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#### Problem 1 Determine the Laplace transform of 3

###### Electrical Engineering

Problem 1 Determine the Laplace transform of 3.5cos(5t-45°). Problem 2 Determine the Laplace transforms of these functions: (a) f(t) = (t - 4) u?t - 2) (b) g(t) = 2e-4 u?t - 1) (c) h(t) = 5cos (2t-1) u(t) (d) p(t) = 6 [ ult - 2) - u(t - 4)] Problem 3 Find the Laplace transform of the signal in Fig. 1. f(t) 25 0 2 6 t Figure 1 Problem 4 Let F(s) = 18(5+1) (s +2)(+3) (a) (b) Use the initial and final value theorems to find f(0) and f(-). Verify your answer in part (a) by finding f(t) using partial fractions. Problem 1 Find the inverse Laplace transform of the following functions: a) F(s) = 10(8+3) 8(32 +48+10) b) FS 1082 +1008+40 (3+3)(8+5)(8+9)2 c) F(s) = \$2+5 33 +682 +258 Problem 2 Find the convolution integral of the following figure pairs: a) z(t) = e(t) * h(t) x(t) h(t) 1 1 -1 0 1 t 0 1 t b) z(t) = x(t) * h(t) h(t) 2 x(t) 1 2et -1 0 1 t 0 t z(t) = 2(t) * y(t) y(t) 3 X(t) 0.5 1 2 3 4 0 1 2 3 44 -1 Problem 3 Find i(t) for t > 0 for the circuit in the following figures. Assume ig(t) = [Cu(t) + 38(t)] mA. + 62 10 is(t) 62 ell ?? Problem 4 Obtain v(t) for t > 0 in the circuit of the following figure. + V 4 H 512 t=0 25 V 1 F Problem 1 For the ideal transformer circuit in the following figure, determine i(t). w 1:3 r + -24 ? Toe utt) o ell ele TF ? - ???? Problem 2 For the op amp circuit in the following figure, find vo(t) for t > 0. Take v, = 1024 u?t) V. Joke 20PF 100 kr + Vs Problem 3 Find v.(t) a) Using convolution integral b) Using transfer function c) when v, = u(t) i bi VC ge 1 + n ele 15 45 2 Problem 4 Using superposition, find v.(t) when it (0) = 2 A and v.(0) = 5 V 2H + loe utt 1F + voll) { 81 56(4A Problem 5 The transfer function of a system is H(\$) input of 5e-t/4 u?t). Find the output when the system has an 48+1 Question 1 6 pts Find the z parameters in the following two-port network when w=2 1F 14 r ? 1 im o 5H 2H z_11 = z_12= z_21= Z_22= 6 pts Question 3 In the following circuit, R=1 Ohm, L=0.25 H, and C=1F. t=0 L mm + . C ult) ??? ? lo A a) Find the initial and final value for i, [1 point] iz (0) = i (0)= b) Obtain IL (s) and ii (t) for t>O [5 points) IL (s) = 11 (t) = Question 2 6 pts Find the convolution integral of the following figure pairs z (t) = x (t) *y(t)and sketch the z(t). Also, report the below values of z(t): y(t) X16) Hitt at z(1) z(3)= z(5)= Question 4 0.5 pts In order for the final-value theorem to hold, all poles of F(s) must be located in the left half of the s plane; that is, the poles must have negative real parts. True O False Question 5 0.5 pts The initial conditions of a capacitor can also be represented either as a voltage or as a current source. O True O False Question 6 0.5 pts In the Ladder method, the transfer function is the complex conjugate of the input. Question 6 0.5 pts In the Ladder method, the transfer function is the complex conjugate of the input. O True O False Question 7 0.5 pts The standard way to represent a state variable is to arrange them as a set of second-order differential equations. O True O False Question 7 9 pts In the following circuit, R=1 Ohm, L=0.25 H, and C=17. t=0 L m (4) R lo 4 C a) Find the initial and final value for iz [1 point] i (0) ii (0) b) Obtain I? (s) and ii (t) for t>0 [7 points] IL (s) ii (t) = u(t) c) Bonus [1 point] Obtain Vc (t) = u(t) Question 6 7.5 pts For the RC circuit in the following figure, use the convolution integral to find the response v.(t) due to the excitation shown in the following figure. Please make sure to sketch the final function and insert the requested values here. V ) A vo 11 G a s Al IF Hun ?? 2 vo(1) = v.(2) V.(3) Question 5 7.5 pts Find the z parameters in the following two-port network when w= 1 r FH 1 0- 1H YE TH ell 9 Z_11 = Z_12= z_21= Z_22= Question 6 7.5 pts For the RC circuit in the following figure, use the convolution integral to find the response v.(t) due to the excitation shown in the following figure. Please make sure to sketch the final function and insert the requested values here. V ) A vo 11 G a s Al IF Hun ?? 2 vo(1) = v.(2) V.(3)

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