Laplace Transform Calculator

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laplace transform calculator
Constant: CPower: t^nExponential: e^(at)Sine: sin(at)Cosine: cos(at)Hyperbolic Sine: sinh(at)Hyperbolic Cosine: cosh(at)t * e^(at)
Laplace Transform F(s):



function updateFields(){var type=document.getElementById('func_type').value;document.getElementById('row_c').style.display=(type==='constant'||type==='power'||type==='exp'||type==='sin'||type==='cos'||type==='sinh'||type==='cosh'||type==='teat')?'table-row':'none';document.getElementById('row_a').style.display=(type==='exp'||type==='sin'||type==='cos'||type==='sinh'||type==='cosh'||type==='teat')?'table-row':'none';document.getElementById('row_n').style.display=(type==='power')?'table-row':'none';}function fact(n){if(n===0)return 1;var res=1;for(var i=2;i<=n;i++)res*=i;return res;}function calculateLaplace(){var type=document.getElementById('func_type').value;var c=parseFloat(document.getElementById('input_c').value)||0;var a=parseFloat(document.getElementById('input_a').value)||0;var n=parseFloat(document.getElementById('input_n').value)||0;var showSteps=document.getElementById('steps').checked;var resText='';var stepText='';if(type==='constant'){resText=c+' / s';stepText='Formula: L{C} = C/s';}else if(type==='power'){var f=fact(n);resText=(c*f)+' / s^'+(n+1);stepText='Formula: L{t^n} = n! / s^(n+1)';}else if(type==='exp'){resText=c+' / (s – '+a+')';stepText='Formula: L{e^(at)} = 1 / (s-a)';}else if(type==='sin'){resText=(c*a)+' / (s^2 + '+(a*a)+')';stepText='Formula: L{sin(at)} = a / (s^2 + a^2)';}else if(type==='cos'){resText=(c === 1 ? 's' : c + 's') + ' / (s^2 + '+(a*a)+')';stepText='Formula: L{cos(at)} = s / (s^2 + a^2)';}else if(type==='sinh'){resText=(c*a)+' / (s^2 – '+(a*a)+')';stepText='Formula: L{sinh(at)} = a / (s^2 – a^2)';}else if(type==='cosh'){resText=(c === 1 ? 's' : c + 's') + ' / (s^2 – '+(a*a)+')';stepText='Formula: L{cosh(at)} = s / (s^2 – a^2)';}else if(type==='teat'){resText=c+' / (s – '+a+')^2';stepText='Formula: L{t * e^(at)} = 1 / (s-a)^2';}document.getElementById('resultFormula').innerHTML='F(s) = '+resText;if(showSteps){document.getElementById('stepDetails').innerHTML=stepText;document.getElementById('stepDetails').style.display='block';}else{document.getElementById('stepDetails').style.display='none';}}function resetCalc(){document.getElementById('answer').style.display='block';document.getElementById('resultFormula').innerHTML='';document.getElementById('stepDetails').style.display='none';setTimeout(updateFields,10);}

Laplace Transform Calculator Use

This laplace transform calculator is an essential tool for students and engineers working with differential equations, signal processing, and control systems. The Laplace transform is an integral transform that converts a function of a real variable (usually time, t) to a function of a complex variable (frequency, s).

To use this calculator, simply select the type of function you are working with from the dropdown menu, enter the necessary coefficients and parameters, and click "Transform".

Function Type
Choose from common functions like constants, power functions, exponentials, or trigonometric functions (Sine/Cosine).
Coefficient (C)
The multiplier in front of your function (e.g., in 5sin(2t), the coefficient is 5).
Parameter (a)
The frequency or exponent value (e.g., in e^(3t), the parameter a is 3).
Power (n)
The exponent value for power functions (e.g., in t^2, n is 2).

How It Works

The laplace transform calculator uses the standard definition of the unilateral Laplace transform, which is defined by the following integral:

F(s) = L{f(t)} = ∫ [0 to ∞] e^(-st) f(t) dt

Instead of performing the integration manually every time, engineers use a table of standard transforms. Our calculator applies these verified formulas instantly:

  • Linearity: The transform of a constant times a function is the constant times the transform.
  • Frequency Shifting: Multiplying by an exponential in the time domain results in a shift in the frequency domain.
  • Power Rule: Transforming t^n involves factorials and increasing the power of s in the denominator.

Calculation Example

Example: Find the Laplace transform of the function f(t) = 3 sin(4t).

Step-by-step solution:

  1. Identify the coefficient: C = 3
  2. Identify the parameter: a = 4
  3. Select the Sine formula: L{sin(at)} = a / (s² + a²)
  4. Apply linearity: L{3 sin(4t)} = 3 * [4 / (s² + 4²)]
  5. Calculate constants: 3 * 4 = 12, and 4² = 16
  6. Result: F(s) = 12 / (s² + 16)

Common Questions

What is the "s" variable in the Laplace transform?

In the context of the laplace transform calculator, s is a complex frequency parameter, defined as s = σ + jω. It represents a domain where differential equations become simple algebraic equations, making them much easier to solve.

Why is the Laplace transform used in control systems?

Control systems use Laplace transforms to determine the "Transfer Function" of a system. This allows engineers to analyze stability and transient response without solving messy time-domain integrals.

Can this calculator handle inverse Laplace transforms?

This specific tool is designed for the forward transform (Time to Frequency). To go backwards (Frequency to Time), you would typically use partial fraction decomposition to match the result back to these standard forms.

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