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HFSS and Planar EM Simulators >
Planar EM Technical Notes >
   S-Parameters >
       Renormalized S-Matrices           


Renormalized S-Matrices

Before a structure’s generalized S-matrix can be used to compute the reflection and transmission of signals, the generalized S-matrix must be normalized to the appropriate impedance. For example, if a generalized S-matrix has been normalized to 50 ohms, it can be used to compute reflection and transmission directly from signals normalized to 50 ohms, as in

where the input signals, Vii, and output signals, Voi, 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.

• ZW and YW are matrices with the desired 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 ZW in the previous equation, the result would be the generalized S-matrix again.

Note

S-matrices for modes other than the dominant one cannot be renormalized or used in circuit simulators.




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