WEBVTT 00:19.720 --> 00:28.780 So, in this lecture will start, we will see circular wave guide. Now this is the circular 00:28.780 --> 00:33.640 waveguide in our lab you see that flanges are still rectangular, but the wave guide 00:33.640 --> 00:39.770 inside that in instead of any rectangular pipe it is a circular pipe. 00:39.770 --> 00:47.690 Now, this is circular wave guide cross section, so the, this wall is metallic, and again since 00:47.690 --> 00:53.540 the structure is conformal with the cylindrical coordinates. So, we will switch over to row 00:53.540 --> 00:59.120 pi n z coordinate or cylindrical coordinate. 00:59.120 --> 01:08.229 Now, again since it is a wave guide. This is a single conduct no TE m mode, and wave 01:08.229 --> 01:12.330 propagation in z direction TE modes. 01:12.330 --> 01:23.570 So, e z is 0, we will express the Helmholtz equation in terms of h z. and this is the 01:23.570 --> 01:29.720 web equation or Helmholtz equation. Web equation and Helmholtz equation as same name, sometimes 01:29.720 --> 01:36.970 we interchange when we use them. Again here you apply separation of variables bracket 01:36.970 --> 01:44.460 into f row and g pi, already we have done that in case of field analysis by coax, when 01:44.460 --> 01:48.410 we saw the coax field analysis similar thing. 01:48.410 --> 02:01.150 So, to those things, and again that k pi and k rho square things, again the solution is 02:01.150 --> 02:09.899 similar. As I said that h z must be periodic in pi. So, h z thing is if you change pi the 02:09.899 --> 02:15.680 function, that time it was potential function, this time it was converts relate magnetic 02:15.680 --> 02:23.840 field pressure that should be same. So, k pi should take discreet values, discreet integer 02:23.840 --> 02:31.150 values. So, that is why we give it a name instead of k pi, we are calling it n pi to 02:31.150 --> 02:35.010 remind us that it is integer values. 02:35.010 --> 02:44.120 Now, for other one, the row variation part or row function part you can write like this. 02:44.120 --> 02:50.200 Now this equation is an important equation in engineering mathematics, this is Bessel 02:50.200 --> 02:57.260 function. So, this is second order differential Bessel function, it is solution was given 02:57.260 --> 03:08.090 by the scientist Bessel. So, that is. There are first kinds of Bessel function j, which 03:08.090 --> 03:15.300 generally denote by the first kind of Bessel function j n and these as second kind is y 03:15.300 --> 03:16.710 n. 03:16.710 --> 03:25.320 So, now we want the fields to be finite inside the entire wave guide including the center; 03:25.320 --> 03:31.989 that means, inside the whole circular wave guide we want the field to be finite, but 03:31.989 --> 03:42.150 the Bessel function of second kind y n that has an infinity when at the center. 03:42.150 --> 04:02.800 So, y n x cannot be supported as a solution . So; that means, these cannot be. So, this 04:02.800 --> 04:10.260 d then should go to 0. So, that this part does not come here. So, when d goes to 0 we 04:10.260 --> 04:16.250 write like this; that means, only c is there, but c we are absorbed, because a constant. 04:16.250 --> 04:24.620 So, a and b has absorbed that c. So, now this is a general solution. This will subject to 04:24.620 --> 04:35.500 the boundary condition. So, this is the whole physical part we have enforced, and got one 04:35.500 --> 04:41.540 or you can say two way constant deduction; that is why we now need to come boundary condition 04:41.540 --> 04:53.620 find this a b. So, as I said consult notes for this manipulation. Again in cylindrical 04:53.620 --> 05:00.500 coordinate will have to manipulate Maxwell’s equations, so that you can express the transverse 05:00.500 --> 05:03.190 field components in terms of longitudinal components. 05:03.190 --> 05:10.510 If you remember it equation 15 was that for condition coordinate, the similar thing we 05:10.510 --> 05:18.710 have not till now done, because though we have seen the coaxial line analysis, but that 05:18.710 --> 05:25.900 analysis was not in terms of these, because of the quasi static field structure in coax, 05:25.900 --> 05:34.270 in two conductor circular thing there we took the help of potential function, and took the 05:34.270 --> 05:42.041 solve only Laplace’s equation, but here in TE m mode or in circular wave guide you 05:42.041 --> 05:49.400 do not have that luxury or that convenient; that is why we need to again manipulate Maxwell 05:49.400 --> 05:56.511 equations in the notes that will shown that how to write it in; that is why I am writing, 05:56.511 --> 06:00.820 or I should have written equation 15 counterpart for cylindrical coordinates. 06:00.820 --> 06:14.360 So, if you do that the four transverse components e row e pi h row h pi can be written in terms 06:14.360 --> 06:21.980 of the longitudinal components instead indicial coordinate; that is e z and h z e z and h 06:21.980 --> 06:32.919 z e z and h z. Now in case of t modes you put that e z 0 in case of TE m modes you put 06:32.919 --> 06:40.770 h z 0. So, this 4 field gets expressed in terms of one longitudinal component only. 06:40.770 --> 06:46.000 And boundary condition for hollow metallic cylinder what is that; obviously, the any 06:46.000 --> 06:52.970 metal or any conductor that tangential electric field is 0. What is tangential field here, 06:52.970 --> 07:04.070 e pi? So, e pi should go to 0 at where on the metallic structure. So, that is why at 07:04.070 --> 07:12.460 row is equal to a; that means, from this whole outer surface it should go to 0, enforce that. 07:12.460 --> 07:24.530 So, for e modes you get that e pi is like this we already knew the, is a thing that 07:24.530 --> 07:32.400 we just put that e pi. So, boundary condition demands; finally, that this Bessel function. 07:32.400 --> 07:45.370 This is actually the j n dash k c row so; that means, the derivative of Bessel function, 07:45.370 --> 07:54.920 that function should be equal to 0. So, this is the demand. Now, this is an equation so 07:54.920 --> 08:12.199 there are number of roots to this equation. So, let the m-th root is called p n m dash. 08:12.199 --> 08:21.490 Now remember that in case of rectangular wave guide, our structure was m and n modes were 08:21.490 --> 08:30.710 t or TE m n, where m stands for the number of variations along x direction. The way it 08:30.710 --> 08:37.419 was defined that k x is equal to m pi by a. So, it was variation along x direction number 08:37.419 --> 08:48.180 of variations how many variations we have along x, and n stands for along b direction 08:48.180 --> 08:58.369 or how many go variation, but here the order is different, these known that here n comes 08:58.369 --> 09:10.179 first m now. So, p n m dash is. Suppose it is a m-th root of j n dash this equation. 09:10.179 --> 09:20.239 So, if that is a root; that means, we can say that k c into a is equal to p n m a, because 09:20.239 --> 09:27.980 this is a root; that means, if I put in this equation that value; that means, in place 09:27.980 --> 09:32.399 of row if I put that I am getting that. 09:32.399 --> 09:41.180 So, here I have written n refers to the number of circumferential variation; that means, 09:41.180 --> 09:49.779 azimuthal variation, and m refers to the number of radial variation. So, this, that Bessel 09:49.779 --> 09:58.170 function is derivative that equation the roots are well tabulated. So, if you see the root 09:58.170 --> 10:07.810 for various values of m and n if you see. So, and find out the cut off frequency. For 10:07.810 --> 10:16.480 cut off frequency first step will have to determine what this beta n m. So, k c already 10:16.480 --> 10:23.449 in the previous one, we have seen k c in the cut off wave number once you know that. 10:23.449 --> 10:30.480 So, you can find beta, once you know beta you can c f c cut off frequency, and that 10:30.480 --> 10:40.670 will be given by this. So, you see cut off frequency is the, denominator is some constant, 10:40.670 --> 10:49.290 but the numerator that depends on the number of modes, the value of n m that you choose 10:49.290 --> 10:58.069 p n m. So; that means, the lowest value here that will be the lowest thing, so that will 10:58.069 --> 11:04.420 be the dominant mode. Now here if you see this stable it becomes clear that what are 11:04.420 --> 11:12.300 these, these one is the lowest, and this is for what. this is for n is equal to 1 m is 11:12.300 --> 11:24.480 equal to 1; that means, I need a TE 1 1 mode TE 1 1 mode, that will give me the minimum 11:24.480 --> 11:33.439 cut off frequency amongst the modes, and so we can say that dominant circular wave guide 11:33.439 --> 11:42.610 mode is t e, till now TE mode, so that will be TE 1 1 mode. 11:42.610 --> 11:53.020 So, dominant TE mode is this. The dominant TE mode is clearly TE 1 1, for circular wave 11:53.020 --> 12:00.129 guide m is at least one, because circular wave guide the solution requires that m is 12:00.129 --> 12:09.119 equal to 0 is not possible. So, no t n 0 mode, and hence no counter part of TE 1 0 mode here, 12:09.119 --> 12:20.689 it starts from TE 1 1. So, if you see the field distribution of TE 1 1 mode that you 12:20.689 --> 12:28.670 can see this is the TE 1 1 mode, this is the dominant mode. 12:28.670 --> 12:35.889 But one problem if you see the field, you see the field is not symmetry. So, this side 12:35.889 --> 12:42.370 it is going like these, this side this is going like these, where as you see this fields 12:42.370 --> 12:49.529 structure is symmetry. These are something like our coax field structure etcetera. So, 12:49.529 --> 12:55.639 one problem is generally we prefer this type of field structure. So, dominant mode field 12:55.639 --> 13:02.379 structure is known at least electric field structure, is not shown good. Now you can 13:02.379 --> 13:08.679 see other things field structure etcetera. So, that why sometimes in circular wave guide 13:08.679 --> 13:15.970 the dominant mode is not used. So, they would need to be able to design convert as to convert 13:15.970 --> 13:22.110 the dominant mode to derive that; obviously, some power is loss, but to have some other 13:22.110 --> 13:25.459 advantage, sometimes engineers do that. 13:25.459 --> 13:31.870 And what is the wave impedance, again from that basic definition of e field by h field. 13:31.870 --> 13:39.639 Who carry power, e e row cross h pi that will carry power? So, that is why you are taking 13:39.639 --> 13:45.179 that ratio, not e row by h row, because they do not carry power that we have already explained 13:45.179 --> 13:53.040 earlier. So, this will be the things, depending on beta you can find that value, also conduct 13:53.040 --> 13:58.350 a loss is given by this and. 13:58.350 --> 14:17.630 Now, t m modes h z is equal to 0, so the is e z and e z solution, something. Again you 14:17.630 --> 14:24.679 see Bessel function has come and sine and cosine variation as before. So, you put the 14:24.679 --> 14:32.819 boundary condition, tangential electric field is 0. So, in terms of here we can say that 14:32.819 --> 14:45.000 e z is 0; that mean this is also z, e z is also a transverse field, e z is also is a 14:45.000 --> 14:52.519 tangential field at tangential electric field. So, if you put that then you get the similar 14:52.519 --> 15:00.839 equation that characteristic equation that j n is this. So, you remember that, that time 15:00.839 --> 15:12.810 it was j n dash which is k c. the cut off wave number, if again we assume that p n m 15:12.810 --> 15:23.309 that is m-th root of this equation, then we got k c in terms of this constant, and this 15:23.309 --> 15:28.690 is a dimension that is also constant. 15:28.690 --> 15:35.639 Now, from the table you can determine which is adding the lowest one. So, phase constant 15:35.639 --> 15:43.389 is given like this, and cut off frequency is like this. so; that means, again you can 15:43.389 --> 15:49.819 see the cut off frequency will be lowest, for the mode whose p n m is lowest, whose 15:49.819 --> 15:59.449 p n m is lowest let us see, that amongst these this is lowest, what is that mode; that 0 15:59.449 --> 16:00.739 and 1. 16:00.739 --> 16:12.929 So, that t m 0 1 mode will be the lowest mode. Dominant t mode is t m 0 1. Now this thing, 16:12.929 --> 16:19.130 but then amongst TM together who is dominant. 16:19.130 --> 16:25.939 Then we will have to compare the dominant circular wave guide. So, f c amongst TE this 16:25.939 --> 16:32.989 is dominant TE 1 1; that is this is the expression for their f c, this is the expression for 16:32.989 --> 16:45.679 t m’s 1. So, this is this. So, you see clearly t 1 1 is a dominant mode, so t 1 1 is called 16:45.679 --> 16:52.869 the dominant circular wave guide mode. It is status is same as t 1 0 mode in rectangular 16:52.869 --> 16:53.869 wave guide. 16:53.869 --> 17:02.519 So, these are again field lines. So, you can see that, whatever I have said that sometimes 17:02.519 --> 17:11.250 this type of symmetric structures they are preferred over this. Where is t 1 1 you see 17:11.250 --> 17:22.049 this one. So, not very symmetric, where sometimes people have problem with this dominant mode 17:22.049 --> 17:27.949 e field is not circularly symmetric, other lower order modal fields are circularly symmetric, 17:27.949 --> 17:35.519 some other, not all other, some other. 17:35.519 --> 17:47.240 TM modal field distribution, then attenuation characteristics you can see that TE 1 1 cut 17:47.240 --> 17:56.890 off etcetera, but the attenuation wise TE 1 1 has quite lower cut off after you have 17:56.890 --> 18:03.360 one of the certain frequency then this cut off is quite small. 18:03.360 --> 18:14.080 Now as I said TE 0 1 is of interest for very low loss. So, you see from these, where that 18:14.080 --> 18:22.809 TE 0 1 you see that it is coming after a certain time, because it is cut off is per, but it 18:22.809 --> 18:31.610 has a sudden it goes to at higher frequency, it is slow, it is value is very small. So, 18:31.610 --> 18:39.210 it can give you very low loss propagation. So, for high power application where you want 18:39.210 --> 18:46.530 to, do not want to lose power it is preferred, but it is fields structure is also symmetric; 18:46.530 --> 18:57.550 that is fortunate that TE 0 1 if you locate, that t 0 1 you see, that it is a very, just 18:57.550 --> 19:04.029 like circular symmetric is there in its electric field; that is why these field was once very 19:04.029 --> 19:09.009 popular among the structures that how to extract that. 19:09.009 --> 19:15.149 So, in many applications circular wave guide is used as over moded guide, because of that 19:15.149 --> 19:24.740 I want to extract TE 0 1, not the dominant t 1 1. So, that time it is used as over moded, 19:24.740 --> 19:30.159 and mode converters are used for converting dominant and other modes to the desired mode. 19:30.159 --> 19:37.070 As I said; obviously, that will give rise to certain amount of loss, but to get the 19:37.070 --> 19:43.389 benefit of the field structure circularly symmetric, we needed. Also if you do not have 19:43.389 --> 19:48.269 circular symmetric field structure, there are problems that at certain point’s fields 19:48.269 --> 19:55.559 may be high. So, electric breakdown may occur. So, that is why in circular wave guide breakdown 19:55.559 --> 19:58.720 is a problem if you do not have a circular symmetric wave guide. 19:58.720 --> 20:05.360 Here we are showing that you know that, actually if you have a breakdown thing. Breakdown occurs, 20:05.360 --> 20:11.590 because if you have inside a wave guide some air now some air. Now, air at eleven mega 20:11.590 --> 20:17.919 volt per meter it breaks down. So, if you give a field higher than that, then it will 20:17.919 --> 20:25.240 break down. So, there will be spark produced and the power cannot flow, because power will 20:25.240 --> 20:35.230 be then taken to the short at thing, short at plates. So, to avoid that, suppose if you 20:35.230 --> 20:41.460 were, nowadays there are applications coming up where people are trying to send huge power 20:41.460 --> 20:47.850 to wave guides. So, in that case the breakdown etcetera may come. So, to prevent that what 20:47.850 --> 20:57.370 they do if, you put a dielectric inside, generally in the form of gases it is put, then suppose 20:57.370 --> 21:03.990 you put the wave guide in a gas chamber, then depending on the dielectric if the, whatever 21:03.990 --> 21:10.629 the dielectric based on that square root of that factor it will be pushed up. 21:10.629 --> 21:16.911 So, that is being shown that if you go on increasing the pressure, the electric break 21:16.911 --> 21:23.270 down value is coming down. So, this is also another thing that in high power application, 21:23.270 --> 21:30.660 these circular wave guide, you want to have circular symmetric thing, because then break 21:30.660 --> 21:37.740 down will not, the chances of occurrence of break down will be less, and then you can 21:37.740 --> 21:45.940 also would gas, generally c f 6 is used to do that in high power microwave people they 21:45.940 --> 22:00.389 use c f 6, to at least increase the break down values two to three times; that is circular 22:00.389 --> 22:14.000 wave guide. So, we have seen these two wave guides, and again. So; that means, we have 22:14.000 --> 22:22.290 seen how TE m mode propagates, how TE m mode propagates in rectangular wave guide and circular 22:22.290 --> 22:25.760 wave guide. Thank you.