S-parameter container with additional reference port

The Y-parameters of a multi-port component defined by its S-parameters required for a small signal AC analysis can be obtained by converting the S-parameters into Y-parameters.

Figure 9.13: S-parameter container with noise wave correlation matrix

In order to extend a $m - 1$-port to have a S-parameter device with $m$ ports assuming that the original reference port had a reflection coefficient $\Gamma_m$ the new S-parameters are according to T. O. Grosch and L. A. Carpenter [12]:

$$S_{mm} = \dfrac{2 - \Gamma_m - m + {\displaystyle\sum_{i=1}^{m-1}}\, {\d... ..._m - {\displaystyle\sum_{i=1}^{m-1}}\, {\displaystyle\sum_{j=1}^{m-1}} S'_{ij}}$$ (9.227)

$$S_{im} = \left(\dfrac{1 - \Gamma_m\cdot S_{mm}}{1 - \Gamma_m}\right)\cdot \left(1 - \sum_{j=1}^{m-1} S'_{ij}\right) \textrm{ for } i = 1,2 \ldots m - 1$$ (9.228)

$$S_{mj} = \left(\dfrac{1 - \Gamma_m\cdot S_{mm}}{1 - \Gamma_m}\right)\cdot \left(1 - \sum_{i=1}^{m-1} S'_{ij}\right) \textrm{ for } j = 1,2 \ldots m - 1$$ (9.229)

$$S_{ij} = S'_{ij} - \left(\dfrac{\Gamma_m\cdot S_{im}\cdot S_{mj}}{1 - \Gamma_m\cdot S_{mm}}\right) \textrm{ for } i,j = 1,2 \ldots m - 1$$ (9.230)

If the reference port has been ground potential, then $\Gamma_m$ simply folds to -1. The reverse transformation by connecting a termination with a reflection coefficient of $\Gamma_m$ to the $m$th port writes as follows.

$$S'_{ij} = S_{ij} + \left(\dfrac{\Gamma_m\cdot S_{im}\cdot S_{mj}}{1 - \Gamma_m\cdot S_{mm}}\right) \;\;\;\; \textrm{ for } i,j = 1,2 \ldots m - 1$$ (9.231)

With the S-parameter transformation done the $m$-port noise wave correlation matrix is

$$C_m = \left\vert\dfrac{1}{1 - \Gamma_m}\right\vert^2 \cdot \left(... ...left\vert 1 - \left\vert\Gamma_m\right\vert^2\right\vert\cdot D\cdot D^+\right)$$ (9.232)

with

$$K = \begin{bmatrix}1 + \Gamma_m\left(S_{1m} -1\right) & \Gamma_m S_... ...m S_{mm} - 1 & \Gamma_m S_{mm} - 1 & \ldots & \Gamma_m S_{mm} - 1 \end{bmatrix}$$ (9.233)

$$D = \begin{bmatrix}S_{1m}\\ S_{2m}\\ \vdots\\ S_{(m-1)m}\\ S_{mm} - 1 \end{bmatrix}$$ (9.234)

whence $T_s$ denotes the equivalent noise temperature of the original reference port and the $^{+}$ operator indicates the transposed conjugate matrix (also called adjoint or adjugate).

The reverse transformation can be written as

$$C_{m-1} = K'\cdot C_m\cdot K'^+ +T_s\cdot k_B \cdot\dfrac{\left\v... ...rt^2\right\vert}{\left\vert 1 - \Gamma_m S_{mm}\right\vert^2}\cdot D'\cdot D'^+$$ (9.235)

with

$$K' = \begin{bmatrix}1 & 0 & \ldots & 0 & \dfrac{\Gamma_m S_{1m}}{1 -... ...& \ldots & 1 & \dfrac{\Gamma_m S_{(m-1)m}}{1 - \Gamma_m S_{mm}}\\ \end{bmatrix}$$ (9.236)

$$D' = \begin{bmatrix}S_{1m}\\ S_{2m}\\ \vdots\\ S_{(m-1)m} \end{bmatrix}$$ (9.237)