WEBVTT 00:19.550 --> 00:29.490 So, welcome to the 14th lecture of this series. Now here we will see another basic building 00:29.490 --> 00:42.080 block that, you saw in your electronics throughout electronics engineering course. And LC circuit 00:42.080 --> 00:49.250 is fundamental. Because it is the only circuit which can produce from noise it can produce 00:49.250 --> 01:01.969 ac signals of any frequency. And then if you want to resonate the circuit at a particular 01:01.969 --> 01:09.350 frequency you can tune it. So, by changing the c value you can tune it as already the 01:09.350 --> 01:14.970 receiver does. You can amplify; obviously, it is a resonance circuit. So, at resonance 01:14.970 --> 01:21.410 signal is very high power is very high. So, you can use it. Suppose you are produce some 01:21.410 --> 01:29.330 small signal that you want to sustain, that you want to give some good amount of value 01:29.330 --> 01:34.570 that you can use it as this. So, this resonated circuit how we do it in micro wave that we 01:34.570 --> 01:36.010 will see today. 01:36.010 --> 01:47.410 So, microwave resonator is a tunable circuits, used in oscillators, oscillators need because 01:47.410 --> 01:54.770 if you have some signal. Now you need to sustain that that is called oscillator. Then amplifiers 01:54.770 --> 02:02.209 you need to make these LC circuit tuned amplifiers you know. They are very sharp is there cut 02:02.209 --> 02:11.300 off. So, that is why their q is very high. Then in wave meters which measures the frequency 02:11.300 --> 02:18.080 of the wave that all of you have seen in your microwave benches there is a generally you 02:18.080 --> 02:27.420 cylindrical wave meter. So, if you want to find out frequency what you do, you put a 02:27.420 --> 02:35.790 cavity or a basically a resonator circuit and go unchanging it is tuning frequency. 02:35.790 --> 02:42.071 Now, if there is any applied signal at some frequency, when the frequency of the signal 02:42.071 --> 02:47.380 and the frequency of the resonating frequency of the cavity is equal suddenly, the whole 02:47.380 --> 02:52.840 circuit will show that it, it has picked up the signals because it will amplify the signal. 02:52.840 --> 03:03.080 So, that is the principle of wave meter, then any filter you want to tune, I want to amplify 03:03.080 --> 03:09.690 certain signal you want to reject certain signal. So, you can use these LC devices to 03:09.690 --> 03:15.420 amplify or reject. That all of we know that whether a various types low pass high pass 03:15.420 --> 03:19.500 and all those types. So, they require all these LC things. 03:19.500 --> 03:26.030 At the tune frequency we know. That impedance is real the reactants part goes away. So, 03:26.030 --> 03:33.860 impedance which is actually is some of resistance plus reactants. That reactants part of the 03:33.860 --> 03:39.461 impedance goes away at the tune frequency. That is the definition of resonance. And that 03:39.461 --> 03:45.480 time average energy stored in electric field becomes equal to average energy stored in 03:45.480 --> 03:53.880 magnetic fields and. So, total energy is twice of electric energy stored in a resonator. 03:53.880 --> 04:00.460 Total energy is maximum at resonant frequency. That is why you get very high sound when you 04:00.460 --> 04:11.550 tune a, suppose you going in a car you are finding the FM radio. You go on distribute 04:11.550 --> 04:17.040 moving the knob. Nowadays you put a push button, but inside there is actually is changing the 04:17.040 --> 04:22.210 c value by changing the knob. Actually I in earlier days you used to change the c value 04:22.210 --> 04:29.860 suddenly, you get a very high song or very high lecture. So, you know that you are tuning 04:29.860 --> 04:34.689 is perfect. Now you have getting that signal because energy is maximum. 04:34.689 --> 04:37.680 . Now, what are the parameters of a resonance 04:37.680 --> 04:43.529 circuit of the resonator; obviously, the resonant frequency fr at which energy of resonator 04:43.529 --> 04:51.789 becomes maximum. Then also there is Q. Q of a resonator is what. Q of a resonator is the 04:51.789 --> 05:00.210 how much energy it maximum you stored, by energy dissipated per cycle. 2 pi is the total 05:00.210 --> 05:11.069 length of which it is going and. So, Q means that I will normally require that in a resonator 05:11.069 --> 05:19.819 energy stored will be high and energy dissipated will be very low. So, Q we demand very high. 05:19.819 --> 05:26.830 And it can be prove that all resonators they have a curve which is very sharp. 05:26.830 --> 05:30.139 . So; that means, their frequency response is 05:30.139 --> 05:37.300 something like this. This is frequency and this is the amplitude of the signal. So, they 05:37.300 --> 05:46.409 are we know these at the maximum amplitude comes at the f r frequency and these. So, 05:46.409 --> 05:53.580 this sharpness is also defined by from this if I fall down to the amplitude falls down 05:53.580 --> 06:04.180 to 1 by root 2 and of this peak. So, if this peak is 1. I am falling to 1 by 06:04.180 --> 06:12.219 root 2 both side now this difference is called the 3 dB band width or delta f. So, definition 06:12.219 --> 06:21.960 of Q is f r by delta f. So, if I have a ready high sharply tuned value; that means, my delta 06:21.960 --> 06:29.400 f is very small, 3 dB band width. So, that is another factor that what is the Q of this 06:29.400 --> 06:35.309 circuit; that means, how good is this resonator circuit it should have quite a good amount 06:35.309 --> 06:41.229 of Q. Also what is the input impedance of the resonator? 06:41.229 --> 06:47.960 Because you see that we are saying that it is impedance becomes real now of a pure LC 06:47.960 --> 06:55.270 circuit there is the impedance is 0. Because at resonance the, but you cannot have a pure 06:55.270 --> 07:00.409 LC circuit. So, there should be some resistant now that resistant is important because when 07:00.409 --> 07:08.900 resonance takes place that time that resistance is to the outside or to the connected circuit 07:08.900 --> 07:16.879 that impedance posed is this resistance value of the whole network. So, for impedance matching 07:16.879 --> 07:23.599 your other parts they should also have that same resistance then only they can match their. 07:23.599 --> 07:30.149 So, the input impedance of the resonator that indicates matching performance what matching 07:30.149 --> 07:32.080 is to be done that depends on that. 07:32.080 --> 07:39.869 So, if you specify these 3 parameters of resonator. Then it is you know for engineering purposes 07:39.869 --> 07:51.379 all the information is known. Now in micro wave region we or lower the microwave region, 07:51.379 --> 07:57.270 we have coaxial resonators it is frequency range is hundred mega hertz to up to 1 gigahertz 07:57.270 --> 08:02.899 as I said. That up to 1 gigahertz you will see is the normal low frequency electronic 08:02.899 --> 08:09.539 circuits, but generally after 100 mega hertz you cannot have the lumped elements you have 08:09.539 --> 08:14.770 the distributed element that is why coaxial line resonators; that means, transmission 08:14.770 --> 08:21.120 line concept and above 1 gigahertz microwave concept wave guide concept that will come. 08:21.120 --> 08:33.250 Now, typical Q of those 1000, you know I think in the your in low frequency engineering electronics 08:33.250 --> 08:44.340 classes, the making Q of an amplifier 10 is easy, but after making Q 10 it is quite tough 08:44.340 --> 08:52.709 to make a amplifier with Q more than 10. Those requires colpitts scillator Hercules oscillator 08:52.709 --> 08:59.440 that type of concept, but here you see the demands is your Q should be quite high. Because 08:59.440 --> 09:06.270 then only your because microwave or high frequency this mega hertz signals they are not. So, 09:06.270 --> 09:10.089 abundantly available like low frequencies signals. So, their amplitude is poor. So, 09:10.089 --> 09:19.330 you need to push up that Q and in that in coaxial resonators we have also seen that 09:19.330 --> 09:27.290 in coaxial line higher order modes also come, but we need to prevent them. So, that prevention 09:27.290 --> 09:35.449 criteria I am saying that b and a, are the inner radius diameters and diameter radius 09:35.449 --> 09:42.360 of the inner conductor and outer conductor. So, these should be made pi into b plus a 09:42.360 --> 09:49.101 that should be miss much less than lambda, so that the TE/TM modes cannot come, because 09:49.101 --> 09:55.820 TE/TM modes has the cut off. So, you can prevent them provided you stick to this answer. They 09:55.820 --> 10:00.269 are stick to this equation this inequality. 10:00.269 --> 10:05.540 . Now, how much you know that one benefit actually 10:05.540 --> 10:12.980 a transmission line is already an LC circuit. Because it is equivalent circuit is you see 10:12.980 --> 10:20.290 it is a series resistant in shunt with a per unit length capacitance. So, LC is there. 10:20.290 --> 10:30.370 Only you need to find out for your case that what is the value of the length of the thing. 10:30.370 --> 10:39.529 So, that length of the resonant length of a coaxial cavity circuit you need to do like 10:39.529 --> 10:46.220 this. So, think of that you have a transmission line that is a reference plane on a line. 10:46.220 --> 10:54.410 Now, this side if you look you are looking at a susceptance of B1, loaded susceptance 10:54.410 --> 11:01.290 loaded, we are saying because this line let us say way is terminated by a load of Z1, 11:01.290 --> 11:07.769 real load similarly here you have a load of Z2, from this reference plane if you look 11:07.769 --> 11:12.519 at the susceptance you are looking at a loaded susceptance. 11:12.519 --> 11:20.620 Now, at resonance we know that total susceptance of the thing will go to 0. So, B1 plus B2 11:20.620 --> 11:34.149 should go to 0. So, now, to design the coaxial resonator, what you do take this reference 11:34.149 --> 11:42.580 plane to the right end. Extreme right of the line then enforces this condition you will 11:42.580 --> 11:50.050 get a resonators length. So, 3 types of configurations are possible. 11:50.050 --> 11:54.339 . And popular the first one is basically gives 11:54.339 --> 12:03.360 you a LC circuit. Where you see this is the transmission line it is left end is shorted; 12:03.360 --> 12:12.600 that means, with reference to the previous one basically this z1 is equal to 0. And z2 12:12.600 --> 12:22.720 is taken as infinity that is the first configuration. So, this is short this side is open. So, you 12:22.720 --> 12:30.210 see this is the line. So, this side is open circuit this side is short circuit the inner 12:30.210 --> 12:38.139 conductor is radius a, outer conductor radius b. So, let us enforce that condition, what 12:38.139 --> 12:56.060 will be a shorted transmission line. So, what will be the shorted transmission line, I think 12:56.060 --> 13:03.129 just let me show you once. This you should be able to do that I have a transmission line 13:03.129 --> 13:06.480 this is shorted or it is. 13:06.480 --> 13:09.060 . First let me write what is the input impedance 13:09.060 --> 13:21.300 of a transmission line is z naught, z l, plus j tan. Loss less line, j tan beta, l z naught, 13:21.300 --> 13:38.910 divided by z naught plus j z l tan, beta l. Now what is if I put this z l is equal to 13:38.910 --> 13:51.350 0. That is the shorted line. This Z in will be how much. z naught then 0 plus, j z naught 13:51.350 --> 14:08.370 tan beta l by z naught. So, that will be j z naught tan beta l. Let show every tan and 14:08.370 --> 14:16.819 for the right side you think that there is a, this is the reference plane right side 14:16.819 --> 14:32.529 is open; that means, z l is infinity and. So, what will be the Z in? Z in is z naught, 14:32.529 --> 14:50.000 infinity plus j, z naught, tan beta l, by z naught plus j infinity tan beta l ok. 14:50.000 --> 15:16.670 So, you can divide throughout by or generally you can write. So, this is z naught, 1 plus 15:16.670 --> 15:34.550 that division will make it 0. By 0 plus j tan beta l. That is how much minus j z naught. 15:34.550 --> 15:48.920 Cot beta l; and in this case length is right side, what is the length, length is equal 15:48.920 --> 15:59.529 to 0. So, that will become minus j z naught cot of 0. What is cot 0? Cot 0 is infinity. 15:59.529 --> 16:11.389 So, minus j z naught into infinity so; that means, b what is that B1 or 2, B2. So, I can 16:11.389 --> 16:20.810 say B2 is infinity. That is why it is written that is infinity. You solve these. So, tan 16:20.810 --> 16:29.610 beta l is infinity. So, the condition is l is equal to lambda by 4, 2 n minus 1. This 16:29.610 --> 16:35.689 is parallel resonant cavity to understand. How to do it that both side you assume transmission 16:35.689 --> 16:42.129 line that is why initially do not assume it be at right side. You write down depending 16:42.129 --> 16:49.389 on the terminations what is the B1 you are seeing susceptance value. B2 you are seeing 16:49.389 --> 17:02.569 from this side of the line then you make the appropriate 3. So, this is called quarter 17:02.569 --> 17:10.470 wave cavity, because it is lambda by 4 into some odd number. So, this is gives you a parallel 17:10.470 --> 17:11.470 resonance circuit. 17:11.470 --> 17:12.900 . So, if you want to create a parallel resonance 17:12.900 --> 17:18.650 circuit. Take a quarter wave cavity; then configuration 2 half wave coaxial cavity. 17:18.650 --> 17:27.270 Here what you do you this side short this side also short. So, if as a why this previous 17:27.270 --> 17:43.750 one was we called LC. Why because if you see the graphs this v and i graphs here; obviously, 17:43.750 --> 17:50.820 this is open means b will be high this is low now these graph comes about that of LC 17:50.820 --> 18:00.880 circuit similar parallel LC circuit similarly series LC circuit, you get the b that both 18:00.880 --> 18:07.410 sides shorts. So, this this is wrongly said e this will be b correct it as this, instead 18:07.410 --> 18:10.840 of H, this will be b, this is e b this is H I. 18:10.840 --> 18:17.550 So, voltage will be like this. Current will be like this; obviously, this is a LC circuit 18:17.550 --> 18:25.510 graph and you can write this. So, this is series resonance circuit; if you want to make 18:25.510 --> 18:29.240 you make it a lambda by 4 cavity. 18:29.240 --> 18:32.970 . Now, there is another that you want to make, 18:32.970 --> 18:40.820 some other value. So, what you do that here you give a gap. So, this transmission line 18:40.820 --> 18:47.820 here you put a short, here also you put a short, but here you make a gap in the line, 18:47.820 --> 18:53.650 that will you give a capacity of gap that is, why you see that this side will be like 18:53.650 --> 19:03.700 this, but this side will be some capacitance omega. So, from that you find out, what is 19:03.700 --> 19:10.810 the length of the transmission line these d. So, so you give a d here z 0, is C is the 19:10.810 --> 19:17.440 gap capacitance between the central conductor, and the shorting terminal. So, depending on 19:17.440 --> 19:25.980 what dielectric, these there you can find out this C value, and then you can design 19:25.980 --> 19:28.090 this capacity when derivative. 19:28.090 --> 19:31.300 . Now, performance if you compare. Now conductor 19:31.300 --> 19:40.410 loss is we have already seen that when we saw the coaxial connectors, that conductor 19:40.410 --> 19:50.820 loss in a coaxial line is minimum when b by a is takes the value 3.6. So, Q is nothing, 19:50.820 --> 19:58.400 but Q's denominator is that loss, or energy dissipated so; obviously, highest Q will get 19:58.400 --> 20:07.150 in a coaxial line, when b by a is 3.6. So, we can choose the value of b by a to be 20:07.150 --> 20:12.110 3.6. That will be your highest Q. So, from any particular design you can find out how 20:12.110 --> 20:18.130 much Q you can get. Now for quarter wave cavity one problem is there that since it is one 20:18.130 --> 20:23.970 end is open. So, energy may leak out from there. So, microwave radiation may takes place 20:23.970 --> 20:34.280 from open end. So, that loss is becomes more. So, that is why Q suffers, but that is why 20:34.280 --> 20:39.340 half wave cavity is generally preferred over quarter wave cavity because it is both ends 20:39.340 --> 20:45.060 are shorted. So, the wave is confined there the leakage of microwave radiation is not 20:45.060 --> 20:48.110 there. Now, after 1 gigahertz of problem with this 20:48.110 --> 20:54.980 coaxial cavities is that, skin effect etcetera becomes more and, that is why we have seen 20:54.980 --> 21:05.790 yesterday, that conductor loss will be more and radiation loss will be more. So, you see 21:05.790 --> 21:11.550 what we have written that skin effect will increase the conduction loss, also radiation 21:11.550 --> 21:17.670 loss will increase both of these effect will decrease the Q of coaxial cavity. So, that 21:17.670 --> 21:21.860 is why at higher frequencies people do not use Q coaxial cavity. 21:21.860 --> 21:25.370 . So, microwave resonators are made of waveguide 21:25.370 --> 21:32.380 resonator, in waveguide we can easily form the cavity. You see you have a wave guide 21:32.380 --> 21:42.820 just this is a wave guide, now this side is open 21:42.820 --> 21:50.930 another side is open it is a transmission line, but if you put on these 2 sides, 2 electrical 21:50.930 --> 21:57.870 walls that becomes a cavity. So, already in a wave guide 4 metallic walls are there the 21:57.870 --> 22:01.260 top wall bottom wall side wall left side and right side. 22:01.260 --> 22:06.101 . Now, also you make these 2 fronts and what 22:06.101 --> 22:13.960 is that front and rear these 2 walls if you make you make a cavity. So, in a this is called 22:13.960 --> 22:20.080 rectangular cavity similarly in a circular wave guide circular wave guide this total 22:20.080 --> 22:29.150 side wall is metallic single conductor now top and bottom you make 2 metal plates that 22:29.150 --> 22:30.820 will make a circular cavity. 22:30.820 --> 22:36.130 . So, both of these are the RLC circuit. So, 22:36.130 --> 22:47.980 you see this is the cavity. In micro wave laboratory all of you used this. Now this 22:47.980 --> 22:54.460 is called frequency meter the on these there are frequency marks are there. What happens 22:54.460 --> 23:01.330 when you move this cavity we move this cavity now basically the length of the cavity; that 23:01.330 --> 23:10.640 means, top and bottom. So, this cavity it is top that is getting changed. So, the length 23:10.640 --> 23:15.220 of the cavity is getting changed the resonance frequency depends on these d that you will 23:15.220 --> 23:24.590 see and. So, the frequency is getting changed. Now, the signal is coming this device probably 23:24.590 --> 23:32.660 clipstone. This clipstone this is creating this some signal. Now that signal is generally 23:32.660 --> 23:38.730 cavity is not passing. So, here the waves w r is meter is showing that not much deflection, 23:38.730 --> 23:46.090 but when the resonance frequency of the cavity matches with the signal it takes power and 23:46.090 --> 23:53.910 that time that power. So, here there is a deflection and you understand that. So, from 23:53.910 --> 23:59.490 the graduation on the frequency meter you find out. So, now, the clipstone is producing, 23:59.490 --> 24:04.120 so and so signal; so and so frequency signal. 24:04.120 --> 24:07.740 . Now, this is the rectangular cavity dominant 24:07.740 --> 24:14.210 mode. You see similar to TE 10 mode, but here since it is a 3 dimensional structure. Is 24:14.210 --> 24:21.400 not a 2 dimensional structure, because wave guide means we have that inside in z direction 24:21.400 --> 24:25.750 it has infinite direction, but the moment you have the z direction instead of z direction 24:25.750 --> 24:32.630 you have 2 separate 2 metal plates separated by d. 2 metal plates separated by d you see 24:32.630 --> 24:39.190 field structure is almost same transverse plane a b, a is like this h is like this. 24:39.190 --> 24:48.420 And here you see that this there is a full the magnetic field is like this. So, this 24:48.420 --> 24:56.580 is TE 1 0 1 mode. 1 0 is the last part is another 1 that is a dominant rectangular field 24:56.580 --> 24:57.900 cavity configuration. 24:57.900 --> 25:01.780 . Now, circular cavity this is the field configuration. 25:01.780 --> 25:09.830 You see circular thing dominant mode was T11. So, this was the mode similar mode is T 1 25:09.830 --> 25:15.950 1 1, but; obviously, these 2 are also preferred as we said that they have circular symmetric 25:15.950 --> 25:17.980 field structures. 25:17.980 --> 25:23.100 . So, this you see that they are better field 25:23.100 --> 25:29.400 structure. That why a circular wave cavity also that non dominant mode also some times, 25:29.400 --> 25:41.860 used ow rectangular cavity the resonance frequency of TE or TM mode that is given by this, this 25:41.860 --> 25:50.000 is c means this, is on the sorry this should not be equal capital c, this is small c which 25:50.000 --> 25:55.110 is the velocity of light, 3 into 10 to the power 8 meter per second. These are and they 25:55.110 --> 26:00.210 again depend on this. So, you here you put that 1 0 1. 26:00.210 --> 26:10.100 . So, basically depends on a, sorry, a and d 26:10.100 --> 26:22.340 because 1 0 dominant mode so n, so b will not matter a and d they matter a by pi and 26:22.340 --> 26:38.740 d by pi. So, that will matter the frequency and. So, 26:38.740 --> 26:44.950 you need to choose the generally rectangular cavity, they have dimensions a and d. So, 26:44.950 --> 26:52.090 you need to choose to give it what frequency you need to do generally the d; that means, 26:52.090 --> 26:59.400 if you vary the d, but distance between the 2 walls you change the rectangular cavity 26:59.400 --> 27:07.210 also you can calculate what is the Q of the cavity, for conductor loss by doing the similar 27:07.210 --> 27:16.160 exercise, as we have done for coaxial line. You can find out this is the Q for conductor 27:16.160 --> 27:23.220 loss. This is the Q for dielectric loss this is simple as 1 by tan delta. So, find out 27:23.220 --> 27:29.870 dielectric find out it is specification. So, you know tan delta tan delta is generally 27:29.870 --> 27:38.660 specified. So, q d finding is not a. So, if you have dielectric let us say that generally 27:38.660 --> 27:45.660 it is tan delta is point 0 0 1 so; that means, you will get thousand Q with that. 27:45.660 --> 27:53.250 Now, you see you find out depending on frequency it matters because of this k presence. So, 27:53.250 --> 27:58.560 as you go higher and higher Q c starts becoming significant, but generally in the microwave 27:58.560 --> 28:04.850 region 10 to l gigahertz it is not that much, that time Q d is the main deciding, but if 28:04.850 --> 28:10.820 Q c is there you the when both conductor loss and dielectric loss are present this is the 28:10.820 --> 28:15.710 formula. So, if Q c is significant you take. Otherwise 28:15.710 --> 28:21.230 knowing the dielectric and delta you can find what will be the Q of the rectangular cavity. 28:21.230 --> 28:28.610 So, you can make by choosing proper dielectric, you can make or here if you can find out what 28:28.610 --> 28:32.120 will be the thing here. 28:32.120 --> 28:35.440 . So, similarly circular wave guide is dominant 28:35.440 --> 28:44.920 mode is instead of TE 11, in circular wave guide TE 111 used in frequency meters as I 28:44.920 --> 28:53.830 said often, TE 011, mode is used as it is Q is higher than TE 111 frequency resolution 28:53.830 --> 29:00.060 of the frequency meter depends on Q cavity is loosely coupled to the guide, it is top 29:00.060 --> 29:06.540 wall is tunable generally as I said that top wall you vary that varies the d, that will 29:06.540 --> 29:13.370 vary the frequency of the thing. Resonance frequency is given by this formula this all 29:13.370 --> 29:15.130 you know that all the things. 29:15.130 --> 29:21.150 So, you see basically resonance frequency is depending on d. These are constant a is 29:21.150 --> 29:30.830 the cross section. So, for a given for broad length this is 1 this is circular a is the 29:30.830 --> 29:40.590 radius, and this is the oh d. So, l value will be your mode number. So, generally for 29:40.590 --> 29:43.170 dominant mode it is l is 1. 29:43.170 --> 29:55.250 So, d is that the determining factor. This TE and TM the change is on in that p n m dashed 29:55.250 --> 30:05.660 or p n m, Q is there. Equations are given Q d is again that one by tan delta these formulas 30:05.660 --> 30:14.790 are just similar that is for, so that by this way of land the resonator circuit, how to 30:14.790 --> 30:22.350 make resonators LC various RLC devices, in the microwave region. Next, in the next lecture 30:22.350 --> 30:28.390 we will see another important thing, that how to make attenuators in microwave region. 30:28.390 --> 30:29.520 Thank you.