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:

[attachimg=1]

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?