a current transformer (the common name for your "Rogowski Coil") when properly selected produce a voltage proportional to the current flowing through the middle. What is all this fancy match about?.

Current Transformer and Rogowski coils are not same they differ a lot.

I know a CT produces a secondary current proportional to the current flowing through the conductor [primiary] in middle. When the terminals of a CT are connected across a known resistor the voltage produced across the resistor terminals is proportional to the current flowing in the middle.

Rogowski coils however do not work on above principal they produce an EMF at the terminals and this in proportion to di/dt where i is the current in the middle conductor.

Regards,
Prahlad Purohit