VSWR and displacement

 This article explores the influence of displacement on transmission line VSWR, voltage, current, impedance, loss and power. It is a theoretical analysis based on a practical model of a RG58C/U transmission line.

VSWR

Introduction

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.

Determinants of VSWR

The VSWR at any point on a uniform transmission line is determined entirely by:

• the line's characteristic impedance, velocity factor and attenuation;
• and distance of the point from the load.

Apart from a very special case that hams are unlikely to encounter, the VSWR on a transmission line decreases smoothly from the load end to the source end as a result of line attenuation.

The decrease is predictable from line loss. Alternatively, line attenuation can be determined from the decreased VSWR.

VSWR is not in any way dependent on the equivalent source impedance of the generator.

The model

RLGC representation

A common method of modelling transmission lines is the RLGC model (see Telegraphers Equation). The transmission line is characterised as having series resistance and inductance and shunt conductance and capacitance per unit length. These values may vary with frequency. The HF model used for this article considers:

• inductance per unit length as constant (due partially to skin effect and a fully effective outer conductor));
• capacitance per unit length as constant;
• resistance per unit length is subject to skin effect and is proportional to the square root of frequency;
• conductance per unit length is due to dielectric loss and is proportional to frequency.

This article performs a steady state analysis of the transmission line (line) using the RLGC model. A steady state analysis is quite adequate for applications where the bandwidth is narrow relative to the propagation time of the line.

This article uses the RLGC model, and all plots are scaled with displacement in metres shown from the load, negative sign means towards the source. The source is on the left and the load is on the right, energy flows from left to right.

Measuring VSWR

VSWR if often measured by inserting an instrument that is calibrated for a nominal impedance, commonly 50+j0Ω.

Conditions on the line are determined by the actual Zo, not the nominal Zo of an instrument and care must be taken to distinguish between the two. Whilst the indication on a 50+j0Ω may be very close to actual Zo for a practical nominal 50Ω coaxial line, it is different and some effects that may be observed are due to that difference.

To illustrate this point, Fig 1 shows an extreme case. It is RG58C/U at 250kHz with a load of 5+j50Ω. At this frequency, the modelled Zo is 50.5-j4.6Ω. Note the indicated power levels for forward and reflected waves calculated based on a directional wattmeters calibrated to nominal Zo and actual Zo.

IndFwd and  IndRev are the power levels that would be indicated by a typical directional wattmeter that has detectors that respond to the voltage at a point z on the TL +/- a voltage proportional to the current at that point on the line, such that those two voltages are equal in amplitude and 0/180 deg phase if Z(z) = nom.

Fig 2 shows the case at 10MHz with a load of 50+j50Ω where the modelled Zo is 50.4-j0.7Ω. Note the indicated power levels for forward and reflected waves calculated based on a directional wattmeters calibrated to nominal Zo and actual Zo. Although the difference is smaller than in the case for Fig 1, there is still a difference.

So, what does this mean for VSWR?

Fig 3 shows the actual and indicated VSWR for the case in Fig 2.

IndVswr is the VSWR that would be indicated by a typical reflectometer that has detectors that respond to the voltage at a point z on the TL +/- a voltage proportional to the current at that point on the line, such that those two voltages are equal in amplitude and 0/180 deg phase if Z(z) = nom.

The red line is the actual VSWR and it decreases smoothly from the load to the source as a consequence of line attenuation. The variation shown on an instrument calibrated for nominal Zo is due to instrument error when applied to a line with different Zo.

So, back to the statement "Apart from a very special case that hams are unlikely to encounter, the VSWR on a transmission line decreases smoothly from the load end to the source end as a result of line attenuation", Fig 3 shows that, but also shows that using an instrument calibrated for a different Zo will cause measurement error.

What is the "very special case"? It is a lossy line at low frequency (Zo contains a large negative reactance term), and a very inductive load (eg 1+j50). This is unlikely to be observed with practical RF transmission lines above 1MHz, and it will NEVER be observed with a directional wattmeter calibrated for Zo purely real. Telephone engineers understand this concept, radio engineers usually ignore it. Hams can safely ignore it for HF and above.

Voltages and currents

Fig 4 shows the magnitude of voltage and NomRo*current along the line. The current has been multiplied by the nominal Ro to put the plot on scale against the voltage (sadly, Mathcad does not support two independently scaled Y axes). The voltage at any point is the resultant of the forward wave and the reflected wave at that point, it is the voltage that would be indicated by a voltmeter at that point. Similarly, for current. These are the voltage and current standing waves. The ratio of the maximum voltage to minimum voltage near the load is about 145/58 by eye, or VSWR=2.5 (which is a good estimate for the VSWR at the mid point of the measurements).

Fig 5 shows the phase in radians of the voltage and current along the line wrt the forward voltage at the load. The green trace is the phase of V wrt I along the line, it is this phase that is the angle of the impedance at points along the line.

The impedance at any point along the line is the ratio of the complex voltage and current at that point

Impedance

In the case where VSWR>1, the impedance at a point varies with displacement along a line.

Fig 6 shows the real and imaginary parts of impedance, or resistance and reactance along the line for the scenario in Fig 2.

Fig 7 is a polar plot of impedance, and shows the impedance spiralling inwards with displacement from the load as a result of line loss. The impedance will converge on Zo as line length increases to infinity.

So whilst VSWR in Fig 3 has changed a little, impedance is changing radically, and the effect is more pronounced the higher the VSWR.

Fig 8 shows the real and imaginary parts of the ratio of reflected voltage to the forward voltage, it is the locus that would be plotted on a Smith Chart and VSWR is a radially scaled parameter. Note the spiral shows the decreasing VSWR that is a result of line loss with distance from the load.

Loss

Loss or attenuation is affected by VSWR.

The attenuation in dB on a line with VSWR=1 is constant per unit length, so attenuation vs length is a straight line.

The effect of VSWR is to change the ratio of voltage to current along the line. Most practical transmission lines at HF have higher series resistance loss than shunt conductance loss, so loss per unit length is sensitive to the changing ratio V/I.

Fig 9 shows the loss in dB from the load to a point x on the line, the slope of this line is the loss per unit length . Note that it is not a straight line, but has a wave superimposed. The loss per unit length is highest where the current is a maximum, and it is lowest where current is a minimum because in this scenario most of the loss is due to I2R.

Fig 9 also shows the loss for VSWR=1, the matched line loss (MLL). Note that loss under mismatched conditions cannot simply be calculated by multiplying matched line loss by some factor dependent on VSWR. There is a formula for calculating "additional loss due to VSWR" from VSWR, it is an approximation and depends on assumptions that are not usually stated. Graphs of the formula that can be found in some handbooks are subject to the same error.

Power

Power decreases from the source to the load as a consequence of line attenuation and standing wave.

Power on a line with VSWR=1 is an exponential decay from source to load. On a line with VSWR>1, power will decrease in accordance with the varying attenuation shown in Fig 9.

Fig 10 shows Power decreasing from the source to the load, and influenced by the varying line loss as a result of the standing wave.

Traps for players

Measured VSWR changes with line length

Measured VSWR does change with line length, but the change is small for low loss transmission lines, and the change is predictable from line loss.

If the change in measured VSWR is not exactly accounted for by known line loss, something else is also happening.

An explanation that often applies is that the transmission line is carrying significant common mode current in addition to the desired differential mode current. The existence of significant common mode current means that the line contributes to radiated power, and that contribution influences the feed point impedance of the antenna, and hence VSWR.

Changing the length of the transmission line under those circumstances is effectively changing the length of one of the radiating elements, the feed point impedance and VSWR. If the VSWR at the load end of the line changes when feed line length is changed, the feed line has become part of the load.

Disturbing the thing being measured

Care must be taken when making measurements of any thing to not disturb the thing that is being measured. This requires an understanding of the thing being measured and the instrument.

A common way in which an antenna system is disturbed during measurement is change in the lengths of conductors that carry current that contributes to radiation.

An example is disconnection of the coax feed line from an antenna so as to connect a hand held analyser to the feed point. If the coax feed line in its normal connection carries significant common mode current, disconnecting it significantly changes the antenna system and the load impedance presented to the feed point. In such a case, it is important to maintain the common mode current path, and in the case of coax, that means maintaining the connection of the shield at the feed point whilst making measurements with the analyser.

Another common way in which the transmission line length is changed during measurement is insertion of an instrument with patch lead. Again, if the line carries common mode current, then such a change is a significant configuration change. It is sometimes suggested that a VSWR meter should have a patch lead to build out the line to an electrical half wave. This measure will not necessarily counteract the configuration change caused by an increase in the length of the outer conductor when it carries significant common mode current.

Where to measure VSWR

The question "should VSWR be measured at the antenna or radio" is often asked.

The purpose of the measurement and other knowledge is relevant to the question, and shapes a strategy for reaching the desired goal.

If the loss in the feed line is known, and that means recently measured, then a measurement of VSWR at one point can be referred to another point. In that case, the measurement can be made at a convenient point and referred to the point of interest. The VSWR calculator can perform the necessary calculations.

For example, if a 144MHz antenna is specified to have a maximum VSWR of 1.5, does it comply with specification if you measure VSWR=1.4 at the source of a line with known loss of 1.6dB? Using  the VSWR calculator , VSWR at the antenna is 1.63, so the antenna does not comply with spec and may be faulty.

The question "should VSWR be measured at the antenna or radio" is often asked in relation to using a hand held antenna analyser such as the MFJ-259B. The answer is the same, but beware of the trap of changing the configuration when substituting the analyser for the existing feed line, equipment etc as discussed in the previous section.

Transmission line error

Transmission lines are not perfect. In the first instance there are manufacturing tolerances, and some lines are infamous for departure from nominal specifications. Further, faults may develop in a line through life. Validate the line loss, and the line performance on a known termination.

Instrument error

VSWR meters are not perfect, and some are quite poor. The article VSWR measurement discusses instrument error and tests to validate an instrument.

Application to other line types.

Although the examples used in this article are for a specific coaxial transmission line, the principles apply to all transmission lines, just key parameters vary from one line to another.

The real world

The above is based on a theoretical analysis of a practical model of a RG58C/U transmission line. It assumes instrument error is zero, and that the tranmission line complies exactly with its nominal parameters.

In the real world there are sources of error which add noise to experimental results, significantly in this case:

• error in line parameters (Zo, attenuation, velocity factor, lack of uniformity);
• instrument error (nominal calibration impedance, absolute calibration, scale linearity, loading / disturbance);
• measurement error.

All of these frustrate the measurement task, but all too frequently, observed results are attributed to transmission line behaviour which cannot be explained by the classic RLGC model when there are probably other reasons that should be discovered.