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Motor Selection Basics: Inertia and Power and Torque Requirements

February 16, 2023

A variety of motor manufacturers provide online tools and calculators to assist users in selecting an ideal miniature motor, including our own MotionCompass™, a web-based application designed to provide mini motor solutions based on unique motion requirements for various applications. However, these tools shouldn’t replace having a fundamental understanding of the factors that go into the motor selection process! The reason is simple: choosing a miniature motor involves having a comprehensive knowledge of the application’s specific parameters, an awareness of the factors that the application prioritizes, and a technical understanding of the motion solution options that are best suited to meet these requirements.

Let’s explore two of the key motor selection factors, inertia matching and torque/power requirements, which will help illustrate the importance of understanding these concepts while simultaneously allowing users to appreciate and better utilize online tools to build on their initial calculations.

What is Inertia Matching?

Rotational inertia is a property of a body or system that tells us how much something is going to resist being angularly accelerated, or being sped up or slowed down in its rotation. If a system has a large rotational inertia, it is going to be difficult to get it to accelerate; if the system has smaller rotational inertia, then it should be relatively easy to get it to accelerate.

Inertia matching refers to matching of load inertia & motor inertia. A low load to motor inertia ratio is particularly desired in applications that require either high acceleration and deceleration or extremely precise positioning to achieve desired performance. Application examples such as high speed, high precision pick & place assembly and motors for textile yarn guides will require the motor to be run in closed loop feedback system. In theory, an inertia ratio close to 1 is desired. However, in real-world applications, this may not be achievable (nor practical). In fact, striving for an inertia ratio close to 1 can result in oversized components, higher system costs, and increased energy consumption.

The adverse effects of Inertia mismatch, or a high load to motor inertia, can become worse by “lack of stiffness” (load compliance) in the system that leads to increased response times, lower system bandwidth & resonance. This means that in demanding applications, it is advisable to start with the possibility of a direct drive system, keep number of power transmission elements to minimum possible, and design stiff machine/couplings. To counter the effect of load compliance and high load to motor inertia, you can use the following methods:

Gear Reduction. Employing a reducer will reduce the load inertia by a power of 2 over the gear ratio
Rigid Coupling/Rigid System/Rigid Machine. Using these will help keep load compliance at the bare minimum.
Servo Tuning. The problem of inertia mismatch can be mitigated by utilizing modern control systems to a huge extent, due to their fast processors that use advanced control algorithms combined with high resolution feedback devices
A. Direct Drive. With a direct drive, a motor is directly coupled to a load (Figure 1). By using the above explanation of rotational inertia, let’s apply Newton’s Second Law of Motion in rotary terms: T = Iα. This fundamental equation shows us that the less inertia a system has, the less torque it will take to meet the desired acceleration rate.

Direct Drive Motor

Figure 1: Direct Drive Motor

If we consider the torque required to accelerate the motor shaft (T + IMo = I + IM x α + IM) and the torque required to accelerate the load (T + IL = I + IL x α + IL)), we realize the motor shaft and the load will have the same angular acceleration and the same angular velocity for a direct drive system, with a perfectly rigid coupling connecting the motor to the load (α + IM = α + IL = α & ω + IM = ω + IL = ω).

This means that the power needed to accelerate the system as a whole can be found by using this equation: P = (IM+IL) x α x ω (Equation 1). This power is produced by the motor torque (TM) and thus the power P = TM x ω (Equation 2). By equating equations 1 and 2, we get the motor torque, or the torque required to accelerate the system as a whole: T + IM = (I + IM+I + IL) x α (Equation 3).

Note that the torque required to obtain a given acceleration will be minimized when I + IM = I + IL. Thus, for maximum power transfer, the rotational load inertia should be close to the motor inertia.

B. Indirect Drive. An indirect drive is where the motor is not directly coupled with the load but is connected through one or more power transmission elements such as gears, pulley belt systems, chain drives, or ballscrews. To simplify the system, it is useful to consider everything beyond the motor shaft to be a black box that features some reflected rotational load inertia (IL). A good understanding of the system is required to calculate the system inertia reflected on the motor shaft; this can be done by using a motor manufacturer’s online tools, CAD software, or standard formula available for standard geometry.

Torque and Power Requirements in Focus

Figure 2 shows the operating curves for a typical motor. To continuously run, the maximum continuous torque value should not be exceeded, as this is the maximum torque value at which overheating will not occur. For intermittent use, however, greater torques are possible. As the angular speed is increased, the ability of the motor to deliver torque diminishes. Thus, if higher speeds and torques are required than can be given by a particular motor, a more powerful motor needs to be selected.

Motor Operating Curves

Figure 2: Motor Operating Curves

Motor Powering a Drum Hoist

Figure 3: Motor Powering a Drum Hoist

Let’s take an example of a motor that is used to operate a drum-type hoist and lift a load, as shown in Figure 3. The maximum torque we must allow for is the torque required to lift the load at a constant velocity, plus the torque needed to accelerate it to this velocity from rest. The torque (T + IM) required from the motor that is needed by the load (T + IL or T + IL /n for a geared load) and to accelerate the motor (I + IM x α + IM) can be written as: T + IM = T + IL/n + I + IM x α + IM (Equation 4). The angular acceleration of the load α + IL is given by α + IM = n x α + IL (Equation 5), and considering the additional torque (T + If ) required to overcome the load friction, the torque used to accelerate the load will be T + IL = I + IL x α + IL + T + If (Equation 6).

If we substitute for T + IL & α + IM from Equations 5 and 6 into Equation 4, we can write the General Expression for Motor Torque for Geared Drives as T + IM = 1/n [ T + If + αL x (I + IL + n2 x I + IM)]
Considering the additional torque (T + If) required to overcome the load friction from Equation 3, we can write the General Expression for Motor Torque for Direct Drives as T + IM = T + If + α x (IL + IM),

The motor needs to be able to run at the maximum required velocity without overheating. The total power (P) required is the sum of the power required to overcome friction and that needed to accelerate the load. As power is the product of torque and angular speed, then the power required to overcome the frictional torque (T + If ) is T + If x ω, with the power required to accelerate the load with angular acceleration (α) is (I + ILα) ω, where I + IL is the moment of inertia of the load. Thus, the General Expression for Motor Power is P = T + If x ω + I + IL x α x ω.

To summarize the above, for all motor selections, the key factors to consider are torque and power requirements. For demanding applications that require either high acceleration and deceleration or extremely precise positioning to achieve the desired performance, inertia matching also becomes essential. Get in touch with us here for more details!