Coefficients in DIY calibration kit for VNA

[jmw]

I'm trying to model surface mount SMA connectors as cheap 1-port VNA calibration standards for work below 3 GHz. I'm starting with the Molex 73251-1352 which has manufacturer-done drawings and CAD models:



This is how I'm thinking of modeling the connector to determine the coefficients to enter on the VNA - hoping some metrology expert can follow along and chime in if I'm doing things right or wrong. First, I'll consider the connector as a transmission line that's terminated by some load representing the open or short standard:



The load is either a frequency-dependent capacitance or inductance that is approximated by a third degree polynomial, whose the coefficients will be plugged into the VNA's calibration file:

\[Z_L(f) = \frac{1}{i 2\pi f C(f)}\qquad C(f) = C_0 + C_1 f + C_2 f^2 + C_3 f^3 \qquad \mathrm{(open)}\]
\[Z_L(f) = i 2\pi f L(f)\qquad L(f) = L_0 + L_1 f + L_2 f^2 + L_3 f^3 \qquad \mathrm{(short)}\]

From transmission line theory, the reflection coefficient at the point of load (\(z = 0\)) is:

\[\Gamma(0,f) = \frac{Z_L(f) - Z_0}{Z_L(f) + Z_0}\]

Since we're measuring the connector, the S11 we would see on a VNA is the reflection coefficient seen at the end of the transmission line. At a distance \(\ell\) away through a transmission line, the reflection coefficient is given by (same as following the "toward generator" guide on a Smith chart):

\[S_{11}(f)=\Gamma(-\ell,f) = \Gamma(0,f) e^{-i2\beta\ell}\]

Here, the wavenumber \(\beta\) is given by \(\beta = \frac{2 \pi}{\lambda} = \frac{2 \pi f}{c/\sqrt{\epsilon_r}}\).

Combining these formulas and solving for C(f) and L(f), I get:

\[C(f) = \frac{1}{i 2 \pi f}\frac{1}{Z_0}\frac{1 - S_{11}(f) e^{i2\beta\ell}}{1 + S_{11}(f) e^{i2\beta\ell}}\qquad\mathrm{(open)}\]
\[L(f) = \frac{Z_0}{i 2 \pi f}\frac{1 + S_{11}(f) e^{i2\beta\ell}}{1 - S_{11}(f) e^{i2\beta\ell}}\qquad\mathrm{(short)}\]

So if I take the S11 data from either a calibrated VNA or a EM simulation model, I should be able to apply those two formulas and extract the Cn and Ln coefficients by doing a least-squares polynomial fit. For the simulation, I modeled the geometry of the 73251-1352, without the screw threads or center pad chamfer, but including the tiny air gap seen above where the insulation isn't flush with the housing. For the short standard, I modeled a flat plane of metal capping the end with the same dimensions as the square base.

I put this geometry through openEMS and after some trial and error the results seemed quite reasonable - the Smith chart shows them as starting at either open or short at DC and then walking clockwise along the unit circle as you'd expect as the frequency increases. The group delay/electrical length as measured from the slope of the S11 phase also seemed close to the real dimensions.

When I solved for the open standard, again everything was really quite good and within a factor of 10 of the numbers I've seen for a commercial SMA kit, which I assume would have similar physical dimensions: (electrical length = 11.57 mm, C0 = 37 fF, C1 = .98 fF/GHz, C2 = -2.4 fF/GHz^2, C3 = .21 fF/GHz^3)

When I solved for the short standard, things got a little weird: the best fit required a negative inductance for L0! It seems the calculated electrical length was a bit too long. Correcting S11 with this length rotated things into the lower half of the Smith chart. Adjusting the meshing of the simulating finer would lead to a shorter electrical length, but not quite enough to get a positive inductance L0.

So, I'm here with a few questions:
1. Is this approach sound for modeling open and short standards? What kind of limitations are there to doing this?
2. It seems the like the electrical length in the equations above is not critical, because if you chose a shorter length (within reason!), everything happening past that length gets lumped into the characteristics of the load, and would be modeled in the Cn or Ln coefficients. So should I just use the length as determined from the CAD drawings (I would use the length of the insulation as the transmission line length, adjusted for the dielectric) and solve for the coefficients with that length? That should yield more sensible inductance coefficients at least. I read somewhere it is ideal for the length of the short to be slightly longer than the open, but I'm not sure why?

[OwO]

Ideally you would use the kit S parameters directly for calibration. I think most proper VNA software accepts S parameter calibration kit characterization.

[jmw]

Well, what I've got is just length + C0, C1, C2, C3 (or L). I guess that started with HP VNAs but yeah it would be nice if it took S parameter data.

[radiolistener]

2. It seems the like the electrical length in the equations above is not critical

electrical length depends on distributed capacitance and inductance, because wave propagation speed is:

v = 1 / sqrt( L * C )

[hendorog]

I think the decision on length depends upon where you want the reference plane to be.

So using the CAD length makes sense.
The electrical length for the short would include the distance between the centre pin and the outer shield/ground so it would be longer.

There is a thread on here which uses an Octave script (credit Claudio IN3OTD) to do the fit.
https://www.eevblog.com/forum/rf-microwave/vna-cal-kit-modelling-script-install-guide/msg2459808/#msg2459808

Might be useful for comparison with yours.

[mark03]

Well, what I've got is just length + C0, C1, C2, C3 (or L). I guess that started with HP VNAs but yeah it would be nice if it took S parameter data.

There is always the option of performing your calibrations on a connected PC using python and scikit-rf.  This makes the vintage UI on your VNA fairly useless, but it does have some advantages, like giving you an "audit trail" of the raw measurements.  Also, if your VNA doesn't support direct S-param calibration, then it also probably doesn't come with many options for calibration.  Offline, you can choose between SOLT, TRL, and fancier things.

Characterizing 100% off-the-shelf standards is a neat idea though.  It would spare hobbyists from needing to derive their own unique cal coefs as @hendorog alluded to.  I can see the pictured connector used as-is for an open.  What about the short and load?

[virtualparticles]

You are doing this correctly. Yes, it is normal to have negative values for some of the capacitance or inductance chararacterization. As long as the curve fits the actual response, you're golden. For a commercial calibration kit, the delay is most important. The capacitance or inductance make very small differences and may not even be distinguishable below a few GHz. The curves of delay vs capacitance or inductance can be fairly different so least error will be achieved with the right choice of delay. To some extent you can trade one off against the other.

[jmw]

I finally got around to completing this idea. I modeled the geometry of a Rosenberger SMA connector for full wave simulation in openEMS, and then constructed physical versions. A forum member was generous and characterized my standards on a N5222B (26.5 GHz Keysight VNA) and 8714ES. Here are the results.

These are the constructed standards. I had brass fittings 3D-printed by Shapeways to shield the terminations. The open standard is just the connector itself, the short standard has a piece of copper sheet soldered at the end to short the center pad to the housing, and for the load standard I used 2x 0805 100 Ω resistors soldered at the end.


Here's a animated view of the simulation, a wave propagating through the load standard. Since it's a matched resistive load, the E and H fields are in sync.


Here's the open standard, from the model, and characterized on the N5222B. The phase difference corresponds to about a 0.25mm difference in electrical length between the simulation and measured result. There's some loss in the real version creeping in after 1 GHz, but it stays below 0.1 dB.


The short standard shows a little more deviation from the simulation, in this case the phase difference corresponds to about 1.5mm. Still pretty good.


The load standard matches the predicted return loss curve pretty closely but the phase response on the N5222B shows an inductive load, where the simulation predicted a capacitive one. On the 8714ES, the phase response is closer to the simulation, so  probably needs a remeasurement to break the tie, but I don't have a $100K VNA handy.


8714ES measurement:


This is a pretty satisfying result - at these frequencies it seems reasonable to use off-the-shelf hardware, carefully constructed into standards, and run with them using modeled s-parameter data. Quite a bit cheaper than a commercial kit, if you're willing to trust the physical version matches the simulated one.