 3rd lecture on the course on engineering electromagnetics. In the last lecture we put down a distributed parameter circuit model for transmission lines and we saw that the variables voltage and current satisfy an equation which we named as 1 dimensional wave equation because the solutions of that equation represent propagating waves. We saw that for voltage the solution was of the form v equal to f 1 of z minus v t plus f 2 of z plus v t and consistent with our argument that the function f 1 with an argument z minus v t represents waves propagating in the positive z direction we can write this as v plus plus v minus so that the function f 2 of argument z plus v t represents waves which are traveling in the negative z direction. This is the general solution and as we have been saying depending on the actual terminations or boundary conditions they will have different values I mean the incident wave or the forward traveling wave and the reflected wave. Similarly by considering the equations that v and i satisfy on the transmission line model we were able to write that i is equal to 1 by l v times f 1 of z minus v t plus f 2 of z plus v t. Is that all right? There should be a negative sign here and we will have some comments to make on this negative sign. Now we write this quantity l v we give it a separate name let us say we call it z naught 1 by z naught as we will soon learn this is a very important quantity in connection with transmission lines. So, we write this as 1 by z naught times f 1 of z minus v t minus f 2 of z plus v t and corresponding to the wave we have written the voltage in terms of a summation of a forward traveling wave and a backward or reflected wave. We write current also in a similar manner as i plus plus i minus. So, that similar to the expression for the voltage the current also is available as a summation of two waves and the moment we write this you can see that the negative sign appearing in front of the backward traveling wave is absorbed in i minus. Now one can compare the expressions for the current and the voltage and one immediately sees that we have v plus upon i plus equal to z naught and v minus upon i minus equal to. Now comparing and seeing the ratio of the second term in the two expressions we see that v minus upon i minus is equal to minus z naught. Now what is happening is that the voltages and voltages of the incident wave and the reflected wave, but current by it is very nature is a directed quantity it will flow in one direction or in the other direction. So, the reflected wave current should be subtracted from the incident wave current. So, the negative sign appears here, but to be able to write i similar to v we have written it as i plus plus i minus. However, that negative sign appears here effectively the result is the same it is just a matter of notation convention. We could equally well have written i plus minus i minus with a positive sign here. You would find text books using that notation and there is nothing wrong with it as long as we are consistent, but majority of text books follow this notation this is what we are going to follow here and we should keep this in mind that only as a matter of convention we are writing i as i plus i minus, but actually that i minus is travelling in the opposite direction to i plus is taken care of by this relation. Now, z naught deserves some more currents. What is z naught we have written z naught equal to l v that is the substitution we have made what is v l times 1 upon root l c. So, that this is equal to square root of l by c for the lossless case that we are dealing with to begin with z naught is a real quantity. If we consider the units of z naught something very interesting turns up we say that the units of l are henry per meter and so that units wise l is going to be volt seconds per ampere inductance is flux upon current magnetic flux upon current. So, flux as units of volt seconds divided by amperes and then c has units of farads per meter this should have a per meter unit also, but that will cancel out with c. And therefore, here it will be volts upon charge or say coulombs whole square units wise z naught will have these units. And one sees immediately that this is going to be volts upon amperes these will be the units of z naught which are the units of impedance or resistance that is ohms. And therefore, from this point of view as well as the way z naught is the ratio of voltage and current for a particular wave we call z naught the characteristic impedance of that particular transmission line. This is a very important quantity z naught is the characteristic impedance. This is such an important quantity its role will be further clarified when we move on to the next topic that it is been given some standard values standard values are 300 ohms or 75 ohms or 50 ohms. 300 ohms is the characteristic impedance of the transmission line that connects the television antenna to the television receiver the parallel wire transmission line. We can also use a coaxial cable there that will have a characteristic impedance of 75 ohms. The antenna has an impedance of close to 300 ohms this will be talking about in the later part of the course. And therefore, an impedance converter is required when you use a coaxial cable along with your television antenna. The 50 ohms characteristic impedance is utilized at RF or microwave frequencies. The impedance of course, it comes out from the parameters L and C the inductance per unit length and the capacitance per unit length. And is therefore, related to the geometry and the materials utilized in the construction of the transmission line. And therefore, it is an important design parameter for transmission lines. It will have a bearing on the kind of power that can be transmitted what will be the properties in terms of attenuation and distortion etcetera. So, Z naught the characteristic impedance is a very important quantity and very fortunately in a particular type of wave the voltage and current have a very simple relationship. This is what we were aiming at in this part of the lecture. So, we have got a relation between the voltage and current of a particular wave. The reflected wave voltage and current also have a similar ratio, but a negative sign comes in for the reasons we just discussed. We move on to the next topic in this lecture that is reflection and transmission. What would be the necessity of discussing this topic? When we have waves propagating on an infinite transmission line there is no problem. Wave would continue to travel, but an infinite transmission line is not going to do as much good. We need to put the transmission line to some use. We will connect it to some different transmission line perhaps we will connect it to some load or we will connect some solid state device for whatever purpose. So, usually the transmission line will be terminated in one thing or the other. It can be shown typically in the following manner. Whatever is it that terminates the transmission line we call it Zl and the transmission line that one is dealing with has the characteristic impedance Z naught and as I just mentioned Zl could represent several types of entities. Now, there will be let us say an incident wave with a V plus and I plus associated wave. We have already seen that V plus and I plus are intimately related. Their ratio must be characteristic impedance of the transmission line that is Z naught. When this wave incident wave reaches the load, the load will have its own voltage and current relationship which it must maintain which may or may not match with the voltage and current relationship for the incident wave. And therefore, in order to be able to maintain its own V i characteristic the load impedance or the termination or the discontinuity here will send in general a reflected wave which we may designate or show in this manner V minus and I minus with the relations of V and I within a particular wave incident wave or reflected wave given by the previous relations. And therefore, in general there will be some part of the incident wave which is reflected, some part will be absorbed in the load and it is of considerable interest to determine how much will be reflected back, how much will be transmitted or accepted by the load. And therefore, we come to this next topic of reflection and transmission. Now, since this causes some confusion many times, we will spend a little while on this aspect. Let us focus attention first on the incident wave. We have shown that it has a V plus and I plus associated with it. What is the kind of current flow on the conductors? The transmission line shown schematically in terms of two conductors. Is it that the incident wave is travelling on the top conductor and the reflected wave travelling on the lower conductor? Not at all, the current associated with the incident wave will have an equal and opposite component in the two conductors. We go a little deeper into this is it that the current for the incident wave will travel in the forward direction in the upper conductor and backward direction in the lower conductor. We cannot say that it will depend upon what is the voltage V plus. If it has a negative value, the current direction will be different. Therefore, the current direction is dissociated from the direction of the travel of the wave. The direction of travel of the wave is given by these functions, how Z and T are appearing together and there will be equal and opposite currents on the two conductors. Their direction will depend upon specific situations. If we put down this kind of arrows that is only notional and similarly following that notion there will be equal and opposite currents associated with the reflected wave and we could put them down like this consistent with our writing i equal to i plus plus i minus. Actually what is the value of i minus will come out from this V minus upon i minus equal to minus z naught and what is the value of V minus and so on. So, notionally this is how we put down the incident wave and the reflected wave and if you follow these simple relations systematically we will not go wrong. So, I hope that explains the nature of the incident wave and the reflected wave and the associated currents. Next we see how we can obtain the reflection and transmission and we start with the load impedance Z L equal to V L upon i L. This is how you would define the impedance Z L. It is the ratio of the voltage across it and the current flowing through it. So, this is just straight forward. What will be V L and i L? They will be the summation of the voltages of the incident wave and the reflected wave and the currents at the load. Normally this does not come out in the symbols because it would just become a little more cumbersome. But starting from this it is very clear that what we write next will be the values of the various quantities at the load which will be V plus plus V minus and as I just said we could say V L plus V L minus. But as long as we are clear it does not matter divided by i plus plus i minus consistent with the general solution for i that we wrote earlier. Now, i plus and i minus are not independent of V plus and V minus. They have a specific relationship and therefore, this becomes Z naught times V plus plus V minus upon V plus minus V minus. We just rewrite things V plus plus V minus upon V plus minus V minus equal to Z L by Z naught. And we already have an expression relating the reflected wave voltage at the load to the incident wave voltage at the load. One can manipulate this using simple component and C that we will have V minus upon V plus equal to Z L minus Z naught upon Z L plus Z naught. Since it gives us the reflected wave and the incident wave voltage is the fraction of the incident wave voltage that is reflected back it is called the reflection coefficient. Now, just as men and women why for equality they could be a reflection coefficient expression for the current also equally valid. We would just need to write V's in terms of i's. So, following that procedure we can see that i minus upon i plus is going to be Z naught minus Z L upon Z L plus Z naught. And to distinguish the two coefficients from each other we could write this as the voltage reflection coefficient and this becomes the current reflection coefficient. Similarly we could write these in terms of symbols rho subscript V and rho subscript i. However, as a matter of convention usually one has the voltage reflection coefficient in mind when one simply says reflection coefficient. If the other one is intended one would say specifically that current reflection coefficient. And therefore, rho is used without the subscript V and simple rho means voltage reflection coefficient. It would be of interest many times to find out what is the total voltage at the load that also can be found out from this without any difficulty or from this we just add the numerator and the denominator that becomes a total voltage at the load. And as a fraction of the incident wave voltage it becomes V L upon V plus equal to twice Z L upon Z L plus Z naught which is called the transmission coefficient. What is the voltage transmitted to the load transmission coefficient given the symbol tau. Yes please. Twice Z L from this you can get this. Now here you see that the reflection or the transmission is a function of the relative values of Z L and Z naught. So characteristic impedance is playing an important role in this phenomenon. One can consider some typical values of the Z L and Z naught and see what kind of values we get. The simplest value we could take is perhaps Z L equal to Z naught. What is the reflection coefficient for that? Rho is 0. What is the transmission coefficient for that? That is 1. What is happening now? The incident wave is completely accepted or absorbed by the termination by the load impedance because it is equal to the characteristic impedance. It holds the same VI relationship as that of the incident wave and therefore there is no requirement to initiate a reflected wave. And in fact as far as the rest of the line is concerned it appears as if the line is continuing to infinity. In practical circuits this is a very desirable situation. You would like that the load accepts whatever power you have managed to generate. In such a situation we say that the load is matched to the transmission line or matched to the characteristic impedance of the transmission. So it is many times called a matched load. Match load is not something which is physically matched necessarily. It should be matched impedance wise. It should cause no reflection. Then it is a matched load. In other values one could try out say ZL equal to 0. What would that correspond to? A short circuit. And you see that a short circuit leads to a reflection coefficient value which is minus 1 and a transmission coefficient which is 0. This reflection coefficient equal to minus 1 could be seen by considering the behavior of the total voltage at the load. And in our next example it will come out very clearly. And the short circuit cannot allow any voltage across itself. So the transmission coefficient is 0. What is happening is that the entire incident wave is being completely reflected back. With rho equal to minus 1 what will be the value of V minus? V minus will be equal to minus V plus. With a change of sign the entire incident wave is getting reflected back because of the short circuit. What would also be interesting is to consider what happens when we connected open circuit at the other end. In this case we would write that ZL is infinity. Open circuit is something which would not allow the passage of any amount of current. So ZL is infinite. What is the reflection coefficient for this? It will be 1. And correspondingly V minus will be equal to V plus. How the open circuit is enforcing its terminal conditions becomes clearer if we consider the currents. And then one would see that I minus and I plus are equal and opposite. So there is 0 current across the open circuit as required. However what is the transmission coefficient? It is 2. And that may raise some eyebrows. However the transmission coefficient has been obtained through the voltage. And we know that transformers can step up voltage without violating any power conservation. And therefore perhaps if we consider the power things will be all fine. However as far as the reflection coefficient and the transmission coefficients are concerned these are some typical values and these are usually expected to be remembered by the students. They are so simple and straight forward. How would we deal with the power? We have slight amount of difficulty here because we have not specified the time variation yet. We are keeping things general for the moment. But power should be related to the product of V and I. That is in any case there. And therefore following that let us say that power is given by product of V and I. And we say that the power carried by the incident wave V plus is equal to the magnitude of V plus times the magnitude of I plus. In any case since we are going to take ratios in the following it should not matter if some constant factor is not appearing here. Now V plus and I plus are related to each other through the characteristic impedance Z naught. So they could be expressed either in terms of V plus alone or in terms of I plus alone. So doing that would give us magnitude of V plus squared by Z naught which is also equal to magnitude of I plus squared Z naught. And as I just stated this is the power carried by the incident wave. Normally many times V I is the power loss. In a resistor. Now is this the power loss? There is no loss mechanism in our transmission line yet. So this is not power loss. This is the power transmitted quite similar to the power carried by the power transmission lines voltage into current. That is not the power dissipated that is the power transmitted. Same thing here. Similarly one can say that the power carried by the reflected wave comes out to be V minus magnitude squared by Z naught or I minus magnitude squared times Z naught. Since the relation between V minus and I minus magnitude wise is again through Z naught. So that we can write the power in the reflected wave as a fraction of the power in the incident wave P minus upon P plus which will be V minus magnitude squared upon V plus magnitude squared. So that it is equal to magnitude rho squared. We are using magnitude because the reflection coefficients could be complex depending upon Z L. Therefore, using magnitude helps us. What would be the power delivered to the load? It should be the power carried by the incident wave minus the power carried by the reflected wave. The difference of the two powers is what is being provided or supplied to the load. Therefore, it becomes P plus minus P minus and one can write this as V minus as 1 by Z naught into V plus magnitude squared minus V minus magnitude squared. And actually it is the fraction of the incident power absorbed by the load that we are interested in many times. So that P L upon P plus can be seen to be simply 1 minus magnitude of rho squared as can be seen in a straight forward plan. And now one can check what is the fraction of the power delivered to the load for both short circuit and for open circuit and in either case that fraction is 0 as expected. If you have any questions up to this point I will be happy to answer those. P L is the power dissipated in the load or carried to the load? It depends upon the nature of the load. As I mentioned Z L could be another transmission line connected to it. Then it could be the power carried by that transmission line. But to another load. So as far as this transmission line is concerned this is the power delivered to whatever entity is serving as the load. But it has to be dissipated at the end somewhere. It will depend upon the terminal conditions. If it is just a straight forward load here then it is dissipated here. If it is a transmission line then depending on the termination there things will take place. So even if it is connected to a transmission line at the end of that transmission line there has to be some this is the line. It will be the final termination which will decide. They are also similar considerations will hold good for that last leg of the transmission line. We cannot have a power just carried and not dissipated anywhere. That is what we are saying depending on the last leg termination one would see what is happening. Now it is time for us to consider some simple traveling wave situations. Now that we have got some basic framework of reflection and transmission etc and we will start with very simple circuits or situations. Some simple traveling wave situations. The circuit that we take is the following. It is a transmission line with characteristic impedance Z naught and we connected to a DC voltage source through a switch. The source voltage is V naught. We consider for the sake of simplicity that the voltage source is an ideal one so that its source impedance is 0. So that when this voltage is applied to the transmission line the entire voltage appears across the transmission line that is the simplicity we are looking for. And we say that at t equal to 0 the switch is closed. We say that initially there is no current on the transmission line and the line is not charged. We say that initially no charge or current on the transmission line. It is a very simple situation. It should be possible to make out what is going to happen. From t equal to 0 onwards a voltage wave V plus equal to V naught starts off. Starts traveling to the right. We could say that to the towards the positive Z direction. And it will keep on charging so to say this capacitor formed by the two conductors of the transmission lines. There will be an associated current I plus equal to V plus by Z naught that is V naught by Z naught which will become the charging current which will be equal and opposite in the two conductors of the transmission line. Now if we take a snapshot let us say at t equal to 0 t equal to t 0 t 0 being greater than 0. What kind of situation do we expect? We expect that this voltage wave this forward wave with associated voltage V naught and associated current V naught by Z naught would have traveled up to a certain distance depending on the velocity of wave propagation on this transmission line. We can plot this voltage on the transmission line. It will be V naught up to a distance Z equal to V t naught where V is equal to V naught. E equal to 1 by square root of l c depending on the distributed parameters l and c of the transmission line. Let us say that t naught is such that the voltage wave has reached up to this point. The wave has reached up to this point with corresponding voltage V naught and corresponding current V naught by Z naught. What happens to KCL at that point? There is current up to some point no current beyond that. So KCL is not applicable in such a situation. This is what we have been saying. One could also consider the current which will have a similar behavior except for its value that is I will be V naught by Z naught up to a distance V naught by Z naught up to a distance Z equal to V t naught. And this is all a consequence of time delay. The signal is taking a finite amount of time in travelling from one part of the circuit to the other. We take another example and this time we do not leave the transmission line infinitely long. We consider what is connected at the other end. Yes please. Sir, will the current rise sharply to that point? Sir, at the edge it will be smooth at the end at Z equal to V t naught. It is instantaneously rising to a value V naught by Z naught. After all the electrons do take some time to move. In this case, we are saying that the current is suddenly jumping to non-zero value and actually there is no mechanism to prevent this build up of current. We can think about it and see what can be said about this later. I have not thought about this aspect earlier. See normally one says that across a capacitor voltage cannot change suddenly or across an inductor current cannot change suddenly. However, here what is being seen by the source is an extremely small capacitance or extremely small inductance. And therefore, that kind of argument perhaps is not applicable. We continue with our simple travelling wave situations and we consider a similar circuit, but this time it is terminated. We say that at the other end the transmission line is terminated in a short circuit and the transmission line has a characteristic impedance Z naught. The short circuit is placed at Z equal to L and the other assumptions remain the same. That is it is an ideal voltage source. So, that the source impedance is 0. At t equal to 0 the switch is closed. Initially there is no charge on or current on the transmission line. Now, the time that the signal will take in travelling from the source end the input end to the load end is going to be t equal to L by V which let us call capital t. We will use this in what we discussed in the following. So, we give it a symbol. Now, as long as t is less than capital t as long as the forward wave or the incident wave has not reached the short circuit. This situation will be identical to what we saw in this example. The line will keep on getting charged and there will be a corresponding charging current which is equal and opposite in the two conductors. The line will keep on getting charged to a voltage V naught. The velocity of course is 1 by square root of L c as usual and we have a forward wave V plus equal to V naught with the corresponding current I plus which is V plus by Z naught. So, that it is V naught by Z naught and that is the situation as long as time is less than capital t with t equal to L by V. Forward wave is initiated which starts travelling towards the other end. Now, what happens at t equal to t capital t? We have connected a short circuit here. The short circuit cannot allow any voltage value other than 0. The incident wave comes with the voltage V naught. So, a reflected wave with V minus equal to minus V naught will be initiated which will start travelling back so that the total voltage at the load V plus plus V minus is equal to 0. The same thing we could have obtained not in terms of this kind of simple logic, but through the reflection coefficient expressions that we have written. For a short circuit reflection coefficient rho will be minus 1. So, V minus will be minus V plus, but this makes very clear the origin of the reflected wave. In an attempt to establish its own voltage current condition the termination sends out a reflected wave in general. As this what is the current associated with this? I minus which will be minus V minus by Z naught. We have to be consistent. So, that it will be once again V naught by Z naught. So, that the total voltage is 0 and the total current is equal to twice V naught by Z naught. This is the situation that continues when small t is between capital T and twice capital T. The incident wave in this part charged the transmission line. The reflected wave is now discharging the transmission line and as the reflected wave travels back it brings back with it 0 voltage condition. So that when this reflected wave reaches the source which happens at t equal to 2 t. In an attempt to establish its own voltage the voltage source sends out a second time around forward wave which is equal to V naught. When the reflected wave reaches the source it brought back with it a 0 voltage condition. It is a voltage source it must maintain a voltage of V naught and therefore a forward wave is initiated with magnitude equal to V naught. With the corresponding current equal to V naught by Z naught. The total voltage and total current can be looked at now. We have V plus plus V minus plus V double plus all this sums to V naught which is the source voltage fine. The current similarly sums to I plus plus I minus plus I double plus so that it is twice V naught by Z naught and this process now one can continue and one can plot the current through the source. For example if that is what we are interested in and we find that as a function of time the current through the source to begin with it is V naught by Z naught. Till a time which is capital T till a time which is twice capital T because up to this time the source is not aware of the reflected wave. At 2 t the current will jump to a value as we have just seen which is twice V naught by Z naught will continue for another twice t seconds and then it will change to 5 V naught by Z naught is that all right where on the transmission line wherever the reflected wave is reaching as far as the source is concerned it is continuing to supply I plus. So, one has to be careful this is the source. If one plotted the load current it will be similar in nature but the time and time instance will be different there will be t 3 t 5 t the source current changes at t 2 t 4 t and 6 t etcetera. This will jump to value which is 5 times V naught by Z naught this simple arrangement can be used for generating very high current pulses and that the current is building up in steps is a direct consequence of the time delay that we have built in in our analysis. One can compare it with the low frequency behavior of things and what would say that instantaneously subject to some limitations the current rises to very large value when the source is short circuited. But here we see that it is going to rise in steps because of the finite time delay effects such an arrangement can also be used for generating high voltage pulses if the short circuit is replaced by some resistance if that can be done all right. So, even the simple arrangement can be very useful. Will this voltage magnet current magnification continue till what extent will it continue? That is a good question this will continue in the loss less model but it will be limited in the lossy model which is what is going to happen in practice. As the current increases I squared R losses are going to increase in the various parts of the circuit and therefore, they will limit the maximum value of the current. So, I suppose the ideality of the voltage source will also come into question. Certainly there will be losses everywhere in practice and once the power that the source can deliver matches the power that is being dissipated the current build up will stop that is only reason. So, we will stop this lecture with a question. What will happen if instead of a short circuit the termination is an open circuit? You think about it and maybe we can discuss the answer in the tutorials.