Transmission Line Equations and Parameters

Ordinary circuit theory makes a silent assumption: that a change in voltage appears everywhere on a wire at once. That holds when the wire is electrically short — far shorter than a wavelength — because the signal crosses it in a time too brief to matter. But a signal travels at roughly the speed of light, so its wavelength shrinks as frequency rises: \(\lambda=u/f\). A one-metre cable is negligible at \(50\,\text{Hz}\) (\(\lambda=6000\,\text{km}\)) yet is several wavelengths long at \(1\,\text{GHz}\) (\(\lambda\approx0.3\,\text{m}\)).

Once the line is a meaningful fraction of a wavelength, the voltage genuinely differs from one end to the other at a given instant. The signal becomes a wave on the line. We could attack this with the full field equations of Part 5, but that throws away the comfortable language of \(V\) and \(I\). The transmission-line model keeps that language and recovers the wave behaviour by smearing the circuit elements continuously along the line.

Any two-conductor line is described by four primary constants, each defined per unit length: the series resistance \(R\) (\(\Omega/\text{m}\)) of the conductors, the series inductance \(L\) (\(\text{H/m}\)) from the magnetic field around them, the shunt conductance \(G\) (\(\text{S/m}\)) of the imperfect dielectric between them, and the shunt capacitance \(C\) (\(\text{F/m}\)) of that dielectric. Their values depend only on the cross-section geometry and the materials, not on the signal:

The standard closed forms for the three workhorse geometries are collected below, with \(R_s=\sqrt{\pi f\mu_c/\sigma_c}\) the conductor's surface resistance (the skin-effect resistance of one square of metal):

Two relations cut across every row and are worth keeping in mind: the product \(LC=\mu\varepsilon\) and the ratio \(G/C=\sigma/\varepsilon\) hold for all these lines. They say the wave's speed depends only on the dielectric, and that the loss is fixed by the dielectric's conductivity — geometry cancels out.

Take a slice of line of infinitesimal length \(\Delta z\). It carries series resistance \(R\,\Delta z\) and inductance \(L\,\Delta z\) along the conductors, and shunt conductance \(G\,\Delta z\) and capacitance \(C\,\Delta z\) across the gap. This is the elementary cell whose repetition is the line:

Kirchhoff's voltage law around the top loop says the voltage drop across the section equals the series impedance times the current; Kirchhoff's current law at the right node says the current lost across the section equals the shunt admittance times the voltage. Dividing by \(\Delta z\) and letting it shrink to zero gives a coupled pair — the telegrapher's equations:

It is convenient to name the series impedance and shunt admittance per unit length, \(Z=R+j\omega L\) and \(Y=G+j\omega C\), so the pair reads \(dV_s/dz=-Z I_s\) and \(dI_s/dz=-Y V_s\). Voltage and current are now locked together, each feeding the other along the line.

Differentiate one telegrapher equation and substitute the other to decouple them. Each variable then obeys a one-dimensional wave equation identical in form to the field equations of Part 5:

The complex propagation constant splits into a real attenuation constant \(\alpha\) (nepers per metre, how fast the wave decays) and an imaginary phase constant \(\beta\) (radians per metre, how fast its phase advances):

The general solution is the sum of two travelling waves — one moving in \(+z\), one in \(-z\) — just as on a plane-wave problem at a boundary:

Substituting the voltage solution back into a telegrapher equation gives the current, and the ratio of voltage to current within a single travelling wave turns out to be a constant of the line — its characteristic impedance:

With it, the current wave mirrors the voltage wave, the backward term carrying a minus sign because it propagates the other way:

The meaning of \(Z_0\) is physical, not just algebraic: it is the impedance a forward wave "sees" as it advances, and the ratio of voltage to current it sets up. A line terminated in a load equal to \(Z_0\) launches only a forward wave — there is nothing to reflect — which is why \(Z_0\) is the target value for matching, the subject of the next two chapters. The propagation constant \(\gamma\) and the characteristic impedance \(Z_0\) together are the line's secondary constants, derived from the four primary ones, and between them they describe everything the line does to a signal.

At high frequencies a good line has negligible loss: \(R\ll\omega L\) and \(G\ll\omega C\), so set \(R=G=0\). The secondary constants then become beautifully simple — the attenuation vanishes and the characteristic impedance is purely real:

Because \(LC=\mu\varepsilon\), the phase velocity \(u=1/\sqrt{LC}=1/\sqrt{\mu\varepsilon}\) is exactly the speed of a plane wave in the same dielectric — the line carries the wave at the medium's own velocity, slowed from \(c\) only by the relative permittivity. For an air-filled coaxial line, combining \(L\) and \(C\) from the table gives the often-quoted design formula \(Z_0=\frac{1}{2\pi}\sqrt{\mu/\varepsilon}\,\ln(b/a)\approx 60\ln(b/a)\,\Omega\). Almost all practical RF and microwave work assumes the lossless line; the small real loss is added back afterward as a perturbation.

A line distorts a signal when its different frequency components travel at different speeds or fade by different amounts, so the pulse shape spreads as it goes. Heaviside found the condition that prevents this on a lossy line: arrange the parameters so that the series and shunt branches lose in the same proportion,

Under this single condition the propagation constant and characteristic impedance collapse to the clean forms below — the attenuation \(\alpha\) is a constant independent of frequency, and the velocity \(u=1/\sqrt{LC}\) is the same for every frequency, so every component arrives together and the waveform is preserved (merely attenuated):

Real cables have too little series inductance to satisfy the condition naturally, so early telephone engineers added it deliberately — "loading" lines with series coils at intervals to raise \(L\) until \(R/L\) dropped to meet \(G/C\). It was the practical triumph that made long-distance telephony intelligible. Note that a lossless line is just the special distortionless case with \(R=G=0\).

For each item, decide first whether the line is lossless, distortionless, or fully lossy — that sets which formula for \(\gamma\) and \(Z_0\) applies. Difficulty rises down the list.