Technical Report on FFT Analyzer 2

Let's observe the movement of the ball when a string is attached and it is spun. As you can see, it will follow a trajectory like that shown in Figure 2-4.

In this diagram (Figure 2-4), the trajectory waveform of the shadow of the ball projected onto the Y-axis is called a sine wave, while the trajectory waveform of the shadow projected onto the X-axis is called a cosine wave. The length of the string (r) corresponds to the amplitude, and the time it takes for the ball to complete one rotation corresponds to the period (T). When expressed in radians, one complete rotation of the ball is 360 degrees = 2π radians, so the number of rotations in time t can be calculated using the following formula (2-2).

(Equation 2-2)

Additionally, the frequency f, which indicates how many rotations occur per second, is given by the following formula 2-3.

Equation 2-3

So, how can we express the trajectories of each shadow after t hours in mathematical terms?

(Equation 2-4)

Did you remember that it is represented by :? This equation 2-4 represents the position of the ball at time t in XY coordinates (x, y).
Here, considering the time when the ball is at an angle φ as shown in figure 2-5, from equation 2-4:

Equation 2-5

In addition, a, b, and r are the sides of a right triangle, so the following equation 2-6 holds.

(Equation 2-6)

In this way of thinking, it becomes clear that the position of the ball can be determined by (a, b) in equation 2-5 or by (r, φ) in equation 2-6. Since the ball is in circular motion, both equations will hold true as long as it reaches the position of φ, regardless of how many rotations it makes.

Physical phenomena possess a periodicity that can be described as wave phenomena. Sine (sin) and cosine (cos), along with tangent (tan), are used as trigonometric functions to explain these periodic physical phenomena.

Now, let's consider the moment when the position of φ is taken as the starting point (at this time, φ is referred to as the initial phase). At this time, equation 2-4 is:

Equation 2-7

Equation 2-8

can be expressed as.

The equation representing the motion trajectory of the ball can be either equation 2-7 or equation 2-8, but the sine wave can be considered as a cosine wave that is phase-shifted by π/2. Additionally, due to the physical background, equation 2-7, which is based on the cosine wave, is generally used.

Next, I will explain important terms related to waveform representation that often come up during waveform analysis using FFT.

When you call out towards the mountain, after a while, you can hear the echo of your voice. Let's think about how to express the time difference between the emitted sound and the returning echo in terms of phase difference.

Let the period of the waveform be T (frequency f), the time difference be τ, and the phase be φ:

Equation 2-9

For example, with the waveform of "A" mentioned earlier, if the frequency of "A ₁" is 106 Hz and the time difference τ = 0.001 seconds until the echo returns, the phase will have a delay of approximately 0.21 π = 38.1 degrees. In the case of "A ₂" with the same time difference τ = 0.001 seconds, if the frequency is 212 Hz, the delay will be 76.3 degrees. It is important to note that even with the same time difference, if the period T (frequency) is different, the phase will also differ.

It is also important to understand the differences between electrical angle and mechanical angle in relation to amplitude, frequency, and phase. Consider a gear with 60 teeth equipped with an electromagnetic detector, rotating at 600 revolutions per minute (r/min). When the gear completes one rotation, the signal from the electromagnetic detector outputs a sine wave corresponding to 60 cycles. Here, the phase represented by one cycle of the signal sine wave, taken as 2π rad (360 degrees), is called the electrical angle, while the phase taken for one rotation of the gear, also as 2π rad, is called the mechanical angle.

As shown in Figure 2-6;

The frequency of the signal sine wave f = 600 r/min / 60 s × 60 gear teeth = 600 (Hz)

The frequency of one rotation of the gear f ₀ = 600 r/min / 60s = 10 (Hz)

In terms of angle, one cycle of the sine wave signal and the gear is represented as the same 2π radians, but in terms of time:

The period of the detector signal = The period of the electrical angle representation = 2π = 1/600 (seconds)

The period of one rotation of the gear = the period of mechanical angle representation = 2π = 1/10 (second)

is.

The electrical angle is based on one cycle of an electrical signal, while the mechanical angle is based on one cycle of an object's rotation. When analyzing phenomena of rotating bodies, it is necessary to pay attention to the relationship between the electrical angle and the mechanical angle in order to evaluate frequency analysis data. An FFT analyzer uses the electrical angle.

The term 'rotational frequency' is often used when analyzing based on the number of rotations. Considering the rotation of gears mentioned above as the reference (fundamental frequency), the signal sine wave has a frequency that is 60 times the fundamental frequency of the gear. This is referred to as the '60th harmonic'. In particular, in rotating machinery, a gear's rotational frequency of 10 Hz is referred to as the first rotation, and a signal sine wave of 600 Hz is referred to as the 60th rotation, using the term 'rotation'.

The term harmonic refers to frequencies that are integer multiples of the fundamental frequency. For example, when the frequency of the fundamental wave "A" (A ₁) is 106 Hz, the frequency of the second harmonic "A" (A ₂) is 212 Hz. Recently, the analysis of harmonic distortion in power supplies has been conducted, which also analyzes the extent of harmonic components that are integer multiples of the fundamental power frequency, whether it is 50 Hz or 60 Hz, in terms of amplitude and phase at different frequencies.

Let's consider the sum (composition) of the next waveform.

(Equation 2-10)

Each composite waveform, as shown in figures 2-7 (③, ④), has the same frequency but different amplitudes and phases.

First, consider x ₂. x ₂ is from the trigonometric function formula;

Equation 2-11

This means that when a cosine wave and a sine wave with the same frequency and phase are synthesized, the resulting wave is identical to a cosine wave with an amplitude of √2 times and a phase shift of π/4. This can also be extended to consider the following.

Equation 2-12

If we take the coefficient a on the cos axis and b on the sin axis, it will look like Figure 2-8.

This is the same as Figure 2-5 above, so from Equation 2-6;

Equation 2-13

That is to say;

Equation 2-14

is established. In this way, a single frequency waveform (hereafter referred to as a simple waveform) has coefficients a and b at the same frequency.It can be expressed in a formula, and when the coefficients a and b at this time are known, it becomes possible to determine the magnitude r of the composite amplitude and the phase difference φ. Similarly, the composite waveform of x ₁ is,The coefficients can be considered as a = 2, b = 0, so the magnitude of the composite amplitude r = 2 and the phase difference φ = 0. Each component "A ₁", "A ₂", "A ₃"... analyzed from the voice of "A" is a simple waveform, so each waveform is,will be represented by. Furthermore, the reference for the phase difference φ here isPlease pay attention to the fact that...

simple waveformI explained how it can be expressed. In this section, I will explain how to determine the coefficients a and b of this expression.

Let's consider the product of the following waveform.

(Equation 2-15)

Figure 2-9 shows the waveforms for a ₁ = 2, a ₂ = 3, and b = 1.

As shown in this figure, the product of waveforms with periodicity will also be a waveform with periodicity. Let's integrate one period of the waveform of x ₁ (from 0 to 2π) and examine its area. The waveform of x ₁ has equal areas on the positive side ● (yellow circle) and the negative side ● (red circle) with respect to zero, so the total area will be zero. x ₂ represents the product with the second harmonic component, and the total area of this waveform will also be zero. However, x ₃ has no negative side, so the total area = amplitude a × T/2.

That is:

Equation 2-16

This means that by multiplying any waveform by (n = 1, 2, 3, ...) and dividing the area of one period by T/2, the original components are extracted, and their amplitude a _n can be determined. Similarly,By multiplying and finding the area of one cycle, and then dividing that value by T/2, you can determine the amplitude b _n.