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. 

Let's review some definitions 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.

Acceleration and deceleration ramps of a motion profile

As opposed to load torque, 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.  Therefore, 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

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.

Calculation Example

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

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

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

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

4.  Acceleration Torque

Fill in the blanks for the variables.  To determine J0, you first need to tentatively select a motor based on load inertia (as mentioned previously), then you can find the rotor inertia J0 for that motor. 

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

5.  Total Required Torque

Add up the load torque and acceleration torque for total required torque.  

Total torque equation

For stepper motors, it's important not to use the "maximum holding torque" specification to select a motor since that is measured at close to zero speed.  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 correctly, can potentially extend life or improve performance of the motor.

Stepper motors: speed torque curve

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

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]

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 e-mail notifications of new posts.

Learn more about motor sizing calculations

Is There An Easier Way to Size Motors?

For the most common applications, motor sizing calculators are offered from motor manufacturers where you can simply plug in values to quickly calculate the results.  A deeper understanding into motor sizing equations is still necessary for more complex applications.  

The thing to remember about motor sizing is that the result is only as good as the data.  Make sure the values used for the calculation is as accurate as possible.  The more guessing you do, the larger safety factor you'll need to use at the end.  Just like in the real world, there will be some unknowns.

Motor sizing for specific applications

Try Our Motor Sizing Calculators


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.