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Unformatted text preview: Problems Section 6.1: Fourier Analysis, Filters, and Transfer Functions P6.1. What is the fundamental concept of Fourier theory? P6.2. The triangular waveform shown in Figure P6.2 can be written as the infinite sum 8 Vt(t) = 1 + rr 2 cos (2000rrt) 8 + 2 cos ( 6000rrt) + ... (3rr) 8 + 2 cos (2000nrrt) + (nrr) in which n takes odd integer values only. Use MATLAB to compute and plot the sum through n = 19 for 0 :::: t :::: 2 ms. Com- pare your plot with the waveform s hown in Figure P6 .2. 1 Figure P6.2 2 t (ms) P6.3. The full-wave rectified cosine wave shown in Figure P6.3 can be written as 2 4 Vfw = - + cos(4000rrt) rr n(1)(3) 4 - cos(8000rrt) + n (3) (5) 4(-1) (n /2 +1 ) + n(n _ 1 )(n + 1 ) cos(2000nrrt) + in which n assumes even integer values. Use MATLAB to compute and plot the sum Problems 341 through n = 60 for 0 &lt; t &lt; 2 ms. Com- pare your plot with the waveform s hown in Figure P6.3. Vr wCt) = Ieos (20001Tt) l 1 0.5 1.0 Figure P6.3 t (ms) P6.4. The Fourier series for the half-wave rectified cosine shown in Figure P6.4 is 1 1 2 vhw(t) = rr + 2 cos(2rrt) + rr( 1 )( 3 ) cos(4rrt) 2 --- cos(8rrt) + n(3)(5) 2(-1) (n / 2+1) + cos(2nnt) + n(n- 1)(n + 1) in which n = 2, 4, 6, etc. Use MATLAB to compute and plot the sum through n = 4 for -0.5 :::: t :::: 1.5 s. Then plot the s um through n = 50. Compare your plots with the waveform in Figure P6.4. 0.5 1.0 Figure P6.4 t (s) * Denotes that answers are contained in the Student Solutions fi les. See Appendix F for more information about accessing the Student Solutions. 342 Chapter 6 Frequency Response, Bode Plots, and Resonance P6.5. Fourier analysis shows that the sawtooth waveform of Fig ure P6.5 can be written as Vs t(l) = 1 - ~ sin(2000rrt) 7f - 2 sin( 4000nt) - 2 sin(6000rrt) 2rr 3rr 2 . - - sm (2000nnt) - nn Use MATLAB to compute and plot the sum through n = 3 for 0 &lt; t &lt; 2 ms. Repeat for the sum through n = 50. 2 1 2 Figure P6.5 3 t (ms) P6.6. What is the transfer function of a filter? De scribe how the transfer function of a filter can be determined using laborator y method s. P6.7. How do es a filter process an input signal to produce the output signal in te rm s of sinusoidal components? *P6.8. The transfer function H (f) = V out!Vin of a filter is shown in Figure P6.8. The input signal is given by IH (f) I 2 1 Vin (t) = 5 + 2 cos(5000rrt + 30) + 2 cos(15000nt) Find an expression (as a function of time) for the steady-state output of the filter. 5 10 5 10 f -~.::--------.----.---- (kHz) f -180 ( kH z) Figure P6.8 P6.9. Repeat Problem P6.8 for the input voltage given by Vin (t) = 4+ 5 cos(10 4 rrt-30 ) + 2 sin(24000nt) P6.10. Repeat Problem P6.8 for the input voltage given by Vi n (t) = 6 + 2 cos(6000nt) - 4 cos(12000nt) *P6.11. Th e input to a certain filter is given by and the steady-state output is given by Determine the (complex) value of the tran s- fer function of the filter for f = 5000Hz....
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- Spring '09
- Mechatronics, ... ..., Low-pass filter, nd Resonance