Using the Laplace transform and showing the details, find...

Using the Laplace transform and showing the details, find the current $i(t)$ in the circuit in Fig. 128 , assuming zero initial current and charge on the capacitor and:
$L=1 \mathrm{H}, \mathrm{C}=10^{-2} \mathrm{F}, \mathrm{v}=-9900 \cos t \mathrm{V}$
$\pi < t < 3$ and 0 othenvise

Key Concepts

Heaviside (Step) Function

The Heaviside step function is used to represent signals that are defined over a specific time interval, effectively 'turning on' or 'off' at predetermined times. In circuits with piecewise or time-limited inputs, the step function facilitates the modeling of these time dependencies in the Laplace domain by multiplying the transformed input by exponential terms, which is essential for properly accounting for the input signal's behavior over time.

Transfer Function

In the context of circuit analysis, the transfer function is the ratio of the output variable (such as current or voltage across a specific element) to the input source expressed in the Laplace domain. It characterizes the dynamic response of the circuit and allows for a systematic approach to determining the output response using algebraic techniques after transforming the input through the Laplace transform.

Inverse Laplace Transform

After solving the circuit equations in the Laplace (s-domain), the inverse Laplace transform is used to convert the solution back into the time domain. This process involves techniques such as partial fraction decomposition, convolution, or using standard transform pairs, and is crucial for obtaining the actual time-domain response of the circuit components, like the current or voltage as functions of time.

Laplace Transform

The Laplace transform is a powerful integral transform method used to convert differential equations governing circuit behavior into algebraic equations in the complex frequency domain. This method simplifies the analysis of circuits by transforming time-domain functions (which include derivatives and integrals due to circuit elements like inductors and capacitors) into functions of a complex variable, making it easier to solve linear differential equations subject to given initial conditions.

Initial Conditions

In the analysis of circuits using the Laplace transform, initial conditions (such as initial current in an inductor or initial voltage across a capacitor) are incorporated directly into the transformed equations. When the initial conditions are zero, the Laplace-transformed circuit equations become simpler because the terms arising from the initial states vanish. This greatly simplifies the algebra involved in solving the circuit.

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