Shunt matching a loaded HF whip with just a VSWR meter

A question was asked on one of the popular online forums:

How to get the most out of an 80 mobile antenna?…I am using a hustler antenna and I had the swr down to 1.3:1. I started researching how to make the antenna better and it seems that maybe an inductive shunt at the base of the antenna to ground would help. I don’t have the equipment to analyze the antenna and the shunt reactance. I made a 9 turn coil 1″ in diameter and 1″ long using n0. 12 awg thnn wire. I installed the coil at the base of the antenna and now the best swr that I can get is 1.8:1. So is there a way that I can set up the coil and antenna using only an swr meter?…

After 50 responses, none of the online experts have offered a direct answer or explanation.

Direct answer

The coil inductance is too low, try a solenoid of 13 turns, 40mm diameter and 40mm length.

Explanation

An antenna of this type at minimum VSWR will have a feed point impedance of near zero reactance and resistance equal to 50 divided by the measured VSWR, so in this case 39Ω.

A characteristic of this type of antenna is that near resonance (ie near zero reactance), reactance changes with frequency much more quickly than resistance, so much so that we can treat the resistance as approximately constant for the purposes of solving the matching network.

A strategy for matching is to create an L network being a capacitive reactance in series with the base of the antenna, and a shunt inductance. The capacitive reactance can be obtained by ‘detuning’ (shortening) the whip if it has a suitable adjustment.

The chart above from one of my articles provides help in designing a suitable shunt inductor. With series R of 39Ω, the required shunt reactance is around 98Ω. At 3.7MHz, that is an inductance of 4.2µH. The calculator at http://hamwaves.com/antennas/inductance.html can be used to explore coil parameters to get in the ball park of 4.2µH.

Adjustment is done by connecting the 4.2µH coil and adjusting the whip tip shorter until minimum VSWR is found. Iteratively changing the inductor a little and re-trimming the whip should allow finding a perfect match (ie VSWR=1).

So is there a way that I can set up the coil and antenna using only an swr meter?

QED.

I might mention that K0BG recommends 0.5-1.5µH, the OP’s coil should have been almost 1.5µH but clearly inadequate in this case. You will see from the graph above that higher values of X are needed as the R component approaches 50Ω.