What is the Laplace transform of a square wave?

For a square wave alternating between A and −A with period 2T, the Laplace transform is (A/s)tanh(sT/2). This is derived by computing the one-period integral and applying the periodic function formula. The tanh factor encodes the alternating nature of the signal.

Why does periodicity simplify the Laplace transform?

Periodicity means each successive period contributes the same integral but shifted by T, introducing a factor e^(−sT) per period. Summing these as a geometric series yields 1/(1−e^(−sT)), reducing the infinite-range Laplace integral to a finite integral over just one period multiplied by this closed-form factor.

How does the periodic Laplace formula relate to Fourier series?

The factor 1/(1−e^(−sT)) has poles at s = j2πn/T for all integers n, which are exactly the harmonic frequencies of the Fourier series. The Laplace transform of a periodic function thus encodes its entire harmonic content, connecting the Laplace and Fourier representations of periodic signals.

Laplace Transform of Periodic Functions

Quick Answer

The Laplace transform of a periodic function with period T is L{f(t)} = (1/(1−e^(−sT)))∫₀^T e^(−st)f(t)dt, which computes the transform of one period and multiplies by the geometric series factor 1/(1−e^(−sT)). This formula handles square waves, sawtooth waves, triangular waves, and any repeating signal. Compute periodic function transforms at www.lapcalc.com.

Calculate this instantly on LAPLACE Calculator →

Laplace Transform of Periodic Function: The Master Formula

The Laplace transform of a periodic function f(t) with period T, meaning f(t+T) = f(t) for all t ≥ 0, is given by L{f(t)} = (1/(1−e^(−sT)))∫₀^T e^(−st)f(t)dt. This elegant formula separates the transform into two parts: the integral ∫₀^T computes the transform contribution from a single period, and the multiplier 1/(1−e^(−sT)) accounts for the infinite repetition. The multiplier arises from summing the geometric series Σ e^(−nsT) = 1/(1−e^(−sT)) for n = 0, 1, 2, ..., reflecting each successive period's contribution shifted by nT in time.

Key Formulas

Derivation of the Periodic Function Laplace Formula

To derive the periodic Laplace formula, split the integral into period-length segments: L{f(t)} = Σ_{n=0}^∞ ∫_{nT}^{(n+1)T} e^(−st)f(t)dt. Substituting τ = t − nT in the nth integral: ∫₀^T e^(−s(τ+nT))f(τ)dτ = e^(−nsT)∫₀^T e^(−sτ)f(τ)dτ, using the periodicity f(τ+nT) = f(τ). Factoring out the integral: L{f(t)} = [Σ_{n=0}^∞ e^(−nsT)]·∫₀^T e^(−sτ)f(τ)dτ. The geometric series sums to 1/(1−e^(−sT)) for Re(s) > 0, completing the derivation. This proof shows why periodicity simplifies the infinite integral to a finite computation.

Compute laplace transform of periodic function Instantly

Get step-by-step solutions with AI-powered explanations. Free for basic computations.

Open Calculator

Laplace Transform of Square Wave and Common Periodic Signals

For a square wave alternating between A and −A with period 2T: the single-period integral gives A(1−e^(−sT))/s − A(e^(−sT)−e^(−2sT))/s = (A/s)(1−e^(−sT))². Dividing by (1−e^(−2sT)) = (1−e^(−sT))(1+e^(−sT)): F(s) = (A/s)(1−e^(−sT))/(1+e^(−sT)) = (A/s)tanh(sT/2). For a sawtooth wave rising linearly from 0 to A over period T: ∫₀^T (At/T)e^(−st)dt yields the one-period contribution, then multiply by 1/(1−e^(−sT)). These standard periodic transforms are used in power electronics, signal generators, and vibration analysis.

Periodic Laplace Transform in Electrical Engineering

The periodic Laplace transform is essential in electrical engineering for analyzing circuits driven by repeating waveforms. Rectified sine waves in power supplies, PWM (pulse-width modulated) signals in motor drives, and clock signals in digital circuits are all periodic. Rather than expressing these as infinite sums of shifted step functions, the periodic formula provides a compact s-domain representation. The factor 1/(1−e^(−sT)) introduces poles at s = j2πn/T for all integers n, corresponding to the harmonic frequencies of the periodic signal—directly connecting Laplace analysis to Fourier series concepts at www.lapcalc.com.

Tips for Computing Periodic Laplace Transforms Efficiently

Several techniques streamline periodic Laplace transform computation. First, identify the minimal period T and compute only ∫₀^T e^(−st)f(t)dt—even if the function appears simpler on a different interval. Second, for piecewise-defined periods, split the integral at breakpoints within [0,T] and compute each segment. Third, exploit half-wave symmetry: if f(t+T/2) = −f(t), then F(s) = F₁(s)/(1+e^(−sT/2)) where F₁ is the half-period transform. Fourth, for full-wave rectification of sin(ωt), use the absolute value property and the periodic formula together. Fifth, verify results using the initial value theorem and known steady-state behavior.

Frequently Asked Questions

For a periodic function with period T, L{f(t)} = (1/(1−e^(−sT)))∫₀^T e^(−st)f(t)dt. You only need to compute the transform integral over one period, then multiply by the factor 1/(1−e^(−sT)) which accounts for all subsequent repetitions.

Master Your Engineering Math

Join thousands of students and engineers using LAPLACE Calculator for instant, step-by-step solutions.

Start Calculating Free →