WEBVTT 00:17.699 --> 00:27.630 Welcome to the second lecture Mathematical Model of Modes. So, as I said that we have 00:27.630 --> 00:40.159 an engineering model and we need to understand basic modes that are TEM TE TM. Now, we need 00:40.159 --> 00:48.730 to develop a mathematical model of those three modes and then only we will be able to go 00:48.730 --> 00:55.770 further in the analysis of this whole transmission process. So, let us start with the mathematical 00:55.770 --> 00:58.250 model first. 00:58.250 --> 01:05.180 Modes are basis of EM signal and I have already said that all EM signals are basis set of 01:05.180 --> 01:12.880 the modes. So, if we have waves propagating in z direction the basis set we call TEM to 01:12.880 --> 01:20.300 z, TM to z, TE to z. Similarly, if the wave is propagating in a 01:20.300 --> 01:31.100 x direction, we call this basic set TEMx, TMx, TEx and similarly for y direction TEMy, 01:31.100 --> 01:39.530 TMy, TEy. Now, simply if we analyze any of these, suppose generally we all make that 01:39.530 --> 01:41.670 the direction of propagation is z. 01:41.670 --> 01:50.020 So, that is why we will make the model only for TEM, TM and TEz models if the propagation 01:50.020 --> 01:57.969 derives in others you can easily change over to this nomenclature. Now, the generic field 01:57.969 --> 02:07.689 distribution of TEMz is a 3D vector field, you know the difference between vectors and 02:07.689 --> 02:17.090 field, the vector is directed across theta z direction and also it has an magnitude and 02:17.090 --> 02:26.230 the field means it also has a special variation. So, we can break any 3D vector field into 02:26.230 --> 02:35.159 two components, one is along the line of propagation because you see our mode definition that time 02:35.159 --> 02:39.950 we have distinguished that what is the direction of propagation and always we are talking in 02:39.950 --> 02:45.299 terms of direction of propagation. So, that we call longitudinal direction and the wave 02:45.299 --> 02:53.680 is propagating in z direction means the longitudinal direction will be z and to that we have a 02:53.680 --> 03:00.569 plane that is called transverse plane. For z directed propagation, we have transverse 03:00.569 --> 03:06.969 plane will be x-y plane, longitudinal is z. Similarly, if we have x directed propagation 03:06.969 --> 03:12.670 and then longitudinal means x direction and transverse means y z direction like that. 03:12.670 --> 03:24.040 So, for TEMz, TEM to z we should have both the electric field and magnetic field which 03:24.040 --> 03:29.579 are transverse that means, they do not have any longitudinal component and definition 03:29.579 --> 03:38.400 of TEMz is nothing but Ez is equal to 0, the electric field does not have any longitudinal 03:38.400 --> 03:43.799 component. So, that is why Ez is equal to 0 and magnetic 03:43.799 --> 03:51.989 field does not have any longitudinal component Hz is equal to 0 and definition of TEM to 03:51.989 --> 03:58.610 z is mathematical definition. The boundary condition you can say or the condition for 03:58.610 --> 04:04.499 being a TEMz where is Ez is equal to 0 and Hz is equal to 0. 04:04.499 --> 04:13.519 Now, is this information sufficient to find generic field distribution of TEMz? The answer 04:13.519 --> 04:21.440 is yes if we can write all the transverse components of electric as well as magnetic 04:21.440 --> 04:29.590 fields in terms of the two longitudinal components Ez and Hz, then this is possible because by 04:29.590 --> 04:39.130 being TEMz, we know the values of Ez and Hz. So, if we can write all other field components 04:39.130 --> 04:50.280 that means, Ex Ey and Hx Hy in terms of Ez Hz, then we can solve. So, this will be tried 04:50.280 --> 04:58.500 and obviously we will have to start from Maxwell’s equation because all solution and all electromagnetic 04:58.500 --> 05:04.910 fields are solutions of Maxwell’s equation. We will have to try to manipulate Maxwell’s 05:04.910 --> 05:23.170 equation so that, we can write the Ex Ey Hx Hy in terms of Ez and Hz. We will see the 05:23.170 --> 05:26.000 definition of Te wave later. 05:26.000 --> 05:36.070 But now I am trying introducing Te wave, it means that the electric field is transverse. 05:36.070 --> 05:42.000 Electric field transverse means it will have Ez is equal to 0 and it does not have magnetic 05:42.000 --> 05:46.620 field transverse. So, magnetic field can have Hz. So, Ez is 05:46.620 --> 05:53.380 equal to 0 and Hz not equal to 0, because if Hz is equal to 0 it will again become TEM 05:53.380 --> 06:11.830 wave. So, this is the definition of TE. Now, the definition of TM, TM will be that Ez not 06:11.830 --> 06:26.490 equal to 0 and Hz equal to 0 also let me write TEM here. So, TEM is Ez is equal to 0, Hz 06:26.490 --> 06:38.550 is equal to 0 and actually these are all z. In all these cases, basically I know the condition 06:38.550 --> 06:48.490 on Ez and Hz and if I can do this manipulation and write all transverse fields in terms of 06:48.490 --> 06:54.260 longitudinal fields that will help me to find out the fields for all these three cases. 06:54.260 --> 07:02.210 So, that we will spend some time here in this lecture probably we will totally take this 07:02.210 --> 07:12.610 time to manipulate Maxwell’s equation and to write the transverse components. 07:12.610 --> 07:18.720 Obviously, there will be some mathematics, but you know without mathematics you cannot 07:18.720 --> 07:29.590 learn in engineering. So, if you are not getting any trouble in this mathematics, we will be 07:29.590 --> 07:37.690 posting you some of the written version of this doc files for these things and you can 07:37.690 --> 07:45.690 brush up your knowledge if not, you can always discuss in the forum, we will try to show 07:45.690 --> 07:50.810 you how to do that and these are all simple and are taken from mainly David Pozer’s 07:50.810 --> 08:02.570 book on microwave engineering, it is a very famous book. So, still if you have any problems 08:02.570 --> 08:06.360 understanding, you come to the forum we will discuss that. 08:06.360 --> 08:11.190 So, first let us see the Maxwell’s equation in differential form and the Maxwell’s equations 08:11.190 --> 08:17.590 are in both forms either differential form or in the integral forms, but here we are 08:17.590 --> 08:23.230 starting because we are a source. So, source means that we will have to start from points 08:23.230 --> 08:29.490 and that show in the differential form you know all this. Here, one thing in the first 08:29.490 --> 08:37.339 equation you see we have a magnetic current. Now, no one has still detected magnetic current, 08:37.339 --> 08:44.399 but if we do not put it actually if there is any aperture type of source that means 08:44.399 --> 08:53.880 there is some conductor then there is an aperture then that generates electric fields and so 08:53.880 --> 09:05.320 that, we equivalently defined a magnetic current. Here, you see that in this one, you see there 09:05.320 --> 09:12.250 are the field quantities at any point we will get these fields and these are also field 09:12.250 --> 09:19.400 quantities but these are the impress sources that means, the source comes here explicitly. 09:19.400 --> 09:25.930 Now, in this one the conduction current source, although we know that when there is a really 09:25.930 --> 09:39.550 a time varying charge that is flowing then we have a conduction current and Maxwell shows 09:39.550 --> 09:49.510 that there is a magnetic current also that means as I said the dou the charge uncharged 09:49.510 --> 09:55.680 flow, they are the till now detected only at, but unless and until we do this we do 09:55.680 --> 10:02.120 not get a symmetry. So, we have M which is a fictitious current, but it can be actually 10:02.120 --> 10:13.611 represented in terms of electric fields. So, we have that and write this and these 10:13.611 --> 10:21.190 are the Maxwell’s equation and E D H B J M molar time varying vector fields which are 10:21.190 --> 10:29.360 all functions of all the special coordinates in rectangular coordinates we are saying this 10:29.360 --> 10:37.500 x, y, z and t, t is the time without time varying we do not get electric magnetic signal. 10:37.500 --> 10:46.960 Others are plane things, now here we have listed all the definite physical quantities 10:46.960 --> 10:54.840 and there units. So, that later there would not be any confusion because in the earlier 10:54.840 --> 11:00.180 books these are definitions of Pozar, but in earlier books there where many times the 11:00.180 --> 11:09.560 H and B etc their interchangeably was used and this notations now almost in scientific 11:09.560 --> 11:16.640 community, this is the standard notation by which you expressed this magnetic current 11:16.640 --> 11:26.470 density which is volts per meter square. As the unit suggest you know that in electric 11:26.470 --> 11:31.200 J conduction current this is electric current density, this conduction electric current 11:31.200 --> 11:39.240 density and there is a magnetic current density as I said is example created by any aperture 11:39.240 --> 11:45.200 antenna like a horn antenna. So, in that horn antenna there is on the aperture, that means 11:45.200 --> 11:52.080 on the horn mount there is a electric field which we can equivalently define and magnetic 11:52.080 --> 11:59.380 current for that and that we put in Maxwell’s equation so that, sources become symmetric 11:59.380 --> 12:04.670 otherwise the magnetic current is not there. So, on the electric current is there which 12:04.670 --> 12:12.630 does not makes the things symmetric, also you know that Electromagnetic wave apart from 12:12.630 --> 12:15.310 Maxwell’s equation if you want to solve the electromagnetic wave because electromagnetic 12:15.310 --> 12:18.050 wave propagates to medium. 12:18.050 --> 12:25.400 So, it interacts with matter and matter all material properties also should be factored 12:25.400 --> 12:34.040 into field quantities which get modified by matter. For simple material, simple means 12:34.040 --> 12:42.360 in M theory you learn that simple material means it should be linear isotropic and homogeneous. 12:42.360 --> 12:49.770 The relation between the two quantities are shown in or the four quantities that are shown 12:49.770 --> 12:58.860 in the Maxwell’s equation. Here we have E, here we have D, similarly here we have 12:58.860 --> 13:06.300 H here we have B. So, we need to relate them when we will reduce them to one equation. 13:06.300 --> 13:15.220 So, that relation is called constitutive relation, B is related to H by the material property 13:15.220 --> 13:23.940 permeability. The electric flux density D is related to electric fields by permeability. 13:23.940 --> 13:30.450 Now, these are materials property which really materials get characterized by this epsilon 13:30.450 --> 13:35.550 mu and also another one which is coming here the conductivity. 13:35.550 --> 13:41.320 So, these three are any electrical characterization of any material and you can distinguish between 13:41.320 --> 13:49.390 two materials by finding these values. Now in free space that means, you know the concept 13:49.390 --> 13:57.870 of free space which is a standard material. So, there this mu naught that value you know, 13:57.870 --> 14:04.779 but you should always remember this value and it is a unit and similarly the free space 14:04.779 --> 14:12.570 permeability epsilon naught that is 8.854 etc 10 to the power minus 12 Fahrenheit per 14:12.570 --> 14:19.800 meter. Also, you require ohms law and that means the relation between conduction current 14:19.800 --> 14:27.040 density and the electric field that is J is equal to this. So, this you see that wave 14:27.040 --> 14:34.990 propagation to matter you need to know mu epsilon and sigma. This J will tell you that 14:34.990 --> 14:39.940 how matter are interacting with the electromagnetic wave. 14:39.940 --> 14:46.230 So, now let us try to find the time harmonic EM signal, time harmonic means most of time 14:46.230 --> 14:56.720 we are interested in steady state and in steady state actually Fourier has proved that any 14:56.720 --> 15:05.800 electrical type of signal that can be broken into various exponential basis thing. So, 15:05.800 --> 15:13.330 we say that e to the power j omega t that is the Fourier basis. So, in the time variation, 15:13.330 --> 15:18.020 we have already seen that if the z variation of any z propagating wave that is e to the 15:18.020 --> 15:25.420 power minus k z, but time variation now we are saying that in the steady state, not in 15:25.420 --> 15:30.649 the transient state, in transient state it maybe something else, but in the steady state 15:30.649 --> 15:37.589 all signals can be expressed as time variation of e to the power 1 j omega t, e to the power 15:37.589 --> 15:43.040 2 j omega t. We assume that our field has time variation 15:43.040 --> 15:49.290 e to the power j omega t that is called Time Harmonic EM signal that means, my variations 15:49.290 --> 15:57.529 will be e to the power j omega t, e to the power 2 j omega t, e to the power c j omega 15:57.529 --> 16:05.240 t all are harmonic that harmonic to omega. So, in such cases if the time variation is 16:05.240 --> 16:10.790 this, we know that we can drop the time variation by switching over to phasor notation. So, 16:10.790 --> 16:17.040 you see that Maxwell’s equation now we can write by noting the time variation is this, 16:17.040 --> 16:24.210 that is why there are two del del t operations in the Maxwell’s equation. This in the first 16:24.210 --> 16:32.160 two equations there are two time derivative that will become J omega so that you see the 16:32.160 --> 16:41.089 del cross c is this. So, in phasor form all this tilde means phasor. Later, many times 16:41.089 --> 16:49.740 will drop this tilde, but always it will be phasor because time derivation generally we 16:49.740 --> 16:57.279 do not deal with, we drop it here, but if we require it can be always a phasor, this 16:57.279 --> 17:04.630 is phasor and it can be always written, sorry this is phasor and this is the actual real 17:04.630 --> 17:08.470 signal. So, actual real signal is real part of this 17:08.470 --> 17:13.530 phasor into e to the power j omega t. You multiply e to the power j omega t with the 17:13.530 --> 17:22.649 phasor and then take the real part that will be your actual v. We will solve Maxwell’s 17:22.649 --> 17:27.489 equation in phasor form, then whatever solution we will get of that phasor electric field 17:27.489 --> 17:35.929 phasor, we will just multiply it with e to the power j omega t, the whole thing real 17:35.929 --> 17:45.730 part will take and we say that is the real electric signal, all signals are real and 17:45.730 --> 17:53.730 there cannot be any imaginary signal, but we deal with for convenience to get rid of 17:53.730 --> 17:59.570 the time variation we come to phasor. So, there is no time variation, then when you 17:59.570 --> 18:01.950 finish we come there. 18:01.950 --> 18:09.549 Now, this is a thing that for TEMz mode, the wave propagates in z direction. 18:09.549 --> 18:17.070 So, that is true, but to make our thing generally we have already seen in the beginning that 18:17.070 --> 18:22.100 wave may propagate in x direction also, wave may propagate in y direction also. So, to 18:22.100 --> 18:32.399 make the thing most general, we will now say that the wave is propagating in k direction, 18:32.399 --> 18:39.850 k in most of our cases will be z, but there can be instances of x directed field also, 18:39.850 --> 18:46.249 y directed field also or in any general field is breakable or can be expressed in terms 18:46.249 --> 18:56.019 of z component, z directed field, y directed field and x directed field and these vector 18:56.019 --> 19:04.460 k is called propagation vector, also sometimes called wave vector. So, vector means we should 19:04.460 --> 19:13.289 define its magnitude and its direction, its magnitude is the wave number k which equals 19:13.289 --> 19:21.499 k which is color quantity wave number and its direction is in the direction of wave 19:21.499 --> 19:22.749 propagation. 19:22.749 --> 19:35.249 Now, in Transverse and Longitudinal Components, any field quantity electric filed phasor can 19:35.249 --> 19:41.230 be broken into a longitudinal and transverse component. The longitudinal component with 19:41.230 --> 19:48.429 this new propagation vector terminology, it is along k vector, though you see we have 19:48.429 --> 20:00.010 still writing here e z, but along k vector and the other component is the transverse 20:00.010 --> 20:10.799 component which we write as e x, y. So, similarly the magnetic phase phasor also can be broken 20:10.799 --> 20:20.029 into the transverse component h x, y and longitudinal component along k direction that is h z x, 20:20.029 --> 20:21.029 y. 20:21.029 --> 20:30.509 So, we can write these phasor which will be having E x, y, z that is breakable into this 20:30.509 --> 20:40.539 is the transverse component plus k e z ex y then we know, but this is the two direction 20:40.539 --> 20:47.950 and what is the z variation of this electric field e to the power minus j beta z because 20:47.950 --> 20:55.769 the wave is propagating in the z direction. Similarly, h is also breakable into this and 20:55.769 --> 21:02.440 this equation numbers also will maintain so that it is similar to the nodes that we will 21:02.440 --> 21:08.489 upload. So, please you can refer to that how they are coming and that is why we are taking 21:08.489 --> 21:20.879 some numbers either you see that 5 6 7 8 then 9 10. 21:20.879 --> 21:27.840 Now, we are trying to model the micro wave transmission and let us assume that we are 21:27.840 --> 21:36.499 far from the source which is away, but we are several wave lengths away from the source 21:36.499 --> 21:42.799 so that transmission is taking place. In time also we had studied in, special direction 21:42.799 --> 21:48.129 also we are studied that we are far away from the source which is not effecting us because 21:48.129 --> 21:56.700 near the source something else happens, but when the wave has started taking energy. So, 21:56.700 --> 22:04.519 that is why we assume that source pre means J and M that we introduced, those are not 22:04.519 --> 22:11.100 there because those where the sources, now we get rid of them and say that del cross 22:11.100 --> 22:21.450 E is equal to this simple, so 11 12. Now, to see del cross E is related to H magnetic 22:21.450 --> 22:28.690 field. Similarly, magnetic field is related to electric field. So, this equation 11 if 22:28.690 --> 22:40.330 you break let us in Cartesian coordinates we get these. So, if we take this then there 22:40.330 --> 22:45.020 will be three components, you can write them as these. 22:45.020 --> 22:52.129 This equation gives you equation 13 which is three in parts so you have three, but here 22:52.129 --> 23:00.789 also we see that we have not succeeded in doing what was our objective, we will manipulate 23:00.789 --> 23:07.730 so that, the transverse component all are expressible in terms of a longitudinal component. 23:07.730 --> 23:19.590 But, you see that here this equation if I take here I will get these, but here you see 23:19.590 --> 23:30.739 that this is what I can do all the electric and magnetic field jumped up transverse fields 23:30.739 --> 23:32.989 can be expressed here. 23:32.989 --> 23:41.510 Similarly, magnetic field you can again break that into three parts, but still it is not 23:41.510 --> 23:46.890 doable we have not got our answer because here you see electric and magnetic field together. 23:46.890 --> 23:54.200 Suppose, I want to solve this first equation, then the problem is I have E x and H y and 23:54.200 --> 23:58.960 that is expressible in terms of H z, I can put the H z condition because longitudinal 23:58.960 --> 24:05.610 condition I can put, but then E and H are mixed up. This is the problem of Maxwell’s 24:05.610 --> 24:06.639 equation. 24:06.639 --> 24:16.580 So, here we have shown that if you eliminate then H x from these two equations 13a and 24:16.580 --> 24:32.140 14b and if you eliminate y from 13a you get H x. So, now, by this we can know that H x 24:32.140 --> 24:39.029 is a transverse component and it is written in terms of longitudinal component. 24:39.029 --> 24:48.129 We will see that still it is a problematic of getting a solution, suppose this E and 24:48.129 --> 24:54.840 H if they are together, now both are unknown to me. So, we need to do something later, 24:54.840 --> 25:05.650 but now first we express like this one by one, we will express this H x H y H z H x 25:05.650 --> 25:13.059 H y E x E y that means, transverse component in terms of E z and H z. So, this is an equation 25:13.059 --> 25:27.359 E z and here this denominator, you see these are all constants omega is a frequency, these 25:27.359 --> 25:31.600 two are material parameter and beta is the propagation constant. 25:31.600 --> 25:42.879 So, that also depends on the materials and these will give some name and that name or 25:42.879 --> 25:47.359 wave number and cut off wave number, wave number already we are familiar that is the 25:47.359 --> 25:56.919 propagation vectors magnitude so that is, omega into this and will also define a cut 25:56.919 --> 26:03.850 off wave number k square minus beta square. So, we write all transverse components in 26:03.850 --> 26:06.940 terms of longitudinal components and these two constants. 26:06.940 --> 26:14.929 These two will be introducing there. So, for later simplicity the physical meaning of k 26:14.929 --> 26:24.309 is the magnitude of propagation vector and this k c will see that it will give us the 26:24.309 --> 26:32.019 cut off wave number and from that will be able to find out some cut off frequency, actually 26:32.019 --> 26:40.610 will see that in cases there will be propagation will be stopped in certain cases that though 26:40.610 --> 26:47.509 depending on the geometry of the structure, there will be some cut off and from these 26:47.509 --> 26:54.779 cut off we can calculate that which frequency electromagnetic signals, they cannot propagate 26:54.779 --> 27:01.200 and that is why it is call cut off wave number, but that meaning we will attach later. Here, 27:01.200 --> 27:07.529 that meaning is not obvious, but I know that is why I am telling you. So, these are the 27:07.529 --> 27:09.610 two constant which we defined. 27:09.610 --> 27:14.549 So, in terms of these two constants if we write then there are four such equations, 27:14.549 --> 27:21.129 but basic philosophy is all transverse fields which I am writing in terms of the longitudinal 27:21.129 --> 27:26.249 components, you see that I have already said these in the nodes you would know that and 27:26.249 --> 27:30.220 these you will have to do so that, this is a very simple mathematic by that you will 27:30.220 --> 27:39.659 get this four fields I can write. Now, my job will be I will put one by one TEM TEMz 27:39.659 --> 27:45.399 condition TEz condition and TEMz condition and find out the field structure. 27:45.399 --> 27:54.039 So, this equation 15 is the part that we have manipulated somewhat and got you. So, choose 27:54.039 --> 27:58.499 appropriate longitudinal component value, equation 15 expresses the four transverse 27:58.499 --> 28:06.169 will components you can put the TEM, TEz values needless to say that choice of z axis for 28:06.169 --> 28:12.169 signal propagation direction is arbitrary, but usual practice one can as well have x 28:12.169 --> 28:17.309 directed propagation etcetera, all is needed to put appropriate longitudinal component 28:17.309 --> 28:23.629 to 0 and put that in equation 15. So, this was our mathematical model we have 28:23.629 --> 28:33.299 developed for the all modes, this model is valid that we could express from Maxwell’s 28:33.299 --> 28:41.669 equation that all four transverse components possible in terms of longitudinal component. 28:41.669 --> 28:48.539 Next, we will be putting the values and that will be specific to that mode which will be 28:48.539 --> 28:55.720 starting from the next class. The next lecture we will see that if we have TEM, what is the 28:55.720 --> 29:00.979 field? If you had TEM, what does the four fields becomes? If you have TE what does the 29:00.979 --> 29:05.730 four fields become? Thank you.