How important is coupler Directivity?

Let's discuss what the term means, and the uncertainty of measurement of DUT VSWR or ReturnLoss due to coupler Directivity.

## Coupler performance parameters

Consider the above diagram, when terminated in a matched load, the key performance characteristics are:

- Coupling Factor: Relates input power (at P1) and that delivered to the coupled port, P3;
- Directivity: This is a measure of the coupler's ability to separate waves propagating in forward and reverse directions, as observed at the coupled (P3) and isolated (P4) ports;
- Isolation: Relates input power (at P1) and that delivered to the uncoupled port, P4; and
- Insertion Loss: This is the ratio of power in a matched load without and with the coupler inserted.

The values of these characteristics in dB are:

Coupling \(C = \frac{P1}{P3}\)

Directivity \(D = \frac{P3}{P4}\)

Isolation \(I = \frac{P1}{P4}\)

InsertionLoss \(L = \frac{P1}{P2}\)

Note that Directivity can be calculated from Coupling and Isolation: \(D = \frac{P3}{P4}=\frac{I}{C}\) .

These values are often expressed in dB by taking 10log of the value, but lets work in simple numeric ratios to explain the ‘problem', we will graduate to dB a little later.

Let's say we have a coupler with Directivity=1000, then for 1V in the Coupled Port P3 from the forward wave, there will be \(\sqrt{\frac{1}{1000}}=0.032 \text{ V}\) at some unspecified phase in the Isolated Port P4.

So, lets say we have a DUT with VSWR=1.5, ReturnLoss=14.0dB or 25, so that causes a voltage of \(\sqrt{\frac{1}{25}}=0.2 \text{ V}\) at some unspecified phase in the Isolated Port P4.

So. the superposition of both voltages with unknown phase relationship will have a magnitude between \(0.2-0.032=0.18 \text{ V}\) and \(0.2+0.032=0.23 \text{ V}\). The Directivity of the coupler is a measure of its imperfection and it gives rise to error in the measured result, albeit moderate small in this case. In fact, displayed VSWR will be somewhere in the range 1.4 and 1.6.

That range might seem pretty acceptable, but if you tried to measure a DUT with VSWR=1.1 with the same coupler, the results are poorer.

Above, the VSWR result would range from 1.03 to 1.17 (depending on the relative phase of the two components).

For some applications, the phase error might be important, eg it may be used to inform the algorithms of a auto-tune ATU.

## deciBels

As mentioned, the four performance parameters can be expressed in dB:

Coupling \(C_{dB} = 10 log \frac{P1}{P3}\)

Directivity \(D_{dB} = 10 log \frac{P3}{P4}\)

Isolation \(I_{dB} = 10 log \frac{P1}{P4}\)

InsertionLoss \(L_{dB} = 10 log \frac{P1}{P2}\)

Note that Directivity can be calculated from Coupling and Isolation: \(D_{dB} =I_{dB}-C_{dB}\).

## Critical value of Directivity

For any given value of VSWR or ReturnLoss, we can readily calculate the error on magnitude and phase of P4 wrt P3 given Directivity. Likewise, we can find the minimum value of Directivity that delivers given uncertainty of magnitude and phase P4 wrt P3.

Let's look at two examples.

### VSWR meter

Let's start with an assumed coupler Directivity of 30dB, and explore the VSWR uncertainty at VSWR=1.2 using Calculate uncertainty of ReturnLoss and VSWR given coupler directivity.

The range of VSWR indicated is from 1.13 to 1.38 for DUT VSWR=1.2. You can try different inputs to explore the effect and choose a minimum Directivity to suit your applications needs.

### Auto ATU

In this case, the amplitude and phase errors are important, typically the algorithm at least in part does a binary search for a solution that meets a threshold VSWR value, and inputs to that process are both the magnitude and phase of P4 wrt P3.

Let's start with an assumed coupler Directivity of 30dB, and explore the VSWR uncertainty at VSWR=1.5 (the tune intended tune threshold) using Calculate uncertainty of ReturnLoss and VSWR given coupler directivity.

This delivers a VSWR range of 1.40 to 1.60 which might be quite fine for the purpose, but we also need to look at the phase error as that may inform the tuning algorithm and phase error much more than 10° is likely to result in tune failures when the ‘wrong turn' is taken. In this case, the phase error is just below 10°, so probably acceptable and worth testing a prototype not so much to see if it works, but if it fails very often when it should have worked.

## Conclusions

- Directivity is a key performance parameter of a directional coupler.
- Whilst there are lots of published ham designs for directional couplers, it is rare to see published Directivity measurements.