Search Coil (Magnetic Flux Measurement)
Theory: changing flux becomes voltage
Overview
I want to measure stray field around a transformer. Is a home-made search coil good enough?
For AC stray fields, a search coil is the most straightforward method. Changing flux through the coil induces a voltage by Faraday's law, and with turns and area known, voltage gives field strength. It is simple and, unlike semiconductor sensors, doesn't saturate or drift much with temperature. But output is proportional to how fast the field changes, so it is small at low frequency and zero for DC, and the coil's own resonance sets an upper frequency limit.
Output voltage
$N$ is turns, $A$ coil area and $\omega$ angular frequency; for sine waves output is proportional to frequency. With the coil small compared with the field's spatial variation, it measures the average field over its area.
Recovering flux density
To recover the waveform, integrate the output back to flux density, with an analog integrator or numerical integration — but any small offset is integrated too and drifts over time.
Self-resonance and upper frequency
The coil's inductance $L$ and interwinding or cable capacitance $C$ resonate, above which output no longer tracks the field. The first formula approximates a long solenoid; short coils have somewhat less. More turns raise sensitivity but increase inductance with the square, lowering the resonant frequency.
From geomagnetic variations to space
Measuring field changes with coils began with Faraday's discovery of induction in 1831. In the 19th century coils observed fine geomagnetic variations; in the 20th they served mineral exploration and electromagnetic surveys of subsurface structure. Satellites and space probes carry search-coil magnetometers for plasma waves from a few Hz to kHz, and instruments combining very many turns with high-permeability cores reach about 1 pT — one fifty-millionth of the Earth's field.
Worked examples
Example 1: output versus frequency
A 1,000-turn air-core coil 20 mm in diameter (3.14 cm²) measuring a 1 μT sinusoidal field:
| Frequency | Output voltage |
|---|---|
| 1 Hz | 2.0 μV |
| 10 Hz | 19.7 μV |
| 50 Hz | 98.7 μV |
| 1 kHz | 1.97 mV |
| 10 kHz | 19.7 mV |
About 100 μV at 50 Hz — hard to measure without amplification.
Yes. Near a transformer, stray fields are often tens to hundreds of μT, giving tens to hundreds of times this output. For weak or low-frequency fields, add turns or area, or use a ferrite core. The core's effect is its apparent permeability, set by shape and larger for slender rods; at 50, the output at 50 Hz and 1 μT becomes about 4.9 mV. A core concentrates and disturbs the surrounding field and saturates in strong fields, so choose by field magnitude.
Example 2: self-resonance
Estimating the same coil's inductance with the long-solenoid approximation gives about 19.7 mH (the real value is lower). With 50 pF of winding and cable capacitance, resonance is about 160 kHz; with 200 pF, about 80 kHz. Keep the measurement band well below resonance by shortening cables or reducing turns.
Example 3: integrator drift
For this coil (turns × area = 0.314 m²), a 1 μV integrator offset drifts the flux density about 3.2 μT per second, and 10 μV about 32 μT per second. For long records, block DC or correct during intervals known to have zero field.
Measurement and analysis cautions
Cautions
- Orientation: a coil measures only the component normal to its plane; use three orthogonal coils for all components.
- Lead loops: leads forming a loop act as a coil too; twist them.
- Electric-field pickup: near high voltage, capacitive coupling adds signal; use an electrostatic shield with a gap.
- Comparing with analysis: extract the field averaged over the coil area from simulations; point values won't match.
“Analysis and measurement differ twofold”
Stray field near a transformer differed nearly twofold between analysis and measurement. The analysis had read flux density at a point, while the real coil was 30 mm across and the field varied strongly over that span. Near a source the field changes quickly, so only coil-averaged values compare. Orientation matters too: a 15° tilt reads the normal component about 3.4% low. Reproducing the measurement conditions in the analysis is the precondition for any comparison.
Common mistakes
Mistakes and fixes
| Mistake | Effect | Fix |
|---|---|---|
| Trying to measure DC fields | No output | Use a fluxgate etc. |
| Too many turns | Resonance enters the band | Check resonance |
| Ignoring integrator offset | Continuous drift | DC blocking, correction |
| Untwisted leads | Extra flux pickup | Twist the leads |
| Comparing with a point value | Large errors near sources | Compare area-averaged values |
I'd like to try induction and coil calculations myself.
Try the Faraday's law tool and magnetic induction simulator for induced voltage, the solenoid coil tool for coil field and inductance, and the RLC resonance simulator for resonance. Compare other magnetic sensors in Rogowski coils, fluxgate magnetometers, Hall effect sensors and magnetoresistive sensors.
Related Topics
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