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Trigonometric waveforms
Published in John Bird, Bird's Higher Engineering Mathematics, 2021
Amplitude is the name given to the maximum or peak value of a sine wave. Each of the graphs shown in Figs. 11.12 to 11.15 has an amplitude of +1 (i.e. they oscillate between +1 and −1). However, if y=4sinA, each of the values in the table is multiplied by 4 and the maximum value, and thus amplitude, is 4. Similarly, if y=5cos2A, the amplitude is 5 and the period is 360∘/2, i.e. 180∘.
Noise and vibration
Published in Sue Reed, Dino Pisaniello, Geza Benke, Kerrie Burton, Principles of Occupational Health & Hygiene, 2020
As sound is transmitted through the elastic medium of air, the air is compressed and rarefied to form a pressure wave like the ripples that appear on a pond when a pebble is thrown in the water. The number of pressure variations per second is called the frequency of sound and is measured in Hertz (Hz) (see Figure 12.1). The wavelength is the distance between two similar points on the sine (curved, peak-and-trough) wave. The velocity of sound (wavelength × frequency) depends on the mass and elasticity of the conducting medium. In air, sound propagates at about 344 m/s at 20°C. In water, sound propagates at about 1500 m/s, and through steel it propagates at about 6000 m/s. The spectrum of good human hearing ranges between about 20 and 20 000 Hz, and everyday sounds contain a wide mixture of frequencies. Speech communications rely on frequencies ranging between 100 and 5000 Hz. Audible sound pressure variations are superimposed on the atmospheric air pressure (about 100 000 Pa) and normally range between 20 μPa and 100 Pa.
The Frequency Domain: Bode Analysis
Published in Richard J. Jagacinski, John M. Flach, Control Theory for Humans, 2018
Richard J. Jagacinski, John M. Flach
This is a complex number in exponential form. This makes it simple to determine the magnitude (1) and the argument (−aω). Note the magnitude does not depend on frequency. The amplitude of the output from a pure time delay will be equal to the amplitude of the input, independent of frequency. The phase response does depend on frequency. Phase lag will be directly proportional to frequency. The proportionality constant, a, is the magnitude of the delay in seconds. This makes good sense because the phase response is expressed in proportions of the sinusoidal cycle, as noted earlier. The frequency of a sine wave determines how many cycles are completed in a fixed time (e.g., cycles per second). A faster (higher frequency) sine wave will be a larger portion of the cycle behind the input after waiting a seconds than a slower (lower frequency) sine wave. This is illustrated in Fig. 13.2.
PTF-based control algorithm for three-phase interleaved inverter-based SAPF
Published in International Journal of Electronics, 2019
Vijayakumar Gali, Nitin Gupta, R. A. Gupta
The above error differential equation has (2N+1) roots, called PTF poles. These PTF poles located in the left side of the s-plane. Therefore, the magnitude of the individual error components will vanish exponentially. The sum of output variables of the various blocks of the PTF, i.e., is unify with the dc component and N-1 harmonic components. Therefore, the elicited harmonics are sine waves with different frequencies. The individual block would be controlled and tuned to various harmonic components. The harmonics are fixed at (p*f) Hz, p = 2, 3 … N. The PTF will be tracking these harmonic signals. The parameters are selected after observing the steady-state behaviour of the supply fundamental voltage, which are fully controllable and observable. The block diagram of PTF, as shown in Figure 3, is used for the extraction of fundamental frequency voltage signal from the distorted three-phase supply voltages. To obtain the closed-loop transfer function of PTF, one has to use mason’s gain formula as follows:
Influence of surface distresses on smartphone-based pavement roughness evaluation
Published in International Journal of Pavement Engineering, 2021
L. Janani, V. Sunitha, Samson Mathew
Generally, Fourier series represents the approximation or expansion of a periodic function, and it makes use of the orthogonality relationships of the sine and cosine functions. The general formulae for Fourier Series are shown from Equations (13)–(16).
Scalar speed control of induction motor with curve-fitting method
Published in Automatika, 2022
Özcan Otkun, Faruk Demir, Selçuk Otkun
The Fourier Series is the trigonometric expression of functions in terms of sine and cosine functions. Most of the single-valued functions that occur in practice can be expressed as a Fourier series in terms of sines and cosines.