Editing Rogowski coil (section)

==Formulae==
[[File:RC PULSE.png|thumb|Example waveform of RC output for [[Switched-mode power supply|switched-mode load]]. As explained above, the output waveform CH4 (green) represents the [[derivative]] of the current waveform CH2 (blue); CH1 (yellow) is 230&nbsp;V AC mains waveform]]

The voltage produced by a Rogowski coil is
:<math>v(t) = \frac{-AN\mu_0}{l} \frac{dI(t)}{dt},</math>

where 
*<math>A = \pi r^2</math> is the area of one of the small loops, 
*<math>N</math> is the number of turns, 
*<math>l = 2 \pi R</math> is the length of the winding (the circumference of the ring),
*<math>\frac{dI(t)}{dt}</math> is the rate of change of the current threading the loop, 
*<math>\mu_0 = 4 \pi \times 10^{-7}</math> [[Volt|V]]·[[Second|s]]/([[Ampere|A]]·[[Meter|m]]) is the [[permeability of free space|magnetic constant]],
*<math>R</math> is the major radius of the toroid,
*<math>r</math> is its minor radius.

This formula assumes the turns are evenly spaced and that these turns are small relative to the radius of the coil itself.

The output of the Rogowski coil is proportional to the derivative of the wire current. The output is often integrated so the output is proportional to the wire's current:
:<math>V_\text{out} = \int v \,dt = \frac{-AN\mu_0}{l} I(t) + C_\text{integration}.</math>
In practice, an instrument will use a lossy integrator with a time constant much less than the lowest frequency of interest.  The lossy integrator will reduce the effects of offset voltages and set the constant of integration to zero.

At high frequencies, the Rogowski coil's [[inductance]] will decrease its output.

The inductance of a toroid is<ref>{{cite web | url=http://www.nessengr.com/technical-data/toroid-inductor-formulas-and-calculator/ | title=Toroid Inductor Formulas and Calculator }}</ref>

:<math>L = \mu_0 N^2 \left(R - \sqrt{R^2 - r^2}\right).</math>