EC-22.1 Impedance, Matching, and the Smith Chart

What this is and why it exists

The transmission line topic derived the reflection coefficient. This one turns it into a working method.

At low frequency a mismatch is a nuisance. At high frequency it decides whether a circuit works at all. The reflected wave interferes with the incident one, and the power never reaches the load.

The Smith chart is the tool for fixing that, and it deserves to be learned properly. It is not a piece of old draughtsmanship. It is a picture of the complex reflection plane, and every high-frequency component you will specify is described in terms of it.

The vocabulary

• Reflection coefficient — the ratio of the reflected wave to the incident wave.
• Standing wave ratio — the ratio of maximum to minimum voltage along a line.
• Return loss — the reflected power, expressed in decibels below the incident power.
• Characteristic impedance — the impedance a line presents to a wave travelling along it.
• Normalised impedance — an impedance divided by the characteristic impedance.
• Matching network — components added to make a load look like the system impedance.
• Stub — a short length of line, open or shorted, used as a reactance.
• Bandwidth of a match — the frequency range over which the match holds.

The mental model

Three numbers describe the same thing, and three different documents will quote three of them. The reflection coefficient is a complex number. The standing wave ratio is a real number derived from its magnitude. Return loss is the same magnitude in decibels. Converting between them by eye saves a great deal of confusion.

The chart is the unit circle in the reflection coefficient plane. Onto that circle are drawn the curves of constant resistance and constant reactance. Once you see it that way, the chart stops being arbitrary. A point is a reflection coefficient, and the curves tell you which impedance produces it.

Two moves matter. Travelling along a lossless line rotates a point around the centre without changing its distance from it. That is the single most used property of the chart, and it is why the chart has angle markings around its edge. Adding a series or shunt element moves a point along one of the drawn curves.

Matching is then a route. From wherever the load sits, add a series element to slide along one family of curves. Then add a shunt element to slide along the other, and arrive at the centre. That is an L network, and there are usually two routes with different bandwidths. Doing it once on the chart and once in algebra makes the chart trustworthy rather than magical.

Above a few gigahertz, components stop behaving like components. A short piece of open or shorted line has a reactance instead, and can be etched rather than soldered. This is the point where board layout becomes circuit design, and it is why microwave boards look the way they do.

Finally, when does a mismatch matter? A modest mismatch reflects a small fraction of the power and is often irrelevant. The same mismatch at the input of a low-noise stage can dominate the design, because it changes the noise figure. The consequence depends on where in the chain the mismatch sits.

What you should now be able to explain or do

• Convert between reflection coefficient, standing wave ratio and return loss.
• Explain what the chart is a picture of, and what its curves represent.
• Say what moving along a line does to a point on the chart.
• Design a two-element matching network, on the chart and in algebra.
• Explain why a length of line can serve as a component at high frequency.
• Say when a mismatch matters and when it can be ignored.

Check yourself