 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 phenomenon 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 have 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 minimum and the maximum will be more pronounced or the difference this standing wave will be more shallow. Let us say that the voltage minimum 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 minimum or in principle between voltage maximum. 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 S minus j tan beta D where as already mentioned beta is twice pi by 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 a 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. 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 loss less 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 alright. 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. Some what better material from the point of view of insulation are going to be required at even higher frequencies at microwave frequencies something like polystyrene alright. 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, aluminium, copper, silver being a material with the highest conductivity as far as pure metals are concerned alright. And that is why since silver is expensive most of good low loss circuits will utilize try to utilize copper or aluminium. 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 the 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 in 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 dz 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 different 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 alright. Now in this circuit we can identify the current and the voltage at the input end and current and 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 alright 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 drop across such a section would depend upon the resistance and the inductance. 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 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. And 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 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 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. These 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 phasor notation 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 one then in one 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 line. Sir in this waveform there is a component of e minus. So 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 of the air 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. 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 unit 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. The 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 alright. 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 loss less 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 the 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 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 too 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 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 and 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 seen 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 up to 0. Upon z naught cos beta l plus j z l sin beta l or 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.