Coaxial Cable Attenuation & Power Loss Calculator

1. The Physics of Attenuation in Coaxial Waveguides

A coaxial cable is a transverse electromagnetic (TEM) transmission line consisting of two concentric cylindrical conductors separated by a continuous dielectric insulating medium. As a high-frequency RF voltage wave propagates along the axis $z$, its total attenuation constant $\alpha$ (expressed in Nepers/meter or converted to decibels/meter via $1\text{ Np} \approx 8.686\text{ dB}$) is the algebraic sum of two independent physical dissipation mechanisms:

In the transmission line lumped-element equivalent model ($R, L, G, C$ per unit length), the attenuation constant under low-loss conditions ($R \ll \omega L$ and $G \ll \omega C$) is governed by: $$\alpha \approx \frac{R}{2 Z_0} + \frac{G Z_0}{2}$$ where $Z_0 = \sqrt{L/C}$ is the characteristic line impedance. Notice that the first term ($R / 2Z_0$) accounts for the series conductor resistance, while the second term ($G Z_0 / 2$) accounts for the shunt dielectric conductance.

2. The Two-Term Empirical Attenuation Formula: α(f) = k₁√f + k₂f

Because pure analytical formulas require microscopic knowledge of conductor surface roughness, braid weave optical coverage, and microscopic dielectric inclusions, RF cable manufacturers characterize line loss using the classic two-term empirical formula:

The physical origin of each term reveals why transmission lines behave differently across spectrum bands:

• The √f Term (k₁ × √f — Conductor Skin Losses): At alternating radio frequencies, the magnetic flux inside the metal forces electrical current to concentrate in a thin outer boundary known as the skin depth: $$\delta = \sqrt{\frac{\rho}{\pi f \mu_0 \mu_r}}$$ Because effective cross-sectional conducting area shrinks proportionally to $\delta \propto 1/\sqrt{f}$, high-frequency AC resistance increases in direct proportion to $\sqrt{f}$. At HF and lower VHF frequencies (below $150\text{ MHz}$), the $k_1 \sqrt{f}$ term accounts for more than $90\%$ of all cable loss.
• The Linear f Term (k₂ × f — Dielectric Dissipation): The insulating dielectric substrate between inner and outer conductors has an alternating electric displacement field running through it. The dielectric loss factor is governed by its loss tangent $\tan\delta = \epsilon'' / \epsilon'$. The shunt dielectric conductance per unit length is: $$G = \omega C \tan\delta = 2\pi f C \tan\delta$$ Because each cycle of the RF sine wave forces polar molecules to re-orient and dissipate kinetic heat, dielectric loss scales linearly with frequency $f$. As frequencies push above $1\text{ GHz}$ (Wi-Fi, 5G, and satellite Ku-band), the linear $k_2 f$ term grows rapidly, eventually rivaling or exceeding conductor skin losses in solid polyethylene cables.

3. Coaxial Geometry Optimization: The 50 Ω vs. 75 Ω Paradigm

For a coaxial geometry with inner conductor outer radius $a$ and outer conductor inner radius $b$, the characteristic impedance is: $$Z_0 = \frac{138}{\sqrt{\epsilon_r}} \log_{10}\left(\frac{b}{a}\right) = \frac{60}{\sqrt{\epsilon_r}} \ln\left(\frac{b}{a}\right)$$ Analytical optimization demonstrates three conflicting physical maxima for air-dielectric lines:

1. Minimum Attenuation ($b/a \approx 3.59$ → $Z_0 \approx 76.7\ \Omega$): By differentiating total conductor resistance with respect to the radius ratio $b/a$, minimum RF skin loss occurs when $b/a = 3.59$. This is the mathematical reason why video broadcast, cable television (CATV), and master antenna satellite systems standardized on 75 Ω coaxial lines (such as RG-6/U), where minimizing microvolt signal attenuation across kilometer-long residential trunk drops is paramount.
2. Maximum Voltage Breakdown ($b/a \approx 2.72$ → $Z_0 \approx 60\ \Omega$): The peak radial electric field at the inner conductor surface is $E_{\text{max}} = V / [a \ln(b/a)]$. Maximizing dielectric breakdown voltage occurs at $b/a = e \approx 2.718$, yielding $Z_0 \approx 60\ \Omega$.
3. Maximum Continuous Power Handling ($b/a \approx 1.65$ → $Z_0 \approx 30\ \Omega$): Maximizing the transfer of raw RF power before thermal core destruction occurs at approximately $30\ \Omega$.

The standard 50 Ω line represents the historical IEEE and military compromise: it sits squarely between the peak power handling of $30\ \Omega$ and the minimum attenuation of $77\ \Omega$, while matching quarter-wave ground-plane monopole radiation resistances ($36.8\ \Omega$) and folded dipole transformations exceptionally well.

4. Thermal Power Dissipation & High-Power Derating Factors

All power lost in a transmission line ($P_{\text{loss}} = P_{\text{in}} - P_{\text{out}}$) is converted into internal thermal heat. In high-power commercial transmitters ($100\text{ W}$ to $50\text{ kW}$), feeder heat generation creates serious engineering failure modes:

• Core Melt and Dielectric Migration: Excessive inner conductor temperatures cause solid or cellular polyethylene to soften. Under gravitational pull or mechanical bending radii, the center conductor slowly migrates off-center toward the outer shield, altering $Z_0$, creating severe local impedance lumps, and eventually forming an electrical dead short.
• Ambient Temperature Derating: Cable manufacturer power ratings are universally specified at an ambient room temperature of $+20^\circ\text{C}$ ($68^\circ\text{F}$) under sea-level convection. In outdoor rooftop deployments where ambient solar loading elevates feeder jacket temperatures to $+50^\circ\text{C}$ or $+60^\circ\text{C}$, maximum allowable transmitter power must be derated by $30\%$ to $50\%$ to prevent catastrophic jacket thermal runaway.
• VSWR Multiplier Penalties: When an antenna load is mismatched (e.g., VSWR $> 2.0:1$), the standing wave creates voltage maximums ($V_{\text{max}} = V_{\text{inc}}(1 + |\Gamma|)$) and current maximums ($I_{\text{max}} = I_{\text{inc}}(1 + |\Gamma|)$). Local dielectric heating increases by $(1 + |\Gamma|)^2$ at voltage antinodes, while localized $I^2 R$ conductor heating spikes dramatically at current antinodes, creating intense localized hot spots along the feeder run.