Steven S. Holland

Came across a link for some free math texts.

Link!

Every now and then I write or modify a matlab function that I need for my research, and will post them to this blog here and there, with the hopes that they will be useful to someone doing antenna/microwave design.  I know, there is the Matlab File Exchange (which is very useful!) , but hey, this is my blog, I want to post my content here.

Below is a function I used to plot smith chart results in matlab.  I used basic plotting code to generate the chart itself, and added a simply plot function to add the impedance locus and constant VSWR circles.  Hope this is helpeful!  Please let me know if anyone finds a bug.  I will note that there is no safeguard in the function for s-parameters that are greater than 1, which if you are plotting the active S-parameters of a multiport device, are possible.  This isn’t a big deal when the S-parameter goes slightly above 1, but if it swings well above 1 you end up with a tiny smith chart and these erratic line segments…it’s a mess.  For passive applications I haven’t come across a bug yet.

smithchart.m

I have been running some Ansoft HFSS simulations of infinite arrays of antennas, recently.  A peculiar thing I noticed is that some of the impedance data, which is calculated from the S-parameter matrices, had sizable spikes (upwards of 300Ohms) inside a frequency band that, from the S-parameters, looked smooth and well matched.  Clearly something was wrong.  A low VSWR and S11 imply impedance levels close to the source impedance, yet these spikes in the impedance were there.

So I decided that, being able to compute all the data I need from S-parameters alone (as HFSS does), I exported the entire S-matrix to MATLAB and took a look at the impedance values I calculated.

HFSS gives the equation for finding the Z matrix as :

Z = sqrt(Zo*I)*inv(I-S)*(I+S)*sqrt(Zo*I)

where I=identity matrix, and S is the full S-parameter matrix

from this I plotted the exact same thing HFSS was giving for the impedances.  So I started plotting different chunks of the equation (e.g. inv(I-s) alone, (I+S) alone…) and found that the inv(I-S) term was spiking at the exact places my impedance was spiking, while the other terms were well-behaved.  Well, looks like that inversion is messing things up…indicating a singular, or more correctly a nearly-singular matrix (singular matrices have NO inverse defined).  Sure enough, using the function:

cond()

in MATLAB revealed huge condition numbers for the matrix (I-S) where the impedance spikes occurred, 2 orders of magnitude larger than the condition numbers calculated over the rest of the frequency sweep.  So I need to invert a matrix that is nearly-singular….now what??  I need single-value-decomposition (SVD) to take out those values that make the matrix singular and do a quasi-inversion, basically approximate the inversion.  MATLAB includes just such a function:

pinv() — where it’s use is  pinv(matrix, tolerance)

The tolerance can be set to an arbitrary value.  For my purposes, I can stand to lose some accuracy, so I chose a relatively large tolerance (around 0.02 or so) to take out those spikes but preserve the overal shape of the impedance waveforms.  Usually you will want to keep this very small to keep the inversion close to the true value.  But this solves my problem.

But it leads to another conclusion — HFSS, apparently, doesn’t test if its S-parameter matrices are nearly-singular (or at least I haven’t seen a way to have it do so).  Certainly it is using all sorts of inversion approximations in the solver the software uses (you can even do iterative matrix solutions in v11 of HFSS…), I don’t think it is likely that they have skimped on such checking on the data output front end.   Please post comments if you konw of any solutions in HFSS for this.

I passed my oral thesis defense today, so I currently have a masters!

In celebration, watch this:

Today! 10AM. I have put a lot of work over the past few months into putting together my thesis, and I am hoping that will pay off.

I found this xkcd comic strangely appropriate — hopefully I am not subconsciously doing something like this during my presentation:

I guess there aren’t imaginary antennas…

A lot of people in RF/Microwaves/Electromagnetics can probably trace back a path of teachers to Maxwell…and here is mine:

Maxwell taught emag to Jeans (at Cambridge),

Jeans taught emag to Smythe (at Princeton)

Smythe taught emag to Rusch (at CalTech)

Rusch taught emag to Strangeway (at University of Southern California)

Strangeway taught emag to me (at Milwaukee School of Engineering) – that’s 5th generation.

Hopefully someday I can add to the list:

Holland taught emag to _________ (at ___________) …

Antennas!  I have spent the last few days learning to use our milling machine, practicing on cheap substrate, fixing the alignment here and there…but I think I have the hang of it now.  I built my first antenna today, which I was pretty happy about.  One problem I ran into was the ceramic substrate I used isn’t very easily cut, but I figured out a way to cutout substrate pieces using a router bit for the mill.

Now I need to find a big ground plane so I can measure it and see how it compares to my HFSS simulations.  Currently it is a 31X31X3mm susbtrate with a single patch on it that resonates at 1.5GHz, to provide a benchmark for my simulations.  I plan on building a number of these to get some experimental verification of some trends, and potentially piece together one of the stacked patch configurations that comes within 4mm of meeting my original design goals.  Assuming this goes well, I will be defending my thesis come early April…