 Welcome to the second lecture in this course on Engineering Electromagnetics. In the last lecture, we described a number of phenomena which involved propagating waves. In fact, we can say that for most of those phenomena, the common theme was propagating waves. These phenomena have important applications particularly in power transmission and communications. And therefore, as we said last time, one of the major objectives of this course will be to learn about wave propagation or propagating waves. As far as wave propagation is concerned, the simplest way to understand this is to consider what are called transmission lines. As mentioned as the title, why transmission lines are the simplest vehicle to understand wave propagation? It is the simplest vehicle because they afford a very simple analysis. Analysis is simple because of two reasons. First is, it is possible to define uniquely voltage and current on transmission lines. We say that it is possible to associate unique values of voltage and current at each point along the length of a transmission line. Many other structures for example, wave guides which we will study later on do not afford this simplification. It is not possible to define uniquely a voltage or current at each cross section as we move along the length of the wave guide, but it is possible on the general class of structures called transmission lines. That is one reason because of which the analysis is quite simple. Secondly, once we have voltage and current available at least as unique quantities, then with some care it is possible to work in terms of Kirchhoff's laws, which are quite familiar to us and therefore, analysis based on these laws will be more easily understood. This needs a bit of explanation. We have said earlier that situations where time delay is important, where time delay is significant in terms of time period and we have seen in other discussions alternative ways of putting the same thing in other words. We have said earlier that when the time delay is significant in terms of the time period Kirchhoff's laws are in general inadequate and many times it is not possible to associate unique values of voltage and current and therefore, we need to work in terms of fields. While in general that is quite fine, this is one important exception that is transmission lines, where it is possible to associate unique values of voltage and current and we can apply with some care Kirchhoff's laws. The care will be required in dealing with the time delay. How we do it will become clear as we proceed. But before we proceed to the analysis, let me just show you first what are the types of structures we are talking about to put things in proper perspective. What are the various types of transmission lines we have in mind when we say that we want to learn about transmission lines. I am sure you can name some simple types of transmission lines readily. One of the most common type of transmission line is what is called the parallel wire transmission line, which may be arranged in free space for example, in the case of power transmission or may be embedded in a dielectric supporting medium. For example, the wires that connect or the transmission lines that connect the television antenna to the television receiver, both will be called parallel wire transmission lines. These can transmit DC as well as AC signals and as we have just mentioned, they find common application in power transmission, telecommunication systems or telephones and more recently in computer networks. We will simply write networks, computer networks, data networks, whatever. So even this simple transmission line has application in a large number of situations. Another transmission line that once again should be familiar to you is the coaxial transmission line, where one has an inner conductor surrounded by an outer conductor and in this case almost invariably you have some dielectric material filling the region between the two conductors. This is the coaxial transmission line not necessarily, the inner conductor is quite thin usually and therefore, it is solid making it hollow would be more difficult. The coaxial transmission line can also be used at DC, AC and with proper construction it can be used for transmitting even microwave frequencies and therefore, in general we call those radio frequencies and call it RF. The applications of this type of transmission line, the coaxial transmission line would be more or less identical. It is somewhat more expensive and somewhat better in quality. In addition to these applications it can also be used in microwave systems because it can transmit RF or radio frequencies. While these are two fairly well known and conventional transmission lines, there is a large number of transmission lines which have been proposed for work at higher frequencies, microwave frequencies, millimeter wave frequencies etcetera. One typical transmission line which is used at microwave frequencies is the microstrip line and I will try to show you the construction of the microstrip line. It consists of a dielectric or an insulating substrate which is usually quite thin and it is backed by a conducting plane on one side. This will be throughout the extent of the microstrip line and on top there is a thin conducting strip which runs like this. This also is the conductor. This as we said is the microstrip line and it is basically used for transmission of radio frequencies and has applications in microwave systems. This one, this is the dielectric or the insulating substrate and depending on the frequency of application, one will choose suitable materials for constructing these and we shall come to that aspect next. What are the basic criteria which one would use to say whether a transmission line any of these types is good or bad? So, that takes us to what we may call fundamental criteria of performance. What could be these? Anybody has any idea? Very good. We say that the fundamental criteria of performance are attenuation which is a more standard term taking into account losses and distortion. Cost will certainly be an important criterion but is usually not a part of academic considerations. What should be these? These should be low. In fact, we can club all the three and one would like that these are as low as possible. You have some idea already what we mean by attenuation. Attenuation is the continuous decrease of the signal level or signal power as one moves along the transmission line. What would cause this decrease? Losses as mentioned by you. This will be caused by losses in the system and what would cause losses in turn? Even if there is no radiation there could be losses because of absorption or power dissipation in the materials that we use to construct these transmission lines. Therefore, I said that depending on the frequency of application one would construct the transmission line with appropriate materials. So, this is where good quality materials would come into picture caused by losses in the materials used for the construction. We shall have more to say about this when we discuss lossy transmission lines. What is the second term distortion? Distortion could be caused by many factors. Noise is one such factor. Distortion. Yes, very good. Distortion can be caused by noise and dispersion but essentially what does distortion mean? Literally as the word implies if the wave shape or the signal shape that we start with changes as we propagate along the transmission line then we say that distortion has set in. And our objective would be to transmit this wave shape as faithfully as possible with as little distortion as possible and therefore a transmission line which has high distortion is not very good. A transmission line that we would aim at will have low distortion. Therefore as a corollary one could say that we will be learning as we proceed how to make lines with low attenuation or how to estimate the attenuation of a given transmission line and similarly how to make lines with low distortion. It will not be possible to go into the cost aspect of the transmission lines. So having said these general remarks and having shown the typical transmission lines that we have in mind for the rest of the consideration. Now we can see how we can go about the analysis of these transmission lines in a unified general manner. And that takes us to what we may call representing these transmission lines in the form of a circuit or writing down a circuit model for these transmission lines and simply put we will go into what is called modeling or representation circuit representation to be more complete. For this purpose we consider a very small length of any of these transmission lines and as you notice these are all two conductor structures and therefore when we take a small length of any of these transmission lines we can represent this by just two wires like this and we can say that this is representing any of the transmission lines shown earlier. We consider a length d z we call it the differential length and infinitesimally small length of the transmission line is what we consider where we have a potential difference v between the two conductors and a current i which is equal and opposite in the two conductors flows. So far we have just put down a small length of the transmission line a two conductor structure with some potential difference and equal and opposite currents in the two conductors. How will we write down the circuit equivalent of this that is the question and that is done by considering that there is an inductance L per unit length no matter how small a length of the transmission line we consider it will have an inductance depending on the inductance L per unit length per unit length in henry per meter. How does this inductance arise it arises because of the magnetic flux associated with the current flowing in the wires and therefore flux linkage per unit current and therefore some inductance which will oppose any change of current and when current tries to change it will cause an induced voltage. So this inductance per unit length L will take into account that back induced emf in addition there will be a capacitance per unit length represented by C in farads per meter which will account for the charge stored per unit length. We have two conductors separate from each other and they will act as a capacitor by our experience we know that and this charge storage will be represented in terms of capacitance per unit length C yes please the question is that do we find such high capacitances that the unit farad is justified typically the transmission lines that we have shown will have capacitances which are of the order of nano or a few hundreds of micro farads per meter but farad is the standard unit for capacitance in mk system of humans. So is henry so that is what we are. Are there going to be any other circuit parameters associated with this length of the transmission line yes there will be other elements which will be there in practice in general but right now let us restrict our attention to lossless transmission lines which is an ideal situation but is convenient for understanding the basic aspects of wave propagation. So knowing fully well that a practical transmission line will always have some small amount of loss present to begin with we consider a lossless transmission line so that there are no elements in the circuit which will account for losses as we progress we will take loss into account so do not worry on that. Also how do we determine these L and C the inductance per unit length and the capacitance per unit length you have done the calculations for these for some typical structures and here it should be sufficient to say that one can determine these parameters L and C per unit length given a transmission line and later on in a separate lecture we will consider how this can be done. So once we have these parameters we draw what is called the distributed parameter circuit representation of the transmission line, how would it look like we have a small length of the transmission line it will have associated with it and inductance which is L d z this inductance is L d z since L is the inductance per unit length and it has a capacitance which is similarly C d z. So physically we consider a short section of the transmission line of length d z but its circuit representation in terms of a distributed parameter circuit is in terms of an inductance L d z and capacitance C d z and once we have a circuit we can proceed further. We can mark various quantities on this this is the voltage difference or the potential difference between the two conductors this is the current I and an equivalent opposite current in the lower conductor. We could consider that these are the input ports for this small circuit at the output port we need not have the same voltage in current they are likely to change depending on these circuit parameters and therefore in general what we put down here is a current at the output which is I plus any change in the current that may have taken place how would we put down that we say that let the rate of change of current be del I by del z and this be multiplied by the distance that we have traveled and therefore the second term represents the change in the input quantity I as long as the section is short this approximation is quite all right capacitance is always between the two conductors we can do that the basic idea is that for this section of length d z there is an inductance L d z which is associated you may write L by 2 d z here L by 2 d z here effectively it will be inductance L L d z for this section same thing goes for the capacitance in fact we are calling it distributed parameters whatever length you may consider it will have the corresponding L or C associated with it all right similarly we may put down the voltage at the output side as V plus the rate of change of voltage as a function of distance z multiplied by the distance d z which also should be equally well acceptable you would notice that we are taking a very small section of the transmission line where time delay is going to be small or negligible and now we can apply Kirchhoff's laws which are very familiar to us for example we can write the change of voltage del V by del z times d z equal to the inductance times the rate of change of current and notionally we are writing the output voltage more than the input voltage but actually it is an inductor so it will cause a voltage drop and therefore taking care of that we write this as minus L d z del i by del t we are using the symbol del for differential operations because we are considering partial differentials with respect to the distance z or with respect to the time variable t both V and i are functions of both z and t hence the need for this symbol similarly one can write down what is the total change in the current del i by del z times d z should be equal to capacitance time the rate of change of voltage fairly familiar relations so this is equal to once again minus d z into del V by del t. Now let us go on to the next side now the way we have written these two equations these would hold good as long as d z is small within that restriction the actual value of d z does not matter and d z is not equal to 0 therefore we can strike of d z from both sides and arrive at simpler relations which read as del V by del z equal to minus L del i by del t and del i by del z equals minus c del V by del t which are called telegraphist equations which relate the space variation and time variation of voltage and current to each other. Now these are first order differential equations but they are coupled what do we mean by that each equation involves both quantities of interest V and i therefore these are coupled first order differential equations. It is simpler to deal with simple differential equations in one variable and by suitable manipulation it is possible to derive an equation which involves only V or only i. For example we may take the z derivative of equation one we can write that here so del 2 V by del z squared is equal to minus L del 2 i by del t del z and if we can substitute for the right hand side in terms of some quantity involving V our purpose will be served and therefore we take the time derivative of this second equation giving us del 2 i by del z del t equal to minus c del 2 V by del t squared. One can substitute for the right hand side term from the second equation here and it is easy to see that we will get equations reading as del 2 V by let me just check since this is an important equation del 2 V by del z squared equal to L c times del 2 V by del t squared. It is easy to see that one can get an equation reading similarly involving the variable i which can also be written as del 2 i by del z squared equal to L c times del 2 i by del t squared just by interchanging the manipulation of these two equations that can be done. We do not need these anymore and these equations that we have got involving only V or only i are called wave equations further since they involve only one of the coordinates of let us say orthogonal coordinate system Cartesian coordinate system out of x, y, z it involves only z derivatives therefore these are called one dimensional wave equations. Now a question can come up naturally why are these called wave equations the answer is straight forward these are called wave equations because their solutions represent propagating waves how that is so let us see in a minute first let us put down the solutions of let us say equation 3 from 3 we can say that in general del 2 V by del z squared is 1 by V squared del 2 V by del t squared so that V squared is 1 by L c or V is 1 by root L c the significance of this V will appear in a minute right now this is just a substitution and written in this form one can see in a fairly straight forward manner that V will have a solution of the form f of z minus V t we are right now not stating the actual form of the function f in specific problems that form will appear automatically mathematically right now all we are saying is that the solution is of the form such that the solution involves argument z minus V t z and t should appear in this manner as an argument in the function then by back substitution one can see that the second order differential equation will be satisfied well you can find out the derivative with respect to time find out the derivative with respect to z put it on the two sides and you will find that the both sides will be equal so that this equation is satisfied what is the solution we are looking for a solution which satisfies this equation alright and functions which are of this kind of an argument will satisfy this this is just by looking at this equation and if there is any doubt one can back substitute and satisfy that RHS is equal to LHS this indeed is a mathematical solution of the equation we have at hand right now it does not appear to be saying much but we will see that important conclusion can be drawn from this what is equally well precisely you have taken words out of my mouth equally well one can have a solution which reads as f of z plus V t by identical arguments and therefore we say that the general solution is of the form of f 1 plus f 2 where f 1 and f 2 are functions with arguments z minus V t or z plus V t alright what is the significance of functions of this kind of argument this is the crucial thing so let us go to the OHP and try to see how we could infer things from this we focus attention on the first part of the general solution say f 1 and we consider its behavior at a fixed instant of time say t equal to t 1 when we fix at the time variable at t 1 then this function f 1 is essentially a function of z variable only and we could plot this function if we knew the shape with respect to the z axis we assume some arbitrary shape for the sake of argument and say that it looks like this and depending on t t equal to t 1 let us say that it starts at z equal to t 1 it is an assumption as long as we are consistent with it it should not affect our general conclusion alright at t equal to t 1 this function f 1 looks like this at this location starting at this location let us say z equal to V t 1 alright we look at the same function at some other instant of time t 2 greater than t 1 by the same arguments what would it look like as far as the z variation is concerned as far as the shape is concerned it should remain the same only its location because the argument has changed from V t 1 to V t 2 is going to be different so if this point was z equal to V t 1 this point should be z equal to V t 2 by continuing with our assumption consistently now what has happened this was let us say some disturbance some signal which appeared here occurred here at this location of z and at a later instant of time it has moved forward in the z direction by what distance V times t 2 minus t 1 okay and that fits in very nicely with what we may call a wave or how a wave is going to be defined let me read out this you need not note it down it is the understanding which is more important if a physical phenomenon that occurs at one place at a given time say here is reproduced at other places at later instance of time the time delay being proportional to the space separation from the first location then the group of phenomena constitute a wave this is precisely how the behavior of the function F 1 is and therefore we say that these solutions represent propagating waves and hence the original equation named as wave equation there are some important points to be noticed note that a wave is not necessarily a repetitive phenomenon in time it could repeat it may not repeat even if it is a one-time occurrence but if it behaves in this manner it will constitute a propagating wave also occurring at one location at one instant of time and at other locations at other instance of time is also an inbuilt feature of a propagating wave if there is something which is constant for all values of z for all time that you would not call part of a propagating and the space separation between the various occurrences is proportional to the time delays t 2 minus t 1 is the time delay it is it is not the yes precise but how it is time delay this is the time taken by this disturbance or this signal in travelling from one location to the other so this is the time separation of the time delay so from this point of view we see immediately that v is the velocity of wave propagation and it has come out very nicely in terms of the circuit parameters of the transmission line L and C and depending on these values of L and C there will be a certain value of v that one will arrive at in different situations now just as f 1 represents waves propagating in the positive z direction it should be easy to see that the second part of the general solution f 2 is going to represent waves in negative z direction and therefore one can say that this general solution for v that we have got can be written as a summation of two parts one travelling in the positive z direction the other travelling in the negative z direction and we may say that v is equal to v plus plus v minus we need to clean up the board in a similar manner one can write down the general solution for i starting from equation 4 which also would represent propagating waves it will if they have an equal amplitude see this is the general solution and as you are aware that one gets particular solution depending on the boundary conditions so depending on the boundary conditions depending on the termination at the other end of the transmission line there may be a reflected wave there may not be a reflected wave alright so this is the general solution and the incident and the reflected waves in general will combine in different manners that also we are going to discuss as we proceed but these are voltage and current on the same transmission line and we will like to see if there is any relation between these v and i alright so for that purpose we use these equations we have written equation one for example del v by del z is minus l del i by del t by considering that this is the general solution for v we can obtain a solution for i so assuming this we say that minus l del i by del t which should be del v by del z and knowing the solution for v we write this as f 1 prime z minus v t plus f 2 prime z plus v t where the symbol prime represents the differentiation of the function with respect to the whole argument alright and now one can divide by l and integrate with respect to time. So, we are using as a solution for i which will read as minus 1 by l and then we have minus 1 by v of f 1 z minus v t on one hand and plus 1 by v f 2 of z plus v t on the other hand good. Therefore, we may write a term which is constant with respect to time and therefore we say that this term could be a function of z at the first time we shall discuss this function next. The nature of this can be examined by considering the second equation taking the z derivative of the solution of i and comparing it with the right hand side minus c del v by del t. So, let us do that we have del i by del z equal to minus c del v by del t and therefore we have 1 by l v times f 1 prime of z minus v t minus f 2 prime of z plus v t plus f prime z that is what the left hand side is. The right hand side can also be written since we know the solution for v it will be c times v f 1 prime z minus v t minus f 2 prime of z plus v t. We are just substituting the solutions for i and v in this equation left hand side and right hand side. And now we compare the left hand side and the right hand side keeping in mind that v is 1 by square root of l c. So, you find that these two terms are equal 1 by l v is going to be equal to c v the terms within the brackets are also equal and therefore we conclude that this f z is such that its derivative is equal to 0 derivative with respect to z. Therefore, f z is something which is constant it was already a constant with respect to time. From this consideration we find that it is a constant with respect to z also and something which is constant with respect to z at all values of z it is existing is not a part of a propagating way. So, on that basis we say that this is equal to 0 since it is not going to be a part of propagating way. We shall continue this discussion next time and we stop here.