The Smith Chart and Impedance Matching

Impedance is unbounded: a resistance can run from zero to infinity, and a reactance from minus to plus infinity. You cannot plot that on a finite page. The reflection coefficient, by contrast, is bounded — for any passive load \(|\Gamma|\le1\), so it always sits inside a circle of radius one. The Smith chart is built on this single observation: work in the \(\Gamma\)-plane, where everything fits, and carry the impedance information along as a printed grid.

The bridge between the two is the relation from Chapter 24, written in normalized form by dividing every impedance by the line's \(Z_0\). That normalization is what makes one chart serve every line, whatever its characteristic impedance.

Define the normalized impedance \(z=Z/Z_0=r+jx\). The load reflection coefficient and its inverse are then a matched pair of bilinear (Möbius) transformations — each a smooth, one-to-one map between the impedance plane and the \(\Gamma\)-plane:

Three anchor points fix the whole picture. A matched load \(z=1\) maps to \(\Gamma=0\), the centre of the chart. A short \(z=0\) maps to \(\Gamma=-1\), the far left. An open \(z\to\infty\) maps to \(\Gamma=+1\), the far right. The entire right-half impedance plane (all passive loads, \(r\ge0\)) folds neatly into the unit disc, and the rim \(|\Gamma|=1\) is the locus of pure reactances — loads that reflect everything.

The magic of the map is that the rectangular grid of constant \(r\) and constant \(x\) in the impedance plane becomes a grid of circles in the \(\Gamma\)-plane. Substituting \(\Gamma=u+jv\) and separating real and imaginary parts gives two families:

Every constant-\(r\) circle passes through the right-hand point \(\Gamma=+1\); larger resistances are smaller circles bunched toward the right. Every constant-\(x\) arc also springs from \(\Gamma=+1\), curving up for inductive (\(+x\)) loads and down for capacitive (\(-x\)) ones, with the real axis itself being \(x=0\). Where an \(r\)-circle crosses an \(x\)-arc is the unique point for that impedance.

To place a load: normalize it to \(z_L=Z_L/Z_0\), find the constant-\(r\) circle for its real part and the constant-\(x\) arc for its imaginary part, and mark where they meet. That point is \(\Gamma_L\): its distance from the centre (as a fraction of the chart radius) is \(|\Gamma_L|\), and the angle it makes from the positive real axis is \(\theta_\Gamma\). Reading backward — from a plotted \(\Gamma\) to the impedance — you just note which \(r\)-circle and \(x\)-arc the point sits on.

Here the chart earns its keep. From Chapter 24, the reflection coefficient at distance \(\ell\) from the load is \(\Gamma(\ell)=\Gamma_L e^{-2j\beta\ell}\). On a lossless line the magnitude is fixed, so the point simply rotates on a circle of constant radius about the centre. Moving toward the generator decreases the phase, which is a clockwise turn; moving toward the load is counter-clockwise:

So the input impedance at any distance is found by spinning the load point clockwise through an angle \(2\beta\ell\) and reading the new grid location — the whole \(\tan\beta\ell\) calculation, done with a compass. The rim of the chart even carries a "wavelengths toward generator" scale so the rotation can be measured directly.

The constant-radius circle the point travels on is the SWR circle. Because it crosses the positive real axis where the impedance is purely resistive and maximal, that crossing reads the standing-wave ratio directly: there \(z=s\). The opposite crossing gives \(z=1/s\), the voltage-minimum point. Drawing this one circle therefore captures the match quality and both reference points at a glance:

Many matching problems are easier in admittance, because shunt elements add admittances. The normalized admittance is \(y=1/z=(1-\Gamma)/(1+\Gamma)\), and on the chart taking the reciprocal is just a 180° rotation through the centre — the point diametrically opposite \(z\) on the same SWR circle. Read with the same grid, the chart now gives conductance circles and susceptance arcs for free.

Matching means driving the operating point to the centre of the chart (\(z=1\), \(\Gamma=0\)), where there is no reflection and all the power reaches the load. Two methods dominate, and both are natural on the chart.

The quarter-wave transformer of Chapter 24 handles a real load: a \(\lambda/4\) section of impedance \(Z_0'=\sqrt{Z_0 R_L}\) inverts the load onto \(Z_0\). On the chart this is the half-turn from a real load on one side to \(z=1\) on the other.

The single-stub tuner matches any load, on the admittance chart, in two moves. First, run along the line (rotate toward the generator) until the point lands on the \(g=1\) circle — at that distance \(d\) the admittance is \(y=1+jb\), the right conductance but a leftover susceptance. Second, connect a shunt stub — a short- or open-circuited length of line — chosen to add exactly \(-jb\), cancelling the susceptance and dropping the point onto the centre. The stub contributes only susceptance because, terminated in a short or open, it is purely reactive (Chapter 24). Two lengths, no lumped parts, perfect match.

For each item, normalize first (\(z=Z/Z_0\)); the chart only knows normalized values. Remember that distance along the line is an angle (\(2\beta\ell\)), clockwise toward the generator. Difficulty rises down the list.