EP 214 Midterm 2022

3. (15 marks total) This question deals with the inhomogeneous series RL circuit discussed in the lectures. The circuit is connected to an input voltage source given by an arbitrary function v ( t ). A switch is closed at time t = 0. There is no initial current in the circuit. Assume variable R and L values. (a) Starting from Kirchhoff’s voltage law, write down the differential equation describing the current in the circuit as a function of time. (3 marks) (b) Derive the transfer function of the current in the circuit. (3 marks) (c) Derive the unit impulse response for the voltage drop across the inductor. (3 marks) (d) Derive the voltage drop across the resistor as a function of time, where the input is a contant voltage source v ( t ) = V 0 . (3 marks) (e) Sketch the voltage drop across the inductor as a function of time, where the input is a contant voltage source v ( t ) = V 0 . (3 marks) 4. (18 marks total) These are six short questions that do not relate to each other but relate to material from the lectures and assignments. (a) Find the Laplace transform of f ( t ) = sin( ωt + φ ). (3 marks) (b) Find the inverse Laplace transform of F ( s ) = 2 + 6 s 4 + e - s s + 1 . State any identities that you used to find your answer. (3 marks) (c) Find the partial fraction decomposition of F ( s ) = - s 2 + 2 s - 3 s 2 + 1 . (3 marks) (d) If R = 20 Ω and L = 5 mH are two components in a series RLC circuit, what is the range of values for C that would result in an overdamped circuit? (3 marks) (e) If L = 1 mH, R = 16 Ω and C = 10 μ F are the three components in an underdamped series RLC circuit with no input voltage source, what is the period of oscillation for this circuit? (3 marks) (f) Sketch the voltage drop across the resistor as a function of time for a homogeneous underdamped RLC circuit, where the initial charge on the capacitor is non-zero and the initial current is zero at time t = 0. On this plot, clearly label the time interval that is related to the quantity p β - α 2 . (3 marks) End of Exam! 2