 Welcome to the 24th lecture in the course Engineering Electromagnetics, we continue with our discussion on parallel plane wave guides which we started in the last lecture and the topics we consider today are going to be the transverse electric waves, the field distribution associated with this kind of waves and finally we see how the field of the TE waves can be expressed as a superposition of plane waves. Before we take up these topics let us have a brief and quick recapitulation of the material we discussed last time. We considered a parallel plane wave guide of this kind which consists of conducting planes which are infinite along the z and the y directions. Therefore they impose boundary conditions such that the tangential electric field at these planes is 0 which is the same thing mathematically as saying that the normal component of the magnetic field is 0 at these conducting planes. With this basic configuration and the boundary conditions identified we wrote the Maxwell's equations for the region between the perfectly conducting planes and for the sake of simplicity we assume that this medium is a perfect conductor and we wrote the components of these Maxwell's equations in Cartesian coordinates with these associated wave equations. These equations were simplified by making the following considerations. We assume that the wave propagates in the z direction with the propagation constant gamma bar so that the derivative in the z direction can be replaced by minus gamma bar and since the structure is infinite along the y direction no particular conditions are imposed in that direction. We considered that the derivative with respect to y of all the field components can be replaced by 0. So with these considerations we obtained the simplified versions of the Cartesian components of the Maxwell's equations. These 6 equations and the associated wave equations for the magnetic field and for the electric field. Then we expressed the transverse field components H x and H y and E x and E y in terms of the longitudinal field components E z and H z. Once we reach this stage it is possible to bring in a further order into this major equations. We say that in general either H z or E z must be non-zero for all field components to be so that it does not remain a trivial situation. We will see an important exception to this later on but right now we say that in general one of these two longitudinal field components E z or H z must be non-zero. So depending on which one is non-zero we get a transverse electric field kind of solution which is referred to as either a T e wave or an H wave that is when H z is non-zero and we get T m waves or alternatively E waves when E z is non-zero. From this point onwards we started considering the solution of these equations subject to the boundary conditions identified earlier for the T e type of waves. For the T e waves we were able to see quite simply that the field components which would exist would be H z H x and E y. The other three field components become zero. E z is as it is zero by virtue of the very assumption that we are talking about the T e wave solution and the other two field components also become zero because of that. Therefore we wrote the y component of the wave equation which the electric field satisfies. Essentially what we are doing is we are picking up a field component which is tangential to the conducting plates in the parallel plane waveguide. So that we can in a simple manner apply the boundary conditions namely the tangential electric field is zero at the conductors. So we pick on E y and write the y component of the wave equation in this manner. Recalling that we introduced a symbol H where H squared was omega squared mu of silent plus gamma bar squared. This simplifies to this kind of second order partial differential equation. And then we consider the nature of E y. It has no variation with respect to y. It has this kind of variation along the z direction since it is propagating with the propagation constant gamma bar in the positive z direction and the only variation that is not specified yet is the variation with respect to the x direction. Recognizing this, this partial differential equation simplifies to this ordinary second order differential equation. Once we reach this stage it is quite easy to write the solution for the part of E y which varies with respect to x. That is E y not x is equal to C 1 sin of H x plus C 2 cosine of H x. Now one can look at the structure that we are dealing with once again. This is the structure alright. And we have got a mathematical solution for E y. And E y is tangential to these conducting planes. And therefore at x equal to 0 as also at x equal to a this tangential electric field component must be 0. That is how we have to choose the constants of integration and the parameter H. So, once we apply the condition that it should be 0 x equal to 0. We straight away make out that the constant that we have introduced C 2 must be equal to. Similarly it is also required to be 0 at x equal to a the upper conducting plane. And what would that imply? That would imply that H a should be equal to an integral multiple of pi. Where m may take on values 1, 2, 3 etcetera. As regards the choice m equal to 0 is concerned. We will see that this will lead to all field components becoming 0 a trivial situation. And therefore that is excluded from the possibilities of values that m can take. Which implies that the parameter H must be equal to m pi by a. That is it is related to the spacing between the parallel conducting planes. And also an index m has been introduced. The role of this index will be mentioned. We will have something to comment on this as we proceed. Therefore, now since we have got the x variation specified. We can now write the expression for the y component of the electric field. The complete expression E y is equal to C 1. And then sin of m pi by a x e to the power minus gamma bar z. So, except for this amplitude constant C 1. We have got now the complete solution for E y for the particular structure that we are dealing with. And the boundary conditions that this structure imposes. As far as this amplitude constant is concerned. It will depend upon how much power one has transferred to this particular field solution from the exciting source. And therefore we need not worry about it too much right now. This is as far as the y component of the electric field is concerned. What are the other field components that exist in this case? The other field components that exist are H z and H x. How are we going to get the expressions for these other field components? We go back to our expressions for the transverse field components in terms of the longitudinal field components. And we see that as far as H z and H x are concerned. They are simply related to E y. H z can be obtained from here. And H x can be obtained later on in terms of H z using this expression. So, using these expressions that have been derived earlier. We are now in a position to write down the complete field expressions. We are going to get H x equal to minus gamma bar upon j omega mu c 1 sin of m pi by A x e to the power minus gamma bar z. And H z turns out to be minus m pi by j omega mu A c 1 cosine of m pi by A x e to the power minus gamma bar z. Now since these field expressions are obtained in terms of the index m which governs the order of variation along the x direction. And there is no variation with respect to y direction because of the particular structure that we are dealing with. We can call the solution as the solution for T e m 0 modes. Here new term is being introduced that is the mode. A mode corresponds to a particular choice of m in these field expressions. And it is a very well defined particular kind of field variation with respect to the x and y, x, y and z directions that a particular mode would refer to. A wave may consist of more than one mode. That is the fine distinction between the mode and the wave. One can compare it to the general situation that we could have let us say an electrical signal which in general has any kind of time variation. And the specific types of electrical signals would be once which have specific types of time variation. For example the sinusoidal time variation. And in fact the analogy can be extended further just as an electrical signal with any general time variation can be expressed in terms of components which have sinusoidal time variation. Any general field for a particular wave can be decomposed into the fields corresponding to specific types of modes. So a time variation is now being replaced by a spatial variation. With that distinction modes serve the same purpose as let us say the sinusoidal time variation serves in the case of time variation. So it should be sufficient here to say that a mode refers to a very specific type of field configuration which of course is a solution of the problem at hand. And in a general situation the wave that propagates on the structure could be composed of more than one mode of one kind or perhaps more than one mode of different kinds. So that distinction between a wave and modes can be kept in mind. So depending on the choice of m we are going to get solutions for different types of modes and these will be referred to as let us say T e 1 0 mode, T e 2 0 mode and so on. The first index here the first subscript here refers to the order of variation along the x direction. And the second subscript here refers to the order of variation along the y direction. And you can see that we are choosing a terminology which is going to serve us even when we use waveguides which have confinement in both x and y directions. Now let us consider the parameter gamma bar. Gamma bar we call the propagation constant and it is equal to h squared minus omega squared mu epsilon. If you look up the expression for h squared it will be consistent with that and of course the square root of this entire expression. In general we say that gamma bar is complex and it has a real part say alpha bar and an imaginary part beta bar. Now the case that we have considered in that case also we can note down the fact that h has been obtained as m pi by a. So that on the right hand side both terms are real under this simplifying assumption that the medium that is filling the region between the conducting planes is a perfect dielectric. So that it has a real permittivity. So since both these terms are real gamma bar can either be completely real depending on their relative magnitudes or it can be completely imaginary. It cannot be complex in that sense. By virtue of the simplifying assumption that we have made we are considering a perfect conductor and we are considering a perfect dielectric. So losses are not appearing in this solution and therefore gamma bar can either be real or can only be completely imaginary. When gamma bar is completely imaginary then we will have gamma bar going to j beta bar and correspondingly e to the power minus gamma bar z. The factor that is taking care of the z variation in these field expressions is going to change to e to the power minus j beta bar times z where further beta bar is going to be equal to omega squared mu epsilon minus m pi by a whole squared whole squared root. And one can make out the fact that it is only above a certain frequency where beta bar will be real or alternatively gamma bar will be completely imaginary. Below that frequency the situation will be the other way round and gamma bar will be completely real. Which case is going to correspond to an attenuating situation as far as the positive z direction is concerned. Whereas this situation where gamma bar is completely imaginary is equal to j beta bar is going to correspond to a propagating wave with associated phase shift as it propagates and with no change in the amplitude. Ordinarily this is the range of frequencies where we will be interested in. This is the range above let us say a certain cutoff frequency where the wave would propagate without attenuation under the simplifying assumptions of no losses. And here an important distinction is appearing between the types of waves that are going to be supported on this parallel plane wave guide and uniform plane waves. There there was no such thing as a cutoff frequency. But here we see that depending on the value of frequency the behavior is going to completely change. And only above a certain frequency we will get propagation without any attenuation. This we will comment on a little later once again. So in this propagating frequency range and let us say for one particular mode say m equal to 1 we can write down the field expressions for the TE10 mode that is m is going to be taken to be 1. And in this situation when gamma bar is equal to j beta bar that is above the cutoff frequency that we referred to. We can write down the various field expressions that we have obtained for the TEM0 modes and they will be appearing as follows. We will have E y equal to C 1 sin of pi x by A. And then here we have e to the power minus j beta bar z. The other two field components can also be written. We have H x equal to substituting gamma bar equal to j beta bar in the earlier expression for H x. The expression that we get for H x is as follows. It is minus beta bar by omega mu and then C 1 sin of pi x by A. And then pi x by A e to the power minus j beta bar z. And finally H z expression turns out to be j m pi by omega mu A C 1 cosine of pi x by A e to the power minus j beta bar z. They all correspond to a particular mode which is propagating in the z direction with a certain phase shift constant. So the z variation is identical in all but their amplitudes can be different. Their variations with respect to x direction in this case and with respect to x and y directions in a general case can be different. But since they correspond to a particular mode their z variation must be identical. Now what we need to consider many times is the way the field looks like with reference to the waveguide the field configuration. How if you if you wanted to visualize the field it will look like. It is relatively simple for us to visualize how the y component of the electric field is going to look like. But the magnetic field components we have more than one magnetic field component and therefore they are going to combine in a manner which right now is not easy for us to predict. So for that purpose we consider the corresponding actual time varying expressions for these phasor expressions. Because after all the complex notation is only a mathematical convenience. There is nothing complex about the fields otherwise. So we get the actual time varying expressions corresponding to these field components. The time varying field components can be obtained by using our standard procedure. We multiply by e to the power j omega t and consider the real part. And when that is done we get e y which is going to be c 1 sin of pi x by a and then instead of the exponential factor we get cosine of omega t minus beta bar z. Similarly h x turns out to have an expression which is minus beta bar by omega mu c 1 sin of pi x by a and once again cosine of omega t minus beta bar z. And then we get the it is not difficult to see that h z is going to have an expression which is minus m pi by omega mu a c 1 cosine of pi x by a and then here we have sin of omega t minus beta bar z and then here we have sin of omega t minus beta bar z. And it clearly brings out something to demonstrate our statement earlier that there is nothing complex about the fields in reality. These factors j etcetera that appear they basically refer to a particular phase. Since h x and h z are in phase quadrature therefore they have a phase therefore they have a difference of a factor j. So this one has a cosine kind of variation with respect to time this one has a sin kind of variation if we take the same time reference alright. So with these time varying expressions available to us we can now consider how the fields look like with reference to the parallel plane guide that we have taken up for analysis. The parallel plane guide here for example on the left side is shown in the x y cross section. And therefore we have this as the x direction and this is the y direction x equal to 0 is this plane and x equal to a is this plane. So one can clearly visualize that at some z equal to constant plane we are considering the x y plane cross section of the parallel plane wave kind. Now how would one represent let us say the electric field in this. Electric field has only one field component E y therefore its representation is fairly straight forward. But even here one can choose two different types of representations or notations. For example one kind of notation will be where the magnitude is shown very clearly magnitude is varying only with respect to x. And it is varying in a sinusoidal manner with respect to x at x equal to a it is 0 and at x equal to 0 it is 0. And therefore round the plane x equal to a by 2 we can consider a sinusoidally spatially sinusoidally varying field. And we will have to mark this as E y because otherwise there is no direction included in this representation. And also since E y is a function of z also and so also for omega t one will have to specify the value of this argument omega t minus beta bar z equal to so and so at that particular time and at that particular value of z will be the variation of E y like this with an amplitude corresponding to C 1 and a multiplication by this factor. As we deviate from that value of z or from that value of time the sinusoidal variation nature will be maintained, but the amplitude is going to change. And therefore strictly speaking this representation should be a three dimensional representation with the amplitude E y coming out let us say out of this plane if it is of a positive sign and going into the plane if it is of a negative sign. So that kind of visualization we will have to carry out ourselves. Now this is a magnitude representation of the fields and we see that while it clearly shows the magnitude variation the direction has to be specified by put labeling this curve if it corresponds to E y we have to say that this is for E y. This restriction can be relaxed by going in for a slightly different type of representation by using directed line segments representation where we may have a magnitude and direction kind of representation. Here we draw directed line segments if it is E y that we want to represent that these line segments will be directed in the y direction and using the same axial notation that is this is y and this is x and x equal to 0 and x equal to a as noted in the figure above we may consider this kind of representation. Let us say like this then how do we represent the fact that as we move away from the middle midpoint x equal to a by 2 plane the field is decreasing we increase the spacing between these directed line segments and therefore let say one representation that is possible is like this directed line segments which are spaced closely together where the field strength is more and then the spacing is decreasing where the field strength is less. Obviously it does not very clearly or accurately show the magnitude but it does show the direction and it gives an idea about the manner in which the magnitude is varying. So one can choose either of these two notations to denote field components when we go to a field where more than one component exists then we will find that this notation is somewhat more it is more communicative of the actual overall field. So let us do that and we first use the magnitude representation for the magnetic field magnetic field components and we now since we have the x and z components in the magnetic field we choose the other cross section of the parallel plane guide such that it is the x z plane that we are talking about and of course these planes at x equal to 0 and at x equal to a remain as they were. And now we put down the values of these cross sections at different values of z. We could say that we are taking a snapshot of the field at a particular instant of time so that t is fixed and only z is varying. And the way z appears in this argument as we go to increasing values of z this argument should decrease. And therefore the value of this argument omega t minus beta bar z may be taken to be equal to let us say pi here and then pi by 2 here and then 0 and so on may be minus pi by 2 here. That is the value is a matter of what kind of scale we choose along this z direction. But let these be the planes omega t minus beta bar z equal to pi pi by 2 0 minus pi by 2. Why we have chosen this kind of planes will be clear in a minute. For example at omega t minus beta bar z equal to pi at z is completely 0. So at such a plane we have only h x. And as far as this argument is concerned it has a maximum value and the argument has the value minus 1. Maintaining the same notation that is the positive values are indicated by drawing occur to the right of the reference plane h x can be shown here in this manner h x has a sinusoidal variation. And these two negative signs cancel. So overall it is a positive sign and therefore we are drawing it to the right of this reference plane. And as I mentioned earlier actually it should be considered as something projecting out normal to the plane of the paper. We must clearly understand that this is the variation at omega t minus beta bar z equal to pi. Where we have only h x which exists. If we shift the value of this argument slightly from pi h z will also appear. And the value of this component itself is going to be altered. So these things we have to keep in mind. Similarly when we go on to the value of this argument equal to pi by 2. Then we find that only h z exists at this specific location. Sign omega t minus beta bar z equal to pi by 2. And h z has a negative sign in front. The variation is governed by the cosine function at x equal to a it is cosine of pi. So another negative sign is going to appear there. So keeping these things in mind the h z component is going to have a cosine sinusoidal kind of variation. And maybe it can be represented in this manner. Once we have considered these two curves. And we have to be very careful in going from the phasor notation to the time varying notation. And in transferring these expressions into this kind of curves. We have to be very consistent in our notation. For example positive sign must be represented by in a particular manner consistently. Here we are representing the positive values by drawing the curve to the right of the reference plane. You can choose some other notation also but you have to be consistent in following that notation. Once we have drawn these the curves and the field components that exist at other planes can be drawn just by inspection. For example here once again we will have only h x and it can be represented by a curve which is like this. And at this point we will have only h z which is going to have a variation which is like this. As for the case of the y component of the electric field this is the magnitude representation of the components of the magnetic field. Now actually it is just for the sake of convenience that we talk about these components separately. What exists in the waveguide as the T10 mode is a composite magnetic field with the components being like this at different points. And therefore we will like to combine these in a certain way and see what is the overall magnetic field looking like. And that can be done by going in for the other kind of representation which we call the representation in magnitude and direction. Now please look at this carefully. At this plane where this argument has the value pi we have only h x existing with maximum amplitude anywhere on the guide and it is x directed. So using this kind of notation we can represent this by directed line segments which are oriented along the x direction. And we do not take these up to the conducting planes because at the conducting planes h x also is 0. And as we move away from this reference plane on either side along the z direction the h x component value itself is going to reduce as we noticed earlier. And therefore as we move away from this we draw similarly directed line segments but shorter in length. And with an increased spacing to signify the fact that as we move away from this reference plane the h x value is going to reduce according to the cosine of the argument omega t minus beta bar z. These are all x directed in some manner they represent the magnitude they surely represent the direction very clearly. But whether it is positive x direction or the negative x direction now following our convention since positive sign was represented to the right of this reference plane we can put an arrow sign which is pointing towards the positive x direction. The same procedure can be repeated at this reference plane where this argument has a value 0. The situation is going to be identical with the reversal in the direction. And therefore we can draw closely spaced minus x directed line segments here and with somewhat increased spacing and reduced length lines segments here. These must not extend up to this reference plane because there h x is 0 and only h z has the maximum value. The same representation is now extended to the z component of the magnetic field. And we see that the z component of the magnetic field is maximum at this reference plane. And following our notation of sign consistently we can draw this kind of line segment here and here. And noting the sign the arrow directions must be like this positive z here, negative z here. As we move away from these conducting planes h z is reducing in value. And therefore at a slightly greater distance from these conducting planes we may draw h z in this manner. Once again the directions are going to be like this which representation can be transferred to corresponding planes with a reversal in the sign. And therefore we may draw h z like this. What will be the arrow directions here? It will be negative z here and positive z here and in a similar manner we will have the z component of the magnetic field represented by this kind of arrows properly identified like this. This is in a way this skeleton showing the important features of the overall magnetic field. And now one can very simply visualize that actually the field is of this form giving us the magnetic field in closed loops in this manner. And if we have proceeded correctly using the notation consistently the arrow directions will not clash otherwise they will clash which obviously is going to be an error. And proceeding consistently and correctly we will have the same arrow direction on a particular magnetic field line and this is the representation in magnitude and direction. Depending on the purpose and the patients one can fill it up appropriately taking care that the spacing between the lines or this the number of lines per unit width in a certain direction represents the magnitude of the field line at different points. If you have any clarifications to ask you are welcome. So this is where we like to stop today. Today we have considered the field components for the transverse electric waves and we have introduced the notion of modes. And therefore we have got the expressions for TE M0 modes. We considered in detail the field expressions for the TE 10 mode. We have considered its time variation and we have seen how these fields can be represented to indicate what kind of field lines are there associated with these fields. Thank you.