Constant-B Circle
How the Constant-B Family Shapes Shunt Matching
The Smith chart is a conformal map of the reflection coefficient plane, and the same physical disk carries two complementary grids. The impedance grid resolves a normalized impedance z = r + jx into constant-resistance circles and constant-reactance arcs, which is convenient when you add components in series. The admittance grid, obtained by rotating the chart 180 degrees, resolves a normalized admittance y = g + jb into constant-conductance circles and constant-susceptance arcs. Each constant-B arc is the full set of admittances whose imaginary part equals a chosen b, and it is exactly this arc that a designer references when deciding how much shunt susceptance to inject at a given plane along a line.
The reason the admittance view is so useful for parallel elements comes straight from circuit theory: admittances in parallel add. A shunt stub or lumped reactor presents a purely imaginary admittance jB at its connection plane, so attaching it to a load admittance Yload = G + jBload produces Ytotal = G + j(Bload + B). The real part G never moves, which means the operating point slides along a constant-conductance circle while crossing successive constant-B arcs. A practical L-match exploits this by first moving along the line (a rotation around the chart) until the point lands on the g = 1 conductance circle, then adding a shunt susceptance that walks the point along that circle straight to the chart center, where y = 1 + j0 and the match is perfect.
Constant-B circles are not merely a drafting aid. Modern simulators such as ADS, AWR Microwave Office, and the open-source scikit-rf still expose the admittance overlay so engineers can verify by eye that a synthesized network is taking the intended path. At RF Essentials we trace these loci when tuning the bias and matching networks inside millimeter-wave converter and amplifier assemblies, where a few millisiemens of stray shunt susceptance from a bondwire or via can shift the realized match by a noticeable fraction of a VSWR point.
Susceptance Normalization and the Governing Relations
y = Y / Y0 = g + jb, where Y0 = 1 / Z0 = 0.02 S for Z0 = 50 Ω
A constant-B circle (fixed b):
{ y = g + jb : g ≥ 0, b = constant }
Parallel (shunt) combining rule:
Ytotal = Yload + Yshunt ⇒ btotal = bload + bshunt, g unchanged
Shunt element susceptance:
Bcap = +ωC, Bind = −1 / (ωL), b = B / Y0
Example: a 0.32 pF shunt capacitor at 10 GHz gives B = 2π × 1010 × 0.32 pF ≈ 20 mS, so b ≈ +1.0 on a 50 Ω chart.
Constant-B Versus the Other Smith-Chart Circle Families
| Circle family | Chart overlay | Fixed quantity | Moves point when you add | Geometry on disk | Typical use |
|---|---|---|---|---|---|
| Constant-B (susceptance) | Admittance (Y) | b = B/Y0 | Shunt reactance (crosses these arcs) | Arcs tangent at y = ∞ | Shunt stub / lumped shunt tuning |
| Constant-G (conductance) | Admittance (Y) | g = G/Y0 | Slides along when adding shunt B | Circles tangent at y = ∞ | Landing on g = 1 for L-match |
| Constant-X (reactance) | Impedance (Z) | x = X/Z0 | Series reactance (crosses these arcs) | Arcs tangent at z = ∞ | Series stub / series L or C tuning |
| Constant-R (resistance) | Impedance (Z) | r = R/Z0 | Slides along when adding series X | Circles tangent at z = ∞ | Landing on r = 1 for L-match |
| Constant-VSWR | Either | |Γ| constant | Lossless line length (rotation) | Circle centered on chart center | Tracking mismatch along a line |
Frequently Asked Questions
What is the difference between a constant-B circle and a constant-X circle?
They sit on different overlays. A constant-X (reactance) arc lives on the impedance Smith chart and is the natural grid for series elements; a constant-B (susceptance) arc lives on the admittance chart, the impedance chart rotated 180 degrees, and is the natural grid for shunt elements. On the combined Y-Smith chart, constant-B arcs curve in the mirror sense to constant-X arcs, and a designer switches overlays each time the network alternates between series and parallel parts.
Why does adding a shunt element move the point along a constant-B circle?
Parallel admittances add: Ytotal = Yload + Yshunt. A pure shunt stub, inductor, or capacitor contributes only jBshunt with no conductance, so total susceptance b changes while conductance g stays fixed. The point therefore slides along a constant-conductance arc and crosses successive constant-B arcs. That crossing is the design lever in a two-element L-match: hop from a constant-conductance arc onto the b value that lands you at y = 1 + j0.
How do you read normalized susceptance from a Smith chart?
Normalize to Y0 = 1/Z0, which is 0.02 S in a 50 Ω system, giving y = g + jb. On the admittance overlay the upper half holds positive b (capacitive shunt) and the lower half holds negative b (inductive shunt); the outer real axis is b = 0 and the center is y = 1 + j0. To convert from impedance, invert: z = 1 + j1 gives y = 0.5 − j0.5, placing the point on the b = −0.5 constant-B circle in the inductive half.