##### 1 February 2002
 MOTION FUNDAMENTALS:

# Relating Accuracy, Resolution, and Linearity

### How does a design engineer compare rotary position feedback products, given a budget for rotary error, with all other factors being equal?

The purpose of rotary position feedback (RTF) is to improve the efficiency of your automated process by minimizing the motor's rotary error. Encoders, resolvers, and potentiometers all generate this type of data. Each has inherent benefits, the examination of which gives rise to product selection. But how does a design engineer compare products, given a budget for rotary error, with all other factors being equal?

Let's review the intended use of these relevant terms:

• Accuracy is used by resolver manufacturers to express the mechanical and electrical rotary error introduced while manufacturing and assembling a resolver. It includes factors such as shaft and bore tolerances, bearing runout, and winding perpendicularity. Typically, its units are arc-minutes.
• Resolution is the term encoder manufacturers use to express the finite number of measurements that can be made in a single revolution. It's usually expressed in lines per revolution for incremental encoders (and in number of bits for absolute encoders), a reference to the number of "opaque lines" (for an optical encoder) etched onto a given radius of the glass disk, which indicate a unique angular position. We also use resolution, in number of bits, to rate an analog-to-digital (A/D) signal converter's measurement capability. Formulae assume that the converter is accurate to one least significant bit.
• Quadrature is an optional feature for incremental encoders; it refers to the 90° phase relationship of the channel A and B signals, each of which has both a rising and a falling edge. By counting both channels concurrently, this feature enhances the encoder's resolution.
• Instrument Error is an encoder manufacturer's term expressing the mechanical rotary error introduced during an encoder's manufacture and assembly. It includes shaft and bore tolerances, bearing runout, and alignment between the emitter, disk, and detector; it's typically given in arc-minutes.
• Quadrature Error is the signal phase error between channels A and B, commonly expressed as a percentage of the electrical cycle. For example, the distance between the leading edge of channel A and the leading edge of channel B is L/4 ±L/10, where L is the cycle duration. Thus, the tolerance divisor (i.e., 10) is the important variable. When encoder manufacturers don't specify instrument error separately, this tolerance is typically the sum total of both the instrument and the quadrature errors.
• Electrical Angle is how potentiometer manufacturers express the device's measurement range. For single-turn potentiometers, this number is slightly less than 360° and is applicable for each input shaft revolution. It exists to allow for the termination of the resistor coil/resistive material. (Use formulae in this article for single-turn potentiometers only.)
• Linearity is the mechanical and electrical rotary error introduced while manufacturing and assembling a potentiometer. Its factors include shaft and bore tolerances, bearing runout, wiper positioning, and element concentricity; it's typically given as a percentage of the output value.

Our task is to place the encoder, resolver, and potentiometer on a level playing field in terms of rotary position error. Because each device can be mounted directly to the rotating axis, we'll consider only the inherent component errors. In typical applications, we'd use an A/D converter for the resolver and potentiometer, converting the analog signal to its digital counterpart for processing. Thus, we must also consider its resolution for these two devices.

## Step 1: Component Resolution Limitations

An RTF device can be only as accurate as the separation between its measuring increments. The formulae show how to calculate this separation in units of arc-minutes. Note that common controllers available today have integrated A/D converters with resolutions of 10, 12, or 16 bits. When considering an analog controller for your resolver or potentiometer, any resolution contributor would be negligible.

 (1)

For resolvers, potentiometers, and absolute encoders:
 CPR = 2BITS (2)

For incremental encoders with quadrature:

 CPR = 4 x LPR (3)

For incremental encoders without quadrature:

 CPR = LPR (4)

where

 RC resolution contribution CPR resolution (counts per revolution) BITS A/D converter resolution or encoder resolution (bits) LPR resolution (lines per revolution)

## Step 2: Mechanical and Electrical Error of Each Component

As an assembly, each RTF device has built-in error due to subcomponent allowances and assembly tolerances. The following formulae convert these errors into arc-minutes. For potentiometers, this value will reflect a worst-case condition that may occur at one extreme of the electrical angle. (Don't confuse the potentiometer's linearity with its resistance tolerance—the latter is the tolerance on the peak resistance value occurring at the end of each revolution; both values are expressed as percentages.) These formulae are applicable for ambient conditions. Note that changing environmental conditions will introduce significant changes in encoders and potentiometers, due to thermal expansion of dissimilar materials, optical alignment changes, etc.

 For resolvers: For potentiometers: ME = ACY (5) ME = EA x LNP x 60 (6) For encoders: ME = IE + QE (7) (8)

where

ME Mechanical and electrical contribution
 ACY Resolver accuracy (arc-minutes) EA Electrical angle for potentiometer LNP Potentiometer linearity IE Instrument error QE Quadrature error K Divisor constant

## Applying the Concept

It isn't my intention to insinuate that a position feedback device's rotary position error should be the only consideration for product selection. There are many other conditions that may dictate the use of one over the others, based on product strengths and weaknesses or material, installation, and maintenance costs. A good example of this exercise's usefulness, however, is when comparing potential feedback devices based on several key design factors. Let's look at the following example.

A product engineer must design automated equipment to make accurate and repeatable rotary position measurements on a cost-sensitive program. The design will be subjected to low-duty usage in an environment where temperature variation could affect the device's repeatability. The engineer has access to an existing digital controller with digital inputs, as well as analog inputs that use 12-bit conversion resolution. Additionally, homing sequences performed by the equipment aren't problematic, making the benefits of an absolute position feedback device over an incremental one transparent for this application. Thus, the principal design factors are rotary position accuracy, cost, and environmental influences on repeatability.

Our engineer has located the following position feedback devices in his inventory:

• A resolver with 3 arc-minute accuracy (signal is unaffected by the environmental variations)
• An incremental encoder with a 1,000-line resolution and quadrature output, with an instrument error of 0.5 arc-minute and a quadrature tolerance of ±10%
• A potentiometer with 0.25% linearity and a minimum electrical angle of 350° that encompasses all required data points for the automated equipment

The preceding procedure permits an accuracy comparison among the available components:

Resolver:

 RPE = RC + ME (9)
 (10)

ME = 3 arc-minutes

RPE = 8.27 arc-minutes

Potentiometer:

 (11)

 ME = 350 x 0.0025 x 60 = 52.5 arcd-minutes (12)

RPE = 57.77 arc-minutes at its extreme position

Encoder:

 (13)

 (14)

RPE = 8.06 arc-minutes

As I've stated, a second major consideration is environmental influence on repeatability. While the resolver signal can withstand this influence, its effect on the other devices yields unsatisfactory performance. To overcome this, our engineer can construct an enclosure to nullify these effects while adding cost to his program.

The final comparison is cost-based, where in this analysis:

Cost to Program = component cost + installation cost + maintenance cost + enclosure cost (where applicable)

In general, the resolver's component cost will be highest and the potentiometer's will be lowest. The installation cost should be similar for all three items. Maintenance costs among the three components would also be similar. Because the application is low duty, the maintenance cost attributed to the shorter operating life of the potentiometer won't be a factor. Lastly, the enclosure cost applies only to the encoder and the potentiometer and includes its material, assembly, and amortized design costs.

In this application, the resolver appears to be the best-suited component, due to both the enclosure cost and the relative proximity of the resolver and encoder accuracies. Regardless, this exercise's purpose isn't to promote one component over another, but rather to demonstrate a tool for comparing one of the basic attributes of an RTF device. Now, based on weighted severity of the design factors involved, engineers can make a more informed product selection. MC

#### Make Contact!

Mike Stegmann is a product manager for Thomson Micron. Contact him at 50 Alexander Court, Ronkonkoma, NY 11779; tel: (888) 554-6684.