Question

2. A pneumatic control system uses a differential pressure sensor to measure differential pressure across a valve actuator. The single cycle of a ramp function of voltage versus time has the form f(t)=100t . In preparation for doing a spectral analysis of this signal, you decide to calculate the Fourier series for the signal to get a qualitative understanding of the frequencies that contribute to give the overall pressure versus time signal. Pressure (kPa) 0.5 Time (s) 0.03 -0.03 02 -001 0.bi 05 3. a) Determine the period, I, the fundamental frequency, (,, and the a, coefficient of the Fourier series for the trace in this diagram. (Note: show your work leading to the final answer)(5 points) b) Evaluate the ai and a2 coefficients of the Fourier series by direct integration. (Note: show your work leading to the final answer)( 5points)

Answer 1

9) AS can وا you see that this signal cycle repeating after time period I I= 0.01 0.01 -10.01) IT = 0.02 sec So, fundamental frequency wo = 21 CD, – 314.16 rady T L-T= 0,02 # Forier series f(x) = a + 4 (ancos (297 x) + bn sin (2017 2919_)) nal L/2 -L -42 f fead de fid) mate. (equation of line) f(o)=0 & f60.01) = 1 [cao ( f(-o)=0) 0 = o ac m 10.01) ( 120.01) =) m=10o I foy= 100%

0,0) ao = 1 0.02 (ঠ) সুন fo -0.01 0,0) ao = 5000 DC da -0.0) 2,01 2 ao- sood a 5000 (2001)*-(-0.012] -0.0) 2 ao mns b) ² an = 2 L f fox) cos(21x) dx -42 0.07 anta al 2 ana 2 cos ( I 100L 0.02 0.02 -0.0) 0.01 4 10 an= dx a cos/loon poonda) -0.01 2.0) 4 sin (loonax).da), de loont dx xsin 100 n512) toont an= 10 0,0)

0.01 4 an= 10 cos (loonga asin (1005 nx) & toon (toona)² -0.0) o an 4 -10 Tooresinent) 0.01*singÝT) + Cos (not -0.01 Sin (NTI) + 100 na (loon) ² 10ont + oslona) (loonit)? 4 an=10 cos (nI) cosina) (loon) 2 (luont) 2 an=0 Sinint)=0 cos (-a)= cos(a) a = 92 =92=0 Ans

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