A correspondent reading Designing a Gamma Match – Simsmith design tool and confirmation of as-built antenna referred me to (Healey 1969) and questioned my method.
I have tried several times to reconcile built and tuned antennas and NEC models with Healey and failed, leading me to think of the problem and devise a good approximation that did reconcile (for me).
This article attempts to reconcile the example given at Designing a Gamma Match – Simsmith design tool and confirmation of as-built antenna, an example where the two built antennas reconcile well with the ARRL published design article and NEC model.
Example for reconciliation
The example antennas are 4 element 144MHz Yagis built around 1970. They were originally designed with a 50Ω split dipole feed, or the option of a folded dipole with 4:1 half wave coax balun.
Gamma arm
Gamma dimensions:
- dipole 12mm dia;
- gamma arm 12mm dia;
- gamma arm spacing 38mm;
- gamma arm length 160mm.
From this we can calculate using RF Two Wire Transmission Line Loss Calculator Zo=218Ω and length=28°.
Tapping point impedance – Healey eqn 1
Impedance at the tapping point:
\(Z_2=\frac{Z_1}{\cos^2 \theta}\\\) \(Z_2=\frac{50}{\cos^2 28}=64\)Transform Z2 by the gamma transmission line – Healey eqn 2 & 3
Using RF Two Wire Transmission Line Loss Calculator:
\(Z_2’=80+\jmath 103\)Calculate shunt reactance of the short circuit stub – Healey eqn 4
Using RF Two Wire Transmission Line Loss Calculator:
\(X_p=\jmath 115\)Find ZΓ – Healey Fig 4
\(Z_\Gamma=\frac1{\frac1{\jmath X_p}+\frac1{R+\jmath X}}\\\) \(Z_\Gamma=\frac1{\frac1{\jmath 115}+\frac1{80+\jmath 103}}\\\) \(Z_\Gamma=19.6 + \jmath 61.5\)Apply step up – Healey Fig 3
Since the tubes are equal size, the step up from Fig 3 is 4:1.
\(Z_{in}=78.5 + \jmath 246.1\\\)The calculated reactance does not reconcile with the Xc=86Ω to tune the built antenna.
The reactance would be nulled by a series Xc=246.1Ω leaving Z=78.5Ω which does not reconcile with the measured Z=50Ω.
Conclusions
This attempt to reconcile the example antennas failed.
References
- Healey, D. Apr 1969. An examination of the gamma match In QST Apr 1969.