S-Parameter Basic Definitions

Scattering parameters (S-parameters) represent the element behavior in matrix form. The S matrix indicates what fraction of an incident voltage wave is transmitted or reflected at each port. The voltage wave is normalized with respect to the square root of the characteristic impedance of the port, for example, a = V*Z₀^-0.5. If the characteristic impedance is completely real, the elements of the S matrix are equivalent to the square root of the fraction of power transmitted or reflected. An S-matrix for a three-port structure is:

where:

• All quantities are complex numbers.
• a_i represents the incident voltage wave at port i.
• b_i represents the transmitted or reflected voltage wave at port i.
• The phase of a_i and b_i is the phase of the incident and of the transmitted or reflected voltage wave at t = 0:

Za_iis the phase angle of the excitation voltage wave on port i at t = 0 (by default, it is zero).

Zb_iis the phase angle of the transmitted or reflected voltage wave with respect to the excitation voltage wave.

• S_ij is the S-parameter describing how much of the incident voltage wave at port j is reflected back or transmitted to port i. Thus, S₃₁ is the amount of power on the port 1 excitation that is transmitted to port 3. The phase of S₃₁ specifies the phase shift that occurs as the voltage wave travels from port 1 to port 3.