الاثنين، 5 أكتوبر 2020

Fourier series.

 Fourier series.

The Fourier series is named after Jean Baptiste Joseph Fourier (1768–1830). In 1822, Fourier’s genius came up with the insight that any practical periodic function can be represented as a sum of sinusoids. Such a representation, along with the superposition theorem, allows us to find the response of circuits to arbitrary periodic inputs using phasor techniques.

Fourier discovered that a nonsinusoidal periodic function can be expressed as an infinite sum of sinusoidal functions. Recall that a periodic function is one that repeats every T seconds. In other words, a periodic function f (t) satisfies

                                       

 

Where n is an integer and T is the period of the function.

According to the Fourier theorem, any practical periodic function of frequency ω0 can be expressed as an infinite sum of sine or cosine functions that are integral multiples of ω0. Thus, f (t) can be expressed

                            

                             

, Where ω0 = 2π/T is called the fundamental frequency in radians per second. The sinusoid sinnω0t or cosnω0t is called the nth harmonic of f (t); it is an odd harmonic if n is odd and an even harmonic if n is even.

The Fourier series of a periodic function f (t) is a representation that resolves f (t) in to a dc component and an ac component comprising an infinite series of harmonic sinusoids.

And to find a0, an, bn:

                                                   

                                                 

                                                 

The Fourier series allows us to model any arbitrary periodic signal with a combination of sines and cosines so we can us tis source in any electrical application and we find that in practice, many circuits are driven by nonsinusoidal periodic functions. To find the steady-state response of a circuit to a nonsinusoidal periodic excitation requires the application of a Fourier series, ac phasor analysis, and the superposition principle so Fourier series is important.

4. Fourier series.

The Fourier series is named after Jean Baptiste Joseph Fourier (1768–1830). In 1822, Fourier’s genius came up with the insight that any practical periodic function can be represented as a sum of sinusoids. Such a representation, along with the superposition theorem, allows us to find the response of circuits to arbitrary periodic inputs using phasor techniques.

Fourier discovered that a nonsinusoidal periodic function can be expressed as an infinite sum of sinusoidal functions. Recall that a periodic function is one that repeats every T seconds. In other words, a periodic function f (t) satisfies

                                       

 

Where n is an integer and T is the period of the function.

According to the Fourier theorem, any practical periodic function of frequency ω0 can be expressed as an infinite sum of sine or cosine functions that are integral multiples of ω0. Thus, f (t) can be expressed

                            

                             

, Where ω0 = 2π/T is called the fundamental frequency in radians per second. The sinusoid sinnω0t or cosnω0t is called the nth harmonic of f (t); it is an odd harmonic if n is odd and an even harmonic if n is even.

The Fourier series of a periodic function f (t) is a representation that resolves f (t) in to a dc component and an ac component comprising an infinite series of harmonic sinusoids.

And to find a0, an, bn:

                                                   

                                                 

                                                 

The Fourier series allows us to model any arbitrary periodic signal with a combination of sines and cosines so we can us tis source in any electrical application and we find that in practice, many circuits are driven by nonsinusoidal periodic functions. To find the steady-state response of a circuit to a nonsinusoidal periodic excitation requires the application of a Fourier series, ac phasor analysis, and the superposition principle so Fourier series is important.

 References.

[1] Charles Alexander, “fundamental of electrical circuit 2nd edition”.

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