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Servo Motor Sizing Calculator

Peak torque, RMS torque, inertia ratio and speed checks over a full motion cycle.

Check whether a servo motor suits your axis. Enter the load inertia and torques at the load shaft, the gear or belt ratio, and a trapezoidal motion profile with acceleration, constant speed, deceleration and dwell times.

  • Live results
  • Full step-by-step working
  • PASS / FAIL checks
  • Free PDF report

Inputs

From motor catalogue (with brake if fitted)

Results

Peak torque
3.331 N·m
PASS
RMS torque
1.934 N·m
PASS
Inertia ratio
3.33
PASS
Motor speed
3000 rpm
PASS
RMS power
607.7 W
Step-by-step calculation (11 steps)
  1. 1Motor speed
    Formulanm = nL·i, ω = 2π·nm / 60
    Solutionnm = 300 × 10, ω = 2π × 3000 / 60
    Resultnm = 3000 rpm, ω = 314.2 rad/s (≤ 5000 rpm)PASS
  2. 2Load inertia reflected to motor
    FormulaJL' = JL / i²
    SolutionJL' = 0.05 / 10²
    ResultJL' = 5×10⁻⁴ kg·m²
  3. 3Inertia ratio
    Formulak = JL' / Jm
    Solutionk = 5×10⁻⁴ / 1.5×10⁻⁴
    Resultk = 3.33 ≤ 10PASS
  4. 4Angular acceleration
    Formulaα = ω / ta
    Solutionα = 314.2 / 0.2
    Resultα = 1571 rad/s²
  5. 5Acceleration torque at motor
    FormulaTacc = Jm·α + JL'·α / η
    SolutionTacc = 1.5×10⁻⁴ × 1571 + 5×10⁻⁴ × 1571 / 0.9
    ResultTacc = 1.108 N·m
  6. 6Load and friction torque at motor
    FormulaTload = (TL + Tf) / (i·η)
    SolutionTload = (15 + 5) / (10 × 0.9)
    ResultTload = 2.222 N·m
  7. 7Torque in each segment
    FormulaT₁ = Tacc + Tload, T₂ = Tload, T₃ = Tload − Tdec
    SolutionTdec = (Jm + JL′·η)·ω/td = 0.9425 N·m
    ResultT₁ = 3.331, T₂ = 2.222, T₃ = 1.28 N·m
  8. 8Peak torque check
    FormulaSF·Tpeak ≤ Tmax
    Solution1.2 × 3.331 = 3.997
    Result3.997 N·m ≤ 7.16 N·mPASS
  9. 9RMS (effective) torque
    FormulaTrms = √[(T₁²·ta + T₂²·tc + T₃²·td) / tcycle]
    SolutionTrms = √[(3.331² × 0.2 + 2.222² × 1 + 1.28² × 0.2) / 2]
    ResultTrms = 1.934 N·m
  10. 10Continuous torque check
    FormulaSF·Trms ≤ Trated
    Solution1.2 × 1.934 = 2.321
    Result2.321 N·m ≤ 2.39 N·mPASS
  11. 11Power
    FormulaPpeak = Tpeak·ω, Prms = Trms·ω
    SolutionPpeak = 3.331 × 314.2, Prms = 1.934 × 314.2
    ResultPpeak = 1046 W, Prms = 607.7 W

Engineering takeaways

  • Selected motor meets speed, peak, continuous torque and inertia criteria with the chosen margin.
  • Inertia of couplings, pulleys, gearbox input and screws must be included in JL. Missing inertia is the most common sizing error.
  • Check regenerative energy during deceleration and size the brake resistor if the drive needs one.

The PDF report contains every input, formula, substitution and check shown above, in LTM calculation-sheet format.

Worked Example

The calculator reflects inertia and torque to the motor, finds the torque in each segment, and compares peak torque, RMS (continuous) torque, inertia ratio and speed with the motor catalogue data you enter.

The calculator above opens with this example. Change any input to run your own case.

Inputs

Load inertia at load shaft JL0.05 kg·m²
External load torque at load shaft15 N·m
Friction torque at load shaft5 N·m
Gear / belt ratio i (motor rev per load rev)10
Transmission efficiency η0.9
Load speed300 rpm
Acceleration time0.2 s
Constant-speed time1 s
Deceleration time0.2 s
Dwell time0.6 s
Motor rotor inertia Jm0.00015 kg·m²
Motor rated torque2.39 N·m
Motor peak torque7.16 N·m
Motor maximum speed5000 rpm
Torque safety margin1.2
Inertia ratio limit10 general automation

Results

Peak torque3.331 N·m PASS
RMS torque1.934 N·m PASS
Inertia ratio3.33 PASS
Motor speed3000 rpm PASS
RMS power607.7 W

Servo Motor Sizing Formulas Used

Reflected inertia and ratio

JL′=JLi2,k=JL′JmJ_L' = \frac{J_L}{i^2},\qquad k = \frac{J_L'}{J_m}
  • iratio (motor revolutions per load revolution)
  • Jmmotor rotor inertia

Acceleration torque

Tacc=Jm α+JL′ αη,α=ωtaT_{acc} = J_m\,\alpha + \frac{J_L'\,\alpha}{\eta},\qquad \alpha = \frac{\omega}{t_a}

Load torque at motor

Tload=TL+Tfi ηT_{load} = \frac{T_L + T_f}{i\,\eta}

RMS torque

Trms=T12ta+T22tc+T32tdta+tc+td+twT_{rms} = \sqrt{\frac{T_1^2 t_a + T_2^2 t_c + T_3^2 t_d}{t_a + t_c + t_d + t_w}}
  • T₁, T₂, T₃torque during acceleration, constant speed and deceleration

Selection criteria

SF⋅Trms≤Trated,SF⋅Tpeak≤Tmax,nm≤nmaxSF\cdot T_{rms} \le T_{rated},\quad SF\cdot T_{peak} \le T_{max},\quad n_m \le n_{max}

Basis: Rigid-body dynamics; trapezoidal motion profile; RMS torque method. Results are for preliminary design and should be confirmed against supplier data and the governing standard.

Servo Motor Sizing Calculator FAQ

What inertia ratio is acceptable for a servo motor?

Below 5:1 gives crisp, high-dynamic response. Up to 10:1 suits general automation. Belt drives and low-response axes can run 20:1 or more with careful tuning. Lower ratios settle faster and tolerate compliance better.

What is RMS torque and why does it matter?

RMS torque is the thermally equivalent continuous torque over a full cycle. The motor heats according to RMS torque, so it must stay below the rated torque. Peak torque must separately stay below the motor maximum.

How do I reduce the inertia ratio?

Increase the gear ratio: reflected inertia falls with the square of the ratio. Alternatively, choose a medium or high inertia motor, or reduce load inertia with lighter pulleys and couplings.

Where do I get the load inertia?

Take it from CAD mass properties, or calculate it: a ball screw axis has J = m·(l/2π)² for the moving mass plus the screw inertia. Include couplings, pulleys and gearbox input inertia, which are often forgotten.

Why add a safety margin?

Friction, misalignment and payload changes are rarely known exactly. A 20% margin on torque leaves room for these and for future cycle time improvements.

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