Q5Electromagnetics Waves

Construction of the Smith chart: the Smith chart is built on the complex reflection-coefficient plane Γ = Γr + jΓi, restricted to the region |Γ| ≤ 1 (a unit circle, since a passive load cannot produce |Γ| > 1). The bilinear (Möbius) transformation relating Γ to normalized load impedance z = Z/Z0, z = (1+Γ)/(1−Γ), maps every constant-resistance line (r = constant) in the z-plane to a circle in the Γ-plane, and every constant-reactance line (x = constant) in the z-plane to a circular arc in the Γ-plane. Plotting the family of these constant-r circles (all passing through the point Γ = +1, i.e. z = ∞, the open-circuit point) and constant-x arcs (all passing through the same point Γ = +1, with the r=0 axis — the |Γ|=1 outer boundary — running vertically as the reactance axis) inside the unit Γ-circle produces the Smith chart grid. The chart's centre (Γ=0) is z=1 (matched, Z=Z0); the left extreme (Γ=−1) is z=0 (short circuit); the right extreme (Γ=+1) is z=∞ (open circuit).

Application 1 — VSWR and reflection coefficient: once the normalized load zL is plotted as a point on the chart, drawing a circle centred at the chart's centre through that point gives the constant-|Γ| (constant-VSWR) circle; the radius of that circle, read directly off the horizontal (resistance) axis where the circle crosses it on the right side, equals VSWR directly (since on the real axis z = r = VSWR when Γ is real and positive) — avoiding any separate numerical computation of |Γ| or VSWR.

Application 2 — impedance along the line: moving along the constant-VSWR circle corresponds exactly to moving along the actual transmission line (a physical distance d toward the generator corresponds to rotating clockwise around the chart by an angle 4πd/λ, since a full 360° rotation corresponds to d = λ/2). This lets the impedance (or admittance, on the same chart interpreted with a half-turn rotation) at any point along the line be read off graphically without repeatedly evaluating Zin = Z0(ZL+jZ0tanβd)/(Z0+jZLtanβd) by hand.

Application 3 — impedance matching design: single-stub and double-stub matching networks (see Part B, Q.5 of this same paper) are designed almost entirely graphically on the Smith chart: converting zL to yL (a rotation by exactly 180° on the chart, since 1/z at Γ maps to −Γ), finding the intersection of the constant-|Γ| circle with the g=1 circle to fix the stub tap point, and reading the stub length directly from the angular position on the outer (|Γ|=1, purely reactive) rim of the chart — the same graphical procedure generalizes to quarter-wave transformer design and to reading off input impedance/admittance for arbitrary combinations of line lengths and reactive loads, making the Smith chart the standard practical tool of RF/microwave engineers for matching-network design without needing repeated complex-number calculation by hand.

Beyond manual design, the same underlying Γ-to-z bilinear mapping is what modern RF CAD software (network analyzers, circuit simulators) still displays as its default impedance-plotting format, because engineers trained on the graphical Smith chart find it far easier to judge match quality, bandwidth, and stability visually on this chart than from a raw table of complex impedance numbers — making the Smith chart as much a standard visualization convention today as it originally was a manual calculation aid.