A Rogowski coil is a toroidal coil of wire (like a solenoid bent to form a donut) used to measure non-DC currents. The conductor carrying the current to be measured passes through the opening of the toroid. This is similar to the current transformer used in industry. Instead of measuring the short circuit current through the coil directly, the measurement is instead the integral of the open circuit voltage. The Rogowski coil does not use a magnetically permeable core like a standard current transformer, making it of low inductance. Since it has no permeable core to saturate it can respond linearly to extremely large currents. Being of low inductance it can also respond to very fast frequency pulses. A standard current transformer can have its core saturated at very high currents, and the inductance limits its frequency response. The closer in form to a perfectly symmetric toroidal coil of wire the Rogowski coil is, the less susceptible it is to external electromagnetic interference. Although it can deviate significantly without much loss in performance. So, a Rogowski coil is the sort of transducer one could use to accurately measure the current of lightning strikes, railguns, normal power circuits, and probably just about anything at high frequency and/or high current. I first ran across the idea of a Rogowski coil while reading "Instrumentation for EM Launcher Systems" [1].
Each turn of the Rogowski coil produces a voltage proportional to the rate of change of the magnetic flux through the turn. Assuming a uniform magnetic field density throughout the turn, the rate of change magnetic flux is equal to the rate of change of magnetic field density times the cross-sectional area of the turn.
(1)
for a coil with n turns,
(2) 
The magnetic field due to a long straight conductor carrying current I is
(3) 
where r is the perpendicular radial distance from the conductor to the point at which the magnetic field is calculated. The direction of the magnetic field being perpendicular to the current and to the radius r, and determined by use of the right hand rule.
The voltage of the whole coil is then
(4) 
In order to get a voltage proportional to current an integrator -either active or passive- must be used. An active integrator, using an operational amplifier is a common solution. Among other things the Op-Amp needs to have sufficient frequency response and current sourcing and sinking capability to drive the capacitor at the expected frequency. Depending on the output voltage of the Rogowski coil, using an Op-Amp that does not meet the frequency response and current requirements will result in an incorrect integration and possible destruction of the Op-Amp if the output of the Rogowski coil is high enough.
The integrator needs a resistor placed across the capacitor in order to be made into a leaky integrator. The resistance placed across the capacitor should be just small enough to leak off the capacitor and keep it zeroed but not so small that it interferes with the integration performance in the frequencies of interest.
Ignoring any leaky resistance added to the integrator of figure 1 the output is
(5) 
After integrating the signal of equation (4) the total output voltage is
(6) 
[1] K. E. Nalty, R. C. Zowarka, L. D. Holland.; Instrumentation for EM Launcher Systemst, IEEE Transactions on Magnetics, Vol 20, no 2, pp 328-331. March 1984.
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