## Resistor tolerance and bridge directivity

Another short note related to the RF bridge I was writing about previously. The PCB has four resistors soldered on. Two pairs of 100 Ω in parallel. Each pair forms one of the two fixed impedances in the two branches of the bridge circuit. The two variable impedances in the bridge are the device under test (left) and the termination on the REF terminal (right). The black component on the bottom is the RF transformer of the balun circuit.

My father pointed out the fact that these resistors don't look particularly high precision. The "101" marking (10 times 10^{1} ohms) is typical of 5% tolerance resistors. 1% parts more often have "1001" or the EIA-96 character code. Unfortunately I can't simply measure them in circuit with a multimeter, because the balun forms a DC short circuit across them. I don't want to desolder them. Still, I was wondering how much variances in these resistors would affect the bridge directivity.

Following is the result of a Monte Carlo simulation showing three histograms for bridge directivity. Each was calculated for one possible tolerance class of the 4 resistors used. The assumption was that individual resistor values are uniformly distributed between their maximum tolerances. The effect of two parallel resistors on the final distribution was included. The peak on each histogram shows the value for directivity that is most likely for a bridge constructed out of such resistors.

Each tolerance class defines the lowest possible directivity (where the two resistors are most mismatched). On the high end the histogram isn't limited. In any tolerance class there exist some small possibility that the resistors end up being perfectly matched, however the more you move away from the average directivity the less likely that is, as the probability asymptotically approaches zero.

This is the same data shown as an estimate of the cumulative distribution function. The annotations on the graphs show the 90% point. For example, for 5% resistors, 90% of the bridges would have higher than 32.1 dB directivity. You gain approximately 20 dB in directivity each time you reduce the resistor tolerance by a factor of 10.

It's important to note that this was calculated using a low-frequency bridge model. In other words, none of the high-frequency effects that cause the real-life directivity to fall as you go towards higher frequencies are counted. Any effects of the balun circuit and the quality of the REF termination were ignored as well. So the directivity numbers here should be taken as the best possible low-frequency case.

Anyway, I thought this was interesting. Similar results apply to other devices that use a resistor bridge circuit as a directional coupler, such as the NanoVNA and its various variants. Also somewhat related and worth pointing out is this video by W0QE where he talks about resistor matching for calibration loads and how different SMT resistors behave at high frequencies.

This post led me to think about the reason for having two 100 ohm resistors instead of one 50 ohm. Does this result in a lower overall tolerance? The average of two gaussians has a 30% smaller standard deviation (1/sqrt(2)). However, if the distribution of resistances is actually uniform (or worse, "binned" resistors where those that fall inside ±1% are removed) then I think using two should be nearly as bad as a single 50±5%. (I'm not up to figuring out the statistics for reciprocal addition of two random variables, so maybe there's some benefit.)