Smith chart from the impedance–reflection map

Choose a real positive reference impedance Z_ref and define normalized impedance z=Z/Z_ref=r+jx. The associated voltage-wave reflection coefficient is Γ=(z−1)/(z+1).

The inverse map is z=(1+Γ)/(1−Γ). The chart is therefore a one-to-one map between finite normalized impedances and points in the Γ-plane, except for the open-circuit point Γ=+1 where z tends to infinity.

The simple unit-disk Smith chart assumes a real positive Z_ref. Complex reference impedances require a more careful power-wave framework and should not be hidden inside the standard chart notation.

A normalized match z=1 maps to Γ=0 at the chart center. A short z=0 maps to Γ=−1 on the left edge. An open z→∞ maps to Γ→+1 on the right edge.

If the load is passive and Z_ref is real positive, Re(z)=r≥0. Chapter 01 already proved this condition maps inside or on |Γ|≤1. Purely reactive loads r=0 lie on the unit circle.

Thus the chart boundary has physical meaning: it is the locus of lossless one-port terminations under the stated normalization, not merely the edge of a plotting template.

Write Γ=u+jv and solve the inverse map z=(1+Γ)/(1−Γ) for the real part r. Algebra gives r=(1−u²−v²)/[(1−u)²+v²].

Holding r constant and completing the square produces a circle centered on the real Γ-axis: [u−r/(1+r)]²+v²=[1/(1+r)]².

As r increases, the circle shrinks toward Γ=+1. The r=0 circle is the unit circle itself; r=1 passes through the chart center because a normalized resistance of one is matched when x=0.

The imaginary part of z=(1+Γ)/(1−Γ) is x=2v/[(1−u)²+v²]. For nonzero x, completing the square gives (u−1)²+(v−1/x)²=(1/x)².

Positive reactance x>0 lies in the upper half of the impedance chart; negative reactance x<0 lies in the lower half. This sign follows from the exp(+jωt) impedance convention used throughout AoN, where inductive reactance is +jX and capacitive reactance is −jX.

All constant-reactance circles are tangent to Γ=+1 because an infinite normalized resistance overwhelms any finite reactance and approaches the open-circuit point.

RF-01 chapter 01 gave Γ(d)=Γ_Lexp(−j2βd) for distance d measured from the load toward the generator on a lossless line. Therefore |Γ| stays constant while its angle decreases.

On the usual complex Γ-plane, increasing d toward the generator is a CLOCKWISE rotation. A distance λ_g/2 changes the phase by −2π and returns to the same point; λ_g/4 rotates Γ by −π and produces the quarter-wave impedance inversion.

The chart therefore turns reference-plane translation into a simple rotation, but the direction is not a convention to memorize independently: it follows from the previously declared coordinate and propagation signs.

Define normalized admittance y=Y/Y_ref=1/z=g+jb when Y_ref=1/Z_ref. From z=(1+Γ)/(1−Γ), inversion gives y=(1−Γ)/(1+Γ).

If one plotted an admittance reflection map using the same circle family, the corresponding point is obtained by Γ→−Γ: a 180° rotation about the chart center. This lets the same paper chart support both impedance and admittance calculations.

Series additions are naturally described in impedance coordinates: adding a series reactance changes x at fixed r, so the point moves along a constant-resistance circle. Shunt additions are naturally described in admittance coordinates: adding susceptance changes b at fixed g, so the point moves along a constant-conductance circle.

A practical chart workflow must name the physical operation behind every motion. Translating along a uniform lossless line rotates on a constant-|Γ| circle. Adding series reactance moves along a constant-r circle. Adding shunt susceptance is handled in admittance coordinates along a constant-g circle.

A successful match ends at Γ=0, equivalently z=1. But many trajectories can reach the center; the chart alone does not decide which topology, component values, bandwidth, loss or fabrication constraints are best.

This is why RF-01 ends with the chart as a reasoning tool, while RF-06 will treat matching-network synthesis and tradeoffs as a design problem.

Take a normalized load z_L=2+j1 on a real reference impedance. Direct evaluation gives Γ_L=(z_L−1)/(z_L+1)=0.4+j0.2, so |Γ_L|≈0.447 and argΓ≈26.6°.

Move a distance d=λ_g/8 toward the generator. The phase rotation is −2βd=−π/2, giving Γ(d)=0.2−j0.4. Mapping back with z=(1+Γ)/(1−Γ) gives z(d)=1−j1.

A further λ_g/8 reaches λ_g/4 total distance, rotating Γ by another −π/2. The resulting impedance is the quarter-wave inversion z=1/z_L=(2−j1)/5=0.4−j0.2. Every numerical step agrees with the tangent-form line equation from chapter 03.

1. Starting from z=(1+Γ)/(1−Γ) with Γ=u+jv, derive the expression for normalized resistance r.
2. Complete the square to derive the constant-r circle center and radius.
3. Derive the constant-x circle equation for x≠0 and identify which half-plane corresponds to inductive reactance under exp(+jωt).
4. Map z=1, z=0 and z→∞ into the Γ-plane and explain their chart locations.
5. For z_L=0.5+j0.5, calculate Γ_L and verify that the point lies inside the unit circle.
6. A lossless line has Γ_L=0.6∠40°. What is Γ at d=λ_g/8 toward the generator?
7. Why does moving λ_g/2 along a lossless line return to the same point on the Smith chart?
8. Show algebraically that converting normalized impedance to admittance corresponds to Γ→−Γ.
9. Explain the chart motion produced by adding a series inductive reactance and by adding a shunt capacitive susceptance, without choosing component values.
10. For z_L=2+j1, verify the λ_g/8 and λ_g/4 transformed impedances given in the numerical audit.

RF-01 is complete. Reflection coefficient, standing waves, local input impedance, quarter-wave transformations and the Smith chart now form one continuous derivation. The next RF course can build two-port and scattering descriptions without treating the chart as a substitute for the underlying voltage/current physics.