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

Using the Laplace transform and showing the details, find the current if in the circuit in Fig. 129 , assuming zero initial current and charge and:
$$R=2 \Omega, L=1 \mathrm{H}, C=0.1 \mathrm{F}, v=255 \sin t \mathrm{V}$
if $0 < t < 2$ w and 0 if $\prime > 2 \pi$$

Key Concepts

Laplace Transform

The Laplace transform is a mathematical tool that converts functions from the time domain into the complex frequency domain (s-domain). In the context of electrical circuits, it simplifies the analysis of circuits by turning differential equations into algebraic equations. This makes it easier to solve for circuit responses, especially when dealing with circuits that include elements like resistors, inductors, and capacitors.

RLC Circuit Analysis

An RLC circuit, which consists of resistors (R), inductors (L), and capacitors (C), is typically governed by second-order differential equations due to the energy storage elements (L and C). The analysis of these circuits involves deriving the differential equation that describes the current or voltage behavior over time and then using methods, such as the Laplace transform, to solve these equations in a systematic manner.

Zero Initial Conditions

Zero initial conditions imply that the initial values of current through the inductor and the charge on the capacitor are zero. This assumption simplifies the Laplace transform process because it removes additional terms related to initial conditions in the transformed equations. It is a common starting point for analyzing transient responses in circuits.

Piecewise Forcing Functions

The forcing function in a circuit can be piecewise-defined, meaning it has different expressions over different time intervals. This is often handled by incorporating unit step (Heaviside) functions in the Laplace transform domain, which allow for the correct representation of time constraints in the response. This concept is crucial when analyzing circuits subject to inputs that are not continuously active over time.

Inverse Laplace Transform

Once the circuit equations are solved in the s-domain, the inverse Laplace transform is used to convert the solution back into the time domain. This step recovers the actual temporal behavior of the circuit element, such as current or voltage, and is essential for understanding how the circuit responds to a given input over time.

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