for the TEM mode or the TEM wave. This is there are no modes in the TEM case. This line which asymptotically touches this at high frequencies, but otherwise goes to infinity as we approach the cutoff frequency. This curve is the curve for the TE waves. In terms of the cutoff frequency it is quite universal. I mean it is applicable to various types of TE modes. And similarly this curve represents the wave impedance variation with respect to frequency for the transverse magnetic waves. This is where we stop today. In the lecture today we have considered the parallel plane guide and the characteristics of TE and TM waves. How they are different from those of the uniform plane waves that we were dealing with earlier. And we also saw how transverse electric and magnetic waves can be supported on the parallel plane guide. And then we developed the concept of wave impedance. Thank you.