This measurement column addresses frequently asked questions received by our customer support center and provides answers to those questions.
When measuring the magnitude of sound and vibration with a sound level meter or vibration level meter, or when performing real-time octave analysis (RTA analysis) with an analysis device, one of the setting parameters is the "time constant."
The instantaneous waveforms of sound and vibration fluctuate rapidly (both positive and negative), so it is not possible to determine the magnitude of sound or vibration from the instantaneous waveform itself. To evaluate the magnitude of sound or vibration, the "root mean square value (RMS)" is calculated from the instantaneous waveform, and then the "RMS" is converted to a level (decibels) and evaluated.
There are several methods for determining the "effective value" from an instantaneous waveform, and one of them is to use an effective value detection characteristic circuit. The "time constant" is one of the parameters that define the characteristics of this circuit.
What is the effective value?
The effective value, also known as the square root of the mean square, is the value obtained by averaging the squared values of a time waveform over a certain period of time and taking the square root of the mean value.
If the time waveform is x(t) and the mean time is T, the effective value X is given by Equation 1.
(Equation 1)
If the data sampled at the sampling interval τ is given by x i = x(iτ) and the number of data points N = T/τ, the effective value can be calculated using Equation 2.
(Equation 2)
When evaluating the magnitude of sound or vibration, the obtained RMS value is converted to a level (decibels),
The value L is evaluated in dB units. Conversion to decibels is done using Equation 3 or Equation 4. Effective
Equation 3 is the formula for calculating the decibel value from the value X. However, in Equations 1 and 2, the square root is taken to obtain the RMS value.
We are looking for the squared value of the effective value before taking the square root X 2 It is better to use Equation 4 to find it.
The calculation becomes easier. Here, X 0 This is the dB reference value, which is 20 μPa for sound. (Evaluating vibration)
The dB reference value for this varies depending on the standard.
(Equation 3)
(Equation 4)
When determining A-weighted and C-weighted sound pressure levels for sound, or vibration levels for vibration, the effective value and decibel value are calculated using equations 1-4 on the time waveform obtained by applying filters with the respective frequency weighting characteristics to the sound and vibration signals. Note that details of the frequency weighting characteristics will be omitted in this measurement column.
Calculation of effective value
If you have a time waveform obtained by sampling (A/D conversion) a signal such as sound or vibration, you can calculate the RMS value from Equation 2. However, if you calculate the RMS value from the entire time waveform, you will know the RMS value of the entire waveform, but you will not know how the RMS value changes over time.
One simple method that comes to mind for determining the time variation of the RMS value is (1) time division. This method involves dividing the time waveform into, for example, 125 ms intervals, calculating the RMS value from the 125 ms time waveform, and plotting it on the horizontal axis with time.
(2) is a method for determining the RMS value using the RMS detection characteristic circuit described later. This method is only an approximation, but it is widely used in practice. The "time constant" is a parameter that determines the characteristics of this circuit. Common values for the time constant include 125 ms for measuring sound pressure levels and 630 ms for measuring vibration levels.
Figure 1 shows the results of determining the time variation of the RMS value from the time waveform of the symphony sound using two methods: (1) a time division method and (2) a method using an RMS detection characteristic circuit. The time division was set to 125ms intervals, and the time constant of the characteristic circuit was set to 125ms.
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Figure 1. Time waveform and RMS value of symphony sound.
Top panel: Time waveform
Middle section: (1) Effective value obtained by time division method (division interval: 125 ms)
Bottom row: (2) RMS value obtained from the RMS detection characteristic circuit (time constant: 125 ms)
In method (1), the time waveform is sampled (A/D conversion) to obtain numerical data, and the RMS value is calculated from that numerical data. This is a simple process now that A/D converters have appeared and digital signal processing has become widespread, but to implement this process using only analog circuits would require a fairly complex circuit.
The method using the RMS detection characteristic circuit described in (2) is an approximate method, but it is widely used because it can be easily implemented with analog circuits using resistors (R), capacitors (C), etc. Even when calculating from numerical data obtained by sampling (A/D conversion) a time waveform, the method of determining the RMS value by performing numerical calculations equivalent to the RMS detection characteristic circuit is widely used because it is a method specified in the standard, it is compatible with the results measured with analog circuits, and it matches human intuition better.
RC series circuit
The basic circuit for the RMS detection characteristic circuit is an RC series circuit consisting of a resistor (R) and a capacitor (C).
This is a circuit with the following connected in series (Figure 2). Here, τ = RC is the value called the time constant of this circuit.
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Figure 2 RC series circuit
Figure 3 shows the response waveform of an RC series circuit when a single-period square wave pulse is input to it. After a time constant τ [seconds] following the pulse input, the value becomes approximately 0.63 and approaches 1 exponentially. After the pulse is interrupted, the value becomes approximately 0.37 after τ [seconds] and approaches 0 exponentially.
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Figure 3. Response waveform (red) of an RC series circuit to a rectangular wave pulse (blue).
RMS detection characteristic circuit
By combining the aforementioned RC series circuit with a circuit that squares the time waveform, the RMS value of the input signal can be obtained.
You can create circuits that exert power.
The process involves using an RMS detection characteristic circuit to determine the RMS value and converting it to a level (decibels).
The lock diagram is shown in Figure 4. When the time waveform of sound or signal is squared and passed through an RC series circuit, the effective value is obtained.
The squared value is obtained. In Figure 4, the squared value of the RMS is fed into a logarithmic arithmetic circuit and converted to a level (decib).
(Converting to a value)
To find the RMS value using an RMS detection characteristic circuit, take the square root of the output of the RC series circuit.
It is necessary. However, when converting to a level using a logarithmic arithmetic circuit, do not take the square root and use the RMS value 2
Since inputting the squared value directly into the logarithmic calculation circuit simplifies the circuit, the squared value is used directly in the logarithmic calculation circuit.
Input into the numerical calculation circuit.
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Figure 4: RMS detection wave characteristic circuit and logarithmic calculation circuit
RMS detection characteristic circuit and output waveform
Figure 5 shows the output waveform of the RMS detection characteristic circuit for a 25 Hz sine wave. The display range is 0.5
This is in seconds. The top panel shows a sine wave waveform at 25 Hz with an amplitude of 1 Pa, the middle panel shows its squared value, and the bottom panel shows the RMS value detection.
This is the output of the dynamic characteristics circuit, representing the square of the RMS value. The time constant of the dynamic characteristics circuit was set to 125 ms.
Figure 5: RMS detection characteristic circuit and output waveform
Top row: 25Hz sine wave, Middle row: Sine wave squared, Bottom row: RMS squared
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Figure 5: RMS detection characteristic circuit and output waveform
Top row: 25Hz sine wave, Middle row: Sine wave squared, Bottom row: RMS squared
The calculation of the RMS value using the RMS detection characteristic circuit is an approximate method. The input signal frequency is low.
The resulting RMS value will fluctuate slightly. In Figure 5, the input waveform has a single amplitude of 1 Pa,
The effective value is 0.707 Pa, and the square of the effective value is 0.5 Pa. 2 The sound pressure level should be 90.97 dB.
The output waveform is approximately 0.487-0.513 Pa 2 It fluctuates within this range. When converted to sound pressure level values, it fluctuates.
The range is 90.86–91.08 dB.
This fluctuation becomes smaller as the frequency of the input signal increases. The example in Figure 5 is a very low-pitched sound.
This is an example using a sine wave with a frequency of 25 Hz. Normal sounds have much higher frequency components.
Therefore, fluctuations in the values obtained by the dynamic characteristic circuit are not much of a problem.
Furthermore, increasing the time constant of the dynamic characteristic circuit reduces the fluctuations. For example in Figure 5, the time constant
When the number is set to 1 s, the variation in sound pressure level is 90.88 to 90.93 dB, compared to the case of 125 ms.
It has become smaller.
summary
This time, we will discuss measuring the magnitude of sound and vibration using sound level meters and vibration level meters, and how to analyze them using rear-view devices.
The RMS detection characteristic circuit used when performing time-to-time octave analysis (RTA analysis),
I introduced one of those setting parameters, the "time constant."
Furthermore, the method of determining the RMS value and decibel value of a time waveform using an RMS detection characteristic circuit is an approximation.
We will also explain that if the signal frequency is low, the RMS value and decibel value may fluctuate.
Yes, this fluctuation decreases as the time constant of the dynamic characteristic circuit is increased.
In this instance, we introduced the parameter of the RMS detection characteristic circuit as the "time constant,"
In standards for sound, vibration, and other similar phenomena, terms such as time-weighted characteristics and dynamic characteristics are used.
(Excerpt from the email newsletter issued on June 21, 2017)
