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

All this being said, here’s a sign it’s really starting to mess with my life: when I woke up this morning I asked myself what the probability of me getting out of bed when my snooze alarm went off for a third time (it’s set for 10 min intervals).  So I told myself, given how freezing it was in my apartment, and that I didn’t feel particularly awake, that I’d rate the likelihood of me getting out of bed on a given snooze alarm as 2/5, assuming I feel the same way independant of which snooze alarm is going off.

SO to get out of bed at the third snooze would be the probability of 2 failures to get out of bed, again assuming independence to make my just woken life easier, and one success:

P(I get out of bed on third snooze) =  (1-P(I get out of bed on first snooze))*(1-P(I get out of bed on second snooze))*P(I get out of bed on third snooze) = (1-2/5)(1-2/5)(2/5) = 18/125 ~ 20/120 = 1/6 ~slightly below 0.2, I’d say around 0.18 = => 18%

Then I went to sleep again and got out of bed on the 4th snooze alarm.

btw: The probability of me ever getting out of bed on my snooze alarm, found by summing over all probabilities from 1 alarm going off to infinitely many snooze alarm events, is 1, as you’d expect.  So it was bound to happen, eventually…

If you are interested in taking a look at what I spent the last two years working on…you can find it  here.

The thesis was on a project aiming to design GPS antennas that fit into a volume of approximately 1.25X1.25X0.2″.

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.

 http://www.falstad.com/antenna/

You can readily see the field pattern distortions that occur from things like making a dipole or loop longer than half a wavelength in size, the multiple lobes that form, and how the current distribution forms on the structure.  You can for from a triangular current distribution on the dipole, to 10 or 20 sine wave variations along the length.  Really easy way to see what happens to the pattern.  There are also some array examples of end-fire arrays and schelkunoff polynomial alrrays.

http://billwhirity.com/milo/videos/microwave_implosion.mov

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