# VSWR

Voltage Standing Wave Ratio is the ratio of the voltage maximum (antinode) to the adjacent voltage minimum (node) on a transmission line.

The standing wave is produced by the superposition of a forward travelling wave and a reflected travelling wave when the transmission line is terminated in other than its characteristic impedance. The reflected wave is created to reconcile the conditions at the end of the transmission line, including the load impedance and the natural ratio of V/I in the forward and reflected travelling waves. Γ is the ratio of Vr/Vf, and VSWR on a sufficiently long lossless line can be predicted as (1+|Γ|)/(1-|Γ|) from the conditions at the termination or some other point. Γ, Vr and Vf are phasor quantities, ie can be expressed as complex numbers.

Direct measurement of VSWR requires observation of the voltage at a number of points along the line, but VSWR is more often estimated from observed conditions at a point, but such estimates typically assume a sufficiently long lossless line.

# Indirect measurement of VSWR

Indirect measurement of VSWR means observation of some other parameters than the voltage standing wave itself (eg using a voltage probe).

Indirect measurement has become the most common way of measuring VSWR, the term VSWR is often used to mean the "notional" result of an indirect measurement wrt a nominal  Zo rather than that which would be measured on a practical transmission line. That gives rise to the use of the term VSWR as a means of qualifying a tolerance range for a load impedance, eg a transmitter might be specified for a nominally 50Ω load with VSWR<1.5.

There are many methods of indirect measurement of VSWR, but one of the most common, and quite an ingenious device in its simplicity and usefulness is the Breune VSWR bridge. This article will use the Bruene circuit as a vehicle for explaining principles that are common to similar instruments.

## Bruene style VSWR meter

Fig 1 shows a typical VSWR meter based on Bruene circuit. The circuit samples voltage and current in a very small region of the transmission line, sufficiently small to consider it a point sample at the frequencies of interest.

The circuit contains two directional samplers, one to indirectly sense the forward travelling wave, and the to indirectly sense the reflected travelling wave.

A sample of the line voltage is obtained by the voltage divider formed by the 10pF and 330pF capacitors.

A sample of the line current is taken by the current transformer, and the secondary current is passed through the 82Ω resistor to develop a voltage at each end of the secondary wrt the centre, that is proportional to the current in the main line, and opposite in phase at each end of the winding.

Since the voltage sample is connected to the centre of the secondary, the ends of the secondary will be a voltage that is proportional to the the line voltage plus line current times Zn (the impedance for which the meter is calibrated),  and the line voltage minus line current times Zn for forward and reflected detectors respectively. The 10pF capacitor is adjusted so that in the reflected position with applied RF power, the meter shows zero deflection with a Zn load (typically 50+j0Ω).

The following describes the circuit in detail in the general sense. V1 is the RF voltage at the lower end of the secondary (wrt ground) in the circuit above,  V2 is the voltage at the upper end of the secondary (wrt ground), Zx corresponds to half of the 82Ω resistor. The description assumes the components (capacitors, transformer, diodes) are ideal, and that the VSWR meter loading of the through line is insignificant.

$V_1=\left | k_1 V + k_2 I Z_x \right | \tag{1}$

$V_2=\left | k_1 V - k_2 I Z_x \right | \tag{2}$

The instrument is calibrated by adjustment of the value of $$k_1$$ so that $$V_2$$ is zero when $$Z_1=Z_n$$. It can be seen that for $$Z_2$$ to be zero, $$k_1 V=k_2 I Z_x$$, therefore $$Z_x=\frac{k_1 V}{k_2 I}$$, and since $$\frac{V}{I}=Z$$ then $$Z_x=\frac{k_1}{k_2} Z_n$$. So, the two expressions can be rewritten as:

$V_1=\left | k_1 (V + I Z_n ) \right | \tag{3}$

$V_2=\left | k_1 (V - I Z_n ) \right | \tag{4}$

Fig 2 shows the equivalent circuit, a voltage sample of $$k_1 V$$, and voltages proportional to current of $$k_1 I Z_n$$.

Since $$V=V_f + V_r$$, and $$V_r=\Gamma V_f$$ and $$I=I_f-I_r$$, and $$I_r=\Gamma I_f$$ eqn (3) can be rewritten as

$V_1=\left | k_1 (V_f (1+ \Gamma)+ I_f (1-\Gamma) Z_n ) \right | \tag{5}$

Substituting $$\frac{V_f}{Z_n}$$ for $$I_f$$

$V_1=\left | k_1 (V_f (1+ \Gamma)+ V_f (1-\Gamma)) \right | \tag{6}$

$V_1=\left | k_1 V_f ( (1+ \Gamma)+ (1-\Gamma)) \right | \tag{7}$

$V_1=\left | k_1 2 V_f \right | \tag{8}$

Similarly, eqn (4) can be transformed to

$V_2=\left | k_1 2 \Gamma V_f \right | \tag{9}$

So

$\frac{V_2}{V_1}=\left | \Gamma \right | \tag{10}$

Substituting ρ for |Γ|

$\rho = \frac{V_2}{V_1} \tag{11}$

VSWR can be calculated from ρ , VSWR=(1+ ρ)/(1- ρ), so the indication of V2 relative to V1 can be scaled in VSWR. The instrument can be used to directly measure ρ at a point (being that of the instrument's sampler) and that knowledge can be used to predict the VSWR that would be observed on a sufficiently long length of adjacent lossless transmission line of the same Zo as the instrument's calibration impedance Zn

To analyse the circuit of Fig 1 at 100W connected to a 50+j0Ω load, for a current transformer of 1:25+25 turns bifilar, the peak current in the 82Ω resistor is 2^0.5*(100/50)^0.5/50 or 0.04A, yielding a peak half winding voltage of 0.04*82/2  or 1.65V. The 10pf capacitor would be adjusted for 0V on the REF switch terminal which will correspond to about 5.5pF for a voltage sample of 1.65Vpk on the 330pF capacitor. This would give a DC voltage about 2.7V (allowing for diode voltage drop) at the FOR switch terminal and 0V at the REF switch terminal.

It is key to note that the accuracy of the instrument in measuring ρ is dependent of the calibration impedance Zn and less dependent on the Zo of the sampler line section where is is relatively short (as it is usually).

## Other circuits

There are other circuits of VSWR meters, the most common other type uses a pair of short loosely coupled transmission lines with detectors in the sampler line section. Differently to the Bruene circuit, the coupling is frequency dependent although some designs achieve compensation over a range of frequencies (eg Bird 43 elements). Frequency dependent samplers cannot be simply calibrated as directional wattmeters. Despite this difference, the same principles apply as explained for the Bruene circuit, in that the circuit samples voltage and current in a very small region of the transmission line, sufficiently small to consider it a point sample at the frequencies of interest. From there on, the explanation is the same (though k1 and k2 may be frequency dependent).

There are other methods for indirect measurement of VSWR that would not be classed as a VSWR meter, eg a true directional transmission line or waveguide coupler with power meters on the coupled ports. This configuration comes into its own on microwave frequencies where it is difficult to construct other types of samplers that are small enough to not disturb the system being measured.

# Directional Wattmeter

Since the forward and reflected detectors in the instrument described above responds to Vf and Vr respectively, the question arises as to whether the instrument can be calibrated in power.

The power passing any point in a transmission line is given by P=real(V*conjugate(I)). When Zn is real (lossless lines and distortionless lines), this can be simplified to Vf*If-Vr*Ir, often expressed loosely as forward power less reflected power.

Instruments are usually calibrated to a nominal Zn that is real (eg 50+j0), and in that case, with better quality instruments, the calibration of the scale(s) in Watts (where Pf=Vf2/Zn and Pr=Vr2/Zn), and calculation of power as P=Pf-Pr is sound. The term directional wattmeter is a bit of a misnomer in that the values of Pf and Pr are not of themselves meaningful, but the difference of the two is the power at that point.

Beware, these comments apply to the type of directional detector described above, and not to a simple line RF voltmeter that is incorporated in some instruments and calibrated only for power when VSWR=1.

## Example

Fig 3 shows the conditions a a length of RG6/U terminated in a load of 60-j26Ω. The model is based on a detailed lossy model of the line.

• The magnitude of line voltage and line current show the classic standing wave patterns.
• The green line is the calculated VSWR on the line (which has a characteristic impedance of 75-j0.8Ω).
• The VSWR(50) line shows the VSWR that would be indicated on a short VSWR meter calibrated for 50+j0Ω, but that should not be interpreted as the VSWR on the actual RG6/U transmission line.
• Note also how the real VSWR falls smoothly from 1.54 at the load to 1.48 at a distance of 20m from the load, a result of line loss.
• Power increase smoothly from the load to the source end, but the slope of the line changes minutely in this example with line current, the slope (meaning the attenuation per unit length) is highest approximately where current is highest, meaning the main loss is I2R loss in the centre conductor.

The VSWR meter captures ρ which is the magnitude of Γ, the phase of Γ is lost and so the VSWR meter is unable to be used to predict the value of Z at a point, except in the special case where VSWR=1 and therefore Z is Zn . Fig 4 above shows the impedance along the line for the example. It can be seen that impedance continually changes along the line, and that is true in the general case except where VSWR=1.

# Errors

There are many sources of errors in VSWR meters, the most common ones are:

• accuracy of the impedance at which the detectors null;
• scale shape (ie the relationship between scale markings and RF power);
• diode voltage drop (related to previous item);
• balance of both detectors;
• length of the sampler;
• power calibration;
• frequency sensitivity / compensation accuracy;
• insertion VSWR (the VSWR caused by insertion of the sampler element in a line, related to the coupling and the accuracy of Zo of the sampler section;
• loss (loss in the sampler section);
• detector / meter response to modulated waveforms;
• application - what to measure, and how to extrapolate that measurement to another place (eg adjacent transmission line or a far end load, and the errors inherent in that extrapolation).

The greatest error is probably the last. VSWR meters are widely used, and ingenious as they are, the results are often incorrectly interpreted.

It is possible to achieve a good null (ie low VSWR) on a nominal dummy load, even though an instrument might itself cause higher VSWR because of an inaccurate line section or excessive coupling.

Mismatch loss or loss due to standing waves can be determined accurately knowing the propagation constant (γ) of the line and the complex reflection coefficient (Γ) at a known point on the line. An approximation of the mismatch loss can be made using the propagation constant (γ) (in fact just the attenuation component) and VSWR (which depends only on the magnitude of the complex reflection coefficient (Γ) that is reasonably accurate only on medium length lines with low VSWR and low loss. This has affects the accuracy of a directional wattmeter of this type for assessing the matched line loss of a s/c or o/c length of line, but for most purposes the effect is small and the test gives good results (providing the VSWR meter gives accurate readings at high VSWR).

# Testing a VSWR meter

The tests here need to be interpreted in the context of whether the device under test (DUT) has only calibrated power scales, or a VSWR Set/Reflected mode of measurement, and whether directional coupler scales are identical for both directions.

1. Connect a calibrated dummy load of the nominal impedance on the instrument output and measure the VSWR at upper and lower limit frequencies and some in between frequencies. The VSWR should be 1. (Checks nominal calibration impedance);
2. Repeat Test 1 at a selection of test frequencies and for each test, without changing transmitter power, reverse the DUT and verify that repeat the forward/set and reflected readings swap, but are of the same amplitude (checks the symmetry / balance of the detectors under matched line conditions).
3. Connect a s/c to the instrument output and measure the VSWR at upper and lower limit frequencies and some in between frequencies. The VSWR should be infinite. (Discloses averaging due to excessive sampler length);
4. Connect an o/c to the instrument output and measure the VSWR at upper and lower limit frequencies and some in between frequencies. The VSWR should be infinite. (Discloses averaging due to excessive sampler length);
5. Connect a calibrated wattmeter / dummy load of the nominal impedance on the instrument output and measure calibration accuracy of power / ρ / VSWR scales at a range of power levels in both forward and reflected directions (Checks scale shape and absolute power calibration accuracy).
6. Repeating Test 1additionally with a calibrated VSWR meter connected to the input to the DUT, and measure the VSWR caused by the DUT at a range of test frequencies (Checks Insertion VSWR).

It is not unusual for low grade instruments to pass Test 1, but to fail Test 6 (and some others, especially Test 3 and Test 4) towards the higher end of their specified frequency range.