Introduction
When planning for an amateur radio installation or when purchasing such items as an antenna tuner, coax, or balun, it is useful to know what we may be up against for transmission line voltages under conditions of antenna mismatch. This article explores what the worst-case standing wave voltage might be on a transmission line when the only mismatch is due to the antenna. Furthermore, the transmitter power is deliberately increased to compensate for the mismatch.
Calculation
When computing the magnitude of the maximum peak voltage on a transmission line under conditions of increasing VSWR, it has been stated without proof [1][2] that this voltage is proportional to the square root of the VSWR. This factor may be derived by assuming that the amount of power delivered to the load will remain constant under conditions of increasing mismatch, which should be the worst imaginable case. This operating condition should be possible if a) there is sufficient forward transmitter power to compensate for reflected power and b) a matching network such as an antenna tuner is placed between the transmitter and the load to protect the transmitter.
Define:
Numerical Example 1
A 100W transmitter feeds an antenna with a 3:1 VSWR. By how much must the transmitter power be increased to maintain 100W into the antenna?
The magnitude of the voltage reflection coefficient is 0.5. The fraction of power returned from the load is given by,
The amount of power reflected from the antenna is 25%. Next, we must calculate the amount by which we need to increase the transmitter power to maintain 100W at the antenna input,
So, the amount by which the transmitter power must increase is 33.33W. Let’s check our answer to see how much power is reaching the load after we increase the power,
where
Pantenna is the amount of power reaching the antenna for a 3:1 VSWR after compensating accordingly.
Thus, we have demonstrated that for a 3:1 VSWR with 133.33W input power, the power incident on the antenna will be 100W when the only mismatch is in the antenna.
Continuing on, from the voltage standing wave definition,
where,
The reflected voltage may be written in terms of a fraction of the forward voltage,
Solution:
The corrected, forward power generated by the transmitter is given by,
and if we set S = VSWR for convenience,
The RMS magnitude of the forward voltage wave is given by,
The peak magnitude of the forward voltage wave is given by,
and the magnitude of the maximum peak voltage on the line is given by,
Finally, replacing S with VSWR yields,
Thus, it has been shown that the magnitude of the maximum peak voltage on the transmission line for the case of constant transmitter power to the antenna load is proportional to,
Numerical Example 2
What would be the maximum peak voltage on the transmission line for the case when the transmitter power is 1 kW and the antenna VSWR is 3:1?
The maximum peak voltage on the transmission line for the case where the transmitter power is 1 kW and the VSWR is 3:1 would be 547.7 Volts.
References
[1] https://owenduffy.net/blog/?p=11773
[2] ARRL Antenna Handbook, 41st Ed., 2007, Chapter 24 Transmission Lines, Line Voltages and Currents, p. 10, Equation 17.