Following on from our previous article, "On the RMS Value and Power of Signals," we will address another frequently asked question from FFT analyzer users.
Why are the mean square or RMS values used to represent the magnitude (strength) of a time signal? Also, why is the power average used as the average?
Time signals from physical sensors such as sound and vibration are AC signals, and their amplitude, which represents their magnitude (strength), is a waveform that fluctuates instantaneously. Therefore, the magnitude (strength) of the signal is usually determined by time averaging. If you simply average an AC signal that does not have a DC component, the result will be zero, and it will not be possible to calculate the magnitude. Therefore, the averaging methods usually involve taking the absolute value and averaging it (hereinafter referred to as absolute value averaging), or squaring the signal, averaging it, and then taking the square root (hereinafter referred to as RMS value). The most basic AC signal is a sine wave, so we will try to calculate this using a sine signal as an example.
First, consider a sine wave with fundamental period T (frequency f = 1/T);
.................................（1）
Expressed as such, it can be easily calculated as follows:
Absolute mean
RMS value
[Note] Regarding equation (1), the definition formula for finding the average of the two methods is as follows:
Absolute mean
................................（2）
RMS value
.................................（3）
In this case, either method can be used to calculate the average value that corresponds to (is proportional to) the amplitude A. Alternatively, you can simply use the amplitude A itself as the representative value without needing to calculate the average.
Next, consider two combined sine waves as shown in equation (4) below.
................................（4）
Since it is difficult to analytically calculate the absolute mean of equation (4), we will try to perform a numerical calculation using specific values in a spreadsheet program.
(4) We will calculate an example where the frequency and phase are changed by setting A1 = 3 and A2 = 2.
Example 1: "When the frequency is changed (assuming ϕ = 0°)"
① When f1 = 1 and f2 = 2 (Figure 1)
Absolute mean 2.220
RMS value 2.550
-
Figure 1
② When f1 = 1 and f2 = 3 (Figure 2)
Absolute mean 1.869
RMS value 2.550
-
Figure 2
Example 2: "When the phase is changed (assuming f1 = 1 and f2 = 2)"
① When ϕ = 0° (Figure 1)
Absolute mean 2.220
RMS value 2.550
-
Figure 1
②When ϕ=90° (Figure 3)
Absolute mean 1.988
RMS value 2.550
-
Figure 3
This example shows that the absolute mean of a composite time signal of two single sine waves with constant amplitude changes significantly depending on the frequency and phase in equation (4).
In contrast, the RMS value is constant and does not depend on the frequency or phase of the combined components. The RMS value can be easily calculated using equation (3).
From equation (3), the effective value is:
........................（5）
And it can be calculated from the amplitude alone.
The calculation method involves taking the square root of the sum of the squares of the RMS values of each sine wave, which matches the calculation result above. Thus, the RMS value of the composite wave is a quantity that depends only on the amplitude of the sine waves contained in the signal at any given time, making it suitable for representing the magnitude (strength) of the signal.
Next, let's consider the physical meaning of the effective value.
In the field of electricity, magnitude is expressed in terms of RMS (Range Value), such as a 100V AC power supply. In this example, 100V represents the RMS value (please refer to the measurement column in the reference materials for an explanation of RMS values).
Generally, when an AC signal with an effective value of V (volts) is applied to a resistor R, the energy is consumed as heat, and the power consumed is V² /R (in watts: W). In other words, the power consumed is proportional to the square of the effective value (this is called the mean square).
Here is an example of calculating power from the effective value.
-
Figure 4 Power of the composite waveform
In the figure on the right (Figure 4):
....... （6）
Therefore, the power P3 of the third stage composite signal is the sum of the powers of the first stage P1 and the second stage P2;
....... （7）
Thus, the power of the combined waveform of the sinusoidal waves of effective values V1 and V2 can be calculated from the sum of the mean squares, just as in equation (6).
Up until now, we've been using the combined waveform of two sine waves, but the same principle holds true for the combined waveform of more sine waves.
In general, any time signal x(t) with a fundamental period T (fundamental frequency f0 = 1/T) can be expressed using the Fourier series expansion as follows:
.....................（8）
【Note】
(1) Strictly speaking, it should be a sum up to infinity ∞, but here we are using a series expansion up to a finite number N.
(2) The right-hand side is described as a combination of a sine wave and a cosine wave,
It can also be written as follows, so it is equivalent to the synthesis of only sine waves.
(8) If we square both sides and take the average, the right-hand side can be easily calculated due to the orthogonality of trigonometric functions;
..................... （9）
The above relationship is called Parseval's theorem.
It will be.
The left side of equation (9) is the mean square (square of the RMS value) itself, and the right side represents the sum of the squares of the RMS values of each sine wave. In other words, the RMS value of any time waveform can be calculated solely from the RMS values of the sine waves that constitute it.
Now, equations (8) and (9) are very important relationships that show the relationship between the time axis (left side) and the frequency axis (right side), so I will explain them again, although I mentioned them two lessons ago.
In equation (9), the value on the left side, the mean square, is called the power of the time signal, especially in the field of signal processing. This is because, of course, it is a value proportional to the power in the field of electricity, as explained earlier. Next, the right side of equation (9) represents the power of the sine waves obtained by decomposing the time signal into frequency components, and this is called the power spectrum.
In other words, equation (9) means:
Power of the time signal = Σ (Power of each sinusoidal component)
In an FFT analyzer, the right-hand side of equation (9) is called the overall value (OA value).
These explanations explain why the "RMS value" is used to represent the magnitude (strength) of a signal at any given time, and why it can be applied to even the most complex waveforms. Similarly, the magnitude (strength) of AC signals such as sound and vibration is also expressed in terms of power, and is averaged (overall average or time average) in terms of power level.
The "power spectrum averaging" commonly used in FFT analyzers is also calculated on power values. If we let the nth spectral value be denoted by the right-hand side of equation (9), then the m-time average of the n spectral values is:
................... （10）
This can be calculated as follows. By taking the square root of equation (10), the linear value, or effective value, can be obtained.
The effective value is abbreviated as RMS (Root Mean Square), which is why the summation average of the power spectrum is also called the RMS average.
The following link will take you to our company's website.
Measurement Column: Fundamentals of Digital Measurement - Part 3: "Time Waveforms and RMS Values"
"Signal Processing" by Iwao Morishita and Hidefumi Obata, published by the Society of Instrument and Control Systems.
(Excerpt from the email newsletter issued on November 25, 2011)
