How to Size a Motor

Proper sizing is a crucial aspect of motor selection. If a motor is undersized, it will not be able to control the load, leading to overshoot and ringing. If the motor is oversized, it may control the load but it will also be larger and heavier, as well as more expensive in terms of price and cost of operations. Everybody agrees on the importance of proper sizing and yet all too frequently, vendors simply get a call asking for a motor of a certain horsepower. The engineer may be just buying a motor the same size as that of a previous platform. They may have added a hefty safety margin to compensate for changes. They may have used a 10:1 or 5:1 ratio of load and inertia to motor inertia…or some mixture of the above. 

At best, these approaches lead to oversized motors that squander money. At worst, they lead to sloppy systems that fail to perform to expectations, leading to substandard product, increased downtime, and reduced productivity. The goal is to specify a motor that provides the speed, acceleration, and torque required to position the load at the designated location and the desired time. It may include a safety margin designed to compensate for motor-two-motor variation or expected changes in the running condition of the machine. The safety margin should be added on top of an informed calculation, however.

Accurate measure sizing is a critical process that requires knowledge about the details of the mechanical system, the operating parameters, and the circumstances under which the equipment will be used. Although numerous better sizing applications exist, it can be hopeful to run through the process.

Start with Inertia 
Inertia – the tendency of an object to resist changes in acceleration – is one of the primary challenges in motion control. “Inertia is not our friend in the motion control world, especially on assembly machines where the motion cycles are short but very high speed.” Paul Bobel, Senior Product Engineer for Inverters, Mitsubishi Electric Automation (Vernon Hills, Illinois). The motor needs to be able to apply sufficient force (in a linear system) or torque (in a rotational system) to change the acceleration of the load, and do so in a controlled fashion.

To effectively size the motor, we need to calculate the load inertia (J_L). For purposes of this article, we will focus on rotational motion although analogous calculations can be made for linear motion. A rotating mass has a moment of inertia, which describes its tendency to resist the application of torque. The simplest closed-form expression of moment of inertia J as

J = mr^2

which describes the moment of inertia for a point mass m at some distance r from the rotation axis. The simple expression is used to build up moment of inertia for a variety of complex shapes such as cylinders, hollow cylinders, disks on shafts, spheres, blocks, and more. 

Load inertiaJ_L should more properly be termed reflected inertia (J_R), as in the inertia reflected back to the motor shaft from the load and all components in between. It should include any additional mechanical elements in the system that the motor will be responsible for moving, such as lead screws, pulleys, belts, couplings, etc. The higher the performance demands of the system, the more detailed this analysis should be. The moment of inertia of a lead screw, for example, is different than the moment of inertia of a simple cylinder it high enough speeds this distinction will affect the performance of the machine.

Determining appropriate analytical expression and calculating mass and position for these complex mechanical systems can be quite challenging. In addition, these calculations need to be based on weight (factoring in acceleration due to gravity) rather than simply mass. In addition, the motor sizing process also needs total system inertia, which includes both J_L and motor inertia J_M (basically, the inertia of the rotor). 

It can easily become a daunting process. “Normally, people include the actual load, the gearbox, and the motor and leave belts, pulleys, and other mechanical things out of the equation,” says Bobel. “They just move to the next major size or use the same frame size but one that produces more torque. This is where the whole 10% oversize approach comes from.”

The ratio of load inertia to motor inertia (essentially, rotor inertia) gives a measure of how effectively the motor can control the load. A high inertia ratio indicates the system that will have difficulty controlling the load. A low inertia ratio (e.g. 4:1 or 1:1) indicates that the motor will do a very effective job of controlling the load but it also reveals that the motor may be oversized for the system, representing more cost, size, and weight than necessary. That said, there are no absolutes.

“When you start getting into very high performance machines, if the mechanics are good, those rules of thumb go out the door,” says Bobel. “We’ve done inertia mismatches in servo and motion control that are much higher – 40:1, 60:1. It depends on the machine design and how well you can tune the system.” The latest crop of auto tuning drives can very effectively compensate for machine resonances and vibration, supporting accurate performance even at very high speeds.

Compliance (torsional flex) is an important factor to take into account. Even a motor sized for a 1:1 inertia ratio will have difficulty controlling a load if there is too much looseness and compliance in the system. Depending on the situation, the mechanical system may need to be tightened up or the operating parameters of the machine may need to be relaxed. 

Gearboxes provide an important tool for managing inertia. A gearbox reduces inertia by the square of the gear ratio. The trade-off is that gearboxes also cut motor speed. That can be a problem with stepper motors, which typically only run at several hundred RPM. Most servo motors operate at speeds between 2000 RPM and 6000 RPM, which enables them to operate at a useful speed even when used with a high-reduction-ratio gearbox.

Add Application Requirements
Once the system inertia has been obtained, it’s time to consider the operating parameters of the application and use them to determine torque. Start by defining the motion profile for the load. Motion profiles vary from application to application. The most basic form is that of a trapezoid: a period of acceleration followed by a period of constant velocity, followed by a period of deceleration (see figure). Acceleration and deceleration (which is just negative acceleration) can be determined using

A =V/t

The motion profile shown in the figure displays symmetric acceleration and deceleration. As a result, there acceleration can be expressed as:

a= Vmax/t₁
-a = V max/t₁ 

Torque Requirements
Now that we’ve calculated the load inertia and the desired acceleration and deceleration, we can determine the amount of torque required to position the load. Total torque T_T is the sum of acceleration torque (T_acc) and a quantity called load torque (T_L). Load torque is just the sum of mechanical losses in the system, typically friction and gravity. As with inertia, it varies depending upon shape, mass, and configuration.

Friction is a factor easily forgotten in the motor sizing process. Any two surfaces sliding together will counter the applied torque. The standard expression for frictional force F is given by

F = µN

where µ equals coefficient of friction for the sliding surface of any moving element and N equals the normal force applied to that element. This expression can be used to estimate frictional force or a measuring instrument such as a torque wrench can be used.

We can define acceleration torque T_acc as

T_acc=J_T (a)+T_L

where J_T  is total inertia (load plus motor). 

The deceleration torque required to stop the load is similarly given by

T_dec=J_T (-a)+T_L

We also need to account for the run torque T_run that maintained constant velocity of the load throughout the run phase and a brief span of idle time at the end of the move. We combine these quantities together to determine the RMS torque (T_RMS) needed from the motor. That expression is given by

This is the torque required from the motor by the application.

Speed-torque Curves
Just finding a motor with the appropriate torque isn’t enough.  It needs to be rated for use at the required speeds. Manufacturers provide speed-torque curves for motors that describe their performance across the operating speed range. These data plots provide an easy reference to determine whether the motor will work for the conditions of the application.

Infrastructure and Environment
Motors don’t operate in a vacuum. An underlying consideration throughout the sizing process should be the voltage and frequency characteristics of your power source. Also consider environmental conditions. Will the motor be exposed to extreme temperatures? In extreme cold, lubricants become more viscous. A motor that would normally operate just fine in a system may stall as a result of resistant lubricant when the temperature plummets, so size accordingly.

Moisture and contamination can likewise be problems. Consider whether you need an IP-rated device – and be sure you choose the level that correctly represents operating conditions.

Physical Size
Although increasing motor power can improve its ability to effectively control the load, there are real practical limitations. In some cases, there simply isn’t room for a motor with the desired frame size. If the primary issue is diameter, some motor manufactures increase output power for a given frame size by stacking more magnet laminations in the design. The motor generates more power at the expense of being longer but not necessarily wider.

Even this approach has limitations, however. Past a certain point, the system designer needs to either reduce speed requirements or redesign the mechanical system in order to fit a bigger motor.

Stepper Motor Nuances
Sizing for stepper motors involve some special considerations. Stepper motors are designed with very high pole counts – on the order of 50 or more. As a result, they can be commanded to advance in discrete steps rather than in continuous motion. This enables them to be operated open-loop in many applications. In this mode, they can be inexpensive yet effective. The problem arises when the motor is placed in an over torque mode and stalls. Because there is no feedback device monitoring the motor shaft, this behavior could go undetected. 

As a result, oversizing became the rule. “Historically, people would do a times two factor when sizing stepper motors,” says Clark Hummel, Flex Center Manager, Schneider Electric Motion USA (Marlborough, Connecticut). If they thought they needed X amount of torque to do the work they needed to do, they would get a motor capable of twice that. It’s a big safety margin.” It was particularly appealing given the low price point of stepper motors – around an order of magnitude less than for a stepper motor.

Today, for the right application, running a stepper motor closed-loop allowance the benefits of both worlds. The money saved on right sizing can be used for the encoder. “Sizing wise you’re more back in the servo world where there are going to be variations in motors,” he says. “You want to give yourself a 15 to 20% safety margin to compensate for variation of motors coming out of the factory or variations in process but it makes the sizing much closer to what you do with a servo motor.”

Getting Past the Basics
Once the basic size has been determined, there are several techniques that seasoned motion engineers apply to reduce the size and cost of the motor. The first is adding a gear reducer, as discussed above. The type of gearbox used can also reduce motor requirements. A worm gearbox is only about 30% efficient, for example, while a planetary gearbox is around 85% efficient. That substitution can potentially reduce the load on the motor. The tradeoff is higher cost, however, which may reduce some of the cost-benefit of going to a smaller motor.

Stepper motors offer inherently more torque at low speeds than do servo motors. For an application operating at low speed, a stepper motor may provide a more economical solution. “If you don’t need the speed, you can get much higher torque in the same envelope at lower speeds so there are a lot of advantages for the right application in using a stepper motor over a servo motor,” says Hummel. “For a given frame size, you might not need to add a gearbox and that is a huge cost difference.”

Motors and motor drives are specified for two modes: continuous-duty mode and peak, or overload mode. The motor drive combination can operate at peak torque/current for brief intervals without harming the motor windings for the drive electronics. A common mistake is to choose a motor with a continuous-duty torque equal to the maximum torque requirement of the application (typically seen during extreme accelerations/decelerations). Motion control applications frequently consist of brief, rapid moves. To choose a motor rated to generate this torque continuously means essentially paying for more motor than necessary.

If the moves take place over time intervals that fall within specifications, then the motor can perform them in overload mode. “People don’t use the overload capability of the motor to accomplish these high accelerations/decelerations,” Bobel says. “They might end up sizing their motor for 100% current when that 100% current peak only happens for milliseconds at a time. Instead, they could be using the overload capability or the extended capacity of the drive and motor to get through those peaks. The approach enables the use of a smaller motor overall, reducing not just physical size but cost and energy consumption of both motor and drive, alike.”

Although this is a valid design technique, it is important to pay close attention to duty cycle. The intervals of peak current draw need to be sufficiently short to fall under specifications but also take place at low enough frequency to enable the motor windings and electronics to cool off. It’s also important to consider duty cycle from the standpoint of the gearing in order to prevent premature aware and failure of the gearbox.

As with all things in engineering, motor sizing involves balancing conflicting demands. “Everything has a trade-off,” says Ed Tullar, Sales and Marketing Manager for Groschopp (Sioux Ctr., Iowa).” You can trade efficiency and then you lose your back torque, you pick up your back torque and then you lose power. Everything is very stressful on the system in a motor powertrain.” The best approach is to determine as much as possible about the application. Use sizing tools available to you to simplify calculations. Share the details with your vendors. They can help you narrow down potential candidates and may be able to direct you to a more economical system that will still meet your needs.