EC-2.4 Transmission Lines

What this is and why it exists

A transmission line is where circuit theory and field theory meet. It is described by voltages and currents like a circuit, but only because the wave picture has been folded into the model.

One idea organises the whole topic. A reflection happens when the line's impedance and the load's impedance disagree. Everything after that is either measuring that reflection or removing it.

The vocabulary

• Distributed elements — inductance and capacitance spread along a line rather than lumped at points.
• Characteristic impedance — the ratio of voltage to current for a wave travelling with nothing to reflect from.
• Propagation constant — how the wave decays and how its phase advances per unit length.
• Reflection coefficient — the fraction of the arriving wave that returns.
• Standing wave — the fixed pattern formed when forward and returning waves add.
• Standing wave ratio — the ratio of the pattern's maximum to its minimum.
• Quarter-wave transformer — a quarter wavelength of line used to match two resistances.
• Stub — a short length of line, open or shorted at its end, used to cancel a reactance.
• Smith chart — a disc on which every impedance is a point and travel along a line is a rotation.
• Scattering parameters — a description in terms of waves reflected and transmitted.

The mental model

Start with why a wire stops being a wire. In an ordinary circuit the voltage at one end of a connection equals the voltage at the other. Once the connection is a noticeable fraction of a wavelength, that is false. The voltage differs along its length, so the inductance and capacitance must be treated as spread out rather than lumped. Two coupled equations follow, and they are the wave equation again in different clothes.

Characteristic impedance is the ratio a wave sees while travelling along the line with nothing to reflect from. It is a property of the geometry and the materials, not of the length. A lossless line has a purely real one, which is why a cable can be specified by a single number such as fifty ohms.

Now the central idea. If the load impedance equals the characteristic impedance, the arriving wave is absorbed entirely and nothing returns. If they differ, part of the wave comes back. The reflection coefficient is the ratio of the returning wave to the arriving one. It is set by the difference between the two impedances, divided by their sum. A short circuit and an open circuit both reflect everything, with opposite signs.

The forward and returning waves then add. Because they travel in opposite directions, their sum is a pattern that stands still in space, with fixed maxima and minima. The ratio of the maximum to the minimum is what an instrument measures, and it is one number reporting how badly matched the load is. A perfectly matched line has a ratio of one.

Impedance transformation follows from the same picture. What a source sees depends on how far it is from the load in wavelengths. The two waves add with a different relative phase at each point. The pattern repeats every half wavelength, so a half wavelength of line shows the load unchanged. A quarter wavelength inverts it, so a short circuit looks like an open circuit and a large resistance looks like a small one. Those two special cases solve a surprising share of practical problems.

The Smith chart is not a relic. It maps every impedance onto a disc so that moving along a line becomes a rotation about the centre. Matching becomes a drawing problem rather than an algebra problem, and you can see what a change does rather than only evaluate it.

Matching has two standard tools, and both aim at the same goal of stopping the reflection. A quarter-wave transformer is a quarter wavelength of line whose impedance is the geometric mean of the two resistances being joined. A single stub is a short piece of line, open or shorted at its far end. Placed at the right distance, it cancels the reactance the load presents.

Scattering parameters close the topic. The two-port sets from circuit theory need ports opened and shorted, and at high frequency neither condition can be created reliably. So networks are described instead by how much of an incident wave reflects and how much passes through. Those are the numbers a network analyser actually reports.

What you should now be able to explain or do

Say when a connection must be treated as a line and why. Compute characteristic impedance and reflection coefficient, and relate the reflection to the standing wave ratio. Transform an impedance along a line, using the half and quarter wavelength cases directly. Read an impedance and a rotation on a Smith chart. Design a quarter-wave transformer or a single stub match, and say why scattering parameters are used at high frequency.

Check yourself