WEBVTT 00:20.330 --> 00:35.550 So, in this lecture we will now see a new thing that instead of the three-dimensional 00:35.550 --> 00:48.100 structure we will see planar transmission line. Actually, nowadays there is a need to 00:48.100 --> 00:55.690 have conformal structure planar structures so that the antennae or the transmission line 00:55.690 --> 01:12.760 they are on the surface and they are also on a, suppose on a source and load and power 01:12.760 --> 01:20.920 transmission circuit all are on a single plane, so that you can reduce the size of the object. 01:20.920 --> 01:28.640 So, that is why there is a need for planar transmission line. We will see two very popular 01:28.640 --> 01:35.780 planar transmission lines. There are number of planar transmission lines, we will only 01:35.780 --> 01:41.920 see the two which are easy to analyze and visualize. 01:41.920 --> 01:52.770 The first we will see is called a strip line. Now strip line is the geometry shown here, 01:52.770 --> 02:01.640 you see that there is a ground plane on the top, there is a ground plane on the bottom 02:01.640 --> 02:14.530 and between the ground planes there is . But there is another thing sandwiched in the inside 02:14.530 --> 02:29.470 this , that is a metal. So, there is a, this metal which is infinite in another transverse 02:29.470 --> 02:37.380 direction; that means, if we call it y sorry that is y. So, z direction in z direction 02:37.380 --> 02:47.980 in this, there is a strip of this of width W metal. So, this metal sandwiched or embedded 02:47.980 --> 02:53.280 inside this dielectric, top is ground bottom is ground. 02:53.280 --> 03:01.170 So, this is a side view of the structure that which is there is a metal, this is top is 03:01.170 --> 03:12.030 a metal bottom is a metal in between all are dielectric. So, what will happen generally 03:12.030 --> 03:20.040 you can which is ground plane is grounded. So, if potentially 0, this ground plane is 03:20.040 --> 03:28.980 grounded and this has got some potential. So, field lines will start from here and will 03:28.980 --> 03:38.970 end up here. Similarly, field lines will end up here, the whole thing is this, and this 03:38.970 --> 03:50.300 length is quite long. So, that this strips, line this central conductor that sees that 03:50.300 --> 04:00.099 throughout they are dielectric. Now, the E field is like this that the H field 04:00.099 --> 04:08.030 will be something like this. Now this is a structure of strip line, it is a bit difficult 04:08.030 --> 04:15.470 to fabricate because inside this you will have to plate this. So, generally it is that 04:15.470 --> 04:23.740 the half portion of this is fabricated and then on that another top portion is put. 04:23.740 --> 04:31.690 Now, it is a planar transmission line; obviously, you see that the whole structure is plane 04:31.690 --> 04:39.300 or not that like that 3D things like wave guide and etcetera by photolithographic fabrication 04:39.300 --> 04:47.050 it is made. As I already said that since it is planar and it is amenable to integrated 04:47.050 --> 04:52.759 circuits so microwave integrated circuit we used this strip lines as the transmission 04:52.759 --> 04:54.050 lines. 04:54.050 --> 05:11.720 Now, as I said this, so now if we look carefully that, sorry, if you look carefully this is 05:11.720 --> 05:17.759 a planar version of coax. I can say because coax is also like this there is a central 05:17.759 --> 05:26.050 conductor and covering that circularly throughout coax actually is the outer conductor. Here 05:26.050 --> 05:33.039 instead of that coax actual thing, it is a rectangular structure but the central conductor 05:33.039 --> 05:40.639 is all the sides it is seeing the all the side means that at least at the top and bottom 05:40.639 --> 05:47.879 it is seeing the outer conductor. So, that is why you see the field lines starting 05:47.879 --> 05:54.229 from the central conductor are all going to the either up ground plane or lower ground 05:54.229 --> 06:07.330 plane. So, it is something like that, but there is a difference that in a coax the field 06:07.330 --> 06:14.240 lines are all radial to the structure, here it is a rectangular structure but the field 06:14.240 --> 06:23.430 lines are not I will say they are not either in horizontal or vertical direction, it is 06:23.430 --> 06:32.039 a mixture. So, that is why the analysis of this strip line is not as straightforward 06:32.039 --> 06:44.879 as the coaxial line. In coaxial line because of the circular symmetry the equation will 06:44.879 --> 06:50.149 take a nice form. Here also you can apply Laplace’s equation to analyze that, but 06:50.149 --> 06:59.189 one thing is since it is a quite large distance here and theoretically a strip line is extending 06:59.189 --> 07:09.389 infinitely in you can say z direction so and it has finite thing here. 07:09.389 --> 07:17.530 So, you will have to apply the Laplace’s thing here. Also, the structure and the coordinate 07:17.530 --> 07:24.699 system they are not properly confirming. So, that is why you need to have some special 07:24.699 --> 07:31.319 function to analyze that. So, Laplace’s equation is used but special 07:31.319 --> 07:41.490 permissions are needed to handle that and also so that is why people try that with the 07:41.490 --> 07:49.669 another way of attack is people try by having some conformal transformation so that you 07:49.669 --> 08:00.839 get a better conformal structure by transforming it in some another plane and their you can 08:00.839 --> 08:10.349 get a good. But one thing we will say that we need to 08:10.349 --> 08:18.649 as an engineer, we need to this transmission line means if we know the it is characteristic 08:18.649 --> 08:25.180 impedance, particularly these lines as the thing suggest that since we have a something 08:25.180 --> 08:37.050 like line. So, and here the whole, the field lines are existing between the two conductors 08:37.050 --> 08:47.410 so TEM wave is supported here. So, since TEM wave is supported so the thing is we can have 08:47.410 --> 08:57.540 characteristic impedance defined for this structure and that if we know, basically engineers 08:57.540 --> 09:03.980 engineering design requires that knowledge of the characteristic impedance of this transmission 09:03.980 --> 09:11.910 line in cases where TM things propagate. Now that can be done. So, people have come 09:11.910 --> 09:20.080 out with some approximate method of analysis of strip line and also people have come out 09:20.080 --> 09:26.941 with some empirical formulas for this strip lines so that we will see one by one. One 09:26.941 --> 09:34.450 is two conductors and homogenous dielectric, so as I said TEM wave should be there also 09:34.450 --> 09:40.640 like coax or parallel plate waveguide higher order modes TE and TM also should be there. 09:40.640 --> 09:49.130 Now, higher order modes can be suppressed by shorting screws at the points where their 09:49.130 --> 09:57.310 maximum takes place so that people have time and variable to suppress the thing. Now as 09:57.310 --> 10:04.250 I said that we can consider it as a planar version of flattened out coax, both have center 10:04.250 --> 10:11.870 conductor completely enclosed by outer conductor uniformly filled by dielectric board. But 10:11.870 --> 10:22.360 as I said the reason that due to the non-conformal, non-conformance between the structure of the 10:22.360 --> 10:30.750 transmission structure and the coordinate system and the field things we are forced 10:30.750 --> 10:36.520 to do conformal mapping is standard procedure for solution of Laplace's equation in case 10:36.520 --> 10:38.480 of strip line. 10:38.480 --> 10:49.000 Now, in any TEM mode we know the phase velocity is given by the velocity of light by epsilon 10:49.000 --> 10:54.980 r provided we are assuming and that is true for strip line that there is no magnetic material. 10:54.980 --> 11:02.611 So, mu not, mu you can always take to be mu not. So, in that case v p will be simply c 11:02.611 --> 11:13.480 by epsilon r, beta we know it is same as k but in case of dielectric it will root over 11:13.480 --> 11:23.460 epsilon r k and z not if we take that l by c, l and c are the per unit length induct 11:23.460 --> 11:29.310 and capacitance then it is 1 by v p; that means, phase velocity into c. 11:29.310 --> 11:35.190 Now, knowing the dielectric we can find the phase velocity. So, once we know the phase 11:35.190 --> 11:44.070 velocity it can be seen that to find impedance I need to find the per unit length capacitance. 11:44.070 --> 11:51.070 So, both capacitance and inductance determination is not necessary. So, the standard procedure 11:51.070 --> 11:58.120 for all these planar transmission lines including strip lines is find out per unit capacitance. 11:58.120 --> 12:04.110 Now how per unit in capacitance can be found out? Now capacitance you know that, if we 12:04.110 --> 12:10.890 give some if you can calculate some charge distribution or if we can find some charge 12:10.890 --> 12:21.500 distribution and find the total charge and if we can find the potential function then 12:21.500 --> 12:28.740 q is equal to c v is the root by which we can find the c. So, that is done for all the 12:28.740 --> 12:32.650 planar transmission lines. 12:32.650 --> 12:47.010 So, you see that as I said that since it is TEM field so it will obey Laplace’s equation. 12:47.010 --> 12:53.920 But the problem is another strip line extends to plus minus; that means, infinity in both 12:53.920 --> 13:02.180 these directions x and this z direction. So, that is a problem that is called micro Laplace’s 13:02.180 --> 13:11.280 equation in a large domain if you want to apply if the computational its create problem. 13:11.280 --> 13:19.380 So, what people have done that if you see the field lines; obviously, far away from 13:19.380 --> 13:27.660 these strip lines, the fields will die down the fields will gradually become field in 13:27.660 --> 13:36.480 fields and after certain distance away their own field. Because, of the structure that 13:36.480 --> 13:44.040 the strip line is existing only at the central option of this whole structure. So, people 13:44.040 --> 13:53.430 have found that for confining the field we will say that at a distance plus a by 2 and 13:53.430 --> 13:59.260 minus a by 2, I have 2 more walls. Obviously, there are 1 ground plane then another ground 13:59.260 --> 14:09.980 plane here, but I will put sorry 2 more, 2 more ground planes or we call it electrical 14:09.980 --> 14:20.300 walls here that conducting walls. So, that the structure is now in confined. 14:20.300 --> 14:26.950 And by doing this when we are putting this thing means this is far away so obviously 14:26.950 --> 14:33.580 I am putting in discontinuity here because from a epsilon r I am putting a metal. But 14:33.580 --> 14:39.750 then the equation expected from that that will be write down here there is a discontinuity 14:39.750 --> 14:45.460 there will be higher even as an modes etcetera, but that will be a, so here it would not change 14:45.460 --> 14:51.650 much this is the idea. So, now we can put Laplace’s equation to 14:51.650 --> 14:57.380 our use that this is the Laplace’s equation this is the potential function. So, we are 14:57.380 --> 15:06.340 in this region we have the Laplace’s equation converse laplasion then put boundary condition 15:06.340 --> 15:15.820 again that at x is equal to plus minus a by 2; that means, that 2 sides there are electrical 15:15.820 --> 15:25.030 walls. So, the potential function will be 0 and also top and bottom ground plane where 15:25.030 --> 15:27.670 the potential is 0. 15:27.670 --> 15:33.610 So, by this whole boundary condition then put separation of variables again everything 15:33.610 --> 15:40.610 will be given in the notes, but the idea is this the central conductor will have a surface 15:40.610 --> 15:50.500 charge density because it has a potential non-zero potential. So, there will be a discontinuity 15:50.500 --> 15:58.750 of the potential function around that strip line central conductor. And we know that if 15:58.750 --> 16:05.480 we have a potential if we take the gradient we get electric field, negative gradient of 16:05.480 --> 16:11.280 the electric of the potential function gives us electric field, electric field multiplied 16:11.280 --> 16:18.550 by the appropriate permittivity gives us the displacement vector. 16:18.550 --> 16:27.440 Now, displacement vector if I know then from Maxwell’s equation we can find out the magnetic 16:27.440 --> 16:36.270 field etc. so field analysis becomes straightforward from there. Also, here what we do once we 16:36.270 --> 16:44.150 know the displacement vector from that we relate it to the Maxwell’s the or process 16:44.150 --> 16:52.279 law that del dot b is equal to rho and from that we find out relate the rho and this potential 16:52.279 --> 16:59.740 function and ultimately manipulate it to find out that q is equal to c b. So, c we get from 16:59.740 --> 17:06.169 that because rho is the charge distribution, rho gives me q and this potential function 17:06.169 --> 17:16.870 gives me potential difference and since they are related here. So, we can always find out. 17:16.870 --> 17:23.959 So, that is the approximate analysis people do. So, here now it is true that since there 17:23.959 --> 17:29.570 is a discontinuity so; that means 2 sides will have 2 different solutions. 17:29.570 --> 17:42.230 So, potential function, the potential function, the idea is within this zone; that means, 17:42.230 --> 17:51.600 from 0 to b by 2 there will be one function 1 type of potential distribution and then 17:51.600 --> 17:59.960 b by 2 plus 2 b there is another potential function because of the discontinuity of the 17:59.960 --> 18:10.499 surface charge density on this. So, that is why it is broken into 2 and then first a n 18:10.499 --> 18:16.879 and b n the constants are determined then total charge can be written from the potential 18:16.879 --> 18:27.860 function knowledge as I said by applying Gauss’s law. So, you have rho x. So, that rho s is 18:27.860 --> 18:36.480 related to the d by applying Gauss’s law so that means once from the field you know 18:36.480 --> 18:44.320 this rho s then q will be get that along x direction and get that total charge. And from 18:44.320 --> 18:53.379 potential function you get the voltage between conductors by since I know E y, E y is known 18:53.379 --> 19:00.830 again from the gradient of the voltage potential function. So, calculate V, now you can find 19:00.830 --> 19:02.370 capacitance per length . 19:02.370 --> 19:11.580 So, by that method it comes like this that the capacitance depends on the width of the 19:11.580 --> 19:19.440 strip line then a is the place where you are how far away you are putting the electrical 19:19.440 --> 19:31.450 walls; w is already there the width the width of the thing and b is the strip line width 19:31.450 --> 19:36.910 that means, the separation of the 2 ground plane the top and bottom, we have separated 19:36.910 --> 19:42.549 a already known. So, now once you have that then this c is 19:42.549 --> 19:50.379 known there you put it c, this is the constant this the velocity of light and epsilon r is 19:50.379 --> 19:56.690 the known dielectric constant. So, you can find the characteristic impedance. Once characteristic 19:56.690 --> 20:02.389 impedance of the device is known all the engineering per applications can be done like how much 20:02.389 --> 20:08.330 power it will take, how to impedance match it etcetera all those things will come. 20:08.330 --> 20:13.880 So, now people have also come out with empirical formula for characteristic admittance but 20:13.880 --> 20:20.710 there they need to have some effective width of the center conductor and that effective 20:20.710 --> 20:28.100 width is given by this formula. So, these are basically if you look at these formulas, 20:28.100 --> 20:34.860 no one expects you to remember these formulas, but by refereeing to these you can find approximately 20:34.860 --> 20:42.159 the up to accuracy of 1 percent this formula is all correct. But this formula assumes the 20:42.159 --> 20:48.179 strip line the central conductor its thickness to be 0. There are other better formulas also 20:48.179 --> 20:56.320 people come out with that for a thick strip lines there also some formulas. And one thing 20:56.320 --> 21:06.999 is to be noted that from this that if actually you see W e is related to W and so; that means, 21:06.999 --> 21:17.369 if the width increases strip line width it extends width in the Z 0 conduct increases 21:17.369 --> 21:25.409 then the characteristic impedance that decreases. So, that is one. 21:25.409 --> 21:35.580 So, now you people have if any design strip line is to be designed then the job is actually 21:35.580 --> 21:41.720 to find the width from the given value because generally the characteristic impedance of 21:41.720 --> 21:47.869 any transmission line is specified. So, strip line what characteristic impedance is required 21:47.869 --> 21:53.940 that is specified and b is the separation of the 2 ground planes and epsilon r these 21:53.940 --> 22:00.190 will be given. So, from that by putting into those formulas we will have to find out what 22:00.190 --> 22:06.210 is W. Now, the either by computer programming (Refer 22:06.210 --> 22:13.369 Time: 22.06) you can find that or the same formula which we have shown before these formula 22:13.369 --> 22:21.220 people have inverted and this is the form that if you are this epsilon r and z not if 22:21.220 --> 22:26.749 they are less than this 120 you take this or if it is more than that you take this. 22:26.749 --> 22:33.399 So, based on that you can find out what is the value of W. 22:33.399 --> 22:42.169 And if this is the value given for it alpha b that is the attenuation constant due to 22:42.169 --> 22:49.700 dielectric plus and this is the attenuation constant due to the conductor loss. Complicated 22:49.700 --> 22:58.119 structure but you see that we need to find out what is W and what is the thickness of 22:58.119 --> 23:04.860 the line that? Means, thickness of the strip line, if you know that people can find out 23:04.860 --> 23:07.159 from these formulas. 23:07.159 --> 23:15.070 Now, another popular thing as I said that in strip line the problem is it is sandwiched 23:15.070 --> 23:21.759 between two they are sandwiched or embedded inside the dielectric, its fabrication is 23:21.759 --> 23:27.249 a bit difficult. So, people have thought that there is a line of symmetry about the strip 23:27.249 --> 23:33.730 line. So, if you remove the top dielectric and the top ground plane then it becomes that 23:33.730 --> 23:43.299 you have a ground plane, you have a dielectric on that there is a metal thing. So, you always 23:43.299 --> 23:50.149 put that metal, metalize some portion of the dielectric and that is very easy to fabricate. 23:50.149 --> 23:56.110 So, that is why that is called micro strip, it is very popular as a transmission line 23:56.110 --> 24:02.730 is popular. It also can radiate and behave as an antenna we will see here only the transmission 24:02.730 --> 24:09.100 line part. So, this micro strip it looks like this. That means it has the ground plane is 24:09.100 --> 24:19.460 the metal and then on the top there is a metallization not extending fully it is partly extended 24:19.460 --> 24:27.520 and between that it carries the power and between the 2 metals it is the dielectric 24:27.520 --> 24:34.539 that is taking power that is epsilon r is the dielectric constant of the dielectric 24:34.539 --> 24:45.950 d is the you can see with our the height and depth of the thing W is the width etcetera. 24:45.950 --> 24:53.999 Field lines if you see that definitely this is the just like micro strip lines fields 24:53.999 --> 24:58.429 from the center conductor which is at you can think now this is the central conductor 24:58.429 --> 25:06.350 this is the outer conductor. But one thing is it is not a coax or strip line that stops 25:06.350 --> 25:13.821 the central conductor is not central to the outer conductor, here you can say this can 25:13.821 --> 25:24.629 be somewhat like a 2 conductor, but 1 conductor is central another we are calling outer. So, 25:24.629 --> 25:30.200 field lines will come from here because, these is that this two are not at same potential, 25:30.200 --> 25:36.950 generally this is grounded and this is at a potential. So, field lines will come here 25:36.950 --> 25:45.870 and this side is here. So, some field lines will also go to here; but obviously due to 25:45.870 --> 25:53.830 dielectric constant of this dielectric by choosing it to bit higher most of the filed 25:53.830 --> 26:00.250 lines can be confined inside this dielectric, but some will definitely go to air. 26:00.250 --> 26:07.000 Now, these are the n H lines as you see. So, its fabrication is again photo lithography 26:07.000 --> 26:14.710 easily integrable, it is already said it is not new. 26:14.710 --> 26:22.070 Now, the point is there is a discontinuity. Unlike strip line you see we are creating 26:22.070 --> 26:33.730 a discontinuity medium that we have a dielectric here we are making it here dielectric here. 26:33.730 --> 26:43.970 So, suppose for argument sake let us assume that there is TEM wave propagation. So, TEM 26:43.970 --> 26:51.489 wave is propagating here and its inside this dielectric is velocity, phase velocity is 26:51.489 --> 27:05.590 given by c it is this c means that e m waves the velocity of light and that divided by 27:05.590 --> 27:11.111 square root of epsilon r, but when it will got to here its v p will be simply c. 27:11.111 --> 27:22.169 Now, these two at interface the 2 phase velocities cannot match. So, definitely since we have 27:22.169 --> 27:30.389 a discontinuity of medium here we this structure cannot support a TEM wave. Strip line was 27:30.389 --> 27:36.879 fully the whole thing was inside dielectric. The central conductor was fully inside dielectric 27:36.879 --> 27:43.539 that is why TEM wave there is no problem both sides seen same surface velocity in both the 27:43.539 --> 27:50.700 sides could have matched. But here TEM not possible and obviously due to this the field 27:50.700 --> 28:00.340 structure will be a hybrid of TE and TM modes But definitely that analysis is not so simple. 28:00.340 --> 28:08.789 So, people have come out that if we make and particularly the substrate is made very thin; 28:08.789 --> 28:18.179 that means, if we make this very, very less than lambda people will say that you have, 28:18.179 --> 28:26.840 since you have a very thin layer of dielectric though there is discontinuity here ignore 28:26.840 --> 28:35.019 that. So, that is the field lines it has been seen that the field lines also behave like 28:35.019 --> 28:41.179 you can see that more or less if you have very thin layer and by good amount of dielectric 28:41.179 --> 28:47.190 here then the strip line, field lines and these field lines are almost same in this 28:47.190 --> 28:55.020 region not above (Refer Time: 28.49). So, that people say that this also is a quasi 28:55.020 --> 29:02.590 TEM field similar to t e m. So, whatever we do for TEM analysis we will do that; that 29:02.590 --> 29:11.820 means we will apply the Laplace’s equation. So, this I said somewhere hybrid needed to 29:11.820 --> 29:18.769 support them. Now, quasi TEM approximation for thin substrate 29:18.769 --> 29:27.409 and people find out an effective dielectric constant. Effective dielectric constant because 29:27.409 --> 29:35.789 as we said that there is a discontinuity in the medium, but people say so that total field 29:35.789 --> 29:41.859 lines etcetera are in an equivalent simply assume that instead of two dielectric c r 29:41.859 --> 29:48.941 and that a there is an effective dielectric homogeneously distributed; obviously, that 29:48.941 --> 29:50.210 effective. 29:50.210 --> 29:56.649 So, that is attributed effective dielectric constant, it value will be something between 29:56.649 --> 30:04.289 one and epsilon r. There are various formulas people have come out with by based on the 30:04.289 --> 30:13.909 certain some empirical reasoning the values this epsilon, epsilon effective that can be 30:13.909 --> 30:21.239 found out then the entire structure becomes homogenous, the field becomes similar to TEM, 30:21.239 --> 30:30.049 quasi TEM. And this epsilon effective is a function of d by separation of the height 30:30.049 --> 30:37.649 of the micro strip from the ground plane and width of the micro strip. These are for TEM 30:37.649 --> 30:44.600 waves we can write beta will be something like this and v p is something like this. 30:44.600 --> 30:50.970 So, equivalent geometry of the quasi TEM micro strip line you see here also after some distance 30:50.970 --> 30:58.760 we will terminate this up to electrical walls. This is our micro strip patch and this is 30:58.760 --> 31:06.320 the same. So, this is the equivalent thing that here we have a discontinuity, here we 31:06.320 --> 31:13.389 say that throughout we have a new dielectric. So, there is no discontinuity between a epsilon 31:13.389 --> 31:20.039 r and a so everything is inside this. So, this is something like a strip line sort of 31:20.039 --> 31:25.710 structure, but here there are no ground planes, but this structure that I have a whole thing 31:25.710 --> 31:27.820 surrounded. 31:27.820 --> 31:37.970 So, and that effective dielectric constant various formulas people have found and compared. 31:37.970 --> 31:44.999 And like in some book I think it is book in this equation they have given some effective 31:44.999 --> 31:51.559 dielectric constant values. So, if you compare that with the actual dielectric constant calculated 31:51.559 --> 31:57.570 by a CAD program you see that almost they are for almost practical cases can be said 31:57.570 --> 32:02.869 to be same. So, those formulas are quite good, empirical formulas. 32:02.869 --> 32:09.440 Now, approximate analysis follows like what we have done for the strip lines. You put 32:09.440 --> 32:16.639 Laplace’s equation inside this confined zone. So, again there will be this boundary 32:16.639 --> 32:22.919 condition that in the side walls the potential function should go to 0 and the top and bottom 32:22.919 --> 32:30.110 potential function goes to 0. So, again there is that is this micro strip will have some 32:30.110 --> 32:33.850 charged distribution. So, there will be a charged discontinuity. 32:33.850 --> 32:43.840 Also, there will be the discontinuity, displacement vector. So, from that you can find out by 32:43.840 --> 32:49.309 Gauss’s law what are the surface charges? From surface charges you go to the charge, 32:49.309 --> 32:56.139 total charge. And also from solution of this potential function by applying boundary condition 32:56.139 --> 33:02.700 you find out what is the potential function from that you find out what is the voltage 33:02.700 --> 33:09.789 between this center conductor and the ground and then you find out the capacitance and 33:09.789 --> 33:12.850 you find out the characteristic impedance. 33:12.850 --> 33:26.099 So, that is shown here. So, finally, you get this is the value for capacitance with quite 33:26.099 --> 33:31.929 tough looking lot of expressions. But the idea that is why you try to understand that 33:31.929 --> 33:39.179 idea is simple I try to show it, value is definitely no one expects you to remember 33:39.179 --> 33:41.409 this. 33:41.409 --> 33:51.539 And now the empirical formulas that an effective should be put use. You see that depends on; 33:51.539 --> 33:58.350 obviously, the dielectric constant of the micro strip, but also on this d by w ratio. 33:58.350 --> 34:08.220 This d by w ratio is important or d by w or w by d. So, depending on whether it is less 34:08.220 --> 34:15.609 than 1; that means, w is less than d w greater than d less than that the whole formula that 34:15.609 --> 34:20.530 determines these are the empirical formulas that characteristic impedance will depend 34:20.530 --> 34:26.090 on this w by d and this. 34:26.090 --> 34:36.220 So then to find out for design purposes when z not is given and epsilon r is known, you 34:36.220 --> 34:42.730 need to determine what is w or w by d then d is known then people will ultimately find 34:42.730 --> 34:48.230 w. So, people have inverted that this is the investigation where these are again constant 34:48.230 --> 34:50.179 etcetera. 34:50.179 --> 34:58.610 And alpha d the attenuation constant due to dielectric clause is given by this and here 34:58.610 --> 35:07.180 this micro strip people create define a filling factor. That means how the fields are part 35:07.180 --> 35:13.790 linear and partly in dielectric that you have seen. So, how much they are filling the homogeneous 35:13.790 --> 35:21.700 thing that is given by this filling factor and the conduction part is simpler to understand 35:21.700 --> 35:36.100 we pick the and that also. And r is surface resistance, surface resistance of this conductor 35:36.100 --> 35:42.690 that is given by this. So, once you know the conductivity of the material you can find 35:42.690 --> 35:55.530 that is and. So, losses etcetera micro strip that also now people have paying attention 35:55.530 --> 36:00.240 and micro strip. 36:00.240 --> 36:08.270 Some common substrate material that is used for micro strip is PTFE glass. So, you can 36:08.270 --> 36:14.760 see and tan delta it is important because we have seen that it directly affects the 36:14.760 --> 36:24.060 dielectric loss. So, people try to go for better and better low dielectric loss things. 36:24.060 --> 36:34.330 So, RT Duroid is one such thing 0.001 loss. Then these are things but these also gives 36:34.330 --> 36:43.270 you epsilon r around 2 3 so that is also designable. So, these are planar lines, their analysis 36:43.270 --> 36:51.600 is almost quasi TEM or TEM analysis and they are not under heavily use. But one thing is 36:51.600 --> 36:56.810 they cannot carry much power. So, these are for low power application, but for high power 36:56.810 --> 37:02.890 applications you will have to still use that web guide or metallic things which can carry 37:02.890 --> 37:03.609 much power. Thank You.