Brushless Motor Winding Currents and the Principle of Induced Voltage Occurrence

Contents of Episode 4:

• ・Blushless Motor Characteristics
• ・Actual Signal Waveforms in Blushless Motor Driving
• ・Brushless Motor Winding Currents
• ・The Principle of Induced Voltage Occurrence in a Brushless Motor
• ・Induced Voltage Waveforms in Brushless Motors
• ・Brushless Motor Winding Currents That Consider Induced Voltages

Here I’ll explain the U phase current in ① which I talked about the last time, when I pointed out that you should be wondering about the actual signal waveforms in brushless motor driving.

• ① Why is the U phase current waveform shaped this way?
• ② I understand the UH and UL signals and waveforms, but when I zoom in, the UH signal has a pulsed shape (repeated on/off switching). Why?
• ③ Where the UH waveform is pulsed, the U phase voltage (the voltage across the U phase winding terminals) has a pulsed waveform, like the UH waveform, and where the UL signal is turned on, I understand why it is at GND level. But what is the diagonal waveform of the U phase voltage where UH and UL are both turned off?
• ④ What does the power supply current represent? This current also has a pulsed waveform.

The “U phase current” is a current in windings. In order to understand the waveform of this current, you have to understand winding currents and induced voltages, and so I’ll explain these things, dividing them into the following four topics.

• ・Winding Currents in Brushless Motors
• ・The Principle of Induced Voltage Cccurrence in Brushless Motors
• ・Induced Voltage Waveforms in Brushless Motors
• ・Brushless Motor Winding Currents That Consider Induced Voltage Waveforms

Brushless Motor Winding Currents

The currents flowing in motor windings are important factors that determine the size of the winding magnetic fields, and through this, the magnitude of the motor torque. And so you need to have a clear understanding of how the winding currents determine these quantities. Here I will use current waveforms for 120 degree conduction, which I explained in “Creating a Timing Chart for a Brushless Motor”.

In 120 degree conduction, the current is switched six times over an electrical angle of 360 degrees (one electrical cycle). For example, the diagram below indicates that a current that had been flowing from the U phase to the V phase is switched to flow from the U phase to the W phase; such operations are performed six times.

In such switching intervals, a current which has been flowing may drop once and then recover. And, when a current is flowing in the other phases (for example, when current is flowing from the V phase to the W phase), current is not flowing (in this case, not flowing in the U phase), and so the current value is zero. Similar things occur on the negative current side.

This is one reason why the current waveforms in 120 degree conduction look like they do in the diagram; but this alone does not explain other places. For example, in the U→V interval in the diagram, the current becomes large, then falls, and then increases again. Why does such a waveform occur?

In order to answer that question, we have to review the theory relating to coil currents. An ideal coil is represented by an inductance, but actual coils also have a resistance value; hence the coil circuit is represented as shown below. From this circuit, when a voltage is applied, the current waveform should be as shown in the diagram, with no decrease in current.

In fact, the factor causing these rising and falling current waveforms can be said to be one reason why control of brushless motors is so difficult. That factor is the generation of electricity by the coils.

Earlier I stated that a motor converts electrical energy into mechanical energy. But if a motor that uses a permanent magnet is rotated by an external force, that mechanical energy can be converted into electrical energy. That is, the motor can act as a generator. And this electricity generation phenomenon doesn’t just occur when the motor is run using an external force; in fact, it also occurs when the motor is running “by itself”. The voltage generated in this way by a permanent magnet and a coil is called an induced electromotive force (emf), an induced voltage, or a back emf (bemf). Here, I’ll be calling it an induced voltage. This induced voltage is the cause of the increases and decreases in the winding currents.

The Principle of Induced Voltage Occurrence in a Brushless Motor

In a brushless motor, induced voltages occur in the coils (windings) due to rotation of the rotor. These induced voltages affect the motor characteristics in various ways, and, like the winding currents, they must be understood.

To begin with, recall what you learned about electromagnetic induction in physics class. A coil tries to prevent changes in a magnetic field (magnetic flux). When, in the diagram below, the N pole approaches the coil, the amount of magnetic flux passing through the coil increases, and so a current flows in the coil so as to generate a magnetic field that opposes this. Because of this current, a voltage appears across the ends of the coil. Conversely, when the N pole is moved away from the coil, the coil acts so as to maintain the magnetic field, and so a current and a voltage appear that are in the opposite direction relative to when the N pole is brought near. This phenomenon in which a voltage appears is called electromagnetic induction. The current that flows at this time is called an induced current, and the voltage is called an induced voltage, or may be called an induced emf or a back emf.

※The arrows of the magnet magnetic fields in the diagram indicate that the strength of the magnetic field changes; they do not mean that unevenness proportional to the size occurs within the coil.

In a brushless motor, as a magnet approaches and recedes, the magnetic flux passing through the coils changes with the rotation of the rotor, and induced voltages occur according to the amounts of the change (upper-right diagram).

When we consider the amount of change in the magnetic flux passing through a coil (teeth), we have to know that the amount of magnetic flux at the rotor surface varies depending on the place. In some cases the magnetic flux density is substantially uniform; in other cases, the flux density is distributed sinusoidally in the circumferential direction, as shown in the lower figure below. Such a rotor is called a “sinusoidally magnetized rotor”. In the case of such a rotor, the magnetic flux amount changes as the rotor rotates (upper-right diagram).

Induced Voltage Waveforms in Brushless Motors

An induced voltage in a brushless motor is a voltage that appears due to a change in the magnetic flux that passes through a coil. The magnetic flux passing through a coil is the magnetic flux that enters the teeth around which the coil is wound.

First of all, think about this “magnetic flux that enters the teeth”. The magnet and the winding coils in a motor are arranged in the positions shown below. So the magnetic flux of the part of the magnet that opposes a set of teeth can be thought of as entering the teeth.

Let’s think about the amounts of entering magnetic flux, taking as examples the two surface flux patterns in the diagrams above.

The graph shows the amounts of surface magnetic flux of the rotor opposing the “point for which the surface flux is shown” over one rotation of the rotor, that is, 360 degrees. The total flux amounts entering the teeth were calculated and are also shown, similarly over 360 degrees. The total magnetic flux amount is calculated by adding the magnetic flux of the magnet portions opposed to the teeth (in the diagram, the range of magnetic flux entering the teeth). So as the rotor turns and the position changes, the flux amount will change.

For example, at the 0 degree rotor position, the N and S poles are equally opposed to the teeth, so that the teeth flux amount is zero. At the 90 degree position, the center part of the N pole is opposed to the teeth, and the teeth flux amount is maximum. When such calculations are performed over 360 degrees, if the rotor has a sinusoidal magnetization, then the teeth flux amount (the total flux) is also sinusoidal.

This change in the teeth flux amount (the function derivative) gives us the induced voltage (when the derivative is taken, a minus sign is added). And so the picture we get is of an induced voltage that is shifted by 90 degrees from the opposing surface flux.

Broadly speaking, magnetization waveforms are classified as sinusoidal, square wave, and trapezoidal. Here I showed an example of trapezoidal magnetization. I’ll also note that the square wave and the trapezoidal shapes differ with respect to the gradients (slopes) when switching between the N and S poles, but there is no clear-cut definition. In general, a shape that has a more gentle (less steep) slope or gradient is called trapezoidal.

In the case of trapezoidal magnetization, the waveform of the magnetic flux entering the teeth is not sinusoidal, and the waveform of the induced voltage that we can calculate by differentiating is different, as shown in the figure. One thing we should notice when we look at the induced voltage that results for trapezoidal magnetization is that the features are clearly different from those of the induced voltage waveform for sinusoidal magnetization. For example, even though in trapezoidal magnetization the slope of the parts where the surface flux changes between N and S is steeper than for sinusoidal magnetization, the parts of the induced voltage where there is a change between positive and negative (called zero-cross points) are more gentle than for the sinusoidal case.

Differences in the induced voltage waveforms affect the characteristics and control of a motor in various ways. However, waveforms have many different shapes from one motor to another, and so do not match the magnetization waveforms (except when the magnetization is sinusoidal), which is troublesome. As a first step in controlling a motor, it is vital that the engineer determine the shape of the induced voltage waveforms for the motor that is to be controlled.

Next time, I will talk about the effects that induced voltages have on winding currents. This will be the final topic in “Brushless Motor Characteristics, Part 1”.

Next page: Brushless Motor Winding Currents That Consider Induced Voltage Waveforms