Electromagnetic Fields and Waves: Notes, Solved Problems and Interactive Lab

Conductors. The surface current on each conductor is $∣ J_{S} ∣ = ∣ H_{ϕ} ∣$ at its surface, $∣ I ∣/ (2 π a)$ on the inner one and $∣ I ∣/ (2 π b)$ on the outer one. With the surface resistance $R_{s} = π f μ_{0} / σ_{c}$ the power lost per unit length is

P_{L}^{'} = \frac{R _{s}}{2} [2 π a (\frac{∣ I ∣}{2 π a})^{2} + 2 π b (\frac{∣ I ∣}{2 π b})^{2}] = \frac{R _{s} ∣ I ∣ ^{2}}{4 π} (\frac{1}{a} + \frac{1}{b}), (7.5.12)

and with $P = \frac{1}{2} Z_{0} ∣ I ∣^{2}$ the attenuation constant $α_{c} = P_{L}^{'} / (2 P)$ follows.

Dielectric. A filling with the loss tangent $tan δ$ gives the plane-wave attenuation of the medium (Section 6.2), $α_{d} = k tan δ /2$ , since the TEM wave travels with $τ = k$ .

Attenuation of a coaxial line

α_{c} = \frac{P _{L}^{'}}{2 P} = \frac{R _{s}}{2 η} \frac{1/ a + 1/ b}{ln ( b / a )}, α_{d} = \frac{k tan δ}{2}, α = α_{c} + α_{d} [Np/m] . (7.5.13)

In circuit terms $R^{'} = \frac{R _{s}}{2 π} (\frac{1}{a} + \frac{1}{b})$ , $G^{'} = ω C^{'} tan δ$ , and $α = \frac{R ^{'}}{2 Z _{0}} + \frac{G ^{'} Z _{0}}{2}$ .

$α_{c} \propto f$ (skin effect) and $α_{d} \propto f$ : copper dominates at low frequencies, the dielectric at high ones.
The inner conductor, the $1/ a$ term, is responsible for most of $α_{c}$ .
Bigger cables lose less, $α_{c} \propto 1/ b$ at fixed $b / a$ , but see the frequency limit of Section 7.5.6.

Common mistake · Using the dc resistance at high frequency

Computing $R^{'}$ from $1/ (σ_{c} π a^{2})$ , the dc resistance of the inner conductor. At $1$ GHz the skin depth in copper is about $2$ μm, and the current flows only in that skin on the outer surface of the inner conductor and on the inner surface of the outer one. The correct $R^{'}$ uses $R_{s}$ and the two circumferences, Eq. (7.5.13), and grows as $f$ .

Real cables

Braided outer conductors, stranded inner conductors and rough surfaces raise $α_{c}$ . Manufacturers' data are often $1.5$ to $2$ times the smooth-copper formula for flexible braided cables.

7.5.4 Attenuation of the coaxial line