EECE 290 Exam, Spring 2008

-1- Consider the two signals : F(t) = 2+3Cos(100nt)+4Cos(200nt)+6Cos(400nt) G(t) = Cos(100nt).Sin(300nt) - Sin(100nt).Cos(300nt) Sin(200zt) Find the period of each of the signals, TF and TG. (p25) a) TF=1/100 TG=1/100 b) TF=1/200 TG=1/300 c) TF=1/50 TG=1/50 d) TF=7/200 TG=1/200 a 2) None of Pfe 9. bave -2- The signal F(t) given in (1) is an approximation of the real signal f(t) with average power equal to 50Watt. What is the %average power error in the approximation?(take R=1Q) (pes) @ 3% b) 35% c)65% d)39% e) None of the above a ~3- Consider the trigonometric Fourier series representation of f(t) as given over the interval (-2,2): f(t)= t+1 -1ist<0 -t+1 Ost<i 0 elsewhere The Fourier series is also a representation of the periodic signal F(t) obtained by repeating f(t) periodically. The period of F(t) is: oo, (p25) a)3 b)2 c)i id)4 ;€) None of the above KY -4- Two periodic functions of period 6 seconds each are given by: f(t) = -t -3<t<0 g(t)=1 O<t<3 t Ost<3 -1 3<t<6 Find the ratio of the amplitude of the 3 harmonic present in f(t) to that present in g(t). (n25/ (a) 17 b) 3/n c)2/3x d)zx e) None of the above enna 1 Se piolt ECR ieo lov folev -5- Calculate the rms value of the (vy) () current flowing in the circuit shown. (p24) a) 8.58 b) 54.85 - (c))42.58A d) 25A___e)None of the above -6- For the wave shown, calculate Ao, As and Bs. (726) a) Ao=1.5 , As=1.5 (oJAs=1.15 c) Bs=1.5 d) As=1.5, Bs=1.15 e) None of the above w= o, ol ee -7- Find the average power in a resistance R=2Q , if the voltage is v= 32Sin(3t)Cos2(t/2) (p26 ) a) 256W b)5i2W c)i92W Ox: We) None of the above -8- In a circuit : v(t) = 5Sin(t) +10Sin(3t), I(t) = 7Sin(2t) + 50 Sin(8t) What is the average power ? (pre ) a) 267.5 W (o)o W c)250W d)35W _ e) None of the above -9- Two periodic voltages V1(t) and V2(t) of the same period (T=2s)are applied to the circuit. Find the current in this circuit due to the first harmonic. (726) LANE) WA De a iOmH v(t) (*) | 4 . C= 2p NAG) (t) 0) Cos(at) 6b) 12 Sin(at) c) 12Cos(nt)+12Sin(xt) d) 0> e) None of the above -10- The function e* (0<x<1) is to be represented by a cosine series, find a2. ( 726/ a)-0.684 b)-0.0827 | c)0.0848) d)0.0424 e) None of the above GOV -Li- Consider the wave form f(t). Find the amplitude and phase of the 3™ harmonic components of this waveform. (227) : Ns ¢ NG (6 cm) 5 . a) C3= 5.1 63=+90° b) C3= 2.57 03=+90° Cc) C3= 4.8 63=+180° d) C3= 5.1 03=#180° (¢))None of the above -12- A rectifier system with input f(t)and output g(t) is described by: g(t) = |f(t)|. For an input of f(t) = (a/4)Sin(wot), the coefficient of the exponential Fourier series of g(t) with n even is: (p27) On 2(1-n?) b)i/(1-n2) c)2/(1-n?) d)1/(m-nn2) e)None of the above -13- Let the signal f(t) be a signal defined between -x and x. F(t) is zero outside the following exponential Fourier series: (p27) ® int y; Cc Gl n=-o The series represents the periodic extension of f(t) with period T. Find T. a) 8x b) 2x c) 6x (a))4n e) None of the above -14- Consider the following signal : F(t) = 2Cos(100xt)+3Cos(300nt)+6Cos(500nt)+9Sin(300nt) Find the coefficients of the exponential Fourier series of F(t). (p22) a= a)Cy=C1=1 5; C3=1.5-4.5] C3=1.54+4.5j 5 Cs=C5=3 b)C,=C.4=-1; C3=4.5-1.5j C3=4.54+1.5j ; Cs=C.5=-3 c)Cy=C4=2 ; C3=2.5-4.5j C3=2.544.5j) ; Cs=C5=3 ; d)Cy=C.14=2 ; C3=2.5-4.5) C3=2.54+4.5j 5 Cs=C5=3 ; e) None of the above -15- Consider a resistance R=1Q across which the following voltage is induced : V(t) =e)20*4 (2447) e)2t+ (34.9j) e300 Find the % energy contained in this sum when taken as an approximation of the signal f(t) defined over (1/50 s; 2/50 s). f(t) has an energy E=2.5 joules. (727/ a) 78.8% b) 82% (c) 88.8% \d) 98.8% e) None of the above -16- A voltage v(t) = Cos(wit)+2Sin(wat) with f,=4KHz and f2=8KHz is passed through a non-linear system having a transfer function of the form : Vo(t)=v+0.1v2 How many distortion (harmonic and inter-modulation) components are present in the output voltage? (2g) a) 2 b) 4\c)6 d)8 e) None of the above -17- If two filters of 300Hz passband are used at each frequency , how many distortion components will remain?(referring to -16-) (p28) (@)2) b) 4 c)6 d)None e) None of the above NN. -18- Referring to -16-, what is the amplitude and phase distortion of the output voltage? (p2¢) a)A=0.8, 0=2.9° b) A=1, 0=45° “S=> c)A=1.02, 0=11.3° d)A=1.2 , 0=90° e)None of the above 1. Find the rms value of v(t) in Fig. 1 over the time interval (0, 5). (2 70) A. Vans = 0.54 mouAw Vims = 0.98 Vims = 0.86 Vems = 0.68 None of the above v(t) 10. iL. Find the expression for the Fourier coefficients C, for the periodic function shown in Fig. 9.(p7z ) A. 1/2mn,n# 0: 1/2, n=0 B. j/nn, n# 0; 0,n=0 C. j/2nn,n#0; 1/2,n=0 ? D. j/3nn,n¥0; . 1,n=0 E. None of the above . A periodic function is represented by: +co 120 t= = v, el On nt n=—00 Fig. 10 shows the plot of the magnitude of the coefficients Vn. Find the average and the fundamental frequency of v(t). (2 72 )) 2: > 5: 2 1Hz 10 Hz 0; 10 Hz Figure 9. n V, LL. WAM 5; 100 Hz + None of the above / HOO pS -2 . * Figure 10. A 5 3 ! -- 0 1 5 3 4 Se 0 14. Calculate the power dissipated in the resistor in Fig. 12 if vi(t) = 10cost and v2(t) = 10cos3t. (p 93) iF an A 12.7 W ||-— ~— B. 50.7 W C. 60.8 WwW v(t) C) 1Q CC) y,(t) D. 70.5 W Vain ~~“ V E. None of the above : Figure 12 4. A series RL circuit in which R = 5Q and L= 20 mH has an applied voltage v=100+ 50sin wt + 25sin 3at V, with w=500 rads/s. Find the instantaneous current. Cp /04) . =20+4.47sin(wtt+63.4)+0.822 sin(3wt+80.54), A b. i=20+4.47sin(wt-63.4)+0.822 sin(3at-80.54), A C. i=8.96+4.47sin(wt-63.4)+0.822 sin(3wt-80.54), A d. i=sin(wt-63.4)+0.822 sin(3ut-80.54), A e. None of the above 5. Determine the power dissipated in the resistor of problem 4. /? ( (00) ~50.1 W . ~51.79 W ~2000W . ~ 2053 W None of the above \ ooo op -— ~pamunrRer 11. The figure below shows the triangular waveform of a voltage source operating at frequency f= 1kHz. Find the amplitudes of the fundamental (1; ) and the second order harmonic (lz) current that flows through an inductor of value L= 1mH when it is supplied by this source.. (Answers are rounded to 2 digits after the decimal point) (2106) ult a) =2.31A,12=1.10A — b) 1,=1.34A,L=0A c) h=0A,b=1.10A ; [e) oa) | =1.29A,l=0A e) None of the above Test ID 1000 4/7 6. Consider a periodic function f(t), described by the following sequence during one period of time: f(t) = 0 for -4s1t<-3 ft)=-Vm for -381t<-] f(t) =0 for -Isi<+]l ft)}=+Vm for +151<+43 f(t) =0 for +3575 +44 where Vm = 20. Find the amplitude “A” of the third order harmonic in the Fourrier series, expressed by Acos(3at — @). (p /23 ) a) A=3 b) A=4 c) A=5 d) A=6 €) None of the above