Math Problem Solver

Circuit Analysis with Laplace Transform

1.1 Given a series RLC circuit with specific initial conditions, derive the expression for the current

through the circuit using the Laplace Transform. Discuss the transient response of the circuit and

how it varies with different damping conditions (underdamped, critically damped, and

overdamped). (13)

1.2 A practical RL series circuit is used in an audio amplifier, with a resistor R and an inductor L. The

circuit is subjected to an input voltage 𝑣𝑖𝑛(𝑡) = 10𝑢(𝑡) 𝑉, 𝑤ℎ𝑒𝑟𝑒 𝑢(𝑡) is the unit step function.

1.2.1 Derive the expression for the output current 𝑖(𝑡) using Laplace Transform (7)

1.2.2 Determine the time-domain expression for 𝑖(𝑡). (6)

1.2.3 Evaluate the initial and final values of 𝑖(𝑡) using the Initial and Final Value Theorems (4)

QUESTION 2 [25]

Introduction to Graph Theory

2.1 Consider a connected planar graph G with vertices V = {v1, v2, v3, v4} and edges E =

{e1, e2, e3, e4, e5, e6}. The graph G is given as follows:

• e1 = (v1, v2)

• e2 = (v2, v3)

• e3 = (v3, v4)

• e4 = (v4, v1)

• e5 = (v1, v3)

• e6 = (v2, v4)

2.1.1 Determine the incidence matrix A of the graph G. (5)

2.2.2 Find a cut-set matrix and a circuit matrix for the graph. (10)

2.2.3 Verify the orthogonality condition between the cut-set matrix and the circuit matrix.