Perhaps this might be of interest :

https://www.mwrf.com/technologies/test-measurement/article/21277729/what-is-the-12-term-vna-calibration-model

Here's a derivation of S-parameters measurement equation:

1. Scattering Parameters (S-parameters):

• Describe the behavior of electrical signals in a network when encountering discontinuities or mismatches.
• Represented as complex numbers, indicating amplitude and phase changes of reflected and transmitted waves at each port of a network.

2. Concept of Incident, Reflected, and Transmitted Waves:

• Incident wave (aᵢ): Signal entering a port.
• Reflected wave (bᵢ): Signal reflected back from a port due to impedance mismatch.
• Transmitted wave (aⱼ): Signal transmitted through a port to another port.

3. Network Representation:

• A network with N ports can be represented by an N x N S-parameter matrix ([S]), where each element Sᵢⱼ is an S-parameter relating aᵢ to bⱼ.

4. S-parameter Measurement:

• Typically performed using a vector network analyzer (VNA), which measures incident and reflected power at each port.
• S-parameters are calculated from these power measurements using the following equation:

Sᵢⱼ = (bⱼ / aᵢ) | aₖ = 0, k ≠ i

Explanation:

• Sᵢⱼ represents the S-parameter at port j for an incident wave at port i.
• bⱼ is the reflected wave at port j.
• aᵢ is the incident wave at port i.
• aₖ = 0, k ≠ i indicates that all other incident waves at ports k (k not equal to i) are set to zero, ensuring isolation of the specific port interaction being measured.

Essential Points:

• S-parameters are frequency-dependent and vary with the network's internal characteristics.
• The S-parameter matrix is symmetric for reciprocal networks (where Sᵢⱼ = Sⱼᵢ).
• S-parameters are widely used in microwave engineering for designing and analyzing high-frequency circuits and components, such as filters, amplifiers, and antennas.

Hi all,

Finally I was able to derive the equations,

Thanks to everyone for the references,

All the books regarding this equations have not derivation of these equations in it.

I have attached it with the message the derivation for it,hope it helps somebody who is in the same phase like me;

Kind Regards,

Jojo Varghese