Engineering Notes by Oriental Motor

Motor Sizing Basics Part 3: Acceleration Torque (and RMS Torque)

[fa icon="calendar"] Feb 27, 2020 9:05:40 PM / by Johann Tang

Johann Tang

Now that we understand the calculations behind load torque and load inertia, we're two steps closer to motor selection.  You might be wondering why I separated load torque and acceleration torque calculations.  That's because in order to calculate for acceleration torque, load inertia and speed must be calculated first. 

TIP: Let's Review First

In Motor Sizing Basics Part 1 - Load Torque, load torque is defined as the amount of torque constantly required for application and includes friction load and gravitational load. 

In Motor Sizing Basics Part 2 - Load Inertia, load inertia is defined as the resistance of any physical object to any change in its speed from the perspective of the rotational axis.

Here we show a typical motion profile with an acceleration, constant speed, and deceleration region.

  • Start from zero speed
  • Accelerate with t1
  • Constant speed at Nm for a duration of t0-t1-t1
  • Decelerate with t1
  • Stop at zero speed


Acceleration and deceleration ramps of a motion profile


Acceleration/Deceleration Torque

As opposed to load torque (which is constant), acceleration torque is the required torque to accelerate (or decelerate) an inertial load up to its target speed.  It is only present when accelerating (or decelerating) an inertial load. 

Total Required Torque

The total required torque is the sum of the load torque and acceleration torque as shown below (with a safety factor to cover for what we don't know).


Required total torque equation


Mathematically, acceleration torque is made up of load inertia and acceleration rate as shown below.  This is the most common equation used to calculate acceleration torque for all types of motors.  


acceleration torque equation


Stepper motors and servo motors can use a different formula since they deal with pulse speed (Hz).  There are two equations available for 2 types of motion profiles: with or without acceleration/deceleration.


Motion profiles: with or without acceleration/decelerlation


TIP: Why use acceleration/deceleration at all? 

That's because even though immediately starting at the target speed might seem easier, but it results in a lot of acceleration torque, and thus requires a bigger motor.  A bigger motor also equals higher cost and bigger footprint, which are not the most desirable for machine designs.


Example: Calculation of Load Torque and Load Inertia

In the following example, let's try to calculate load torque, load inertia, and acceleration torque using what we've learned so far.  For me personally, I calculate load inertia first, then load torque, then speed, then acceleration torque.  Information below describes the motor mechanism and given parameters.


Ball screw motor sizing calculation example


Step 1:  Load Inertia

Calculate the load inertia for the screw, then the table and load separately, then add them up.  The load inertia can be used for a tentative motor selection, which I will explain in a bit.


Load inertia calculation for ball screw


Step 2:  Load Torque

Use the load torque equation for screws and fill in all the blanks for the variables.  Make sure to use the right equation for the specific application.


Load torque calculation for ball screw


Step 3:  Speed (RPM)

The required speed is calculated with the following equation.  Use the pitch/lead of the screw PB in order to convert linear speed to RPM.


Speed conversion from linear to rotary speed


Here are some formulas for stepper motors when dealing in pulse speed (Hz).


Acceleration torque formulas for stepper motors

Step 4:  Acceleration Torque

Here's a common formula for acceleration torque for all motors.


Common acceleration torque formula for all motors


We just need to fill in the blanks for the variables.  To calculate for acceleration torque Ta, tentatively select a motor based on load inertia (as mentioned previously), then plug the rotor inertia value J0 for that motor into the acceleration torque equation.

TIP: How To Tentatively Select a Motor Based on Load Inertia

For AC constant speed motors, AC speed control motors, and brushless speed control motors, you will need to look at the permissible load inertia values.  For stepper motors or servo motors, you will need to know the allowable inertia ratio each type of motor can handle.   

For stepper motors, the general guideline is to keep the inertia ratio (load inertia or reflected load inertia divided by rotor inertia) under 10:1, and 5:1 for faster motion profiles or smaller frame sizes than NEMA 17. 

For closed-loop stepper motors, up to a 30:1 inertia ratio is recommended. 

For auto-tuned servo motors, the inertia ratio increases to 50:1.  For manual-tuned servo motors, it can increase to 100:1.  

After you make a tentative motor selection based on load inertia, find the motor rotor inertia in the specifications, then plug the value for J0 in order to complete the acceleration torque calculation.


Acceleration torque equation


Just for kicks, here is another equation for acceleration torque using Hz units.  "i" is for gear ratio.


Acceleration torque calculation using Hz


Step 5:  Total Required Torque and Safety Factor

Add up the load torque and acceleration torque for total required torque.  We will need a stepper motor that can output 0.85 Nm torque at the very least.  


Total torque equation


However, this is without a safety factor.  If you use a safety factor of 2, then we will need a stepper motor that can output 1.7 Nm torque at about 1200 RPM; depending on the acceleration/deceleration rates.  Safety factors are determined based on the accuracy of the variables.

TIP: Don't Use Maximum Holding Torque to Size Stepper Motors

For stepper motors, it's important not to use the "maximum holding torque" specification to select a motor since that is measured at zero speed and at full current. 

Since the torque produced by a stepper motor decreases as speed increases, you will need to look at the speed-torque curve to determine if the stepper motor will work at that speed or not.  Typically, selecting a motor based on the total required torque and max required speed is a safe bet even though the motor may not need that torque at its max speed.  A little bit of oversizing, when done properly, can extend life or improve performance of the motor.

Stepper motors: speed torque curve


Step 6:  RMS Torque

For servo motors, there is another calculation that must be done, which is RMS torque.  Root mean squared torque, or RMS torque, refers to an average value of torque that considers all the varying torque values used during operation as well as the time duration each torque value is needed.  RMS torque is used to determine if the motor is properly sized to avoid thermal overload.


Servo motors: required torque vs RMS torque


For servo motors, the required torque must be below the motor's peak torque, and the RMS torque must be below the motor's rated torque.  Since peak torque requires a high level of motor current, it cannot be sustained continuously without overheating the motor.

Let's now look at the equation for RMS torque and visualize the variables in a motion profile pattern.


Effective Load (RMS) Torque Formula

TIP: More about RMS torque
For more information about RMS torque, here's a good article from Linear Motion Tips (Design World), Why RMS torque is important for motor sizing.



For this application, we require a motor with high positioning (stop) accuracy, which would be either stepper motors or servo motors. 

For a stepper motor, we would need to meet or exceed the following requirements.

Load Inertia = 5.56 × 10−4 [kg·m2]

Total Torque = 0.85 [N·m]*

Maximum Speed = 1200[r/min]

For a servo motor, we would need to meet or exceed the following requirements.

Load Inertia = 5.56 × 10−4 [kg·m2]

Total Torque = 0.85 [N·m]*

RMS Torque = 0.24[N·m]

Maximum Speed = 1200[r/min]

*Calculated torque does not include safety factor.


With a required torque, load inertia, and a required speed, we have sufficient information for motor selection.  However, there is another important criteria to consider in order to maintain long term life.  HINT: it has something to do with bearings.  Please subscribe to receive new posts.


Technical Reference - Sizing formulas & Selection Examples

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Next: Motor Sizing Basics Part 4 - Radial and Axial Loads


Topics: AC Motors, Stepper Motors, Alphastep Hybrid Control, Linear & Rotary Actuators, Speed Control, Motor Sizing, BLDC Motors, Servo Motors

Johann Tang

Written by Johann Tang

Johann Tang is a Product Specialist at Oriental Motor USA Corp. with over 15 years of knowledge and experience supporting applications of various types of electric motors, gearheads, actuators, drivers, and controllers.