A Rogowski coil senses AC current.
This is a current transformer, and transformers couple...
This is certainly a transformer (one magnetic circuit, with two or more
windings), but 'current transformer' implies a high coupling (nearly 1),
and a low impedance on the secondary, both of which do not apply here.
Why can a coil no just be in parallel with a wire and measure current?
[as in, a helical coil with the helix axis parallel to the primary wire]
The coupling in a Rogowski coil comes from the inner semicircle parts
being close to the sensed wire, while the outer semicircle of the wire
is farther away. Both semicircles are roughly parallel to the sensed wire, so they couple, but the inner half couples more strongly than the
outer half. Thus, the positive coupling on the inside and negative
coupling on the outside do not quite cancel entirely. The 'return' wire is not parallel to the primary, and its path inside the R-coil is
insensitive to the current being measured.
The coupling of the R-coil is
determined by that coil geometry without regard to the return wires
going to the integrator. The 'helix-in-parallel' would have coupling
due to the wires from the ends
of the solenoid as they are routed to the integrator. Coiling (other
than as it moves the wire path closer/farther from the primary) of
the solenoid is not productive.
The R-coil does NOT have full flux coupling, so an open circuit
on the Rogowski coil only loads the primary winding with the
same series inductance (maybe a microhenry per meter) as any wire in
space; a closed circuit on the Rogowski coil would only change
that by a few parts per million.
The EMF in the Rogowski coil is produced as a side effect of
the self-inductance of the primary wire. Using the side-effect
means one can measure with no significant energy cost.