"Waveforms and FFT-3" 3. Time Difference and Phase (continued)

3. Time difference and phase (continued)

Torque meters are one example of products that utilize phase difference.
Let's look at this waveform. For the principle, please refer to the following.

The torque sensor has a switch that changes the rotation direction of the attached motor to match the rotation direction of the shaft (torsion bar).
The included motor rotates an internal gear (with the gear located on the inner diameter side), a design feature that allows measurements to be taken even when the shaft (torsion bar) is stopped.

Figure 1. Initial phase (without torsion)

Assume the torque sensor switch is set to CW and the load-side shaft is locked (fixed). The waveforms of signal outputs E and F when the drive-side shaft is twisted slightly left and right in this state are shown below. The red waveform is signal E (reference), and the blue waveform is signal F.
Looking at the waveform before twisting (no load), we can see that the initial phase difference is adjusted to approximately 180 degrees.

Figure 2: When twisted CW

The torsion bar twists slightly in proportion to the torque, and this is detected as a phase difference between E and F in the signal output. The initial phase difference is set to zero, and this setting is called zero adjustment.
The magnitude of the torque applied to the torsion bar is detected as the magnitude of the phase difference. When twisted to the right (CW), the phase difference becomes large, and the torque meter displays a positive polarity (+). When twisted to the left (CCW), the phase difference becomes small, and the torque meter displays a negative polarity (-). The polarity indicates the direction in which the torsion bar is twisted.

Signals E and F are generated by regenerative power, and are output as approximate sine waves due to the change in magnetic flux caused by the rotation of the internal gear and gears A and B. If the internal gear and gears A and B (torsion bars) rotate in the same direction and at the same rate, the relative speed difference between the internal gear and gears A and B disappears. Consequently, the change in magnetic flux also disappears, the magnitude of the signal becomes small or no signal is generated at all, and a dead zone (a range in which operation is not possible) is created. Therefore, the internal gear must rotate in the opposite direction to the axis. Note that when the switch is set to CW, it rotates in the opposite direction to the axis.

When the detector switch is set to CCW (Critical Control Wave), the rotation direction of the internal gear is reversed.
As explained with rotary encoders, the E and F waveforms progress from left to right in CW mode, but in CCW mode, they progress from right to left, changing the initial phase difference. Therefore, torque meters are instructed to perform zero adjustments separately for CW and CCW modes.

Figure 3: When twisted in a CCW direction

In the case of CCW (Clockwise-Clockwise) configuration, the polarity is indicated as + (larger phase difference) when the load-side shaft is locked and the drive-side shaft is twisted CCW, and - (smaller phase difference) when twisted CW (clockwise). Please note that the twist direction and polarity indication of the torsion bar change when using CW configuration.
The polarity indication is unclear, but the switch is either CW or CCW.

• If the load side is the brake, the polarity is positive (+).
• In situations where the load side is the drive side, such as engine braking, the polarity is negative.

(Even if the rotation direction of the torsion bar is the same, the direction of twisting will be reversed.)
This is how it works. Please visualize the twisting direction of the torsion bar to understand it.
While the principle of a torque meter is based on phase difference, the actual signal processing involves further refinements to improve accuracy.

Up until now, we've considered phase differences based on a reference signal, but let's consider the phase difference when there is only one sine wave.
Phase difference is defined by taking a specific point in time as the reference point, setting it as time zero, and measuring the difference from that point. In the case of a rotating body, the reference point is determined by placing a reflection mark and triggering the analysis with a signal detected by the reflection mark. If there is no reference signal, an FFT analyzer will simulate a cosine wave of the same frequency and use this as the reference to display the phase difference.
The following shows the phase difference of a sine wave triggered at 0V. To view the waveform in detail (to achieve high temporal resolution), it is better to take smaller samples. Therefore, the sine wave is set to 100Hz, the frequency range to 40kHz, and the sample length to 8192. The X-axis has also been magnified for better viewing. The first half shows the sine wave, and the second half shows the cosine wave. In the measurement, the phase difference is approximately -90 degrees and approximately 0 degrees, respectively. Frequencies other than 100Hz are noise waveforms, so please focus on the phase difference at 100Hz.

Figure 4-1 Phase difference when there is one sine (cosine) wave

Figure 4-2 sin waveform (magnified view of the X-axis)

Figure 4-3: Note the phase of the cosine waveform at 100Hz (magnified view of the X-axis).

If we express a sine wave as an equation,

sin(2πft−Φ) Φ: ±phase difference
f=1/T T: Time of one cycle (s) f: Frequency
2πf = 360/T

As you already know,
Electrical angles can be measured in radians or degrees (deg), and the unit Φ is used in either radians or degrees (deg) depending on the context.

If the phase is delayed by 90 degrees,
sin(2πft−Φ) Φ=90 degrees, 2π/4 radians

If the phase is advanced by 90 degrees,
sin(2πft+Φ) Φ=90 degrees, 2π/4 radian

Phase lag is represented by a negative number, and phase lead by a positive number.
As you know, a sine wave becomes a cosine wave when it advances by 90 degrees, and conversely, a cosine wave becomes a sine wave when it lags by 90 degrees. Since both sine and cosine waves are related by a 90-degree phase difference, it seems that either approach would suffice. I initially said that the power spectrum of an FFT analyzer should be considered as the amplitude of a sine wave, but actually, it's more convenient to consider it as a cosine wave. I'll explain why next time.

point

A sine wave is represented by amplitude A, frequency f, and phase φ.
Asin(2πft + φ) φ = 0: No lag
φ = positive value: progress
Value of φ = -: delay

(Excerpt from the email newsletter issued on February 22, 2007)