# QEX on SWR dependence on output impedance #2

(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.

QEX on SWR dependence on output impedance set out an analysis if his model that showed why his conclusions were invalid. This article derives a simple mathematical expression for the solution of the VSWR detector Wright used.

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.

The purpose of the VSWR detector formed by R1, R2, C1, C2, R3 and their indicating circuits (not shown) is to respond to Z, the ratio V/I seen looking to the right into RL (node 4) (Bruene 1959).

Let’s apply a bit of high school maths and basic circuit analysis.

If we assume that the current through the capacitor branch is insignificant (which it should be in a good design), we can write a simple expression $$Z=\frac{V}{I}=\frac{V_L+I R_L}{I}$$ where $$I=\frac{V_S-V_L}{R_S+R_1+R_2+R_L}$$

Substituting for I: $$Z=\frac{V}{I}=\frac{V_L+\frac{V_S-V_L}{R_S+R_1+R_2+R_L} R_L}{\frac{V_S-V_L}{R_S+R_1+R_2+R_L}}$$ so it can be seen that Z is a function of RS.

Little wonder that Wright proved that indicated VSWR depended on RS in his scenario. One wonders if there is confirmation bias at play here. He actually hinted the problem when he observed that this only occurs when VL is not zero.

So, lets solve the previous expression for VL=0 $$Z=\frac{V}{I}=\frac{0+\frac{V_S-0}{R_S+R_1+R_2+R_L} R_L}{\frac{V_S-0}{R_S+R_1+R_2+R_L}}=R_L$$.

The use of VL does not simulate a mismatched load independent of the source, but using test cases for RL will properly simulate mismatched loads independent of source.

## References

• Bruene, W. Apr 1959. An inside picture of directional wattmeters. In QST Apr 1959.
• Wright, M. Sep 2021. SWR dependence on output impedance. In QEX Sep/Oct 2021