Renormalized S-Matrices in HFSS
Before a structure's generalized S-matrix can be used in a high frequency circuit simulator to compute the reflection and transmission of signals, it 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 that are normalized to 50 ohms.
To renormalize a generalized S-matrix to a specific impedance, HFSS first calculates a unique impedance matrix Z, associated with the structure defined as follows:
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(1) |
- S is the n x n generalized S-matrix.
- I is an n x n identity matrix.
- Z0is a diagonal matrix having the characteristic impedance (Z0) of each port as a diagonal value.
The renormalized S-matrix is then calculated from the unique impedance matrix using this relationship:
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(2) |
where
- Z is the structure's unique impedance matrix.
- ZW and YW are diagonal matrices with the desired impedance and admittance as diagonal values. For example, if the matrix is being renormalized to 50 ohms, then ZW would have diagonal values of 50.
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 above equation, the result would be the generalized S-matrix again.
HFSS needs to calculate the characteristic impedance of each port in order to compute a renormalized S-matrix.
The S-matrices initially calculated by HFSS are generalized S-matrices that have been normalized to the impedances of each port; however, you can compute S-matrices that are normalized to specific impedances, such as 50 ohms.
To convert a generalized modal S-matrix to a renormalized modal S-matrix, HFSS first needs to compute the characteristic impedance at each port. There are several ways to compute characteristic impedance. Two methods — the Zpv and Zvi methods — require an impedance, or integration, line.
HFSS will always calculate Zpi impedance, the impedance calculation using power and current, which are well-defined for a port because they are computed over the area of the port. Zpv and Zvi are not calculated by default. This is because v is computed by integrating along a user-defined integration line. If Zpi is very small (less than 10^-14) HFSS uses Zpv (if it is non-zero). In this case, a warning says that the design has too few conductors touching the lumped port. To renormalize the solution to a Zpv or Zvi characteristic impedance, you must define an integration line.