[Episode 6] Test showdown! A serious battle between Ichinose and Ninomiya!

2025.09.08

table of contents

From this point forward I’ll be talking about the output characteristics of brushless motors, splitting my discussion into talks on the following four topics. Output characteristics are something you need to know about if you want to control a motor—whether you’re starting it up or stopping it, or controlling the torque, rotation rate, or other parameters.

Contents of Episode 6

Brushless Motor Output Characteristics: Torque and Rotation Rate

A motor outputs mechanical energy; torque and rotation rate are two quantities that are used as indices to measure that output. Here I’ll explain how these two quantities are determined, and how they are related.

Torque of a brushless motor

A torque is a force that a motor creates to rotate a shaft. The magnitude of the torque of a motor is also related to the motor diameter and length, but let’s simplify these quantities as a constant A. What we want to know in order to design a motor driver is how the torque depends on the magnitude of the magnetic force of the electromagnet, on the magnitude of the magnetic force of the permanent magnet, and on the relative angle θ between them.

This is expressed as an equation below. Here the electromagnet magnetic force is the product of the coil windings and the current flowing in the windings, multiplied by a constant B. The coil windings are expressed as the number N t/c of windings in a single coil. The constant B is a value determined by the teeth size and the material properties of the core. The magnetic force of the permanent magnet depends on the magnet performance, which in turn depends on the magnet material, methods used to work the material, and other factors.

Magnitude of torque

In order to simplify the equation, let’s assume that the motor characteristics are constant. Then among these relationships, the relation between the permanent magnet magnetic force and the number of windings is constant. If we use a constant C to represent them, then the output torque becomes a function of the current and sinθ. This equation is important. I’ll also mention that the equation T=A×sinθ presented in “Creating a Timing Chart for a Brushless Motor” was obtained by holding the current constant in this equation (here A is different from the A of the equation above).

Rotation rate of a brushless motor

The rotation rate of a motor is determined by the relative magnitudes of the output torque, which I just mentioned, and the load torque that is needed to rotate a fan or some other part. This much is clear from the two physics equations shown below. The first equates the torque with the product of the moment of inertia and the acceleration (this is the rotational version of F=ma). The second one says that velocity is obtained by integrating the acceleration. From these two equations, we see that if there is a difference between the output torque and the load torque, an acceleration occurs, and the velocity changes. This velocity change imparts changes to the output torque and the load torque, and when they become equal the acceleration becomes zero, and there the rotation rate stabilizes.

Motor acceleration and velocity

When a rotation rate change causes a change in the output torque in this way, let’s think about what happens when, for example, there is no load torque (no loading). In the above equations, if the load torque is zero, the acceleration a is not zero, and the velocity rises. Of course, the acceleration can’t go on forever. This is related to how the current that determines the torque is itself determined. As the equation for the current shown below indicates, the current is determined by the winding impedance and by the difference between the applied voltage and the induced voltage. As the rotation rate increases, the induced voltage Vbemf increases and the current decreases, ultimately reaching zero—that is, the acceleration becomes zero, and the rotation rate stabilizes. The motor rotation rate is determined by the interplay of these factors.

Equation for current

Brushless Motor Output Characteristics: Driven States of Actual Motors

There are three main driven states for a motor. The first is the state in which the motor is turning with no load applied; at this time, the output torque is essentially zero. The second is the state in which there is a load and the motor is rotating. This could be called the normal state of motor use. Finally, there is the locked (fixed) state, in which the motor is not turning, even though a voltage is being applied. Apart from these, there is a state in which the motor is being turned by an external force (a regenerating state or electricity-generating state), a reverse rotation state, and a state in which no voltage is applied.

No-load state

This is a state in which no output torque is required (strictly speaking, there actually is a load, due to friction in the bearings or other causes, so a torque is needed). In this case, the windings current is effectively zero. A zero current means that the applied voltage and the induced voltage are essentially the same. Put another way, this can be called a state in which the rotation rate has risen until the induced voltage has become the same as the applied voltage.

In cases where the applied voltage has a square-wave (rectangular-wave) shape while the induced voltage is sinusoidal, as in the diagram below, it may be argued that in such a state the values are not the same. But when the net current value is zero, we will call them the same voltage.

No-load state

Loaded state

In this state, a torque is needed to rotate a fan or other load body. Since a winding current is needed to generate the torque, the induced voltage is smaller than in the no-load state, as in the diagram below. Therefore the rotation rate is reduced compared with the no-load state.

Loaded state

Locked state

In this state the motor does not turn, and an induced voltage does not appear. Among the three main states, the current is greatest in this state, and the torque is also the greatest.

Locked state

Next time, I’ll talk about S-T characteristics and I-T characteristics so that we can reach a deeper understanding of changes in these driven states.

Next page: S-T Characteristics and I-T Characteristics of Brushless Motors

    Teacher Sugiken’s Motor Library

    Teacher Sugiken’s Motor Driver Dojo

    An Introduction to Motors

    Brushless Mortor Driver

    Motor Q&A

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