# Rogowski coil

A Rogowski coil is a toroid of wire used to measure an alternating current i(t) through a cable encircled by the toroid. The above picture shows a coil encircling a current-carrying cable, with its output terminals v(t) connected to an integrator circuit, to obtain a voltage Vout(t) proportional to i(t).

A Rogowski coil, named after Walter Rogowski, is an electrical device for measuring alternating current (AC) or high speed current pulses. It consists of a helical coil of wire with the lead from one end returning through the centre of the coil to the other end, so that both terminals are at the same end of the coil. The whole assembly is then wrapped around the straight conductor whose current is to be measured. Since the voltage that is induced in the coil is proportional to the rate of change (derivative) of current in the straight conductor, the output of the Rogowski coil is usually connected to an electrical (or electronic) integrator circuit to provide an output signal that is proportional to the current.

## Contents

This type of coil has advantages over other types of current transformers.

• It is not a closed loop, because the second terminal is passed back through the center of the toroid core (commonly a plastic or rubber tube) and connected along the first terminal. This allows the coil to be open-ended and flexible, allowing it to be wrapped around a live conductor without disturbing it.
• It has an air core rather than an iron core, it has a low inductance and can respond to fast-changing currents.
• The full-scale output of a properly designed Rogowski coil is typically a few hundred millivolts. Although a correctly formed Rogowski coil, with equally spaced windings, is largely immune to electromagnetic interference, the integrator circuitry it requires may not be.

This type of coil also has some disadvantages over other types of current transformers.

• The output of the coil must be passed through an integrator circuit to obtain the current waveform. The integrator circuit requires power, typically +/-15Vdc. Traditional split-core current transformers do not require integrator circuits. The integrator is lossy, so the Rogowski coil does not have a response down to DC; neither does a conventional current transformer (see Néel effect coils for DC).

Due to these disadvantages, Rogowski coils are primarily used when ease-of-installation is a high priority. Traditional or split-core current transformers are used when accuracy, noise immunity, and currents during power quality events are a high priority.[citation needed]

## Formulae

Example waveform of R.C. output for switched-mode load. As explained above the output waveform - CH4 (green) - represents the derivative of the current waveform - CH2 (Blue). CH1 (Yellow) is 230V AC Mains waveform

The voltage produced by a Rogowski coil is

$V = \frac{-AN\mu_0}{l} \frac{dI}{dt}$,

where $A= \pi a^2$ is the area of one of the small loops, N is the number of turns, and $l=2 \pi R$ is the length of the winding (the circumference of the ring). $\frac{dI}{dt}$ is the rate of change of the current threading the loop. This formula assumes the turns are evenly spaced and that these turns are small relative to the radius of the coil itself. At high frequencies, the Rogowski coil's inductance will decrease its output. The inductance of a toroid is

$L = \mu_0 N^2 (R - \sqrt{R^2-a^2})$

$\mu_0 = 4 \pi \times 10^{-7}$ V·s/(A·m) is the magnetic constant, $R$ is the major radius of the toroid, and $a$ is its minor radius.

## Similar devices

A device similar to the Rogowski coil was described by Arthur Prince Chattock of Bristol University in 1887.[1] Chattock used it to measure magnetic fields rather than currents. The definitive description was given by Walter Rogowski and W. Steinhaus in 1912.[2]

More recently, low-cost current sensors based on the principle of a Rogowski coil have been developed.[3] These sensors share the principles of a Rogowski coil, measuring the rate of change of current using a transformer with no magnetic core. The difference from the traditional Rogowski coil is that the sensor can be manufactured using a planar coil rather than a toroidal coil. In order to reject the influence of conductors outside the sensor's measurement region, these planar Rogowski current sensors use a concentric coil geometry instead of a toroidal geometry to limit the response to external fields. The main advantage of the planar Rogowski current sensor is that the coil winding precision that is a requirement for accuracy can be achieved using low-cost printed circuit board manufacturing.