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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