Hall Voltage Explained: Formula, Calculation, and Real-World Sensor Applications

Hall voltage is the measurable transverse electrical potential difference that appears across a current-carrying conductor or semiconductor when it is subjected to a perpendicular magnetic field. When charge carriers (usually electrons) flow through a material and encounter a magnetic field, the Lorentz force pushes them toward one edge of the material. This accumulation of charge on one side creates a measurable voltage gradient perpendicular to both the current flow and the magnetic field.

The Physics and Math Behind Hall Voltage

To design or troubleshoot circuits relying on the Hall effect, you need to understand the variables that dictate the magnitude of the generated voltage. The fundamental equation for Hall voltage ($V_H$) is:

V_H = (I × B) / (n × q × t)

• I = Control current flowing through the material (Amperes)
• B = Magnetic flux density perpendicular to the current (Tesla)
• n = Charge carrier density of the material (carriers per cubic meter)
• q = Charge of a single electron ($1.602 \times 10^{-19}$ Coulombs)
• t = Thickness of the material in the direction of the magnetic field (meters)

A Worked Numeric Example: Doped Silicon vs. Copper

Let us calculate the expected Hall voltage for a doped N-type silicon slab used in a commercial sensor, and compare it to a standard copper PCB trace.

Scenario A: Doped Silicon Sensor Element

• Control current ($I$): 5 mA (0.005 A)
• Magnetic field ($B$): 0.1 T (1000 Gauss, typical for a small neodymium magnet)
• Carrier density ($n$): $1 \times 10^{21}$ carriers/$m^3$ (typical for moderately doped silicon)
• Thickness ($t$): 100 μm (0.0001 m)

Plugging these into the formula:

$V_H = (0.005 \times 0.1) / (1 \times 10^{21} \times 1.602 \times 10^{-19} \times 0.0001)$
$V_H = 0.0005 / 0.01602$
V_H = 31.2 mV

A 31.2 mV signal is robust and easily readable by a standard 12-bit ADC on an ESP32 or an external ADS1115 module.

Scenario B: 1 oz Copper PCB Trace

If we run 5A through a 1 oz copper trace (35 μm thick) in the same 0.1 T field, the carrier density of copper ($8.5 \times 10^{28}$ carriers/$m^3$) is so massive that the resulting Hall voltage is roughly 0.05 μV. This is buried in thermal noise and practically impossible to measure without specialized laboratory equipment. This is exactly why commercial Hall sensors are built from semiconductors like silicon or indium antimonide, not copper.

What Hall Voltage Changes in Power Electronics

In practical circuit design, the existence of Hall voltage fundamentally changes how we approach galvanic isolation in current sensing.

Historically, measuring high DC currents required a shunt resistor. A shunt works, but it introduces a voltage drop, generates heat ($I^2R$ losses), and ties your measurement circuit's ground directly to the high-power ground. If a high-side shunt is used, you need complex, expensive differential amplifiers to reject the high common-mode voltage.

Hall effect current sensors (like the Allegro ACS724 or Texas Instruments TMCS1100) solve this. The high current flows through a primary copper leadframe, generating a magnetic field. A secondary silicon Hall element sits across an isolation gap, generating a Hall voltage proportional to the current.

While Hall sensors do not eliminate $I^2R$ heating at high currents (the internal leadframe still has ~1.5 mΩ of resistance), they completely eliminate the need for a shared ground, allowing a 3.3V microcontroller to safely monitor a 400V DC solar inverter bus.

Where You Meet Hall Voltage in Practice

If you work with electromechanical systems or power electronics, you are already relying on Hall voltage. Here is where it shows up on the bench:

• BLDC Motor Commutation: Brushless DC motors in e-scooters, drones, and CNC spindles use three internal Hall sensors to detect the rotor's permanent magnets. The controller reads the changing Hall voltage to know exactly when to fire the next MOSFET phase. If your e-bike motor stutters on startup, a failed Hall sensor (often due to a broken 5V pull-up or melted signal wire) is the first suspect.
• DC Clamp Meters: Unlike AC clamp meters that use current transformers, a DC clamp meter has a physical gap in its ferrite core. A Hall sensor sits in that gap. When you clamp it around a wire, the magnetic field concentrates across the gap, generating a Hall voltage that the meter scales to display Amperage.
• Throttle Position Sensors: E-bike twist throttles and industrial joysticks use a linear Hall sensor (like the SS49E) and a rotating magnet. Because there is no physical contact like a carbon track potentiometer, they do not suffer from wiper noise or mechanical wear.

Common Confusion: Hall Voltage vs. Faraday Induction

The most frequent mistake makers and junior technicians make is confusing the Hall effect with Faraday's law of induction. Both involve magnetic fields and voltage, but the physics are entirely different.

Faraday's Law (Induction) requires a changing magnetic field ($d\Phi/dt$). A standard AC current transformer (CT clamp) only works on AC because the alternating current creates a collapsing and expanding magnetic field, which induces a voltage in the secondary coil. If you put DC through a CT clamp, the magnetic field is static, $d\Phi/dt$ is zero, and the output voltage is zero.

The Hall Effect responds to the presence of a magnetic field, regardless of whether it is moving or static. A Hall sensor will output a steady DC Hall voltage if you hold a permanent magnet perfectly still next to it. This is why Hall effect clamp meters can measure DC battery current, while standard inductive clamps cannot. For a deeper look at the underlying physics, Georgia State University's HyperPhysics provides an excellent breakdown of the Lorentz force mechanics involved.