Effect of the Antenna Wire on the Antenna Efficiency

Gerald Schuller, DL5BBN, 2014-06-14

Introduction

Another member of our club station DK0TU, Andreas, DL7JAS, told me that my steel wire on my vertical antenna might lead to losses of about 5 dB. Since this seemed to be a lot, I calculated the theoretic loss compared to copper, to see if exchanging the wire would be worthwile (it was).

Goal: maximize the Antenna efficiency

Problem to solve: Which wire is better to build an antenna?

Approach: compute the Skin effect for different materials, compute a resulting effective conductivity for a given frequency, and use an antenna simulation program (Xnec2c) to compare the computed antenna gain.

Computation of the Skin effect:

See: http://en.wikipedia.org/wiki/Skin_effect

The Skin effect depth ds is (the depth of the conductive material at the "skin" of the wire):

ds=sqrt((2*rho)/(Omega*mu_r*mu_0))
with
\rho: resitivity=1/conductivity
\mu_r : relative magnetic permeability
\mu_0: permeabiltiy of free space, magnetic constant mu_0= 4pix10^-7 H/m

The full area of a cross section of the antenna wire is:

F_a= pi*r_a^2

The area inside, without skin-depth, is:

F_i=pi*r_i^2
inside readius:

r_i=r_a-ds
The area of the conducting ring along the "skin" of the wire is:

F_a-F_i=pi*(r_a^2-r_i^2)=pi*(r_a^2-(r_a-ds)^2)=pi*(r_a^2-(r_a^2+ds^2-2*r_a*ds))=pi*(-ds^2+2*r_a*ds)
Fraction of conducting area:

pi*(-ds^2+2*r_a*ds)/F_a=pi*(-ds^2+2*r_a*ds)/(pi*r_a^2)=(-ds^2+2*r_a*ds)/(r_a^2)=
        =(ds*(2*r_a-ds))/(r_a^2)

This is the factor that we need to multiply the conductivity with to obtain the effective conductivity.

Examples for 7 MHz and steel and copper cable with 1mm diameter:

(values from http://en.wikipedia.org/wiki/Electrical_conductivity, and http://en.wikipedia.org/wiki/Permeability_%28electromagnetism%29)

Steel Wire

Steel conductivity : 1/rho=6.9x10^6
relative permeability of "electric steel" mu_r= 4000
mu_0= 4pix10^-7 H/m
cable diameter: 1mm, r_a=0.5e-3 m

Skin effect depth ds:

ds=sqrt((2*rho)/(Omega*mu_r*mu_0))

= sqrt((2/6.9e6)/(7e6*2*pi*4000*4*pi*1e-7))

Factor for conductivity from skin effect:

(ds*(2*r_a-ds))/(r_a^2)

=(1.1450e-06*(2*0.5e-3-1.1450e-06))/(0.5e-3^2)
        =0.0045748)

Effective conductivity: 6.9e6 * 0.0045748=3.1566e+04 1/Ohm

Simulation with xnec2c of a vertical antenna on 40m: Max gain at 7MHz: -0.9dB

Copper Wire:

Conductivity: 1/rho=5.96×10^7
relative permeability mu_r= 1

Skin effect depth ds:

ds=sqrt((2*rho)/(Omega*mu_r*mu_0))
= sqrt((2/5.96e7)/(7e6*2*pi*1*4*pi*1e-7))
        = 2.4640e-05

Factor for conductivity from skin effect:

(ds*(2*r_a-ds))/(r_a^2)

=(2.4640e-05*(2*0.5e-3-2.4640e-05))/(0.5e-3^2)
        =0.096131
Effective conductivity: 5.96e7 * 0.096131=5.7294e+06 1/Ohm

Simulation with xnec2c of a vertical antenna on 40m: Max gain at 7MHz: +4.5 dB!

Comparison:

This means the steel cable antenna looses about 5.4 dB gain compared to the copper cable antenna, and you can gain 5.4 dB gain, almost an S-Step, by replacing a steel wire for the antenna by a copper wire!

Experimental Verification

I used a beacon on 40 m at my antenna location in Erfurt to compare the reception strength. The same difference in strength is also to be expected for my transmit signal for the 2 different wire forms. On April 10, 2014, I received OK0EPB - 7 039,4 kHz with my 9 m long vertical antenna with S6-7, from 18:20 until after 23 hours. The next day I exchanged the wires in the afternoon from steel to copper. The I listened again for the beacon at the same time of day to compare the strength. On April 12 after 18:20 hours I received OK0EPB with S7-8, so indeed 1 S-Step or about 6 dB more! Although this is not a very precise measurement, it gives some indication that theory and practice correspond to each other.