PERIODIC FUNCTION:
A function f(t) is said to be periodic if,
f(t+T)=f(t); for all values of t
where T is some positive real number. This T is interval between two successive repetitions and is called the period of f(t).
A sine wave having a period of [tex]T = \frac{{2\pi }}{\omega }[/tex] is common example of periodic function.
DIRICHLET CONDITION:
For a given function f(t),
- f(t) is periodic having a period of T.
- f(t) is single valued everywhere.
- In case it is discontinuous, f(t) has a finite number of discontinuities in any one period.
- f(t) has a finite number of maxima and minima in any one period.
- The integral, [tex]T = \frac{{2\pi }}{\omega }[/tex] exists and is finite.
The function f(t) may represent either a voltage or current waveform.
TRIGONOMETRIC FOURIER SERIES:
According to Fourier theorem, a function f(t) which satisfy the Dirichlet condition, may be represented in trigonometric form by the infinite series.
\[\begin{array}{l}
f(t) = {a_0} + {a_1}\cos {\omega _0}t + {a_2}\cos 2{\omega _0}t + .... + {a_n}\cos n{\omega _0}t + {b_1}\sin {\omega _0}t + {b_2}\sin 2{\omega _0}t + .... + {b_n}\sin n{\omega _0}t\\
= {a_0} + \sum\limits_{n = 1}^\infty {\left( {{a_n}\cos n{\omega _0}t + {b_n}\sin n{\omega _0}t} \right)}
\end{array}\]
since [tex]{\omega _0} = \frac{{2\pi }}{T}[/tex] above equation can be written as
[tex]f(t) = {a_0} + \sum\limits_{n = 1}^\infty {\left( {{a_n}\cos \frac{{2\pi n}}{T}t + {b_n}\sin \frac{{2\pi n}}{T}t} \right)} [/tex]
where, [tex]{\omega _0}[/tex] is the fundamental angular frequency, T is the period and [tex]{a_0}[/tex],[tex]{a_n}[/tex] and [tex]{b_n}[/tex] are constant which are depend on n and f(t).
FOURIER ANALYSIS:
Evaluation of fourier constant called fourier analysis.
1. Value of [tex]{a_0}[/tex]:
[tex]f(t) = {a_0} + \sum\limits_{n = 1}^\infty {\left( {{a_n}\cos n\omega t + {b_n}\sin n\omega t} \right)} [/tex]
Integrate over a period t=0 to t=T.
[tex]\int\limits_0^T {f(t)} dt = \int\limits_0^T {{a_0}} dt + \sum\limits_{n = 1}^\infty {\int\limits_0^T {({a_n}\cos n\omega t + {b_n}\sin n\omega t} } )dt = {a_0}T[/tex]
Hence, [tex]{a_0} = \frac{1}{T}\int\limits_0^T {f(t)} dt[/tex]
= mean value of f(t) between the limits 0 to T i.e. over one cycle or period.
2. Value of [tex]{a_n}[/tex]:
[tex]f(t) = {a_0} + \sum\limits_{n = 1}^\infty {\left( {{a_n}\cos n\omega t + {b_n}\sin n\omega t} \right)} [/tex]
multiply both sides of the fourier series by [tex]\cos k\omega t[/tex] and integrate between limits t=0 to t=T.
[tex]\int\limits_0^T {f(t)\cos k\omega t} dt = \int\limits_0^T {{a_0}\cos k\omega t} dt + \sum\limits_{n = 1}^\infty {\int\limits_0^T {({a_n}\cos n\omega t\cos k\omega t + {b_n}\sin n\omega t\cos k\omega t} } )dt[/tex]
=[tex]{a_k}\frac{T}{2}[/tex]
Therefore,
[tex]{a_k} = \frac{2}{T}\int\limits_0^T {f(t)\cos k\omega t} dt[/tex]
3. Value of [tex]{b_n}[/tex]:
[tex]f(t) = {a_0} + \sum\limits_{n = 1}^\infty {\left( {{a_n}\cos n\omega t + {b_n}\sin n\omega t} \right)} [/tex]
multiply both sides of the fourier series by [tex]\sin k\omega t[/tex] and integrate between limits t=0 to t=T.
[tex]\int\limits_0^T {f(t)\sin k\omega t} dt = \int\limits_0^T {{a_0}\sin k\omega t} dt + \sum\limits_{n = 1}^\infty {\int\limits_0^T {({a_n}\cos n\omega t\sin k\omega t + {b_n}\sin n\omega t\sin k\omega t} } )dt[/tex]
=[tex]{b_k}\frac{T}{2}[/tex]
Therefore,
[tex]{b_k} = \frac{2}{T}\int\limits_0^T {f(t)\sin k\omega t} dt[/tex]
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