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The Beta or Hairpin Match is a simple and robust form of matching a lower impedance Yagi to the transmission line.
The diagram above adapted from the ARRL Antenna Handbook shows a Beta Match in the centre of a Yagi Driven Element (DE).
The system operates by shortening the DE so that the parallel equivalent impedance at the DE terminals is the desired load resistance in shunt with some capacitive reactance. The hairpin offers an inductive reactance that is connected in parallel with the DE to cancel that capacitive reactance.
Let’s explore an example design for a 5el DK7ZB 144 array using 10 mm diameter DE and 5 mm diameter for other elements on a nonconductive boom, to be matched to 200 Ω ohms to suit a 4:1 balun and 50 Ω coax.
An NEC model of the unmatched antenna give the feedpoint Z as 32+j3 Ω, Y=0.031j0.0029 S.
The Beta or Hairpin match depends on shortening the DE to change the feedpoint Y to 1/200+j/X S, or 0.005+j/X S .
That occurs with the DE shortened about 42mm at both ends, and Y=1/200+j/74 S, or 0.005+j0.0135 S.
For simplicity, we will treat the hairpin as a lossless transmission line. The hairpin need to provide a shunt admittance of j/74 S, or a reactance of 74 Ω to cancel the capacitive susceptance 1/74 S at the feed point.
So, this short circuit stub (the hairpin) needs to have a high Zo to keep its losses low. Lets choose a spacing / diameter ratio of 10:1 for Zo=120*acosh(10) or 360 Ω, and make it from 5 mm aluminium or copper, with centre to centre spacing of 50mm. The length needed will be about atan(74/360) or 11.5 ° or 11.5/360*300/144.1 m or 66.5 mm.
Alternatively, the stub can be designed interactively using TWLLC.
Parameters  
Conductivity  3.800e+7 S/m 
Rel permeability  1.000 
Diameter  0.0050 m 
Spacing  0.0500 m 
Rel permittivity  1.000 
Loss tangent  0.000e+0 
Frequency  144.100 MHz 
Length  0.067 m 
Zload  1.000e8+j0.000e+0 Ω 
Yload  1.000e+8+j0.000e+0 S 
Results  
Zo  359.19j0.08 Ω 
Velocity Factor  1.000 
Length  11.59 °, 0.032 λ, 0.067 m 
Line Loss (matched)  4.01e4 dB 
Line Loss  65.148 dB 
Efficiency  0.00 % 
Zin  0.03+j73.64 Ω 
Yin  0.000006j0.013580 S 
Above are TWLLC results for an aluminium stub. The indicated loss conductance of the stub is 0.000006 S, so efficiency of the Beta Match is 0.005/(0.005+0.000006) or 99.9 %. In reality, connection resistance will degrade that slightly, but constructed of appropriate materials, Beta Matches are robust, reliable, and efficient. They are used to good effect on production antennas.
Let’s explore the same example design for a 5el DK7ZB 144 array using 10 mm diameter DE and 5 mm diameter for other elements on a nonconductive boom, but to be matched to 50 Ω to suit a 1:1 current balun and 50 Ω coax.
An NEC model of the unmatched antenna give the feedpoint Z as 32+j3 Ω, Y=0.031j0.0029 S.
The Beta or Hairpin match depends on shortening the DE to change the feedpoint Y to 1/50+j/X S, or 0.02+j/X S .
That occurs with the DE shortened about 15mm at both ends, and Y=1/50+j/56 S, or 0.02+j0.0179 S.
For simplicity, we will treat the hairpin as a lossless transmission line. The hairpin need to provide a shunt admittance of j/56 S, or a reactance of 56 Ω to cancel the capacitive susceptance 1/56 S at the feed point.
So, this short circuit stub (the hairpin) needs to have a high Zo to keep its losses low. Lets choose a spacing / diameter ratio of 10:1 for Zo=120*acosh(10) or 360 Ω, and make it from 5 mm aluminium or copper, with centre to centre spacing of 50mm. The length needed will be about atan(56/360) or 8.8 ° or 8.8/360*300/144.1 m or 52.0 mm.
Alternatively, the stub can be designed interactively using TWLLC.
Parameters  
Conductivity  5.800e+7 S/m 
Rel permeability  1.000 
Diameter  0.00500 m 
Spacing  0.05000 m 
Rel permittivity  1.000 
Loss tangent  0.000e+0 
Frequency  144.100 MHz 
Length  0.051 m 
Zload  1.000e8+j0.000e+0 Ω 
Yload  1.000e+8+j0.000e+0 S 
Results  
Zo  359.19j0.07 Ω 
Velocity Factor  1.000 
Length  8.83 °, 0.025 λ, 0.051 m 
Line Loss (matched)  2.47e4 dB 
Line Loss  63.074 dB 
Efficiency  0.00 % 
Zin  0.02+j55.77 Ω 
Yin  6.684e6j1.793e2 S 
Above are TWLLC results for a copper stub. The indicated loss conductance of the stub is 0.000007 S, so efficiency of the Beta Match is 0.02/(0.02+0.000007) or 99.97 %. In reality, connection resistance will degrade that slightly, but constructed of appropriate materials, Beta Matches are robust, reliable, and efficient. They are used to good effect on production antennas.
Version  Date  Description 
1.01  06/08/2010  Initial. 
1.02  13/10/2013  Added example 1. 
1.03  
1.04  
1.05 
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