Calculator Form
Formula Used
The main identity is the second shifting theorem. When a function is delayed by a unit step, its transform receives an exponential delay factor.
L{u(t - a) f(t - a)} = e^(-as) F(s)
For a rectangular pulse, the calculator uses the difference of two unit steps.
L{u(t - a) - u(t - b)} = [e^(-as) - e^(-bs)] / s
The tool also applies common base transforms. These include constants, powers, exponential terms, sine terms, cosine terms, and polynomial sums.
Example Data Table
| Mode | Input | Delay | Base Transform | Final Transform |
|---|---|---|---|---|
| Single step | u(t - 2) | 2 | 1 / s | e^(-2s) / s |
| Power | u(t - 3)(t - 3)^2 | 3 | 2 / s^3 | 2e^(-3s) / s^3 |
| Pulse | u(t - 1) - u(t - 4) | 1 to 4 | 1 / s | [e^(-s) - e^(-4s)] / s |
How to Use This Calculator
- Select single shifted step or rectangular pulse mode.
- Choose the base function type for the delayed signal.
- Enter delay, scale, and any needed function values.
- Use pulse end only when pulse mode is selected.
- Enter an s value for a numeric estimate.
- Press calculate to show results below the header.
- Use CSV or PDF buttons to save the result.
About Unit Step Laplace Calculations
Purpose
A unit step Laplace calculator helps study delayed signals. These signals appear in control systems, circuits, vibration models, and process design. The step function turns a signal on at a chosen time. That makes piecewise behavior easier to describe. Instead of writing many separate time intervals, you can write one compact expression.
Practical Value
The Laplace transform changes a time function into an s domain expression. This change can simplify differential equations. Delays need special care. A delay in time becomes an exponential factor in the transform. This factor is important because it keeps the start time visible after transformation.
Supported Inputs
This calculator supports constants, powers, exponential functions, sine functions, cosine functions, and polynomial forms. It also supports rectangular pulses. A pulse is built from two step functions. The first step turns the signal on. The second step turns it off. This is useful for finite duration inputs.
Result Meaning
The result table shows the time form, base function, base transform, and final transform. The base transform is the transform before delay. The final transform includes the delay multiplier. A numeric estimate is also shown at your selected s value. This estimate helps compare cases quickly.
Accuracy Notes
Symbolic outputs use standard transform rules. Polynomial entries are read as descending coefficients. For example, 2,0,5 means 2t squared plus 5. The calculator does not replace formal proof. It gives structured help for routine engineering and mathematics work.
Best Workflow
Start with a simple function. Confirm the delay. Then test more complex inputs. Use the example table as a pattern. Export the result when you need a record for reports, homework, or design notes. Clear inputs give clearer transforms. Always check units and timing assumptions before final use.
FAQs
What is a unit step function?
A unit step function is zero before a chosen time and one after that time. It is often used to model delayed switching.
What does the delay value mean?
The delay value is the time where the shifted signal starts. In u(t - a), the delay is a.
Which shifting theorem is used?
The calculator uses the second shifting theorem. It places an exponential multiplier on the base transform.
Can this calculator handle pulses?
Yes. Pulse mode uses two step functions. One turns the signal on, and the other turns it off.
How are polynomial coefficients entered?
Enter coefficients in descending order. For example, 1,0,3 means t squared plus 3.
What is the numeric estimate?
It is the final transform evaluated at your chosen s value. It helps compare different delayed signals.
Why does e^(-as) appear?
It appears because a time delay changes the transform by an exponential factor. The value a is the delay.
Can I export the result?
Yes. Use the CSV button for spreadsheet records. Use the PDF button for a simple printable summary.