Calculator Inputs
This engineering tool supports real poles, repeated poles, damped oscillation terms, and optional pure delay.
Example Data Table
These examples show common engineering response templates handled by the calculator.
| Case | F(s) | f(t) | Typical interpretation |
|---|---|---|---|
| Real pole | 4/(s+2) | 4e-2t | First-order decay in thermal or RC systems. |
| Repeated pole | 3/(s+1)2 | 3te-t | Repeated dynamics in higher-order transient models. |
| Damped oscillation | (2(s+1)+6)/((s+1)2+32) | 2e-tcos(3t)+2e-tsin(3t) | Underdamped vibration or control response. |
| Delayed response | e-2s[5/(s+4)+2/(s+1)2] | u(t-2)[5e-4τ+2τe-τ] | Transport delay in process or signal systems. |
Formula Used
The calculator applies standard inverse Laplace pairs and the time-shift theorem to residue-based engineering models.
1) Real first-order pole
L-1{ A/(s+a) } = Ae-at
2) Repeated pole
L-1{ R/(s+b)n } = R · tn-1e-bt / (n-1)!
3) Damped oscillatory pair
L-1{ [C1(s+σ)+C2] / [(s+σ)2+ω2] } = e-σt[C1cos(ωt)+(C2/ω)sin(ωt)]
4) Time shift theorem
L-1{ e-LsF(s) } = u(t-L)f(t-L)
The complete calculator combines all active terms into one delayed time response, then evaluates the resulting waveform over the chosen time grid.
How to Use This Calculator
- Enter up to three simple residue terms in the form
A/(s+a). - Enable the repeated-pole block when your model contains
R/(s+b)^n. - Enable the oscillatory block for underdamped terms with
((s+σ)^2+ω^2). - Add a delay if the model includes a factor like
e^-Ls. - Choose the time window and number of plot points.
- Press Submit and Compute to show the inverse result, metrics, graph, and downloadable table above the form.
FAQs
1) What kind of inverse Laplace problems does this tool solve?
It solves structured engineering forms built from real poles, repeated poles, damped oscillatory terms, and optional pure delays. That covers many transfer-function responses used in controls, circuits, vibration, and process systems.
2) Can it handle arbitrary symbolic algebra?
No. This version is designed for practical residue-based engineering models rather than unrestricted symbolic manipulation. It is strongest when your Laplace function is already decomposed into common inverse-transform building blocks.
3) Why do I see a unit-step term in delayed responses?
A factor of e-Ls in the Laplace domain shifts the time response by L. The unit-step function keeps the signal at zero before the delay begins.
4) What does the repeated pole order mean?
A repeated pole of order n produces a polynomial factor in time. As n increases, the response includes higher powers of time before the exponential decay term.
5) How is the oscillatory block interpreted physically?
It represents damped sinusoidal motion, common in second-order control systems, mechanical vibration, and RLC circuits. Sigma sets decay and omega sets oscillation frequency.
6) What does the stability note tell me?
It checks whether the decay parameters are positive. Positive values usually produce bounded decaying behavior, while negative values may indicate growth, instability, or an unrealistic modeled response.
7) What is the approximate response area?
It is the trapezoidal numerical integral of the plotted time response over your selected window. This helps compare transient magnitude and cumulative signal effect.
8) Why should I increase the plot points?
More points improve the smoothness of the graph and the accuracy of displayed peaks, minima, and area estimates. Sharp transients and oscillations especially benefit from denser sampling.