How to Design and Simulate PCB Antenna

I had no idea we can actually simulate antennas in Matlab you're not the only one lots of people don't know that you can do electromagnetic analysis with matrub it's a fairly new capability what they need to add or birthday where they can start this where can they start good good good good um so they install Matlab the install antenna toolbox and the place where I always recommend to start is from the apps and so I'm gonna open up this the this this panel here and you can see that I have all the products from MathWorks and if you browse down you see here in the area of signal processing and communication you have the the antenna designer app the array designer app you have the machine Network designer app the PCB designer app talking about antennas so you have lots of lots of different features I put a star here so they are pinned at the top of my tool strip so the place where I would like to start is always from the antenna designer app it's um it's an app so it's interactive it's point-and-click and of course it opens up on my other screen so bear with me for a second I put it here how does it work here you essentially you start with a new session in this case we open up a new session and you have available a catalog of antennas that are sort of predefined so you see you have a horn antenna so you have cone antennas type of dipoles sorry you have a different monopole patch antennas even dielectric resonator antenna spirals all sort of different options um when we are going to use PCB antenna is that like special container for PCB antenna or um we will use PCB antennas but later not from the catalog because the PCB antenna is not predefined in Geometry it's a custom maintenance so I will show you how to start with the predefined shape and then modify the shape to create your own PCB antenna so in this case you know we took inspiration from um uh this specific antenna from from the eye from this application node so we will start with a very simple inverted F antenna coplanar inverted F antenna and then we will first design it in free space then we will put it on a dielectric substrate and then we will Meander the radiator to shrink the electrical dimension of the antenna and as well essentially make it more compact essentially increase the the electrical lengths of them so if someone needs to use an antenna for their designs then these are exactly the steps what they would be doing to be sure the antenna on their PCP is going to work properly and they can also check the all the fields and parameters of the antenna during the simulations so what will be the result what will be the results um so in this case for example let me just pull it up this is our Meander antenna inverted F uh Meander antenna and for example what we can get is the pattern of the antenna at the resonant frequency the S parameters the impedance uh different with the um charge distribution for example the current distribution so all these properties can be analyzed using electromagnetic analysis on your specific antenna this one also like this is just maybe side question do you have also like 3D viewer of the field or can you rotate this yeah absolutely there you go oh wow oh cool also PCB is rotating this is really this is really nice because very often I see this field and I have no idea how it is related to the PCB and right this is really cool right and you can also you can you have different type of options you can also overlay your antenna on top of the pattern you see it here so essentially this is gonna it's quite small because we shrink it with the dielectric but you can essentially play around with different options essentially to show the PCB and the pattern you could also cut the pattern through different at the Moodle or elevation Cuts essentially and look at the metrics actually so the PCB would be like in the exact center because it wasn't like pointing out from the field or or that would really look why is that because the feed point of this antenna is actually on this leg here if you see there where my mouse is pointing so with that respect the radiation happens along the radiating arm so it's not exactly in the center and we can also I think see the whole graph the whole graph we can see the S parameters we can look at the uh the S parameters or the impedance so I haven't don't have the figure here but we can regenerate it um I can just look here at the S parameters for example so let me just I click now F9 I just selected my lines of code and I click F9 and then this one will execute in the background what we see is um I have the image here 11 frequency points over which we computed the S parameters so the electromagnetic analysis is running on the Fly and it's Computing and solving the structure is meshing the structure so it applies um a planar mesh on the metal and the volumetric mesh on the on the dielectric I think maybe we should explain what does it mean as parameters I I created a video about this but uh some people they may not really know what s parameters mean and how they can be useful basically this means that uh in this 2.55 all the energy goes out basically that's what is happening that's why the uh it's why the graph goes so low um so s parameters is um s stands for scattering parameters and it's uh essentially a fancy way to look at the impedance of the antenna let's call it like that let's let's try I'm just going to do this I'm gonna copy this I'm gonna put it here and instead of calling these parameters I'm going to type in impedance and it's going to be super fast because essentially we already analyzed the structure and what do we see we see two traces the reactants and the resistance so the reactance is the imaginary part of the impedance the resistance is the real part of the impedance and what you see here is that the 255 gigahertz our reactances crosses zero zero reactance means essentially that the structure is resonant which is great because an antenna is a piece of metal that one is resonant irradiates energy and if you look at the impedance we did uh made a fantastic job straight on 50 ohm no needs for impedance man impedance or matching Network this antenna is uh is well done actually kudos to TI that did a good design now the question is okay I would like to design my own PCB antenna how can I do it here right so I'm gonna first of all I'm gonna clear my workspace so I'm gonna remove all my variables and close all my figures so that you can transmit that um they don't have I can say eases in my sleeves or things like that so let's uh clean up everything and then I'm gonna go to my um app the one that we were launching before the antenna designer app I have a question because I I used to work in Matlab like maybe 20 years ago so I have no idea what we what do we have here in this window not in this one the other one uh the this is the main matter of um uh environment when you launch Matlab this is what you find on the left hand side we have the current folder so I'm now in this particular folder these are all the files that are listed here um this is the command window so here is where you type in your matrub commands this is the workspace this is where your variables are defined and this is the editor essentially actually I'm going to close it because we are going to create a new one that's what we are going to create exactly exactly that was the end result this is our our I'm going to say the script or the programs that creates the antenna so actually what we will see is to design an antenna and then generate and create a script that you can re-execute over and over again again and and modified so that you can automate a lot of the steps for the design of them okay okay so now we can go back to the app let's go to the app okay so the we talked briefly about the catalog so we have all our antennas but let's say that you know nothing and you want to say give me an antenna that has a normally directional pattern so the radiates energy everywhere in space that is not directional so you can imagine already intuitively the whole antenna will be directional a dipole will be omnidirectional and in fact if I open here all the horn the aperture antennas of the horn antennas are grayed out so you cannot select them because they're not omnidirectional and if you browse down you find other antennas that are on Mid Direction so the inverted F antenna is omnidirectional because if you remember the radiation pattern had a shape of a sort of a donut so it sends the power all around and so I'm going to select my inverted antenna and now um the first question is of course on what could be what should be the geometric properties of this antenna so how long and large each pieces of of this antenna should be to make it internal resonant at a specific frequency so in our case we are working at 2.4 gigahertz so I just type here 2400 megahertz and when I click on accept the geometry of this antenna gets automatically scaled to be resonant at the gradient frequency wow nice I love this feature because honestly I don't remember all the rules of thumb that makes the antenna resonant and by the way this is not no electromagnetic analysis is done at this point in time this is really Gathering all the knowledge of the antenna designers over the years we give you this rapid calculation to make the antenna resonant so let's verify this so I'm gonna plot the impedance of this antenna in or over this frequency range this is very fast the antenna is a perfect electrical conductor I know it because you see it here that the metal is p e c so essentially it doesn't have any any losses and atomic losses um and um so it's in free space it's a pure piece of metal in free space and we can see that indeed the reactance is zero 2.4 and the resistance is not bad it's something we can verify it I think it's around the 70 ohm or something like that 67 ohm there you go okay so the antenna is resonant we can verify again yes parameters we can see what is the bandwidth in general the boundaries of a of an inverted F I believe is around seven percent but in this case we can count it so you can see here this is 50 Mega 100 megahertz 150 megahertz I'm looking at them where the S11 crosses minus 10 DB which means that it's essentially well matched so if I and now I'll show you the my favorite usage of Matlab so if I do 150 megahertz let's say divided by 2.4 gigahertz this gives me six percent of boundaries approximately ballpark which is pretty much what I was expecting or what the literature gives you um okay let's go back to our things other things that we can look at is for example the current distribution so the surface property of the antenna where most of the essentially which is which is not surprising because the the inverted F antenna you can actually see it as um a quarter wave monopole bent over and so this is where the excitation comes over but essentially this is a quarter wave monophone and this is the ground plane over which the excitation is so what does it mean a quarter wave monopole is essentially a it is so when you have a dipole essentially you have Alpha wave you can take a half of um of the length of the dipole and put it on a ground plane and you get a monopole essentially because the ground plane reflects the energy and this does the same only that this planner instead of being sticking up okay yeah so it means on the left there is basically some kind of like uh zero or the lowest current and correct that's the maximum correct correct and this extra leg that you see here this is actually a short to the ground plane and this is useful to improve the impedance matching of our quarter wave monopole then there's one way at least in which I thought it you can intuitively look at an inverted F antenna okay what else can we do we can also look at the 3D pattern and we will see our donut so here the gain is a 3.9 the the DBI actually we're not looking at the game we are looking at the directivity because there are no losses because it's perfectly electrical conductor no um no dielectrics so it's good all the energy that we give gets radiated the ones that doesn't get reflected of course and then what else and then we can look for example uh I'm curious rotating yeah yeah let's rotate it because this looks a little bit different from what we've seen in the beginning right there is this right there is a little bit of a bump uh it is actually smaller in power than than our big donut let's put let's say let's say it but we will see how this big bumper changes when you start um putting the antenna on a dielectric as well as adding the Meander so effectively you are changing the design of the antenna so the real F antenna like this one will have also this bump or directing like uh ahead of the PCB no no it gets smaller actually okay there is the there is still the bump but it's a little bit smaller so this is cause only because this is all like uh perfect antenna that's fine okay exactly exactly and we can actually look at the a different cuts of the pattern as well I find it quite cool and so for example we can look at the at the mood cut and then we can put but here is no pcbc and I have no idea what directions we are looking at right right but we can still do something cool we can still look at the antenna Matrix so for example we can look at the what is the main logo what is the the frontal backlash uh front to back ratio the side lobe level the the the beam width and so forth so it's it's it's it's cool I always go back to this one because this one gives me what is atom with zero and elevation zero as a reference point and then I go back here and then I say oh yeah okay see where I am okay in the in the in the block of things that that you can do um let me just do one thing and I want to simplify a little bit what we are seeing I just want to put the four plots now on our um uh on our um on our screen let me make it a little bit bigger and then what I always like here and I think what um tell me you can you can adjust the numbers and you will see all the changes there exactly sensitivity analysis that's essentially what any engineer would do is uh go there and say if I change a little bit this number what happens okay is my design going crazy or is it still doable um so all of these here are the geometric properties so you see it's a catalog it's an antenna from the catalog because the shape is predefined but all the geometric properties can be tuned so let's say that I want to increase the width of the radiator arm and the feeder arm and you might ask me what are these things how do I know what properties these are related to well of course you can change it you can try it and if you have good eyes you will see the difference that this one has gone thicker and this one has gone thicker but of course we also have of course the documentation so I can pull up the Matlab documentation uh I'm gonna add it here and for example I can go to the antenna catalog these are all our antennas and for example let me look for patch antenna this is all our catalog and I'm gonna have and I'll do it nothing where are my inverted antennas I don't see them so what do I do I go here and I say inverted f that's probably the easier way to use the matter of documentation always search and then here oh this is an inverted F it's not a coplanar investment so let's go for planner in the third F okay that's it invert the desk planner I don't remember exactly how it is called but here you can see that there are all the geometric properties so this one tells me that this is the height of the inverted f um this one is the length to the open uh yeah I can see it I can see all of the lengths of the shorts and then you have all the width and the ground plane size so you have all the properties and I can say mapping between the naming of the properties to the geometric properties of the actual antenna so what happened here when I increased the thickness of our radiator and feeder arm I increased it by 50 percent let me just apply the 1.5 Factor so you see that essentially the resonance point moved a little bit to the left let's move it a little bit more let's put a two percent let's increase it by 100 percent let's see what happens as soon as I say okay you will see the resonance moving to lower frequencies essentially we are increasing the capacitive coupling between the radiator and the ground plane um but this is also I can do the other way around I can make it thinner and see if the resonance frequency moves up as I would expect and we can try around you see that this gives you a good indication of what what matters essentially okay so we know about the radiator arm and the feeder arm that essentially that's what fits our resonance the question can you tweak it some kind of automatically oh yes we will see it a little bit later so you can see here that we have an optimized tab so in the optimization Tab and we'll talk a little bit later about this you can give a bounce to your variables and say between I don't know 0.1 and 10 or whatever you like and then there is an optimization algorithm that will optimize especially specific objective function okay function would be the gain or the impedance or different properties so so my next question would be like we have this uh theoretical antenna but we still have not included uh PCB for example because there will be influence from the PCB material so how do we do that how do we do that I tell you what's my favorite way of doing it um you could for certain antennas that are in the catalog you could put a reflector and then add a dielectric in between the reflector and your catalog and your antenna essentially but in this case we don't have a reflector so what I do what I would recommend is to export this antenna design as a script now that I have a script um I can essentially modify or everything in my antenna step by step so let me just do this so I'm gonna remove this one and put it here at the bottom because it's going to be helpful later on so here you see what do I do I let me remove a little bit of the commands that are helpful but I just want to have the code a little bit more compact here okay so uh this code was now automatically generated from what we write from the design what we have just created yep okay and this is the antenna described yep so I'm gonna say for example I like to have it parameterized so I say F0 is my Center frequency I say 2.49 in that way it becomes parameterized so if I want to redesign my antenna at a different frequency I can redesign design is the same function that we actually invoked at the beginning when we scale the antenna so if I execute this piece of code what I will get is my antenna with the default properties to be resonant at let's put it here on the side so that we don't we don't lose it anymore um let's simplify things here as well so the antenna analysis is curious how you generate the code and then I end up a removing half of it so I want a lean space I wanted between 2.2 gigahertz maybe and 2.6 gigahertz so this will speed up the simulation this one is is the range not necessarily this is the the range of frequencies over which I want to analyze my antenna so if I want to have more points then the analysis will be slower yeah so this will speed up the simulations a little bit because we are only looking at the place what we need exactly so here we look at the impedance here we look at the s-parameters the reference impedance is always 50 ohms so I can remove it and then and let me do this and what I get here on my other screen is actually a bunch of figures so let me dock it so I get the impedance 2.4 gigahertz resonant DS parameters a good match at 50 ohm and then we also get oh this one made a mistake I have zero all right I have a question again I have a question uh so you run this uh kind of command like design inverted F coplanar and then figure will draw the picture figure will open a new figure and then you can show antenna object that's what draws the picture exactly okay I understand and if you wonder what can you do with an antenna object um what can you do in general I say what are the methods that I can apply to my antenna object and then it tells me all the things that I can compute so you see we have as parameters we have pattern this is one of the ones that we just used we have impedance we can also look at current and we can look at the efficiency of the antenna at the enh field so if you want to look at the near field or if you have antennas that are couples actually you can look at the mesh of the antenna if it gets analyzed you can look at how much memory more or less it will take to analyze this is a very simple antenna you see how quick it is and by the way my computer is five years old so it's really end of life it's not a super computer just to be very very honest um so these are the type of things that you can do with antenna objects and and you can all the elements of the catalog supports this method and then you can essentially visualize it or compute different properties here so and again I see then uh then you have like frequency range line space figure impedance right you don't have their show line again because impedance if I just execute impedance it will draw impedance in the last few years that you have opened the river okay I understand now and then yeah so so whenever I type in figure I say give me a new figure and then I get a new figure here if you go it's named the tab is named figure so that's basically the empty correct okay correct correct and um but I haven't answered yet your question that is uh I just repeated everything that we did from the app from the command line but we haven't put this in a PC okay so let's put it on a PCB so what I'm gonna do is I'm gonna create a new object that is called PCB ant this is my name and we can we can give it a for Robert or whatever whatever name you prefer and I'm gonna say PCB stock and I'm gonna pass in here my antenna object and I'm gonna do again the show PCB and okay um so let's execute this to code this these two lines and what do we say um nothing has changed again so if I look at them what is it sorry I browse in the wrong direction if I look at my figure here PCB and it looks the same as it was before but if I look at the object the object you see it it's here is printed here the PCB and object is a different object it's a PCB stack so I sort of loaded my geometry on a PCB stack and this geometry got imported as the single layer so it's a single PCB layer and this layer one is our antenna polygon so it's a polygon that has all these properties here um the name it inherits the name from the antenna type it has a certain board shape that is a rectangle that includes the there's a boundary that is larger than our antenna it doesn't have vs it it has a feed V it is a feed on the feed of the our inverted F antenna but it was it has not dielectrics essentially because it only has one layer and the conductor is is PEC so if you do PCB and daughter and this is very helpful with matrub you can always use them tab completion so you can go to the elements so it tells me that the conductor is PEC so we can change it for example to Copper or something else we will do it in a second actually but first let's put the antenna on top of that electric okay let's do it right away yeah okay so what unless you have a question no no this was exactly my question like I would like to correctly arrange the ground in in the antenna in the PCB right right right right right right so I'm going to say D1 is going to be a dielectric again top completion mat raise your mind sometimes sometimes it reads a little bit my mind a little bit too much we have different options I'm going to choose dielectric fr4 I'm going to say let me execute D1 so D1 is a dielectric fr4 with a certain Epsilon R is a certain nostalgent and a certain thickness and by the way we have a catalog of dielectrics we do like catalogs and you can see that these are all the dielectric materials that are supported out of the box so this is the name it's an air of course in the default but you have fr4s you have different Rogers material and if you want to Define your own dielectric you just have to specify what is the relative permittivity and the loss tangent and there you go um you can always rewrite the uh the values yeah okay yep yeah so let's say that I want a D1 dot Epsilon R to B 4.5 okay maybe we can have a look into the Texas instrument document um we need to ideally we would like to find the stack up what they use for the reference board right right so here there is the the reference design here and if you browse down so there is a ZIP file there you go no not this one the other one I think about this one sorry the driver rev C I think and this is a ZIP file that if we open it up then we find this the extension text description with the thickness of the PCB so it's a the layers are either 0.25 or 0.5 millimeter okay and the copper is 35 micrometers okay so let's use these numbers so we need to use 0.25 millimeter thickness of the the electrical yeah and and the 35 micrometers for the for the copper and so they have it like for layer PCB uh I think that the PCB is a little bit more complicated than just the antenna in the sense that they have other stuff put on that you can actually see so maybe it's uh plus uh 0.25 plus 0.5 plus 0.25 I think that that's the bottom layer for the the for the dielectric that you put below the ground plane oh okay so the ground plane is on Layer Two that's why we would like to use 0.25 correct okay but let's let's go back to our antenna so let's say that we put this one is all on top it's all on top so we can just sum them all so zero twenty five zero five zero twenty five so we can have a one millimeter of thickness for our dielectric okay to start with and then we want to say that PCB and Dot layers okay I understand so we put that the thickness of the whole PCB but we put the second layer 0.25 millimeter correct correct so we have just one single slab of dielectric right now that represents all the dielectric layers because now right now we were putting in our antenna on top of it okay and let's see what happens here when as I do this I execute it this tells me a couple of things that the electric thickness is updated with the board thickness so this is not good I need to change the board thickness of our PCB stack so let me just go here and say PCB and dot board thickness we're gonna set it to the same one e minus three okay in that way we can have a board with multiple layers of the electric and each dielectric can have a different thickness but sometimes we use semicolon and sometimes not it's very simple if I don't use semicolon when I execute a line the output of the line comes in the command window okay if I put semicolon you don't see the end of it so okay straightforward there what else can we do the other thing that we can do is I'm sorry again for interacting how did you specify where the second layer is D1 that's I put I said so we know that the layer one it's our metal inverted antenna now we put D1 in place of Layer Two so if I now look at pcbient you will see that now it has two layers the top layer is a metal polygon but we in our shape and the second layer is the dielectric so now we have a PCB stock with two layers and what I want to do now is to trim a little bit of PCB board so that the borders coincide with our ground play so and this is a I like it because it also gives you an idea on how you can start modifying the the structure start adding a different type of geometric shapes for example so how I'm gonna do it I want to modify what I want to modify the board shape and so I'm going to say PCB and dot board shape is going to be a rectangle so I'm going to say it is a rectangle and the rectangle length is a rectangle length is equal to our antenna object daughter ground plane length why did you use antenna dot rectangle it because um it's a way to define the geometries so in the antenna toolbox provides geometry so you have to use okay you have to use the antenna toolbox to generate it okay I understand right because then you can apply Boolean operations so if you want to sum two rectangles um essentially it does the The Logical sum of the two if you want to do the intersection because these are shapes that come from the antenna package you recognize that there are certain operators that can be applied to that um what else do we do we'd want the board shape width to be the antenna object a DOT ground plane not length but width but we also want to add the height of the antenna so we're going to say antenna object dot height and I'm gonna do it two times because I want a little bit of space on um two times this is cool because uh basically you really will be always simulating the real PCB because if you don't combine these parameters together then if you change the PCB size then you would not be changing the antenna ground plane but we have to do it because then use it exactly exactly you can always keep it synchronized you can always if you synchronize your your PC beam to your antenna shape now what's the problem now I need to offset a little bit background plane so I'm gonna add one more parameter that is the center and I don't I'm not going to apply any offset on the X Direction I want it centered but I'm going to apply one offset on the um on the Y Direction so I'm going to move it up by height and you see how I I keep on essentially modifying my PCB looking at it changing it playing around and this is very Interactive but the good news is that after I have a script and if I want to change it everything is parametrized you make it look so simple but how do you know you know all these parameters which you can put inside of the rectangle is like because we looked at it before so this is our ground plane with our ground plane length and our height of our work okay so we are using the parameters from the antenna design exactly exactly so when you look at the geometry when you look a little bit of the sensitivity analysis you get acquainted with what are the geometric shapes that matter and you you now use the geometric properties of the antenna to Define what is our PCB okay still we have all the copper we have on one layer we need to separate right so let's let's do let's put a let's do the top layer to be called okay okay let's do that uh so I don't remember anymore how to do it so I do PCB and so pcb1 says that there is a conductor a PCB and Dot conductor is PEC so I'm gonna do I'm gonna Define um T1 let's call it let me do it actually let me do it here from the command line C1 is equal to the con duck does it work doesn't work okay let's look at the help what does it say so I go back to Matlab I go back to uh let me just go here signal processing and I want to find my antenna toolbox okay not signal processing but under Arafat mixed signal of course antenna toolbox material catalog okay metal catalog oh this looks about right so it's not conductor it's metal [Music] and so I'm gonna say C1 is metal material okay so it wasn't conductor this was a bit because it was metal and I want to have copper let's give it a try if it works yes it works specified in the materials correct correct is one of the material catalog and now I want to change C1 dot thickness because we want to use the same number that we saw in the reference design it was I think 35 micrometers we had it there you go so we put it here on the side the thickness is a 35e minus 6. I updated that's the right value now I need to do something else I need to make sure that this is part of my script so I go back here I go back there see one meta copper and C1 thickness listen and let's do it again oh I made a mistake I should Define first the copper and then the thickness and then I need to assign it so I'm going to say PCB ant dot conductor is equal to C1 because I created C1 but I didn't assign it okay now I'll re-run it and what do I see I see it's slightly darker you see also here in the legend that it says that is copper so looks about right there is nothing on the bottom this is our PCB good shall we analyze it now but in real PCB the ground would be on a little bit deeper now and the chip would be on the top and the antenna feed would be correct correct but before we do that I would say let's analyze it because the dielectric will change the properties okay it will shorten the wavelength but we don't know how much is going to shorten the wavelength so let's look if the resonance move because of the dielectric okay um so let's do this and I'm gonna do I'm gonna go back actually I'm simply gonna copy this one as it is I'm just interested in looking at the impedance um I'm Gonna Keep it like this I'm moving it down there putting it here and I know that the rest I know that essentially the wavelength will become shorter uh wavelength will become I know that the frequency will go down essentially the resonance frequency will go down so let me plot it essentially up to I don't know 2.4 gig or something like that and see where we go so now we get the S parameters oh let me just do 0.5 this and we need to analyze now our PCB ant not our antenna object but our PCB antenna over this okay let me just run this analysis and see where we are going 40.5 okay so I get the figure here the analysis takes a little bit longer now because we have a thicker dielectric we still have 21 points so we need to remember maybe next time to reduce the number of points so we speed up the analysis and we make it a little bit faster but let's see what is our where is our impedance gone there we go oh it's gone very low right away we're very low I mean let's let's go we start from 1.6 Maybe and we can go to definitely 2.3 something like that would be to be 11 points let's reanalyze it and let's see where we are going so now we have 11 points for the analysis what is it there we go so we are now resonant exactly at two gigahertz so we were resonant at 2.4 now we are resonant at two so now we can scale all the geometry of our antenna by the ratio of these two numbers so let's do that I would say what do you think I don't know how you are going to tweak it back to 2.4 so what I'm gonna do I'm going to start with our antenna object that we started with and I'm gonna say antenna object Dot and then go through go through all the properties of this antenna object so what do I have I have the radiator arm width I'm gonna say the disk we want to scale it sorry I'm gonna scale it by how much and I'm gonna do the ratio between 2 divided by 2.4 this is what I'm going to scale it okay and would not this be better use the optimize feature you can you can definitely but in the end I'm not sure that it takes more time or less time in the sense that the optimization is something that will explore all possibilities well in this case what I'm simply doing it here is if I do this and divide by scale I do have also with specified down there in the next line how you are going to delete it okay yeah yeah I just wanted to show what I'm gonna do it and actually to save you a little bit of time I can just go and copy what I've already done so you can see it here okay and I scale them all like this this is just to save a little bit of editing okay perfect so you figure out how to actually adjust the uh geometry of the antenna to move the resonance frequency back to 2.4 especially I'm shrinking the antennas by this amount and if I say what how much is this I mean 20 percent so you made everything smaller by twenty percent okay and then the next part is just copy again what we have done and just repeat it okay so I'm just after I scale it and so I'm gonna let's design our antenna just to make sure that we start with we don't rescale it two million times so now we Scala we design our antenna we scale it down but our scaling Factor we put it on a PCB stack we look at it and let me see if we have an older figure here and let's do this I'm gonna put it x y and then I'm gonna go here and then I'm gonna put it X Y let's see if we were successful so you are running both and we are going to compare them right so this one was the older one and this is the newer one so you see that is a little bit smaller [Music] um 20 percent how do you see it just look at the so this one is uh this one is the old one that is you see here is 35 millimeter I think I think I see now 30 millimeters this one is 35 this is 30. so we went we lost a centimeter essentially but just putting it on the substrate let me put back where it was and let me put it down there good now we have Shrunk the antenna and now we can reanalyze it let's see how we are doing the only thing is that now we are back to the frequency range that is around 2.4 gigahertz hopefully so let me put it to 0.5 and 2.3 something like that and let's see how we are doing it's kind of exciting because every time you want to check I mean am I doing the right things every step every step you check it you modify it you check it you modified actually I would say you modified you check it but I realize that 2.36 gigahertz which is actually not too bad for just the first possible or something now we could apply an optimizer and by the way look also how good the impedance is very very close to 250 ohm this is actually from our application node from TI not our but from our application node from TI and here essentially this is an inverted F this is exactly the same structure that we have worked with you see with the Via to the ground and the feed point however the radiator arm is not straight but it is meandered I'm not gonna do it programmatically because otherwise I literally you were worried so I can see that you were worried so I'm gonna do it with another app and so you might have noticed here we have a PCB antenna you can draw it okay I was worried that if if you don't have anything to draw the right right right I was gonna make you sweat no no you can you can draw it and it's better than that we can draw it using all the parameters that the ti application note gives us so that we can draw it and at the same time um draw it in a parametrized way so what can we do let me just do something for my convenience do you have something uh so simple also for the stack up or Stack Up has to be done programmatically a stack up can be done here in the app good so what do we see here you have the first thing you have a actually now you have an empty session here you define your stock up so you define your board shape and then you will add layers you can add a metal and dielectric layers and you can Define the material properties for the metal conductor and dielectrics and you can add variables and actually what I would recommend in this case is to start with the variables okay so here the application node from TI gives us this table here with L1 L2 L3 and all their meanings there is one thing that I when I looked at it I didn't quite like is that actually L3 can be expressed as the sum of W1 D5 W2 D6 and again W2 so what I uh what I can do is simply to type in all these values and then use a except for L3 that is a dependent variable okay and so I can at least start with that I'm going to put it on my other screen because it makes it a little bit easier for me to type in but here I can say I want to add the design variable it's going to be L1 and actually here you see the canvas settings are by defaulty millimeters um so I'm just gonna say L1 value is a 3.94 for example and then I'm going to add a second variable that is going to be L2 2.7 and and so forth um I have a done all this work already so if you don't mind I'm gonna load how to I'm just gonna load the state in which I just have the variables look prepared just to save the time of the editing um and so let's look at it so I still haven't done anything here there is nothing on my canvas okay but let me just collapse this one and you will see here these are my uh these are my variables and one two and four five two oh L3 okay L3 you don't have L3 okay three there you go this is the okay part so L3 let me just put back the PowerPoint L3 is equal to yeah and you see the value here that is five millimeter that is the same value that comes up here so being good and then we have uh form other variables I have lbs and WBA let's start with this wgp and lgp are the ground plane uh Dimensions from the PCB and actually wgp is a three centimeters on the length side and lgp is the sum of D1 um L3 yeah I see I see two times L5 uh two times L2 and D3 yeah so this is the width of my PCB so here I created my variables then what do I want to do I'm going to add the board shape so now let me maximize this one um oh and maybe a little bit I'm gonna draw a rectangle and this is gonna be my board shape so I call it board shape and what my board shape is gonna be it's gonna be length we have these two variables here and Bs length board shape wbn width of the board shape there we go Let's uh let's fit to view so this is gonna be and then we're gonna Center it so the way we want to center it at um essentially we want to put the center in the Middle with the X Direction so it's I'll be lbs divided by two that makes sense draw it but you can also parameterize everything and from the y direction I'm gonna say WBS divided by 2 . yes this is Center the vertical Center but I want to offset it by the ground plane w g d wgp is it right okay why is that because if I look at my diagram I want this point here essentially actually here this point here is going to be my coordinate 0 0. okay that's that's essentially here and that that will help me position it because everything is uh um all the dimensions are referenced to that point exactly so that's that's a that's a convenient way of looking at it okay so now I have my board shape and now we're gonna add here we can already start defining my layers it's going to be copper with thickness the 35e minus 3 because it was 35 Micron the units are millimeters so there you go and then I'm going to add a layer I'm going to add a metal layer that's good and I'm gonna I want the metal layer to be bright and yellow so that it reminds us of metal it can be any color nope uh what did we go here we go good now we have a metal layer what do we do on the metal layer once we are on the metal layer we start drawing shapes and I'm gonna draw let me make an example I'm going to draw a couple of shapes here like this and then I go another rectangle like that mm-hmm and then I go another rectangle like this okay and so this is how you build the antenna you just add the dimensions of this will be the parameters would be okay exactly so rectangle this is going to be my first rectangle um let's look at this so the first rectangle is going to be number one here it is W1 D in the L6 in terms of with a length and width so I'm gonna so that the length is going to be W1 and the width is going to be at 6. 6 okay I was a little bit generous with the dimension when I draw it by hand and then we have to put it um where so we want to put it with respect to our offset with the center let me do it like this it's always W one half the half of W1 that is the width to check the center Double one divided by two and then we want to have it plus D1 to have it okay the center of this rectangle here on the X on the x-axis plus D1 and on the y-axis I'm gonna have a L6 divided by 2 that is the center of our rectangle but we want to remove this little bit this is below the minus D4 is it right so let's look at it okay this is our first rectangle I guess you could still draw it directly on the canvas but that would not be like uh you can draw it by hand if you are a confident drawer let's put it like that but the point is that we want to have it parameterized in Geometry why because if we want to scale it down I just go and change W1 of W2 and then all the geometry will be automatically opening up and closing down depending on my design um I fast forward again if you don't mind or I can do a couple more rectangles but I mean I already prepared a session where I just drew all the I think there are 12 or 13 rectangles so here I just jump forward and we get just all the rectangles nicely nicely put together in good order and now we will see it in a second there you go so you see the the uh this was on this was my work today so what do we see here we have all the rectangles so you said rectangle one rectangle two rectangle three and you see the rectangle one is exactly what we did right now if I go back here we recognize the the the the position the coordinates that's all good now um we do something really cool so I just um we need to connect them somehow we need to connect them exactly so I'm gonna collapse this one I'm gonna pull down here I'm gonna to select all my rectangles and I'm gonna say just ah so now I'm gonna get a polygon that is going to be um sorry I didn't select it let's go to path okay so we're gonna get a polygon that has all the vertices of our rectangles and this is going to be our shape on the top layer and now let's let's see if what if I did a good job they say that I change W2 let's remember that it was 0.5 and I change it to 1. then automatically everything gets will get fatter perfect and 2.5 or I go back to the to the right number because you know I did it a bunch of times and then I got all sorts of results between here and the Sun or the W1 or we can do 1.5 or something like that and and you can see that how things are changing automatically also the board shape everything is nicely parametrized you spend a little bit of time with the parameterization but once you do this you can optimize your number because all of these are variables for optimization you know you know um when I was starting with 3D software like mechanical software I didn't parameterize anything and then when I wanted to change something I spent so much time like redoing stuff that I learned that always when I'm drawing something just use parameters because it will save you a lot of time in future right right now that's I think that we're all been there Robert it all started that way and then and still nowadays I said oh but just do five seconds and they do it on the Fly and I will never do it again and you know when you started isn't gonna work it's um it's that's it um so I don't remember anymore 0.9 I mean you see there I mean what was it sorry W1 0.9 yes I remember correctly good good good let me go back to the app okay what's next so now we have a only one layer that is our top metal layer and now we start adding a dielectric layer for example I have a dielectric layer one and I move it down and what properties so let's expand it let's move it a little bit down is it dielectric we want to have fr4 let's see but our Epsilon R was 4.5 so we change it to 4.5 and our thickness was um if I remember correctly 0.25 millimeters I think and we change the color because we we make it green green yeah that it looks a little bit better okay cool sorry and and now you see here so this is my dielectric layer that really doesn't look particularly exciting because there is nothing on it but if I look at my 3D view of my antenna this is um this is the whole PCB right so let me just try to try to rotate it there you go so you see my metal layer in yellow and my dielectric layer okay here and I'm gonna add another layer I'm gonna add another metal layer okay and then I'm going to add another dielectric layer you could also copy and paste the layers if you want to if you have structures that are very similar and that's also possible so let's start with the electric layer that is the easier one I'm gonna make it again green because it makes it a little bit easier to follow a darker green and we're gonna do it fr4 um 4.5 and the thickness was zero point um uh 75 so it's one millimeter in total so the total thickness is one millimeter right and then we have this metal layer where we need to add our ground plane and so let's uh let's do our ground plane so I'm gonna again put a rectangle that I can do it by hand if I feel uh if I feel good about it and then I can also stretch it up here I can also rotate you can also rotate shapes you can also as you can see you also have other shapes in this case everything is regular is a rectangular so it's fine but let's call it gnd and I have here a couple of variables that are helping me with the ground plane so I have the W the L the length of the ground plane and then I have the width the sorry the width of the ground plane that's good and then the offset is going to be half of it essentially so it's going to be the length of the ground plane divided by two so hopefully it will be nice and centered in the middle of our board and this one is gonna be a ground plane with divided by two but as we subtract it if you remember from the board shape the ground plane we just put a minus in front of it and there we go so now we have a our PCB board and just the Via and the and the feed Point yeah are missing exactly so at least now we have defined our stack up so we're good with the stock up so we can just um collapse the layers essentially that's those are essentially are done Let's uh put the other feed that's how the feed point it just puts it somewhere there [Music] where do we want the feed Point let's let's look back to our PowerPoint the feed point was here yeah sorry it's probably a little bit yeah it is yeah so the feed point is D1 plus W1 plus D5 plus a W2 half that's the X Direction and the y direction is for example minus minus very good so let's do that so let me do that so let's see if we remember what we said I'm looking I'm looking on my other screen when I have the where I have the the the the power point essentially so what do I have here on the PowerPoint we have a D1 D1 and then we have plus W1 plus D5 85 Plus W2 divided by 2 and the X Direction was minus D for half is it right yeah no not bad not bad the result something else here there is that we want to from a metal layer from the top metal layer oh I still have metal Layer Two here I don't like metal Layer Two I like ground so that I remember what we are talking about so feed one is now between a top metal layer and ground it's called feed one if hit voltage is one but I want to have a square and I want a diameter that is maybe W two maybe divided by two sorry is half of the so that is the diameter becomes half of the width of our strip okay and now we see it that here it's I don't know if you can see it but is right there yeah I can see yeah it's it's right there it's right there so our feed point is perfect about right now we also at the VM so let's add a VM same story as before um between top layer and the ground layer that's good and where is the location of our fee of our of the our via is D1 plus W1 divided by 2 and minus D4 divided by 2. okay that's easy to remember so it's um D1 plus W1 divided by 2 and this one is minus D4 divided by 2. is it right looks about right let's look this one and we do the width to be W1 maybe divided by four because W1 is a little bit thicker then W2 okay people look at it so I made it a little bit smaller so they are more or less the same dimension good I think that uh are we forgetting something let's check it we can validate our design okay everything passed it's good it did a good job so now what can we do now and now we need to export it I guess well we can analyze it oh you can analyze it directly here yeah I didn't know that so you can set okay I want to analyze it around 2.4 gigahertz but I want to I'm going to do lens space what was it 2.2 and the two point six gigahertz 11 points something like that let's set this is our frequency range as soon as I provide the frequency range I can look at the impedance let's see how it's going so it takes a little bit more time than before every time we're making things a little bit more complicated it takes a minute or so it's not too bad and then we see if we did a good job or if we messed up something hopefully it's good it's it's always good to do something but if I do something verify and check it out but you see how we really we built a bit of confidence in our design starting from zero where are we we are resident at 2.55 gigahertz there are two resonance here actually there is a parallel residence and but this one has an impedance that is really high so now we don't want to use this resonance frequency here because it will make it very very hard to match but this resonance here is fantastic because it's 50 ohm so it's really good the other thing that we can do is um lesson a check um you can look at impedance we can also look at the as parameters but we can also play around with the mesh and the mesh is a very important parameter because like we were saying before the mesh is what determines the quality of the results so right now you see here I have the the mesh that is automatically managed so I have a minimum Edge length so that is around five millimeters the maximum Edge length is 36 millimeters I can also go to a manual mesh so for example I can reduce it a little bit to half of it and this will give me an idea if if my results are accurate or not so you can make the mesh finer and finer this will take longer in the analysis results but at a certain point in time you can it doesn't make a difference anymore and that is a good point where to be where you want to be essentially where the mesh doesn't influence the results anymore now I knew because I did it before that the mesh was a bit coarse and we are off because of that okay a little bit it's actually not we will see there is a little bit of difference but it's not a massive difference so I refined a little bit the mesh and with the refined results of the mesh we actually reproduce exactly the same simulation results exactly gave us because for this current measure for the previous measurement the measure was maybe too big so you didn't get like exact uh exactly exactly especially on the on the on the borders essentially because there are you can see here that essentially because of the meander you have a lot of details that need to be captured by the Mansion uh before we had a much more linear design it was easier to analyze there is less coupling in between the antenna elements um so now with the with a more complex design very refine a little bit the mesh so you see that we are it moved a little bit but it's not a it's not a dramatic change I mean it's not that we were massively worse the resonance is still there we can look at the S parameters and we will see there is a nice Notch essentially that is pretty much the same notch that we find in our application node and it's like 2.52 and I what we are comparing against is in free space that is around 2.5 essentially I think that this is what we're looking at 2.5 and we are actually here around 2.5 I mean I don't have a lot of points but essentially yeah it can it can even go there because there is one point is 45 something yeah exactly so we are very really a little bit further away we can find a little bit and yes parameters but that's the idea now we export export as a matrav script uh so the I mean this is it so you see all the variables that automatically generated or the dependency the PCB stock and then all the geometric properties let me make it a little bit bigger of the rectangle so with our name and then they get summed up together and then they get you recognize all the syntax because it's exactly what we've done before so we added dielectric layer one the electric layer to the ground metal layer at the feed location the Via location um oh we didn't change the we did change it and so we did change it let me check it maybe we forgot that one oh you see we don't have the losses of the okay of the copper probably because I changed it but then I reloaded the 35e minus 3. and ground no this is there is only one metal okay okay for everything okay so we can re-update the plot and then we recompute it so as you see how the script helps you to double check things as well to find if there is something uh not right and then essentially we create our PCB stack putting everything together and then we are good actually the script is not as complicated as I would uh think it's readable yeah it's readable it's readable and it's uh human readable I would say because um frankly said we could also generate just an object that encapsulates all the properties but then it makes it a lot harder to modify it because then you always have index the Field properties of the object well in this case because we are constructing the object with all its properties it is easier to rebuild it differently so that's a that's a that's the thing and so here we have a 6 over 11 simulation points we'll keep on running simulations 7 over 11 times one will change yeah it's perfect well we should be should have some a little bit more losses it's very very thin the metal so we should have a little bit of more loss so a little bit lower frequency Maybe let's see that's what we need there we go now it's higher frequency huh no very same I pretty much a lot of things I just thinking pretty much nothing changed or not a lot okay good good to check it at least okay uh so expert okay again babes we can generate Gerber files there are a couple of things that can be interesting here um I can generate a gerber file for my aunt I call it like that I can use a default connector or I can use another connector there are a family of connectors that are predefined again we have a catalog of connectors but if you want to Define your own connector that's also possible so this is basically to test this PCB correct to fabricate it once you get the the once you get the Gerber files you can fabricate it and once you get it home you can imagine you can measure it yeah this is a feature that we really used pretty much initially to test our designs yeah to verify that the solver was doing the right things and then you can choose different PCB Services one that I like is the Mayu writer for example uh and then what does it mean PCB service Services essentially is a the type of viewer that you can use to visualize the Gerber files essentially okay so as soon as I say okay let me just move this one here on the side if it allows me it's working in the background so it's like a software correct yeah exactly so here I have my Gerber files I don't see them yet here on the left okay okay you have not opened them yet so these are my Gerber files and then if um if I find my web browser in my web browser here this is uh the menu Labs Gerber services so what I'm gonna do is I'm gonna select all of these and let me just do it like this just do it like this and then drag and drop them done and this is our PCB I can just show the outline because we're not interested in the silk screen is there also the connector or you didn't add any connector I believe that the connector is at the back oh a little bit big but it's already there so it's um that's our PCB I was expecting the connector to be where the field on the top yeah it's it got stuck on the back if it would be on the front it would be really big on top of it I I believe that actually we have the possibility to control the position of the connector in this case probably you would need to use something else anyway because that would influence the antenna yeah it's so big awesome yeah compared to the to the size of the antenna that's probably not the best option yeah you just selected randomly something exactly okay that's a can you export this uh to I don't know like dxf or something generic what you can import to other um the except I have to come back to you to that with that question could that answer simply because I know that it's something that definitely the exact input and Export is something we are working on I'm not sure if it is out in release 22b okay but I know for example that we can import STL files we can Import and Export STL file I don't think the step is supported yet right now we're supporting Gerber and STL step in the XF I think is work in progress if I if I'm saying the right things okay because I think dxf would be something what more people could import into the pcbs all right right right right and by the way we also do allow the import of Gerbers so which is also quite cool because if you want if you have a design you want to modify it import the Gerber you you get the boundaries of your geometric shapes and then you operate on that one and we actually also have an example where the antenna is taken from a photo so we extract the boundaries of the antenna just from a photograph and the image recognition which is also quite I I okay I wonder I like it what can I say okay there is one more feature what I would like to see the optimization the optimization yes so you've seen it in the antenna designer app in the array designer app and now also the PCB designer app we have the optimize feature um this is a really this is a very cool feature because it um it uses um not any optimizer but it actually uses an Optimizer well do you have options out of the box between two different optimizers that are both surrogate optimization methods good so what do we have here now the optimization pane is open you have two Optimizer like I was saying this idea and the surrogate optimization method they are both surrogate optimization methods Sadia is a meant for antenna design and it was developed by a professor at the University of Glasgow Professor bolu it's a very sophisticated algorithm very fast I would say the state-of-the-art front-end optimization so what does it mean or it's going to be fast so you don't kind of simulate every correct possibility but you somehow iterate to the correct solution or something like that it's a I have a slide actually that might be but no you don't need to even I just wanted to know because normally what would I expect is like okay you change something you simulate it again you change something you smell it but I guess that would take ages now what it does is that essentially you sample your design space with a number of uh pseudoranium values and then you build a sort of um surrogate model a behavioral model of your analysis results and then you perform the optimization using this surrogate model so every time that you do the calculation you use the model and once you get closer to your Optimum it verifies that the models is correct so it performs again an electromagnetic simulation checks if the model is right or wrong and then it might update it so it's an iterative method but a great benefit here is that it doesn't run an electromagnetic analysis every time that the new value is tried so that's really the the the simple idea of surrogate optimization method how the surrogate model gets constructed it's really complicated because you need to have a certain surface model essentially that that that represents the behavior of your structure um so that's really surrogate methods in general um so here we have our variables so you see here we have the variables that we defined before in our design so for example we could optimize for W2 and say yeah I want it between 0.2 and we would like to optimize it for frequency 2.45 I would say this is a so we are now optimizing the geometry okay so here you set the limits for the optimization what do you correct what it may change exactly so these are the boundaries of your independent variable so we change the geometry this is just one example I just put one honestly just one random variables and then what do we want to optimize we can for example maximize the gain this is our our objective function so we want to have a higher directivity or maybe we want to maximize the bandwidth or minimize the area frequency we would like to optimize frequency you know uh what is optimizing the frequency um it should be 2.45 let's say um we okay we can do that we can minimize the bandwidth uh which means at 2.45 gigahertz okay and then it will become as narrow as possible at 2.45 gigahertz and then um essentially it's um um and then essentially we can apply a constraint function because of course when you do an optimization you can potentially you could get a a very narrow bandwidth at 2.45 gigahertz but you would could get a very bad gain for example okay I understand so for example you could say I want again that I wanted a larger than 1.5 okay something like that so you can apply different constraints that's what I'm trying to say one of the typical things to do is maximize for gain maximize the gain and the constrained the S11 to be below minus 10 DB for example so that's that's the type of things that we can we can change um the frequency range I wanted 2.14 and let's say let's just add three points so we can do 2.0 I don't know 40 uh 45 uh 2.5 2.55 no I think I would do 2.40 then 2.45 now yeah we can do that and 50. yeah just to have three points the fewer points you have the better it is and then you have to choose what you want to optimize so I don't know what can move the length some of these right so if you remember when we started we looked at the sensitivity of of the of the antenna so we played with a with the radiator arm length for example or the radiator arm width so these are all properties that essentially or knowledge that we got at the beginning when we started the simple inverted coplanar F to have an idea now what are the variables that are really irrelevant what are the variables that really make a difference and so one of the variables that I chose now was w 2 is the is the width of the trace so by making the with the um essentially uh making it with the narrower with a thicker radiator arm we were moving the frequency um lower and with the thinner radiator arm we were moving the frequency higher so those are the type of things we can try and yeah now we can run the simulation I just have one variable here we can have multiple variables one will be faster I guess yeah but I'm also running 100 iterations which means that it will take in any case a long time so I don't think that we will be able to see the results but at least we can see the construction of the model and so what's running now essentially as well sorry instead of getting a warning there okay good now I applied everything I think so I just I need to apply this one okay good everything is applied and now I can run it and if you have a a better computer than mine you can also use parallel Computing so you can parallelize the simulation and essentially what happens now is that you will see now in a second that the the surrogate model gets built that's the part that takes the most time because essentially you need to do the electromagnetic analysis of your circuit on all these frequency points for a range of value for for a number of values for our variables now we chose only one and then we can the surrogate model will be used for the optimization and back and forth we will have some electromagnetic analysis in between to validate the results we didn't talk about how much it cost or you know this kind of because I know uh people they always ask like how much this is going to cost me or this is only for very uh Rich companies and so I would like to maybe cover this if you can talk about this so the math works that is the makers of Matra Basin a wonderful company and actually the price list is published so if you go to the MathWorks website you will just see the price actually if I go to the here we can do it you can stop this if it's going to take two hours if it's well down your computer or something hopefully will not do much but um if you go to Once Works external and then I go to products and I browse down to my antenna toolbox there you go view pricing there are different prices so this is a commercial license this is I'm currently I'm I'm in Europe so that's why you see the price in euros uh there are prices in dollars is equivalent essentially this is commercial this is Perpetual which means you buy it now you can use it forever it's for you so you would just go here you would pay 1 680 click on buy now ah there is required product Matlab right so does it mean you need to buy it separately you need to have a matter of license yes and how much is Matlab license and let's look into that um I honestly I don't know it by heart okay plus this and Perpetual license it means there are no updates Perpetual license means that you get essentially two updates for the first year and then and after that you don't get updates okay to our license usually it's the same for all the kind of other software so you get one year subscription or support or something like that right right exactly the same and then after that you I think is 18 of the price if I remember correctly okay for uh for the maintenance the majority of schools they have um they have an agreement with mass works and everybody it's head count so it's campus-wide so all the students have all the products and they don't have to pay period okay okay so next time we'll be talking about the other um Electronics uh apps which you have in your signal Integrity again I had no idea you have seen the Integrity in Matlab that's new there's even newer than the antenna and uh that's everything thank you very much to Georgia for helping me to create this video and thank you very much to you for watching by the way we are preparing some very interesting tutorials so if you don't want to miss them hit the Subscribe button if you want you can also check out our federal online courses where you will find everything important from basic board design up to Advanced Hardware design and PCB layout the link is in the description that's all for this video thank you again don't forget to leave your comments and see you next time bye

Channel: Robert Feranec

Views: 43,934

Keywords: pcb antenna design, pcb antenna design tutorial, pcb antenna simulation, pcb antenna design software, pcb antenna tuning, pcb antenna wifi, pcb antenna design sofware, matlab tutorial, matlab antenna toolbox, matlab antenna design, matlab antenna simulation, matlab antenna toolbox tutorial, matlab antenna modeling, matlab antennas, matlab antenna design toolbox

Id: Rke7d9MKOLs

Length: 97min 46sec (5866 seconds)

Published: Thu Oct 13 2022