Matching to imperfect antennas

[Bret Foreman]

Due to some constraints in the manufacturing process of my product, there is substantial mechanical variation in the BLE antenna. I can constrain this variation to a degree, but it gets more expensive the higher tolerances I want to hit.

I can simulate my antenna impedance at a given frequency using HFSS and I can include random mechanical variations to yield a large set of simulated impedances. I can then average those impedances to get a "canonical" antenna impedance.

Now, assuming my matching network is a perfect complex conjugate of the canonical antenna, what would be the VSWR for each of the randomly generated variations in the above simulation?

It seems that the reflection coefficient, gamma, is (Zo-ZL)/(Zo+ZL). And then the VSWR is simply (1+mag(gamma))/(1-mag(gamma)). Where Zo is the output impedance of my matching network and ZL is the impedance of each (simulated) antenna.

A number of problems pop out immediately when trying to apply this approach:

First, it's quite easy to generate combinations of Zo and ZL that produce a mag(gamma) greater than 1. This seems physically impossible because it implies a reflected power higher than the incident power. Is the formula wrong or am I misinterpreting the meaning of mag(gamma)>1?

Second, when you plug ZL = conjugate(Zo) into the equation for gamma, you don't get zero. In fact, you get j*Im(ZL)/Re(ZL).

I'm guessing that the classic formula for VSWR does not use mag(gamma) but rather the mag(Re(gamma)). That would give zero when you plug in the conjugate.

Can any of you RF wizards help clarify my math?

[Bret Foreman]

From my perspective, the important behavior is power transfer, not reflection. And I think we can all agree that a conjugate match will optimize power transfer from the output of the matching network to the (perfect) antenna. The question, then, is what figure of merit can I use to evaluate the degradation of power transfer as the impedance of the manufactured (imperfect) antenna varies? VSWR seems like the wrong metric, since the magnitude of gamma can easily be bigger than 1, which results in nonsensical (often negative) values for VSWR.

At the end of the day, I want to numerically evaluate how many dB of matching loss I will suffer for a given variation in antenna impedance, and then I want to calculate the statistics of this loss for hundreds of simulated impedances at a given frequency.