QEX on SWR dependence on output impedance

(Wright 2021) sets  out to prove a dependence of VSWR on source impedance, a common ham assertion.

Wright gives the schematic of the minimal VSWR detector he simulates in SPICE.

The schematic is sparse, it does not show where the forward and reflected signals are measured.

So, the crux of his proof depends on this test case:

Solution

Wright’s circuit is trivial to solve by hand. Lets assume that the values of R1 and R2 are much much less than RL, it makes the maths a little simpler. This is not an impractical assumption, a good VSWR detector, one with very low InsertionVSWR will have R1 and R2 are much much less than RL.

Lets solve this using Python as a complex number calculator.

I will ignore the very small current flowing in the capacitor branch.

If we calculate the voltage impressed from Vs to Vl, we get 1.5.

We can calculate the current through the series path, I get 0.018015162531799005-j0.008215654291790999.

We now know the current in RL and can calculate the voltage looking into RL from the left 1.4007581265899502-j0.41078271458954996.

So, at that node, we know V and I, we can calculate Z looking right into RL: 72.976+j10.478.

Here is the Python code:

>>> import cmath
>>> rs=17.928+31.434j
>>> rl=50
>>> vs=2
>>> vl=0.5
>>> r1=0.5
>>> r2=0.5
>>> zt=rs+r1+r2+rl
>>> zt
(68.928+31.434j)
>>> i=(vs-vl)/zt
>>> i
(0.018015162531799005-0.008215654291790999j)
>>> v=vl+i*rl
>>> v
(1.4007581265899502-0.41078271458954996j)
>>> z=v/i
>>> z
(72.976+10.478j)
>>> z0=50
>>> rho=abs((z-z0)/(z+z0))
>>> rho
0.20460301944607484
>>> vswr=(1+rho)/(1-rho)
>>> vswr
1.5144676795317642

Knowing Z looking into RL, we can calculate VSWR(50).

The answer is VSWR=1.51 which reconciles with Wright’s calculation.

The serious flaw in his thinking is that the ratio V/I seen by the VSWR detector is 50Ω, it is in fact 72.976+j10.478Ω.

The presence of VL means that the ratio V/I at the VSWR detector depends on the entire series circuit. This is a contrived circuit that does not represent usual implementations.

The circuit is not a clear representation of a fixed mismatched load, and the conclusion he draws that indicated VSWR depends on source impedance is wrong.

Accepted theory

Heaviside gave us the well accepted theory that VSWR is a function of load impedance and characteristic impedance of the line. Source impedance does not enter into the equation.

More at QEX on SWR dependence on output impedance #2 .

References

  • Wright, M. Sep 2021. SWR dependence on output impedance. In QEX Sep/Oct 2021