Calculation of characteristic impedance from S parameters.

08 Jan 2024

Introduction

Suppose one wants to measure or simulate the characteristic impedance of something similar to a transmission line. This can be for example a 75 Ω coax cable or a via structure on HFSS. The characteristic impedance can be calculated form its measured or simulated S parameters references to 50 Ω.

Suppose also that the structure is “symmetric enough” to have the same characteristic impedance on each side (see https://en.wikipedia.org/wiki/Image_impedance)…

Recommended method using ABCD parameters

According to https://en.wikipedia.org/wiki/Image_impedance, and assuming the symmetry hypothesis which allows to simply discard the second result, the characteristic impedance can be calculated as:

Z_0 = sqrt((A B)/(C D))

The ABCD parameters can be obtained from S parameters with https://www.microwaves101.com/encyclopedias/network-parameters:

{: ( A = ((1 + S_(11))(1 - S_(22)) + S_(12)S_(21))/(2S_(21)) , B = 50 Omega ((1 + S_(11))(1 + S_(22)) - S_(12)S_(21))/(2S_(21)) ), ( C = 1/(50 Omega) ((1 - S_(11))(1 - S_(22)) - S_(12)S_(21))/(2S_(21)) , D = ((1 - S_(11))(1 + S_(22)) + S_(12)S_(21))/(2S_(21)) ) :}

Calculation is made as follows:

Z_0 = 50 Omega sqrt( ( ((1 + S_(11))(1 - S_(22)) + S_(12)S_(21)) ((1 + S_(11))(1 + S_(22)) - S_(12)S_(21)) ) / ( ((1 - S_(11))(1 - S_(22)) - S_(12)S_(21)) ((1 - S_(11))(1 + S_(22)) + S_(12)S_(21)) )

These formulas can be conveniently entered into an Excel spreadsheet.

Alternative non recommended method using transfer S parameters

Transfer S parameters can also be used for this calculation. However, this method is NOT recommended because the calculations are cumbersome.

Expressing the characteristic impedance as a reflection coefficient from 50 Ω, and recalling that by definition its invariant through the system, the following can be written:

[[b_1],[a_1]] = [[T_11,T_12],[T_21,T_12]] \ [[a_2],[b_2]] Gamma = b_1 / a_1 = (T_11 Gamma b_2 + T_12 b_2)/(T_21 Gamma b_2 + T_12 b_2) = (T_11 Gamma + T_12)/(T_21 Gamma + T_12)

Rearranging:

Gamma (T_21 Gamma + T_12) = (T_11 Gamma + T_12) T_21 Gamma^2 + (T_12 - T_11) Gamma - T_12 = 0

Solving by the usual methods:

Delta = (T_(12)-T_(11))^2+4*T_(21)*T_(12) Gamma = (T_(11) - T_(12) +- sqrt((T_(12)-T_(11))^2+4*T_(21)*T_(12)))/(2*T_(21))

So, the procedure, can be outlined as follows:

1. Convert S parameters to T parameters using the previous formulas or scikit-rf (https://scikit-rf.readthedocs.io/en/latest/api/generated/skrf.network.s2t.html)

2. Calculate the Gamma of the characteristic impedance references to 5 Ω.

3. From the Gamma, calculate the characteristic impedance.