Renormalized S-Matrices

Before a generalized S-matrix can be used to compute the reflection and transmission of signals, the generalized S-matrix must be normalized to the chosen impedance. For example, if a generalized S-matrix has been normalized to 50 ohms, it can be used to compute reflection/transmission directly from signals normalized to 50 ohms:

where:

• The input signals, Vi_i, and output signals, Vo_i, are both normalized to 50 ohms.
• To renormalize an S-matrix to a specific impedance, the system calculates a unique impedance matrix associated with the structure.

The unique impedance matrix, Z, is:

where:

• S is the nxn generalized S-matrix.
• I is an nxn identity matrix.
• Z₀ is a matrix with the characteristic impedance of each port as a diagonal value.

The renormalized S-matrix is then calculated from Z using this relationship:

where:

• Z is the structure’s unique impedance matrix.
• Z_W and Y_W are matrices with the chosen impedance and admittance as diagonal values.

Visualize the generalized S-matrix as an S-matrix that has been renormalized to the characteristic impedances of the structure. Therefore, if a diagonal matrix containing the characteristic impedances of the structure is used as Z_W in the previous equation, the result is the generalized S-matrix again.