Unformatted text preview: Exam 1 Questions/Solutions Yi Ma 1 Short question Find the exponential form for the following complex numbers (using radians for the angle): (i) (ii) (iii) 2 Long question PSfrag replacements Consider the following firstorder RL circuit: Figure 1: A switching firstorder RL circuit. (i) What is the current for , assuming the circuit was in steady state? (ii) Suppose we switch at initial condition ? Solution: . Derive the differential equation that . The differential equation is satisfies for 1 U2 T 672 4 C & R D P H S E2 QI% G( 0 A & ' F2 4 C 6 8 Solution: At steadystate, & )( & ' Y R `6 4 a72 53 A %' 72 4 C %' 0 F2 2 4 X C X 6 6 A 9 8 B @8 6 72 4 C 2 0 1 )' E2 D & % V H I% 6 " 4 C 6 W 4 C V W 26 4C 0 6 72 4 C 6 4 72 53 " """ $ # # # ! ; ; . . What is the (1) (iii) Firstly, assume , find the solution of for Solution: You need to solve A particular solution is then , and the homogeneous part is after taking into account the initial condition (iv) Instead of , if we assume , what is the solution of Solution: Now, you need to solve You already know a particular solution to the equation (2) is particular solution, say , to the following differential equation: 2 g R 6 H 4 " h H I% 6 VS 4 C 2 ( G% 6 0 2 ( 4 % is of the form you should get differential equation (3) is is of the form . Substitute into the above equation and compare the coefficients, and . Then, using superposition, a particular solution to the original . Then the total solution . Taking into account the initial condition , you have T C 6F2 4 C H I% 6 P 6 H 4 h g " g " f h i H 672 4 aC R % 6 V 4 C P A %'
. for A 6 q F2 4 c C R 6 6 % 6 V 4 C P F2 ( A 4 %' 72 4 C %' 0 F2 2 4 X C X 6 R 6 F2 ( 4 %' 72 4 C %' 0 F2 2 4 X C X 6 6 b2 T % 2 ( G% 72 4 C % 6 g " % hi H 0 2 ( G% A 0 B 72 4 # vcC 0 672 4 vc C 6F2 4 rcC 6 02 t ( u0 2 H t u0 2 H 2 ( 72 53 6 4 V 4 C A 6 p F2 4 C 6 72 4 C 72 4 C ' 0 2 X 6 % F2 4 C X 6 A 6 e F2 4 dcC 6 72 4 # rcC 72 53 6 4 . (2) . The total solution is 72 53 6 4 ? (3) . Now you need to find first a (4) 6 F2 4 # scC ...
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 Spring '07
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 Thermodynamics, Steady State, 2 g

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