 Welcome to the 17th lecture in the course Engineering Electromagnetics. We have been discussing the topic of wave propagation. Continuing with the same topic, the concepts for discussion today are the following, the surface impedance, the power loss in a conductor and the reflection at a perfect conductor at normal incidence. Considering the first concept listed for today that is surface impedance, we proceed as follows. We introduced the concept of the depth of penetration in the last lecture and we saw that the depth of penetration is the reciprocal of the attenuation constant alpha. When we consider the depth of penetration for good conductors for which alpha is very large, it is quite clear that the wave is going to penetrate only within a very short distance in a good conductor. And therefore, the wave phenomena are restricted to more or less the surface in a good conductor. And therefore, when we talk about the concept of surface impedance, the concept may be introduced in the following manner. We consider a large block of a conducting medium. Let this be the x direction, let this be the y direction and the block extends in the z direction. We consider that a wave is propagating inside the medium in the y direction. We consider it to be a plane wave and then we recall that for a plane wave, the field components are in a plane normal to the direction of propagation. And therefore, we expect the field components electric and the magnetic field components either in the x direction or in the z direction. Therefore, let us say that this is the electric field vector say oriented along the x direction and therefore, for this y propagating wave, we will have a corresponding magnetic field intensity vector which is in this direction. And therefore, we can write this as h. Let us consider that these are the field values for the electric and the magnetic field vectors at the surface at y equal to 0. And this wave propagates inside this conducting medium with the propagation constant gamma, gamma the propagation constant of the medium. And therefore, all field quantities are going to vary according to the factor e to the power minus gamma y. A question can be raised that why not e to the power plus gamma y. Then we say that we are considering that the medium is of a very large extent along the y direction. So, that there are no discontinuities in the y direction and a wave which is propagating in the negative y direction is not generated. So, considering that the medium is of sufficient thickness along the y direction, we can restrict our attention to a wave which is propagating in the positive y direction. Let us look at the kind of current that is going to flow in the conducting block. Let us say that we plot it in the x y plane at the surface depending on the electric field and the conductivity of the medium. We are going to have some value of the conduction current density J naught. And as we go down along the y direction, the electric field vector is going to change in this manner. And accordingly everywhere the conduction current density vector is going to change in the same manner. And therefore, if we try to plot it, it may have a plot which is like this where this plot is J naught e to the power minus gamma y. That is at different values of y at different depths from the surface the current density vector is going to have different magnitudes which is a situation is going to be different in let us say a perfect dielectric. Now, we may consider say a unit width along the z direction. This current density vector is going to be in amperes per meter square. If we consider the cross section y z and if we consider a unit area in that cross section, then the current in that unit cross section is J naught. That is what these units and this notation would mean. Now, we consider a unit width along the z direction. And we try to consider what is the current flowing in the direction of the electric field in this case in the x direction in a unit width along the x direction. Because after all the variation along the y direction is very well precisely put down. Also in a conducting material in a medium which is a good conductor while this is shown extending to a large depth it is highly exaggerated. This will be only to a very small depth depending on the frequency and the conductivity of the medium. So, this entire phenomenon will be restricted to a very close to the surface. From this point of view, now one can define what is called the surface impedance of the medium. Denoting it by the symbol z s it is the ratio of the tangential electric field to the linear current density J s. The meaning of these symbols E tan is the electric field parallel to the surface the y equal to 0 surface in this case and at the surface. That will be the interpretation for E tan and similarly J s is the linear current density that is it has units amperes per meter. Somehow we will try to integrate out the variation of the current along the y direction along the depth. And then we will be able to say what is the current in the x direction per unit width along the z direction. Which will become the linear current density or the current per unit width. Now how will we find out J s? J s is going to be equal to the current density J integrated along the y direction from y equal to 0 to infinity. This will be the value of the conduction current density conduction current which flows through a unit width along the z direction. And therefore we substitute the expressions for the current density and we get J is equal to J naught e to the power minus gamma y d y integrated from 0 to infinity. Which is going to be J naught by gamma with a negative sign and then we evaluate the expression e to the power minus gamma y between the limits 0 to infinity. Which is going to have a value which is J naught by gamma where gamma as we have said already is the propagation constant for this conducting medium. Further J naught is also going to depend upon the electric fuel intensity at the surface that is e tan. And therefore we can write J naught equal to sigma e tan and they are going to be collinear they are going to be in the same direction. And therefore we see that Z s is going to be gamma by sigma where gamma is the propagation constant for the medium sigma is the conductivity of the medium. And if we consider the units now the units will be volts per meter here and amperes per meter here that is units of volts. And therefore the name for this quantity Z s surface impedance is quite appropriate. The value of gamma by sigma is going to be 1 by sigma and we put down the expression for gamma which in the general case is J omega mu into sigma plus J omega epsilon. However the concept is particularly applicable to good conductors. And therefore we can use the expression of expression for gamma in the specialization that the medium is a good conductor. In which case we are going to have Z s as approximately in the approximation that we are dealing with a good conductor J omega mu sigma whole square root by sigma which is J omega mu upon sigma whole square root which can be split into real and imaginary parts as omega mu by 2 sigma whole square root into 1 plus J. That is it has a real part and an imaginary part the magnitudes are equal. In fact this is just the expression for the intrinsic impedance of the medium using the approximation that it is a good conductor. And now having done this kind of home work it is perhaps easy to see that why the impedance seen here should be equal to the intrinsic impedance of the medium. In fact we are considering the effectively an infinite medium there is no other discontinuity and therefore the impedance seen is the characteristic or the intrinsic impedance of the medium. The utility of the concept is what we say in the following. The R S or the real part of the surface impedance is going to be omega mu by 2 sigma whole square root which can be written as 1 by sigma times delta where delta is the depth of penetration or alternatively the skin depth where delta has the expression 2 by omega mu sigma. One can see easily that R S equal to 1 by R S equal to omega mu by 2 sigma whole square root is equal to 1 by sigma delta. This is the real part of the surface impedance considering the entire depth of the medium and considering that the medium extends for a few skin depths that is how we have integrated from 0 to infinity. However when we write it like this 1 by sigma delta it goes to show that we are considering the medium with the decaying current density along the depth equivalent to a medium which let us say extends up to this depth only which depth is equal to the skin depth delta and supports a uniform current density. In that case one can readily see by applying Ohm's law and trying to find out the resistance of this much portion in terms of conductivity and the cross section normal to the direction of propagation of the direction of flow of the current and one will find that per unit width along this direction this will be the resistance of this much portion. Therefore this concept helps us in simplifying this decaying current phenomenon into an equivalent uniform current density flowing through a skin depth flowing through a depth of the medium equal to the skin depth or the depth of penetration delta. If you recall this is the way we had calculated the primary constant R resistance per unit length for a coaxial cable. We consider the depth equal to skin depth for the conducting media and then in this manner this was multiplied by the circumference of the two cylindrical conductors and that gave us the resistance per unit length. So this is the convenience offered by the concept of surface impedance and the actual time varying decaying phenomenon can be simplified to a uniform current density phenomenon at least conceptually for the purpose of calculations etcetera. The same concept can be utilized for estimating the power that is dissipated in the conductor. We can calculate the pointing vector at the surface it will have a direction into this conducting block y direction. When we go sufficiently deep into this the electric field and the magnetic field vectors are going to decay to a 0 value. So at that point there will be no the value of the pointing vector will approach 0. So where has this power gone that was being fed into this conducting block that has got dissipated in this conducting media. Therefore, by considering the power that flows into this conducting block at y equal to 0 we can determine the power loss per unit area in this media. So let us try to do that we write h equal to say h naught e to the power minus gamma y and let us say that this is in the z direction. What will be the corresponding value of e in a plane wave then we use the result that e x by h z should be equal to the intrinsic impedance of the medium that is minus eta considering the cyclic order of the subscripts of the field components. Which we have seen is equal to the surface impedance z s and therefore, e is going to be equal to minus z s h naught e to the power minus gamma y x cap. In what notation are we writing these field expressions we are writing these field expressions in the phasor notation because the time variation is not appearing in these explicitly. So how can we find out the pointing vector or the average power flow we write this equal to half real of e cross h star substituting the expressions for e and h what are we going to get. And we further say that let us say that we evaluate this at the surface that is at y equal to 0 without this average pointing vector will have different values of y because of this factor. But as we argued out earlier let us find out the pointing vector at the surface and therefore, this is going to be half r s h naught square in what direction in the y direction which can be written as half r s magnitude h tan square where h tan stands for the magnetic field intensity vector at the surface tangential to the surface which also takes care of the y equal to 0 condition. So this way knowing the tangential magnetic field magnitude and the surface resistance the real part of the surface impedance we can estimate the power loss and the units of this will be watts per meter square. If we consider a unit cross section in the x z plane normal to the direction of propagation of the wave into the conducting block within that unit cross section. So many watts of power are being fair into this and eventually being dissipated in this medium as oh make loss. Now this is one situation where one can make out that we are taking the real part here is making an effect on the final result. And from here one can conclude that if e and h are not in phase then this product will be complex and we will have to consider the real part as far as the power flow is concerned. This aspect will come up again and I will point it out again at that time. Sometimes the phasor notation in a way masks the actual time varying phenomenon although the phasor calculations are straight forward and simple. So let us consider the corresponding time varying quantities. How are we going to be able to do that? We will have the time varying magnetic field intensity vector h equal to real of the phasor h multiplied by e to the power j omega t. That is the rule that we have specified earlier for going from phasor notation to time varying notation. And therefore, this is going to be h naught e to the power minus alpha y and then cosine of omega t minus beta y recognizing that gamma has a real part alpha and an imaginary part beta of course, in the z direction. Similarly, the corresponding time varying electric field can be written it will be real of the phasor electric field intensity vector times e to the power j omega t, which we write as minus r s h naught times real of 1 plus j e to the power minus comma y e to the power j omega t which simplifies to minus r s the direction is going to be x cap. So, it is minus r s h naught e to the power minus alpha y and then we will have here cosine of omega t minus beta y minus sin of omega t minus beta y a direct consequence of the fact that the intrinsic impedance of the surface impedance is not a real quantity here it is a complex quantity. We can continue on this side and now we try to find out the instantaneous pointing vector p equal to e cross h and we try to evaluate it at y equal to 0 at the surface and that would give us r s h naught squared the alpha y term will drop out since we are working at y equal to 0 and then within the brackets we will have cos squared omega t minus cos omega t sin omega t. Instantaneous pointing vector and if we calculate the average value only the first term will contribute a factor half the second term will integrate to a 0 value over a full period. So, eventually the average of this quantity is going to be half r s magnitude h tan square. No you consider this is in the y cap direction this is x cap. So, x cap cross z cap with a negative sign will give you plus y cap and as we argued out earlier this is the pointing vector at the surface and this is the power being dissipated per unit area in the conducting medium as well. And therefore, usually one employs the engineering approximation that is one finds out the field intensity vectors for a particular structure or for a particular problem assuming that the conductor is perfect then one will be able to find out the tangential magnetic field and then one says that for the real conducting medium this is the way the power loss will be per unit area of the medium. So, this concept of surface impedance and surface resistance therefore, helps us in calculating the power loss in a conducting medium. And we need not really worry or consider in detail what is going on along the depth just by using the surface impedance we can work at the surface and calculate correctly the various quantities associated with the conducting medium. We have discussed the first two topics listed for the day if you have any questions we can try those now. Assume that the wave was going in the y direction. Yes. So, that means that along the wire say when we have the power lines or whatever the electromagnetic waves that are actually going into the wire. See the power that is being transmitted along the transmission line or along some other structure that will go along the axis of the structure. But we use lossy conductors the real conductors are lossy which will require some power flow into the conductor in this manner. So, this part will account for the power that is being dissipated or being lost into the conductor. The resistance loss in a wire. You could say that. Sir, in other words if we consider a super conductor there would be no electric field or magnetic field inside the material. Yes, that is true. So, wherever we are interested in making devices or circuits where this kind of power loss is to be reduced further or is to be avoided to the extent possible. One will try to use better conductivity materials and logical sequence will be to use super conductors if they are economically and otherwise feasible. So, let us take up the next topic and that is the reflection at a perfect conductor. Now, we are switching the medium that we will consider to a perfect conductor. Let us first try and see what is it that we expect on the basis of a physical argument. Let us say that this is the interface between say air and a perfect conductor. Let this be at x equal to 0 say this is the x direction, this is the y direction into the plane of the blackboard and this is the z direction. We consider a uniform plane wave incident on this perfectly conducting surface from say some negative value of x that is it is incident in this manner. This is the incident way. Now, as it impinges on the surface we know that there is no electric field in a perfect conductor. Now, so the electric field here must be 0. What can we say about the magnetic field? If we consider the Maxwell's equation which reads as del cross E equal to minus mu del h by del t. Then it says that there cannot be any time varying magnetic field in a space that is in a perfect conductor where electric field is nil. There could be some d c magnetic field. So, time varying magnetic field must be 0 which means there cannot be any propagating wave in the region x greater than 0 which is occupied by the perfect conductor. So, no power can be propagated further we are assuming a perfect conductor. So, conductivity is infinity therefore, there is no loss mechanism present either. There can be no power loss also therefore, whatever power is incident has to be completely reflected back because no power can be taken up by a propagating wave. There can be no power loss and therefore, whatever is the power which is incident has to be completely reflected back from the point of view of conservation of power. And we are considering the case of normal incidents to keep things simple in the beginning. Now the next question that comes up is what are going to be the field vectors in the reflected wave. One simple assumption will be that the field vectors in the reflected wave have identical magnitude to those in the incident wave and the direction is appropriately adjusted so that the wave propagates in the reverse direction. But on the basis of these arguments all that we could conclude was that whatever power is incident must be completely reflected back we cannot really say what will be the field amplitudes. For that purpose we make considerations similar to what we made for the transmission lines and we consider what will be the boundary condition say for the electric field on this interface between two different media. Let us consider that the incident wave has an electric field which is oriented in the y direction and E i it is equal to E naught E to the power minus j beta x and we say that it is in the y direction. And we need to find out what is the electric field vector in the reflected wave and we say that the boundary condition that is applicable to this electric field intensity vector is that E tan on the surface x equal to 0 is equal to 0. Because just inside the surface electric field is 0 and therefore, by applying the boundary condition that tendential electric field components are continuous across a boundary between two different media at the surface and just above the surface also the tendential electric field must be 0. Now the tendential electric field is at x equal to 0 it is E i plus E r which must be 0 and therefore, E r must be equal to minus E i and you can readily see the analogy with the transmission lines. And therefore, E r is going to be minus E naught E to the power plus j beta x y tan because the reflected wave is propagating in the reverse direction in a direction opposite to that of the incident wave. The total electric field can now be considered which will be a superposition or a combination of the incident electric field and the reflected electric field and it will be E naught into E to the power minus j beta x minus E to the power minus j beta x in the direction y cap which is minus twice j E naught sin of beta x y cap this must be plus thank you and then on combination we will get this result. Now we want to look at this in some more detail because this has a character which is different from that of a propagating wave which can be brought out very clearly if we consider the corresponding time varying expression. We have the corresponding total time varying electric field as real of the phasor E t multiplied by E to the power j omega t and this comes out to be twice E naught sin beta x cosine omega t. In particular when we compare it with the expressions for propagating waves we see that there are no functions with arguments omega t minus beta x or t minus x by v. So, therefore, this is something which is different from a propagating wave and as we shall see this kind of expressions represent standing waves. We can consider the behavior for different time instance and if this is the x equal to 0 surface then for say cos omega t equal to 0 we are going to have variation which is going to look like this where this is twice E naught and so is this. This is the situation for omega t equal to 0 for omega t equal to pi by 2 there will be no field 0 field and at different time instance in this manner you can visualize what will be the picture of the electric field intensity the total electric field intensity. For example, for omega t equal to pi we are going to have a situation which is exactly symmetric, but the field values will be negative of those that we had earlier and therefore, this is the situation for omega t equal to pi and all other possible values of omega t will fill the space in between. So, as time changes the at a particular location at a particular value of x the electric field intensity changes sinusoidally with time. There are certain fixed locations where the electric field is always 0 and also the locations of the field maximum are fixed and therefore, it is truly a standing way. Yes, please. Thank you that would have been a mistake because of the j factor it should be sin. Please make this correction and we correct these values also this will be omega t equal to pi by 2 and this will be omega t equal to minus pi by 2 is that all right now. The spacing between the nodes can be looked at wherever beta x is equal to n pi we are going to have a node. And therefore, the values of x this is twice pi by lambda x equal to n pi and therefore, values of x which are n lambda by 2 will be the node points. Since we are dealing with negative values of x and negative sign can be incorporated that should not change the effect of this result. So, these points the adjacent node points have a separation which is lambda by 2 a result which is quite in parallel with the result that we got for standing waves on transmission lines. This being a perfect conductor acting like a short circuit is a 0 field point the maximum electric field becomes twice that of the incident wave electric field. And there are fixed node points and fixed anti node points. In a similar manner one can determine the behavior of the total magnetic field. We have seen that the incident and the reflected fields are in this manner the reflected electric field has changed the direction it is just the opposite of the incident wave. We have already made out that the reflected wave has to propagate in a direction opposite to that of the incident wave. And it must carry the same power as brought in by the incident wave. And therefore, we can make out that if h i is equal to h naught e to the power minus j beta x z cap then h r must be equal to h naught e to the power plus j beta x z cap. So that the cross product of e r and h r has the same magnitude as the cross product of e i and h i and has a direction which is opposite to that of the incident wave. On this basis one can find out the total magnetic field which will be twice h naught cosine beta x z cap h naught cosine beta x z cap and the time varying total magnetic field which will be twice h naught cosine beta x cosine omega t. One can consider the behavior of the total magnetic field and one can say show it here. One will find that the behavior is as follows these become the node points and these become the anti node points. And we have a plot which looks like this. This will be the situation at omega t equal to 0 and at omega t equal to pi the situation will be in this manner. And similarly for all other values of this is the situation for omega t equal to pi. And similarly for all other values of omega t the total magnetic field intensity vector will show variation between these two limits. And once again the node points are going to be fixed the anti node points are going to be fixed. At the perfectly conducting surface there will be maximum magnetic field and the node points are going to be midway between the previous node points between the node points for the electric field. But the standing wave nature is quite clear here also. There is one small point which remains the total magnetic field intensity at say x equal to 0 plus is like this. At x equal to 0 minus since there can be no time varying magnetic field the total magnetic field intensity must be 0. And applying the boundary condition satisfied by the tangential magnetic field at an interface between two media we find that there must be a linear current density J s which is equal to h t or which is equal to twice h naught. By considering that the tangential magnetic fields at an interface are discontinuous by the linear current density at the interface. So, that gives us this result. This is where we like to stop today. In the lecture today we introduced the notion of surface impedance which is applicable to good conductors. We also saw how we can estimate the power loss in a conductor and then we considered what happens when a uniform plane wave is incident normally on a perfect conductor. Thank you.