Laplace Transform

The Laplace transform turns differential equations into algebra. Learn the definition and region of convergence, common transform pairs, properties (derivative, integral, shift), transfer functions, poles and zeros, stability from pole locations, first- and second-order responses (ω_n, ζ, overshoot), and initial/final value theorems.

Circuits and control systems are described by differential equations. Laplace transforms convert them into polynomials in s, so responses, stability and frequency behaviour can be found quickly — the language of analog circuit and control design.

An analogy

Laplace is like translating a hard poem into a language where grammar is simpler: solve it there, then translate the answer back.

Why Laplace?

Derivatives become multiplication by s, integrals division by s; initial conditions are included automatically. Circuit elements become impedances: R, sL, 1/(sC).

Poles and zeros

Poles (roots of the denominator) determine natural response: left-half-plane poles → decaying (stable); right-half-plane → growing (unstable); complex poles → oscillation. Zeros shape the response.

Second-order behaviour

Underdamped (ζ < 1) oscillates with overshoot; critically damped (ζ = 1) is fastest without overshoot; overdamped (ζ > 1) is sluggish. Series RLC: ω_n = 1/√(LC), ζ = (R/2)√(C/L).

Frequency response and design

Setting s = jω gives the frequency response (Bode plots). Pole placement, compensation and filter design all work in the s-plane. Partial-fraction expansion returns to time domain.

Formulas

• Definition: X(s) = ∫₀^∞ x(t) e^(−st) dt
• Key pairs: u(t) ↔ 1/s, e^(−at) ↔ 1/(s + a), sin ωt ↔ ω/(s² + ω²)
• Derivative: dx/dt ↔ sX(s) − x(0)
• Second-order system: H(s) = ω_n² / (s² + 2ζω_n s + ω_n²)
• Overshoot & settling: M_p = e^(−πζ/√(1−ζ²)), t_s ≈ 4/(ζω_n)
• Final value: x(∞) = lim_{s→0} s X(s)

Worked example: Overshoot

Given: ζ = 0.5

Find: M_p.

• M_p = e^(−π × 0.5/√0.75) = 0.163

Answer: 16.3 % overshoot

Worked example: Series RLC

Given: R = 10 Ω, L = 1 mH, C = 1 µF

Find: ω_n, ζ, overshoot, settling.

• ω_n = 1/√(10⁻⁹) = 31 623 rad/s
• ζ = 5 × √(10⁻³) = 0.158
• M_p = 60 %, t_s = 4/(ζω_n) = 0.8 ms

Answer: Lightly damped: 60 % overshoot, 0.8 ms settling

Common mistakes

• Wrong: Applying the final value theorem to unstable systems. — Right: Valid only if all poles of sX(s) are in the left half-plane.
• Wrong: Forgetting initial conditions. — Right: Include x(0) terms in derivative transforms.

Applications

• Circuit transients: RLC and filter responses.
• Control systems: Transfer functions and stability.
• Filter design: Pole placement for Butterworth etc.

Key takeaways

• Differential equations → algebra in s.
• Poles decide stability and response.
• ω_n, ζ describe second-order systems.
• Final value = lim sX(s).

Going further

• Partial fractions: Inverse transforms of rational functions.
• Routh–Hurwitz: Stability without solving roots.
• Bode asymptotes: From poles/zeros to frequency plots.