Torque Pulsation in Sinusoidal Driving, Sinusoidal Current Timing and Phase Changes, Advance Angle Control, and Sinusoidal Driving Waveforms

This time, in the second half of Episode 7, I’ll start by explaining torque pulsation in sinusoidal driving.

Contents of Episode 7

• ・What Is Torque Pulsation?
• ・Sinusoidal Conduction
• ・Waveforms in Sinusoidal Driving
• ・Synchronous Rectification
• ・Power Supply Currents in Sinusoidal Driving
• ・Torque Pulsation in Sinusoidal Driving
• ・Sinusoidal Current Timing
• ・Sinusoidal Current Phase Changes
• ・Advance Angle Control of a Brushless Motor
• ・Sinusoidal Driving Waveforms

Torque Pulsation in Sinusoidal Driving

In the previous article, I talked about sinusoidal driving as a conduction method in which there is no occurrence of the Torque Pulsation that we see in 120 degree conduction. But this time, let’s think about torque pulsation in sinusoidal driving. We’ll use equations similar to those for 120 degree conduction to find the torque.

First let’s consider the phase torques. The current flowing in the U phase windings is sinusoidal, and so the U phase electromagnet magnetic force changes accordingly. Then we find sinθ from the positional relationship θ with the rotor (permanent magnet) and calculate the torque for the U phase, and we get the sinusoidal waveform shown below. The waveform period is one-half of the current period, and the overall waveform is raised (the center points are not at zero; the minima are close to zero).

In sinusoidal driving as well, when we look at only the torque due to one phase, we see that this kind of pulsation occurs, but when we take the sum of the torques for all three phases, we find that the total torque is constant. We see this from the fact that when we add the three sine waves, shifted by 120 degrees from each other, we end up with zero. The components that change sinusoidally add up to zero, and only the components that are higher for each of the phases add up to result in a constant torque.

Phase torque =A⋅magnetic force (=current)⋅sin(θ)
　　　　　\( = A \cdot \sin(\theta) \cdot \sin(\theta) \)
　　　　　\( = A \cdot \left(-\frac{1}{2} (\cos(\theta+\theta) – \cos(\theta-\theta))\right) \)
　　　　　\( = -\frac{A}{2} (\cos(2\theta) + \frac{A}{2} \)
　　　　　\( \rightarrow \sin(2\theta) + B \)

※Regarding the line indicated by “→”:
・The distinction between cos and sin is not important, and so sin2θ is substituted.
・-A/2 is likewise not important here, and so is omitted.
・A/2 is replaced with B.

Total torque = U torque + V torque + W torque
　　　　　\( = (\sin(2\theta) + B) + (\sin(2(\theta+120)) + B) + (\sin(2(\theta+240)) + B) \)
　　　　　\( = \sin(2\theta) + \sin(2(\theta+120)) + \sin(2(\theta+240)) + 3B \)
　　　　　\( = \sin(2\theta) + \sin(2\theta+240) + \sin(2\theta+480) + 3B \)
　　　　　\( = \sin(2\theta) + \left(-\frac{1}{2}\sin(2\theta)\right) + \left(-\frac{\sqrt{3}}{2}\cos(2\theta)\right) + \left(-\frac{1}{2}\sin(2\theta)\right) + \frac{\sqrt{3}}{2}\cos(2\theta) + 3B \)
　　　　　\( = 3B \)

In the article “Sinusoidal Conduction”, I described a concept on which, if the three-phase composite winding magnetic field rotates while maintaining a constant magnitude and a constant angle with the rotor, then torque pulsation does not occur. And we see that the process I just described confirms that for such a case, torque pulsation is zero.

I’ll also mention that when calculating torque, where the phases of the winding currents are concerned, the relative angle with the rotor is 90 degrees at the winding current peak positions, and when the current is zero the angle θ is either 0 or 180 degrees. That is, as with 120 degree conduction, it is assumed that a voltage is being applied to the winding terminals such that the current is as shown at the rotor magnet position. But if it should happen that that’s not the case, then what?

Sinusoidal Current Timing

The sinusoidal current phase in the last discussion was such that the current peaked where the angle with the magnet was 90 degrees. Let’s examine how the torque changes if the angle deviates from this position.

As assumed conditions, let the phases of the winding currents for each of the phases U, V, W change uniformly. That means the 120-degree shift between the phases is maintained.

Let’s calculate phase torques like the way we did before. If a current phase shifts by C degrees, we see from the equations below that the amount by which the torque is raised changes (from A/2 to AcosC/2). Here cosC has a value of 1 or lower. Hence if the phase shifts, the amount of rise of the torque is decreased.

Phase torque =A⋅magnetic force (=current)⋅sin(θ)
　　　　　\( = A \cdot \sin(\theta (\theta – C)) \cdot \sin(\theta) \)
　　　　　\( = A \cdot \left(-\frac{1}{2} (\cos(\theta – C + \theta) – \cos(\theta – C – \theta))\right) \)
　　　　　\( = -\frac{A}{2} (\cos(2\theta – C) + A \cdot \cos(C)/2) \)
　　　　　\( = \sin(2\theta) + D \)

This becomes the waveforms shown below. If the angle shifts by 30 degrees, D in the upper equation is decreased with respect to B in the previous equations by the amount cos30°, or 0.866. And so the total torque is also reduced by the amount 0.866.

This can also be confirmed by calculating the total torque from the composite magnetic force. If the currents are shifted by 30° each, then θ in the equations above is 60°. Relative to the case in which θ is 90° so that sin90°=1, we have sin60°, or 0.866.

In a comparison of the two total torque values with θ at 90° and 60° as in the figure above, even though the currents are almost unchanged, the output torque is smaller when the angle is 60°. Put another way, we can say that the motor efficiency is worsened. Also, the I-T characteristic, and also the S-T characteristic, are affected by this change in current phases.

And so we can represent the cause of the change in total torque in various different ways. For example, since the angle between the composite magnetic force and the magnet magnetic force is not 90°, the fact that sinθ is less than 1 is also a factor. And the fact that there are places where phase torques go negative is also a cause.

This change in motor efficiency when there are shifts in the sinusoidal current timing is an important phenomenon influencing motor performance. Another important thing, however, is the fact that no change in torque pulsation occurs.

Sinusoidal Current Phase Changes

In sinusoidal driving, the phases of the sinusoidal currents are relatively easy to control if the phases of the applied voltages are the same as the current phases. However, the actual sinusoidal current phases change relative to the applied voltage phases due to various factors. The reasons why the phase of the sinusoidal current is not the same as the phase of the applied voltage are shown below.

First, voltage and current are expressed by:

\( V – V_{\text{bemf}} = R \cdot I + L \frac{dI}{dt} \)

From the equation we see that the current phase is affected by the inductance and the induced voltage. This complicates motor current control.

<Phase lag due to inductive components>

Because windings are inductive, the phase of a sinusoidal current tends to lag according to the difference between the applied voltage and the induced voltage. This phase lag changes depending on the voltage frequency, that is, the motor rotation rate, and so it cannot be treated as a constant value, which is a problem. This lag causes a drop in motor efficiency.

<Current phase changes with the applied voltage phase>

The applied voltage phase is the phase difference with the induced voltage. If the applied voltage phase advances (to the left in the figure below), the voltage difference also advances, and so the current phase advances. Using this characteristic, the phase of a lagging current can be returned by means of the inductive component I mentioned, so that declines in motor efficiency can be avoided. Such an operation to advance the applied voltage phase is called advance angle control in a motor driver.

<Current phase changes with the applied voltage amplitude>

Using the advance angle control I just described, the current phase can be advanced. But if the optimum state for motor efficiency is to be maintained, the amount of phase adjustment, called the advance angle amount (advance angle value), cannot be treated as a constant value. This is not only because the lag amount of the current phase changes with changes in frequency, but also because the difference voltage phase changes with changes in the applied voltage amplitude, affecting the change in the current phase. Similar changes occur when the induced voltage changes.

So the current phase greatly affects the motor torque and efficiency, but the current phase also deserves our attention because of the complexity of these adjustments.

Advance Angle Control of a Brushless Motor

A motor driver detects the position of the rotor and applies a voltage according to the rotor position to turn the motor. The timing with which this voltage is applied must take into consideration the strength and direction of the winding magnetic field formed by the winding current (the electromagnet magnetic field). One method of adjusting this timing is advance angle control.

Before explaining the concept of advance angles, I want to clarify the definition of the voltage application timing. In sinusoidal driving, the reference position is the position of the same phase as the induced voltage. In 120 degree conduction, the reference position is the position at which the center of the induced voltage and the center of the 120 degree conduction waveform coincide.

In advance angle control, the phase of the applied voltage is advanced with respect to this reference phase (position) to output the voltage. And so the reference phase is the position at which the advance angle is zero, and the amount of advance from this is called the advance angle amount (advance angle value). This advance angle control can be applied to 120 degree conduction as well as to sinusoidal driving.

The figures below show an example of how the torque of a motor changes when advance angle adjustment is performed. The upper row compares advance angles of 0° and 15° in 120 degree conduction, and the lower row compares advance angles of 0° and 13° in sinusoidal driving.

We see that in sinusoidal driving, the average torque is higher at an advance angle of 13° compared with an advance angle of 0°. This is because when the advance angle is 0° the current tends to lag; advancing the angle results in recovery from the lag.

And we see that even when using 120 degree conduction, advance angle adjustment increases the total torque. But at the same time, the larger pulsations are a cause for concern.

These results tell us the following.

• 1) Compared with sine waves, in 120 degree conduction the change in torque resulting from advance angle adjustment is small. In other words, we can say that 120 degree conduction is impervious to current phase shifts (there is little decrease in efficiency). This may be because in 120 degree conduction there are times when current is not flowing in the windings.
• 2) Even when the amplitude of the applied voltage is the same, the torque values are lower for sinusoidal driving (in the above figures, the values are around 40 for 120 degree conduction, but in the low to mid 30’s for sinusoidal driving). This is because the maximum values of the voltage across windings (line voltages) are different for 120 degree conduction and for sinusoidal driving.

Sinusoidal Driving Waveforms

The voltages across windings are an important quantity in motor control. A motor is required to rotate at a desired rotation rate and with a desired torque, and so the magnet magnetic forces and winding specifications of the motor are designed so as to be able to deliver this output (rotation rate and torque). And so it is important that, to the extent possible, we avoid changes in the output range caused by differences in the conduction waveforms such as those I have described (120 degree conduction and sinusoidal driving). I’d like to talk about the reasons for changes in the maximum output, and introduce a number of sinusoidal driving waveforms that result in higher outputs.

First, let’s review the sinusoidal driving waveforms that I explained earlier (the top-right among the figures below). Here sine waveforms are created, without any modification, within the range of the power supply voltage (in the figure, ranging from 0 to 100); these are called pure sine waves. In this case, the sine waves have as maxima a lower limit of 0 and an upper limit of 100; at first glance, they appear to be the same as 120 degree conduction (in which the lower and upper limits are again 0 and 100).

Let’s now consider voltages that determine winding currents. A winding current is the current that flows due to differences in the potentials of phases. And so the sizes of currents, that is, the motor output, are determined by line voltages.

When we again compare the line voltages for pure sine waves and for 120 degree conduction, we find that there are differences in the voltage amplitudes. For 120 degree conduction there is a voltage range from -100 to 100, which is 200, whereas for pure sine waves the range is from -86.6 to 86.6, or a range of 173.2. From this, we can say that conduction using pure sine waves has a narrower output range, and a lower voltage utilization ratio, than for 120 degree conduction. This is the reason for the lower torque in sinusoidal driving that I just mentioned.

Sinusoidal driving is better than 120 degree conduction when it comes to quietness, but the low voltage utilization ratio is a drawback. Because of this, many sinusoidal driving waveforms that raise the voltage utilization ratio have been proposed. The four main proposals appear in the second and third rows of the above figure. Looking at the phase voltages, many of these do not really look like sine waves, but the line voltages are distortion-free sine waves. All four waveforms have maximum line voltages that are comparable to that for 120 degree conduction.

In motor drivers, the “saddle-shaped sinusoidal” waveform shown in the figure is widely used. This is also called a two-phase modulated sine wave with the lower parts fixed. When viewing waveforms with an oscilloscope, it is important to also check the conduction waveforms.

Up to this point, I have explained phenomena and circuit operation that are seen when a motor driver is used to rotate a brushless motor. I have been talking about oscilloscope waveforms, induced voltages, adjustment of motor rotation rates, power supply currents, output characteristics, torque pulsation, and conduction waveforms; these all constitute basic knowledge that should be understood when running a motor.

When developing a motor, naturally you’ll want to design a motor so that you can obtained the desired output characteristics (rotation rate, torque), but at the same time, you’ll be looking for efficiency, quietness, and reliability. The information I’ve provided you is the bare-minimum basic information needed by a motor driver developer. Please take the time to master it.