How does adding a series reactance move an impedance point along a Smith chart?

Adding a series element changes only the reactive part of the impedance while leaving resistance unchanged, so the impedance point moves along a constant-R circle, not a constant-X circle. A series inductor increases x and slides the point clockwise (upward) along its resistance circle; a series capacitor decreases x and slides it counterclockwise (downward). Constant-X circles instead form the reference grid you read the new reactance value from: when you stop adding series reactance, the point lands on whichever constant-X arc corresponds to the total normalized reactance x = (omega L minus 1/(omega C)) divided by the system impedance Z0. No single lumped element drives a point along a constant-X arc, because holding reactance fixed while resistance changes is not what a series or shunt element does; a series element moves along a constant-R circle and a shunt element moves along a constant-conductance circle on the admittance overlay.

RF Design

Constant-X Circle

Q: How do you derive the equation of a constant-X circle on the Smith chart?

Start from the bilinear mapping between normalized impedance z = r + jx and reflection coefficient Gamma = (z - 1)/(z + 1). Writing Gamma = u + jv and separating real and imaginary parts, the locus of points with fixed reactance x satisfies (u - 1)^2 + (v - 1/x)^2 = (1/x)^2. This is the equation of a circle centered at (1, 1/x) in the reflection-coefficient plane with radius 1/x. All such circles pass through the point (1, 0), which is the open-circuit location at the right edge of the chart. As x increases toward infinity the radius shrinks toward zero and the circle collapses onto the open-circuit point; as x approaches zero the radius grows without bound and the arc flattens onto the real axis, which is itself the x = 0 contour.

Q: Why are constant-X circles drawn as arcs instead of full circles on a Smith chart?

Mathematically each constant-X locus is a complete circle of radius 1/x centered at (1, 1/x) in the Gamma plane, but only the portion that falls inside the unit circle |Gamma| = 1 corresponds to passive impedances with non-negative resistance. The part of the circle outside the unit boundary maps to negative resistance, which a passive network cannot present, so it is not drawn. The result is the familiar set of arcs that start at the open-circuit point on the right, curve up into the inductive (upper) half or down into the capacitive (lower) half, and terminate where they meet the outer |Gamma| = 1 boundary. The real axis, where x = 0, is the only constant-X contour that appears as a straight line.

/KON-stuhnt eks SUR-kuhl/

One of the two families of circles that form the grid of the Smith chart, the constant-X circle is the locus of every normalized impedance point that shares the same normalized reactance x. In the reflection-coefficient plane each one is a circle of radius 1/x centered at (1, 1/x), and all of them pass through the open-circuit point at the far right of the chart. Only the arc that lies inside the unit circle |Γ| = 1 is drawn, since the remainder maps to physically impossible negative resistance. The upper arcs (x > 0) represent inductive reactance and the lower arcs (x < 0) represent capacitive reactance, with the horizontal real axis being the special x = 0 contour.

Category: RF Design

Circle center (Γ plane): (1, 1/x)

Circle radius: 1/|x|

Reading Reactance off the Smith Chart Grid

The Smith chart overlays two orthogonal families of circles onto the unit disk of the reflection coefficient. The constant-R circles track normalized resistance, while the constant-X circles track normalized reactance. Together they let an engineer read any complex impedance directly as a point and, more importantly, watch how that point migrates as matching elements are added. The constant-X family is the set of arcs that fan out from the open-circuit point on the right edge, sweeping upward for inductive loads and downward for capacitive loads.

Each arc carries a single reactance label such as x = 0.5, x = 1, or x = 2. Because the loci are referenced to the system impedance (almost always 50 ohms), a point sitting on the x = 1 arc at 5 GHz represents an actual reactance of 50 ohms, equivalent to a 1.6 nH series inductor or a 0.64 pF series capacitor depending on which half of the chart it occupies. The arcs crowd together near the right-hand open-circuit point and spread apart toward the center, which is exactly why high-Q, lightly loaded networks are sensitive to small component tolerances when their impedance sits out near the rim.

Geometry in the Reflection-Coefficient Plane

The shape of every constant-X arc follows from the same bilinear transform that defines the entire chart. Substituting z = r + jx into Γ = (z − 1)/(z + 1) and isolating the locus of fixed x produces a true circle whose center always lies on the vertical line u = 1 and whose radius is inversely proportional to the reactance. Large reactance values give tiny circles hugging the open-circuit point; the x = 0 locus degenerates into the straight real axis, which is why it is the only constant-reactance contour that is not visibly curved.

Bilinear mapping of impedance to reflection coefficient:
Γ = (z − 1) / (z + 1), where z = r + jx (normalized to Z₀)

Constant-X locus (with Γ = u + jv):
(u − 1)² + (v − 1/x)² = (1/x)²

Circle geometry:
center = (1, 1/x) | radius = 1/|x|

De-normalizing to actual reactance:
X = x × Z₀ → L = X / (2πf) or C = 1 / (2πf × |X|)

Example: on a 50 Ω chart the x = 1 arc at f = 5 GHz means X = 50 Ω, i.e. L ≈ 1.6 nH (inductive, upper half) or C ≈ 0.64 pF (capacitive, lower half).

Constant-X vs. Constant-R Circles

Property	Constant-X Circle	Constant-R Circle
Quantity held fixed	Normalized reactance x	Normalized resistance r
Center in Γ plane	(1, 1/x)	(r/(r+1), 0)
Radius	1/\|x\|	1/(r+1)
Common point	Open circuit (1, 0)	Open circuit (1, 0)
Appearance	Arcs above and below real axis	Full circles tangent at right edge
Element that moves a point along it	None directly (it is the reactance grid you read; a series element instead moves the point across these arcs)	Series inductor or capacitor
x = 0 / r = 0 special case	The real axis (straight line)	The outer \|Γ\| = 1 boundary

Common Questions

Frequently Asked Questions

How do you derive the equation of a constant-X circle on the Smith chart?

Begin with Γ = (z − 1)/(z + 1) for z = r + jx, write Γ = u + jv, and separate real and imaginary parts. Fixing x gives (u − 1)² + (v − 1/x)² = (1/x)², a circle centered at (1, 1/x) with radius 1/|x|. Every such circle passes through (1, 0), the open-circuit point. As x grows the radius shrinks toward that point; as x → 0 the radius grows without bound and flattens onto the real axis.

Why are constant-X circles drawn as arcs instead of full circles?

Each locus is a complete circle, but only the segment inside |Γ| = 1 corresponds to passive impedances with non-negative resistance. The part outside the unit boundary maps to negative resistance, which a passive load cannot present, so it is omitted. What remains are the arcs that begin at the open-circuit point, curve into the inductive upper half or capacitive lower half, and end at the outer boundary. The x = 0 contour is the lone straight line, the real axis.

How does adding a series reactance move an impedance point relative to these circles?

A series element changes only reactance, so the point travels along a constant-R circle while crossing successive constant-X arcs. A series inductor raises x and slides the point clockwise (up); a series capacitor lowers x and slides it counterclockwise (down). The constant-X grid is what you read the final reactance from: when you stop, the point rests on the arc equal to the net x = (ωL − 1/(ωC))/Z₀.

Related Terms

← Constant-R Circle Constellation Diagram →