Article

What the Smith chart is actually showing you

The Smith chart looks like an eye test. It is actually one simple idea drawn carefully: the whole right half of the impedance plane, folded into a circle so that infinity fits on the page.

The fold

Impedance is awkward to plot. Resistance runs from zero to infinity and reactance from minus infinity to plus infinity, so any honest graph of it runs off the paper immediately.

So plot the reflection coefficient instead:

gamma = (Z - Z0) / (Z + Z0)

For any passive load, gamma has a magnitude of 1 or less. The infinite half-plane becomes a disc of radius one, and everything fits. The chart is that disc, with the old impedance grid drawn on top of it, bent by the fold.

Three landmarks fall out immediately:

Where the circles come from

The curved grid is not decoration. Lines of constant resistance in the impedance plane become circles under the fold, and so do lines of constant reactance. You can work out exactly where.

A constant resistance circle has centre r/(1+r) and radius 1/(1+r). Every one of them passes through the far right point, which is the open circuit, because any resistance with infinite reactance looks like an open. The r = 1 circle passes through the centre, which is why it matters so much in matching: it is the set of impedances whose real part is already right.

These are the arcs that curve away above and below. Centre (1, 1/x), radius 1/|x|. Above the axis is inductive, below is capacitive, and the axis itself is purely resistive. These are the ones the regression suite checks by arithmetic, because a chart drawn slightly wrong is very hard to see and very easy to trust.

Which is why components move you the way they do

Now the useful part. A component in series adds reactance and cannot touch resistance. So it moves you along a constant resistance circle: up for an inductor, down for a capacitor.

Open a 25 ohm load with a series inductor

A component in shunt adds susceptance and cannot touch conductance. So it moves you along a constant conductance circle, which is the same family of circles mirrored about the centre. Turn on Admittance grid in the Overlays panel and they appear.

Open the same load with a shunt capacitor

That is the entire vocabulary. Series moves you round one family, shunt round the other, and matching is finding a route between them that ends in the middle. Everything else on the chart is bookkeeping.

Feeling it rather than reading it

There is a faster way to internalise this than reading about it. Put a component in the ladder, then grab its node on the chart and pull.

It will not go where you drag it. It goes to the nearest point on its own curve, because that is the only place that component can put it. Drag a series part and you slide round a constant resistance circle. Drag a shunt part and you slide round a constant conductance circle. Ten seconds of that teaches more than a page of this.

And the transmission line, which does neither

A length of line does not add series or shunt anything. It rotates the whole reflection coefficient about the centre, clockwise as you move away from the load, at constant radius: one complete turn per half wavelength.

Constant radius means constant |gamma|, which means constant VSWR. A lossless line does not improve or worsen the match, it just moves it around the clock. That is why the VSWR circle overlay is so useful with lines: your dot travels along it rather than across it.

What the colours are for

In QuickSmith each component draws its own arc in its own colour, from the node before it to the node after. So the path from the load to Zin is attributable: you can see which part did which piece of the journey, and when a network does not do what you expected, you can see which part is not pulling its weight.

If you want a guided version, Help → Guided examples walks through the worked designs one component at a time, saying what each one does and why it goes where it goes. Example 1 is exactly the exercise in this article: one part at a time, watching the trace.