Wave Propagation on Transmission Lines

Take the lossless line of Chapter 23, characteristic impedance \(Z_0\), and connect a load \(Z_L\) at its far end. A source launches a forward wave toward the load; whatever the load cannot absorb it sends back as a reflected wave. We measure position by the distance \(\ell\) from the load back toward the generator, so the load sits at \(\ell=0\) and the generator at \(\ell=\) (line length). The total voltage and current are the sum of the two travelling waves:

At the load the ratio of voltage to current must equal \(Z_L\). Imposing \(V(0)/I(0)=Z_L\) on the expressions above solves immediately for the ratio of reflected to incident voltage — the load reflection coefficient:

This is the same formula as the plane-wave reflection coefficient at a boundary in Chapter 22, with \(Z_L\) and \(Z_0\) playing the roles of \(\eta_2\) and \(\eta_1\) — transmission-line reflection is boundary reflection, dressed in circuit language. Its value tells the whole story of the termination: a matched load (\(Z_L=Z_0\)) gives \(\Gamma_L=0\) and no reflection; a short (\(Z_L=0\)) gives \(\Gamma_L=-1\); an open (\(Z_L\to\infty\)) gives \(\Gamma_L=+1\). For any passive load \(|\Gamma_L|\le1\). A complex \(Z_L\) makes \(\Gamma_L\) complex, with a phase angle \(\theta_\Gamma\) that we will need in a moment.

The reflection coefficient is not confined to the load — we can define it at any point as the ratio of the reflected to the incident voltage there. Moving a distance \(\ell\) back toward the generator, the forward wave gains phase \(e^{j\beta\ell}\) and the reflected wave loses it, so their ratio winds by \(e^{-2j\beta\ell}\):

On a lossless line the magnitude \(|\Gamma(\ell)|=|\Gamma_L|\) never changes — only the phase rotates, completing a full turn every half wavelength (since \(2\beta\cdot\tfrac{\lambda}{2}=2\pi\)). That periodicity is why everything on a line repeats every \(\lambda/2\), and it is the kinematic heart of the Smith chart. On a lossy line the factor \(e^{-2\alpha\ell}\) also shrinks the magnitude as you retreat from the load, spiralling \(\Gamma\) inward.

The impedance seen looking into the line at distance \(\ell\) is \(Z_{\text{in}}(\ell)=V(\ell)/I(\ell)\). Writing it through the local reflection coefficient gives a compact form, and substituting \(\Gamma(\ell)\) yields the celebrated input-impedance equation:

For the lossless line \(\gamma=j\beta\) and \(\tanh(j\beta\ell)=j\tan\beta\ell\), giving the form used in nearly every problem:

This one equation replaces a whole network of reactances: knowing \(Z_0\), the load, and how many wavelengths of line stand between you and it, you can read off the impedance the generator actually drives. It is periodic in \(\ell\) with period \(\lambda/2\) — the same repetition the reflection coefficient showed — and it is the formula behind every length the rest of the chapter chooses on purpose.

Three terminations are worth singling out, both because they are common and because they reveal what a length of line can do on its own. Setting \(Z_L=0\), \(Z_L\to\infty\), and \(Z_L=Z_0\) in the lossless formula gives:

Both the shorted and open lines look purely reactive from the input — no power is dissipated, so all of it returns. By choosing the length \(\ell\), a shorted stub can be made to present any inductance or capacitance you like, which is how reactive elements are built at microwave frequencies where lumped coils and capacitors fail. A neat by-product: measuring the shorted and open input impedances of an unknown line at the same length gives its characteristic impedance directly, \(Z_0=\sqrt{Z_{\text{sc}}\,Z_{\text{oc}}}\).

Wherever a reflection exists, the forward and backward waves interfere. Their sum has a magnitude that varies with position: \(|V(\ell)|=|V_0^{+}|\,|1+\Gamma_L e^{-2j\beta\ell}|\). It peaks where the two waves add in phase and dips where they oppose, producing a fixed standing-wave pattern:

The ratio of these is the standing-wave ratio — identical to the one defined for plane waves in Chapter 22, and the single number most often used to judge a match:

A matched line has \(s=1\) (flat, no standing wave); a short or open has \(s\to\infty\). The pattern carries usable information: at a voltage maximum the line impedance is purely real and equal to \(sZ_0\); at a voltage minimum it is \(Z_0/s\). Sliding a probe along a slotted line to find these extrema was the classic way to measure an unknown load before network analysers — and the voltage minima, sharp and easy to locate, were the preferred reference points.

Two special lengths drop out of the input-impedance formula and are used constantly. A half-wavelength of line (\(\beta\ell=\pi\), so \(\tan\beta\ell=0\)) returns the load impedance unchanged — the line is invisible, repeating whatever it is terminated in every \(\lambda/2\):

A quarter-wavelength (\(\beta\ell=\pi/2\), so \(\tan\beta\ell\to\infty\)) does the opposite — it inverts the load about \(Z_0\), turning a high impedance into a low one and vice versa:

That inversion is a ready-made matching tool. To match a real load \(R_L\) to a line of impedance \(Z_0\), splice in a quarter-wave section whose own impedance is the geometric mean of the two — a quarter-wave transformer:

It is exact only at the frequency where the section is precisely \(\lambda/4\), so it is inherently narrowband — but simple, lossless, and everywhere in antenna feeds and microwave circuits. Matching a complex load, or doing it over a band, is the broader problem the next chapter solves graphically with the Smith chart.

For each item, find \(\Gamma_L\) first — it fixes the SWR and feeds the input-impedance formula. Watch the line length in wavelengths, since \(\tan\beta\ell\) is what does the work. Difficulty rises down the list.