Rogowski Coil

Ampère and Faraday in one device

A Rogowski coil is two laws of electromagnetism in series. Ampère's law says that current flowing in a wire produces a circulating magnetic field, with H · dl = I_enclosed integrated around any closed loop in space. Faraday's law says that a changing magnetic flux through a coil induces an EMF: V_emf = -dΦ/dt. Putting the two together, a helical coil wound around the wire encloses some fraction of the wire's flux, and as the current changes, the induced voltage at the coil's terminals is V = M · dI/dt, where M is the geometry-only mutual inductance between conductor and coil. No magnetic materials are needed, no flux concentrators, no calibration of saturation — just the geometry.

Walter Rogowski described this geometry in 1912 in his paper "Die Messung der magnetischen Spannung" ("Measurement of magnetic tension"). Until the late 20th century the coil was a laboratory curiosity, hampered by the high cost of low-drift integrators and stable amplifiers. Modern electronics make Rogowski-based instruments practical: every utility-scale fault recorder, almost every electric-vehicle DC-link inverter measurement, and most plug-in clip-on current probes above 1 kA use them. The key innovation that turned a 1912 laboratory device into a 2020s production instrument was the cheap precision op-amp integrator (sub-microvolt offset, picoamp bias current) that lets the coil's output ride low and clean for hours.

Why no iron — the saturation argument

The conventional current transformer (CT) uses an iron core to multiply the coil's sensitivity. Permeability μ_r in laminated steel is typically 1,000 to 10,000, so mutual inductance grows by the same factor — a CT produces 1,000 times more output per amp than an air-core coil of the same geometry. That's a huge advantage when measuring tens-of-amp signals at line frequency: a 5 A CT can drive a 1-ohm burden producing 5 V directly, no amplification needed.

The fatal flaw is saturation. Iron's B-H curve flattens around 1.5 to 2 tesla. Once the flux density hits the saturation flux, additional current produces almost no additional flux, so the CT's response collapses. For a CT rated for 1 kA at 50 Hz, a fault current of 25 kA in the first cycle of a short-circuit drives the core deep into saturation, and the secondary side stops faithfully reproducing the primary current. Protection relays based on saturated CTs miss the fault and the upstream breaker stays closed too long, escalating damage. Worse, modern power electronics inject high-frequency harmonics from PWM modulation; CT saturation distorts the harmonic content and corrupts power-quality measurements.

An air-core Rogowski coil has μ_r = 1. The mutual inductance is set by geometry alone, and geometry doesn't saturate. A coil that measures 10 amps faithfully at 50 Hz will also measure 100 kA faithfully at 50 Hz (assuming the wire is big enough and the conductor isn't on fire). The trade is much lower output voltage — milivolts per amp instead of volts per amp — which the integrator and the modern op-amp recover at low cost.

Worked example: sizing a coil for inverter measurement

Design a Rogowski-based current probe for a 100 kW solar inverter outputting 250 A peak at 50 Hz fundamental, with switching ripple at 20 kHz.

Coil geometry:
  Conductor diameter   25 mm (cable)
  Coil major radius r  35 mm (around the conductor)
  Coil minor radius a   4 mm (cross-section)
  Turn count N        500 turns
  Cross-section A     π·a² = π·(4 mm)² ≈ 50 mm² = 50e-6 m²

Mutual inductance:
  M = μ₀ · N · A / (2π · r)
    = (4π · 10⁻⁷) · 500 · 50e-6 / (2π · 0.035)
    ≈ 143 × 10⁻⁹ V·s/A
    ≈ 143 nV·s/A  (or 0.143 µH)

Output voltage at 50 Hz fundamental, 250 A peak:
  I(t) = 250 sin(2π · 50 · t)
  dI/dt at peak: 250 · 2π · 50 ≈ 78,540 A/s
  V_coil = M · dI/dt = 143e-9 · 78,540 ≈ 11.2 mV peak

After integration (op-amp integrator, R = 10 kΩ, C = 100 nF):
  Time constant τ = RC = 1 ms
  V_int = (M / RC) · I = (143e-9 / 1e-3) · 250 = 35.8 mV peak
  Note: integrator gain shapes spectrum; output is proportional to I directly

Output for 20 kHz switching ripple (50 A peak ripple):
  dI/dt at peak: 50 · 2π · 20000 ≈ 6.28 × 10⁶ A/s
  V_coil = M · dI/dt = 143e-9 · 6.28e6 ≈ 0.9 V peak
  The raw coil voltage emphasises high frequencies (+20 dB/decade)
  The integrator flattens this back to 7.2 mV peak from the 20 kHz ripple

Bandwidth check:
  Coil self-resonance (estimate): L_coil ~ N²·μ₀·A/(2πr) ≈ 71 µH;
                                 C_stray ~ 10 pF → f_res ≈ 6 MHz
  Integrator HP corner: f_lo = 1/(2π·R·C) = 159 Hz (acceptable)
  Useful range: 1 Hz to 1 MHz (limited by f_res / 10 for clean response)

The example highlights why integration is needed: raw coil output rises 20 dB per decade with frequency, so a 1 percent switching ripple at 20 kHz produces 80 times more raw coil voltage than the 50 Hz fundamental. The integrator flattens the spectrum and recovers a clean current waveform. The 6 MHz coil resonance is well above the 20 kHz switching frequency, so the response is flat through the harmonics that matter.

Rogowski coil vs current transformer vs Hall sensor

Sensor	Core	Saturation	DC response	Range	Bandwidth	Typical use
Rogowski coil	Air	None	No	mA – MA AC	0.1 Hz – 50 MHz	Fault current, inverter, fast pulses
Iron-core CT	Laminated steel	1.5 – 2 T	No	1 – 5,000 A AC	50 Hz – 10 kHz	Utility metering, protection relays
Hall-effect (open-loop)	Air with ferrite concentrator	Concentrator may saturate	Yes	0.1 – 1000 A DC/AC	DC – 100 kHz	DC bus monitoring, battery
Hall-effect (closed-loop)	Ferrite with feedback winding	Linearized via feedback	Yes	0.1 – 2000 A DC/AC	DC – 200 kHz	Precision DC/AC measurement
Fluxgate	Soft magnetic strip	Drives core into saturation	Yes	µA – 100 A DC/AC	DC – 10 kHz	Geomagnetic, leakage current
Shunt resistor	n/a	n/a	Yes	µA – kA DC/AC	DC – tens of MHz	Direct in-line, low side

The selection logic in the field is straightforward. For a utility revenue meter watching steady 60 Hz current within nameplate rating, an iron-core CT is unbeatable on cost and accuracy. For a protection relay that must read 25 kA fault current cleanly, a Rogowski coil avoids saturation. For DC bus current in an EV traction inverter, Hall-effect or shunt are the only options because Rogowski is blind to DC. For an arc-fault detector that needs to see microsecond-scale di/dt spikes, the Rogowski's high-frequency bandwidth is the natural choice. For a clamp-on probe that an engineer needs to slip around an energized 4-inch bus bar in the field without unbolting anything, the flexibility of a Rogowski cable makes it the practical winner.

Where Rogowski coils win

• Utility fault recorders. Capture fault currents tens of times rated, where a CT would saturate.
• Inverter and motor-drive diagnostics. Wide bandwidth captures both 50 Hz fundamental and 20 kHz PWM ripple cleanly.
• Lightning and surge testing. Megamp pulses at megahertz bandwidth; no other sensor handles both.
• Flexible clip-on probes. The cable opens, wraps around a live bus, and clicks closed — no need to interrupt service.
• Welding equipment. Pulsed currents in the kA range with fast rise times.
• Pulsed-power physics. Tokamak coils, capacitor banks, rail-gun discharges.
• Power quality analyzers. Harmonics through the 50th order on multi-kA distribution feeders.

Common misconceptions

• Position of the conductor doesn't matter. True only for a perfectly closed, uniformly wound coil; real coils show 1 to 5 percent variation with conductor offset.
• Output is the current. The raw output is dI/dt; the user must integrate to recover I.
• Air-core means no shielding needed. External AC fields induce error voltages just like any coil; toroidal closure with a return-loop cable cancels most of this.
• Rogowski coils are slow. A well-built coil easily reaches 50 MHz; the limit is usually the cable's distributed capacitance, not the coil.
• Coil terminals can be open. Like all air-core devices, the coil is safe with open terminals — unlike an iron-core CT, which develops dangerous voltage if its burden is removed.
• Integrators don't drift. Op-amp offset gets multiplied by RC into a slow ramp; modern designs use chopper amps, digital integration, or windowed integration to keep drift bounded.