📘 Laplace Transform – Complete Theory & Worked Examples (Control + Network)
Laplace Transform is one of the most important mathematical tools in Electrical Engineering. It converts time-domain differential equations into algebraic equations in s-domain. This simplifies analysis of circuits and control systems.
🔹 1. Why Laplace Transform?
In network analysis, solving differential equations directly is complex. Laplace Transform converts derivatives into multiplication by s. Thus circuit equations become simple algebraic equations.
🔹 2. Definition
L{f(t)} = ∫₀^∞ f(t)e^{-st} dt
Where:
- s = σ + jω
- f(t) = time domain function
🔹 3. Important Laplace Transform Pairs
L{1} = 1/s L{t} = 1/s² L{e^{-at}} = 1/(s+a) L{sin(at)} = a/(s²+a²) L{cos(at)} = s/(s²+a²)
🔹 4. Properties
Linearity
L{af(t) + bg(t)} = aF(s) + bG(s)
Differentiation in Time Domain
L{df/dt} = sF(s) − f(0)
Second Derivative
L{d²f/dt²} = s²F(s) − sf(0) − f'(0)
🔹 5. Worked Example 1 – Basic Transform
Find Laplace of:
f(t) = 5e^{-2t}
L{5e^{-2t}} = 5/(s+2)
🔹 6. Worked Example 2 – Using Differentiation Property
Given: f(t) = 3t²
L{t²} = 2/s³ Therefore, L{3t²} = 6/s³
🔹 7. Application in Circuit Analysis
Consider RL circuit:
L di/dt + Ri = V(t)
Taking Laplace:
L[sI(s) − i(0)] + RI(s) = V(s)
Solve algebraically for I(s).
🔹 8. Initial & Final Value Theorem
Initial Value Theorem
f(0⁺) = lim s→∞ [sF(s)]
Final Value Theorem
f(∞) = lim s→0 [sF(s)]
🎯 GATE Important Points
- Derivative property frequently tested
- Initial/Final value theorem common
- Inverse Laplace important
- Partial fraction expansion must be strong
Laplace Transform = Bridge Between Time & Frequency Domain
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