Question

Derive the formulas to convert S-parameters to ABCD parameters.

Answer 1

To convert S-parameters (scattering parameters) to ABCD parameters, you can follow a systematic process using the following formulas. First, let's define the parameters:

S-parameters:
S11: Reflection coefficient at Port 1 (incident signal and reflected signal at Port 1)
S12: Transmission coefficient from Port 1 to Port 2
S21: Transmission coefficient from Port 2 to Port 1
S22: Reflection coefficient at Port 2 (incident signal and reflected signal at Port 2)

ABCD parameters:
A: Transmission gain
B: Forward propagation constant
C: Reverse propagation constant
D: Transmission gain

The formulas to convert S-parameters to ABCD parameters are as follows:

Calculate the Forward Propagation Constant (B) and Reverse Propagation Constant (C):
B = √(S21 * S12)
C = √(S21 * S12)

Calculate the Transmission Gain (A) and Transmission Gain (D):
A = S11 + (S12 * S21 * S22) / (S11 - S22 + S12 * S21)
D = S22 + (S12 * S21 * S11) / (S11 - S22 + S12 * S21)

After calculating these values, you can construct the ABCD parameter matrix as follows:

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 [ A  B ]
 [      ]


ABCD=[ C D ]

The ABCD parameter matrix is a 2x2 matrix that represents the characteristics of the two-port network.

Please note that in some cases, you may need to consider the reference impedance (Z0) when performing the conversion. In those cases, the S-parameters should be normalized with respect to the reference impedance before applying the above formulas.

Keep in mind that ABCD parameters are commonly used for two-port networks analysis, especially in transmission lines and microwave circuits. Ensure that the S-parameters and ABCD parameters are expressed in the same reference impedance system when performing the conversion.