Question

13-9 Find the first five nonzero Fourier coefficients of the shifted and offset square wave in Figure P13-9. Use your results to write an expression in the corresponding Fourier series v(t) (V) 0 5- t(ms) 0.25 1.25 2.25 10 FIGURE P13-9

Answer 1

fourier series of V(t) can be represented by V(t) = a_0 + sigma^alpha_0_n = 1 a_n cos n w_0 t + sigma^alpha_n = 1 b_n sin n w_0 t here, T_0 = 2: w_0 = 2 pi/T_0 = 2 pi/2 = pi a_0 = 1/T_0 integral^T_0_0 V(t) dt = area over one period/period = 5(1.25 - 0.25) + 10(2.25 - 1.25)/2 = 15/2 = 7.5 a_0 = 7.5 a_n = 2/T_0 integral^T_0_0 V(t) cos n w_0 t dt = 2/2 [integral^1.25_0.25 5 middot cos n pi t dt + integral^2.25_1.25 (-10)cos n pi t dt] = 5/n pi sin n pi t]^1.25_0.25 + (-10/n pi)sin n pi t]^2.25_1.25 = 5/n pi [sin(1.25n pi) - sin(.25n pi)] - 10/n pi [sin(2.25n pi) - sin(1.25n pi)] = 5/n pi [sin(1.25n pi) - sin(0.25n pi)] - 10/n pi [sin(2n pi + 0.25n pi) - sin(1.25n pi)] = 15/n pi sin(1.25n pi) - 15/n pi sin(0.25n pi) = 15/n pi [sin(n pi + 0.25n pi) - sin(0.25n pi)] \r\n a_n = 15/n pi [sin(n pi)/0 cos(0.25n pi) +