With Keysight 85052D cal kit definitions in hand, how can i mathematically determine the modeled phase of a coaxial open and/or short at the connector, rather than at the end of the offset?

I have Keysight measurement data for my 85052D which gives nominal values for modeled phase, so i know what sort of results I should be getting. However my calculations are a little bit off and I don't know why.

I've attached an excel spreadsheet with my calculations. I'll give a breakdown here of my steps:

Where f is frequency in Hertz, C is capacitance, PI is 3.14...;

Capacitance coefficients are computed like so: C = C0+C1*f+C2*f^2+C3*f^3

Phase from capacitance is computed like so: ATAN(C / ( -1 / ( 50*2*PI*f ) ) ) * 2

Phase from delay is computed like so: -(delay)( 360 * f ) * 2, Where delay is 29.243ps

Total phase that should be measure at the connector end is the sum of phase from capacitance and phase from delay.

...Except that its not... my results using these calculations do not precisely match the nominal values found on the cal cert. My only guess is that the offset loss coefficient should be involved somehow. If that's the case, then how do I determine a phase component from the offset loss coefficient?

I took Dr. Kirkby's advice and contacted Keysight technical support via email. What follows is the response:

Brendan Bono - Inside Applications Engineer:

The answer is that the polynomial model is a simplification of the more complex bessel functions which more accurately characterize the model phase for the standard. More details are provided by Dr. Joel Dunsmore and Ken Wong who are the Principal Engineers in our Metrology and VNA departments:

From Dr. Dunsmore:

In fact we must understand that the capacitance of the open or the value of the short does NOT follow the polynomial equation exactly. It is more complex (a curve) but in the very early days of computers (1978 or so) it was too hard to do the complex math that solved for the Bessel functions that are better at predicting the curve for capacitance. So the 8510 just used a 4th order polynomial which worked pretty well. As instruments got better it was easier to see the difference between the polynomial computation and the actual values, and for kits above 50 GHz, we had to go to data-based calibration kits as the polynomials became too much in error.

So: the short answer is: yes there will be a bit of difference. For 8753 kits, the polynomial is slight different to make the fit best in the region of DC-9 GHz, for 8510 kits, the fit is not as good over DC-9 GHz but better up to 26 GHz, and the PNA and beyond used the 8510 kits. The values for specifications take into consideration this small error.

And follow up from Ken Wong:

The theory behind the cal kit definition model is explained reasonably well in App Note 1287-11(attached). It also provides the equations that relates the various terms in the calibration standards definitions to its complex response (S11).

The kit definition is a generic (nominal) model. All cal standards are tested to meet their deviation from nominal specs. There are three major factors that influence the deviation – curving fitting accuracy of all the terms, measurement uncertainty and manufacturing tolerances. Only authorized service centers within Keysight have the accuracy to measure these cal kit devices. Quite often, the same device may have difference definitions for different bandwidth coverage. This was presented in various S800 special sessions.

As mentioned by Dr. Joel, the simple model falls apart for high frequency connectors, 2.4mm and smaller. Data-based model is recommended. We will introduce data-based models for all the cal kits in the next year.