WEBVTT 00:20.680 --> 00:29.420 Now, in this seventh lecture, we will see the another high frequency transmission structure; 00:29.420 --> 00:37.430 that is wave guide as I already said that, wave guide is, it propagates non TEM mode 00:37.430 --> 00:45.390 that is both TEM and TM modes propagates through wave guides. Now wave guides means it has 00:45.390 --> 00:53.600 a, its cross section is uniform. Now that cross section, actually it is a cylindrical 00:53.600 --> 00:59.480 structure. The cross section may be rectangular, may be circular, and may be elliptic, depending 00:59.480 --> 01:04.309 on that we say whether it is a rectangular wave guide, whether it is an elliptic wave 01:04.309 --> 01:10.670 guide, whether it is circular wave guide etcetera. So, it is a very popular one rectangular wave 01:10.670 --> 01:18.970 guide like a hollow rectangular pi. So, that we will see support some interesting modes, 01:18.970 --> 01:25.710 and that is why it is popular, that its field structure etcetera is very convenient to extract 01:25.710 --> 01:33.450 that type of mode in rectangular wave guide is very popular. So, almost in all the applications 01:33.450 --> 01:39.360 of microwave engineering in high frequency, particular in the Giga hertz range, rectangular 01:39.360 --> 01:46.550 wave guide is used, but will see that sometimes we need a circular cemetery that time circular 01:46.550 --> 01:49.630 wave guide is also used, that will see in the next lecture. 01:49.630 --> 01:54.290 So, rectangular wave guide it is a single conductor transmission line as the wave guide 01:54.290 --> 02:02.259 says. So, your see this, this is a closed metallic structure, conductor boundaries parallel 02:02.259 --> 02:08.149 to z axis. This is our custom you can take it to anyone, but generally we say that wave 02:08.149 --> 02:13.740 will propagate in the z axis direction. So, conductor boundaries they are parallel to 02:13.740 --> 02:20.920 z axis. Now arbitrary cross section, but generally we work with by it can be theoretically any 02:20.920 --> 02:28.930 arbitrary cross section, but generally work with as I said rectangular structure or cross 02:28.930 --> 02:35.850 section or circular cross section etcetera. The structure uniformed z direction, in z 02:35.850 --> 02:40.860 direction I do not have any variation, and they are infinitely long in z direction. infinitely 02:40.860 --> 02:47.790 long means that it is far away from the source, far away from the load, so that those discontinuity 02:47.790 --> 02:55.380 etcetera not there, so a pure mode is propagating through this, and this metals are perfectly 02:55.380 --> 03:01.400 conducting, you know that non real metal is perfectly conducting; that is why we will 03:01.400 --> 03:09.170 see that they will any real metal that, since it is not ideal metal, ideal conductor. So, 03:09.170 --> 03:11.500 there will be some loss in the conductor. 03:11.500 --> 03:20.209 Now, this is a, generally this is a in a standard, wave guide rectangular guide standard is called 03:20.209 --> 03:29.180 WR 90, actually this WR are various things that 36 or WR 70 etcetera. Now this WR 90 03:29.180 --> 03:36.980 is actually used in radar band, x band, x band is a 1 of the radar brand, generally 03:36.980 --> 03:44.440 military radars operate. And also in laboratories etcetera we used this WR 90, because this 03:44.440 --> 03:51.209 x band 8 to 12 Giga hertz, this dimensions of the wave guide there depending on the frequency 03:51.209 --> 03:59.170 they are chosen, and this is quite convenient dimension that is why WR 90 is the standard, 03:59.170 --> 04:05.010 and WR 90 wave guide means an x band wave guide. This is a picture you see within this, 04:05.010 --> 04:11.010 this halo metallic pipe is there, this is a larger one, and actually the rectangular 04:11.010 --> 04:17.690 cross section is this. This is 0.9 inch that is 2.286 centimeter 04:17.690 --> 04:27.090 in length, and 0.4 inch that is 1.016 centimeter in width, but this is called flange, this 04:27.090 --> 04:35.990 is much larger than that, this is also rectangle. So, here two wave guides or two components, 04:35.990 --> 04:41.259 you can connect them together, so there are holes here. So, by this four holes at the 04:41.259 --> 04:48.780 four corner two structures get , so that this mouth of this wave guide, and mouth of the 04:48.780 --> 04:54.020 wave guide which is connecting with it, they are flushed together, so that there is no 04:54.020 --> 05:01.849 discontinuity, no air gap there. This flange helps to make connection with two wave guides, 05:01.849 --> 05:10.350 so that there is no discontinuity created between the mouths of two wave guides. Now 05:10.350 --> 05:18.419 for our analysis this will be the structure you see the generally the length of the rectangle 05:18.419 --> 05:20.690 or that is called a. 05:20.690 --> 05:26.840 Now, we are taking one of the corners, the left hand corner as the origin, so it is a. 05:26.840 --> 05:33.580 You see in some books, or sometimes we takes center as the origin, that time this side 05:33.580 --> 05:42.259 is a by 2, this side is minus a by 2. Similarly plus b 1 plus b by 2 minus b 2, but we have 05:42.259 --> 05:49.449 taking this geometric this convention that this is our 0. So, this is our b; that is 05:49.449 --> 05:56.349 the width of the rectangle or lower side of the rectangle; that is b, and higher side 05:56.349 --> 06:04.910 is a. So, we say that this wall is the broad wall; this wall is the narrow wall. 06:04.910 --> 06:13.629 So, broad wall is of length a, and narrow wall is of length b. So, they are four walls 06:13.629 --> 06:20.669 one is the top wall this is metallic, the bottom wall is metallic, side wall is metallic, 06:20.669 --> 06:29.461 this side wall is metallic, inside there is hollow thing dielectric it may be air, or 06:29.461 --> 06:34.160 may be filled with any completely filled with any dielectric. So, that is why we are saying 06:34.160 --> 06:40.930 that inside thing, is a region through which the wave will go that is the dielectric region, 06:40.930 --> 06:48.740 if it is not dielectric it is free space. Sorry another thing is that, our propagation 06:48.740 --> 06:56.770 direction is z. So, this broad wall that is in the x direction, and the narrow wall that 06:56.770 --> 07:02.389 is in the y direction direct now. 07:02.389 --> 07:10.030 So, we have already seen when we have seen TEM mode, that if you have seen hollow pipe 07:10.030 --> 07:15.050 is a single conductor, wave guide is a single conductor in a TEM single conductor, you cannot 07:15.050 --> 07:22.060 TEM mode, because you do not have any voltage between the two waves, because all are same 07:22.060 --> 07:26.999 structure. So, there potential difference is 0; that is why you can have any transverse 07:26.999 --> 07:34.229 electric and transverse magnetic that we have already seen, but both TE and TM possible, 07:34.229 --> 07:40.990 whatever I said. Now let us start that for TE modes, converse electric, we known that 07:40.990 --> 07:47.789 the longitudinal component of the electric field will be 0. So, if you do that then; 07:47.789 --> 07:53.400 that means, we will have to express all the four transverse fields, if they are present 07:53.400 --> 08:02.900 in terms of the longitudinal magnetic field, because Ez is not 0. So, in TE modes; obviously, 08:02.900 --> 08:11.710 Ez will present, so the transfer component of Ez; that is small hz xy. So, will write 08:11.710 --> 08:17.960 the Helmholtz equation in terms of that, and then will apply separation of variables, this 08:17.960 --> 08:20.029 we have already seen. 08:20.029 --> 08:26.539 So, you know all this mathematics. So, again you are getting two constant, this side also 08:26.539 --> 08:33.250 this function of a f, this is the function of g, but it is said that these two functions, 08:33.250 --> 08:41.130 they are equal to, their sum is equal to a constant, because my kc square is constant. 08:41.130 --> 08:48.760 So, that is possible only if separately they are equal to two constant so that is b. Now 08:48.760 --> 08:57.060 it is solution, you know all these two solutions, again you can see that these d 2 f dx 2; that 08:57.060 --> 09:03.080 means, double differential of the function f with respect to x that will be if you take 09:03.080 --> 09:06.870 this side here. So, that will be constant into the function; that means, again some 09:06.870 --> 09:17.440 cos sin type of thing. So, this is in x, so that is why cosine of and sin of x. Similarly 09:17.440 --> 09:27.040 for g you have y so cosine in y variable sine in y variable. So, these are general form 09:27.040 --> 09:28.370 that you know. 09:28.370 --> 09:38.290 Now, you find out the transverse components once you know. So, you see this is the transverse 09:38.290 --> 09:46.140 component, sorry longitudinal magnetic field component hz, we have solved in this; A B 09:46.140 --> 09:53.770 C D are co constants yet to be determined. So, now, we have already developed remember 09:53.770 --> 10:01.029 equation 15, where we have express all the transverse field in terms of longitudinal 10:01.029 --> 10:08.130 field, in terms of hz, because Ez is 0 in this case. So, in terms of hz we can write, 10:08.130 --> 10:15.380 what is the ex, it will be also involve in this A B C D, and some form of cos and sine, 10:15.380 --> 10:22.160 because that depends on the this differentiation etcetera. Similarly we can write ey in terms 10:22.160 --> 10:27.120 of a b c d and this cos and sine function etcetera. 10:27.120 --> 10:34.339 Now we can also do it for other two things. Now let us put the boundary condition. What 10:34.339 --> 10:43.040 is the boundary condition, that you see in the top wall and bottom wall it is a conductor. 10:43.040 --> 10:51.190 So, in the conductor there are; that means, this plain is which plain, it is y z plane. 10:51.190 --> 10:55.940 So, electric conjuctival electric field on this bottom wall and top wall should go to 10:55.940 --> 11:01.560 0. Now conjuctival electric fill means, on the top wall let us say conjuctival field 11:01.560 --> 11:10.709 means, either it will be an ex field or a ez field, but in a TE mode field is not there. 11:10.709 --> 11:21.360 So, boundary condition is ex at this one is 0 for y is equal to 0; that is the top wall. 11:21.360 --> 11:28.611 Sorry y is equal to 0 is the bottom wall, and y is equal to b is the top wall. Similarly 11:28.611 --> 11:35.560 we can say that what will happen to the side wall; left side and right side. They are also 11:35.560 --> 11:40.890 metallic conductors, so there also tangential thing will be 0. What is tangential? They 11:40.890 --> 11:46.870 are; obviously, one z component, but we do not have any z component. What is the other 11:46.870 --> 11:54.100 component? y component so ey should go to 0 there, where at left side wall and right 11:54.100 --> 12:01.080 side wall. So, put this boundary condition if we now put into the, those solution that 12:01.080 --> 12:06.180 we got; that means, ex ey hx hy whatever we got. 12:06.180 --> 12:15.100 So, in that for ex ey if we put this, then we have put this and force that to that 0. 12:15.100 --> 12:24.709 So, ex x 0 is equal to 0, that implies that this is, this one, it total thing is 0 means 12:24.709 --> 12:34.360 that you cannot have any other thing except saying that, so d is equal to 0. So, by that 12:34.360 --> 12:45.459 you can find out what is ex, and also for b you can put that, it will take this value 12:45.459 --> 12:52.800 ky. So, similarly if put the other boundary condition, that gives you b is equal to 0 12:52.800 --> 13:00.480 and kx takes some discreet values, those are given by this like ky. 13:00.480 --> 13:13.510 So, we can write here all the five fields. Generally we have six fields; two transverse 13:13.510 --> 13:20.661 for it electric field, two transverse magnetic field, one longitudinal electric field, one 13:20.661 --> 13:27.519 longitudinal magnetic field, but in case of key field, one longitudinal e field that is 13:27.519 --> 13:36.850 ez is 0 so we are writing hz ex ey hx hy. So, all this constitute the solution of the 13:36.850 --> 13:44.250 field. So, that is field analysis that you have now come out from that basic wave equation, 13:44.250 --> 13:50.380 you can find out this. So, that is why we have developed those modes and t modes characterization, 13:50.380 --> 13:55.829 because now we can see it becomes very easy to find out the field. 13:55.829 --> 14:01.230 Now, also that time we have developed what is beta. Beta is this. Now we have seen that 14:01.230 --> 14:14.730 this kc, sorry this kc is here that ky square plus kx square is kc, ky square is by boundary 14:14.730 --> 14:19.720 condition they have been fixed to some discrete values, kx also is to discrete value. 14:19.720 --> 14:25.790 That is why beta will be this discrete value that beta will be k square minus this, this. 14:25.790 --> 14:36.579 So, these are the numbers m and n, they are the mode numbers, TEMN mode. So, now, you 14:36.579 --> 14:43.089 see that we won the TE wave to propagate, then beta should be a real quantity, as we 14:43.089 --> 14:48.940 were saying that otherwise it will become an mode. We want the propagating mode so beta 14:48.940 --> 14:56.139 should b real. Now beta should real means, this under square root thing should be positive; 14:56.139 --> 15:03.779 that means, k should be greater than 0 kc, or k should be greater than this values. So, 15:03.779 --> 15:10.079 we know the value of the k that is the wave number that was a constant that was. If you 15:10.079 --> 15:16.000 remember that time I put k is equal to omega root over mu epsilon. So, that should be greater 15:16.000 --> 15:17.530 than this. 15:17.530 --> 15:26.090 Now, this on the other hand if k is less than kc, you see propagating when k is greater 15:26.090 --> 15:33.440 than kc. Now k less than kc that time the condition is this. Now omega is nothing, but 15:33.440 --> 15:41.080 angular frequency so you can find the corresponding frequency. So, see that below, if f is below 15:41.080 --> 15:48.870 this value, this value is given by a means the number, then the wave cannot propagate, 15:48.870 --> 15:54.259 wave will not be able to pass to the wave guide. So, this is the stark constant to the 15:54.259 --> 16:01.939 TEM line, as we have seen that any frequency can pass to a TEM line, but in a wave guide 16:01.939 --> 16:07.680 both for T and TEM mode we see there is a cutoff. So, on that frequency we have calculated 16:07.680 --> 16:13.069 that f is less than equal to this. So, if we take that equal to value, so; that 16:13.069 --> 16:19.290 means, frequency, all the frequencies below this cutoff, there will be wave will not be 16:19.290 --> 16:24.970 able to propagate. So, you need to have a frequency above this, and then only the wave 16:24.970 --> 16:32.220 will propagate. So, that is why this wave guides are called a high pass filter. So, 16:32.220 --> 16:36.990 low frequencies are not pass, but all the high frequencies after the cutoff is passed, 16:36.990 --> 16:41.459 and this is given by this. So, all the signal is frequency is below this cutoff, cannot 16:41.459 --> 16:48.430 propagate to wave guide. All the signals frequencies higher than cutoff frequency can propagate 16:48.430 --> 16:50.100 in a wave guide. 16:50.100 --> 17:00.920 Now, they are many modes depending on the value of m and n, you seen that when we say 17:00.920 --> 17:07.240 this. So, various values of m and n; m and n can be any number, any integer - m and n 17:07.240 --> 17:14.930 both are starting from 0 and integer. So, there can be various modes TE modes, but out 17:14.930 --> 17:22.720 of that, the one with lowest cutoff frequency is called dominant mode. So, let us say that 17:22.720 --> 17:30.300 lowest one will be m and n both are 0, but if m and n are 0, if we go back to the field 17:30.300 --> 17:39.310 equation. Here if we put m and n 0 you see by putting n is equal to 0 ex is 0, by putting 17:39.310 --> 17:45.990 m is equal to 0 ey is 0, by putting m is equal to 0, by putting m is equal to 0. 17:45.990 --> 17:54.910 So, if m and n both are 0, here also you see that, if n is equal to 0. So, sine 0 this 17:54.910 --> 18:07.840 is also 0. So, all the fields ex ey hx hy all are 0. So, that is not a physical solution. 18:07.840 --> 18:16.480 So, we say that this is not acceptable. So, m is equal to n; that means, T 0 0 mode TE 18:16.480 --> 18:25.460 m n, you see the first of script is m, the second script is n, TE 0 0 not possible in 18:25.460 --> 18:33.140 a wave guide. Now, since a is greater than b that is our convention. So, lowest cutoff 18:33.140 --> 18:41.140 frequency occur for m is equal to 1 n is equal to 0, because m is related to a, if you look 18:41.140 --> 18:51.830 at that m pi by a, so that will give you the frequency, and that frequency is nothing, 18:51.830 --> 18:57.970 but 1 by 2 a mu epsilon. Mu epsilon are the parameters of the thing, if it is they are 18:57.970 --> 19:06.790 wave guide, these are the mu epsilon or the free space values, and a is the broad dimension 19:06.790 --> 19:14.120 of the wave guide. Now this dominant mode is called TE 1 0 mode. 19:14.120 --> 19:21.260 Is a very interesting mode, and if you have a wave guide where more than one mode is propagating, 19:21.260 --> 19:28.950 we call it wave guide is over moded. So, in over moded field, power is transported by 19:28.950 --> 19:35.180 all the modes which are propagating. So, you need to separately exit the modes. Similarly 19:35.180 --> 19:41.040 if the modes are taking power, you need to extract the modes separately. So, that creates 19:41.040 --> 19:47.720 an engineering problem. Also, when you have one type of mode extraction the other power 19:47.720 --> 19:54.500 that is carried by other modes that gets wasted, generally we try to make a wave guide non 19:54.500 --> 20:02.580 over moded, now the wave impedance. Remember it is we cannot have characteristic impedance 20:02.580 --> 20:09.040 here because we cannot define p and i. Characteristic impedance is a low frequency concept, but 20:09.040 --> 20:14.950 this wave impedance that is always there, as we have already discussed earlier that 20:14.950 --> 20:22.330 wave impedance is the fundamental property of the wave. So, that will be k eta by beta, 20:22.330 --> 20:28.550 various modes will have various betas. So, actually we should have written beta m n. 20:28.550 --> 20:35.350 Similarly here z TE m n, so that various TE modes T m n modes they will have, different 20:35.350 --> 20:41.240 wave impedance. Now wave impedance ZTE is real when beta is 20:41.240 --> 20:50.270 real, because k is a number its real number for simple material, eta is real number intrinsic 20:50.270 --> 20:58.770 impedance of free space equal to 120 pi or 377 ohms. So, if beta is real ZTE is real. 20:58.770 --> 21:05.040 Now beta dl means we have seen that when, the wave is propagating or we have a propagating 21:05.040 --> 21:10.990 mode. So, by looking at wave impedance you can see that, whether you have a propagating 21:10.990 --> 21:18.530 mode, if you have a beta imaginary, then ZTE will be imaginary and you can say that I have 21:18.530 --> 21:24.539 an evanescent mode. So, I cannot transfer power to the evanescent mode. Evanescent mode 21:24.539 --> 21:29.740 will die down after sometime, because its beta will actually, instead of wave propagation 21:29.740 --> 21:36.400 e to the power minus j beta z, it will be e to the power minus j again j beta z, so 21:36.400 --> 21:41.720 that will be a loss e to the power, it will be contributing to the attenuation of the 21:41.720 --> 21:45.690 wave. So, that is why after sometime it will be propagating. 21:45.690 --> 21:53.280 Now, this is the picture of wave impedance. So, you see x axis is f by fc, if we are above 21:53.280 --> 22:02.360 fc then the both TE and TEM modes, they are impedance, wave impedance is real that is 22:02.360 --> 22:11.280 why it is shown as r naught, but when they are less, then the impedance is reactive, 22:11.280 --> 22:18.990 and for T mode this is the variation, up to here you see that just before the cutoff, 22:18.990 --> 22:28.540 it is shooting up to a very high reactants. Similarly just after cutoff then it is gradually 22:28.540 --> 22:36.270 decreasing, and coming to this value intrinsic impedance, this is eta, so intrinsic impedance 22:36.270 --> 22:44.040 value. So, when you are further away from cutoff; that means, quite high from the cutoff 22:44.040 --> 22:54.930 frequency, you approach the impedance of eta. Guided you see this is from a book, I do not 22:54.930 --> 23:00.040 remember the book; it is characteristic impedance of wave guide mode. Now this wrongly written 23:00.040 --> 23:05.740 this is not characteristic impedance. This is talking of wave impedance; that is why 23:05.740 --> 23:11.630 I have given this heading as wave impedance, it is from some good book, but they are writing 23:11.630 --> 23:19.060 characteristic impedance. This is basically the electric field to magnetic field ratio; 23:19.060 --> 23:23.450 this should be called wave impedance, not characteristic impedance. This is loose stock, 23:23.450 --> 23:29.470 but you should under know that it is the wave impedance concept. 23:29.470 --> 23:35.700 Now, similarly guided wavelength and phase velocity can be easily calculated, we have 23:35.700 --> 23:43.340 already seen. So, since beta is greater than k wavelength inside guide, is larger than 23:43.340 --> 23:49.980 free space wavelength. So, if you a 10 Giga hertz signal that is wavelength is 3 centimeter, 23:49.980 --> 23:58.430 but when its passes through any wave guide in the T E mode, then its wavelength is more, 23:58.430 --> 24:05.430 may be 3.5 centimeter or something. Also the phase velocity you see, we are showing here 24:05.430 --> 24:13.360 that why it is so, because this values you see k actually forty TEM mode beta is equal 24:13.360 --> 24:22.670 to k that we have already seen. And since this beta is less than k; that means, 24:22.670 --> 24:31.750 that in T E mode the phase constant beta is less than this wave number k, we get that 24:31.750 --> 24:39.050 lambda g the guided length is greater than free space wavelength. Similarly for phase 24:39.050 --> 24:45.610 velocity, for the same reason phase velocity is larger than these. Now what is this velocity? 24:45.610 --> 24:51.600 This is if it is free space one by this, actually if you put this value, because these are constant, 24:51.600 --> 25:00.690 this will come to be 3 into the power 8; that is the speed of light. So, inside a wave guide 25:00.690 --> 25:08.670 the t mode phase velocity is larger than the velocity of light. So, you can have a fast 25:08.670 --> 25:14.270 wave, we call it any wave, we is faster than the velocity of light; that is fast wave. 25:14.270 --> 25:19.870 So, inside wave guide you have fast wave; that is why wave guides are called fast wave 25:19.870 --> 25:20.870 structure. 25:20.870 --> 25:30.550 Now, dominant mode if we TE 1 0 mode, if you know it ex is 0 everywhere, that means it 25:30.550 --> 25:37.020 does not have an x component you seen it only have a y component, and that y component is. 25:37.020 --> 25:44.740 There is the mistake it will be sine n pi x by a u i is equal to ez is equal to 0, h 25:44.740 --> 25:54.140 x this also should be in pi x by a. And it does not have any longitudinal component, 25:54.140 --> 26:00.350 similarly h does not have any y component, but h has a hz component. 26:00.350 --> 26:05.770 So, let us see the structure, if we see the structure, the structure looks like this. 26:05.770 --> 26:12.480 This is the 3 d structure, this is red things are, you see if I take this cross section 26:12.480 --> 26:18.350 this, this is the cross section there you see electric field, at the center it is very 26:18.350 --> 26:25.490 high, its electric field is vertical or from top wall to bottom wall it is coming electric 26:25.490 --> 26:30.640 field, and you know field lines when that dense means its value is very high. So, at 26:30.640 --> 26:37.050 the center its value is very high, at the sides its value is very less. Whereas, magnetic 26:37.050 --> 26:42.730 field you see this is on the top wall the magnetic field is like this, you see and in 26:42.730 --> 26:48.280 the central portion there is no magnetic field. Also in the side wall you see at the center 26:48.280 --> 26:54.070 there is the very dense or very high value of magnetic field, gradually at the side it 26:54.070 --> 27:03.270 is there. This whole structure repeats after every lambda g by 2 and currents also we have 27:03.270 --> 27:04.270 seen. 27:04.270 --> 27:10.510 Let us see some better structure for better understanding. So, you see electric fields, 27:10.510 --> 27:16.340 they are from top wall to bottom wall they are flowing whatever I said. The green ones 27:16.340 --> 27:22.440 are magnetic fields. So, magnetic field is like this, and you see that the whole thing 27:22.440 --> 27:27.550 is getting repeated magnetic field in the, along the longitudinal direction after very 27:27.550 --> 27:34.110 lambda g by 2. And current distribution if you see that this is the current distribution 27:34.110 --> 27:40.900 but this one is a bit wrong actually. Central portion there is no current, because you see 27:40.900 --> 27:48.290 that the magnetic field is like this. So, central portion, there are no currents, 27:48.290 --> 27:53.630 longitudinal current is not there at the center, other places they have longitudinal currents; 27:53.630 --> 27:59.020 that is why what happens you see since the magnetic field is not there, we can cut a 27:59.020 --> 28:07.600 slot there. If we cut a slot a very small thing, we can probe inside the wave guide. 28:07.600 --> 28:14.610 Actually that is the way by which the field inside wave guide is measured. We have already 28:14.610 --> 28:19.980 discuses in an earlier NPTEL lecture, that how to measure impedance, how to find fields 28:19.980 --> 28:27.130 etcetera, and you can see that lecture. So, there we all use in a wave guide, in a slotted 28:27.130 --> 28:34.300 web guide, we have a slot cut in the middle that we can do, because of this dominant mode 28:34.300 --> 28:41.120 it does not have any current here. So, we do not disturb anything the magnetic field 28:41.120 --> 28:42.120 is here. 28:42.120 --> 28:55.140 Here you can see in a better way, you see the magnetic field, if I go along the center 28:55.140 --> 29:03.470 they are no magnetic field. So, if I cut a slot, I am not disturbing any magnetic field, 29:03.470 --> 29:08.080 because otherwise you see that if I cut a slot and disturb the field, basically I am 29:08.080 --> 29:13.080 disturbing the whole structure. So, what I am measuring is not the proper, but due to 29:13.080 --> 29:20.380 this fine variation that, at the center portion there is no magnetic field, and also you see 29:20.380 --> 29:28.860 that at this portion there are currents are minimal. So, I can easily cut a slot, and 29:28.860 --> 29:33.590 I can probe what is happening. Another beautiful thing is if I do it at the 29:33.590 --> 29:40.100 center, we have already seen that in this cross section, the electric field is maximum. 29:40.100 --> 29:46.230 So, when I am cutting the slot I am not the disturbing magnetic field here, here; obviously, 29:46.230 --> 29:51.870 electric field cannot be here because it is the T E mode. So, note a longitudinal electric 29:51.870 --> 29:57.590 field, but here at the center the electric field is maximum. So, in all the cross section 29:57.590 --> 30:06.010 the central field is maximum, electric field. So, the beauty is, when I am when I am sensing 30:06.010 --> 30:12.420 the maximum with maximum sensitivity sense, because electric field is maximum there; that 30:12.420 --> 30:19.230 is why the slot is cut at the middle in a rectangular wave guide, which is working with 30:19.230 --> 30:21.840 TE 10 mode. 30:21.840 --> 30:28.370 So, electric field is vertical maximum at center 0 at side walls, and varying cosinusoidally 30:28.370 --> 30:38.470 along x direction this when here. h field has both hx and hz component that we have 30:38.470 --> 30:45.350 already seen, hx is maximum at center, 0 at side walls etcetera. So, the longitudinal 30:45.350 --> 30:50.610 slot at center wont disturb magnetic field, e field is best for sensing at center, because 30:50.610 --> 30:53.240 it is high. 30:53.240 --> 31:01.490 So, these are other modes structure, and there you are that there are magnetic field is, 31:01.490 --> 31:08.210 there are, you see if I put the cut a slot here there will be magnetic field gets disturb, 31:08.210 --> 31:13.580 because this magnetic field is there. Suppose I cut it here there is a magnetic field here, 31:13.580 --> 31:20.010 I cut it here, so there is only for TE 1 0 mode we can cut a slot there; that is why 31:20.010 --> 31:25.170 for all experimental purposes or for any sensing TE 1 0 mode is preferred. 31:25.170 --> 31:32.310 Now, these are some other field structures also you see that, there are some field structures 31:32.310 --> 31:40.790 which are not uniform, but this is a uniform, our dominant mode TE 1 0 field electric field 31:40.790 --> 31:50.140 that is uniform. So, we require uniform fields, and then we can have various other things, 31:50.140 --> 31:56.940 and wave impedance, cutoff wave number is this, phase constant is this. 31:56.940 --> 32:08.700 Wave impedance you see it is a less is this. So, if you do lambda by 2 a. Now wave impedance, 32:08.700 --> 32:15.940 this is intrinsic impedance this 377 ohms, but this TE 1 0 mode its impedance, wave impedance 32:15.940 --> 32:22.540 is much larger than this intrinsic impedance, so for a WR 90 wave guide at 10 gigahertz 32:22.540 --> 32:28.760 if you do, because I know what is the value of a, 10 gigahertz means 3 centimeter. So, 32:28.760 --> 32:38.500 this wave impedance is 505 ohm at 10 Giga hertz. Power flow we have done this calculations, 32:38.500 --> 32:43.950 and for propagating modes p 1 0 is non-zero real power. 32:43.950 --> 32:50.040 So, also dielectric loss, we have already found the expression, this conductor loss 32:50.040 --> 32:55.170 if you do. Actually all this mathematics how to arrive at this expression that will be 32:55.170 --> 33:00.420 given in your notes which will be uploaded, you can see from there, but basic things we 33:00.420 --> 33:05.840 have already in the first five lectures we have given you, how to calculate that these 33:05.840 --> 33:08.430 are all applications of that. 33:08.430 --> 33:16.420 So, if you see that at attenuation of the dominant mode. You see this is the dominant 33:16.420 --> 33:24.450 mode. So, essentially just though the mode starts after this, roughly near about 5 6 33:24.450 --> 33:30.220 gigahertz, but attenuation is changing very fast, but after some time the attenuation 33:30.220 --> 33:37.330 stabilizes and it is quite small; that is why it will use it from 8 to 10 gigahertz; 33:37.330 --> 33:44.280 though the cutoff starts at 6 gigahertz, but near that the attenuation is varying, but 33:44.280 --> 33:51.120 after sometime attenuation stabilizes. So, it can use, but you cannot use it greater 33:51.120 --> 33:56.500 than 12 gigahertz, because after 12 gigahertz you will see that the next higher order mode; 33:56.500 --> 34:04.770 that is TE 0 1 mode, not TE 1 0 mode. TE 1 0 mode also have, this is TE 1 0 mode, this 34:04.770 --> 34:14.649 is the next mode that comes then 1 TEM modes comes. 34:14.649 --> 34:22.940 So, this is the thing that first TE 1 0 cutoff takes place as a I said roughly 6 gigahertz, 34:22.940 --> 34:33.250 then TE 0 actual this are all the thing. This is other modes will start coming, and then 34:33.250 --> 34:43.049 TE 2 0 cutoff that comes, but TM 11 cutoff that comes around 12 gigahertz; that is why 34:43.049 --> 34:51.510 after gigahertz people do not use it. So, this is called, these are structure that you 34:51.510 --> 34:57.450 can have this diagram, to find out what is their attenuation as well as what is the cutoff. 34:57.450 --> 35:04.470 Now, similarly how did you will see TEM mode here hz is equal to 0. So, here the Helmholtz 35:04.470 --> 35:10.460 equation will be in terms of the longitudinal e field ez. Again with the separation of variable, 35:10.460 --> 35:21.950 again the boundary condition is, in the side wall what is the electric field, tangential 35:21.950 --> 35:27.259 component that will be in the side wall? It will be ez, and in the top and bottom wall 35:27.259 --> 35:29.119 it is ez. 35:29.119 --> 35:38.060 So, you force that by that you get those values, and again those kx and ky are some discrete 35:38.060 --> 35:44.500 values. So, transverse components you can write once you determined ez, you can write 35:44.500 --> 35:53.880 ex ey hx hy phase constant, same expression. So, cutoff frequency lambda g v p is same. 35:53.880 --> 36:05.940 Now field expression suggest that if, sorry here it suggests that if either m. Here you 36:05.940 --> 36:15.230 see that this is the big different from the transverse magnetic field, and if either m 36:15.230 --> 36:20.089 or n is 0, all the transverse component are 0. 36:20.089 --> 36:29.231 So, TM 00 TM 0 n and TM m 0 modes cannot exists, because all the fields go to 0. So, only thing 36:29.231 --> 36:35.020 is dominant mode; that means, the fast one that can start is, that it should be both 36:35.020 --> 36:41.671 one. So, dominant modal wave impedance is given by this cutoff frequency this. Now if 36:41.671 --> 36:51.440 we compare with TE mode, because this is dominant TM mode TM 11 and this you compare that TM 36:51.440 --> 37:02.380 11 is this. This; obviously, it has extra package of this, that is why the cutoff frequency 37:02.380 --> 37:11.910 for TE 1 0 mode, is less than cutoff frequency of TM 11 mode. So, among all the combined 37:11.910 --> 37:17.619 Tn TM mode that TE 1 0 is the dominant mode 37:17.619 --> 37:24.779 Now you can see that TE 1 0 mode starts at that is why I was saying, that TE 1 0 mode 37:24.779 --> 37:32.119 for WR 90; that means, this as I said 0.9 inch and 0.4 inch or this centimeter values, 37:32.119 --> 37:43.000 TE 1 0 mode start at 6.56 gigahertz, TE 2 0 mode starts at 13 gigahertz, TE 01 mode 37:43.000 --> 37:50.410 starts at 14 gigahertz; that is why before down set off TE 20 mode, you should not go 37:50.410 --> 37:58.769 beyond that; that means, that is why we generally WR 90 is operated up to 12 gigahertz, it is 37:58.769 --> 38:06.140 operated up to 12 gigahertz. So, that TE 20 mode does not come here. Now you see that 38:06.140 --> 38:19.309 T TE modes come, then the next one that TE 11 and TM 11 they come together, then TM modes 38:19.309 --> 38:20.799 start propagating. 38:20.799 --> 38:30.720 And this are again the, if you change the dimension you see a by b, we call rectangular 38:30.720 --> 38:36.400 wave guide, but what happens if we make a by b 1 that is a square wave guide. Now in 38:36.400 --> 38:45.960 case of square wave guide you will see that, it is over-moded wave guide, when TE 10 is 38:45.960 --> 38:52.109 propagating that time also TE 01 propagates to it. So, that is why there are problems 38:52.109 --> 38:58.390 because this two waves are different orientation of electric filed so, but if you want that, 38:58.390 --> 39:02.470 in some application you want that electric field should be in the both the direction, 39:02.470 --> 39:05.019 there square wave guide also is used. 39:05.019 --> 39:12.450 Now, these are the methods of excitation of various modes. So, as various models are different 39:12.450 --> 39:22.700 field structures you need to excite, the field structures differently like TE 1 0 mode generally 39:22.700 --> 39:31.390 what we do at the central position, we put that coax if central conductor we connect 39:31.390 --> 39:42.839 to the wave guides, wall where electric field is dominant, and then it starts electric field 39:42.839 --> 39:49.540 from there. Whereas the in case of TM 11, we from the side wall which I try to excite, 39:49.540 --> 39:54.200 like that there are if you look at the electric field and magnetic field structure you get 39:54.200 --> 40:00.700 the clue, that how to excite the field. These are just you to know that this type of thing 40:00.700 --> 40:07.880 you see, that at the central portion you put a coaxial cable, and connect the central conductor 40:07.880 --> 40:20.210 to the wave guide bottom layer, you get a you make that potrabation and get the excitation 40:20.210 --> 40:24.450 of modes. So, I think that is all about wave guides, 40:24.450 --> 40:29.200 rectangular wave guides. In the next lecture we will see that circular wave guide geometric. 40:29.200 --> 40:29.609 Thank you.