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) value for evaluation.
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.
RMS detection characteristic circuit and logarithmic calculation circuit
Figure 1 shows a block diagram of the process used in a sound level meter to determine the RMS value from the instantaneous waveform of sound using an RMS detection characteristic circuit, and then convert it to a level (convert it to a decibel value). The square of the RMS value is obtained by squaring the time waveform of sound or a signal and passing it through an RC series circuit. In Figure 1, the square of the RMS value is fed into a logarithmic arithmetic circuit to convert it to a level (convert it to a decibel value).
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Figure 1: RMS detection characteristic circuit and logarithmic calculation circuit
Let's consider applying the same processing to time-series data sampled by an A/D converter. Let x(i) be the instantaneous waveform sampled by the A/D converter, etc. Squaring this is simply the process of finding x(i)². The output y(i) of the RC series circuit (dynamic characteristic circuit) can be found using equation 1. This is the weighted sum of the output y(i-1) of the circuit one sample prior and the squared value of the instantaneous waveform x(i)². The sound pressure level L(i) at time i can be found using equation 2. Since the unit of y(i) is Pa², if we take the square root before calculating, it becomes equation 2'. The calculation results of equation 2 and equation 2' are the same.
𝑦(𝑖)=(1−h0)∙𝑦(𝑖−1)+h0∙𝑥(𝑖)2 Equation 1
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formula 2
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formula 2'
Here, τ is the time constant (s), fs is the sampling frequency (Hz), and p0 is the reference sound pressure value of 20 μPa. h0 is a constant defined by equation 3. When the time constant is 125 ms and the sampling frequency is 48 kHz, h0 will be a small value such as approximately 0.0001667.
Here, τ is the time constant (s), fs is the sampling frequency (Hz), and p0 is the reference sound pressure value of 20 μPa. h0 is a constant defined by equation 3. When the time constant is 125 ms and the sampling frequency is 48 kHz, h0 will be a small value such as approximately 0.0001667.
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formula 3
RC series circuit and its output
The basic circuit for the RMS detection characteristic circuit is an RC series circuit, which is a circuit in which a resistor (R) and a capacitor (C) are connected in series (Figure 2). Here, τ = 𝑅𝐶 is 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 the pulse is applied, the value becomes approximately 0.63 after a time constant τ seconds, 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 square wave pulse (blue).
Calculation of the sound pressure level of a burst signal
Figure 4 shows a burst signal with a duration of 1 second, a frequency of 1 kHz, and a single amplitude of 1.41 Pa (sound pressure level of 94 dB), along with the sound pressure level calculated from it. The top panel shows the instantaneous sound pressure waveform, the middle panel shows the output of the dynamic characteristic circuit (squared sound pressure), and the bottom panel shows the sound pressure level (dB). The time constant is F (125 ms).
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Figure 4. Burst signal and its sound pressure level
The output of the dynamics circuit is almost zero (0.0183 Pa2) about 0.5 seconds after the burst signal stops. The sound pressure level has only dropped by 17.37 dB. However, these are the same values, just viewed in terms of squared sound pressure or decibels. After the burst signal stops, the sound pressure level decreases at a rate of approximately 4.34 dB every 125 ms. It drops by approximately 34.74 dB per second.
Table 1 shows the values of the waveform in Figure 4 at 0.0625-second intervals. A sine wave with a single amplitude of 1.41 Pa (sound pressure level of 94 dB) is applied from second 1 to second 2. As can be seen by comparing with Figure 3, 125 ms (time constant) after the burst signal is applied, the circuit output reaches 0.6320 Pa² (91.99 dB). After 500 ms, it becomes 0.9815 Pa² (93.90 dB), which is almost the same value as the sound pressure level of the signal (94 dB). After the burst signal stops, the circuit output (Pa²) decreases at a rate of 0.3677 times every 125 ms (time constant). Expressed in terms of sound pressure level (dB), this corresponds to a slope of 4.34 dB decrease every 125 ms (time constant).
Table 1 Burst signals and their sound pressure levels
summary
In this article, we introduced a method for calculating the RMS value from an instantaneous waveform using an RMS detection characteristic circuit, and the output value of the circuit when a burst signal is input.
When a rectangular pulse is input to an RC series circuit, the output reaches approximately 0.63 times the pulse amplitude after a time equal to the circuit's time constant. When the pulse is interrupted, the circuit's output decreases at a rate of approximately 0.37 times the pulse amplitude over the same time as the time constant.
The RMS detection characteristic circuits used in sound level meters and other devices also use the same circuitry and therefore exhibit the same characteristics. When viewed in terms of squared sound pressure (Pa²), it decreases by 0.37 times in the same amount of time as the time constant, so it quickly becomes a very small value. Expressed in terms of sound pressure level (dB), it decreases by only 4.34 dB in the same amount of time as the time constant.
Even after a burst signal or impact sound stops, the sound pressure level does not immediately decrease due to the time constant. If the time constant is 125 ms, the sound pressure level will only decrease by 34.74 dB after 1 second. Therefore, to accurately measure the sound pressure of each repeated impact sound, it is necessary to leave an interval of about 0.5 to 1 second between impact sounds. If the measurement can be performed with a shorter time constant than 125 ms, setting the time constant to 10 ms will result in a decrease of 4.34 dB after 10 ms and 43.43 dB after 100 ms, so the sound pressure level will not be affected even if the interval between sounds is short.
(Excerpt from the email newsletter issuedonJuly24,2019)
