Dear Prahlad,
sorry for being late, but I was a bit busy the last days.
What you want to do with the Rogowski coil is not at all trivial! The more as it's hard for you to get special components in Mumbai.

One of the critical behaviours of Rogowski coil is, that signal is dependent on change rate of current. So, theoretically (and practically!) the output signal of Rogowski coil can carry very narrow and very high spikes! A thourough understanding of actual flowing currents to be measured is needed to estimate the height of voltage spikes.

Another issue is speed of integrator. If it's not fast enough, then it cannot capture the narrow spikes and integrate them.

So, two important things the integrator must fullfill: It must be fast and precise. And if the height of voltage spikes is unpredictable, then a protection scheme must also be implemented. This is difficult, because the protection scheme can drastically degrade the precision of integrator, when choosen unproperly.

I know from you the inductance and copper resistance of your Rogowksi coils and I would recommend you the following circuit for further experiments:



From earlier formulas a load resistance of more than 10kOhm is recommended. So, R = 10kOhm was choosen.
To take into consideration manufacturing tolerances of Rogowski coils, 'Ra' was implemented. If you set Ra = 2.5kOhm nominally, then you can choose Ra = 0...5kOhm to adjust the output signal with a span of +-20%. You can put the once found 'Ra' directly near Rogowski coil, perhaps in the plug.

'R' can be choosen much higher than 10kOhm. But then 'Ra' must also be choosen higher, accordingly. 'R' also limits current flowing into the integrator. As the output of OPamp must keep this current flowing on, no more than about 1mA should flow through 'R'. This means a maximum voltage of about 10V at input of integrator.
If you observe much higher voltages here, then you should increase 'R'. Keep in mind, that input voltage at integrator input is allowed to be much greater than supply voltage of OPamp. The only you have to do is to keep the current through 'R' less than about 1mA, which means to increase 'R' accordingly.

It's really important, that the integrator has a protection scheme. This is needed not only for the situation that no supply voltage is applied to OPamp, but also, if the signal height at input of integrator is not yet known accurately.
And it's also important, that the protection circuit is fast acting, without introducing lots of capacitance to ground! For this reason two 1N4151 were choosen, which clamp the voltage at inverting input of OPamp to a sane value within a few nanoseconds.

Unfortunately, leakage current is not very low. The base collector junction of 2N3904, for instance, would provide much lower leakage current, of course, but turn-on time is in the microsecond range, which is not fast enough.

Happily leakage current can be drastically decreased, when using the offset adjusting scheme arround non-inverting input of OPamp: If the trimmer is adjusted to set the potential at inverting input to exactly zero, then no leakage current is flowing, because the voltage drop across 1N4151s is also zero!

But what, if due to drift the voltage across the diodes increases again?
I have measured a leakage current of only about 100pA per diode, when voltage dropping across them is 2mV. (Take care that these diodes are shielded from ambient light!)

This makes clear, how important it is to use an OPamp providing very low offset voltage, providing very low temperature drift and providing very low long term drift! For these reasons, I cannot recommend to use an TL081. Only a TL051 can be used in this circuit. With a TL081, with a bit luck, also good results can be achieved, but that is rather unprobable! The TL051 is the much better choice!!

Integration capacitance 'C' must not be choosen smaller than 10nF. Otherwise influence of 4M7 resistor is no longer negligible.

You can adjust output voltage of integrator by changing 'R' and 'C'. It depends on your Rogowski coils and the actual currents to be measured, which set of 'R' and 'C' will give satisfying results. I cannot give concrete values here, of course.

The biggest output signal is achieved when 'R' and 'C' both are made low, means R = 10kOhm and C = 10nF. If this will not be enough for achieving proper output signal, then either an additional amplifying stage is needed, or Rogowski coil parameters (inductance) must be modified. Additional amplifying in front of integrator and behind is always critical! So, it's best to modify Rogowski coil if a change is needed.

How to adjust the offset voltaqe?
Connect the input of integrator directly to ground by the help of a very short wire. Any existing offset voltage of OPamp will be amplified by the factor (4M7 + 10k) / 10k = 471 then. Now adjust the output of TL051 to exactly 0.0mV. Remove the ground connection. That's it.

Take care, that the whole circuit is properly shielded! This is also valid for the Rogowski coil!
It's not wise to manufacture the coil as you did, means to bend the windings onto a long tube being very near to current carrying wire. Why?
Such an arrengement shows drastical coupling capacitance, and without shielding the Rogowski coil will pick-up the switching noise of current carrying wire. So, using a shield arround the Rogowski coil can be very advantageous. Take care, that this shield comes not too close to the Rogowski coil. Otherwise here also a too high coupling capacitance can result, which can short circuit high frequency components of signal to ground before they can flow into the integrator.

Kai