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Q1. [30 Points] Consider the circuit below and answer the following questions: (a) [18 Pts] Use nodal analysis to analyze the circuit and find the voltages of the nodes V1, V2, and V3. (b) [4 Pts] Calculate the voltage Vo. (c) [4 Pts] Calculate the current Ix. (d) [4 Pts] Calculate the power of the 16 V voltage source. Q2. [20 Points] (a) [10Points]Calculate the equivalent inductance, Leq, for the inductive network in the circuit below where 𝐿1 =32𝐻, 𝐿2 =50𝐻, 𝐿3 =84𝐻, 𝐿4 =84𝐻, 𝐿5 =24𝐻,and 𝐿6 =12𝐻. Show your work in details. (b) [10 Points] Calculate the equivalent capacitance, Ceq, for the capacitive network in the circuit below where 𝐢1 = 40 πœ‡πΉ, 𝐢2 = 40 πœ‡πΉ, 𝐢3 = 70 πœ‡πΉ , 𝐢4 = 10 πœ‡πΉ , and 𝐢5 = 30 πœ‡πΉ. Show your work in details. Q3 [25 Points]: For the circuit below where 𝑅1 = 200 π‘œhπ‘š, 𝑅2 = 160 π‘œhπ‘š, 𝐿 = 40 𝐻 , and 𝐢 = 0.5 𝐹, answer the following questions (show your detailed solution): (a) [6 Points] Redraw the circuit in the phasor domain. (b) [7 Points] Find the total impedance seen by the voltage source (i.e. between terminals a and b). (c) [6 Points] Calculate the current i(t) in the time domain. (d) [6 Points] Calculate the voltage V(t) in the time-domain. Q4 [25 Points]: For the circuit below where 𝑣𝑖(𝑑) = 25 sin (90𝑑 + 10π‘œ), 𝑅1 = 60 π‘œhπ‘š, 𝑅2 = 100 π‘œhπ‘š, 𝐿 = 0.8 𝐻 , and 𝐢 = 80 πœ‡πΉ, answer the following questions (show your detailed solution): (a) [5 Points] Redraw the circuit in the phasor domain. (b) [5 Points] Calculate the total impedance (ZT) seen by the voltage source. (c) [5 Points] Calculate the voltage Vo(t) in the time domain. (d) [10 Points] Using your own words, explain what will be the effect of increasing the frequency of the source on the impedance of the following elements: (1) Resistor, (2) Inductor, (3) Capacitor. Ordinary Differential Equations Part A (12 pts). (True/False Questions). Choose true or false. 1) 𝑦!!! + 5𝑦!! + 3𝑦! + 𝑦 = 2π‘₯ is homogeneous third order linear differential equation. β–‘ True β–‘ False 2) One of the solutions of the differential equation 𝑦!! + 5𝑦! = 0 β–‘ True β–‘ False !!! ! 3) The Laplace transform of 𝑒 sin 2𝑑 is (!!!)!!!. is 𝑦 = 1. β–‘ True β–‘ False 4) If the Laplace transform of 𝑓(𝑑) is 𝐹(𝑠) then the Laplace transform of 𝑑𝑓(𝑑) is ! 𝐹(𝑠) . !" β–‘ True 5) The Laplace transform of 𝛿 𝑑 βˆ’ 6 is 𝑒 !!! β–‘ False β–‘ False β–‘ False !" =2𝑦 6) !" β–‘ True is the same as β–‘ True 𝐷π‘₯βˆ’2𝑦=0 𝐷𝑦 βˆ’ 4π‘₯ = 0 . !" = 4π‘₯ !" . Part B (24 pts). (Multiple Choice Questions). Chose the correct answer. 1) 2) 3) 4) 5) 6) If the roots of the auxiliary equation of a second order linear homogeneous differential equation are given as π‘š!,! = Β±5𝑖, then the general solution is given as a) cos5π‘₯+sin5π‘₯ b) 𝑐!cos5π‘₯+𝑐!sin5π‘₯ c) 𝑐! cos 2π‘₯+𝑐! sin 2π‘₯ d) cos5π‘₯+sin5π‘₯ e) None of them A set of two linearly independent functions is !! !! a) 𝑦! = 𝑒 , 𝑦! = 2𝑒 b) 𝑦! =cos5π‘₯, 𝑦! =2cos5π‘₯ !! !! c) 𝑦!=5𝑒 , 𝑦!=𝑒 !! !! d)𝑦!=𝑒 , 𝑦!=π‘₯𝑒 e) None of them The Laplace transform of 𝑒!!𝑑! sin (5) is a) !"# (!) b) ! !"# (!) c) !!"# (!) !!!! !!! ! !!!! d) !!"# (!) !!! ! e) None of them The inverse Laplace transform of 𝐹 𝑠 = ! !!!! is (!!!!) a) 7sin (𝑑 βˆ’ 4) b) 7𝒰(π‘‘βˆ’4)cos(π‘‘βˆ’4) c) 7𝒰(𝑑 βˆ’ 4)sin (𝑑 βˆ’ 4) d) βˆ’7𝒰(π‘‘βˆ’4)cos(π‘‘βˆ’4) e) None of them 𝑓(𝑑) = 0, 0 ≀ t < Ο€ 4 sint, t β‰₯ Ο€ can be written as a) 𝑓 𝑑 =4sint𝒰(tβˆ’Ο€) b)𝑓 𝑑 =sint𝒰(tβˆ’4Ο€) c) 𝑓 𝑑 = sint 𝒰(t βˆ’ Ο€) d) 𝑓 𝑑 = 4sint 𝒰(t + Ο€) e) None of them !"=𝑦, π‘₯0=2 The solution of the given initial value problem !" is !"=π‘₯, 𝑦0=0 ! !! ! !! !" a) π‘₯=3𝑒 βˆ’3𝑒 , 𝑦=3𝑒 βˆ’3𝑒 !!! !!! b) π‘₯=3𝑒 +𝑒 , 𝑦=3𝑒 βˆ’π‘’ ! !! ! !! c) π‘₯=𝑒 +3𝑒 , 𝑦=𝑒 βˆ’3𝑒 ! !! ! !! d) π‘₯=3𝑒 +3𝑒 , 𝑦=3𝑒 βˆ’3𝑒 e) None of them Part C (64 pts). (Workout Problems). Solve the following problems. Show your work clearly. a) Show that 𝑦! = 5π‘₯ is also a solution to the given DE. In order to construct the general solution, why is it not possible to take 𝑦! = 5π‘₯ as a second solution? Explain. b) Use the method of Reduction of Order to find a second solution 𝑦! and the general solution. 1) Given that 𝑦! = 4π‘₯ is a solution to the DE 𝑦 βˆ’ 𝑦 + !! ! 𝑦 = 0. 2) Use the Laplace transform to solve the given initial value problem ! 0, 0≀𝑑<1 𝑦 + 4𝑦 = 𝑓 𝑑 , 𝑦 0 = 1, where 𝑓(𝑑) = 𝑑 βˆ’ 1, 𝑑 β‰₯ 1 . Do you know another method to solve the above initial value problem besides the Laplace transform? Explain. 3) Given the initial value problem y!! + 16𝑦 = 𝛿 𝑑 βˆ’ 2 , y 0 = 0, y! 0 = 1. Which method would you choose (The Laplace transform or variation of parameters) to solve the above initial value problem? Explain why and solve it. !" 4) Solve the given initial value problem !" = π‘₯ + 5𝑦, 𝑦 0 = 1 . How can you evaluate !" π‘₯β€² 0 and 𝑦′ 0 ? Explain and find it.

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Electrical Engineering Problems
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