Woolly thinking superposes power on transmission lines

Recently on social media, F6AWN cited a “document written by a team of engineers working in a company of 12,000 people devoting to perfecting power” which ended with the following formula:

The paper does not state that the phasors are RMS, RMS phasors, but it appears so… so let’s use like.

Taking the expression \(P_{load}=\frac{|V_f|^2}{Z_0}-\frac{|V_r|^2}{Z_0}\), on the surface of it, it appears to be subtracting one power quantity from another, \(P=P_f-P_r\). Electric field components can be superposed, so can magnetic field components, and in a transmission line, the equivalent voltages and currents can be superposed, but powers cannot be superposed (read up on the Superposition Principle).

So the original quoted formulas do not apply as generally as written.

That said, we often loosely say \(P=P_f-P_r\), but is it always true, sometimes true, never true?

Well, the answer is that it is sometimes true, it is true when Zo is purely real, see Power in a mismatched transmission line.

There are two cases where Zo is sufficiently close to purely real that the error in applying \(P=P_f-P_r\) is small to insignificant:

  1. using practical transmission lines at a frequency high enough that Zo is approximately real; and
  2. measurements with a sampler that transforms a point sample of voltage and current into Pf and Pr for some real calibration impedance.

An example of the latter would be use of a properly calibrated Bird 43 with 50Ω element which is calibrated for Zref=50+j0Ω. Note that if you insert such an instrument into a line with actual Zo=50-j1, whilst the displayed value of Pf and Pr of themselves are slightly in error, calculated \(P=P_f-P_r\) is correct (see Power in a mismatched transmission line which BTW uses amplitude of phasors rather than RMS.)

This brings us to related bit of traditional ham wisdom posted recently on social media, Warren Allgyer stated:

Power readings on 50 ohm instruments like a Bird 43, and SWR meter, or even the output power meter of a transceiver are not accurate unless made at a 50 ohm point.

The only specific instrument mentioned is a Bird 43 (Directional Wattmeter), the other two are non-descript, more later.

In respect of the Bird 43, as explained earlier, the Bird 43 samples V and I in a very small region, approximately at a point, and transforms a point sample of voltage and current into Pf and Pr for some real calibration impedance, usually 50+j0Ω. As explained above, calculated \(P=P_f-P_r\) is correct, even if Zo departs from 50+j0Ω, even if Zo=75-j2Ω. Whilst the net power calculated is correct, the stand alone values of Pf and Pr are only exactly correct wrt 50+j0Ω and VSWR calculated from them is VSWR that would exist on a line of Zo=50+j0Ω.

If you were to use such an instrument in a line where Pr is not equal to zero, whilst Pf and Pr do not have stand alone value, the quantity \(P=P_f-P_r\) is correct, you can measure the ‘net’ power in a line even in the presence of standing waves. The traditional ham wisdom quoted above is wrong.

The only comment that can be made about “SWR meter, or even the output power meter of a transceiver” is that if they transform a point sample of voltage and current into Pf and Pr for some real calibration impedance, usually 50+j0Ω, they behave as explained for the Bird 43. Just because a transceiver incorporates a directional coupler to measure VSWR for display and protection circuits, it does not mean its power meter displays Pf.

Now polite people on social media do not call out errors like these, so they remain unchallenged for the gullible to learn and recite later themselves.