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                                                                                                                                                                                      Circuit Analysis Using Fourier and Laplace Transforms
                                                                                                       ......                                                                                                                                                                                                                                                                                            EE2015: Electrical Circuits and Networks
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           Nagendra Krishnapura
                                                                                                                                                                                                                                                                                                                                                                                                                                     https://www.ee.iitm.ac.in/∼nagendra/
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               Department of Electrical Engineering
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         Indian Institute of Technology, Madras
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   Chennai, 600036, India
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            July-November 2017
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                                                                                                                                        Nagendra Krishnapura https://www.ee.iitm.ac.in/∼nagendra/                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             Circuit Analysis Using Fourier and Laplace Transforms
..                                        Circuit Analysis Using Fourier and Laplace Transforms
                                                                                                       Basedon
                                                                                                                                                                    exp(st) being an eigenvector of linear systems
                                                                                                                                                                                                                                 Steady-state response to exp(st) is H(s)exp(st) where H(s) is some scaling factor
                                                                                                                                                                    Signals being representable as a sum(integral) of exponentials exp(st)
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                                                                                                                                        Nagendra Krishnapura https://www.ee.iitm.ac.in/∼nagendra/                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             Circuit Analysis Using Fourier and Laplace Transforms
..                                        Fourier series
                                                                                                       Periodic x(t) can be represented as sums of complex exponentials
                                                                                                                                                                    x(t) periodic with period T0
                                                                                                                                                                    Fundamental (radian) frequency ω = 2π/T
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        0                                                                                                                         0
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       ∞
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                x(t) = ∑ a exp(jkω t)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                k                                                                                                                       0
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             k=−∞
                                                                                                                                                                                                                                 x(t) as a weighted sum of orthogonal basis vectors exp(jkω0t)
                                                                                                                                                                                                                                 Fundamental frequency ω0 and its harmonics
                                                                                                                                                                                                                                 ak: Strength of kth harmonic
                                                                                                                                                                    Coefficients ak can be derived using the relationship
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                ∫ T
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           a = 1                                                                                                                                                           0 x(t)exp(−jkω t)dt
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           k                                                           T                                                                                                                                                                                                                                                                                           0
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        0                                        0
                                                                                                                                                                                                                                 “Inner product” of x(t) with exp(jkω0t)
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                                                                                                                                        Nagendra Krishnapura https://www.ee.iitm.ac.in/∼nagendra/                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             Circuit Analysis Using Fourier and Laplace Transforms
..                                        Fourier series
                                                                                                                                                                    Alternative form
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           ∞
                                                                                                                                                                                                                                                                                                                                                                                                                   x(t) = a +∑b cos(kω t)+c sin(kω t)
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               0                                                                                                                    k                                                                                                               0                                                                                        k                                                                                                       0
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   k=1
                                                                                                                                                                    Coefficients b and c can be derived using the relationship
                                                                                                                                                                                                                                                                                                                                                  k                                                                                             k
                                                                                                                                                                                                                                                                                                                                                                                                                                                                               b                                                                  =                                                                       2 ∫ T0 x(t)cos(kω t)dt
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               k                                                                                                                  T                                                                                                                                                                                                                                                            0
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   0                                        0
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 c                                                                =                                                                       2 ∫ T0 x(t)sin(kω t)dt
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               k                                                                                                                  T                                                                                                                                                                                                                                                    0
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   0                                        0
                                                                                                                                                                    Another alternative form
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         ∞
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                x(t) = a +∑d cos(kω t +ϕ )
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            0                                                                                                                    k                                                                                                                0                                                                               k
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                k=1
                                                                                                                                                                    Coefficients b and c can be derived using the relationship
                                                                                                                                                                                                                                                                                                                                                  k                                                                                             k
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                d                                                                 = √b2+c2
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                k                                                                                                                                                            k                                                                 k
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           (c )
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   −1                                                                                k
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             ϕ                                                                    = −tan
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                k                                                                                                                                                                                                                                                     b
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                                                                                                                                        Nagendra Krishnapura https://www.ee.iitm.ac.in/∼nagendra/                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             Circuit Analysis Using Fourier and Laplace Transforms