How does a Rogowski coil differ from a current transformer?

A Rogowski coil is air-cored and cannot saturate, so it stays linear for large or very fast currents. Its output is proportional to the time derivative of current, so it needs an integrator and cannot measure DC. A current transformer uses an iron core and gives a large output at low frequency, but saturates at high current or with DC bias.

How large is the output voltage?

A coil with 500 turns, 50 mm inner radius, 60 mm outer radius and 10 mm height has a mutual inductance of about 182 nH. It produces about 57 mV for 1000 A at 50 Hz, but about 182 V for a current slope of 1 kA/µs, so turns and cross-section must suit the application.

Does the reading depend on conductor position?

For a closed coil with uniform winding, Ampère's law makes the output independent of conductor position. In a worked example of a clamp-on coil with a 10° gap at the joint, moving the conductor 30 mm toward the gap lowered the output by about 3.4%, and an external conductor 80 mm away near the gap added an error signal of about 5.9%.

Rogowski coil (current sensor)

Category: Electromagnetic Field Analysis | Revised 2026-09-29

CAE visualization for rogowski coil theory - technical simulation diagram

Rogowski Coil (Current Sensor)

Principle: measuring current with an air-cored coil

Overview

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In an inverter test someone wrapped a thin rope-like sensor around a cable to measure current. Can that really measure hundreds of amps?

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That's a Rogowski coil — a long, thin coil with no core, closed into a loop around the conductor. The conductor's magnetic field threads the coil and induces a voltage proportional to how fast the current changes. With no core there is nothing to saturate, so it stays linear from tens of kiloamps down to nanosecond switching edges in power semiconductors.

Output voltage and mutual inductance

For a uniformly wound coil of $N$ turns with a rectangular cross-section of inner radius $a$, outer radius $b$ and height $h$:

$$ v(t) = -M\,\frac{di}{dt}, \qquad M = \frac{\mu_0 N h}{2\pi}\ln\frac{b}{a} $$

With $N=500$, $a=50$ mm, $b=60$ mm and $h=10$ mm, $M \approx 182$ nH.

Measured current	Coil output
50 Hz, 1000 A rms	About 57 mV rms
Slope of 1 kA/µs	About 182 V

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Same coil — tens of millivolts at mains frequency, but over 100 V on a switching edge.

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The output is set by $di/dt$. For mains-frequency work you want more turns for sensitivity; for power electronics you need a small $M$ or you'll overrun the input of the next stage. The design direction flips completely with the application — that's what makes this sensor interesting.

Why the position of the conductor doesn't matter

$$ \oint_{C} \mathbf{H}\cdot d\mathbf{l} = I_{enc} $$

A closed, uniformly wound coil effectively measures the line integral of the field around its path — the left side of Ampère's law. So the output depends only on the total current passing through, not on where the conductor sits, and conductors outside the loop contribute nothing. The ideal breaks down when the winding is non-uniform or the loop is not closed, as at the joint of a clamp-on coil.

Coffee Break Trivia

Born as a tool for measuring “magnetic voltage”

The Rogowski coil is named after the German electrical engineer Walter Rogowski. His 1912 paper with Steinhaus was not about measuring current: it described a long, thin coil for measuring the “magnetic voltage” — the line integral of the field — between two points in space. Close that coil into a loop and, by Ampère's law, it measures the current passing through. That use later became standard for current sensing.

Design calculations: integrator and bandwidth

The integrator

$$ v_{out}(t) = \frac{1}{RC}\int v\,dt = -\frac{M}{RC}\, i(t) $$

For an output of 1 V per 1000 A, $RC = M/10^{-3} \approx 182$ µs. A passive RC integrator only integrates well above $1/(2\pi RC) \approx 873$ Hz, so measuring 50 Hz needs an active integrator built around an op-amp.

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Active integrators have a weakness too: any small DC offset at the input is integrated and the output drifts away. Practical integrators are therefore designed to stop integrating at very low frequency — one more reason a Rogowski coil cannot measure DC.

Self-resonance and the upper bandwidth

$$ L_{coil} = N M, \qquad f_{res} = \frac{1}{2\pi\sqrt{L_{coil}\,C_{coil}}} $$

The coil above has self-inductance $L = NM \approx 91$ µH. With 50 pF of stray winding capacitance it resonates at about 2.4 MHz, with 200 pF at about 1.2 MHz, and the usable bandwidth lies well below that. To capture fast switching waveforms, reduce the turns to lower the inductance and make the coil thinner and shorter to lower the stray capacitance.

Aspect	Rogowski coil	Current transformer (iron core)
Saturation	None	Saturates at high current or with DC bias
DC	Cannot measure	Cannot measure
Low-frequency sensitivity	Low (needs an integrator)	High
Fast currents	Good, up to self-resonance	Limited by core loss and saturation
Installation	Flexible; easy to retrofit	Often requires threading the conductor

Assessing accuracy with field simulation

Worked example: error from the joint gap

For the same coil (500 turns, $a=50$ mm, $b=60$ mm), a gap with no winding at the clamp joint was modelled by integrating the flux linkage of every turn from the Biot–Savart field of an infinite straight conductor. Values are changes relative to the output with the conductor at the centre.

Gap	Conductor 30 mm toward the gap	Conductor 30 mm away from the gap	External conductor 80 mm near the gap	External conductor 80 mm opposite
0°	0.00 %	0.00 %	0.00 %	0.00 %
5°	−1.72 %	+0.50 %	+3.08 %	−0.58 %
10°	−3.39 %	+1.02 %	+5.87 %	−1.17 %

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With no gap it really is zero. But a 10° gap picks up more than 5% from an outside conductor.

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In a switchboard with three-phase cables side by side, this crosstalk from neighbouring conductors is often the biggest error. The fixes are to orient the joint away from the adjacent phases, centre the conductor in the coil, and overlap the winding at the joint to shrink the gap. Check the position sensitivity by calculation like this, or by calibrating while moving the conductor.

Modelling with FEM

There is no need to model every turn. Treat the coil cross-section as a region that carries no current and only picks up flux, and get the mutual inductance from the summed flux linkage of the turns.
Iron structures nearby (switchboard enclosures, busbar brackets) change the field distribution, but Ampère's law still holds, so a closed coil is barely affected. They matter when there is a gap, and a gap near iron increases the error.
At high frequency, skin and proximity effects skew the current distribution in the measured conductor. With a gap, the error then varies with frequency even for a centred conductor, so run an eddy-current analysis at the frequencies you need.
To assess bandwidth, couple the coil to a circuit model that includes winding capacitance (sometimes treated as a distributed transmission line) and locate the self-resonance.

Coffee Break Trivia

When the “noise” is really the electric field

When measuring power-semiconductor switching with a Rogowski coil, spikes unrelated to the current sometimes appear. Often the cause is capacitive coupling: the conductor voltage swinging at hundreds of volts per nanosecond injects current into the coil through stray capacitance. An electrostatic shield on the coil, a little more distance from the conductor, or a differential input stage all help. It is an electric-field problem, so a magnetic-only simulation will not reproduce it.

Common mistakes

Mistakes and fixes

Mistake	Effect	Fix
Using a passive integrator at mains frequency	No real integration; amplitude and phase are wrong	Use an active integrator
Not checking output voltage for fast edges	The next stage saturates or is damaged	Estimate the maximum of $M\,di/dt$
Pointing the joint at a neighbouring phase	Crosstalk errors of several percent	Turn the joint away from adjacent conductors
Measuring current with a DC component	The DC part disappears and the waveform shifts	Combine with a Hall-effect sensor or similar
Trusting waveforms near self-resonance	Ringing mistaken for current	Check the bandwidth and reduce turns if needed

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I'd like to calculate the coil parameters myself.

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Check coil inductance with the coil inductance calculator and the basics of induction with the Faraday's law simulator. Iron-core sensing is covered in current transformer analysis, DC-capable sensing in Hall-effect sensors, and coils that measure flux directly in search coils.

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