# Stepper motor torque basics

## Stepper motors remain popular for industrial motor applications because they are low-cost, rugged, simple, have high torque at startup and low speeds, require little maintenance, and they can operate in an open loop control system.

By Eric Rice, Applied Motion Products March 26, 2018 A stepper motor divides a full rotation into a number of equal steps, important for many industrial motor and motion control applications. The motor’s position can be caused to move and hold at one of these steps as long as the motor is carefully sized to the application in respect to torque and speed. Holding torque is a measurement of how much rotating force is required to force a stationary stepper motor shaft out of position. Holding torque (T) is the product of a motor’s torque constant (KT) and the current (i) applied to the stator windings.

T = KTi

In most applications, electronic drivers control stepper motors. They employ pulse width modulation (PWM) technology to monitor the stator current and apply the proper voltage to achieve the desired current and torque. When a motor is stationary, the driver only needs to use enough voltage to overcome the resistance of the stator coils (also known as motor phases). This is described by Ohm’s law that calculates voltage as the current in amps multiplied by the resistance in ohms. If voltage increases, so does current, but if resistance increases, current reduces.

V = iR

Because most high-performance stepper motors have low phase resistance, the driver does not need much power supply voltage to hold the motor in position. For real applications, the motor does not remain forever stationary; it is used to move a load. Moving something at a particular speed requires that dynamic torque be available at that speed. Stepper motors do not change instantly from standstill to a given speed. They must accelerate just as a car gradually increases speed when the driver steps on the gas pedal. To accelerate the vehicle faster requires more gas. Stepper motors are similar, following Newton’s famous law F = ma. Below is the formula for Newton’s law expressed in rotational terms, where torque (T) is proportional to the rotor and load inertia (J) and angular acceleration (A):

T = JA To manage a heavier load or accelerate faster requires more torque. However, the dynamic torque of a stepper motor decreases as speed increases because when a motor starts moving, it becomes a generator. As the rotor’s magnetic field moves among the stator coils, a voltage appears on the motor terminals. The driver must apply an extra voltage to the motor to overcome this voltage, known as back EMF, which is a product of motor speed (w) and voltage constant (KE). Also, stator coils, like all coils, have inductance that resists the current change. As the stator current changes to keep the rotor turning, more voltage must be used to overcome inductance (L). The voltage equation for a motor in motion is:

V = KEώ + iR + L(di/dt)

A PWM driver will increase the voltage applied to the stepper motor to keep the current and torque constant. At some speed, the power supply will not have enough voltage, and the motor current will begin to fall. The torque drops with the current. If using a higher voltage power supply, the dynamic torque remains flat to a higher speed (see Figure 1). The process of sizing an application involves calculating the required torque and speed range necessary to move the load. For example, if the application needed 80 oz.-in. of torque up to 10 revolutions/second (rps), this motor could use a 24-V power supply (see Figure 2).

If we need to go farther and faster, one might accelerate to 80 oz.-in. at 20 rps. The stepper motor would require a 48-V power supply (see Figure 3).

Eric Rice is an application engineer at Applied Motion Products. He has worked in the motion control industry for 20 years, specializing in stepper motors, servo motors, drives, and controls. He has a degree in Electrical Engineering from the University of Illinois Urbana-Champaign.

This article appears in the Applied Automation supplement for Control Engineering and Plant Engineering.

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