Scattering Matrices for Adaptors

The adaptor equations for a connection of $M$ ports, in either the series (2.31) or parallel (2.32) case, may be written as

$${\bf b} = {\bf Sa}$$ (2.37)

where ${\bf b} = [b_{1}, \hdots , b_{M}]^{T}$ and ${\bf a} = [a_{1}, \hdots , a_{M}]^{T}$, and where we have

$${\bf S} = {\bf I}_{M} - \mbox{\boldmath$\alpha$} _{s}{\bf 1}^{T}\hspace{1.0in}$$ Series adaptor (voltage waves) (2.38)

$${\bf S} = - {\bf I}_{M} + {\bf 1} \mbox{\boldmath$\alpha$} _{p}^{T}\hspace{1.0in}$$ Parallel adaptor (voltage waves) (2.39)

Here ${\bf 1}$ is an $M\times 1$ vector containing all ones, ${\bf I}_{M}$ is the $M\times M$ identity matrix, and $\alpha$$_{s}$ and $\alpha$$_{p}$ are defined by

$$\mbox{\boldmath$\alpha$} _{s} = \frac{2}{\sum_{j=1}^{M}R_{j}}[R_{1},\hdots,R_{M}]^{T} \mbox{\boldmath$\alpha$} _{p} = \frac{2}{\sum_{j=1}^{M}G_{j}}[G_{1},\hdots,G_{M}]^{T}$$

The sum of the elements of either $\alpha$$_{s}$ or $\alpha$$_{p}$ is 2. For power wave variables, we have a similar relationship,

$$\underline{{\bf b}} = \underline{{\bf S}}\,\underline{{\bf a}}$$ (2.40)

where

$$\underline{{\bf S}} = {\bf I}_{M} - \sqrt{\mbox{\boldmath$\alpha$}_{s}}\sqrt{\mbox{\boldmath$\alpha$}_{s}}^{T}\hspace{1.0in}$$ Series adaptor (power-normalized waves)

$$\underline{{\bf S}} = - {\bf I}_{M} + \sqrt{\mbox{\boldmath$\alpha$}_{p}}\sqrt{\mbox{\boldmath$\alpha$}_{p}}^{T}\hspace{1.0in}$$ Parallel adaptor (power-normalized waves)

Here the square root sign indicates an entry-by-entry square root of a vector (all entries of $\alpha$$_{s}$ and $\alpha$$_{p}$ are non-negative).

Defining the Euclidean norm of a column vector ${\bf x}$ as $\Vert{\bf x}\Vert _{2} = \sqrt{{\bf x}^{T}{\bf x}}$, it is easy to show that a power normalized scattering matrix $\underline{{\bf S}}$ is norm-preserving in either the series or parallel case, i.e., we have

$$\Vert\underline{{\bf b}}\Vert _{2} = \Vert\underline{{\bf a}}\Vert _{2}$$ (2.41)

For voltage waves, we have the preservation of a weighted $L_{2}$ norm, i.e.,

$$\Vert{\bf b}\Vert _{{\bf P},2} = \Vert{\bf a}\Vert _{{\bf P},2}$$ (2.42)

where $\Vert\cdot\Vert _{{\bf P},2} = \sqrt{(\cdot)^{T}{\bf P}(\cdot)}$; in this case, ${\bf P}$ is an $M\times M$ positive definite diagonal matrix simply given by diag $(G_{1},\hdots,G_{M})$. It should be clear that (2.41) and (2.42) are merely re-statements of power conservation at a memoryless, lossless $M$-port.

Note that multiplying ${\bf S}$ or $\underline{{\bf S}}$ by a vector requires, in either the series of parallel case, $O(M)$ adds and multiplies; in particular, it is cheaper than a full $M\times M$ matrix multiply.