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The Beta or Hairpin Match

The Beta or Hairpin Match is a simple and robust form of matching a lower impedance Yagi to the transmission line.

Beta Match

Beta Match

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.

Example 1

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 non-conductive 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=0.031-j0.0029 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.

Beta match example

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 1.000e-8+j0.000e+0 Ω
Yload 1.000e+8+j0.000e+0 S
Results
Zo 359.19-j0.08 Ω
Velocity Factor 1.000
Length 11.59 °, 0.032 λ, 0.067 m
Line Loss (matched) 4.01e-4 dB
Line Loss 65.148 dB
Efficiency 0.00 %
Zin 0.03+j73.64 Ω
Yin 0.000006-j0.013580 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.

Example 2

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 non-conductive 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=0.031-j0.0029 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.

Beta match example

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 1.000e-8+j0.000e+0 Ω
Yload 1.000e+8+j0.000e+0 S
Results
Zo 359.19-j0.07 Ω
Velocity Factor 1.000
Length 8.83 °, 0.025 λ, 0.051 m
Line Loss (matched) 2.47e-4 dB
Line Loss 63.074 dB
Efficiency 0.00 %
Zin 0.02+j55.77 Ω
Yin 6.684e-6-j1.793e-2 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.

Changes

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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