# Traditional directional wattmeter uncertainty due to coupler Directivity

NanoVNA – how accurate does the LOAD need to be – part 1? discussed the importance of the calibration parts to accuracy of the NanoVNA.

Let’s digress for a moment at look at a directional wattmeter, a traditional way of measuring ReturnLoss and VSWR. This article examines the effect of coupler Directivity alone on uncertainty. There are other contributions to uncertainty, they are outside the scope of this article.

## Directivity

Let’s review the meaning of directional coupler Directivity.

Above is a diagram from Mini-circuits.

An important property of a directional coupler is its Directivity, $$Directivity=10 \log \frac{P3}{P4}$$.

If we  take a directional coupler that is perfectly calibrated for Zref=50+j0Ω and use it to measure a perfect load of $$50.05 \text{ Ω}$$ we will measure ReturnLoss that equates to the Directivity of that coupler.

## A Bird 43 example

A 50W forward wave wrt 50Ω has a voltage of 50V.

(Bird 2016) specifies the minimum Directivity of the coupler in a Bird 43 directional wattmeter as 25dB.

That means that for a 50V forward wave, less than ideal Directivity may contribute up to 2.8V to the reflected wave detector.

If the instrument indicated 2W reflected power, that would be a reflected voltage of 10V which indicates VSWR=1.5.

Now the actual reflected wave would be ‘between’ 10-2.8=7.2V and 10+2.8=12.8V because the internal 2.8V and reflected wave of unknown phase relationship combine to result in the displayed 10V (corresponding to VSWR=1.5).

So if the reflected wave was as low as 7.2V, the magnitude of the complex reflection coefficient ρ=7.2/50=0.144 and $$VSWR=\frac{1+\rho}{1-\rho}=1.33$$.

If it were as high as 12.8V, the magnitude of the complex reflection coefficient ρ=12.8/50=0.256 and $$VSWR=\frac{1+\rho}{1-\rho}=1.69$$.

So, although the meter readings indicate VSWR=1.5, the actual VSWR could be between 1.33 and 1.69 as a result of the less than ideal Directivity of the instrument.

Above is the result of a calculator you can try.

An exercise for the reader is to rework the above example using ReturnLoss.

So whilst people talk about measuring VSWR=1.1 with a Bird 43, there is quite wide uncertainty at low indicated VSWR due to the coupler’s minimum Directivity of 25dB, see above.

Above is a graph showing the upper and lower confidence limits vs indicated VSWR.

Above is a zoomed in view of the same graph. Indicated VSWR=1.2 could actually be anywhere between 1.07 and 1.35 due to Directivity alone. There are of course other contributions to uncertainty.

## Other cases

Above is the low end graph for minimum coupler Directivity 30dB (eg LP-100A).

Above is the low end graph for minimum coupler Directivity 40dB (eg well calibrated VNA).

## Conclusions

Coupler directivity is a very important parameter is determining uncertainty of ReturnLoss and VSWR measurements.

## References

Bird Technologies Inc. 2016. RF Directional Thruline Wattmeter Operation Manual 920-43.