Lift · one curve, five views
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Poles × and zeros ○ of Z(s) · axes scaled by asinh
How to read it. Each rung is one level of the fraction. A series element adds impedance, which slides Γ along a red circle of constant resistance or reactance. A shunt element adds admittance, which slides Γ along a green circle. "Walk the rungs" animates that composition of Möbius maps at the bead's frequency, starting from the load.
Why the disk. Stieltjes: f(z) = 1/(a₁z + 1/(a₂ + 1/(a₃z + …))) with every aₖ > 0 is a Stieltjes function. Cauer: that same fraction is the driving-point impedance of a ladder of shunt capacitors a₁, a₃, … and series resistors a₂, a₄, …. Positive components dissipate, so Re Z ≥ 0 on the imaginary axis and |Γ| ≤ 1. Lossless L and C alone give Re Z = 0 and the trace rides the rim. One negative rung is an amplifier, and the trace is free to leave.
Why the disk is not enough. The Smith chart samples Z on one line, s = iω. A positive-real function also has to be analytic in the whole right half-plane Re s > 0, and a Stieltjes function has every pole and zero on the negative real axis, alternating. A negative rung deep in the ladder can satisfy Re Z ≥ 0 on the line while hiding a pole and a zero at positive s. They nearly cancel, so the line never sees them. The s-plane plot sees them directly. Leaving the disk rules a fraction out. Staying inside rules nothing in.
Three ways two views can disagree. A change of chart loses nothing: Z and Γ are the same Riemann sphere, turned a quarter turn about the axis through ±j, because Γ = (z − 1)/(z + 1) has the unitary matrix (1/√2)[[1, −1], [1, 1]]. A projection drops a coordinate: the flat Smith chart is the frequency column seen end-on, so it shows crossings that are not there. A restriction samples a slice: every view of this curve lives on the line s = iω, so no rotation of it can show a pole that sits off the line. The lift walks through the first two. The s-plane plot handles the third.
The lift grew out of Christopher Bryant's Smith-chart animations in What is a Smith Chart? at Arena Physica. Our thanks to them. Designed and built by Claude Opus 5.5 (Anthropic), with Astra-6 (OpenAI) helping with the mathematics and composition. percolate.space · J. Councilman · 2026.