Find the differential equation and express it in proper

Question

Assignment

Figure 1: Circuit for Problem 1.

1. For the circuit shown do the following without using Laplace Transforms.

(a) Find the differential equation and express it in proper form.

(b) Write down the impulse response of this circuit assuming R = 10Ω and L = 1H.

(c) Find the frequency response function H^ (jω) of this circuit assuming R = 10Ω and L =1H.

Figure 2: Circuit for Problem 2.

2. For the circuit shown do the following without using Laplace transforms.

(a) Write the differential equation in proper form.

(b) If R = 1Ω , C = 1F, and L = 4H, what is the impulse response of this circuit?

(c) Find the magnitude of the frequency response function |H^/ (jω)| at ω= 0 rad/s and ω = ½ rad/s.

3. This problem is to be solved WITHOUT the use of Laplace transforms. The step response of a system is y_step (t) = tu (t).

(a) Find the impulse response of this system.

(b) Find the output of this system if the input is x (t) = C^-3tu (t). (C) Is this system BIBO stable? Prove your answer.

4. The impulse response of a circuit is h (t) = te^-tu (t).

(a) Find the s-domain transfer function of this system.

(b) Find the s-domain output Y (s) if the input to the system is x (t) = u (t).

(c) Perform a partial fraction expansion of the s-domain output Y (s).

(d) Find the time domain output y (t) by using inverse Laplace transforms.