Mechatronics & Robotics

Servo Motor Sizing Calculator — RMS & Peak Torque, Duty Cycle

Estimate servo RMS torque, required peak torque, thermal utilization and inertia ratio from load inertia, gearing, motion timing and candidate motor ratings.

Unit-aware inputs Deterministic calculation Engineering interpretation
Calculation workspace

Enter the known values and review the calculated result

Deterministic calculation
01
Parameters

Input parameters

Use consistent values and select the intended engineering units.

Transmission and inertia

Motor-shaft motion profile

External and holding torques

Selected motor verification — optional

02
Output

Results

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Calculation tools Save, export and manage your results

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Engineering reference

Method, application and limitations

Review the calculation method, intended application and engineering assumptions before using the result in a design decision.

01
Method

Formula and calculation method

Servo motor sizing equations

This calculator evaluates a repeated servo motion cycle from motor-side speed, load and motor inertia, gear ratio, transmission efficiency, motion times, external torque and candidate motor torque ratings. Its primary results are RMS torque, required peak torque and thermal utilization.

Angular acceleration

The acceleration magnitude is calculated from the entered maximum motor speed and acceleration time:

α = ωmax / tacc

  • α — required angular acceleration [rad/s2]
  • ωmax — maximum motor-shaft angular speed [rad/s]
  • tacc — acceleration time [s]

Reflected inertia and inertia ratio

Load inertia is reflected through the gear ratio by the square of the ratio:

Jload,ref = Jload / i2

The total inertia accelerated by the motor model is:

Jref = Jmotor + Jload / i2

The reported load-to-motor inertia ratio is:

Inertia ratio = (Jload / i2) / Jmotor

  • Jload — entered load inertia
  • Jmotor — motor rotor plus input-side inertia
  • i — motor/load speed ratio

External torque and phase torque

The JavaScript applies transmission efficiency to the external torque as:

Text,eff = Text / η

The phase torques are then calculated as:

Tacc = Jref · α + Text,eff

Tdec = -Jref · α + Text,eff

Tconst = Text,eff

Tidle = 0

The required peak torque is the largest absolute phase torque:

Tpeak,req = max(|Tacc|, |Tdec|, |Tconst|)

RMS torque over the servo duty cycle

Total cycle time is:

tcycle = tacc + tconst + tdec + tidle

Cycle-equivalent RMS torque is:

TRMS = √[(Tacc2tacc + Tconst2tconst + Tdec2tdec + Tidle2tidle) / tcycle]

The RMS value represents the equivalent torque level used by this calculator for continuous thermal screening of the selected motor.

Motor utilization

Thermal utilization = (TRMS / Tnom) · 100%

Peak utilization = (Tpeak,req / Tpeak,motor) · 100%

  • Tnom — available continuous motor torque
  • Tpeak,motor — available peak motor torque

Motion duty cycle

The reported duty cycle is the fraction of the complete cycle occupied by acceleration, constant-speed motion and deceleration:

DC = [(tacc + tconst + tdec) / tcycle] · 100%

This is a motion-cycle duty factor, not an electrical PWM duty cycle.

Mechanical power indicators

The calculator also reports secondary mechanical-power indicators:

Ppeak = max(|Taccωmax|, |Tconstωmax|, |Tdecωmax|)

ωRMS = √{[ωmax2tacc/3 + ωmax2tconst + ωmax2tdec/3] / tcycle}

PRMS = TRMS · ωRMS

These values are mechanical indicators from the implemented motion model. They are not electrical input-power or drive-loss calculations.

02
Application

When to use this calculator

When to use this servo motor sizing calculator

Use this calculator for preliminary servo motor sizing when the load inertia, motor inertia, gearing and repeated motion profile are already known. It is particularly useful for evaluating whether a candidate motor has sufficient continuous and peak torque capability for an acceleration-cruise-deceleration cycle.

Typical applications include:

  • preliminary servo sizing for indexing and positioning axes;
  • checking RMS torque over a repetitive machine cycle;
  • calculating required peak torque during acceleration and deceleration;
  • comparing RMS torque with the motor continuous torque rating;
  • comparing required peak torque with the motor peak torque rating;
  • evaluating reflected load inertia through a gearbox or reducer;
  • screening the load-to-motor inertia ratio;
  • studying the effect of acceleration time, dwell time, gear ratio or external torque on servo loading.

Inputs required by the current solver

The supplied JavaScript requires positive values for load inertia, motor inertia, gear ratio, acceleration time, deceleration time, maximum motor speed, continuous motor torque and peak motor torque.

Constant-speed time and idle time may be zero. External torque defaults to zero when it is not finite. Transmission efficiency defaults to 1 when the supplied value is missing or outside the implemented valid range.

What this calculator does not determine

This is not a complete manufacturer-specific servo selection tool. It does not derive load inertia from mechanism geometry, select a catalog motor, read a motor torque-speed curve, verify drive current or voltage, calculate regenerative energy, model mechanical resonance, or verify control-loop tuning.

The current solver also does not use the provided motor maximum-speed verification field, so maximum permissible motor speed must be checked separately against the motor and drive data.

03
Decision support

How to interpret the result

How to interpret the servo sizing results

RMS torque

TRMS is the primary continuous-duty result. Compare it with the continuous torque available from the candidate motor under the actual operating conditions. The calculator expresses this comparison as thermal utilization.

  • Thermal utilization = 100% means TRMS equals the entered continuous torque.
  • Thermal utilization above 85% triggers the calculator's warning state.
  • TRMS greater than the entered continuous torque triggers an invalid result.

Required peak torque

Tpeak,req is the largest absolute torque calculated for acceleration, deceleration or constant-speed operation.

  • Peak utilization = 100% means required peak torque equals the entered motor peak torque.
  • Peak utilization above 90% triggers a warning.
  • Required peak torque greater than the entered motor peak torque triggers an invalid result.

Inertia ratio

The inertia ratio compares reflected load inertia with motor inertia. The current decision logic uses:

  • ratio ≤ 10 — no inertia-ratio warning from this criterion;
  • ratio > 10 — inertia mismatch warning;
  • ratio > 30 — inertia-critical limiting factor.

These thresholds should be treated as screening criteria only. The acceptable ratio for a real servo axis must be confirmed from the selected motor and drive documentation and from the required dynamic response.

Limiting factor

The solver can report the following limiting-factor classifications:

  • inertia-critical — inertia ratio is greater than 30;
  • inertia-mismatch — inertia ratio is greater than 10;
  • external-load-driven — the efficiency-adjusted external torque exceeds the modeled inertial acceleration torque;
  • none — none of the above limiting-factor tests is active.

Safe, warning and invalid states

A safe interpretation means the entered peak and continuous motor torque limits are not exceeded and none of the implemented near-limit or inertia warnings is active.

A warning indicates limited margin or elevated inertia ratio. It does not mean the axis has failed, but it indicates that motor selection, motion profile or gearing deserves review.

The current JavaScript treats torque demand above the entered peak or continuous motor rating as invalid. In that state the primary numerical outputs are returned as null rather than being retained as an over-limit sizing result.

Passing these checks does not prove that the complete servo system is suitable. Motor speed capability, torque-speed characteristics, drive sizing, regenerative energy, mechanical stiffness, resonance, thermal environment and control performance remain separate checks.

04
Worked case

Calculation example

Servo motor sizing example

Consider a repetitive servo axis with a gearbox. The objective is to calculate RMS torque and required peak torque and then compare them with the ratings of a candidate servo motor.

Input data

  • Load inertia Jload = 0.040 kg·m2
  • Motor inertia Jmotor = 0.0015 kg·m2
  • Gear ratio i = 4.0
  • Transmission efficiency η = 0.90
  • Acceleration time tacc = 0.20 s
  • Constant-speed time tconst = 0.80 s
  • Deceleration time tdec = 0.20 s
  • Idle time tidle = 0.40 s
  • Maximum motor speed ωmax = 120 rad/s
  • External torque Text = 1.20 N·m
  • Motor continuous torque Tnom = 1.90 N·m
  • Motor peak torque Tpeak,motor = 4.00 N·m

All example values are already expressed in SI units, so no additional unit conversion is required.

1. Angular acceleration

α = 120 / 0.20 = 600 rad/s2

2. Reflected inertia

Jload,ref = 0.040 / 42 = 0.0025 kg·m2

Jref = 0.0015 + 0.0025 = 0.0040 kg·m2

Inertia ratio = 0.0025 / 0.0015 = 1.67

3. Effective external torque

Text,eff = 1.20 / 0.90 = 1.333 N·m

4. Phase torques

Tacc = 0.0040 · 600 + 1.333 = 3.733 N·m

Tdec = -0.0040 · 600 + 1.333 = -1.067 N·m

Tconst = 1.333 N·m

Tidle = 0 N·m

5. Required peak torque

Tpeak,req = max(3.733, 1.067, 1.333) = 3.733 N·m

Peak utilization = 3.733 / 4.00 · 100% = 93.3%

6. RMS torque

tcycle = 0.20 + 0.80 + 0.20 + 0.40 = 1.60 s

TRMS = √[(3.7332·0.20 + 1.3332·0.80 + (-1.067)2·0.20 + 02·0.40) / 1.60]

TRMS = 1.665 N·m

Thermal utilization = 1.665 / 1.90 · 100% = 87.6%

7. Duty cycle

DC = (0.20 + 0.80 + 0.20) / 1.60 · 100% = 75.0%

Interpretation

The candidate motor remains below both entered torque limits: 3.733 N·m is below the 4.00 N·m peak rating and 1.665 N·m is below the 1.90 N·m continuous rating.

However, the calculator returns a warning because peak utilization exceeds 90% and thermal utilization exceeds 85%. The result therefore indicates a relatively small torque margin even though neither hard torque limit is exceeded.

This does not complete servo selection. The selected motor must still be checked at the actual operating speed using manufacturer data, together with drive capability, regeneration, mechanical dynamics and application-specific margins.

05
Model boundaries

Assumptions and limitations

Assumptions and limitations

  • The motion model consists of acceleration, constant-speed operation, deceleration and idle time.
  • Angular acceleration is calculated only from ωmax/tacc. The same acceleration magnitude is used for the deceleration torque calculation. Therefore tdec changes RMS weighting and cycle duration but does not independently change the modeled deceleration magnitude.
  • Load inertia is supplied by the user. The calculator does not derive inertia from mechanism mass or geometry.
  • Load inertia is reflected through the gear ratio as Jload/i2.
  • The input HTML labels the external torque as load-shaft torque, but the supplied JavaScript applies Text/η without dividing torque by the gear ratio. The implemented equation should therefore not be interpreted as a complete physical load-shaft-to-motor torque transformation.
  • Transmission efficiency modifies the external torque term only. It does not modify the modeled inertial acceleration torque.
  • Invalid, zero or greater-than-one transmission efficiency is replaced by η = 1 by the current solver.
  • Idle torque is fixed at zero in the calculation. The holding-torque field present in the supplied input markup is not consumed by the supplied JavaScript.
  • The motor-shaft-travel alternative input is not used by the supplied JavaScript. A positive direct maximum-speed value is required by the current solver.
  • The separate candidate-motor maximum-speed field is not used for a speed-limit check.
  • Motor inertia, continuous motor torque and peak motor torque are required by the current JavaScript even though some of these fields are presented as optional in the supplied input markup.
  • Peak mechanical power is calculated using ωmax with each phase torque. It is a simplified indicator rather than a time-resolved power profile.
  • The model does not include gearbox inertia, backlash, compliance, resonance, variable friction, changing process force or detailed gravity loading unless their effect has already been represented in the entered quantities.
  • Thermal utilization is based only on RMS torque divided by entered continuous torque. Motor thermal time constants, ambient-temperature derating, cooling conditions and drive thermal limits are not modeled.
  • The calculator does not evaluate the manufacturer's torque-speed envelope, regenerative braking capability, DC-bus limits, amplifier current limits or electrical power demand.
  • Final servo selection should be verified using the motor and drive manufacturer's data for the actual speed, duty, installation conditions and required performance.
06
Questions

Frequently asked questions

What does this servo motor sizing calculator calculate?
It calculates required RMS torque, required peak torque and thermal utilization for a repeated motion cycle. It also reports reflected load-to-motor inertia ratio, required angular acceleration, cycle time, peak utilization, motion duty cycle and mechanical power indicators.
What is RMS torque in servo motor sizing?
RMS torque is the root-mean-square torque over the complete motion cycle. In this calculator, acceleration, constant-speed, deceleration and zero-torque idle periods are weighted by their duration. The result is compared with the entered continuous motor torque as a preliminary thermal-duty check.
What is the difference between RMS torque and peak torque?
Peak torque is the highest absolute torque required in any modeled motion phase and is checked against the motor peak rating. RMS torque represents the cycle-equivalent torque used for continuous thermal screening. A short, demanding acceleration can produce high peak torque while a longer dwell period keeps RMS torque relatively low.
How does gear ratio affect the servo inertia ratio?
The implemented model reflects load inertia to the motor by dividing the entered load inertia by the square of the gear ratio. The reported inertia ratio is therefore (Jload / i²) / Jmotor. Increasing the reduction ratio can substantially reduce the reflected load inertia seen by the motor.
What does servo motor duty cycle mean in this calculator?
Duty cycle is the percentage of total cycle time occupied by acceleration, constant-speed motion and deceleration. Idle time is included in total cycle time. This definition describes the motion cycle and should not be confused with PWM duty cycle or another electrical switching quantity.
Does this calculator calculate load inertia from mass and geometry?
No. Load inertia must already be known and entered as an input. The calculator reflects that inertia through the entered gear ratio and combines it with motor inertia. Use a separate mechanism or inertia calculation when load inertia must first be derived from masses, dimensions, pulleys, screws, gears or other components.
How is acceleration torque calculated?
The solver first calculates angular acceleration as maximum motor speed divided by acceleration time. It then multiplies that acceleration by the combined motor-side inertia and adds the efficiency-adjusted external torque. The implemented equation is Tacc = Jref · α + Text,eff.
What does an inertia ratio above 10 mean?
The calculator uses an inertia ratio above 10 as an internal warning threshold and a ratio above 30 as an inertia-critical condition. These are screening thresholds in the calculator, not universal servo limits. Allowable inertia ratio depends on the motor, drive, tuning method, mechanical stiffness and manufacturer requirements.
Does the calculator verify maximum servo motor speed?
No. Maximum motor speed is used as an input for the acceleration and power calculations, but the supplied solver does not compare it with the separate candidate-motor maximum-speed field. Speed capability must therefore be verified independently using manufacturer data.
Is holding torque during dwell included in the RMS torque?
Not in the supplied JavaScript calculation. The implemented model sets idle torque to zero, so dwell time lowers RMS torque without adding holding torque. Applications that require significant standstill torque need an additional thermal check or a revised duty-cycle model.
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Engineering notice. Verify input data, assumptions, applicable standards and required safety factors before using results for design release or safety-critical decisions.