Technical Report: About FFT Analyzers 2

Let's look at the movement of a ball when a string is attached to it and it is spun. As you can see, it follows a trajectory like that shown in Figure 2-4 below.

In Figure 2-4, the waveform of the ball's shadow projected onto the Y-axis is called a sine wave, and the waveform of the ball's 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). The angle of one rotation of the ball is 360 = 2πrad in radians, so the number of degrees it rotates in time t can be calculated using the following equation 2-2.

(Formula 2-2)

Furthermore, the frequency f, which represents the number of rotations per second, is given by the following equation 2-3.

(Formula 2-3)

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

(Formula 2-4)

Do you remember that it is expressed as:? Equation 2-4 is an equation that expresses the position of the ball after time t in XY coordinates (x, y).
Now, considering the case where the ball is at an angle φ as shown in Figure 2-5, from equation 2-4:

(Formula 2-5)

Thus, since a, b, and r are the sides of a right triangle, the following equation 2-6 holds true.

(Formula 2-6)

Considering this, we can see that the position of the ball can be determined by both (a, b) in equation 2-5 and (r, φ) in equation 2-6. Since the ball is undergoing circular motion, both equations will hold true as long as it reaches the position of φ, no matter how many rotations it makes.

Physical phenomena exhibit periodicity to such an extent that they can be described as wave phenomena. Sin and cos, along with tangent (=tan), are used as trigonometric functions in mathematical formulas that explain these periodic physical phenomena.

Now let's consider the case where the position φ is the starting point (at this time, φ is called the initial phase). In this case, equation 2-4 is:

(Formula 2-7)

(Formula 2-8)

It can be expressed as follows.

The equations that represent the trajectory of the ball's motion can be either Equation 2-7 or Equation 2-8. However, since a sine wave can be considered as a cosine wave with a phase delay of π/2, and also due to physical reasons, Equation 2-7, which uses the cosine wave as the reference, is generally used.

Next, we will explain important terms and their meanings that frequently appear in waveform analysis using FFT, in relation to waveform representation.

If you speak towards a mountain, you will hear an echo after a while. Let's consider how the time difference between speaking and the echo returning can be represented by a phase difference.

If the period of the waveform is T (frequency f), the time difference is τ, and the phase is φ, then:

(Formula 2-9)

For example, in the waveform of "A" mentioned earlier, if the frequency of "_A1" is 106 Hz and the time difference τ until the echo returns is 0.001 seconds, then the phase will be delayed by approximately 0.21π = 38.1 degrees. In the case of "_A2", even with the same time difference τ = 0.001 seconds, if the frequency is 212 Hz, the delay will be 76.3 degrees. Thus, it is important to note that even if the time difference is the same, the phase will also be different if the period T (frequency) is different.

It is also important to understand the difference between electrical angle and mechanical angle in relation to amplitude, frequency, and phase. Let's consider a gear with 60 teeth to which an electromagnetic detector is attached, and the gear rotates at 600 revolutions per minute (r/min). When the gear rotates once, the electromagnetic detector outputs a signal equivalent to 60 cycles of a sine wave. Here, the phase expressed by taking one period of the signal sine wave as 2π rad (360 degrees) is called the electrical angle, and the phase expressed by taking one rotation of the gear as 2π rad is called the mechanical angle.

As shown in Figure 2-6;

Signal sine wave frequency f = 600 r/min / 60s × 60 Gear teeth = 600 (Hz)

The frequency of one gear rotation is _f0 = 600 r/min / 60s = 10 (Hz)

This is the result. The signal sine wave and the period of a gear are both expressed as the same angle, 2π rad, but the time is:

Detector signal period = Electrical angle display period = 2π = 1/600 (seconds)

The period of one gear rotation = the period of the machine angle display = 2π = 1/10 (seconds)

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 involving rotating bodies, it is necessary to evaluate frequency analysis data while paying attention to the relationship between the electrical angle and the mechanical angle. FFT analyzers use the electrical angle.

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

The term "harmonic" refers to frequencies that are integer multiples of the fundamental frequency. For example, if the fundamental frequency of "A" (" _A1") is 106 Hz, then the frequency of its second harmonic (" _A2") is 212 Hz. Recently, analyses of harmonic distortion in power supplies have been conducted, which also involve analyzing the amplitude and phase at different frequencies, using 50 Hz or 60 Hz as the fundamental power supply frequency, and determining the extent of the harmonic components at integer multiples of that frequency.

Let's consider the sum (composition) of the following waveforms.

(Formula 2-10)

As shown in Figure 2-7, ③ and ④, the combined waveforms will have the same frequency but different amplitudes and phases.

First, let's consider _x². _x² can be found using trigonometric formulas;

(Formula 2-11)

This means that when a cosine wave and a sine wave with the same frequency and phase are combined, the result is identical to a cosine wave with √2 times the amplitude and a phase lag of π/4. This can also be extended as follows:

(Formula 2-12)

If we plot the coefficient a on the cos axis and b on the sine axis, we get Figure 2-8.

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

(Formula 2-13)

This means:

(Formula 2-14)

This holds true. Thus, a single-frequency waveform (hereinafter referred to as a simple waveform) can be defined by setting coefficients a and b at the same frequency.It can be expressed by an equation, and if we know the coefficients a and b in this case, we can find the magnitude r of the combined amplitude and the phase difference φ. Similarly, the combined waveform of x ₁ is,Since the coefficients a = 2 and b = 0 can be considered, the magnitude of the combined amplitude r = 2 and the phase difference φ = 0. The individual components "_A1", "_A2", "_A3", etc. obtained by frequency analysis of the sound "A" are simple waveforms, so each waveform is,This will be expressed as follows. Note that the reference for the phase difference φ here isPlease note that this is the case.

Simple waveforms areWe explained how this can be expressed as follows. In this section, we will explain how to find the coefficients a and b of this equation.

Let's consider the product of the following waveforms.

(Formula 2-15)

Figure 2-9 below shows the waveforms when _a1 = 2, _a2 = 3, and b = 1.

As shown in this diagram, the product of periodic waveforms results in a similarly periodic waveform. Let's look at the area obtained by integrating the waveform _x1 over one period (from 0 to 2π). In the waveform _x1, the area on the positive side ● (yellow circle) and the area on the negative side ● (red circle) are the same symmetrically with respect to 0, so the total area is zero. _x2 represents the product with the second harmonic component, and the total area of this waveform is similarly zero. However, _x3 has no negative side, so the total area = amplitude a × T/2.

In other words:

(Formula 2-16)

This means that by multiplying any waveform by (n = 1, 2, 3, ...), finding the area of one period, and dividing by T/2, the original components can be extracted, and their amplitude an can be determined.By multiplying by , we can find the area of one period, and then by dividing that value by T/2, we can find the amplitude _bn.