 Welcome to the 27th lecture in the course Engineering Electromagnetics. For the last few lectures we have been considering the parallel plane guide as the last topic in the series we consider the attenuation constant of the various types of waves that we have considered on the parallel plane guide. For this purpose we make the following considerations. You would recall that in general the propagation constant gamma bar for the parallel plane guide was written to have the following expression. We wrote gamma bar equal to m pi by a whole squared minus omega squared mu epsilon whole square root and we said that for the frequency range such that omega is less than a certain frequency called the cutoff frequency. Gamma bar is completely real we may call it alpha bar and for the frequencies greater than this cutoff frequency gamma bar is completely imaginary and we call it let us say j beta bar. And we said that this frequency range does not correspond to propagating waves and the fields would just attenuate exponentially with the attenuation constant alpha bar. On the other hand in this frequency range the fields would propagate without any attenuation in the idealized situation that there is no lost presence. We consider the conductors constituting the parallel plane guide to be perfect conductors and the material filling the parallel plane guide also to be a perfect dilute. And therefore, in that idealization this situation is quite fine. However, when we consider practical materials, practical materials which constitute the parallel planes or practical media which would fill the parallel plane guide they are going to have some loss. And therefore, in the frequency range where in the ideal case we said that there is no attenuation there is going to be a small amount of attenuation present depending on the quality of the material that we utilize for fabricating the wave guide. This small amount of attenuation is very different in quantity and nature from the large attenuation constant that exists in the frequency range below the cutoff frequency. This distinction we must keep in mind and therefore, we say that in practice actually gamma bar would be equal to a small attenuation constant plus j beta bar even in the frequency range in which normally we said that wave propagation takes place. Now, how do we estimate this small attenuation constant that is present in the frequency range in which waves propagate on the parallel plane guide? We discuss a general method for obtaining this. The method for calculating the attenuation constant and so that there is no doubt left we may write here for the frequency range where omega is greater than omega c. And for this purpose for obtaining a general method for estimating this small attenuation constant we once again make use of the transmission line equations. And we write let us say v equal to v naught e to the power minus alpha z e to the power minus j beta z where alpha and beta would depend upon the transmission line on which such a wave is existing. But in general for a practical transmission line there will be some attenuation constant alpha and a phase shift constant beta. Similarly, we are going to have the associated current which may be expressed as i naught e to the power minus alpha z e to the power minus j beta z. And now let us calculate the power that is associated with this wave and we calculate the average transmitted power. And let us use the symbol for power transmitted as W t and we call this average power which is going to be given by half real of v i star using the notion of complex power. Therefore, it turns out to be half real of v naught i naught star e to the power minus twice alpha z. And immediately we see that the transmitted power at different values of z as the wave propagates is decreasing by virtue of the attenuation which is continuously taking place. So, let us consider the change in the transmitted power per unit length or the rate of change of transmitted power as a function of z. And we can estimate this by taking the derivative of the power transmitted with respect to z. So, we write del W t average by del z which is going to be minus 2 alpha times half real of v naught i naught star e to the power minus 2 alpha z or in brief it is going to be minus 2 alpha and W t average itself. Now normally this derivative is interpreted as the rate of increase of let us say W t. Since we have a negative sign appearing we may interpret this as a rate of decrease by associating a negative sign in front of this, which is also quite realistic because of attenuation the total power transmitted is reducing. So, negative rate of change of W t average is going to be positive. Rate of change of rate of decrease of transmitted power is equal to twice alpha times the total power transmitted. What is this rate of decrease of power transmitted? Why is this power transmitted decreasing? It is decreasing because there is power loss. And therefore, this we make call W L average. And that gives us very simple expression for estimating alpha the attenuation constant for the propagating wave that is alpha is equal to W L average divided by twice W t average or in words it is the total power loss total power loss per unit length. This is the W L average per unit length that we have to keep in mind total power loss per unit length divided by twice the total power transmitted. Now somebody may raise an objection that we are not using the qualifying word average in these two terms the numerator and the denominator. Now actually that is not necessary as long as both are calculated on the same basis. It could be the peak powers or it could be the average powers as long as we are using the same kind of calculation for both it does not really matter. And that gives us a very straight forward expression for estimating the attenuation constant alpha. Now therefore, what is our strategy for calculating this attenuation constant alpha? We use what is called an engineering approximation. The steps in this engineering approximation can be listed as follows. We list down so that we can utilize this later on also the method which is known as the engineering approximation for estimating the attenuation constant. What are the steps in this engineering approximation? We first say that estimate the fields in the given structure ignoring losses like we have done for the parallel plane wave kind. Calculate the fields satisfying the boundary conditions imposed by the structure at hand and in the first step we ignore losses. If there is a conducting material which is being used we assume it to be perfect. If there is a dielectric material appearing in the structure we assume it to be perfect again. Now we make the important assumption that even in the presence of losses even in the presence of small amount of losses the basic field structure is not going to change. As long as the losses are small this is a reasonable engineering approximation. So, the second step amounts to assuming that the fields unchanged even in the presence of small amount of losses. Now this is somewhat restrictive it means that if the losses are large then we have to solve the problem taking the losses into account from the beginning. But that is rarely the case mostly the waveguides that we are going to make will have small amount of loss then only they will be of practical utility. Then we calculate the power transmitted and as we just clarified it may be peak or it may be efferent it is not going to really matter as long as the other term is also calculated on the same basis. So, we calculate the power transmitted along the length of the waveguide. For example, if you have assumed the wave propagation in the z direction in the parallel plane guide. So, what is the power transmitted in the z direction that is what we calculate here. Then we estimate the power loss estimate the power loss in whatever material is lossy it may be the conductor or it may be the dielectric or it may be both estimate the power loss in the lossy conductor and the dielectric. And then of course, we have got an expression for alpha saying that alpha is the total power loss per unit length divided by the total power transmitted as we just wrote down. This is the method which is known as the engineering approximation and it works quite well as long as the losses are small. Having obtained an estimate of the attenuation constant alpha this way if one is more interested in arriving at accurate expressions for the fields for the lossy waveguide one can use this in the fields and then undergo another iteration for calculating the attenuation constant. And hence that will be a refined estimate of the attenuation constant and the field expressions can be defined further. However, in practice the second iteration is rarely necessary because we use waveguides which have a low loss low attenuation constant. Now, this method can be demonstrated for the various types of waves that the parallel plane waveguide supports but before that let us see how we may estimate the power loss in the conductor. Usually the dielectric material that is utilized has a smaller relatively much smaller contribution to the total power loss as compared to the conductor loss. Let us see how this can be estimated in practice. You would recall that the waveguide that we are dealing with is the parallel plane waveguide and now we are interested in let us say estimating the power that is lost in the conducting planes per unit length in the direction of propagation. How are we going to be able to do that? We consider the component of power that flows normally into these planes. That is the power which is taken away from the power flow in the z direction and that is the power loss. The power loss has to be considered in both the conducting planes naturally. And for this purpose we assume again not a very restrictive assumption that the thing thickness of the conducting planes is sufficiently large. What would be sufficiently large? A thickness which is several skin depths will be considered a thickness which is sufficiently large and therefore the effect is that we can consider the power that is lost in the conducting medium propagating as a plane wave uniform plane wave in the conducting planes. That is the advantage of these simplifying assumptions and therefore for the power loss in the conducting planes or we may write this simply as conductor loss. We use the expression that P n average. The normal component of the pointing vector flowing normally into the conducting surface. That is what would be associated with the power loss in the conductor. And what would be what would this be equal to? It will be half real of E tan cross H tan star. Now as far as the tendential component of the magnetic field all these are both of these are at the surface at the conducting surface where we want to estimate this normal component of the pointing vector that should be quite obvious. As far as the tendential component of the magnetic field is concerned we have got expressions for the magnetic field for various types of mode supported within this parallel plane guide and we can pick the component which is tendential. Calculate its value at these conducting planes and use that here. What about the tendential component of the electric field? We impose the condition that for these perfect conductors the tendential component of the electric field must be 0. But it is also clear that for the practical case when the conductors do not have infinite conductivity there will be some small value of the tendential electric field which is going to be present. How is this to be estimated? So then we recall the relationship between the fields for a uniform plane wave propagating into an infinite medium composed of the material of the conducting planes. And then these two are going to be related through the surface impedance so that we can write E tan by H tan is equal to the surface impedance of the material the conducting material at hand which also is equal to the intrinsic impedance of this conducting material as we have seen. And it is equal to R s into 1 plus j where R s is omega mu m by 2 sigma m where the subscript m goes to show that these are the quantities for the conducting materials or for metal. Using this therefore we can obtain p n average as half R s magnitude H tan square because that is what H tan multiplied by H tan complex conjugate is going to be and these are going to be mutually perpendicular to the direction of propagation of power into the conducting planes. And therefore the cross product will be simply a multiplication it can be seen easily that this is simply the I squared R kind of loss in the conducting medium the units of this are going to be watts per meter square. The tangential component of the magnetic field is associated with a linear current density on the surface of the conducting material and the relation the vector relation is as follows J s linear current density is equal to n cap cross H tan where n cap is the unit outward normal at the surface at which we want to estimate the linear current density. Magnitude wise it is simply J s equal to H tan and therefore we see that the normal component of the pointing vector is equal to half R s magnitude J s square this is the power loss this is the ohmic loss per unit area of the conducting material. Since you already have the magnetic field it is easiest to apply it in the form half R s magnitude H tan square right. Now this thing can be applied to calculating the attenuation constant alpha for the various types of waves that we have become familiar with on the parallel plane guide. We do this calculation first for the TEM wave attenuation constant for TEM wave. Recalling the steps in the engineering approximation method of estimating the attenuation constant the first step was calculate the fields ignoring losses. This part we have already done and the fields obtained were as follows we have the y component of the magnetic field in terms of an amplitude constant C 4 times e to the power minus J beta z and the other field component present in this case was E x equal to eta times C 4 e to the power minus J beta z where eta is square root of mu upon epsilon the permeability and the permittivity of the medium filling the parallel plane wave guide and beta in this case also had a very simple expression beta was simply omega times square root of mu epsilon. Since this is in the idealization that no losses are present the loss mechanisms the conductivity of the conductor or that of the dielectric is not appearing in these expressions that is quite fine. The next step is to calculate the power loss and as I said while in general the power loss could occur because of the lossy dielectric material as well as the lossy conducting material the dielectric loss is relatively much less and therefore we calculate the conductor loss and the magnetic field that is now present H of y one can visualize this as follows H of y in the y direction it is uniform throughout the space between the conducting planes and if we consider the unit normal to these conducting planes it is going to be in this direction this will be n cap for the top plane and this will be n cap for the bottom plane and H of y is of course in the y direction. So n cap cross H y is going to be in which direction it is going to be in equal and opposite directions and it will be a direction which is like this in the upper plane which we may call J s and a direction like this for the lower conducting plane and if you obtain J s that will be the current per unit width along the y direction that is how it is called linear current density. And if you calculate the loss based on this that will be the power loss per unit area for a unit width along the y direction and for a unit length along the z direction. So with this clarification we can now proceed with the calculation of the power loss we saw that pn average is equal to half rs magnitude h tan square and it is going to have a value which is half rs c4 square assuming that c4 is real. This is the power loss per unit area of each of the conducting planes and therefore the total power loss W L and since this is calculated on the basis of the power loss in the conductor sometimes a subscript is also added here either just a c or conductor. What will be the total power loss in the conductor? It will be twice of this since there are two conducting planes alright times let us say b times half rs c4 square where b is the width of the structure along the y direction. So per unit per for a width b of the structure and for a unit length of the structure this is the total power loss. For calculating the power transmitted also we will use the same width of the structure because otherwise the structure is of infinite width. So to get around that problem we say that let us assume that the structure is of width b which is simply b times rs times c4 square. Similarly we can calculate the pointing vector which will be involved in the total power transmitted. We can calculate p t average as half real of e x cross h y star and we can put vector signs here and say that we consider the z component of this which is the only component appearing in this case which is going to be equal to half eta c4 square which is the power transmitted per unit area and therefore this is in watts per meter square. Now what is it that we require for the attenuation constant what we require is the total power transmitted which can be obtained by integrating this pointing vector over the cross section of the waveguide that is wt average is equal to p t average da integrated over the cross section of the waveguide. For the simple case of the TEM wave since the pointing vector or the z component of the pointing vector is uniform everywhere within the waveguide the integration is trivial. We just consider what is the cross sectional area now how much is the cross sectional area the spacing is a along the x direction and along the y direction we once again use the width b to be consistent and therefore this is simply half ab eta c4 square. So, that now we have everything for calculating the attenuation constant and alpha comes out to be w l by twice wt and as you can see it is in a straight forward manner r s by eta times a. We can substitute the expressions for eta and r s but before we do that a simple comparison is quite instructive one can compare this with the attenuation constant that we obtained for low loss transmission lines if we considered only the resistive term and in that case alpha was half r by z naught and the correspondence between the two expressions is quite clear there we had z naught as the characteristic impedance of the transmission line here eta is the characteristic impedance or the wave impedance for the TEM wave in the parallel plane guide. We can substitute for these expressions r s and eta and we obtain this as 1 by a square root of epsilon by mu and then omega mu m by 2 sigma m once again whole square. So, following our method we have obtained the expression for the attenuation constant. We started with ignoring losses calculated fields and then we assume that the field expressions do not really change even if a small amount of loss is present and then we calculated the power loss and the total power transmitted. If the dielectric were also lossy that dielectric loss can also be estimated in a straight forward manner using the method described in the context of the transmission line. And therefore, it is the total loss that will appear here instead of just the conductor loss and this is the kind of expression that we are going to have. The expression can be plotted as a function of frequency one can see that it is going to vary with frequency as the square root of frequency. This kind of variation in the attenuation constant because of the surface resistance of the skin resistance was pointed out in the case of the transmission lines also. The plot for the attenuation constant as a function of frequency is shown here. The frequency axis is here the abscissa and this is the attenuation constant and for the TEM wave for which we have just calculated the attenuation constant the variation in the attenuation constant as a function of frequency or omega is like this. Since there is no cut off frequency for this wave it starts from 0 frequency onwards and it starts rising as the square root of frequency. If there are no questions then we proceed to calculate these attenuation constant for the TE waves as well. The method of calculation is going to remain the same attenuation constant for let us say TE M0 modes. The starting point would be the field expressions obtained ignoring losses and in the frequency range in which wave propagation takes place these field components are written as follows. E y is C1 sin of M pi by A x e to the power minus j beta bar z and H x is minus beta bar by omega mu C1 sin of M pi by A x e to the power minus j beta bar z. And we also have the z component of the magnetic field in this case H z equal to j M pi by omega mu A C1 cosine of M pi by A x e to the power minus j beta bar z. And as you would recall here beta bar is omega squared mu epsilon minus M pi by A whole squared whole square. And we assume that even in the presence of a small amount of loss this is the kind of fields that are going to propagate. And for power loss we write p n average which is going to be half r s magnitude h tan square. We have two components of the magnetic field here right, but it is quite easy to make out which of these is the tendential field component. This is the parallel plane waveguide and it is the z component of the magnetic field which is going to be tendential to the conducting planes. And that is what is going to be involved in this calculation. And while in general H z is a function of x it has to be evaluated at x equal to 0 or at x equal to a for this purpose. And therefore, it is simply half r s and then M pi C1 by omega mu A whole square. So, that the power loss in the conductor W L average taking into account both conductors both conducting planes is going to be and if you assume that the width is B along the y direction it is going to be B times r s M pi C1 by omega mu A whole square. Next we undertake the calculation of the power transmitted. And for that purpose we first have to estimate the pointing vector in the z direction and calculate its average value. And it is going to be half real of E y cross H x star. And of course, it will have only the z component and one can see that it is going to be the negative sign is going to be nullified it will be beta bar C1 squared by twice mu into omega. And then sign squared M pi x y now in this case because of the variation of the fields along the x direction the pointing vector giving the power flow in the z direction is not uniform at all values of x. It is varying in this manner and therefore an integration is going to be required however the integration is going to be quite straight forward. The total power total transmitted power is going to be p t average integrated over the cross section. And when we integrate this part from 0 to a we are simply going to get a value a by 2 alright. And then of course we are going to consider that the width along the y direction is b. So, keeping these two things in mind we are going to get the average power transmitted as follows it will be beta bar C1 squared by 4 omega mu times a b. So, that the attenuation constant turns out to have the following expression alpha is twice M squared pi squared by omega mu a cube and then r s by beta bar. One can write the expressions for r s and beta bar and obtain the expression for the attenuation constant as twice M squared pi squared and then omega mu M by twice sigma M by omega mu a cube into beta bar which is omega squared mu epsilon minus M pi by a whole square root. Of course it is a function of frequency and the variation that it has as a function of frequency is shown here. The interesting thing is that the attenuation constant now is decreasing with frequency. This is the variation of the attenuation constant for the T e modes. And of course below the cutoff frequency there is no wave propagation and therefore there is no attenuation associated with the wave propagation. The expression for the attenuation constant for the T m M 0 modes can also be obtained in a similar manner and for the sake of completeness it is mentioned here alpha T m M 0 modes and the expression turns out to be twice omega epsilon r s by a beta bar. One can expand this further by writing down the full expressions for the surface resistance and the phase shift constant. This has a different type of variation with frequency and it is plotted here. The attenuation constant as a function of frequency for the T m modes is the one shown in blue. And the interesting thing about this is that it undergoes a minimum. At the cutoff frequency this also rises very sharply and then it starts varying in a manner very similar to that of the T m waves. And at root 3 times f c which can be seen by considering its derivative with respect to omega and considering where it is 0. It can be seen that at this value of the frequency the attenuation constant undergoes a minimum. Now all these observations can be very important when we want to utilize the parallel plane guide in practice. This is where we will stop this lecture. In the lecture today we have considered the attenuation constant for the various types of waves that are supported on the parallel plane guide. We first introduced a general method for estimating the attenuation constant using an engineering approximation and then we obtained the expressions for the T e m and T e waves on the parallel plane guide. We also considered the frequency variation of the attenuation constant. Thank you.