What does s = jω mean?

In control theory, the Laplace variable s is replaced by jω (where j is the imaginary unit and ω is angular frequency in rad/s) to evaluate the steady-state sinusoidal response. This substitution converts the transfer function into the frequency-domain representation H(jω).

How should I enter polynomial coefficients?

Enter coefficients in descending power order, separated by commas. For example, the polynomial s² + 3s + 2 is entered as "1, 3, 2". The constant term (s⁰) is last. Do not skip zero coefficients — include 0 as a placeholder.

What units does the phase output use?

The phase ∠H(jω) is returned in degrees, computed as atan2(imaginary part, real part) of H(jω) converted from radians.

Can this tool handle unstable systems?

Yes — the frequency-domain evaluation works for any rational transfer function regardless of stability. Stability is not assessed here; it only computes the gain and phase at the specified ω.

What is a damped sinusoid in the Laplace domain?

A damped sine, e^(-at)·sin(ωt)·u(t), transforms to ω / ((s+a)² + ω²). The damping coefficient a controls the exponential decay rate, and ω is the sinusoidal frequency in rad/s.

Laplace Transform & Control System Helper

Name: Laplace Transform & Control System Helper
Availability: InStock
Author: Nham Vu

Look up Laplace transform pairs and evaluate transfer function gain and phase at a given frequency.

Transform Lookup

Time-domain signal f(t)

F(s) =

Transfer Function Evaluator

Numerator coefficients (highest power first)

Denominator coefficients (highest power first)

Angular frequency ω (rad/s)

Reference Table — Common Laplace Pairs

#	f(t) (t ≥ 0)	F(s)	Condition
1	δ(t)	1	Unit impulse
2	u(t)	1 / s	Unit step
3	t · u(t)	1 / s²	Unit ramp
4	tⁿ · u(t)	n! / s^(n+1)	n ≥ 0 integer
5	e^(−at) · u(t)	1 / (s + a)	a > 0
6	sin(ωt) · u(t)	ω / (s² + ω²)	ω > 0
7	cos(ωt) · u(t)	s / (s² + ω²)	ω > 0
8	e^(−at)·sin(ωt)·u(t)	ω / ((s+a)² + ω²)	a, ω > 0
9	e^(−at)·cos(ωt)·u(t)	(s+a) / ((s+a)² + ω²)	a, ω > 0
10	t · e^(−at) · u(t)	1 / (s + a)²	a > 0

Summary

Look up Laplace transform pairs and evaluate transfer function gain and phase at a given frequency.

How it works

Select a time-domain signal from the dropdown in the Transform Lookup panel.
The corresponding Laplace-domain formula F(s) is displayed with parameter labels.
Enter numerator and denominator polynomial coefficients (comma-separated, highest power first) in the Transfer Function panel.
Enter an angular frequency ω in rad/s and click Evaluate.
The tool computes H(jω) by substituting s = jω, then returns |H(jω)| and the phase angle in degrees.

Use cases

Quickly look up the Laplace transform of standard control signals without referencing a textbook.
Verify the gain and phase of a designed transfer function at a specific frequency during Bode plot analysis.
Check the frequency response of a PID controller by entering its transfer function coefficients.
Evaluate system bandwidth by sweeping ω and observing where the magnitude drops 3 dB.
Cross-check hand calculations of gain margin or phase margin at crossover frequencies.

Frequently Asked Questions

Last updated: 2026-07-01 · Reviewed by Nham Vu