Do you know your arc-seconds from your gradians?

Paul Boughton

Angle sensors are generally rated and, just as importantly, priced according to their measurement performance. But performance is stated in a variety of ways and some manufacturers confuse matters by using crafty ‘spec-manship’. Mark Howard explains some of the terminology and provides an explanation of how to specify an angle sensor that is right for your application.  

The instrumentation industry seems particularly adept at confusing its customers. For example, take the number of terms used to describe a device that measures angle: rotary sensor, angular position sensor, angle transducer, rotary encoder, shaft encoder, rotation transducer, angle detector, and so on. Each term does have its own specific connotation but for most engineers, the precise meaning of the terminology is secondary to their need to measure angle. For the purposes of this article, the term ‘angle sensor’ will be used.

Over the past 100 years, most of the main physical principles and laws have been used to measure angle – potentiometer (Ohm’s law), magnetic (Hall, magnetostriction & magnetoresistive effects), inductive (Faraday’s laws), capacitive effects, optical and laser. Each technique has its own strengths and weaknesses and, accordingly, some sectors prefer certain techniques. For example, optical and capacitive devices offer high levels of precision but are unreliable in wet or dirty environments, so they tend to be chosen for laboratory or test equipment (where the environment is tightly controlled) but tend not to be used in petrochemical, military or aerospace applications. Similarly, potentiometers are rarely chosen for high vibration environments due to the limited life of their resistive tracks. Unsurprisingly, the way in which the different products are presented and specified on their data sheets tends to accentuate their positives and obfuscate or remain silent on the negatives. This is further complicated as certain industries have their own preferred measurement units and terminology. [Page Break]

The aim of this article is to provide some clarification and to shed some light on the common pitfalls to avoid when choosing an angle sensor.


Measurement performance is quoted in a myriad of different units. Any proper comparison between products should be based on common units.

Pulses per rev (PPR). PPR is commonly quoted for incremental angle sensors, especially optical devices, and describes the number of pulses that the device outputs per revolution. Importantly, the product’s PPR is not necessarily connected to accuracy. A common misconception is that an angle sensor that produces 1,000 PPR is accurate to 1/1000th of a rev. But this is incorrect!

Counts per rev (CPR). Many angle sensors output two lots of pulses – usually referred to as A/B pulse streams (in quadrature) so that direction of travel is indicated.  Accordingly, for each PPR there are two leading edges and two trailing edges, which can then be used to generate CPR. The difference is important – a device with 1024PPR  has 4 times the resolution of a device offering 1024CPR.

Bits. The number of bits in an angle sensor’s output is an increasingly common term due to the increasing use of digital outputs such as RS422, SSI, CANbus, etc. Each additional bit doubles the quoted resolution. Whilst obvious to some, it is counter intuitive to many. For example, a 12bit product will output 4,096 steps over a rev, whereas a 14bit product (it doesn’t sound that different does it?) will output 16,384 steps. A ‘couple of bits’ makes a huge difference to measurement performance. Note also the difference between x bits of resolution and y bits of accuracy. For example, an angle sensor may be specified as 10bits of resolution with 8bits of accuracy – ie 1024 steps per rev with an accuracy of 1/256 of a rev.

Radians.  Radians are still widely used by the military, aerospace and scientific sectors, especially in motion control. A radian is the angle subtended by a circle’s arc whose length is numerically equal to the circle’s radius. There are 2P radians per rev. A milli-radian (usually ‘millirad’) is 1/1000th of a radian and a micro-radian (usually ‘urad’) is 1/1000th of a milliradian. The ‘mil’ is commonly used by military organisations and provides the handy property of subtension – that 1 mil approximately subtends 1 metre at a distance of 1000 metres. It’s a useful unit of measure if you’re lobbing shells onto an enemy position.

Gradians. The gradian is a unit of angle equivalent to 1⁄400 of a revolution. It is also known as a gon, grad, or grade. One grad equals 9⁄10 of a degree. The unit originated in France as the grade, along with the metric system. Although attempts at a general introduction were made, the unit has only been adopted in some countries and specialised areas, such as surveying. Subdivisions of gradian used in surveying are c's (1c = 0.01grad) and cc's (1cc = 0.0001grad).  

Degrees, Arc-Minutes & Arc-Seconds. Wouldn’t life be simpler if everyone used degrees? Well it would, but who said life was going to be simple, especially for a design engineer. So 1revolution = 360degrees (I guess you knew that), but each degree can be divided up into 60arc-minutes and each arc-minute can be divided up into 60arc-seconds. Accordingly, 1degree = 3600arc-seconds.

Per centage. Per centage is often used to describe the accuracy or linearity of lower performance angle sensors and should rightly be (but often is not) specified in terms of per centage of full-scale. Importantly, the full-scale of some angle sensors is not 360 degrees but may be 60, 90, 120 or 180 degrees. Accordingly, a product with a full-scale of 90 degrees and a linearity of 0.1 per cent of full-scale is likely to be more accurate than a device with 360 degree full-scale and 0.05 per centage linearity over full-scale.[Page Break]

You can use the look-up table (Fig 1) to convert between the most common units.

Resolution, repeatability and accuracy

When comparing different products, as well as using common units, it’s also important to have a common meaning of the terminology. The main terms to note are:
* Accuracy – a sensor’s output veracity against a perfect scale.
* Resolution – the smallest increment or decrement in position that a sensor can measure.
* Precision – refers to a sensor’s reproducibility. This is most usefully expressed in terms of the sensor’s Repeatability.
* Linearity – how well a sensor's actual performance across a specified range matches a straight line. Linearity is usually measured in terms of a deviation, or non-linearity, from an ideal straight line. In many cases Linearity and Accuracy are the same in the absence of any offset.

For a surprisingly large number of engineering applications, the critical factors are resolution and repeatability rather than linearity or accuracy. Getting this wrong and over-specifying the accuracy or linearity requirements could be very costly.[Page Break]

When comparing performance factors between angle sensors it is important to read the small print of the product’s data sheet (if there is no small print or clarification notes in an angle sensor’s data sheet, it may be a fair indication that it’s not that good a product). For example, there should be some qualification or basis stated for how measurement performance is expressed. Common pitfalls are:
* Accuracy measured at a specific temperature and/or humidity. This can be a fair indication that the product will drift relative to temperature or humidity, so also look out for its temperature/humidity coefficient (see below). If a temperature coefficient is not stated it may be an indication that it’s inconveniently large. For some capacitive devices there may also be some limitations to the measurement performance relating to humidity or foreign matter – most notably condensation, which will cause measurement to drift.
* Accuracy measured at specified installation tolerances. In some unpackaged or OEM sensors, installation requires a Swiss watch maker with a steady hand and a microscope in order to match the installation tolerances required for the product’s stated accuracy.
* Mismatch between resolution and repeatability. In some products resolution will be stated without any mention of repeatability or the resolution is specified at a much finer level than repeatability. A product with 16 bits of resolution and ±4 bits of repeatability is only really a 12bit product – but it’s been presented so that a less discerning engineer will compare it favourably with a true 16bit product.
* High resolution but slow speed operation. One way for a sensor manufacturer to achieve high resolution cheaply is to slow the measurement rate and carry out lots of averaging. Accordingly, resolution and repeatability need to be considered in light of measurement speed – for example, 16bit resolution with 2bit repeatability at 500Hz.

This can be illustrated with an example. Consider the following angle sensor.

Quoted resolution is 21 bits or 2,097,152 steps per revolution.  Repeatability is stated as +/- 1bit at 1kHz so the quoted resolution is a true rather than inflated representation.  

Accuracy is stated as a linearity (max. deviation from true position) over full-scale of 360 degrees at <40 arc-seconds (so <0.003% of full-scale). Crucially, these parameters are stated with realistically achievable installation tolerances of ±0.35mm axially and <0.25mm radially. Temperature coefficient over the temperature range of -55 or -40 to 80˚C is stated as <0.25ppm/K over full-scale, which equates to a temperature drift of about 2 arc-seconds for every 10 degrees of temperature change (in other words, tiny compared to the likely differential thermal expansion of the host system). Since the devices are inductive rather than optical or capacitive, there will be no deterioration due to humidity or foreign matter.[Page Break]

In summary, if you want a solid comparison of angle sensors:

1. Ensure you have consistent units.
2. Read the small print of any data sheet and make sure you fully understand it, particularly the difference between resolution, repeatability and accuracy.
3. Consider the effect of installation tolerances on measurement performance.
4. Consider the effects of humidity, foreign matter, lifetime and temperature on measurement performance.
5. Run a test and verify the manufacturer’s data yourself.
Mark Howard is General Manager, Zettlex UK Ltd, Newton, Cambridge, UK.