Mean squared
RMS value
When actually trying to find the RMS value of a signal using equation ② (or equation ①), the problem arises of how to determine the average time T. If the time signal x(t) is a periodic signal, the average time is calculated as its period or an integer multiple of that periodic time. For the most basic periodic function, a sine wave:
Therefore, the effective value can be calculated from equation ② as a/√2. Here, a is called the amplitude (or single amplitude) of the sine wave. In simple terms, it is the peak value (or maximum value) of the sine wave. In other words, the effective value of a sine wave is approximately 0.707 times its amplitude, and conversely, the amplitude of a sine wave is approximately 1.414 times its effective value. Incidentally, the ratio of the maximum amplitude to the effective value is called the crest factor, and for a sine wave, it is 1.414.
Calculating equation ② in analog circuits was extremely difficult. Therefore, in the days before digital signal processing was common, testers and multimeters determined the amplitude using absolute-mean detection and then calculated the equivalent RMS value. Nowadays, it is of course common to calculate it using equation ②, but because it differs from the previous method, it is sometimes called the true RMS value.
Next, we will specifically look at how to calculate the RMS values of various time signals based on sine waves. (The following are examples of calculations performed using spreadsheet software.)
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Figure 1. Weights and RMS values of the straight line (0.58)
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Figure 2. Weights of the Hanning window, effective value (0.61)
The graph in Figure 1 shows an example where the amplitude of a sine wave with an effective value of 1 (i.e., amplitude of 1.414) increases linearly, and the average is calculated over 16 periods, which is considered a large period. The effective value is approximately 0.58. The graph in Figure 2 shows an example where the Hanning window, a time window function commonly used in FFT analyzers, is applied to 16 periods of the same sine wave with an effective value of 1. The effective value of this time waveform is approximately 0.61. This value is almost equal to the square root of 3/8, confirming that the reduction in signal power due to applying the Hanning window is 3/8.
In the two examples above, the amplitude of a sine wave at a constant frequency was changed (modulated). However, the graph in Figure 3 shows an example where two sine wave signals with different frequencies and amplitudes (effective values of 1 and 2) are combined (added), resulting in a time waveform with a maximum amplitude of approximately 4.2. As can be easily inferred, the effective value of the combined waveform can be easily calculated from the individual effective values, but it is not 3 (=1+2). It is simply a summation of power. When calculated, the effective value is approximately 2.2, which can be seen as the square root of 5 (=12+22).
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Figure 3. Combination of two sine waves: RMS value (2.2)
So far, we've discussed cases where the period of a periodic function is known in advance. But what if the period of the periodic function is unknown? How do we determine the average time in equation ②?
Two methods are possible. One is to use a sufficiently long average time that includes as many periods as possible, even if the average time is not an integer multiple of the period of the periodic function. The other is to use a window function used in FFT analyzers. In this method, as explained in the example above, the error can be reduced by applying a Hanning window function to the average time and then correcting for the power reduction afterward.
Furthermore, if the time function we want to find is not a periodic function, how should we determine the average time? Roughly speaking, when classifying real-world signals, they can be classified into periodic signals, transient signals, and continuous signals (with infinite period), such as random signals.
In the case of transient signals like those shown in Figure 4, it is not possible to precisely determine the averaging time. Since the signal is temporally finite and unevenly distributed, the value of equation ② changes depending on the averaging time. The magnitude (strength) of a transient signal is generally expressed as an unaveraged quantity, i.e., energy.
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Figure 4
We define this as the energy of the transient signal. The integration interval T is the time the signal exists. For random signals without periodicity, we need to take an infinity average. For random signals, equation ① is:
It is defined as follows. In reality, this will be an approximation by averaging over a finite length of time.
To summarize regarding the effective value of the time waveform:
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It corresponds to the square root of the signal's power.
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The method for finding it is:
Squaring → Addition → Average → Square Root
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For time-waveforms where amplitude fluctuates over time, the power is averaged over its average time. The relationship between the maximum amplitude (peak value) and the RMS value (i.e., crest factor) of a time-waveform varies depending on the signal.
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The RMS value of a composite waveform is the square root of the sum of the power values, not the sum of the RMS values.
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For transient signals, the energy is defined as the energy that is not averaged over the integration interval.
(Excerpt from the email newsletter issued on January 24, 2008)
