Encoder Angular Accuracy: In-depth Conversion Manual for Arcminutes/Arcseconds

The conversion between arcminutes (′) and arcseconds (″) for encoder accuracy is the most basic and error-prone link in industrial precision measurement. Below is a complete guide from core logic and quick conversion to practical applications.

I. Core Conversion Benchmarks (Must Memorize)

Angular units use the sexagesimal system, which is consistent with the logic of 'hours-minutes-seconds' in time.

Unit Relationship	Conversion Formula	Notes
1 Degree (°)	= 60 Arcminutes (′)	Basic Hexadecimal System
1 Arcminute (′)	= 60 Arcseconds (″)	Commonly Used Unit for Encoder Accuracy
1 Degree (°)	= 3600 Arcseconds (″)	3600 is the Core Calculation Coefficient

II. Quick Conversion Formulas and Practical Examples

1. Degree ↔ Arcminute/Arcsecond

This is the step to convert encoder parameter specifications (usually in °) into commonly used accuracy units in the industry.

Degree to Arcminute: Value × 60
Example: What is the accuracy of ±0.1° in arcminutes?

0.1 * 60 = ±6″

Degree to Arcsecond: Value × 3600

Example: What is the approximate accuracy of ±0.00278° in arcseconds?

0.00278 * 360 = ± 10″

2. Arcminute ↔ Arcsecond

This is the most commonly used internal conversion, used to compare indicators of different accuracy levels.

Arcminute to Arcsecond: Value × 60
Example: What is the accuracy of ±5′ in arcseconds?

5 * 60 = ±300″

Arcsecond to Arcminute:Value ÷ 60

Example: What is the accuracy of ±15″ in arcminutes?

15 ÷ 60 = ± 0.5′

III. Encoder-Specific: Conversion Between Pulse Count (PPR) and Arcseconds (″)

The resolution of an encoder is usually expressed in PPR (Pulses Per Revolution), and its corresponding theoretical minimum step is the single-step angle. This is a key conversion to evaluate whether the 'accuracy level' meets the standard.

Core Formula:

Application Quick Reference Table:

Encoder Resolution (PPR)	Theoretical Single-step Angle	Single-step Arcsecond (″)	Applicable Accuracy Level
1024	0.351°	≈ 1266″	General Speed Measurement, Low-grade Positioning
2048	0.176°	≈ 633″	General Servo, Packaging Machinery
8192	0.0439°	≈ 158″	Medium-High Precision Servo, CNC Machine Tools
65536	0.00549°	≈ 19.78″	High-Precision Robots, Precision Machining
131072	0.00275°	≈ 9.89″	Semiconductor Equipment, Ultra-High Precision

⚠️ Important Note:

The values calculated in the above table are theoretical resolution steps. The actual encoder accuracy is usually much worse than the step size.

For example: An encoder with 65536 PPR has a step size of about 20″, but the accuracy marked by the manufacturer may be ±15″ or ±30″.
Selection Principle: The marked accuracy of the encoder must meet the system requirements, and it is usually 3-5 times higher than the system positioning accuracy.

IV. Practical Phrases and Application Scenarios

Accurate expression can reflect professionalism when communicating with customers or writing technical documents:

Explaining Parameters to Customers
'The encoder we selected has a single-turn resolution of up to 65536 PPR, and the corresponding minimum distinguishable angle is only 19.8 arcseconds, which is equivalent to dividing 1 degree into more than 60,000 parts with extremely high resolution.'
Confirming System Compatibility
'Your equipment requires a positioning accuracy of ±0.01° (i.e., ±0.6′). To ensure the reliability of closed-loop control, we need to select an encoder with an accuracy within ±0.2 arcseconds (″), and the corresponding pulse count is recommended to be above 200,000 PPR.'
Installation and Debugging Instructions
'Although the nominal accuracy of the encoder is ±5 arcseconds, if the eccentricity error during installation exceeds 0.1mm, it will directly introduce a cumulative error greater than ±1 arcminute. Therefore, please strictly control the coaxiality.'

V. Pitfall Avoidance Guide

Do Not Confuse: The value converted from PPR is resolution (minimum scale), not accuracy (degree of correctness). High resolution does not mean high accuracy.
Unify Units: When calculating the tolerance zone or performing error superposition, be sure to first unify all units to arcseconds (″) or arcminutes (′) before calculation.
Decimal Place Handling:
High-precision scenarios (within ±1″): Keep 2-4 decimal places.
Medium and low-precision scenarios (above ±10″): Keep 1-2 decimal places or round directly.