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: $$\begin{array}{l} R=2 \Omega, L=1 \mathrm{H}, C-0.5 \mathrm{F}, v(t)-1 \mathrm{kV} \text { if } \\ 0 < t < 2 \text { and } 0 \text { if } t > 2 \end{array}$$

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:
$$\begin{array}{l}
R=2 \Omega, L=1 \mathrm{H}, C-0.5 \mathrm{F}, v(t)-1 \mathrm{kV} \text { if } \\
0 < t < 2 \text { and } 0 \text { if } t > 2
\end{array}$$

Key Concepts

Laplace Transform

The Laplace transform is a mathematical tool used to convert differential equations in the time domain into algebraic equations in the s-domain. This technique simplifies the analysis of linear time-invariant systems, such as electric circuits governed by differential equations, by handling derivatives as polynomials in s. It is particularly useful in dealing with initial value problems and switching functions.

RLC Circuit Analysis

RLC circuit analysis involves studying the behavior of circuits that include resistors (R), inductors (L), and capacitors (C). In the s-domain, the impedance of each element is represented as algebraic expressions, which allows the derivation of a transfer function relating input to output variables, such as voltage to current. This approach eases the process of solving the circuit’s differential equations, particularly when analyzing transient response.

Initial Conditions

Initial conditions refer to the values of the circuit variables, such as current and voltage, at time t = 0. In Laplace transform methods, zero initial conditions simplify the transformed equations because terms corresponding to these initial values vanish, streamlining the algebraic manipulation and solution process for the circuit's response.

Piecewise Functions and the Unit Step Function

Piecewise functions, often represented using the unit step function (Heaviside function), model signals that turn on or off at specific times in time-domain analysis. In the Laplace transform framework, these functions have known transforms that allow the incorporation of non-continuous inputs, enabling accurate representation of time-limited or switched signals in the s-domain.

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