Library / Foundations: Maths / FMA-47
Foundations: Maths
Laplace transform of a decaying exponential
Laplace transform of e^(−at) for t ≥ 0, evaluated at a real value of s.
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- Foundations: Maths
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- F(s)
The Laplace transform of e−at for t ≥ 0 is 1/(s + a). It is defined for s with real part greater than −a. Transforming a differential equation turns it into algebra in s, where this pair appears as the transform of a decaying exponential. Initial conditions must be included when transforming a derivative.
Laplace transform of e^(−at) for t ≥ 0, evaluated at a real value of s. It is one of the standard pairs used to turn differential equations into algebra.
Calculator
- Time constant of the exponential, 1/a1 s
- Pole location, s = −a−1 1/s
- Value of the exponential after one time constant, e−10.3679
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 1/s the transform of e−at with a = 1 1/s is 0.3333. The transform has a single pole at s = −1: the decay rate of the time function appears as the location of the pole.
- A faster decay (larger a) moves the pole further left in the s-plane, which is the picture used for stability in control engineering.
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 | 0.04762 | 0.9091 | 0.8615 | +10% | −6.25 % |
| a Decay rate a | 0.1 to 20 1/s | 0.04545 | 0.4762 | 0.4307 | +10% | −3.23 % |
Sweep of Laplace variable s (real value)
| Laplace variable s (real value) (1/s) | Transform at s |
|---|---|
| 0.1 | 0.9091 |
| 2.587 | 0.2787 |
| 5.075 | 0.1646 |
| 7.562 | 0.1168 |
| 10.05 | 0.0905 |
| 12.54 | 0.07387 |
| 15.02 | 0.0624 |
| 17.51 | 0.05402 |
| 20 | 0.04762 |
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; the transform converges for s greater than −a. Lowers the value in inverse proportion to s + a: a larger s gives a smaller transform. Raising it by 0.5 (to 2.5) moves the result by −14.3 %, from 0.3333 to 0.2857.
Decay rate a
The decay rate a in the time function e−at, in 1/s. Its reciprocal is the time constant. Lowers the value in inverse proportion to s + a, in exactly the same way as s. Raising it by 0.5 (to 1.5) moves the result by −14.3 %, from 0.3333 to 0.2857.
Worked example with the values above
- Add the two rates: s + a = 2 + 1 = 3.
- Take the reciprocal: 1 / 3.
- F(s) = 0.3333.
Assumptions and limits
- The time function is zero before t = 0 (causal) and the transform converges.
- s is in the region where the transform exists, here real and greater than −a.
Common mistakes
- Forgetting that the transform has a pole at s = −a, not s = a.
- Ignoring initial conditions when transforming the equation of which this is a term.
Statistics and uncertainty
The transform is an exact mathematical relation, so it has no statistical content; its use in practice is to model a measured decay, where the decay rate a is the uncertain quantity (read it from a fit, as for any first-order response).
Where it is used
- Building transfer functions of first-order systems.
- Finding inverse transforms by partial fractions.
Maths you need
- Laplace transform: turns a function of time into a function of s
- Causal function: one that is zero before time zero