HFSS15: Renormalized S-Matrices

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:

(1)

where

• S is the n x n generalized S-matrix.

• I is an n x n identity matrix.

• Z₀ is 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:

(2)

where

• Z is the structure’s unique impedance matrix.

• Z_W and Y_W are diagonal matrices with the desired impedance and admittance as diagonal values. For example, if the matrix is being renormalized to 50 ohms, then Z_W 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 Z_W 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.