# Phase of Scattering Parameters

I do not understand the meaning of the phase of scattering parameters.

Let's consider for instance a two port network, and let's focus on S21.

I know that its absolute value represents a ratio between power transmitted at port 2 and power sent in port 1. But what is the meaning of its phase?

• normally -Pd phase = generated = , +Pd = dissipated, and in between 0, 180 is RLC phase shift or time delay Commented Oct 31, 2019 at 12:30

The phase describes how much the signal is delayed in time from the input to the output.

Therefore the the S parameter can describe how much a signal is attenuated AND phase-shifted in time.

A positive phase means that the output signal is leading the input, while a negative phase results in a lagging (delayed) output signal.

• I have another question about the phase of S parameters. Let's consider for instance S21 in a two - port network made by a transmission line. Why is it in general a function of frequency? I'd say that the delay is equal to L/v (and so constant with frequency), where L is the length of the line, and v the speed of light in the medium of that line. Commented Jan 11, 2020 at 8:19
• Because the delay (in seconds) will result in different phase depending on the frequency. E.g. 20° is not the same time delay at two different frequencies. Commented Jan 11, 2020 at 20:36
• Perfect, thank you very much! Commented Jan 11, 2020 at 21:21

The history of circuit theory tells you reactive components have a phase shift relationship between voltage and current as a function of frequency. As well, scattering parameters also have this complex value.

I think the crucial concept you are missing is the concept of phasor.

A phasor represents a sinusoidal signal of a given (constant) frequency. It is commonly represented by a complex number.
Its magnitude represents the amplitude (voltage or current) of the signal; its argument or phase represents its phase difference with respect to some reference signal (here: the incident signal).

If the incident signal and the scattered signal is represented by a phasor the scattering coefficients also represented by a complex number (though not a phasor) gives the ratio between those signals.

The magnitude of the scattering parameter tells you how much the signal will be amplified/attenuated; its argument (=phase) tells you how much the phase of the scattered signal will be shifted with respect to the phase of the incident signal (see multiplication of complex numbers in polar form: magnitudes are multiplied; arguments are added).

Consider a really simple two port network, a short length of lossless matched cable. S11 = S22 = 0, and magnitudes of S12 and S21 are unity.

The delay through the length of cable will cause a phase shift between input and output. This will be represented in the phase of S21.