Question

What does the Laplace transform do the differential equation of an RLC circuit?

• A. It simplifies the equation by transforming it from the frequency domain to the time domain
• B. It retains the differential equation as it is
• C. It converts the circuit parameters into phasor form
• D. It converts a time-domain differential equation into an algebraic equation in the s-domain

Hint:

The Laplace transform simplifies transient circuit analysis because it maps difficult time-domain operations directly into linear algebraic systems: Calculus in \(t\) \(\rightarrow\) Algebra in \(s\).

Answer 1

The Correct Option is D

Concept: The Laplace transform is a powerful integral mathematical tool used to convert functions from the continuous time domain, represented by variable \(t\), into the complex frequency domain, represented by the complex variable \(s = \sigma + j\omega\). The mathematical definition of the unilateral Laplace transform of a time function \(f(t)\) is: \[ \mathcal{L}\{f(t)\} = F(s) = \int_{0}^{\infty} f(t) e^{-st} dt \] A vital operational property of this transform is its treatment of differentiation: \[ \mathcal{L}\left\{\frac{df(t)}{dt}\right\} = sF(s) - f(0^-) \] This property allows differential calculus operations to be swapped out for simple polynomial equations containing the algebraic variable \(s\).

Step 1: Examining the effect of Laplace transform on RLC systems.
An RLC circuit in the time domain contains differential or integral expressions representing voltage-current relationships across inductors (\(v_L = L \frac{di}{dt}\)) and capacitors (\(i_C = C \frac{dv}{dt}\)). When we apply the Laplace transform to the total system equation, the operations of differentiation and integration are systematically mapped to multiplication and division by the complex variable \(s\).

Step 2: Verifying option alignments.
This mathematical transformation completely redefines a complex differential system into a linear polynomial system. Consequently, it maps a time-domain differential equation directly into an algebraic equation within the complex \(s\)-domain. This matches the exact wording of Option (D).