Paper V: Smith coordinates as cyclic signal coordinates

13 Smith coordinates as cyclic signal coordinates

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Let \(Z_0>0\) be a reference impedance and

\[ z=\frac{Z}{Z_0}=r+{\mathrm i} x. \]

The reflection coefficient is

\[ \boxed{ {\Gamma}=\frac{z-1}{z+1}, \qquad z=\frac{1+{\Gamma}}{1-{\Gamma}}. } \]

Classical input 13.1: Cayley geometry

The map sends the finite passive half-plane \(r\ge0\) onto the closed unit disk with \({\Gamma}=1\) omitted. Adjoining the point at infinity in impedance adds that missing boundary point; the open right half-plane maps conformally onto the open disk. Constant resistance and reactance become the familiar Smith circles. This conformal geometry is classical and is not replaced by polyphase refinement.

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Writing \({\Gamma}=u+{\mathrm i} v\), the grid equations are

\[ \left(u-\frac{r}{1+r}\right)^2+v^2=\frac1{(1+r)^2}, \]

and, for \(x\ne0\),

\[ (u-1)^2+\left(v-\frac1x\right)^2=\frac1{x^2}. \]

Fix

\[ \omega_N=e^{2\pi{\mathrm i}/N} \]

and rotate the reflection coefficient by

\[ R_{N,k}({\Gamma})=\omega_N^k{\Gamma}. \]

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Theorem 13.2: Polyphase Möbius action

Regard normalized impedance as the projective line \(\mathbb P^1({\mathbb C})={\mathbb C}\cup\{\infty\}\). For \(N\ge1\), the pullback is a globally defined projective action. In the finite chart, wherever the denominator is nonzero, it is

\[ \boxed{ T_{N,k}(z) = \frac{(1+\omega_N^k)z+(1-\omega_N^k)} {(1-\omega_N^k)z+(1+\omega_N^k)}. } \]

As maps on \(\mathbb P^1({\mathbb C})\) it satisfies

\[ T_{N,k}\circ T_{N,\ell}=T_{N,k+\ell\, (\mathrm{mod}\,N)}. \]

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Proof of Theorem 13.2

Let \(C(z)=(z-1)/(z+1)\). Then

\[ T_{N,k}=C^{-1}\circ R_{N,k}\circ C. \]

For a division-free construction, represent a point by a nonzero pair \([u:v]\), identifying \([cu:cv]=[u:v]\) for \(c\ne0\). With \(w=\omega_N^k\), use the matrix

\[ A(w)=\frac12\begin{pmatrix}1+w&1-w\\1-w&1+w\end{pmatrix}. \]

It has determinant \(w\ne0\), preserves nonzero pairs, and commutes with rescaling, so it induces a projective map. Direct multiplication gives \(A(w)A(w')=A(ww')\) and \(A(1)=I\). Since \(\omega_N^N=1\), this proves the cyclic composition law modulo \(N\). The Cayley coordinate of \([u:v]\) is \([u-v:u+v]\), and applying \(A(w)\) changes this pair to \([w(u-v):u+v]\). This proves the conjugacy also at poles. For \([z:1]\), taking the coordinate ratio gives the displayed finite-chart formula.

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Corollary 13.3: Impedance–admittance is the two-phase split

For \(N=2\) and finite \(z\ne0\),

\[ \boxed{T_{2,1}(z)=z^{-1}.} \]

On the full projective line it exchanges \(0\) and \(\infty\). Thus a half-turn of the reflection disk exchanges normalized impedance and admittance.

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Proof of Corollary 13.3

\(\omega_2=\exp(\pi{\mathrm i})=-1\), so \(A(\omega_2)[u:v]=[v:u]\). The finite ratio is \(1/z\) when \(z\ne0\); the coordinate swap also gives \([0:1]\leftrightarrow[1:0]\). No ordinary division by zero is used to represent infinity.

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Smith reflection disk with sixteen phase rays and four radial seed orbits. — The classical Smith grid equipped with sixteen exact phase rays and four radial seed orbits. The Cayley transform creates the chart; the projector calculus supplies the nested cyclic coordinate system.

13.1 Radial seeds and physical phase

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For \(0\le\rho<1\) and angle \(\theta\),

\[ {\Gamma}=\rho e^{{\mathrm i}\theta} \]

gives

\[ z(\rho,\theta) =\frac{1+\rho e^{{\mathrm i}\theta}}{1-\rho e^{{\mathrm i}\theta}}, \]

with

\[ r(\rho,\theta)=\frac{1-\rho^2}{1-2\rho\cos\theta+\rho^2}, \qquad x(\rho,\theta)=\frac{2\rho\sin\theta}{1-2\rho\cos\theta+\rho^2}. \]

A fixed \(\rho\) is a constant-mismatch circle. Refining \(N\) increases angular resolution without changing \(\rho\).

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The same reflection circle at phase orders two, four, eight, sixteen and thirty-two. — The same reflection circle at phase orders \(2,4,8,16,32\). Each coarse phase ray appears in every finer grid whose order is a multiple of its order. The exact residue-projector refinement law concerns coefficient packets, not sums of impedance points.

For a lossless transmission line,

\[ {\Gamma}_{\rm in}={\Gamma}_L e^{-2{\mathrm i}\beta\ell}. \]

One \(N\)-phase step corresponds to a line-length increment \(\lambda/(2N)\), up to the chosen orientation convention. Hence the phase address has a direct electrical interpretation.

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