Final Exam with Useful Formulas - Electric Circuits | ENEE 205, Exams of Microelectronic Circuits

Download Final Exam with Useful Formulas - Electric Circuits | ENEE 205 and more Exams Microelectronic Circuits in PDF only on Docsity! Name: UNIVERSITY OF MARYLAND Department of Electrical and Computer Engineering ENEE 204 - Basic Circuit Theory Spring 2007 (sections 010x) Exam 3 Monday, May 14, 2007 8:00 am – 10:00 am Instructions • Do not turn over this page until you are instructed to do so. • Please write all of your answers directly on the exam in the space provided. That is the only item that will be turned in and graded. • This is a closed-book and closed-notes exam. • Calculators are permitted, but no other electronic devices may be used. All cellular telephones must be turned off and must remain in your bag throughout the exam. Laptop computers and other handheld computing devices are prohibited. • Explain your answers completely. A correct answer does not guarantee full credit, nor does an incorrect answer guarantee loss of credit. Your grade on each problem will be based upon your understanding of the material as demonstrated through your explanation of the solution. • You are encouraged to use your calculator to perform complex number arithmetic, but you must show your work to receive credit. • Important: Zero credit will be given for matrix equations that are solved directly on a scientific calculator. (You may, however, use your calculator to check your answers.) Honor Pledge I pledge on my honor that I have not given or received any unauthorized assistance on this examination. Signature: E3.1 (11 pts) E3.2 (12 pts) E3.3 (13 pts) E3.4 (13 pts) E3.5 (12 pts) E3.6 (16 pts) E3.7 (13 pts) E3.8 (10 pts) Total (100 pts) ENEE 204, Spring 2007 Exam 3 Useful Formulas Trigonometric Conversion Identities: sin u = cos ( u − π 2 ) cos(−u) = cos u − sin u = cos (u + π 2 ) − cos(u) = cos(u ± π) sin(−u) = − sin u = cos (u + π 2 ) Sum and Difference Angle Relations: cos(u + v) = cos u cos v − sin u sin v sin(u + v) = sin u cos v + cos u sin v cos(u − v) = cos u cos v + sin u sin v sin(u − v) = sin u cos v − cos u sin v Complex Number Identities: e jθ = cos θ + j sin θ, j ≡ √−1 j2 = −1 e±jπ = −1 (−j)(j) = 1 e jπ/2 = j 1/j = −j e−jπ/2 = −j Given z = a + jb = re jφ, it follows that: z∗ = a − jb = re−jφ zz∗ = z∗z = |z |2 = a2 + b2 = r 2 z + z∗ = 2a = 2Re{z} z − z∗ = j2b = j2 Im{z} Polar/Rectangular Conversion: Sinusoidal Function Complex Phasor Calculator Polar: x(t) = Xm cos(ωt + φ) X̂ = Xme jφ (Xm 6 φ) Rect.: x(t) = Xr cos(ωt)−Xi sin(ωt) X̂ = Xr + jXi (Xr , Xi) Xr = Re{X̂} = Xm cosφ, Xi = Im{X̂} = Xm sinφ Xm = |X̂| = √ X2r +X 2 i , φ = { tan−1(Xi/Xr) Xr > 0 tan−1(Xi/Xr)± π Xr ≤ 0 Re Im Xr X m Xi φ X = Xm e jφ = Xr + jXi ^ Page 2/14 ENEE 204, Spring 2007 Exam 3 Problem E3.2 (12 pts) The following circuit is operating in the AC steady state with a frequency of ω = 50,000 rad/s. 40 cos(ωt), ω = 50 krad/s 2 nF 2 kΩ 4 kΩ80 mH i1(t) Find an expression for the current i1(t) in the standard form A cos(ωt + φ). Page 5/14 ENEE 204, Spring 2007 Exam 3 Problem E3.3 (13 pts) In the following circuit, the load resistance R is adjustable. 3 kΩ 2 kΩ 12 V vx is = vx 6000 R (a) How should you adjust the load resistance in order deliver the most power to the load resistor R? (10 pts) Page 6/14 ENEE 204, Spring 2007 Exam 3 (b) What is the maximum power that can be dissipated in the load? (3 pts) Page 7/14 ENEE 204, Spring 2007 Exam 3 (b) Calculate the input impedance of the system, Rin ≡ v1 i1 . (3 pts) Page 10/14 ENEE 204, Spring 2007 Exam 3 Problem E3.6 (16 pts) v1(t) 2 Ω 1 H 3 Ω1 H v2(t) (a) Calculate the transfer function H(s) = V̂2/V̂1 for the circuit shown above. Please express your answer as a ratio of two polynomials in s. (5 pts) (b) Make a diagram showing the poles and zeros of H(s) in the complex plane. (2 pts) Page 11/14 ENEE 204, Spring 2007 Exam 3 (c) Suppose the input signal signal is the unit step function, i.e., v1(t) = u(t) = { 0 t < 0 1 t ≥ 0 Derive an expression for the output v2(t) for t > 0. (9 pts) You may assume that dv2 dt ∣∣∣∣ t=0+ = +2 V/s. Page 12/14 ENEE 204, Spring 2007 Problem E3.1 (11 pts) 10 kQ 4kQ 18 kQ WV WV + Ft Mga a1ma(f) 9K 2108 wy, MW WV 2kQ 10 kg Use the voltage and current divider relations to determine the voltage vzg in the above circuit. ze imy = dy Voerace F + Ve = ‘B46. 4 tq 4 NNT comQing We, 100, % Bla NMéesisroOMs - ' 21.28 Rey = 21 ij (igs 19) = Rigag = WReR. tour qe oe Armen wD A, panna [ 3 + Zak Ds Ren = Mier jo AB Aven. ReR Now, wk use CURnen OWL TO FIND hy » = (2k) = Tok. ‘ " G44 i242 oa on Ny = (Twa) (ren) = BAN EXAM #3 SOLUTIONS Exam 3 Page 4/14 ENEE 204, Spring 2007 Exam 3 Problem E3.2 (12 pts) The following circuit is operating in the AC steady state with a frequency of w = 50,000 rad/s. ~sDen. | QnF po Om ae 24 40 cos(wt), ' | , oases « * mig 44a $100 7 vy = Ve, = 40 Find an expression for the current i: (¢) in the standard form Acos(wt + @). Z- 2 = ~oe ZA =swek = 454 e we ~ Siok. a) ST lee Z, = (4|)09) = 2% 2 24jr2 (ee) “hk 4404 VOLTREE O1iOER ? A Zeg Q+ygh Ve oe fyov) = (40 Log 4 2~ j10 v) 24jla 2s” > A ; . V, = -445R (v). ‘ . 4a . 3189 I, = oa = -{[+ 53 (may, = 3.16¢ = 216 L044 (ma) { ‘ At) = 316 cos (So.0004 + 1.99) mA Ae ans, ah EXAM #3 SOLUTIONS Page 5/14 ENEE 204, Spring 2007 Exam 3 Problem E3.3 (13 pts) In the following circuit, the load resistance R is adjustable. +x + WA WA 3kQ 2kQ oY 12M) t bas az (a) How should you adjust the load resistance in order deliver the most power to the load resistor R? (10 pts) OPE Cri VoUmiee (R= ~) kee @ Pomr kh TES us? Uy Vx ty 3k & =~ 0 40 OG = FV, + Ve ~ : NAA gee enn AAA, wean Use NONE -vO Tete METHOD TO den a Rew Finn Um: tN t Ua I a A Va 4 — Me _ 1a-%, | 3k ae 6k 6k hel . : poe MN LT SOuwt FR Y= +£V 4, fe hee =H = By = Buk. . ve _ Ry = = Sym Ake ook Powke TRANSEER wey R= Yen Cor A EXAM #3 SOLUTIONS Page 6/14 ENEE 204, Spring 2007 Exam 3 Problem E3.5 (12 pts) You may assume that the op-amp in the following circuit is ideal. 3kQ Kh BKQ Zk Ano Nin. Wr AA int LEHSTONS PO I. : ako MOT MATTER HORE | © . vy 7 + 4kQ 9kQ A + 3kQ 7K0 Vp (a) Derive an expression for the output voltage v> in terms of y. {10 pts) Va. Va - Vs ken A? > 4 ACM = oO 4 OR. Va = a PGA. OP-AMP > 2. Yr HY, ty v, kee Al? a” 4 Bk La 3 6 Wind GIGS : VW = 384 ROHS, EXAM #3 SOLUTIONS Page 9/14 ENEE 204, Spring 2007 Exam 3 (b) Calculate the input impedance of the system, Rin = a. (3 pts) 1 uy ~. i= Hee = tbte) ra Sen - _ 3% %, A 1 — =o = a Ban er & Row Me x EXAM #3 SOLUTIONS Page 10/14 ENEE 204, Spring 2007 Exam 3 Problem E3.6 (16 pts) on £ AW ‘ Comeure . 1H ‘ 32 —_—> Za > Bada ‘ : wo) aguy 1H} IMP oanee 1 = 1 \ aoe ’ {a) Calculate the transfer function H(s) = “wM for the circuit shown above. Please express your answer as a ratio of two polynomials in s. (5 pts) _ 2(s43) zy = (3.)|| i. = aS A VOUTRLE DIM OER * U = ee V * Zea As ' His) Ain) His) - FT _ Q(se3) _ Z(o14) alsa) Alaayasl48) Fa Fs46 as Qlses HG) - (546) (o4t (b) Make a diagram showing the poles and zeros of H(s) in the complex plane. (2 pts) Tm 45] . EXAM #3 SOLUTIONS Page 11/14