Constant-R Circle
How the Resistance Locus Maps onto the Smith Chart
The Smith chart is a conformal map of the normalized impedance plane onto the unit circle of the complex reflection coefficient Γ = (z − 1)/(z + 1), where z = Z/Z0 is the load impedance normalized to the system characteristic impedance, usually 50 Ω. When that mapping is applied to a vertical line of constant resistance in the z plane (the set r = const, x varying from −∞ to +∞), the straight line transforms into a circle. That image is the constant-R circle. Because the transform is conformal, the right-angle intersections between resistance lines and reactance lines in the impedance plane are preserved, so constant-R circles cross constant-X arcs orthogonally everywhere on the chart.
Every member of the family is tangent to the outer boundary at the same spot, the open-circuit point Γ = 1. The r = 0 circle is the chart's outer rim, representing a purely reactive load with no dissipation. As r increases, the circles march rightward and shrink: r = 0.5 has radius 0.667, r = 1 has radius 0.5 and threads through the exact center of the chart, and large values of r crowd into a tiny circle hugging the open-circuit point. The center of the chart, Γ = 0, corresponds to r = 1 and x = 0, the perfectly matched 50 Ω condition that every matching network is driving toward.
Series Matching Along a Constant-R Circle
The practical payoff is in matching. A series inductor or capacitor adds purely reactive impedance, so it cannot change r; it can only move the operating point up or down in reactance. On the Smith chart that motion is constrained to the impedance's own constant-R circle: a series inductor walks the point clockwise toward higher inductive reactance, and a series capacitor walks it counterclockwise toward higher capacitive reactance. Designers pair this with shunt elements (which travel constant-conductance circles on the admittance overlay) to reach the center. The constant-R circle that passes through the chart center, r = 1, is especially important because any point landing on it can be matched to 50 Ω with a single series reactance.
Center and Radius Equations
Γ = (z − 1) / (z + 1), z = r + jx (normalized to Z0)
Constant-R circle (in the Γ plane, Γ = u + jv):
(u − r/(r+1))2 + v2 = (1/(r+1))2
Center: (r/(r+1), 0) Radius: 1/(r+1)
Where r = R/Z0 is the normalized resistance. Example: for r = 1 the center is (0.5, 0) and radius is 0.5, so the circle passes through the chart center Γ = 0; for r = 0 the center is the origin and radius is 1, giving the unit boundary. All circles are tangent at Γ = 1.
Constant-R Circle Geometry by Normalized Resistance
| Normalized r | Resistance at 50 Ω | Center (u, v) | Radius | Notable property |
|---|---|---|---|---|
| 0 | 0 Ω | (0, 0) | 1.000 | Unit boundary (lossless) |
| 0.2 | 10 Ω | (0.167, 0) | 0.833 | Large, near outer rim |
| 0.5 | 25 Ω | (0.333, 0) | 0.667 | Common L-network target |
| 1.0 | 50 Ω | (0.500, 0) | 0.500 | Passes through chart center |
| 2.0 | 100 Ω | (0.667, 0) | 0.333 | Right-side, smaller |
| 5.0 | 250 Ω | (0.833, 0) | 0.167 | Tight circle near open |
| → ∞ | open | (1, 0) | 0.000 | Collapses to open-circuit point |
Frequently Asked Questions
What are the center and radius of a constant-R circle on the Smith chart?
For normalized resistance r, the circle is centered on the real axis at (r/(r+1), 0) with radius 1/(r+1). The r = 0 circle is the unit boundary (center at origin, radius 1); r = 1 sits at (0.5, 0) with radius 0.5 and passes through the chart center; and as r grows toward infinity the circle shrinks to the single open-circuit point (1, 0). Every circle in the family passes through that same point at the far right edge.
Why does moving along a constant-R circle correspond to adding series reactance?
A series element adds to the reactive part x while leaving the resistive part r unchanged. Since the constant-R circle is exactly the locus of points with that fixed r, the impedance slides along it: a series inductor moves it clockwise (more inductive), a series capacitor counterclockwise (more capacitive). Series matching steps are therefore always drawn as arcs on constant-R circles, while shunt steps follow constant-conductance circles on the admittance chart.
How do constant-R circles relate to constant-X circles?
Constant-R circles map lines of fixed resistance and constant-X circles map lines of fixed reactance; together they form the orthogonal grid of the impedance Smith chart, intersecting at right angles. Any z = r + jx sits at the unique crossing of its r circle and its x arc. Constant-R circles are full circles tangent at (1, 0); constant-X arcs emanate from that same point and curve into the inductive (upper) or capacitive (lower) half.