Position/Presence/Proximity

# Understanding Integrated Hall Effect Rotary Encoders

November 1, 2006 Sensors

### How Hall-based rotary encoders work.

Linear Hall sensors generate a DC output voltage proportional to the strength of an applied magnetic field and can be used for high-resolution angle sensors when placed near a diametrically magnetized magnet (Figure 1). The rotating magnet generates a sinusoidal waveform, one full wave per revolution. This type of setup can be used only for a limited angular range because the output voltage (in relation to the rotation angle) is ambiguous at angles >90° in both directions from the zero crossing point.

 Figure 1. Simple angular measurement using a linear Hall sensor

In practice, only the "quasi-linear" range up to about ±45° can be used for accurate angle measurements. This setup—being very sensitive to the position of the sensor relative to the magnet and to unwanted external magnetic fields—requires tight mechanical tolerances and, in many cases, magnetic shielding. The temperature of the magnet is also important; magnet strength changes with temperature, resulting in an undesirable output voltage change. To compensate, the Hall sensor must be able to measure the temperature and correct the output voltage according to the temperature coefficient of the magnet.

 Figure 2. Measurement principle of a 360° Hall-based sensor

## Angle Measurements up to 360°

Making angular measurements over a full revolution requires additional measures. Austriamicrosystems' approach uses four Hall elements, rather than one, and spaces them equally underneath a diametrically magnetized rotating magnet (Figure 2) to generate four sinusoidal waveforms, each phase shifted by 90° from its neighbor, as shown in Equation 1:

where:

â = peak amplitude

α = magnet rotational angle relative to the sensor

By using differential amplification of two opposite sensors (H1–H3 and H2–H4), we get Equation 2, which generates two 90° phase-shifted signals with double amplitude:

These two analog signals are digitized by ADCs and processed further in the digital domain.

## The CORDIC Algorithm

A CORDIC (coordinate rotation digital computer) transforms sine and cosine information into angle and magnitude information, using the following relationships:

where:

A= measured angle

α = magnet rotational angle relative to the sensor

â = peak amplitude

`1 2 3 `

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