 Welcome to the fifth lecture in the course on Engineering Electromagnetics. In the last lecture we considered some travelling wave situations and also saw what is the behavior of sinusoidally time varying signals on transmission lines. We had started our consideration on the concept of impedance transformation. So, in the lecture today we shall complete the concept of impedance transformation and then go on to consider the important phenomenon of standing wave formation on transmission lines. Picking up the thread from where we left last time we saw that the general expressions for the voltage and current on a transmission line are like these and then considering that the input impedance at a distance L from the load is going to be the ratio of the voltage and current at z equal to minus L. We saw that this ratio comes out like this. You can notice that we are dealing with sinusoidally time varying signals and it is actually the situation when we are talking about the steady state. It is not the transient situation and therefore, the effect of the transmission line and the load impedance is visible or noticeable at such a distance. We take up from this point onwards and realizing that the ratio of V naught minus upon V naught plus is equal to rho the reflection coefficient. You would recall that reflection coefficient is the ratio of the reflected voltage and the incident voltage at the load. The load here is at z equal to 0 and hence these are the reflected wave and the incident wave amplitudes at z equal to 0. Making the substitution in the expression for the input impedance we obtain the following expression. It is z naught times e to the power j beta L plus rho times e to the power minus j beta L divided by e to the power j beta L minus rho e to the power minus j beta L. However, the reflection coefficient rho is available in terms of the load impedance z L and the characteristic impedance z naught. It is equal to z L minus z naught divided by z L plus z naught and therefore, one can replace rho by the expression in terms of z L and z naught. Then one can rearrange the terms in a simple manner to obtain the following expression z i is z naught times z L cos beta L plus j z naught sin beta L upon z naught cos beta L plus j z L sin beta L. It is a simple rearrangement and therefore, a very common form for the input impedance at a distance L from the load impedance is the following. It reads as z naught times z L plus j z naught tan beta L upon z naught plus j z L tan beta L. While at low frequencies, we would simply say that the input impedance here once these are simple connecting wires is z L. We find that now that we are taking time delays into account which is required typically at high frequencies. The input impedance is not just z L. It is a function of the characteristic impedance z naught. It is a function of the phase shift constant beta which in turn is related to wavelength and also the length of the transmission line small n. On the phase of it a fairly complicated relationship between the input impedance and the load impedance. This is an entirely new phenomenon which we have not seen earlier and is a result of the time delay effects. One can see fairly easily that if the time delay effects are negligible. For example, if beta L is so small that it can be neglected then we get back to the familiar low frequency approximation that z i is equal to z L. For other situations one can make appropriate substitutions and then see what is the behavior of the input impedance. Before we go into that one can also write the same expression in terms of admittances depending on what kind of connections we make on the transmission line parallel or series. It will be convenient to work either in terms of admittances or in terms of impedances and therefore, an expression in terms of admittances is also equally useful. And either one can come back to these starting equations and say that the input admittance y i is i upon v and follow through a similar procedure or one can make substitutions here recognizing that y i is going to be 1 by z i and defining the characteristic admittance as the reciprocal of the characteristic impedance and also the load admittance as the reciprocal of the load impedance. One can see from the expression for the input impedance that y i has an identical form for its expression it reads as y i equal to y naught y L plus j y naught tan beta L upon y naught plus j y L tan beta L and a simple substitution would verify this. For some simple situations for some simple values of the load impedance it is straight forward to see what is the behavior of the input impedance. For example, if z L is equal to z naught the matched condition as we have been saying one sees that z i is simply z naught the transmission line appears to be of infinite length of characteristic impedance z naught if the load impedance equals the characteristic impedance. As I said for other values of the load impedance z L the situation is not so simple or not so readily identifiable. So, let us go to the overhead projector and see the variation of the input impedance for some typical load impedance values. The situation that we discussed just now that is the load impedance equals the characteristic impedance and we are plotting normalized values of the reactance of the input impedance and of the resistance of the input impedance on the y axis and the x axis is displaying the distance L in wavelength from the load and we see that for the load condition that R L is equal to 1 the normalized load resistance is equal to 1 and there is no reactive component in the load the input impedance is purely resistive x is always 0 and it continues to be of value normalized value equal to 1 that is just the case we saw just now from the expression for the input impedance. Other kind of terminations which are possible are for example a short circuit that is the resistive and the reactive parts are 0 for such a case as one can also make out from the expression for the input impedance there is no real component in the input impedance at any distance from the load there is only a reactive component the input impedance seen for a short circuit termination is entirely imaginary reactive and it is basically a tan function as a substitution Z L equal to 0 in the expression for the input impedance will readily show and it is the tan function behavior that we are seeing here and it is periodic in fact any value of reactance can be seen for as we move to different distances from the load this is a very new kind of situation for us similarly one can consider other kinds of terminations for example termination where the resistive part is 0 but the reactive part has a normalized value equal to 1 there again we see that the input impedance has no real part is completely imaginary and once again has a periodic behavior the periodic behavior is there no matter what load condition we consider and therefore these are the various types of variations that one can have depending on the kind of load that one is connect for a situation where the reactance in the load is 0 but the resistance has a normalized value equal to 3 you find that there are both resistive and reactive components in the input impedance they are both periodic the reactance goes through 0s at regular intervals and the resistance what is this resistance it is the real part of the input impedance at different distances from the load is going through maxima minima maxima minima and so on we will have occasion to refer to this behavior later on in the context of the standing waves therefore we have seen that the input impedance seen on a transmission line is not constant its behavior is periodic it may have a real value or a reactive value it is not so directly related to the load impedance and for different load impedance values it is going to have different types of behaviors this is a new phenomenon for us similarly there is another new phenomenon that we are going to talk about and that is the phenomenon of standing waves particularly in the context of circuits this is a new phenomenon so let us make some space on the board and go on to this new phenomena we start from the expression for the total voltage and write the general expression for voltage as we did earlier it is v equal to v naught plus e to the power minus j beta z plus v naught minus e to the power plus j beta z all right we are aware that the first term represents the incident wave or the forward traveling wave or the wave traveling in the positive z direction so that the phase term or the phase is increasingly negative or in absolute terms it has a decreasing phase as z increases the phase behavior for the reflected wave is the other way round for increasing values of z it has an increasing phase and therefore intuitively we can make out that they will be specific values of z where the two waves may interfere constructively so that the voltage is directly add up the amplitude of the incident wave and the reflected wave directly adds up constructive interference by the same amount by the same logic they may interfere destructively where their amplitudes are going to subtract from each other directly and therefore here itself intuitively we can make out that there are going to be voltage maxima and voltage minima on the transmission line as we consider different values of z away from the loop since the phase condition is going to be periodic repetitive these locations of voltage maxima are also going to be periodic so also will be the locations of voltage minima so intuitively it should not be difficult at all to make out that in general they will be standing waves on the transmission what will be the value of the voltage at the voltage maxima and what will be the value of the voltage at the voltage minima that we can work out mathematically that is what we proceed to do next we say that v at z equal to minus l is going to be v naught plus e to the power j beta l plus v naught minus e to the power minus j beta l now we know that v naught minus upon v naught plus is equal to rho the reflection coefficient of the voltage maxima and therefore we can take out the first term as common v naught plus e to the power j beta l is taken out as common and we are left with 1 plus rho e to the power minus 2 j beta l now for the sake of some simplification we are going to consider the magnitude of this total voltage at voltage maxima or voltage minima it will be the magnitude of the voltage which will be maximum anywhere on the line and similarly the voltage minima and therefore let us write rho also in terms of its magnitude and the phase angle theta there should be no difficulty in making out that the reflection coefficient rho is a complex quantity the expression for rho is available here depending on the nature of z l it can be real or complex and therefore we can write the magnitude of the total voltage as v naught plus times 1 plus e to the power minus magnitude of rho e to the power j theta minus 2 beta l we can write the term containing the exponential in terms of the sin and cosine cosine plus j sin and hence find out the magnitude of the overall expression the procedure is fairly straight forward and therefore perhaps one can write the result it will be v equal to v naught plus magnitude into 1 plus magnitude of rho squared plus twice rho magnitude cosine of theta minus 2 beta l whole square because of the magnitude of a real part and imaginary part. Now, because of the appearance of the term cosine cosine of theta minus 2 beta l it is clear that depending on the argument of the cosine theta minus 2 beta l the total voltage v may have a maximum value or the may have a minimum value specifically we see that for theta minus 2 beta l equal to 2 and pi we have v equal to v naught plus magnitude into 1 plus magnitude of rho which is going to be the maximum value of the voltage anywhere on the transmission line. Of course, it is not a single location whatever values of l satisfy this condition there we will have the voltage value maximum and therefore there are going to be a number of voltage maxima along the transmission line provided we have a transmission line which is sufficiently long in a similar manner v is going to be v naught plus into 1 minus magnitude of rho that is a minimum value of the voltage for theta minus 2 beta l equal to 2 n plus 1 pi and this is what we were saying that intuitively one can make out that there are going to be voltage maxima and voltage minima on the transmission line this is what their value is going to be. This is the expression which gives the total value of the voltage and one can put in some simple values of the reflection coefficient rho and then see what is the behavior of the magnitude of v going to be to be able to understand this better. We consider the value say for magnitude of rho equal to 1 and theta equal to pi essentially rho equal to minus 1 when do we obtain this kind of reflection coefficient value when the termination is short circuit. So, we are going to have this kind of situation many times let us see what is the behavior of the total voltage for this. For this we see that v is going to be v naught plus magnitude and then twice cosine of pi minus 2 beta l whole square rho just substituting the value of rho and theta magnitude of rho and theta in the previous expression. Now cosine term can be written in terms of half its angle. So, that the expression is simplified and doing that we see that it is going to be v naught plus magnitude and then twice cosine of pi by 2 minus beta l and since we are considering the magnitude which we would like to remain positive we can put a magnitude sign here or alternatively we choose the positive square root every time. So, that this is equal to v naught plus magnitude twice sign of beta l. So, for a short circuit termination we see that the total voltage v is going to have the behavior of rectified sinusoid. One can incorporate the time variation into this phasor notation explicitly and see what is the overall behavior as a function of l and as a function of time. What is the rule that we have proposed for going from phasor notation to the time varying notation we multiply by e to the power j omega t and take the real part. So, doing that we find that the phasor that we have got here corresponds to a time variation time varying voltage which is equal to twice v naught plus magnitude and then cosine of omega t sin of beta l removing the restriction of positive values only. For time varying signals now we can have negative values as well. These things can be seen a little more clearly on the overhead projector. So, replacing this we find I hope you have no difficulty in understanding this figure in relation to the time variation. The only difference is that the load point is at l equal to 0 that is on the left hand side of this display. For the figure on the board the load point is on the right hand side besides that there is no other difference. The curve that has the maximum amplitude is the one for which we have just considered the example that is rho equal to 1 times e to the power j pi and we are considering the ratio of total magnitude v divided by the amplitude of the incident wave that is v naught plus and we find that at the short circuit this reflection coefficient corresponds to a short circuit termination. At the short circuit total voltage is 0 and then it varies in this kind of sinusoidal fashion. It remains positive because we have considered only the positive value. We want to find out the magnitude of the voltage. One can similarly consider any value of rho and substitute in the expression for the total voltage and find the magnitude of v upon magnitude of v plus ratio for different values of beta l. The abscissa is marked in terms of wavelengths from the load or in terms of the value of the total phase shift beta l. So, it is pi by 2 pi 3 pi by 2 2 pi or 0.25, 0.5, 0.75 or 1 lambda from the load. Once again you see that it is a repetitive phenomenon. The voltage maxima repeat, voltage minima also repeat. In fact they are interspersed between each other and if you consider the separation between adjacent minima or adjacent maxima that is going to be the same. One thing which is very clearly noticeable here is that the minima are very sharp. In fact for this particular value of rho it goes to 0. It is extremely sharp and those of you who have seen this on a microwave bench you would recall that these are really extremely sharp. The voltage maxima are not so sharp. For other values of rho also we have shown the variation of the total voltage upon the magnitude of the incident voltage and you find that as the value of rho the magnitude of rho goes down the overall excursion of the total voltage also goes down. So, for larger values of rho there is a standing wave of a larger amplitude and for smaller values of rho there is a standing wave of a smaller amplitude. In fact for rho equal to 0 there is no standing wave at all. To consider the standing wave nature and to see how it relates to the time varying signal the expression that we have written earlier. We focus attention on the curve for rho equal to 1 into e to the power j pi for which we have written the expression on the board and now that is shown here for different instance of time that is for different values of omega t. For omega t equal to 0 the plot is this as a function of beta l. Now this is a proper sine function with the value which is twice v0 plus at other instance at other values of omega t depending on the factor cosine omega t the amplitude is going to go down at omega t equal to pi by 2 it will be 0. So, if one took a snapshot of omega t voltage at this instance of time there will be nothing no voltage anywhere on the transmission line. But shortly afterwards at omega t equal to 3 pi by 4 there will be a small amplitude sinusoid and as omega t increases to pi there will be a full amplitude sinusoidal variation. Although at a particular location particular value of beta l the amplitude keeps on varying as a function of time and that is what we expect from a sinusoidal time varying signal. The locations of the voltage maxima or voltage minima are fixed this is precisely what we mean by a standing wave. The nodes and anti nodes in a standing wave have a fixed location and that is when we visualize the actual time varying voltage or current that is the behavior of the standing wave voltage or current. Now in terms of this standing wave phenomenon now one defines what is called the standing wave ratio which is an extremely important quantity what is standing wave ratio simply put it is the ratio of the maximum voltage to the minimum voltage and it is given the symbol S and it is equal to V max upon V min and as we worked out already it is equal to 1 plus magnitude of rho upon 1 minus magnitude of rho. One could say that the reflection coefficient and the standing wave ratio are not independent of each other S is available in terms of rho. In fact it only utilizes the magnitude of rho not even the phase of rho but the significance of the standing wave ratio S will become clear to us in a short while. Right now let us consider what are the various possible values for S. Now we also write down the value of rho in terms of Z L and Z naught and now we see that for Z L equal to Z naught that is the simplest situation and therefore we do try to consider it wherever possible. For Z L equal to Z naught what is rho? Rho is simply 0 and S is simply 1 implying that voltage maximum and voltage minimum have the same value or in other words there is rho standing there quite consistent with the fact that the load is a matched load and is not sending out any reflected wave. Similarly one can consider other extreme values of Z L for example Z L equal to 0 a short circuit for which we find that rho is minus 1. However magnitude rho is plus 1 and we find that S the standing wave ratio is infinite for a short circuit. We just saw on the overhead projector that for this situation for the short circuit termination the voltage minimum go to 0 and therefore S is rightly coming out to be infinite. Similarly for an open circuit termination Z L equal to infinity we find that rho is 1 and once again S is infinity and we can say that whereas magnitude of rho varies from 0 to 1 these are the minimum and the maximum limits S varies between 1 and infinity as we have just seen. In fact in most of the practical situations for example when we are designing a high frequency circuit we will attempt to move towards the situation where rho is 0 the reflection coefficient is as low as possible or correspondingly the standing wave ratio is close to 1 as close to 1 as possible because that corresponds to the matched condition and in that condition the power transferred to the load is maximum. Having introduced the concept of standing waves and the term standing wave ratio we should also clarify what is the significance of the standing wave ratio S that is what we proceed to do next. Now typically we are dealing with high frequency voltage and current signals we have seen and we have stated earlier that it is difficult to see these high frequency voltage and current signals with the usual low cost instruments. There are many difficulties they will not have the adequate bandwidth they will not have sufficient amplification at such high frequencies the interconnections that we make for transporting the signal to the display device may not faithfully transmit these high frequency signals. So there are a number of problems in seeing high frequency voltage in current signals. However as we will see shortly it is possible to derive a signal which is proportional to the power corresponding to these high frequency signals. For example if we utilize this same figure and let us say that we connect a non-linear device here and we connect the output let us say to an indicating instrument or say a volt meter. Let this non-linear device semiconductor device be a square law device which is a situation which is obtained quite frequently particularly at low signal levels what do we mean by a square law device we say that the output voltage is proportional to the square of the input voltage. The input voltage that we consider is the high frequency signal let us say it is V naught cosine omega t sin beta z that is it is corresponding to a standing wave for a short circuit as we just saw or with a slight phase shift with a phase shift of pi by 2 radians corresponds to the open circuit. At this voltage incident on the square law device what will be the output like that one can work out V out is going to be proportional to the sin squared beta z term can be taken out and what remains is V naught squared cos squared omega t which can be expressed in terms of twice the angle of the cosine so that it is V naught squared by 2 plus V naught squared by 2 cosine of twice omega t. Now the second term in the brackets is at twice the frequency of the original voltage signal is a very high frequency term and can be say shorted out through an appropriate capacitance leaving us with a voltage which is V naught squared by 2 sin squared beta z and at a particular location at a particular value of z it has a d c value this is what we meant by saying that it is possible to derive a signal which is proportional to the power contained in the input signal this is proportional to the power of the signal incident on the detector diode. Now as one moves this detector diode back and forth this output voltage is going to vary depending on the standing wave and therefore it will be possible to make out the locations of voltage maximum and voltage minimum. Also we will have signals proportional to the voltage maximum or voltage minimum so by considering their ratio it will be possible to make out or quantitatively measure the standing wave ratio. So that is the significance of the standing wave ratio it is a practically measurable quantity with relatively simple equipment. Standing wave ratio can be measured by making this kind of a simple arrangement. What do we do or how do we utilize this information once we measure the standing wave ratio that we shall see as we proceed further but right now the fact that the voltage maxima and the voltage minima their locations can be identified itself is a fact of considerable significance. We can see from here that if we consider the adjacent voltage maxima or equally well adjacent voltage minima and the separation between them then one can make out that twice beta times l adjacent is going to be twice pi whether these are adjacent voltage maxima or adjacent voltage minima either way this product is going to be 2 pi alright and therefore we find that since beta is 2 pi by lambda then 2 pi by lambda times l adjacent is equal to pi therefore l adjacent is equal to lambda by 2 or as obtains in case of wave guides it will be lambda g by 2 and therefore the wave length can be measured very simply if we are able to make out the locations of voltage maxima and voltage minima. We can make our measurement more accurate by considering more than 2 voltage maxima or voltage minima. Correspondingly that factor will come out here and one will obtain a more accurate value of the wave length. Now I have a question in principle whether we consider the adjacent maxima or adjacent minima the separation is lambda by 2 but in practice which one are we going to utilize it will be the voltage minima that we will utilize for the obvious reason that they are more accurately determinable alright theoretically there is no difference between these. We go on to consider some more aspects related to the phenomenon of standing waves yes please. Yes. See normally the power problem will occur when you have very sharp voltage minima as for example for a short circuit or for an open circuit but the moment you deviate even slightly from the voltage minima you will have some indication on your meter. So it will be possible to make out the location alright and to make the measurement more accurate one will adjust the equipment for maximum amplification. So those adjustments will depend upon the skill of the experiment alright. The consideration that we want to make next is that of impedance at the locations of maximum voltage or alternatively at the locations of minimum voltage. Now we have seen that the total voltage V at voltage maximum is V naught plus and 1 plus magnitude of rho and actually one can see that the incident wave amplitude and the reflected wave amplitude are adding directly and that is how this is the maximum voltage anywhere on the transmission line alright. This we have just seen what we need to see next is what is the current at the location of the voltage maximum. Now since the currents in the incident wave and the reflected wave subtract from each other normally when the voltages are adding directly the currents will subtract directly and therefore we say that at this location if we consider the magnitude of the current it will be V naught plus magnitude minus V naught plus V naught minus magnitude upon Z naught which will be the minimum value of the current anywhere on the transmission line. So associated with the maximum voltage at this location the current is minimum and therefore if we were to consider the impedance at this location impedance at locations where V is V max it will be simply V max by V min I am sorry it should be I min which expressions we have already put down and therefore it is going to be Z naught times V naught plus plus V naught minus upon V naught plus V naught minus V naught minus minus V naught minus which is equal to Z naught times S, S is the standing wave ratio since this is Z naught into V max by V min which is the maximum value of the impedance seen anywhere on the transmission line because this is the maximum voltage anywhere on the transmission line except for the fact that it repeats itself and this is the minimum current anywhere on the transmission line. So the ratio is the maximum anywhere on the transmission line and the characteristic impedance is real particularly in the loss less approximation that we have been dealing with so far and therefore the impedance at the maximum location is completely real and it has a value Z naught times S. In a similar manner one could have considered the impedance at the locations of minimum voltage at V equal to V min and it will turn out that it is the ratio of V min upon I max and writing similar expressions it will be seen that it is equal to Z naught by S which again is real and is the minimum value of the impedance anywhere on the transmission. With that we stop this lecture here. Welcome to the sixth lecture in the course Engineering Electromagnetics. In the last lecture we got familiar with the concept of impedance transformation and the phenomena of the formation of standing waves. We saw that on a transmission line because of the interference between the incident wave and the reflected wave there are voltage maxima and voltage minima that are formed on the transmission line. We also considered the impedance at the locations of voltage maxima and at the locations of voltage minima. We start today from this point onwards. We will first consider how impedance can be measured on a transmission line and then we shall go on to consider the effect of losses which are always present on a practical transmission line. The first topic that we pick up today is that of measurement of impedance. Let us say that the problem we are faced with is that we have a transmission line at the end of which a certain load impedance Z L has been connected and we need to find out or determine this load impedance. Normally at low frequencies this poses little problem. We will measure the voltage across the impedance and the current across the impedance and the ratio of these two quantities will give us the load impedance. However at the high frequencies that we have in mind here we have been saying that it is difficult to measure time varying voltages and currents because of the high frequencies involved. In fact we said that in such a situation the quantities that are measurable are the V SWR or simply SWR which is the voltage standing wave ratio. It is the ratio of the voltage maximum to voltage minimum on a transmission line and the other things that we can measure are the features of the standing wave. For example the locations of V min or voltage minimum or V max the voltage maximum and in terms of these measurable quantities we shall see how we can measure the load impedance in a convenient manner. Let us say that this is a transmission line with characteristic impedance Z naught and in general for any load impedance Z L there is going to be a standing wave which we may represent in the following manner for the general case. Depending on the actual value of Z L the minima and the maxima will be more pronounced or the difference this standing wave will be more shallow. Let us say that the voltage minima are formed at various locations and the first voltage minimum is at a distance which we may call. Now we have the general expression for the input impedance on a transmission line at a distance L or D from the load which reads as Z in is equal to Z naught times Z L plus J Z naught tan beta D upon Z naught plus J Z L tan beta D. Now since D is the location of a voltage minimum we know what should be the input impedance seen looking into this plane. What should it be? It should be completely real and it should be the minimum anywhere seen on the transmission line and specifically we have seen that it is going to be equal to Z naught by S where S is the standing wave ratio or the voltage standing wave ratio. We can easily determine the distance of the voltage minimum from the load. We can easily determine beta which is related to the wavelength beta is equal to 2 pi by lambda. Lambda again can be found out from the separations between the voltage minima or in principle between voltage maxima and therefore we know most of the things. In fact everything in this expression and now one can compare the various parts the imaginary part in the real part and get to know the real part of Z L and the imaginary part of Z L. Simple manipulation leads to the following expression for Z L it is equal to Z naught times 1 minus J S tan beta D upon Z naught times 1 minus J S tan beta upon S minus J tan beta D where as already mentioned beta is twice pi by lambda, lambda is the wavelength. So, this becomes the basis for a very simple determination of the load impedance at high frequencies. One can consider the real part and the imaginary part separately also and the expressions can be obtained by so called rationalization of the denominator and we get R L equal to S Z naught upon S squared cos squared beta D plus sin squared beta D plus sin squared beta D. And X L has the same denominator S squared cos squared beta D plus sin squared beta D, but the numerator now reads as minus Z naught into S squared minus 1 times sin beta D times cosine beta D. This is a very common technique utilized in a microwave laboratory for the measurement of load impedance. It requires the measurement of the standing wave ratio measuring the location of the first voltage minimum and the wavelength. Now, it is not really necessary to consider just the first voltage minimum from the load. If we consider the behavior of the input impedance at different distances from the load impedance the behavior is periodic at any other voltage minimum also we will find the same value of the input impedance. The topic that we take up next is that of transmission lines including losses and we call these general transmission line equations. So far for the sake of simplicity we were considering a lossless transmission lines. We ignored the presence of any losses in the transmission line. However in practice loss is always going to be present. The amount of loss may be different, but some loss is inevitable in a practical transmission line. Why do we say that loss is inevitable? When we consider the materials that we are going to use for the construction of the transmission line which are going to be conductors or dielectrics. We immediately realize that these materials are not going to be perfect. The conductors are not going to be perfect. They will have some conductivity which is less than infinity. Similarly, the dielectrics will not be completely insulating. They will also have some conductivity greater than 0 and it is this non-idealness of the materials that one is going to use which is going to show up as losses. One can consider some typical materials which are utilized in transmission lines on the overhead projector and get an idea of what exactly we mean by the non-idealness of the various materials. On this transparency here we are showing the conductivities of various materials. Conductivity sigma in moles per meter or in simons per meter. These are the units of conductivity and these are the conductivities at low frequencies because it so happens that the conductivity does not remain constant with frequency. It changes with frequency. So one has to specify the frequency at which the conductivity is being specified. These conductivities are typically at low frequencies, a few kilohertz at the most. Various materials are listed here and their conductivities are also listed here. In the beginning we have what may be called the insulators with very small conductivity or alternatively very high resistivity. Fused quartz, polystyrene, mica, hard rubber, bakelite, you can recognize some of these materials as being used in various types of circuits or even in transmission lines. Hard rubber is probably utilized in the transmission line that is used to connect the television antenna to the television receiver. Somewhat better material from the point of view of insulation are going to be required at even higher frequencies at microwave frequencies, something like polystyrene. Bakelite is used for making printed circuit boards. As we go down this list, animal body also has some finite conductivity. We could include the human body also here. Then we go on to materials which are typically called semiconductors because they may behave as conductors or insulators depending on temperature or doping. And in the second column we have got materials which are essentially good conductors starting from steel, brass, aluminum, copper, silver being a material with the highest conductivity as far as pure metals are concerned. And that is why since silver is expensive, most of good low loss circuits will utilize trite utilize copper or aluminum. Some low cost circuits may be built for example, waveguides using brass. As we will soon see the loss depends upon the conductivity or the resistivity. And if one wishes to have a loss value which is even better than can be obtained using these good conductors, one could try to utilize what are called super conductors. For quite some time it was the niobium based super conductors which had the highest critical temperature up to which they would remain super conducting. It was about 21 degrees Kelvin and it required liquid helium to maintain this kind of temperatures. However in late 80s and early 90s the high critical temperature super conductors were quite a sensation. And these would be super conducting at comparatively higher temperatures something like 80 degrees Kelvin which kind of temperatures could be maintained easily using liquid nitrogen with a boiling point of 77 degrees Kelvin. And therefore it was quite a leap in maintaining the super conducting nature of these materials. This is vitrium, barium, copper, oxide. There are many other considerations in the super conductors which we cannot go into at present. So having considered the reason for the losses in transmission lines, we now go on to consider what kind of circuit model we are going to utilize for these transmission lines and what changes in the behavior of the voltage and current on such a transmission line are going to take place because of the presence of losses. As we did before for the lossless transmission lines, we consider a small length of the transmission line let us say of length d z, a transmission line which has a potential difference v between its two conductors and supports an equal and opposite current i in the two conductors. And as for the lossless transmission line we make an equivalent circuit for this using the distributed parameter representation that is associating with each small length of the transmission line a certain inductance, a certain capacitance and now when the losses are present a certain resistance and a certain conductance. So the circuit turns out as follows where these are L, R, C and G. The nature of L and C has been mentioned and discussed earlier R and G are new circuit elements in this equivalent circuit. Here R is the resistance per unit length so that the units of R would be ohms per meter arising out of the finite conductivity of the conductors so that it represents the ohmic loss in the conductors. Similarly G is the conductance per unit length with units being Mohs per meter or Simons per meter representing the dielectric loss. There could be difference mechanisms causing loss in the dielectric we are not going into those details here G represents a circuit element taking into account all loss mechanisms present in the dielectric. Now in this circuit we can identify the current and the voltage at the input end and current end voltage at the output end. At the output end we are going to have current as I plus del I by del Z into d Z and similarly the voltage here at the output is going to be V plus del V by del Z into d Z. All right making some space here. Since we have considered a very short section of transmission line we can apply circuit theory laws here and we can write based on our experience earlier the change in voltage del V by del Z equal to minus R i minus L times del I by del T. The voltage at the output end the voltage drop across such a section would depend upon the resistance and the inductance. Similarly the change in the current del I by del Z is going to be minus G V minus C del V by del T which become the general telegraphists equations for the current. For the general transmission line including losses. In the lossless idealization it is very easy to see that R and G are neglected or they drop to 0 and we get back to the telegraphists equations we wrote for the lossless transmission line. From this point onwards it is somewhat easier to consider a particular kind of time variation because then one can utilize for example, if we consider the sinusoidal time variation then one can utilize phasor notation and the time derivative can be very easily handled. And therefore, writing these equations for sinusoidal time varying signals and naturally using phasor notation. In phasor notation the time derivatives would be replaced by a factor j omega and therefore, utilizing that we have del V by del Z equal to minus R plus j omega L into I and del I by del Z is equal to minus G plus j omega C into V. A special case of these general telegraphists equations for sinusoidal time varying signals. Where has the time variation gone? It is been absorbed in the phasor notation. It is still present and it can be recovered whenever we wish by using our familiar formula multiply by e to the power j omega t and take the real power. So, it is just that in the phasor notation the time variation has become implicit. Now, once again these are two partial differential equations in the variables V and I and they are coupled and using the familiar manipulation we can decouple these to obtain equations governing the behavior of just V or just I. When we do that we get for example, del 2 V by del Z squared equal to R plus j omega L into G plus j omega C into V into and a similar equation in current I. It should not be very difficult to see for example, if we take the partial derivative of the first equation with respect to Z and substitute for del I by del Z from the second equation this is what we are going to get. Now, instead of this whole quantity we introduce a new symbol and call it gamma squared. So, that in terms of this new symbol gamma we write del 2 V by del Z squared is equal to gamma squared V and in a similar manner we have del 2 I by del Z squared equal to gamma squared I where what is gamma squared or what is gamma? Gamma is the square root of gamma of R plus j omega L into G plus j omega C. So, these are the equations second order differential equations in V and in I for which the solution can be written more or less by inspection the solutions are of the form V equal to V naught plus e to the power minus gamma Z plus V naught minus e to the power plus gamma Z. And we could have written some other amplitude coefficients here say A here and B here, but anticipating the interpretation for each term we are writing these amplitude coefficients as V naught plus and V naught minus in a similar manner I is going to be I naught plus e to the power minus gamma Z plus I naught minus e to the power gamma Z I think it is all right. When you write these I naught plus and I naught minus in terms of the V's then a negative sign appears otherwise conceptually we say that the total voltage is the superposition of two waves travelling in opposite directions. So is the current it is a superposition of two waves travelling in opposite directions. Now, in the presence of losses what we expect is that there should be a gradual or regular decrease in the amplitude. Some power should be dissipated continuously in the ohmic loss or in the dielectric loss as a result of which the amplitude should decrease regularly as the wave propagates on the transmission line. That behavior is not obvious here also we expect that the propagating nature of the signals will be retained. Those things can be brought in through a very simple means. Let us consider the first term in the general solution for the voltage and write this as V naught plus e to the power minus gamma Z. What is the nature of gamma? It is complex it is written in terms of such an expression. So, let us say that gamma has a real part alpha and an imaginary part beta and in terms of these real and imaginary parts. Now, we can write this as V naught plus e to the power minus alpha Z and e to the power minus j beta Z which of course is in phase of the rotation and we can reconstruct the actual time varying signal using the procedure we stated just a short while ago. And therefore, we say that the actual time varying voltage which will be a function of both distance Z and time t is going to be equal to V naught plus e to the power minus alpha Z and cosine of omega t minus beta Z. The second part is the familiar portion representing a wave propagating in the positive Z direction. What about the first part? Yes, that shows that there is an exponential decay or decrease in the amplitude and this is a direct result of the losses in the system exponential decay or we call this attenuation. So, based on such an interpretation we can now say that alpha which is the real part of gamma is the attenuation constant. What are the units of it? The units are nephers per meter meaning that if alpha has a value 1 then in 1 meter the amplitude would have decreased by a factor of e. That is how one would understand the unit nephers. Similarly, beta is the imaginary part of gamma and it is the familiar phase shift constant. What are the units of beta? The phase shift constant radians per meter. So, this is how the voltage and current would behave on a lossy transmission line on a general transmission. Yes, please. Sir, in this wave function there is a component Z minus. Sir, if we put that value it means that it is exponentially increasing. When we find out alpha the real part of gamma then if you work it out you will see that there will be a square root involved and we are going to choose the appropriate sign in the square root. So, that for a passive transmission line in the direction of the propagation there is only a decrease in the amplitude. This kind of physically meaningful choices are always made as far as mathematical solutions are concerned. Sir, V naught. Yes, please. So, if you notice I said in the direction of propagation of the wave. So, V naught minus is travelling in the negative Z direction in that direction it should decrease. Sir, there is a general solution and there will be a V naught minus component. We focus attention on the first term and we try to see how it behaves. In a similar manner one could have considered the second term and then come to the same conclusion. The next step that we took from this point onwards in the lossless transmission line situation was to find out a relationship between V and I. Same thing can be done here also and without going through so many steps we just state the final result and say that in this case also the amplitudes of the voltage and current amplitudes in the forward travelling wave are related through Z naught. So, that Z naught is V naught plus upon I naught plus and it is equal to R plus j omega L upon G plus j omega C which of course, will go to square root of L by C if R and G are absent. So, these are all more general expressions. These are the general expressions for the transmission lines from which as a special case one can obtain expressions for the lossless situation. It is interesting to see what is the effect of this regular decrease in the amplitude or attenuation on the formation of standing waves. And for that let us go back to the overhead projector. What we are showing here is a transmission line with characteristic impedance Z naught terminated in some load impedance which let us say is 3 Z naught. We consider a wave incident we call it we label it as 1. In general they will be a reflected wave also Z L is not equal to Z naught and we label that reflected wave as 2. In the presence of losses how does the envelope of the incident wave look like? It looks like this there is a continuous regular decrease in the amplitude. Had the losses not been present it would just have been a straight line. Similarly how does the reflected wave look like? It starts from here satisfying the conditions imposed by the reflection coefficient and the relationship between the Z L and Z naught. For this value of Z L you can work out that the reflection coefficient is half. So half of this is reflected back plus half and then as it travels away from the load once again it also suffers a regular continuous decrease in the amplitude. Now their constructive and destructive interference pattern the standing wave pattern is going to look like this which we are calling here labeling here by number 3 which is the standing wave pattern which is looks wise quite different from the standing wave pattern that one sees on a lossless transmission lines. So this is entirely an effect of the losses present on the transmission line. At this point at the load point one can see that the various conditions the reflection coefficient the transmission coefficient etcetera are satisfied. Yes why we are calling it envelope is because actually these are all time varying quantities. So if you consider the time variation they will all be filled up with a very rapidly time varying signal and this will be the envelope of that time varying signal. They are all envelopes. So this is the kind of effect that the losses on a transmission line are going to have. Now while we have considered the general transmission line incorporating losses in practice we are going to make transmission lines which are low loss. We will try to minimize losses. How are we going to do that? We will try to use materials which are as close to ideal or perfect as possible. So we will try to use good conductors and we will try to use good dielectrics and therefore practical transmission lines will be low loss transmission lines and therefore that is a case of considerable importance and that is what we consider next. The question that will be raised immediately will be when do we say that the line is low loss? The answer to that is that the loss is low if r is much less than omega l and correspondingly if g is also much less than omega c then we will say that the line is low loss. We will actually see the expression for the attenuation constant etcetera and where r and g are going to appear and therefore this kind of inequalities will be required if the loss is to be low. Under the low loss conditions things become considerably simple and the various concepts that we have discussed earlier the impedance transformation and the formation of standing waves they do not undergo much of a change therefore this low loss transmission line approximation is quite important. Starting with z naught the general expression is r plus j omega l upon g plus j omega c whole square root which under the low loss approximation will be simply square root of l by c and once again it remains a real quantity particularly under this low loss approximation. Under this approximation one can obtain the expressions for the attenuation constant alpha and the phase shift constant beta. How that can be done? We start with the expression for gamma it has a real part and an imaginary part and gamma is also given a name gamma is called the propagation constant. So the propagation constant gamma has a real part and an imaginary part and its general expression is r plus j omega l into g plus j omega c which we rewrite as j omega square root of l c. We take out factor j omega l from the first term and factor j omega c from the second term and then what we are left with is 1 plus r by j omega l and 1 plus j omega 1 plus g by j omega c. In the low loss approximation r by omega l is much less than 1 and so is g by omega c and therefore, one can use a suitable approximations. We use the binomial approximation and as a result we get here this is approximately j omega square root of l c and then we have 1 plus r by 2 j omega l and 1 plus g by 2 j omega c multiplying out the terms within the brackets we get j omega square root of l c and then 1 plus r by 2 j omega l plus g by 2 j omega c. Do you agree with this? What about a fourth term? That will be too small that is a product of two small terms and within the first order of approximation that can be safely neglected and now one can see that it simplifies to r by 2 square root of l by c plus g square root of l by c by 2 plus j omega square root of l c. One can compare the real and the imaginary parts and one finds that alpha under this low loss approximation turns out to be half of r by z naught plus g times z naught. Recognizing that root l by c is the characteristic impedance under the low loss approximation. So, we have got the attenuation constant under this approximation. It is in terms of r and in terms of g, the higher the values of these the greater will be the loss. One can easily obtain expressions for alpha and beta which are exact without taking recourse to low loss approximation. There also one will find that alpha will be related to r and g. The continuing with this approximation the expression for beta is quite straight forward. It is simply omega times square root of l c which is the expression we had for the phase shift constant beta under the loss loss less approximation. Now, since beta is this the phase velocity or the velocity of the wave which was omega by beta is simply 1 by square root of l c. These are the various quantities of interest on a transmission line the attenuation constant the phase shift constant and the velocity. Many times these will be enough to tell us about the quality of the transmission line and the behavior of the various signals on the transmission lines. For most practical transmission lines even under low loss approximation the contribution to the attenuation constant by the conductance term. The term representing the dielectric loss is much smaller than the term representing the ohmic loss in the conductors. Therefore, usually it is further approximated as r by 2 z naught. This is the expression for attenuation constant alpha under low loss approximation. Further in practice the second term is negligible compared to the first term. Therefore, we write this further as simply r by 2 z naught and where loss is very important and you would like to minimize loss and you are not happy with the loss you are able to achieve with simple good conductors. You could perhaps try to use super conductors where r would be almost 0. So, that is why so much interest in super conductors and there are other aspects of this that we shall have an occasion to discuss as we proceed. Under the low loss approximation we also need to consider what is the behavior of the input impedance. The general expression for the input impedance for lossy transmission lines is going to be z naught. And then z l cos hyperbolic gamma l plus z naught sin hyperbolic gamma l divided by z naught cos hyperbolic gamma l plus z l sin hyperbolic gamma l. This can be done in the scene fairly easily by considering the total voltage and total current in terms of e to the power minus gamma z and e to the power gamma z kind of terms. Now, gamma l is alpha l plus j beta l and for low loss transmission lines the alpha l term is going to be much less than beta l. Particularly, when we consider short transmission lines which are typically utilized at radio frequencies which are just a few wavelengths long. And under such conditions when alpha l is so small compared to beta l the input impedance simplifies considerably. And we have z in approximately as z naught times z l cos beta l plus j z naught sin beta l upon z naught cos beta l plus j z l sin beta l or z naught put alternatively it will be z naught times z l plus j z naught tan beta l upon z naught plus j z l tan beta l the familiar expression for the input impedance. Let us stop this lecture here.