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Please help guide my ignorance on this topic.

From what I understand, S-parameters are reflection coefficients generalized to N-port systems (via an S matrix), but for a 1-port system, these converge to the same thing?

In addition, how does this all relate to the smith chart? How does one get the reflection coefficient out of the chart and how fundamentally does that relate to the normalized load impedance you're plotting on there?

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N-port systems have N^2 S-parameters, so yes, 1-Port has 1-S-Parameter (which is called reflection factor). Here’s an explanation attempt for S-Parameters: When dealing with conductors, most of us know the concept of voltage U and current I, where U/I=R is the conductors resistance. But in RF/HF, there are situations (e.g. a waveguide) where one cannot simply define U or I anymore, but instead we have to deal with waves. S-parameters just put the wave amplitudes into relation. Take for example a laser pointing into a mirror. The relation of incident and reflected laser beam (waves) is the reflection coefficient or S11. S-Parameters are dimensionless complex numbers. The Smith Chart is mainly a (very powerful) drawing aid to put these complex numbers onto paper.

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