Coaxial Cable Loss Budget: Dielectric Alone Is Not Enough
A 10 m length of coaxial cable has inner conductor radius a = 1.0 mm and outer conductor inner radius b = 4.0 mm. The dielectric has ε_r = 2.25 and tanδ = 1.0×10^-3 at 1.0 GHz. The copper conductors have σ = 5.8×10^7 S/m. A student computes only the dielectric attenuation, predicts about 1.36 dB total loss at 1.0 GHz, and is puzzled when a vector-network-analyzer measurement gives about 3.0 dB for the same reel. The connectors are known to contribute roughly 0.3 dB. You are asked to compute the characteristic impedance, the dielectric and conductor attenuation constants, and decide whether the measurement is consistent with distributed loss alone. Use TEM mode and ignore radiation.
1. Compute the characteristic impedance and the dielectric attenuation over 10 m.
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For a TEM coaxial line,
Z0 = (1/(2π)) sqrt(μ/ε) ln(b/a).
With ε_r = 2.25, sqrt(μ0/ε0) = 376.7 Ω, so sqrt(μ/ε) = 376.7/1.5 = 251.1 Ω. Then
Z0 = (251.1/(2π)) ln 4 ≈ 39.97 × 1.386 ≈ 55.4 Ω.
The dielectric attenuation is
α_d = (π f/c) sqrt(ε_r) tanδ = (π × 10^9 / 3×10^8) × 1.5 × 10^-3 ≈ 0.0157 Np/m.
Over 10 m this is 0.157 Np. Converting with 8.686 dB/Np gives about 1.36 dB. That part of the student's calculation is correct.
2. Compute the conductor attenuation using the surface resistance.
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The surface resistance is
Rs = sqrt(π f μ0/σ) = sqrt(π × 10^9 × 4π×10^-7 / 5.8×10^7) ≈ 8.25×10^-3 Ω.
For coaxial TEM, with η = sqrt(μ/ε) = 251.1 Ω,
α_c = Rs (1/a + 1/b) / [2η ln(b/a)].
Here 1/a + 1/b = 1000 + 250 = 1250 m^-1 and ln(b/a) = ln 4 = 1.386. Thus
α_c ≈ 0.00825 × 1250 / (2 × 251.1 × 1.386) ≈ 10.31 / 696.3 ≈ 0.0148 Np/m.
Over 10 m this is 0.148 Np, or about 1.29 dB. This is comparable to the dielectric loss, not negligible.
3. Reconcile the calculated losses with the 3.0 dB measurement and identify the dominant loss.
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The distributed loss is the sum of the two mechanisms:
α_total = 0.0157 + 0.0148 ≈ 0.0305 Np/m.
Over 10 m, α_total L ≈ 0.305 Np, which is 0.305 × 8.686 ≈ 2.65 dB. Adding the known connector loss of about 0.3 dB gives approximately 2.95 dB, consistent with the measured 3.0 dB within experimental error. Neither mechanism dominates at 1 GHz: dielectric and conductor losses are nearly equal. The student's 1.36 dB prediction ignored roughly half the loss. As frequency increases, α_d scales linearly with f while α_c scales as sqrt(f), so dielectric loss becomes increasingly dominant at higher frequencies; at lower frequencies the conductor term becomes relatively more important.