What exactly does line impedance stabilization mean?
What parameters of line impedance are stabilized and how?
How does line impedance / source impedance vary with frequency?
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It is all in the name. A Line Impedance Stabilization Network does just that. AC lines can vary wildly depending on where you are, how your house/outlet is wired, nearby radio stations, etc.
The LISN is basically a large low-pass filter that gives you a "quiet" AC signal to, as Jasen said, measure conducted emissions from a unit. Different standards require different LISNs. For example, the LISN used in CISPR22 is diferent than MIL-STD-461. The end goal is the same though. The LISN ensures that you will read similar conducted emissions levels at different test locations.
When performing a conducted scan you connect your spectrum analyzer to a 50ohm impedance within the LISN. I am attaching an image I did for work not too long ago that shows the gain and phase across frequency. This shows the LISN used for CISPR11/22. R3 is the impedance you would measure across, R1 is where you would attach the unfiltered AC. You can see I'm doing the sweep from the "output" which connects to the unit.
An LISN is used to provide a way to measure power-line borne noise originating from an appliance.
Typical wiring at an installation is nothing like a transmission line, the impedance is all over the places and there are many branches and stubs
inside the box is a large resonant filter built with wide-band air core inductors.