Hall Effect Sensor
Theory: a voltage from the Lorentz force
Overview
I'm building a circuit to measure motor current with a Hall element. I don't know how to choose the element and core.
A Hall element exploits the effect whereby, when field is applied to a current-carrying semiconductor plate, the Lorentz force pushes charge to one side and a transverse voltage appears. The voltage is proportional to field, so it senses field. To measure current, a gapped magnetic core around the conductor concentrates the current's field in the gap, where the element measures it. Element material sets sensitivity and temperature behaviour; gap size sets sensitivity and the current ceiling. First get a feel for each number.
Hall voltage
$n$ is carrier density, $q$ elementary charge, $t$ plate thickness, $\mu_H$ Hall mobility and $W/L$ width-to-length ratio. At constant current, low-density thin elements give more output; at constant voltage, high-mobility ones do.
Field in the core gap
If core permeability is high enough, magnetic-circuit reluctance is set almost entirely by the gap, and gap field is proportional to current. $N$ is turns (1 for a busbar passing through once) and $g$ gap length.
An application that waited a century
The Hall effect was discovered in 1879 by the American graduate student Hall, passing current through a thin gold film in a magnetic field. The electron wasn't yet known, and in metals the effect was tiny and impractical. As semiconductor research advanced in the mid-20th century, low-carrier-density semiconductors were found to give large Hall voltages, and Hall elements were produced. Today they are everywhere: smartphone electronic compasses, motor rotor-position sensing and electric-vehicle current sensors.
Worked examples
Example 1: silicon element sensitivity
A silicon element with carrier density 10²² m⁻³ and 1 µm thickness has sensitivity 624 V/(A·T). At 1 mA and 10 mT, the Hall voltage is 6.24 mV.
Example 2: material comparison (constant 1 V, 10 mT)
| Material | Mobility | Hall voltage | Notes |
|---|---|---|---|
| Silicon | 0.135 m²/Vs | 1.4 mV | Integrates with circuitry |
| GaAs | 0.85 m²/Vs | 8.5 mV | Good temperature stability |
| InSb | 7.8 m²/Vs | 78 mV | High sensitivity, strong temperature drift |
Example 3: current-sensor gap
| Gap | Field per ampere | Field at 100 A | Current at 0.4 T saturation |
|---|---|---|---|
| 1 mm | 1.26 mT | 126 mT | 318 A |
| 2 mm | 0.63 mT | 63 mT | 637 A |
Widening the gap raises the current range but lowers sensitivity. How do I decide?
First, the core must not saturate at the maximum current; then choose a gap giving enough element output against noise and temperature drift. But this formula neglects core reluctance and fringing (field spreading from the gap edges), so the real gap field is smaller and varies with position. Small element misalignment changes sensitivity, so compute the gap-field distribution by magnetic analysis and check the effect of element position and assembly tolerance. For large currents, closed-loop sensors with a cancelling winding avoid core saturation.
Design workflow
- Set the field (or current) range, required resolution and temperature range.
- Choose element material and drive mode (constant current or voltage) from sensitivity and temperature behaviour.
- Size core shape and gap by balancing saturation against sensitivity.
- Use magnetic analysis to check gap-field distribution and crosstalk from neighbouring conductors.
- Design offset and temperature compensation and verify on prototypes.
“The neighbouring phase leaks in”
In an inverter's three-phase current sensors, one sensor's reading picked up several percent of the neighbouring phase's current changes. A neighbouring busbar passed close to the core gap, and part of its field entered the gap. 3D magnetic analysis with current in only the neighbouring busbar reproduced the leakage ratio. Rotating the core 90° away from that busbar and adding a magnetic shield nearly eliminated it. Hall elements faithfully pick up fields other than the intended one.
Common mistakes
Mistakes and fixes
| Mistake | Effect | Fix |
|---|---|---|
| Ignoring fringing | Sensitivity overestimated | Compute distribution by analysis |
| Not checking saturation at maximum | Output clips at high current | Revise gap or core section |
| Ignoring temperature effects | Drift at high temperature | Temperature compensation |
| Ignoring neighbouring conductors | Crosstalk | Review layout and shielding |
| Ignoring element misalignment | Large unit-to-unit variation | Analyse with tolerances |
I'd like to try related calculations.
Try the Hall effect simulator and magnetic field calculator. Other magnetic sensors include fluxgate magnetometers, magnetoresistive sensors, Rogowski coils and magnetic encoders.
Related Topics
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