Lossy transmission line

A lossy transmission line has nonzero $R^{\prime }$ and/or $G^{\prime }$: finite conductor resistance per unit length and/or finite shunt conductance per unit length. As a wave propagates, its amplitude decays exponentially and its phase velocity may depend on frequency. This is the realistic model of any cable carrying RF or high-speed signals over appreciable distances.

A lossless approximation (where $R^{\prime }=G^{\prime }=0$) is convenient, but lossy effects limit cable length, system bandwidth, and receiver sensitivity.

Distributed parameters

The lumped-element model retains all four parameters:

$R^{\prime }\text{ (Ω/m)},L^{\prime }\text{ (H/m)},G^{\prime }\text{ (S/m)},C^{\prime }\text{ (F/m)}.$

• $R^{\prime }$ originates from finite conductor conductivity. At high frequency, skin effect concentrates current in a thin surface layer, increasing $R^{\prime }$ as $\sqrt{f}$.
• $G^{\prime }$ originates from dielectric loss, imperfect insulators dissipating energy. For real polyethylene at room temperature, $G^{\prime }$ scales roughly with $f$.
• $L^{\prime }$ and $C^{\prime }$ are nearly the same as in the lossless case; they describe the magnetic and electric field storage of the line cross-section.

Propagation constant and characteristic impedance

The Propagation constant becomes complex:

$\gamma =\sqrt{(R^{\prime }+j\omega L^{\prime })(G^{\prime }+j\omega C^{\prime })}=\alpha +j\beta .$

$\alpha$ (Np/m) is the attenuation constant, how fast wave amplitude drops with distance. $\beta$ (rad/m) is the phase constant.

The Characteristic impedance becomes complex too:

$Z_0=\sqrt{\frac{R^{\prime }+j\omega L^{\prime }}{G^{\prime }+j\omega C^{\prime }}}.$

In general $Z_0\in \mathbb{C}$, with a small phase angle. A lossless line has real $Z_0$ ($=\sqrt{L^{\prime }/C^{\prime }}$); a lossy line has $Z_0=R_0+j{X}_0$ with $X_0$ small but nonzero.

Forward and backward waves

The wave solutions on a lossy line:

$\tilde{V}(z)=V_0^{+}{e}^{-\alpha z}{e}^{-j\beta z}+V_0^{-}{e}^{+\alpha z}{e}^{+j\beta z}.$

Forward wave amplitude shrinks as $e^{-\alpha z}$. Backward wave amplitude grows in $+z$, because it’s traveling in $-z$ and decaying as it goes. So the standing-wave pattern on a mismatched lossy line is less pronounced near the source (where the forward wave is strongest relative to the reflected wave that has decayed twice) than near the load.

Low-loss approximation

For most engineering lines at moderate frequencies, $R^{\prime }\ll \omega L^{\prime }$ and $G^{\prime }\ll \omega C^{\prime }$. Expanding $\gamma$ to first order in $R^{\prime }/(\omega L^{\prime })$ and $G^{\prime }/(\omega C^{\prime })$:

$\alpha \approx \frac{1}{2}\left(\frac{R^{\prime }}{Z_0}+G^{\prime }{Z}_0\right),\beta \approx \omega \sqrt{L^{\prime }{C}^{\prime }}.$

$\alpha$ splits into a conductor-loss contribution ($R^{\prime }/(2Z_0)$) and a dielectric-loss contribution ($G^{\prime }{Z}_0/2$). Designers minimize one or the other depending on which dominates.

$\beta$ in this limit is just the lossless value — small losses don’t shift the wave speed much. $Z_0\approx \sqrt{L^{\prime }/C^{\prime }}$ remains nearly real, justifying the lossless calculation for many practical purposes.

When losses cease to be “small”

The low-loss expansion fails when:

• $\omega \to 0$ (DC limit): $R^{\prime }$ dominates over $\omega L^{\prime }$, $G^{\prime }$ dominates over $\omega C^{\prime }$. The line transitions to a diffusive regime described by the “telegraph equation” with damping, the case for high-loss audio and power lines.
• $\omega \to \infty$: skin effect makes $R^{\prime }\propto \sqrt{\omega }$. The conductor loss eventually keeps up with $\omega L^{\prime }$ in proportion.

At microwave frequencies (multi-GHz) in a typical RG-58 coax, conductor losses become the bottleneck. Below 1 GHz, both losses are small enough that “low-loss” formulas are accurate to within a few percent.

Power and decay

The time-averaged power in the forward wave decays as

$P_{\text{av}}(z)=\frac{|V_0^{+}{|}^2}{2Z_0}{e}^{-2\alpha z}.$

Twice the attenuation in power because power ∝ amplitude². For a 100 m cable with $\alpha =0.05$ Np/m: $e^{-2\cdot 0.05\cdot 100}=e^{-10}\approx 4.5\times 10^{-5}$. About 90 dB of attenuation, which may or may not be acceptable depending on system budget.

In dB: ${\alpha }_{\text{dB/m}}=8.686{\alpha }_{\text{Np/m}}$. Cables are commonly specified in dB/100m at a few representative frequencies.

Distortion

A pulse on a lossy line spreads because $\alpha$ and $\beta$ are frequency-dependent. Different Fourier components attenuate and delay at different rates. The result:

• Amplitude distortion: high frequencies attenuate more (in most media), low-pass filtering the pulse.
• Phase distortion (dispersion): different frequency components arrive at different times, smearing the pulse temporally.

These are the practical signal-integrity issues that limit data rates on long cables. Equalization (frequency-shaping the transmitter or receiver) is the standard countermeasure: pre-emphasis at the source, equalization filter at the receiver. Used in everything from Ethernet to SerDes to coaxial undersea cables.

When matching matters more

On a lossy line with strong reflections, the multiple-bounce pattern between source and load decays as each round trip loses $e^{-2\alpha l}$ (assuming partial reflection at each end). For very lossy lines, even bad mismatches eventually settle, the reflections dying out before causing trouble. For low-loss lines (the regime we usually want), reflections persist for many round trips and matching matters more.

This is why high-frequency precision instruments use low-loss cables AND careful matching: the lower the loss, the more visible the mismatch.

Versus lossless

	Lossless	Lossy
$R^{\prime }$, $G^{\prime }$	0	> 0
$\gamma$	$j\beta$	$\alpha +j\beta$, $\alpha >0$
$Z_0$	Real	Complex (small imaginary part)
$u_p$	$1/\sqrt{\mu ϵ}$	Slightly frequency-dependent
Attenuation	None	$e^{-\alpha z}$
Use	Idealization, lab standards	Real cables

The lossless model gets used for analysis and Smith-chart work; the lossy model is needed for budget calculations, system simulation, and any practical cable specification.