Appendix 2A: AC Resistance of a Round Wire When the Skin Depth δ Is Comparable to the Radius a of the Wire

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We have shown in Section 2.2 that the parameters α and β are given by δ⁻¹ in a good conductor. The approximations made in arriving at this result assumed that the loss tangent T = σ_c/ωε is large. Another way of stating the approximation is to say that the displacement current density $j ω ε \tilde{E}$ $j ω ε \tilde{E}$ is neglected in comparison with the conduction current density $σ_{c} \tilde{E}$ $σ_{c} \tilde{E}$ . The propagation constant γ = jk may be obtained in this case by neglecting the first term on the RHS of (2.4)

(2A.1)

γ^{2} = - k^{2} = j ω μ σ = 2 π f μ σ e^{j π / 2} .

$γ^{2} = - k^{2} = j ω μ σ = 2 π f μ σ e^{j π / 2} .$

The wave equation for the current density $\tilde{J} = σ_{c} \tilde{E}$ $\tilde{J} = σ_{c} \tilde{E}$ is given by

(2A.2)

\nabla^{2} \tilde{J} - j ω μ σ_{c} \tilde{J} = 0.

$\nabla^{2} \tilde{J} - j ω μ σ_{c} \tilde{J} = 0.$

For the cylindrical one-dimensional problem under consideration in this section

(2A.3)

\tilde{J} = \hat{z} J (ρ),

$\tilde{J} = \hat{z} J (ρ),$

and from Equation 2A.2, we obtain

(2A.4)

\frac{d^{2} J}{d ρ^{2}} + \frac{1}{ρ} \frac{d J}{d ρ} - j ω μ σ_{c} J = 0.

$\frac{d^{2} J}{d ρ^{2}} + \frac{1}{ρ} \frac{d J}{d ρ} - j ω μ σ_{c} J = 0.$

This function is the Bessel equation considered in Section 2.15 and since the currentdensity is finite at the origin, we reject the function Y₀ and choose J₀:

(2A.5)

J = A J_{0} (T ρ),

$J = A J_{0} (T ρ),$

where

(2A.6)

T = \sqrt{- j ω μ σ_{c}} = j^{- 1 / 2} \frac{\sqrt{2}}{δ} .

$T = \sqrt{- j ω μ σ_{c}} = j^{- 1 / 2} \frac{\sqrt{2}}{δ} .$

Since T is complex, J₀(Tρ) is also complex and has a real part and an imaginary part. The following special functions called Ber(x) and Bei(x) are defined and tabulated in many mathematical tables:

(2A.7)

Ber (x) = Re [J_{0} (j^{- 1 / 2} x)],

$Ber (x) = Re [J_{0} (j^{- 1 / 2} x)],$

(2A.8)

Bei (x) = Im [J_{0} (j^{- 1 / 2} x)] .

$Bei (x) = Im [J_{0} (j^{- 1 / 2} x)] .$

In terms of the current density ${\tilde{J}}_{a}$ ${\tilde{J}}_{a}$ at the surface of the conductor, the current density in the wire may be written as

(2A.9)

\tilde{J} (ρ) = {\tilde{J}}_{a} \frac{Ber (\sqrt{2} ρ / δ) + Bei (\sqrt{2} ρ / δ)}{Ber (\sqrt{2} a / δ) + Bei (\sqrt{2} a / δ)}, 0 < ρ < a .

$\tilde{J} (ρ) = {\tilde{J}}_{a} \frac{Ber (\sqrt{2} ρ / δ) + Bei (\sqrt{2} ρ / δ)}{Ber (\sqrt{2} a / δ) + Bei (\sqrt{2} a / δ)}, 0 < ρ < a .$

Figure 2A.1 shows |J(ρ)/J_a| versus the radius for two values of a/δ. The solid lines show the results based on Equation 2A.9. The broken lines are shown for comparison and are obtained assuming a parallel plane formula

(2A.10)

\frac{\tilde{J} (ρ)}{{\tilde{J}}_{a}} = e^{- (a - ρ) / δ}, a ≫ δ,

$\frac{\tilde{J} (ρ)}{{\tilde{J}}_{a}} = e^{- (a - ρ) / δ}, a ≫ δ,$

which is a good approximation when a ≫ δ. As expected, the solid and the broken curves are close for a/δ = 7.55 and differ considerably for a/δ = 2.39.

FIGURE 2A.1
(a) Current distribution in a cylindrical wire for different frequencies. (b) Actual and approximate (parallel plate formula) distribution in cylindrical wire. (From Ramo, S., Whinnery, J.R., and Van Duzer, T., *Fields and Waves in Communication Electronics*. p. 297. 1967. Copyright Wiley-VCH Verlag GmbH&Co. KGaA. Reproduced with permission.)

The magnetic field $\tilde{H}$ $\tilde{H}$ may be obtained from Equation 1.40 and it has only the ϕ component

(2A.11)

{\tilde{H}}_{ϕ} = \frac{1}{j ω μ} \frac{d {\tilde{E}}_{z}}{d ρ}

${\tilde{H}}_{ϕ} = \frac{1}{j ω μ} \frac{d {\tilde{E}}_{z}}{d ρ}$

and

(2A.12)

{\tilde{E}}_{z} = \frac{\tilde{J}}{σ_{c}} = \frac{A}{σ_{c}} J_{0} (T ρ) = \frac{{\tilde{J}}_{a}}{σ_{c}} \frac{J_{0} (T ρ)}{J_{0} (T a)} .

${\tilde{E}}_{z} = \frac{\tilde{J}}{σ_{c}} = \frac{A}{σ_{c}} J_{0} (T ρ) = \frac{{\tilde{J}}_{a}}{σ_{c}} \frac{J_{0} (T ρ)}{J_{0} (T a)} .$

Hence,

(2A.13)

{\tilde{H}}_{ϕ} = \frac{{\tilde{J}}_{a}}{σ_{c}} \frac{T}{j ω μ} \frac{J_{0}^{'} (T ρ)}{J_{0} (T a)} = \frac{- {\tilde{J}}_{a}}{T} \frac{J_{0}^{'} (T ρ)}{J_{0} (T a)} .

${\tilde{H}}_{ϕ} = \frac{{\tilde{J}}_{a}}{σ_{c}} \frac{T}{j ω μ} \frac{J_{0}^{'} (T ρ)}{J_{0} (T a)} = \frac{- {\tilde{J}}_{a}}{T} \frac{J_{0}^{'} (T ρ)}{J_{0} (T a)} .$

From Ampere’s law,

(2A.14)

{2 π a {\tilde{H}}_{ϕ} |}_{ρ = a} = - 2 π a \frac{{\tilde{J}}_{a}}{T} \frac{J_{0}^{'} (T a)}{J_{0} (T a)} = \tilde{I} .

${2 π a {\tilde{H}}_{ϕ} |}_{ρ = a} = - 2 π a \frac{{\tilde{J}}_{a}}{T} \frac{J_{0}^{'} (T a)}{J_{0} (T a)} = \tilde{I} .$

The internal impedance per unit length is

(2A.15)

Z_{i}^{'} = \frac{{{\tilde{E}}_{z} |}_{ρ = a}}{\tilde{I}} = \frac{{\tilde{J}}_{a} / σ_{c}}{\tilde{I}} = - \frac{T J_{0} (T a)}{2 π a σ_{c} J_{0}^{'} (T a)} .

$Z_{i}^{'} = \frac{{{\tilde{E}}_{z} |}_{ρ = a}}{\tilde{I}} = \frac{{\tilde{J}}_{a} / σ_{c}}{\tilde{I}} = - \frac{T J_{0} (T a)}{2 π a σ_{c} J_{0}^{'} (T a)} .$

Equation 2A.15 may be expressed in terms of Ber and Bei functions defined by Equations 2A.7 and 2A.8, respectively, giving

(2A.16)

R^{'} = Re [Z_{i}^{'}] = \frac{R_{s}}{\sqrt{2} π a} [\frac{Ber q Be r^{'} q - Bei q Be i^{'} q}{{(Be r^{'} q)}^{2} + {(Be i^{'} q)}^{2}}] (Ω / m),

$R^{'} = Re [Z_{i}^{'}] = \frac{R_{s}}{\sqrt{2} π a} [\frac{Ber q Be r^{'} q - Bei q Be i^{'} q}{{(Be r^{'} q)}^{2} + {(Be i^{'} q)}^{2}}] (Ω / m),$

(2A.17)

ω L_{i}^{'} = \frac{R_{s}}{\sqrt{2} π a} [\frac{Ber q Be r^{'} q + Bei q Be i^{'} q}{{(Be r^{'} q)}^{2} + {(Be i^{'} q)}^{2}}] (Ω / m),

$ω L_{i}^{'} = \frac{R_{s}}{\sqrt{2} π a} [\frac{Ber q Be r^{'} q + Bei q Be i^{'} q}{{(Be r^{'} q)}^{2} + {(Be i^{'} q)}^{2}}] (Ω / m),$

where

(2A.18)

q = \frac{\sqrt{2} a}{δ}

$q = \frac{\sqrt{2} a}{δ}$

and

(2A.19)

R_{s} = \frac{1}{σ_{c} δ} = \sqrt{\frac{π f μ}{σ_{c}}} .

$R_{s} = \frac{1}{σ_{c} δ} = \sqrt{\frac{π f μ}{σ_{c}}} .$

Figure 2A.1 shows the current distribution for different frequencies.

Figure 2A.2 shows R′/R_s and $ω L_{i}^{'} / R_{s}$ $ω L_{i}^{'} / R_{s}$ versus a/δ. As expected, they reach the horizontal line 1 as a/δ tends to infinity.

FIGURE 2A.2
(a) Solid wire skin effect quantities compared with d.c. values. (b) Solid wire skin effect quantities compared with values from high-frequency formulas. (From Ramo, S., Whinnery, J. R., and Van Duzer, T., *Fields and Waves in Communication Electronics*. p. 297. 1967. Copyright Wiley-VCH Verlag GmbH&Co. KGaA. Reproduced with permission [1].)

Reference

1.Ramo, S., Whinnery, J. R., and Van Duzer, T., Fields and Waves in Communication Electronics, Wiley, New York, 1967.

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Table of Contents for Appendix 2A: AC Resistance of a Round Wire When the Skin Depth δ Is Comparable to the Radius a of the Wire

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Table of Contents for
Appendix 2A: AC Resistance of a Round Wire When the Skin Depth δ Is Comparable to the Radius a of the Wire