 33rd lecture in the course engineering electromagnetic. In the lecture today we continue our discussion on the general properties of electromagnetic resonators and then continue to consider the properties of transmission line resonators. You would recall that last time we considered the example of a parallel resonant circuit and we were able to make some conclusions of general significance which would be applicable to resonators of various types and shapes that are going to be required for work at different frequencies. Continuing with that illustrative example of the parallel resonant circuit what would be the effect if the resonator is coupled to an external agency. Normally the resonator is required to do some useful work it will be coupled to some outside agency which will load the resonator in some way. So what is the effect of this loading or how are we going to take this into account that is what we consider first. And as I said we will continue with our illustrative example of the parallel resonant circuit. In the parallel resonant circuit we have L and C and the loss in the circuit in the circuit elements or the rest of the circuit is taken into account by connecting resistance R in shunt which for a good quality circuit is going to be high. Now if the resonator is connected which is the case in practice to some external agency or some other loss mechanism for some other purpose is introduced in the resonator then that loss because of the additional loss that we may have introduced or because of the loading effect of the external agency can be considered to be represented by another resistance in shunt which we may give the symbol R L. And therefore recalling that for the parallel resonant circuit the quality factor was given as R by omega naught L. Now the R that is going to be required in the numerator in this expression for the quality factor will be the parallel combination of the resistance R representing the losses in the circuit itself resonant circuit itself and the effect of loading that is the resistance R L. So that it will be the parallel combination of R and R L divided by omega naught L which we may write as 1 by R plus 1 by R L whole inverse divided by omega dot L which we could rewrite in a different manner as 1 by Q equal to omega naught L by R plus omega naught L by R L. Now to indicate that the quality factor that we are talking of now in the presence of loading is going to be different from the quality factor in the absence of loading we should use the subscript L here the loaded quality factor which continues through. And now the two terms that appear on the right hand side can also be given different names. For example the first term can be called the unloaded quality factor say 1 by Q sometimes one would use the subscript Q naught or Q unloaded to signify that this term represents the quality factor of the resonator in the absence of any losses right. To keep the notation in the same form the last term on the right hand side which represents the effect of the loading due to various reasons can be considered to be the external quality factor if the quality factor were to be decided only on the basis of this loss resistance and the intrinsic loss of the resonator were absent then what would be the quality factor of the resonator that we call the external quality factor. And represent this by 1 by Q E and therefore we have this general relation which holds good for resonators of various shapes that 1 by Q L is equal to 1 by Q plus 1 by Q E the significance of this expression is that the loaded Q and the unloaded Q they can be measured by measuring the resonant frequency and the frequency separation between the 3 dB points right. And therefore the characteristics of the loading R L due to whatever reasons it may be can be made out. And therefore when we use resonators for testing materials or for some other checking purposes then this relation comes in very handy. While this kind of resonators based on lumped elements can be utilized up to let us say frequencies which are 100 megahertz we have stated earlier that as the frequency of operation increases beyond this there appear problems in using this form of the resonant circuit. The problems are of different types for example because of the parasitic reactances it may be difficult to correctly estimate the behavior the losses may also increase. And therefore as we try to have resonators at frequencies higher than this we have to change the configuration of the resonator change the shape in which the resonator is going to be obtained. And then as we stated earlier we can use what to call the transmission line resonators. It was stated earlier that the transmission line resonators can be operated efficiently over a frequency range which extends from 100 megahertz to roughly say 1000 megahertz and as some of you may be familiar short circuited section of a transmission line of an appropriate length acts like a resonator. The section may look something like this the section is open circuited at one end and is short circuited at the other end. It has a length L which for the sake of illustration we take as lambda by 4 we can generalize this length later on. And let the phase shift constant on this transmission line be beta with an associated velocity of the waves phase velocity of the waves as v. Now how are we going to see or show that such a section acts as a resonator. We should be able to show that under certain conditions under appropriate circumstances there is a resonant build up of voltage or current possible. If we can show that then we can say that yes this section can act as a resonator. So, for this purpose let us connect some time varying source and let us connect it somewhere close to the short circuit with the help of a probe and let us connect a voltage v naught which is varying sinusoidally with time. So, that in phasor notation it may be written as v naught e to the power j omega t. And let us consider the time varying behavior of the circuit for such a such an excitation. Let us say that at t equal to t dot the voltage that the probe provides is v naught e to the power j omega t naught. We have assumed that the probe voltage is connected very close to the short circuit. The short circuit must maintain a 0 voltage condition across itself. And therefore, going through the arguments which should be familiar to us a forward wave is going to be initiated to satisfy the boundary conditions at the short circuit. This forward wave may be represented as v plus where v plus should be equal to minus v naught e to the power j omega t naught which is going to travel to the far end the open circuited end. And let us say that takes a time capital T which is l upon the velocity. So, that at time instant t equal to t naught plus capital T as this forward wave reaches the open circuit that is going to impose its own boundary condition. Namely the total current through the open circuit must be equal to 0. As a result of which we must have a reflected wave v minus which as we all know should be equal to v plus itself that is minus v naught e to the power j omega t naught which reflected wave is going to arrive back at the probe close to the short circuit at time instant small t equal to t naught plus twice capital T. And now let us see what is the total voltage at the probe attempting to be like. And that will be v naught e to the power j omega what will be the probe voltage now it will be t naught plus 2 t because the time instant is changed. So, its phase has to change accordingly this is the probe voltage in addition there is a reflected wave which is come which is minus v naught e to the power j omega t naught. This is what the total voltage at the probe tends to be which is going to be of the nature v naught e to the power j omega t naught plus 2 t we take this factor entire factor is common. And therefore, we are left with 1 minus e to the power minus j omega what is the value of this factor omega 2 t or twice omega t that one can make out twice omega t we write in terms of the phase shift constant beta realizing that v is equal to omega the radian frequency divided by beta the phase shift constant. And therefore, omega is simply let us replace capital T in terms of l and v. So, that is twice omega l by v and then omega by v is going to be equal to the phase shift constant beta. So, that it is twice beta times l beta the phase shift constant is further replace in terms of the wave fling. So, that it is twice into twice pi by lambda into l which let us say is lambda by 4. So, that this is equal to simply pi radians which is going to happen this kind of value of pi omega t equal to pi is going to occur only under very special conditions at a particular frequency and for a particular length of this section. But under those conditions this is going to be equal to pi and therefore, this is going to be twice v naught e to the power j omega t naught plus twice capital T which the short circuit here has to counter to maintain its 0 voltage boundary condition. So, this twice beta l equal to pi does not depend upon frequency. How do you say that? Because l is lambda by 4 is sufficient. Yeah it is lambda by 4 at a particular frequency all right if you want to put it that too as long as the section is lambda by 4 where the frequency is automatically taken into account l is lambda by 4 is for all difference. How can that be? It is lambda by 4 at a particular frequency all right and therefore, the short circuit is going to force a second time around forward wave called v double plus which is going to be equal to minus 2 v naught e to the power j omega t naught plus twice t all right. At t equal to t naught we had a first time around forward wave which was minus v naught of magnitude v naught at t equal to t naught plus 2 t we have a second time around forward wave which has a magnitude twice v naught. We can continue this argument and see that at t equal to t naught plus 4 t we will have a third time around forward wave which is likely to be 4 v naught e to the power j omega t naught plus 4 t and therefore, as we consider this particular instant t naught and as we consider its increments by twice capital T 4 capital T and so on. We see that the voltage on the transmission line is going to increase and since this time is rather small since the frequencies are high the voltage is going to rise in a resonant fashion. For how long is the voltage going to continue to rise till the losses in the transmission line the conductor loss and the dielectric loss match the power fed by the probe to this resonant circuit all right. And a voltage step up of several hundred volts is possible for a good low loss resonant section. One can also see that it is not necessary that the length is exactly lambda by 4 it could be any odd multiple of lambda by 4 also. So, that we may generalize that L is equal to an odd multiple of quarter wave length. So, this is the way we can show that such a section such a quarter wave length short circuit section of a transmission line acts as a resonator. And for a modest voltage connected here they can be a fairly high voltage generated on the resonator which is limited by the losses in the resonant section. Till the losses in the section counter the power that is fed by the probe once they become equal the further rise will be inhibited all right that is this that is the steady state condition. Having seen that such a section can act as a resonator now let us work out the other properties for example, the input impedance and how it behaves as a function of frequency. And we look at the short circuit section in a slightly different way just as we considered transmission line sections actually. This is the short circuit section with primary constants of the transmission line being R L C and in fact strictly speaking G also. But the role of G is going to be relatively minor compared to the role of R. And the secondary constants which would be say the characteristic impedance Z naught and the propagation constant gamma or the phase shift constant beta and the attenuation constant alpha in general. And let us consider the input impedance of this section which is of length L which at resonance we consider to be equal to lambda by 4. But as we deviate this deviate from this resonant frequency the length in terms of wavelength is obviously going to check. As far as the input impedance of a short circuit transmission line is concerned we write this as Z naught the characteristic impedance times tan hyperbolic of gamma L where gamma is the complex propagation constant. So that it is Z naught sin hyperbolic of alpha plus j beta times L divided by cos hyperbolic of alpha plus j beta L which sin hyperbolic and cos hyperbolic function can be expanded further to give us Z naught equal to sin hyperbolic alpha L and j times sorry there is no j here we simply have cosine beta L plus cos hyperbolic alpha L and here it is j times sin beta L divided by cos hyperbolic alpha L cos beta L plus sin hyperbolic alpha L j times sin beta L. Now at resonance beta L as we have just seen is equal to pi by 2 so that as far as the resonant condition is concerned the cosine beta L terms are going to drop out and the sin beta L terms will become simply 1 and we are going to have let us call it Z resonance equal to cosine hyperbolic alpha L divided by sin hyperbolic alpha L with a factor Z naught in front. Now for a good low loss resonance section alpha is going to be small it will cause a small attenuation of the fields per unit length and therefore we can use this small argument approximation for these functions giving us the approximate value of the input impedance at resonance equal to Z naught upon alpha L as long as alpha L is small alpha L could become large for very large lengths for example. So as long as alpha L is small this is the value of the input impedance at resonance which is going to be quite high for low loss sections. What is the value of alpha? Alpha in general is equal to half R by Z naught plus G Z naught but to keep things simple and which is also practically quite alright the second term is quite small compared to the first term and therefore one can use the approximate value of alpha that is R by 2 Z naught. This is the value of the input impedance at resonance what we want to make out is how this input impedance varies as a function of frequency as we change the operating frequency by a small amount around the resonant frequency. Therefore just as we considered such a behavior for the parallel RLC circuit the same thing we are going to attempt here. We say that let F be equal to F naught plus delta F where delta F is typically small particularly for a good low loss resonant circuit. So that the beta L value that we are considering beta L is written as omega by V times L and omega is twice pi F by V times L and now we replace F by F naught plus delta F. So that it is twice pi L by V into F naught plus delta F which is equal to value of beta L at the resonant frequency plus a correction term. The correction term is simply twice pi L delta F by V and the first term the value of beta L at resonant frequency we have already identified is equal to pi by 2. And therefore the sine and cosine function that we have are going to become as follows. Sine beta L will go to cosine of twice pi L by V delta F and similarly the cosine beta L will go to minus sine twice pi L by V times delta F which values of sine and cosine functions can be substituted in the general expression for the input impedance. Carrying out this process we have Z in equal to Z naught then sine hyperbolic alpha L and cos beta L which goes to minus sine twice pi L by V delta F and then we have plus J cos hyperbolic alpha L and sine beta L which goes to cosine of twice pi L by V times delta F that is as far as the numerator is concerned. And the denominator becomes minus cos hyperbolic alpha L sine twice pi L by V delta F and then the last term here becomes J sine hyperbolic alpha L cosine twice pi L by V delta F. And can compare this with the previous general expression for the input impedance and satisfy oneself that no mistake has been made. Now realizing that the arguments of the sine cosine and the sine hyperbolic and the cos hyperbolic functions are small particularly for a good resonator. The attenuation constant and therefore alpha L will be small and the frequency deviation will be small only around the resonant frequency we are going to get some output. And therefore this is going to be approximated as Z naught and then we have here minus of alpha L times twice pi L by V delta F. And the second time in the numerator is simply J the cos hyperbolic and the cosine functions being approximated by 1 in the small argument approximation. And then we have minus twice pi L by V delta F and then J times alpha L. As far as the first term in the numerator is concerned it is a product of two small quantities and in comparison with the second term its magnitude is going to be quite small. And therefore can be safely neglected and on that basis we get the value which is Z naught. We ignore this term in comparison with this and then divide the numerator and the denominator by factor J which gives us Z naught upon alpha L plus J times twice pi L by V times delta F. From where at resonance that is when delta F is 0 we get the same value of the input impedance as we have obtained before that is Z naught upon alpha L. Now therefore we normalize the input impedance with respect to this value at resonance giving us Z in by Z resonance equal to 1 upon 1 plus J 2 pi delta F by V and alpha. Now recalling that V is equal to omega by beta this factor twice pi delta F by V alpha is twice pi delta F and then V gives us beta by omega and alpha which may be written as delta omega by omega and beta by alpha. And now recalling the form of the input impedance as a function of frequency that we got for the parallel resonance circuit. If we write this as 1 upon 1 plus J 2 q into delta omega by omega. If we compare this with this expression then we see that q is simply beta by 2 alpha. In terms of the secondary constants of the transmission line the quality factor the selectivity comes out as beta by 2 alpha where beta is the phase shift constant and alpha is the attenuation constant. This kind of behavior of the input impedance is exactly the same as for the parallel resonance circuit. And we should also notice that at resonance the input impedance value is high Z naught by alpha L where alpha L is small and therefore the resistance the input impedance at resonance is high. This is the distinctive feature of parallel resonance circuits. If one consider the series resonance circuit then we will find that at resonance the input impedance is small and the input admittance will have this kind of behavior. So, therefore by comparison we are able to say that the quarter wave short circuit section that we are considering as a resonator behaves like a parallel resonance circuit. This expression appears in a slightly different form but if we put in the values of beta and alpha in terms of the primary constants of the transmission line it will come out in a very familiar form. For example, we will have q equal to beta which we write in terms of the primary constants as omega naught into square root of L c using the low loss approximation for the transmission line divided by 2 alpha where we neglect the contribution of the dielectric loss in the attenuation constant in comparison with the conductor loss. And therefore alpha is R by 2 Z naught which is going to appear in the numerator in this form using the low loss approximations. And therefore you see that this is equal to omega naught L by R. Now this expression for the quality factor has come out in a slightly different form from the expression for the parallel resonance circuit where we had R upon omega naught L. But if we compare the two carefully then we would notice that R upon omega naught L involves a resistance R which is a shunt resistance and is a high value for a low loss circuit. The R that is appearing here in the denominator is a small resistance indicating the series loss resistance in the transmission line. And therefore the quality factor remains high although it appears in this form. One can see easily that if one were to consider the dielectric loss related attenuation constant then one will have an expression which is quite consistent with R upon omega naught L kind of expression for the parallel resonance circuit. And therefore we find that even for the transmission line resonator we have the behavior which is quite consistent with the illustrative parallel RLC circuit that we had considered earlier. The quality factor can be considered in an alternative manner also. We had written the quality factor in terms of the stored energy and power loss also. So let us apply that concept and see how consistent results we get. Now for a short circuited transmission line let us consider the behavior of current as a function of various locations on the transmission line. It is going to be a standing wave and therefore the current can be written as I equal to some current magnitude I naught times cosine beta z and let us say e to the power j omega t in phasor notation. So that at z equal to 0 which let us say is the location of the short circuit the current is maximum and otherwise it varies in a cosine sinusoidal fashion as the z parameter takes on increasing values away from the short circuit. On this basis now it will be possible for us to calculate the stored energy and the power loss. Let us see how we are going to do that. But first let us put down the definition of the quality factor which will be omega naught W upon P L which is the resonant frequency of the power loss. So this is omega naught times the energy stored in the resonator divided by the power loss in the resonator. And as we have been saying for this kind of expressions earlier both numerator and denominator should be calculated on the same basis either average or P and as a matter of practice if we calculate average values we have less chances of going wrong. The energy stored is calculated as follows. Let us consider the energy stored in the magnetic field associated with the current alright and the energy stored in the magnetic field in this form over a small length d z of the transmission line is going to be 1 by 4 times omega naught times the inductance associated with such a section times i i star. This is the average energy stored in a in an infinitesimally small length d z of the transmission line. So that the energy stored in the magnetic field average energy stored in the magnetic field which should be an integral of this and of this from 0 to lambda by 4 can be calculated. And this gives us an expression which is L by 4 and i naught squared and then integral of cos squared beta z over the limits 0 to lambda by 4. And therefore, one gets lambda by 32 and i naught squared this is the energy stored average energy stored in the magnetic field. What we require is the total average energy stored there will be energy stored in the electric field also. But as we identified last time the total average energy stored w is equal to w m plus w e which is equal to twice w m which remains fixed and distributes itself in varying proportions as a function of time between the energy stored in the electric field and the energy stored in the magnetic field. And therefore, this need not be calculated separately although that can be done by considering the associated energy stored in the magnetic field voltage and then the energy stored in the electric field. And therefore, this is equal to lambda by 16 i naught squared times a. In a similar manner we consider a short section of length dz consider the resistance associated with it and the ohmic loss caused by that resistance. And we ignore the loss because of the dielectric if it is present then it can be taken into a cup. And if we call this loss in the small section as d p l then this is going to be half r dz times i i star. So, that the total average power loss in the resistance is an integral of this or of this from 0 to lambda by 4. The integral is straight forward and the result that we get is lambda by 16 i naught squared r. So, that when we substitute these expressions for p l and w in the expression for the quality factor we get an expression which is quite familiar by now q is equal to omega naught l by r the same expression that we had got by considering the input impedance of the short circuited transmission line section. Now, it is not that it is only an open circuited short circuited section of a transmission line which is lambda by 4 or an odd multiple of lambda by 4 which will act as a resonator. Those of you who have worked on a transmission line demonstrator would recall that one can have other conditions also, but then the length is changed appropriately. For example, open circuited transmission line sections which are half wavelength long would also have similar properties. Analytically they can be looked at in an analogous manner and they will have same kind of properties. For example, the input impedance will be that of a parallel RLC circuit and it will be high at resonance. So, these transmission lines sections which show a behavior input impedance behavior which is analogous to that of the parallel resonant circuits are given a separate name. These are called anti resonant circuits and as you may guess the circuits which behave as a series RLC circuit would be called resonant circuits. The major difference between the two will be the behavior of the input impedance at resonance. The anti resonant circuits have a high value of the input impedance at resonance. Because the resonant circuits have a low value of this impedance at resonance they are rather difficult to use. It becomes difficult to couple sufficient amount of power in a resonant circuit which behaves as a series resonant circuit. And therefore, this kind of circuits which behave as parallel resonant circuits are more useful and you would find that these are applied much more often in practice. The way the quality factor is appearing in terms of the primary constants of the transmission line L and R when we have ignored the shunt conductance G. If you want to estimate the quality factor of the transmission line resonant circuits we will need the values of these parameters in terms of the geometrical and the construction parameters of the transmission line. And therefore, it will be useful to list the values of these primary constants for some commonly used transmission lines. The commonly used transmission lines would be the parallel wire transmission lines and the coaxial transmission line for which we have given the expressions for the primary constants earlier but for the sake of ready reference they can be given here also primary constants. The one commonly used transmission line is the parallel wire transmission line using two identical conductors separated by a certain distance. The separation is called capital D and the diameter of the individual conductors is called small D. And we list the characteristic impedance and the resistance R per unit length for this transmission line. The characteristic impedance for such a transmission line has the value 1 by pi mu naught by epsilon naught epsilon R prime where epsilon R prime is the real part of the relative permittivity of the dielectric medium whole square root times cos hyperbolic inverse of capital D by small D. And similarly the resistance per unit length is twice R m by pi D capital D by small D and then we have D by small D whole square minus 1 and then whole square root of the denominator. The other common transmission line is the coaxial transmission line consisting of an inner conductor and an outer conductor where the radii of these two conductors are B and A. So, that the diameters are 2 B and 2 A. In terms of these parameters the characteristic impedance comes out to be 1 by 2 pi and once again mu naught epsilon naught epsilon R prime whole square root which is simply the intrinsic impedance of the medium the dielectric medium filling the region between the two conductors times natural log of B by A. And the resistance R per unit length is R m by 2 pi and then 1 by 2 pi. 1 by A plus 1 by B and you will recall that R m is the surface resistance we have used R S symbol also for this which is equal to omega mu by 2 sigma whole square root. What we are going to require are the inductance in addition to R for being able to estimate the quality factor. And that is related to the characteristic impedance in the following manner L is equal to mu naught epsilon naught epsilon R prime whole square root times Z naught. If we require the capacitance C per unit length that also has a similar expression, but not exactly the same it is mu naught epsilon naught epsilon R prime whole square root in the numerator. And then in the denominator we have division by the characteristic impedance Z naught. The characteristic impedance expressions are here and therefore L and C can be calculated and hence the quality factor can be calculated since R expression is also given. In practice the quality factors that one can achieve using these transmission lines can range from several hundreds to even up to about 10,000. And this is where we stop today in the lecture today we have considered the behavior of transmission line resonant sections. Thank you.