Library / Foundations: Maths / FMA-48
Foundations: Maths
Laplace transform of a sine
Laplace transform of sin ωt for t ≥ 0, evaluated at a real value of s.
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- Result
- F(s)
The Laplace transform of sin ωt for t ≥ 0 is ω/(s² + ω²). It has a pair of poles at s = ±jω on the imaginary axis, which is how a pure undamped oscillation shows up. With the cosine pair it lets the response to a sinusoidal input be found by algebra. The value here is evaluated at a real s.
Laplace transform of sin ωt for t ≥ 0, evaluated at a real value of s. Together with the cosine pair it gives the response of a system to a sinusoidal input.
Calculator
- Frequency in Hz, ω/2π0.4775 Hz
- Period, 2π/ω2.094 s
- Transform of the cosine with the same ω and s (companion pair)0.1538
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- At s = 2 and ω = 3 rad/s the transform of sin ωt is 0.2308. The cosine pair gives 0.1538 at the same s, and the squares of the two transforms add up to 1/(s² + ω²), which is 0.07692.
- The poles at s = ±jω mean a sinusoid does not decay: a system driven at its natural frequency responds without limit, which is resonance.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| s Laplace variable s (real value) | 0.1 to 20 1/s | 0.007335 | 0.333 | 0.3256 | +10% | −6.07 % |
| ω Angular frequency | 0.1 to 50 rad/s | 0.01997 | 0.248 | 0.2281 | +10% | −3.96 % |
Sweep of Angular frequency
| Angular frequency (rad/s) | Transform at s |
|---|---|
| 0.1 | 0.02494 |
| 6.337 | 0.1435 |
| 12.57 | 0.07756 |
| 18.81 | 0.05256 |
| 25.05 | 0.03967 |
| 31.29 | 0.03183 |
| 37.52 | 0.02657 |
| 43.76 | 0.0228 |
| 50 | 0.01997 |
Sweep one variable
Variables and how each one contributes
Transform at s
What the equation gives you. Everything else feeds it.
Laplace variable s (real value)
The Laplace variable s, shown here as a real value in 1/s. In general s is complex. Lowers the value: a larger s makes the denominator s² + ω² larger. Raising it by 0.5 (to 2.5) moves the result by −14.8 %, from 0.2308 to 0.1967.
Angular frequency
The angular frequency ω of the sine, in rad/s. Raises the value when s is large compared with ω, because ω in the numerator then dominates; it flattens and falls once ω exceeds s. Raising it by 0.5 rad/s (to 3.5 rad/s) moves the result by −6.67 %, from 0.2308 to 0.2154.
Worked example with the values above
- Square both: s² = 4 and ω² = 9, so the denominator is 13.
- The numerator is ω = 3.
- F(s) = 3 / 13 = 0.2308.
Assumptions and limits
- The function is sin ωt for t ≥ 0 and zero before.
- s is in the region of convergence (real part greater than 0).
Common mistakes
- Using the sine pair for cosine, which has s in the numerator instead of ω.
- Forgetting the factor ω in the numerator.
Statistics and uncertainty
The pair is exact. When it is used to model a measured oscillation, the frequency ω is the quantity to estimate, and its error shifts the pole positions in proportion.
Where it is used
- Finding the response of a system to a sinusoidal input.
- Deriving transfer functions of oscillators.
Maths you need
- Laplace transform: turns a function of time into a function of s
- Causal function: one that is zero before time zero