Calculator Inputs
Formula Used
Periodic Laplace transform:
L{fp(t)} = [ ∫0T e-st f(t) dt ] / [ 1 - e-sT ]
For enabled piecewise intervals, the numerator is evaluated as a sum of numerical integrals over each segment inside one period.
Average value:
Average = (1 / T) ∫0T f(t) dt
RMS value:
RMS = √[(1 / T) ∫0T f(t)2 dt]
This page uses midpoint numerical integration, which is practical when your periodic function is piecewise, mixed, or not easy to transform symbolically.
How to Use This Calculator
- Enter the period length T for one repeating cycle.
- Set the target s value where you want the transform.
- Choose chart limits to inspect transform behavior across several s values.
- Enable the segments that describe one full cycle of your signal.
- For each enabled segment, enter start, end, type, and coefficients.
- Interpret formulas with local time u = t minus the segment start.
- Click Calculate to view the transform, averages, waveform preview, and graph.
- Use the CSV and PDF buttons to keep a study or project record.
Example Data Table
| Item | Value | Notes |
|---|---|---|
| Period T | 2.0 | One full pattern repeats every two time units. |
| Segment 1 | [0, 1), constant, a = 2 | This creates a flat level during the first half-cycle. |
| Segment 2 | [1, 2), sine, a = 1.5, b = π, c = 0, d = 0 | This adds oscillation during the second half-cycle. |
| Target s | 1.5 | Use this for a detailed evaluated transform value. |
| Chart Range | 0.2 to 5.0 by 0.2 | Shows how the transform changes as decay increases. |
Frequently Asked Questions
1. What is a periodic Laplace transform?
It is the Laplace transform of a function that repeats every period T. Instead of integrating over all time directly, you integrate over one cycle and apply the periodic formula.
2. Why does the calculator use one-period data only?
A periodic signal repeats exactly. Once one cycle is defined correctly, the periodic Laplace formula extends that cycle to all future repetitions automatically.
3. What does local time u mean?
Local time resets at each enabled segment start. This makes coefficients easier to control because every piece begins from its own interval origin.
4. Can I leave gaps between segments?
Yes. Any uncovered interval inside one period is treated as zero. This is useful for pulse trains, gated signals, and intermittent forcing functions.
5. Why must s be positive here?
Positive s keeps the exponential kernel decaying and avoids unstable or misleading numeric behavior for many practical periodic signals studied in coursework and engineering models.
6. Is the result exact or numerical?
The result is numerical. This approach is very helpful when piecewise segments mix constants, polynomials, exponentials, and trigonometric terms that are tedious to integrate by hand.
7. What does the graph show?
The main graph shows the periodic Laplace transform versus s across your selected range. The waveform preview shows the one-period shape used in the calculation.
8. When should I increase integration steps?
Increase the step count when segments change rapidly, oscillate strongly, or contain steep exponential growth or decay. More steps usually improve numerical stability and smoothness.