View source for Engineering:Antenna equivalent radius

{{Antennas|characteristics}}
The '''equivalent radius''' of an [[Engineering:Antenna (radio)|antenna]] conductor is defined as:<ref>E.A. Wolff, ''Antenna Analysis'', Chapter 3, John Wiley & Sons, New York, NY, Second Edition, 1966.</ref><ref>David M. Drumheller K3WQ, ''The Antenna Equivalent Radius: A Model for Non-Circular Conductors'', QEX, American Radio Relay League, Newington CT, 2017 March/April pp. 10ff.</ref>
<div style="text-align: center;"><math>
r_e = \exp\left\{
             {1\over L^2} \oint_\ell \oint_\ell
             \ln \vert \boldsymbol{x} - \boldsymbol{y} \vert
             \; dx \; dy 
          \right\}
</math></div>
where <math>\scriptstyle\ell</math> denotes the conductor's [[Circumference|circumference]], <math>\scriptstyle{L}</math> is the length of the circumference, <math>\scriptstyle{\boldsymbol{x}}</math> and <math>\scriptstyle{\boldsymbol{y}}</math> are [[Euclidean vector|vectors]] locating points along the circumference, and <math>\scriptstyle{dx}</math> and <math>\scriptstyle{dy}</math> are differentials segments along it. The equivalent [[Radius|radius]] allows the use of analytical formulas or computational or [[Experimental data|experimental data]] derived for antennas constructed from small conductors with uniform, ''circular'' cross-sections to be applied in the analysis of antennas constructed from small conductors with uniform, ''non-circular'' cross-sections. Here "small" means the largest dimension of the cross-section is much less than the wavelength <math>\scriptstyle{\lambda}</math>.

== Formulas ==
The following table lists equivalent radii for various conductor cross-sections derived assuming 1) all dimensions are much less than <math>\scriptstyle{\lambda}</math>, 2) for cross-sections composed of multiple conductors, the distances between conductors are much greater than any single conductor dimension. . Formulas for the square and triangular cross-sections follow from numerical evaluation of the double integral. All other formulas are exact.
{| class="wikitable"
|-
! align="center"| Cross-Section 
! width="250pt" align="center"| Description 
! width="330pt" align="center"| Equivalent Radius
|-
|-
| align="center"|[[File:Two-conductor ant eq rad.png|200px]] 
! align="center"| Two identical circular conductors
| align="center"| <math> \sqrt{rs} </math>
|-
| align="center"|[[File:two-diff-conductor antenna cross-section.png|200px]] 
! align="center"| Two circular conductors<br> with unequal radii 
| align="center"| <math> \exp\left\{ { r_1^2 \ln r_1 + r_2^2 \ln r_2 + 2r_1r_2 \ln S \over (r_1+r_2)^2} \right\} </math>
|-
| align="center"| [[File:three-conductor antenna cross-section.png|200px]]
! align="center"| Identical circular conductors<br> arranged in a triangle
| align="center"| <math> \left( rs^2 \right)^{1/3} </math>
|-
| align="center"| [[File:four-conductor antenna cross-section.png|200px]] 
! align="center"| Identical circular conductors<br> arranged in a square
| align="center"| <math> \left( \sqrt{2}\,rs^3 \right)^{1/4} </math>
|-
| align="center"| [[File:five-conductor antenna cross-section.png|200px]] 
! align="center"| Identical circular conductors<br> arranged in a pentagon
| align="center"| <math> \left( 2.62\,rs^4 \right)^{1/5} </math>
|-
| align="center"| [[File:six-conductor antenna cross-section.png|200px]] 
! align="center"| Identical circular conductors<br> arranged in a hexagon
| align="center"| <math> \left( 6\,rs^5 \right)^{1/6} </math>
|-
| align="center"| [[File:multiple-conductor antenna cross-section.png|230px]] 
! align="center"| <math> N </math> Identical circular conductors <br> uniformly spaced around a circle
| align="center"| <math> \left( N r R^{N-1} \right)^{1/N} </math>
|-
| align="center"| [[File:flat thin conductor antenna cross-section.png|200px]] 
! align="center"| Flat, infinitely thin conductor
| align="center"| <math> \displaystyle{ We^{-3/2} \approx 0.22\, W} </math>
|-
| align="center"| [[File:square conductor antenna cross-section.png|200px]] 
! align="center"| Square conductor 
| align="center"| <math> \displaystyle{ 0.58\, W } </math>
|-
| align="center"| [[File:triangular conductor antenna cross-section.png|200px]] 
! align="center"| Equilateral triangle conductor 
| align="center"| <math> \displaystyle{ 0.41\, W } </math>
|}

== Derivation ==
The equivalent radius is derived by equating the average magnetic vector potential at the surface of a conductor of arbitrary cross-section with the potential on the surface of a cylinder.

Assume a conductor's cross-section dimensions are small compared to the wavelength, current only flows axially along the conductor, the current distribution slowly varies along the conductor's length, and current is approximately uniformly distributed along its circumference (owing to the [[Physics:Skin effect|skin effect]]). Furthermore, only the current in a neighborhood around any point on the conductor significantly contributes to the potential at that point. Time dependence is ignored, as it may be incorporated by multiplying the current distribution by a time-varying sinusoid. These conditions imply that a quasi-static condition exists and that the geometry is, effectively, one of an infinitely long conductor with a constant surface current density <math>\scriptstyle{Q}</math> (current per area), thereby reducing a three-dimensional problem to a two-dimensional one. Also implied is the magnetic vector potential is parallel to the conductor's axis.

First, consider the potential at a fixed point <math>\scriptstyle{\boldsymbol{y}}</math> on the circumference of the arbitrary cross-section. With the circumference divided into differential segments <math>\scriptstyle{dx}</math>, the current distribution may be approximated by placing a vertical line current within each segment, each with a linear density of <math> \scriptstyle{Q\, dx} </math> (current per length). It is well known that the potential of such a line current is <math> \scriptstyle{-(\mu_0 Q dx/2\pi)\ln \vert \boldsymbol{x} - \boldsymbol{y} \vert} </math>, where <math>\scriptstyle{\mu_0}</math> is the permeability constant. The potential at <math>\scriptstyle{\boldsymbol{y}}</math> is the sum of the potentials for all the strips, which is
<div style="text-align: center;"><math>
A(\boldsymbol{y}) = - {\mu_0 Q \over 2\pi} 
                    \oint_\ell
                    \ln \vert \boldsymbol{x} - \boldsymbol{y} \vert
                    \; dx .
</math></div>
The average potential is then
<div style="text-align: center;"><math>
\bar{A} 
        = {1 \over L} \oint_\ell
          A(\boldsymbol{y}) \; dy \, 
        = - {\mu_0 Q \over 2\pi L}
              \oint_\ell \oint_\ell
              \ln \vert \boldsymbol{x} - \boldsymbol{y} \vert
              \; dx \; dy .
</math></div>
Now consider the case of cylinder with the same linear current density as the conductor of arbitrary cross-section. It is also well known that the potential at any point on its surface, which is also equal its average potential, is
<div style="text-align: center;"><math>
A_c = - {\mu_0 Q L \over 2\pi} \ln \left( r_e \right) .
</math></div>
Equating <math>\scriptstyle{\bar{A}}</math> and <math>\scriptstyle{A_c}</math> yields
<div style="text-align: center;"><math>
\ln\left( r_e \right) 
        = {1\over L^2} \oint_\ell \oint_\ell
          \ln \vert \boldsymbol{x} - \boldsymbol{y} \vert
          \; dx \; dy .
</math></div>
Exponentiation of both side leads to the formula for the equivalent radius.

The formula for the equivalent radius provides consistent results. If the conductor cross-section dimensions are scaled by a factor <math>\scriptstyle{\alpha}</math>, the equivalent radius is scaled by <math>\scriptstyle{|\alpha |}</math>. Also, the equivalent radius of a cylindrical conductor is equal to the radius of the conductor.

== References ==
{{Reflist|33em}}

{{DEFAULTSORT:Antenna Equivalent Radius}}
[[Category:Antennas (radio)]]

{{Sourceattribution|Antenna equivalent radius}}