Modes are solutions, of Maxwell s equation applied to a specific device.

Transcription

1 Mirowave Integrated Ciruits Prof. Jayanta Mukherjee Department of Eletrial Engineering Indian Institute of Tehnology, Bombay Mod 01, Le 06 Mirowave omponents Welome to another module of this NPTEL mok sore. Mirowave integrated iruits. In this module we are going to over just the basi introdution of ertain mirowave omponents. Now later on in the ourse, we will of ourse over these omponents or various mirowave omponents of different number of ports with some more detailed mathematial analysis. But in order to understand a flavour of this mirowave engineering, let me show you some of the basi onepts bind these mirowave omponents. So the first thing that we need before going to the mirowave omponent is the onept or modes in mirowave engineering. Now I have not given a solution or disussed about the solution of the Maxwell s equation for various mirowave omponents. I just showed you the basi Maxwell s equation, the homogeneous equation neessary for solving the, solving for the eletri and magneti fields, in the various mirowave omponents. But there is something alled modes, whih we have to deal extensively when we are in mirowave engineering. [Refer Slide Time: 01:55] Modes are solutions, of Maxwell s equation applied to a speifi devie.

2 [Refer Slide Time: 02:25] TEM E H 0 TE E 0, H 0 TM E 0, H 0 There are broadly lassified 3 modes. T or Three modes, types I an say. TEM, TE and TM. TEM modes are where both the eletri and magneti field are perpendiular to the diretion of propagation. So we have EZ is equal to HZ equal to 0, where Z orresponds to the diretion of propagation. TE modes are where Z is not equal to 0, EZ equal to 0, and TM modes are where EZ is not equal to 0, but HZ equal to 0. [Refer Slide Time: 03:30] Now usually, not usually it s always true that there is a single TEM solution. That is we have only one eletromagneti wave whih satisfy the TEM onditions. However there are multiple TE and TM modes. Other interesting thing about these two modes is in single ondutor waveguides, only TE and TM modes exist. So we are at low frequeny like AC frequeny or when we are dealing with a DC battery, we are familiar with eletrial power transfer only by means of two ondutors. But then we have also eletromagneti power transfer to vauum, where there are no ondutors. Then we have our mirowave omponents where we have single ondutors i.e. single piee of metal is able to ondut power. Obviously we annot have flow of harges as we

3 know in onventional two wire iruits. So here these single wire ondutors, they simply at as a guide for eletromagneti waves. So there is no onept of voltage of urrent, there is only the onept of eletri and magneti fields. [Refer Slide Time: 05:05] Whereas the TEM solution is appliable only for two wire ondutors.

4 [Refer Slide Time: 05:50] For example, say let us onsider oaxial waveguides. A oaxial wave line, waveguide an be drawn like this. Sine we have two modes, therefore TEM mode is the primary means of (om) propagation. One other thing I forgot to tell you about this TE and TM modes. [Refer Slide Time: 06:15] If I an go bak to the previous slide, is that for TE and TM modes, they have something alled a ut off frequeny. So ut of frequeny is the frequeny below whih the TE and TM modes annot transmit. For TEM mode no ut off frequeny is there. So they an transmit over any frequeny. Now the mode as I said the TEM and TM, there are multiple solutions possible and

5 that TE or TM mode for a partiular waveguide whih has the lowest ut off frequeny is alled a dominant mode. [Refer Slide Time: 07:10] So then, sine in a oaxial waveguide TM mode an exist, hene it an transmit over the entire range of frequenies. Hene even though there might be some TE and TM solution, TEM mode is the dominant mode.

6 [Refer Slide Time: 09:01] V ˆ d 0 jk H (x, y,) x. e V ˆ d 0 jk E(x, y,) y. e Now an interesting example of a waveguide is what is known as a Parallel Plate Waveguide. A Parallel Plate Waveguide is a waveguide whih has two parallel plates and these plates are infinity long and infinitely wide. The only finite dimension is the distane D, separating the two planes. Now the TEM wave solution for this parallel waveguide is given like this. The Solution for the TEM waves are given like this. Here H X is oriented only along the X diretion and E is oriented only along the Y diretion. E represents the eletri field, H represents the magneti field. Eta is given by square root of mu over E, is alled intrinsi impedane. That is the impedane due to the material properties. So this is the TEM solution and as we an see there is only a single solution. We don t have multiple solutions.

7 [Refer Slide Time: 11:25] k k 2 2 n y E(x, y,) Ax sin e d j n y H x Axos e k d j n y Ey Axos e k d j j j n 2 Ex H y 0, k, k d If we try to find out the other solutions, for example the TM solutions, the solutions are given like this. The reason I am writing expliitly the solutions is beause I want to show you some of the properties of these equations. See, the values of this eletri and magneti fields, first of all H is oriented only along the X diretion, i.e. Perpendiular to the diretion of propagation Z and E has omponents, both along Z and Y. And note there is a variable N in this equation. What this variable N does is, for different values of N you get different solutions of EZ, HX and EY. And these various solutions for every new values of N are alled the various modes. So we don t have a single solution, we have multiple solutions. And then all of these will have ertain ut off frequeny and the one that has the lowest ut off frequeny will be the dominant mode. So we will have modes like say TM 1, TM 2, TM 0 and so on.

8 [Refer Slide Time: 14:15] Now for a retangular, similarly you know, for a retangular waveguide, there is a devie alled, or a omponent alled a retangular waveguide, whih is just a hollow of a retangular pipe, like this. It is a pipe like struture whih is hollow inside and the dominant mode for this is TE 10 that is beause whereas for the parallel plate guide we had solutions that varied only with the single variable N, in the ase of this retangular waveguide, we have two variables M and N. So we will get modes like TE 10, TE 11 and so on. So TE 10 is the dominant mode. That is it has the lowest ut off frequeny. So for TE 10, the field lines, if I an draw them, the magneti field are onentrate like this and the eletri field are like this. So lambda by two separation will ause eletri fields and the magneti fields to rotate. So the magneti field will rotate in the opposite diretions and the eletri fields whih are perpendiular to the diretion of propagation, they will also be oriented in an opposite diretion.

9 [Refer Slide Time: 15:40] k k 2 2 n y E(x, y,) Ax sin e d j n y H x Axos e k d j n y Ey Axos e k d j j j n 2 Ex H y 0, k, k d Now the onsequene of this if you an go bak to the solution for the parallel plate waveguide. Now we see that the propagation onstant Beta is given like this, where K is related to the atual wavelength that we see and KC is a onstant. It depends on N and the value of D. D is the separation between the two plates. Now this is for a parallel plate waveguide, one again. Now what you see is that this propagation onstant Beta, whih is the, whih is kind of the apparent wave length that we observe. Beause Beta is a spatial frequeny in this equation. So this expression will repeat itself after every one, after every beta value, after every lambda dash value, given by two Pi upon beta. So if beta is a spatial frequeny but then the atual real, wavelength that we have inputted into the waveguide is lambda. So there is an apparent dihotomy or something, it is not mathing.

10 [Refer Slide Time: 17:10] k k k Usually as you know the frequeny is related, as you know the speed of, suppose C is the speed of light then we have, that is given by new lambda. Where new is given by C upon lambda. So new and lambda should be inversely proportional to eah other. By this relationship of beta is equal to K square minus KC square, Where beta is the apparent wavelength and K is the real inputted wave length, related to the inputted wavelength is given by this relationships. Now this of ourse is related to the, is diretly proportional to the frequeny of input. So then the simple relationship, simple inverse relationship that exists before doesn t exist anymore. This is the apparent wavelength and this term is related to the frequeny like this. So if we atually plot, in fat this equation is given like this. So we see that beta and omega are related by a omplex equation. Of ourse for high values of omega beta is again proportional to omega. So 2 Pi, initially the relation we were expeting was New equal to C upon lambda, on in other words 2 Pi new is equal to 2 Pi C upon lambda. This implies omega is equal to beta C. So here we had expeted a lender relationship between omega and beta. But what we are atually observing is a relationship like this.

11 [Refer Slide Time: 19:30] Note that as I said, for high values of omega, the relationship will again be linear. But for low values of omega, we get a kind of omplex relationship and if you plot this urve between beta and omega, then we get a urve like this. This is known as the dispersion diagram. Another omponent that is frequently and most widely used in mirowave engineering nowadays is the mirostrip line. So the mirostrip line onsist of a ground plane with a top layer metal with a sub strip omprising of some Dieletri like Teflon or glass epoxy or erami in between. This is the onstrution of a mirostrip line.

12 [Refer Slide Time: 21:52] The problem with mirostrip line, is that, if you an ome bak to our written slides. The problem with the miro strip line is that, it is, sine it onsist of two ondutors, two different dieletri materials. Even though it has two ondutors, the modes are not stritly TEM. So in a mirostrip line, you have, if you take a ross setion of a mirostrip, whih I showed in the previous slide. It is something like this. Two metal layers sandwihed between some dieletri onstant epsilon 1. If we had just air in between, then it would look something like this will all epsilon 0. If air were there, then it would be a two ondutor waveguide and TEM would be the dominant. But sine it is not so, we have another dieletri material in between. What we usual do is we assume as if there is neither air nor the dieletri material present. But sorry, but this entire spae outside and inside of this metal layer is overed with a material having a dieletri onstant epsilon effetive. Is suppose C is the apaitane between these two plates, with just this dieletris present and C 0 is the apaitane between these two plates with air present and C dash is the apaitane with, with this effetive dieletri with dieletri material having epsilon effetive present, then we an, you know we an find out a relationship for this epsilon effetive, in terms of this measured values of C and C0.

13 [Refer Slide Time: 24:05] For example, so I was disussing the three types of ondutor, three types of mirostrip lines for analysis. One where there was a dieletri material with epsilon 1, other where air was present between the lines. And the third where we had an effetive dieletri material having the dieletri onstant epsilon detetive present. Now suppose C is the apaitane for this ase, C 0 is the apaitane for this ase and suppose with the effetive dieletri onstant epsilon effetive also the net apaitane value that I obtain is like this, then I an write C upon epsilon effetive is equal to C 0 upon epsilon ero, from whih I an write epsilon effetive is equal to epsilon 0 C upon C 0. So this is the relationship for the dieletri onstant. This imaginary dieletri onstant that is surrounding the inside and outside of these metal plates and it is related to the values of C and C 0, with this relationship. Now before ending this leture just one small onept, I want to stress is the onept of impedane. As you saw, we have single ondutor wave guides for mirowave propagation but then there is no onept of voltage or urrent. Yet we inherently know that, impedane, the onept of impedane is related to voltage and urrent. However there is something alled an impedane, when we deal with various waveguides. So the question is what is this impedane that we are talking about? Sine voltage and urrent are not effetive onepts we really annot define impedane in terms of voltage and urrent always.

14 [Refer Slide Time: 26:20] E H eff eff x 0 y 0, Z 0 0 eff We have already got a hint of something alled intrinsi impedane whih is related to the property of the material, and that is given by mu upon epsilon. So this is a value of an impedane whih is purely dependant on the properties of the material. In addition we also have something alled eletromagneti impedane whih is the ratio of the eletri to the magneti field. So here instead of voltage divided urrent, we have EX divided by HY and this is also an effetive way of expressing impedane. In fat for air this intrinsi impedane omes out to be 377 ohm. That s why antennas whih are interfae, often onsidered as interfae between air and waveguide. The waveguide has to be basially designed to produe an impedane mathing. The waveguide or the antenna has to be designed to produe and impedane mathing of 377 ohm. Now this is intrinsi impedane. For a mirostrip line that we were disussing we have found out what is an E effetive and for a mirostrip line that is the value of Z 0 is often given by this formula where epsilon effetive as we found was equal to epsilon 0 C upon C 0. Where C is the measure, apaitane between the plates of the atual mirostrip line and C 0 is the apaitane between the mirostrip metal plates, had air been the dieletri medium. So just to summarise this leture, in this leture we have overed some basi onepts about mirowave engineering like impedane, modes, dominant modes, TEM, TE and TM modes and also introdued some of the devies that are ommonly used in Mirowave engineering. We will of ourse over devies in more detail. But as a starting point these onepts are very, it is very important to understand, this onepts. Like what is dominant mode, what are the TM and TE modes and, so with that we ome to an end of this module.

15 Thank you.