Essay about Fourier Transform in Power System Relaying

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Contents
Introduction 3
Fourier Series, Continuous Transform and Discreet Transform 3 it should be noted that the coefficients in the equations above are given as follows. 3
Application of DFT in power system relaying 7 10
Conclusion 10
References 10

Introduction
The use of digital computers for power system relaying has been proposed long time ago in [1]. Discrete Fourier transform (DFT) was one of the first algorithms that have been proposed to be used in digital relaying. DFT has been the focus of many researched due to its simplicity and its relevant properties. Variations of the DFT have been proposed. This includes the short window, long window and the symmetrical component DFT [2]. DFT has many advantages compared
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Given any function that’s periodic as the one given below in figure [1] , Fourier series decomposes the function into an infinite sum of trigonometric functions given in equation f(θ)=a_o+∑_(k=1)^∞▒〖a_k cos⁡kθ 〗+∑_(k=1)^∞▒〖b_k sin⁡kθ 〗
In practical applications the counter cannot be taken to infinity but an upper bound of error is taken so that the series can be truncated after certain number of terms. Let the upper error bound be ε and let the index after which the series can be truncated is m. Let’s define a partial sum S_n≜ a_o+∑_(k=1)^n▒〖a_k cos⁡kθ 〗+∑_(k=1)^n▒〖b_k sin⁡kθ 〗
Then the counter is given by the following relation ε ≤|f(θ)-S_m |≤|∑_(k=m)^∞▒〖a_k cos⁡kθ 〗+∑_(k=m)^∞▒〖b_k sin⁡kθ 〗| it should be noted that the coefficients in the equations above are given as follows. a_k=1/π ∫_0^2π▒〖f(x)cos⁡kx □(24&dx)〗 b_k=1/π ∫_0^2π▒〖f(x)sin⁡kx □(24&dx)〗
This is assuming the period of the function is 2π. If the period is more or less, the integral has to be adjusted accordingly.
The more common form of the Fourier series is the exponential form written as f(θ)=∑_(k=-∞)^∞▒〖c_k e^jkθ 〗
Where c_k=1/2π ∫_0^2π▒〖f(θ) e^jkθ □(24&dθ)〗
Again assuming the period is 2π, if not then the intergral has to be adjusted accordingly.
It should be noted that in the case of the Fourier series we can enumerate the coefficients i.e. we can strike a one-to-one correspondence between the set of

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