We've discussed transfer functions over the past three installments, but this time we'll talk about a different topic: "power and energy." In the fields of acoustics and vibration, terms like power summation, energy summation, power averaging, and energy averaging are often used interchangeably. Here, we'll discuss qualitatively how these two concepts are used differently in the fields of electrical engineering, mechanical engineering, acoustics, and signal processing.
Physically, when a physical system has the ability to do work on other things, this is called the energy of the system. The ability to do work per unit time is called power, and since it is the time derivative of energy, it is also called power. The SI units are J (joule) for energy and W (watt) for power.
In electrical systems, power is, of course, electrical power (unit: W), and energy is electrical energy (practical unit: kWh), which is the integral of electrical power.
The output of rotating bodies such as engines and motors is also called power (work rate), and is commonly referred to as horsepower, with the unit being horsepower (ps), but recently it has been measured in kilowatts (kW). The output (power) of a rotating body can be calculated as the product of torque (rotational force) and rotational speed.
Batteries, such as lithium batteries, have specifications such as energy density (how much electrical energy can be stored per kilogram) and power density (how much power can be output per kilogram). When installed in a car, energy density is an important factor in extending the driving range of an EV, while power density is an important factor in acceleration and other performance aspects.
In acoustic systems, acoustic power is defined as the sound energy radiated from a sound source per unit time, and its unit is watts (W). If acoustic power is denoted as P (W), then its level is called the acoustic power level (L W).
.................................（1）
This is the result. Here, P0 is the reference value of 10⁻¹² (W). Normally, acoustic power is applied to steady-state sound sources as a time-averaged value. Therefore, it cannot be applied to sound sources such as sudden, impactful sounds, so acoustic energy (in units of J), which is the time integral of acoustic power, is used. If acoustic energy is in J (J), then the level obtained by converting it to a level is called the acoustic energy level (L J);
.................................（2）
This is the result. Here, J0 is the reference value, which is 10⁻¹² (J).
Now, in the field of signal processing, the power of a time signal x(t) is the mean square of the signal;
.................................（3）
It is defined as follows. When x(t) is a periodic function, infinity calculations are unnecessary. Also, the energy of the time signal x(t) is:
.................................（4）
It will be.
(Note)
The energy in equation (4) diverges for infinitely continuing signals such as continuous signals, but it is meaningful for finite signals such as transient signals. That is, energy is the total integral of the squared value of the signal, and power is the energy per unit time. If the unit of the time signal x (t) is EU, then the unit of energy is EU² s and the unit of power is EU².
Next, let's consider the frequency analysis of a time signal x(t). When the power or energy of a signal is decomposed into a certain frequency band and its distribution is obtained, it is called the power spectrum (unit: EU2) and the energy spectrum (unit: EU2 s), respectively. Furthermore, even for the same time signal, the spectral value changes depending on the bandwidth being analyzed, so sometimes the analysis is performed per unit frequency to use the power spectral density (PSD) and energy spectral density (ESD). The units for these are EU2/Hz for PSD and EU2 s/Hz for ESD.
Thus, in the field of signal processing, power and energy do not have a physical meaning, unlike in the fields mentioned earlier; they are simply quantities that are processed with dimensions equal to the square of the signal.
Actual operations such as addition, subtraction, averaging, and overall calculations performed by FFT analyzers and octave band analyzers all involve calculations using quantities with dimensions equal to the square of the time signal. In this sense, power addition and energy addition, and power averaging and energy averaging can be said to refer to the same thing.
【keyword】
Energy, power, work rate, joule, watt, electric power, energy, horsepower, ps, output, torque, rotational speed, energy density, power density, acoustic power, acoustic power level, acoustic energy, acoustic energy level, mean square, power spectrum, energy spectrum, power spectral density, ESD, energy spectral density, ESD
[Reference materials]
- "Spectral Analysis," by Mikio Hino, Asakura Shoten (1977)
- "Dictionary of Noise Terminology," edited by the Japan Society of Noise Control Engineering, Gihodo Publishing (2010).
(Excerpt from the email newsletter issued on March 19, 2015)
