 Welcome to the 32nd lecture in the course engineering electromagnetic. In this lecture we start the new topic of resonators and today we go on to consider the general properties of resonators. Before initiating discussion on the resonators it will be useful to quickly recapitulate what kind of applications the resonators have. The resonators have a number of applications some of these applications you may be able to recall readily. For example the resonators are used in oscillators, tuned circuits for example in tuned amplifiers. Resonators have applications in filters also and they can be used as discriminators. They can be many more applications of resonators and therefore we see that there are many important applications of resonators. These applications could span from a few hertz very high frequency say optical frequencies which would be hundreds of terahertz. And depending on the frequency of application as we shall discuss shortly the actual physical shape of the resonator can become quite different. So let us consider these different structures that are used as resonators at different frequencies various let us say configurations which can be used as resonators. Of course the most familiar configuration will be the one utilizing lumped element inductance and capacitance as a parallel resonant circuit or as a series resonant circuit. For example we consider the parallel resonant circuit and we can draw this in the simple and familiar form like this. This is the capacitance C involved in the circuit and this is the inductance L involved in this resonant circuit and we know that the resonant frequency say omega naught is related to the value of these elements as 1 upon square root of L c. And therefore as we try to achieve higher resonant frequencies the values of these elements should reduce fine. This kind of circuit is quite fine up to let us say tens of megahertz at higher frequencies what problems arise will be mentioning in a minute. As we try to have higher resonant frequencies these elements should have smaller and smaller values and perhaps let us say a situation will come where we have very small capacitance and it is connected with a very small inductance may be just a single turn inductor which should have a pretty high resonant frequency. However you would recall that as we try to increase the frequency of operation the problem because of the lead inductance inductance due to the connecting wires and the parasitic capacitances etcetera increases and becomes difficult to predict the behavior of the circuit that we are trying to build. If we want to obtain a circuit at a particular resonant frequency then it may require a large number of trials because of these associated problems which are difficult to analyze and model accurately. And therefore instead of using this kind of a lumped parameter resonant circuit we can go in for resonators which are based on transmission lines. And therefore that leads as to a transmission line resonator which may be a short circuited open circuited section or it could be open circuited open circuited short circuited short circuited section of an appropriate length. With these end conditions we could have the length as lambda by 4 and it can work as a resonator and we shall discuss such a transmission line resonator in more detail later on. Such a configuration serves us pretty well for frequencies from 100 megahertz to about 1000 megahertz. What problems would arise if we try to have such a resonator at even higher frequencies say when frequencies reach gigahertz range the length will become quite small that is one thing. The loss will become rather high because of the skin resistance and because of the dielectric material which is invariably involved in the transmission line. And therefore the performance of the resonator will be affected in a manner which again we are going to talk about shortly. And therefore one has to think of alternative configurations, configurations alternative to these which would work well at even higher frequencies. So, picking up the lead from here that is trying to reduce the capacitance and the inductance further one can get a structure which let us say looks like this. We have parallel plates which are separated by a rather large distance. So, that the capacitance associated is quite small and for inductance we can put very wide and thin straps. So, that these are connected by this kind of what we are calling straps. So, that the associated inductance is quite small similar straps could be put on all four sides. Such a structure is expected to have a resonant frequency which is quite high because the associated inductance and capacitance are quite small. While this is a conceptual structure a structure which can be realized in practice can be derived from this by increasing these straps such that it becomes a completely enclosed cavity. So, that we go to the next physically realizable configuration obtained by connecting six conducting plates in this manner. And actually this is nothing but a portion of a cylindrical wave guide what we have shown is a cylindrical wave guide of rectangular cross section. It is a certain length of the rectangular wave guide where we put short circuiting plates along the length of the wave guide at appropriate separation. So, that this structure resonates this is nothing but a closed cavity resonator which can be used since it is free from any dielectric material in general which is going to have a rather low loss and therefore, a good performance even at high frequencies. Such a structure this kind of closed cavity resonators derived either from rectangular wave guide or from circular wave guide are quite useful at frequencies which range from typically 1 gigahertz to 100 gigahertz. Why should there been upper limit on the frequency range of application that comes in when we consider the associated wave lengths and therefore, the corresponding sizes of these cavity resonators at 100 gigahertz the wave length is 3 millimetres. The guide wave length will be of that order only and therefore, will be will have to make that kind of small small structures which are difficult to fabricate and also as the frequency increases the conductor loss is going to increase according to the skin resistance. So, those problems are anticipated if we continue to increase the frequency further using the same resonator configuration. And then an alternative configuration that means suggested which can work even at higher frequencies with a satisfactory performance is that you just have a pair of parallel conducting walls or reflecting walls between which let us say more or less a plane wave reflects back and forth and therefore, a resonator kind of action can obtain. Such a structure which is not enclosed on all sides but still works as a resonator is called an open resonator. While in principle such as open resonator structure can be used at even these microwave frequencies but only in some special applications it will offer an advantage. But at higher frequencies one has to use such an open resonator configuration particularly at optical frequencies where let us say you would like to stabilize the frequency of an optical source which is generating a wavelength of the order of micron or a fraction of a micron. One cannot think of building such a closed cavity resonator one has to build this kind of open resonator and there will be mirrors separated by orders of microns acting as resonators for that kind of frequency range. So, open resonators for frequencies greater than let us say 100 gigahertz going to optical frequencies alright. So, this is the sequence in which one could consider the various resonator configurations as evolving from the familiar low frequency configuration involving the lumped element inductance and capacitance. Now each one of these configuration can be analyzed rigorously and accurately and they will have their particular methods of analysis suitable to their suitable to that particular configuration. Yet since they are all resonators they share some properties which are common some properties are general irrespective of the configuration and irrespective of the frequency that is utilized. So, it is these general properties that we will like to discuss today and this we do with the help of taking up the parallel resonant circuit. For discussing the general properties as I just mentioned we consider a parallel resonant circuit with let us say a capacitance C here and an inductance L here. We excite this by applying a sinusoidally time varying voltage V alright without time variation resonator action does not take place causing flow of current I in this manner and it is quite a typical to consider that these lumped elements inductor and capacitor are lossless and club their losses which are always present in practice in another item which is a resistor connected in shunt in the parallel resonant circuit. And then we can consider that these elements L and C are lossless for a good resonator where the losses are low this shunt resistance will be high. So, that it affects in a relatively small way insignificant way the performance of the rest of the circuit and the impedance seen looking into these input terminals may be called Z n. Now, let us consider the complex power flow complex power fed to the above circuit what would be that that will be half V I star where I star stands for the complex conjugate of the current which can be written in the in alternative manner as half Z in I I star or alternatively half V times V star Y in star where Y in is the reciprocal of Z in or it is the input admittance. For the parallel resonant circuit it is more convenient to work in terms of admittance and therefore, we try to put down an expression for Y in star that will be straight forward it is 1 by r and then minus 1 by j omega L minus to take care of the complex conjugate and once again minus j omega C. So, that the complex power fed to the circuit half V I star which is half V V star Y in star is equal to half V V star times 1 by r minus 1 by j omega L minus j omega C. Now, wherever we are able to identify a unique value of the inductance L and capacitance C there this kind of an expression is fine, but that will be so only when we have a lumped element circuit for the various other forms that we have shown it is not easy to identify values of this kind of R L and C and therefore, it will be better to put these terms in terms of powers or stored energies which can be identified even for other configurations or in the more general case and therefore, let us try to identify the significance of each of these terms in terms of powers or stored energies we can work here put these terms in terms of powers or stored energies which can be identified even for other configurations or in the more general case and therefore, let us try to identify the significance of each of these terms in terms of powers or stored energies we can work here as far as the average power loss in the resistance is concerned that is simply half 1 by 2 r V V star expressing it in terms of voltage this is the average power loss in the resistance R. Similarly, one can talk of the average stored energies taking up the stored energies in the inductor and calling it W N what is the average energy stored in the magnetic field associated with the inductor we go by the expression half L I squared and extend that to cover the case of sinusoidal time variations. So, that average power becomes 1 by 4 L and then I L I L star which can be written as 1 by 4 L and I L is V times V star upon omega squared L squared which can be seen to be equal to 1 by 4 L squared. V V star upon omega squared L which becomes the average energy stored in the magnetic field associated with the inductor. Similarly, the capacitor will also have some stored energy in the electric field associated with it which can be written in a more simple manner for the parallel resistance circuit as 1 by 4 C times V times V star which becomes the average energy stored in the electric field associated with the capacitor C. Now in terms of these quantities the average power loss and the time average energies stored we can express the complex power fed to this resonance circuit and 1 can see that this is going to be P L and then plus twice j omega into W M minus W E that can be verified in a straight forward manner 1 can just substitute the expression for W M and W E here and see that we get back the previous expression from which if we now want to find out the input impedance we can just equate this to half Z in I I star. So that the expression for the input impedance becomes P L plus twice j omega into W M minus W E divided by half I I star. Now as it turns out quite conveniently that even if we started with a series RLC circuit the input impedance will appear in the same form in terms of of the power loss and the stored energies and therefore we can take this to be a general expression for the input impedance for the resonant circuits and now 1 can proceed further. We define resonance as the condition where the input impedance is completely real and therefore the resonance occurs when the input impedance be completely real when W M equals W E that is the energy stored in the magnetic field of the inductor and the energy stored in the electric field of the capacitor they should be equal at resonance which means the expressions were already written here that 1 by 4 V V star by omega squared L should be equal to 1 by 4 C times V V star at resonance let us say which occurs at omega equal to omega naught. From there we get the familiar expression for the resonant frequency and that is omega naught square is 1 by L C or omega naught is equal to 1 by square root of L C quite a familiar expression. The evolution that we have been able to make is that now we have been able to identify the resonant condition for any resonator configuration the resonant condition is that the energy stored in the magnetic field should equal the energy stored in the electric field which is more generally applicable than to just lumped element resonant circuits. What is the parameter in terms of which we specify a resonator or in terms of which we measure the quality or the performance of the resonator that is the quality factor or the selectivity quality factor denoted by Q how is it defined? It is defined as the average energy stored divided by the average power loss multiplied by the resonant frequency omega naught. Average power loss is the decrease in energy per second and therefore a factor omega naught appears here and we will like to calculate the quality factor for this demonstrative resonant circuit in terms of the parameters of the circuit. As far as the average power loss is concerned we had put down an expression for P L which is the only lossy element resistor is the only lossy element in this resonant circuit. How about the average energy stored? There we have argued that for resonance we have W M equal to W E. Since the time variation is sinusoidal what kind of peak energy do we expect in the magnetic field or in the electric field that will be twice the average value. So peak energy is twice W E or twice W M depending on which one we are looking at. The capacitor or the inductor which means that it is this energy twice W E or twice W M which is redistributing itself differently at different instance of time and otherwise the total energy is remaining constant at 2 W E or 2 W M. One can consider this in a little more detail by considering the variation of the stored energies as a function of time. Each of these let us say W E is proportional to say sin square omega t fine W E is related to V V star and therefore if V is taken to various sin omega t the stored in electric energy is going to various sin square omega t and one can plot this as a function of let us say twice omega t and it will plot as a positive function having this kind of variation with a maximum at twice omega t equal to pi 3 pi etcetera and going to 0 values at 2 pi and 0. The horizontal axis is twice omega t. This is the variation of W E and we know that the maximum value here is twice W E. In a similar manner if we were to plot the variation of the energy stored in the magnetic field associated with the inductor that will vary in a complementary manner and wherever the energy stored in the electric field is maximum the energy stored in the magnetic field will be minimum going to 0 value and one can plot this in the following manner. A very interesting pattern of intertwined sin square functions where the vertical axis can now be converted to W M and the maximum value is once again either twice W E or twice W M. Wherever the two are equal that is this level when drawn accurately we will have W M equal to W E and that will be the resonant condition. At resonance the energies are equal otherwise it is the energy twice W E or twice W M which is redistributing itself in different ways for different instance of time and therefore as far as the average energy stored is concerned that can be taken to be simply twice W E or twice W M. So the numerator here which can be written as W M plus W E where this is the average energy stored in the magnetic field this is the average energy stored in the electric field becomes either twice W M or equally well twice W E. So that for the quality factor we can write omega naught into twice W E by P L at which level this expression for the quality factor will be applicable in general. And now we use the expression specific to the circuit that we are taking up as an example and there we have this as omega naught into twice W E is half C V V star and P L we wrote as 1 by 2 R V V star which comes out as R omega naught C and since omega naught C is equal to 1 by omega naught L at resonance this is equally well R upon omega naught L is this going to be a high value or a low value for a low loss parallel resonance circuit the shunt resistance R that has been put will be a high resistance and therefore for a good resonance circuit good parallel resonance circuit this Q value will be quite high. If one had started with a series resonance circuit then whereas this expression would have remained the same the quality factor would have come out as omega naught L upon R where that R would be the series resistance indicating losses in the circuit and that would be small and therefore once again for a good series resonance circuit one will get a high value of the quality factor. So, these are some very general observations about the resonance circuits while we have an expression for the quality factor in terms of the parameters R L C etcetera in general for example for the other configuration that we mentioned these parameters are not available. So, how are we going to estimate the quality factor for these or even measure the quality factor for that purpose it is quite useful and interesting to see the behavior of this function the input impedance as a function of frequency that gives a very good indication of how one can measure the quality factor in practice. So, that is what we take up next we write the input impedance which is the reciprocal of the input admittance and input admittance is put down in a more straight forward manner for the parallel resonance circuits. So, that we have 1 by R plus 1 by j omega L plus j omega C the reciprocal of this. And now let us say that the actual frequency omega the radiant frequency omega is slightly shifted from the resonant frequency fine for a good quality resonance circuit we will get some output only close to the resonant frequencies. And therefore, we say that let omega go to omega naught plus delta omega. So, that we get here 1 by R plus 1 by j L into omega naught plus delta omega plus j times omega naught plus delta omega into C reciprocal of the whole thing which becomes 1 upon R in the above expression. Now, we make a small approximation binomial approximation since delta omega is taken to be a small deviation from the resonant frequency. And therefore, this is plus j omega naught C plus j delta omega plus j C reciprocal of the whole thing. Now, we are going to get a 1 upon j omega naught L term here and plus j omega naught C term is here. And by comparison one can see that 1 by j omega naught L plus j omega naught C is going to be 0 at resonance these two terms are going to nullify each other completely. So, using that we get this as 1 upon 1 upon R plus j delta omega C and minus delta omega upon j omega naught squared L which can be simplified to read as j omega naught squared R L upon j omega naught squared L plus it will be minus because of two j's coming in minus delta omega C and then we have j omega naught R and then omega naught squared L and finally, minus delta omega R. Once again omega naught squared times L C will be 1 and therefore, we can simplify this expression further as follows we get z in equal to j omega naught squared L plus j omega squared R L upon j omega naught squared L minus 2 delta omega. Now, one can divide through by j omega naught squared L leaving only R in the numerator. So, that it is R upon 1 minus 2 delta omega R minus 2 delta omega R minus 2 delta omega R minus 2 delta omega upon j omega naught squared L recognizing that R upon omega naught L is the quality factor for the parallel resonance circuit that we are considering this becomes R upon 1 plus 2 j q into delta omega by omega naught. This is the behavior in terms of this quality factor the slight shift in frequency from the resonant frequency and of course, the resonant frequency omega naught. One can obtain the magnitude of z in from here which will be R upon 1 plus 4 q squared into delta omega by omega naught whole squared and the square root of the entire denominator and also the phase angle that is angle z in which will be minus tan inverse of 2 q into delta omega by omega naught. So, that one can consider a plot of the magnitude and phase of the input impedance and the behavior comes out as follows. We plot this as a function of delta omega by omega naught when delta omega by omega naught when delta omega by omega naught is 0 then the input impedance magnitude is simply R which of course, is the resonant condition by definition and therefore, here we can consider the magnitude of z in and this value simply equal to R and as we shift on either side by a small amount. So, that this variable acquires small positive or negative values we will get the familiar bell shaped resonant curve going down to very small values once this deviation from the resonant frequency becomes significant. One can identify the three d b points here where the magnitude becomes R by root 2 and those three d b points the frequency difference between these three d b points may be called let us say 2 delta omega. For the magnitude to become root 2 times less than the maximum value what we require here from this expression is that 2 q delta omega by omega naught should become equal to 1. Then only for this value of delta omega only the magnitude will become R by root 2 and therefore, we get q equal to omega naught upon twice delta omega where the numerator is the resonant frequency and twice delta omega is the frequency separation between the three d b points when we consider the variation in the input impedance magnitude as a function of frequency. Both quantities are measurable with what is depends upon the actual value of the quality factor. If the quality factor is very high then this frequency separation will be very small and then it is going to require very stable source of frequency and so on. But otherwise in practice one can say in general that this is measurable and this becomes another working definition for the quality factor based on which one can perform the measurement and measure the quality factor for any given resonant circuit. For the various other shapes that we showed other than the lumped element configuration this definition can be used for the measurement of the quality factor. Coming to the phase of the input impedance you see that for positive values of delta omega the phase is going to be negative and vice versa and as delta omega by omega naught assumes large values the phase angle is going to asymptotically become minus 90 degrees or plus 90 degrees depending on the deviation and therefore, this can be plotted in this manner this is the phase of the input impedance. This goes to minus 90 degrees asymptotically and this goes to plus 90 degrees asymptotically and therefore, if one were to measure the nature of the input impedance then above resonant frequency it will become capacitive and below resonant frequency it will become inductive. However, one note of caution should be made here this is the behavior capacitive and inductive for the parallel resonant circuit. If one looks at the series resonant circuit behavior carefully that behavior will be just the opposite of this. However, the point that we want to make here is that around the resonance the nature of the input impedance changes. This is where we would like to stop if you have any questions we can try out these. So, in the lecture today we have introduced several types of resonators which can operate from very low frequencies to very high frequencies and we have considered the general properties of these resonators by taking the example of a parallel resonant circuit. Thank you.