The best methode doubtless is the bridge, as Steve already mentioned.
Another methode is to parallel the Rogowski coil by a well known capacitance 'C' and to analyze the resonance frequency. 'L' of Rogowski coil can then easily be estimated using the Thomson formula:

fres = 1 / (2 x pi x SQRT(L x C))

But this will only work successfully if the series resistance of Rogowski coil is not too high. Otherwise the resonance will be damped.
I have calculated your situation and found, that if 'L' is greater than about 10mH, a useful resonance should be observed with the following methode:



Use a sinus generator, not a square wave! Amplitude of sinus should be about 1Vs. Use a high quality 10n capacitor.

At L = 10mH a resonance of 38.5kOhm at 15.9kHz should occur. At output of TL081 you will observe a resonance yielding an amplitude of about 0.5Vs.
At L = 100mH a resonance of 380kOhm at 5.05Hz should occur. At output of TL081 you will observe a resonance yielding an amplitude of 5.1Vs.

So, to find the resonance just change the frequency between about 1kHz and 20kHz and search for the resonance. Change the frequency only very slowly!

You can connect a speaker at output of TL081, in series with a resistor of about 1kOhm. Then you will even hear what happens.

Keep in mind, that a Rogowski coil normally does not use any core material showing a relevant permeability 'µr'. So, if you feed any conductor through the Rogowski coil showing a relevant µr (µr > 1), then the inductivity measurement will not provide the actuall inductivity of Rogowski coil. This includes to some degree the existance of magnetic material outside but NEAR the Rogowski coil!

By the way, I have used this methode some years ago to analyze pick-ups of electrical guitars (for instance Fender's Stratocaster). Inductivity is much higher than in your application, about several Henrys and above methode showed very nice results.

Kai