Ball Screw Positioning Resolution: How to Calculate It from Lead, Encoder Counts and Gear Ratio

Ball screw positioning resolution describes the smallest theoretical linear movement represented by one motor command step or one feedback count. It can be calculated from the ball screw lead, the number of motor steps or encoder counts per revolution, and any transmission ratio between the motor and screw.

The calculation itself is straightforward. The difficulty is understanding what the result actually represents. A calculated resolution of 0.5 μm does not mean that the machine can position to an accuracy of ±0.5 μm. Ball screw lead error, backlash, elastic deformation, mounting accuracy, thermal expansion and controller performance can all make the real positioning error much larger.

This guide explains how to calculate ball screw positioning resolution correctly and how to distinguish theoretical resolution from positioning accuracy and repeatability.

What Is Ball Screw Positioning Resolution?

When a ball screw rotates by one complete revolution, the nut moves by one screw lead. A 5 mm lead ball screw therefore moves the nut 5 mm for every complete screw revolution.

The motor and controller divide that revolution into smaller angular increments. These increments may be expressed as:

• Encoder counts for a servo motor
• Full steps or microsteps for a stepper motor
• Command pulses from a motion controller
• Feedback counts from a linear or rotary encoder

Ball screw positioning resolution is the theoretical linear travel corresponding to one of these increments.

Basic Ball Screw Resolution Formula

For a directly coupled motor and ball screw:

Using symbols:

R = L ÷ N

Where:

• R = theoretical linear resolution, normally in mm/count or μm/count
• L = ball screw lead in mm/revolution
• N = effective motor steps, command pulses or encoder counts per revolution

If the ball screw has a 5 mm lead and the system provides 10,000 effective counts per screw revolution:

R = 5 ÷ 10,000 = 0.0005 mm/count

0.0005 mm = 0.5 μm

The theoretical feedback increment is therefore 0.5 μm per count.

How Gear Ratio Changes Linear Resolution

Some axes use a gearbox or timing-belt transmission between the motor and ball screw. To prevent confusion, this article defines the transmission ratio as:

i = Motor revolutions ÷ Ball screw revolutions

Under this definition, a 2:1 reduction means the motor rotates twice while the ball screw rotates once. The complete formula becomes:

R = L ÷ (N × i)

This definition must be checked before using a value from a gearbox or pulley catalogue. Some manufacturers express the ratio in the opposite direction.

Example: 5 mm Lead with a 2:1 Reduction

• Ball screw lead: 5 mm
• Encoder resolution: 10,000 counts per motor revolution
• Transmission ratio: 2 motor revolutions per screw revolution

R = 5 ÷ (10,000 × 2)

R = 0.00025 mm = 0.25 μm/count

The reduction improves the theoretical linear resolution because more motor counts are required for one ball screw revolution. However, it also reduces the linear distance travelled for a given motor speed. Gearbox backlash, belt elasticity and transmission error must also be considered.

Servo Motor Example

Consider a servo-driven CNC axis with the following conditions:

• Ball screw: SFU1605
• Nominal diameter: 16 mm
• Lead: 5 mm
• Effective feedback resolution: 10,000 counts per motor revolution
• Connection: direct coupling

R = 5 ÷ 10,000

R = 0.0005 mm = 0.5 μm/count

If the axis is commanded to move 20 mm, the theoretical number of counts is:

Required counts = Travel distance ÷ Resolution

20 ÷ 0.0005 = 40,000 counts

The same result can be checked using screw revolutions:

• 20 mm travel ÷ 5 mm lead = 4 screw revolutions
• 4 revolutions × 10,000 counts = 40,000 counts

The two methods produce the same result, confirming that the unit conversion is consistent.

Stepper Motor Example

A standard 1.8° stepper motor has 200 full steps per revolution:

360° ÷ 1.8° = 200 steps/revolution

If the driver is set to 16 microsteps per full step:

200 × 16 = 3,200 microsteps/revolution

For a directly driven 5 mm lead ball screw:

R = 5 ÷ 3,200

R = 0.0015625 mm = 1.5625 μm/microstep

This is the theoretical command increment. It should not be interpreted as a guaranteed 1.5625 μm positioning accuracy. Motor torque, load, friction, resonance and the non-linearity of microstep movement can prevent every commanded microstep from producing an equal mechanical displacement.

Do Not Confuse PPR with Encoder Counts

Encoder terminology is not always used consistently. An incremental quadrature encoder may be specified in pulses per revolution, while the motion controller counts the rising and falling edges of both A and B channels.

For example, a 2,500 PPR quadrature encoder may produce:

2,500 PPR × 4 = 10,000 counts per revolution

Using 2,500 instead of 10,000 in the resolution formula would make the calculated linear increment four times too large. Before calculating, confirm whether the drive documentation specifies:

• Pulses per revolution
• Lines per revolution
• Counts per revolution
• Command pulses per motor revolution
• Effective feedback resolution after electronic gearing

The safest approach is to use the effective counts recognized by the drive or controller, rather than relying only on the encoder label.

How Ball Screw Lead Affects Resolution

With the same motor and feedback resolution, a smaller lead produces a finer theoretical linear resolution.

A smaller lead provides finer resolution and greater mechanical advantage, while a larger lead produces higher linear speed at the same screw rotational speed. Lead selection therefore requires a balance among resolution, speed, motor torque and the permissible rotational speed of the ball screw.

For the relationship between lead, rotational speed and travel, see How to Calculate Ball Screw Lead.

Resolution Is Not the Same as Positioning Accuracy

Theoretical resolution only describes how finely the control system divides one revolution. It does not describe how closely the axis reaches the commanded physical position.

For example, an axis may have a theoretical resolution of 0.5 μm/count but use a C7 rolled ball screw whose accumulated travel error is much larger than 0.5 μm. The controller can command very small increments, but the mechanical system cannot necessarily reproduce every increment with the same accuracy.

Ball screw accuracy grades describe permissible travel deviation, not encoder resolution. Learn more in What Is Ball Screw Lead Accuracy? C3, C5 and C7 Explained.

What Determines Actual Axis Accuracy?

After calculating the theoretical resolution, the following mechanical and control factors must still be evaluated.

Ball Screw Lead Error

The actual nut travel may differ from the nominal lead. This deviation accumulates over the travel length and is controlled by the ball screw accuracy grade.

Backlash and Reversal Error

Axial clearance can create lost motion when the direction reverses. Preloading can reduce backlash, but excessive preload increases friction, heat and required motor torque.

Elastic Deformation

The screw shaft, ball nut, support bearings, nut housing and machine frame all deform under load. A high encoder resolution cannot compensate automatically for mechanical deflection that is not measured at the load.

Thermal Expansion

Heat generated by the ball nut, support bearings and motor can change the effective screw length. On a long axis, even a small temperature rise can produce a positioning change much larger than the calculated resolution.

Mounting and Alignment

Misalignment between the ball screw, support bearings and linear guides can increase running resistance and cause uneven deformation. Coupling error, bearing installation and the rigidity of the nut bracket also affect the final result.

Feedback Location

A motor encoder measures motor-shaft rotation. It does not directly measure the final table position. A separate linear encoder mounted on the moving axis can detect more of the mechanical error between the motor and table, although the overall control design becomes more complex.

A Practical Calculation Procedure

1. Confirm the ball screw lead. Use millimetres per screw revolution, not thread pitch unless pitch and lead are the same for that screw.
2. Identify the resolution value. Confirm whether the specification means pulses, encoder lines, quadrature counts, full steps or microsteps.
3. Check the transmission. Determine how many motor revolutions produce one ball screw revolution.
4. Calculate the theoretical increment. Use R = L ÷ (N × i).
5. Check speed and torque. A resolution-focused choice must still meet the required travel speed and axial thrust.
6. Estimate real positioning error. Include lead error, backlash, deformation, thermal growth and mounting conditions.
7. Verify the assembled axis. Use suitable measurement equipment to compare commanded travel with actual table movement.

Motor torque should be checked separately because increasing the transmission reduction or changing the ball screw lead also changes motor speed and load conditions. See Ball Screw Motor Torque Calculation: Formulas and Examples.

Final Selection Principle

Ball screw positioning resolution can be calculated from the screw lead, effective counts per revolution and transmission ratio. The result is useful for checking whether the motor and control system provide a sufficiently small command or feedback increment.

However, resolution is only one layer of axis performance. A machine with extremely fine encoder counts may still have poor positioning accuracy if the ball screw accuracy grade, backlash, rigidity, temperature control or installation quality is inadequate.

For a reliable design, calculate the theoretical resolution first, then confirm the required speed, motor torque, ball screw accuracy grade, preload, support arrangement and structural rigidity. The complete axis-not a single specification-determines the final positioning result.

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