# Rob's web

How does an SWR meter or directional wattmeter work? How does itseparate forward and reflected power?

'SWR' is shorthand for 'Standing Wave Ratio'... but what does that mean? When a transmission line is terminated in a matched load - in other words, a load equal to the characteristic impedance of the line - the RF voltage and current are constant along the line. Losses in a reallife line will mean that the voltage and current will decrease slightly but steadily as you travel away from the transmitter, but let's ignore that for the rest of this discussion. When the line is terminated in a mismatched load, standing waves will appear - the voltage and current vary up and down, going through a complete cycle in each electrical wavelength along the line. By probing along a transmission line at a series of points, you can actually measure the RMS values of voltage and current, and plot out the standing waves as in Fig 1 (1). Note that the standing waves are not travelling along the line - they really do stand there, pinned at either end by the conditions at the transmitter and the load.

Fig 1: Standing waves of voltage and current on a transmission line.

Yet something moves. Power (or more correctly, energy) flows from the transmitter to the load. To understand what is happening, we can visualise the standing wave as the result of two separate waves travelling in opposite directions. Concentrating forthe moment on the voltages, we can visualise a forwardtravelling voltage wave EF from the transmitter, and a reverse-travelling voltage wave ER reflected from the mismatched load. A reactive load determines the initial phase relationship between EF and ER at the load itself, but it is the addition and cancellation between EF and ER in progressively-changing phase that creates the standing wave along the line. Fig 2 shows what is happening at points P, Q, R, S and T marked in Fig 1. When EF and ER are in phase (point P), a voltage maximum occurs; when they are 180° out of phase (point S), a voltage minimum; and so on for the intermediate points.

Fig 2 - How the forward and reverse voltage waves (EF and ER) interact at various points along the transmission line to create the standing wave in Fig 1.

The power flowing forward along the transmission line is:

where Z, isthe line impedance. Like wise the reverse power is given by:

The power delivered to the load is simply:

What the directional wattmeter does is to sense the forward and reflected waves, and indicate the corresponding power levels. It does this by using the important fact that forward voltage and current (EF and IF) are always in phase, everywhere along the line, while ER and IR are always 180° out of phase (2).

Fig 3 - Simplified version of the Bruene bridge directional sensor. Sampled RF voltages proportional to the main-line voltage and mainline current (EV and EI) can be either added or subtracted. The diode detector displays the resultant.

For example, Fig 3 shows a very simple version of the Bruene bridge directional wattmeter. If the instantaneous voltage on the line is V, and the instantaneous current is I, the capacitive voltage divider Cl and C2 takes a sample of the voltage (Ev) and the current transformer T1 takes a sample of the current, developing a voltage E, across the load resistor Rl. The reversing switch S1 allows us either to add the two instantaneous RF voltages E, and Ev or to subtract them. A diode detector detects the amplitude of the resultant RF waveform.

When EI and EV are added in phase, we are sensing the forward wave in the transmission line. When EI and EV are added in antiphase we are sensing the reflected wave. To satisfy the condition that there is no reflected wave from a matched load, the circuit needs to be proportioned to make EI = EV under these conditions, so that they subtract to zero. That is how a directional wattmeter is made, initially by designing the capacitor ratio and transformer turns ratio to make EI and EV approximately equal, and finally by adjusting Cl or C2 to equalise them exactly. Fig 3 is not a very practical form of Bruene bridge, because it's not good design practice to have a reversing switch floating at RF potential - certainly not in something that's intended to be a measuring instrument. Fig 4 shows the alternatives, which are simply different ways of extracting the instantaneous sum and difference of EI and EV and presenting them to a pairof matched diode detectors. You can find these and othervariants in a variety of handbooks and other circuits.

Fig 4 - Some more practical versions of the Bruene bridge.

Fig 5 - Transmission line directional sensor, again sampling both the main-line voltage and the current. In this orientation the sensor samples the forward wave - reverse for reflected.

Another important type of directional coupler is based on a parallel section of transmission line (Fig 5). This looks different from the Bruene bridges in Figs 3 and 4, but the principle is exactly the same. The pickup line samples the main-line current by magnetic coupling (mutual inductance) and simultaneously samples the main-line voltage by capacitive coupling. The capacitively-coupled voltage EV appears across the resistor R1, and the inductively-coupled voltage EI is in series with EV. By suitably proportioning the line dimensions and R1, you can again make EV and EI exactly equal with a matched load. If you physically rotate the whole sensor by 180°, EI will reverse in phase but EV will stay the same, so once again we have made a directional sensor. This is the basic principle of the well-known Bird directional wattmeters, where the sensor is a short section of pickup line contained inside a rotatable 'slug'. Unlike the Bruene bridge, the sensitivity of this type of directional coupler increases with frequency, so for use as a wattmeter it needs to be frequency-compensated.

You may have noticed that none of these so-called 'wattmeters' is truly measuring power. What the meter displays is the rectified RF voltage that is the sum or the difference of EV and EI The conversion to power occurs on the meter scale itself, which is calibrated in terms of power delivered into a matched load. The RF power is ideally proportional to the square of the rectified voltage, but the meter calibration also compensates for the loss of sensitivity at low levels due to the threshold voltage of the rectifier diode.

With a matched load, the reflected power is zero and the power PL delivered to the load is simply the forward power PF. If the load is mismatched, you have lost your power calibration and the indication of PF is meaningless. But if you also calibrate the reverse sensor using a matched load, so that it can read the reflected power PR a very curious property emerges. The total power, PL into a mismatched load is still the simple difference (PF - PR) , even though the readings of PF and PR are meaningless individually (3).

The SWR calibration of a directional wattmeter is very simple. Turning back to Fig 1, Standing Wave Ratio is fundamentally defined as the ratio between the maximum and minimum voltages anywhere along the line. It used to be measured by direct probing along the transmission line to find these two voltages, and the waveguide slotted line still uses this principle. Recalling from Fig 2 that the voltage variation comes from the interaction between the forward and reflected voltages EF and ER it follows that:

If the sampled voltages EV and EI have been adjusted to be equal (and opposite) with a matched load, then SWR is simply EV/EI. It's also quite easy to show mathematically that if you insert the directional meter anywhere along the line, you will measure the same value of SWR (4). As I've remarked in previous columns, an 'SWR meter' can also be considered as measuring the reflection coefficient ρ, where:

When we go through the ritual of adjusting the 'forward' reading to full-scale, and then switching to read 'reverse', we are actually reading (EV - EI) as a fraction of (EV + EI). In otherwords the instrument is displaying lpi on a linearscale. The familiar non-linear SWR calibration on the meter face is simply converting from ρ to SWR using the equation above. I hope this bit of simple maths has tied together all the related quantities, and explained how they are measured by a so-called 'directional wattmeter' or 'SWR meter'.

But what if there isn't a transmission line at all - what if the directional wattmeter is connected directly between the output of the transmitter and a load consisting of lumped components? Can you still have a 'Standing Wave Ratio' when there's no transmission line for the waves to stand on? In strict literal terms the answer has to be 'no', but in everyday engineering practice it's definitely 'yes'. SWR is regarded as just another mathematical way of expressing the quality of an impedance match, one of the set of related quantities including ρ, return loss and Y-, Z- and S- parameters. RF engineers simply use which everone is most helpful for the problem at hand, and convert freely between them.

### Notes and references

1. This explanation is abridged from 'An Inside Picture of Directional Wattmeters', by Warren Bruene, W5OLY, QST, April 1959 - the clearest description ever published, by the man who invented the ubiquitous 'Bruene bridge' SWR meter shown in Figs 3 and 4.
2. The fixed phase relationships between EF and IF (0°) and between ER and IR (180°), are independent of the load impedance. The load impedance only affects the phase relationship between EF and ER (and likewise between IF and IR).
3. This property falls out of the mathematics of these instruments, as described in Reflections, by Waiter Maxwell, W2DU and available from www.arrl.org). Its validity is limited only by the directivity of the directional bridge under mismatched conditions, and the accuracy of the forward and reflected power scales.
4. If your SWR appears to vary according to the position of the meter along the line, there's a problem with the measurement.

G3SEK, g3sek@ifwtech.co.uk