Inverse Laplace Transform using Partial Fraction Calculator

Calculate the time-domain function f(t) from a second-order s-domain transfer function.

Enter Transfer Function Coefficients

F(s) =

s +

s² + s +

What is an Inverse Laplace Transform using Partial Fraction Calculator?

An inverse Laplace transform using partial fraction calculator is a tool designed to convert a function from the complex frequency domain (the ‘s-domain’) back to the time domain (the ‘t-domain’). This process is fundamental in control systems, circuit analysis, and solving linear differential equations. The “partial fraction” method is a crucial algebraic technique used to break down complex rational functions (fractions of polynomials) into simpler fractions. These simpler fractions directly correspond to known inverse Laplace transform pairs, making the conversion manageable.

This specific calculator focuses on second-order systems, which are common in engineering and physics to model behaviors like oscillations and decay. By inputting the coefficients of the transfer function, the calculator automatically determines the system’s poles (the roots of the denominator) and applies the correct partial fraction expansion to find the final time-domain equation, f(t).

The Formula Behind the Calculation

The calculator solves for the inverse Laplace transform of a second-order transfer function, F(s), of the form:

F(s) = (As + B) / (s² + Cs + D)

The core of the method is to find the roots (poles) of the denominator polynomial `s² + Cs + D = 0` using the quadratic formula. The nature of these roots dictates the form of the partial fraction expansion and, consequently, the form of the time-domain function f(t).

1. Distinct Real Roots: If C² – 4D > 0, the denominator has two different real roots, p₁ and p₂. The function is expanded to `F(s) = K₁/(s – p₁) + K₂/(s – p₂)`. The inverse is `f(t) = K₁e^(p₁t) + K₂e^(p₂t)`.
2. Repeated Real Roots: If C² – 4D = 0, there is one real root, p, with a multiplicity of two. The expansion is `F(s) = K₁/(s – p) + K₂/(s – p)²`. The inverse is `f(t) = K₁e^(pt) + K₂te^(pt)`.
3. Complex Conjugate Roots: If C² – 4D < 0, the roots are a complex conjugate pair, `α ± jω`. The function is manipulated to match the standard forms for damped sinusoids. The resulting inverse is `f(t) = e^(αt) * (M cos(ωt) + N sin(ωt))`.

Variables Table

Variable	Meaning	Unit	Typical Range
A, B, C, D	Coefficients of the s-domain transfer function polynomials.	Unitless	Any real number
s	Complex frequency variable (s = σ + jω).	rad/s	Complex plane
f(t)	The resulting time-domain function.	Varies (e.g., Volts, Amps, Position)	Real numbers
p₁, p₂	Poles of the system (roots of the denominator).	rad/s	Complex plane

For more details on the method, consider reviewing resources on the Inverse Laplace Transform by Partial Fraction Expansion.

Practical Examples

Example 1: Overdamped System (Distinct Real Roots)

Consider a system with the transfer function `F(s) = (s + 5) / (s² + 6s + 8)`.

• Inputs: A=1, B=5, C=6, D=8
• Calculation: The denominator `s² + 6s + 8` factors to `(s + 2)(s + 4)`. The poles are s = -2 and s = -4. The partial fraction expansion is `F(s) = 1.5/(s + 2) – 0.5/(s + 4)`.
• Result: The inverse Laplace transform is `f(t) = 1.5e^(-2t) – 0.5e^(-4t)`. This shows a response that is a combination of two decaying exponentials without oscillation.

Example 2: Underdamped System (Complex Conjugate Roots)

Consider a system with the transfer function `F(s) = (s + 2) / (s² + 2s + 10)`.

• Inputs: A=1, B=2, C=2, D=10
• Calculation: The poles are `s = -1 ± j3`. The function is rewritten to match sine and cosine transform pairs: `(s+1)/((s+1)² + 3²) + 1/((s+1)² + 3²)`. For help with this step, you can use a Quadratic Formula Solver.
• Result: The inverse Laplace transform is `f(t) = e^(-t) * (cos(3t) + (1/3)sin(3t))`. This represents a damped sinusoidal oscillation.

How to Use This Inverse Laplace Transform Calculator

Using this calculator is a straightforward process:

1. Enter Coefficients: Input the values for A and B for the numerator polynomial, and C and D for the denominator polynomial, into their respective fields. The visual representation of F(s) will update as you type.
2. Calculate: Click the “Calculate” button to perform the inverse Laplace transform.
3. Review Results: The primary result, `f(t)`, will appear in the green box. You can also review the intermediate steps, including the type of roots, the calculated poles, and the partial fraction expansion.
4. Analyze the Graph: The chart below the results provides a visual plot of the time-domain function `f(t)`, allowing you to see the system’s response over time (e.g., if it oscillates, decays quickly, etc.). For more on system responses, see this article on Second Order Systems.
5. Copy: Use the “Copy Results” button to easily save the calculated function and intermediate values to your clipboard.

Key Factors That Affect the Time-Domain Response

The behavior of the output function `f(t)` is entirely determined by the location of the poles of the transfer function F(s).

• Real Part of the Pole (σ): This determines the rate of exponential decay or growth. A more negative real part leads to faster decay. A positive real part signifies an unstable system where the response grows infinitely.
• Imaginary Part of the Pole (ω): This determines the frequency of oscillation. If the imaginary part is zero, the system does not oscillate. A larger imaginary part corresponds to a higher frequency of oscillation.
• Poles on the Real Axis: Lead to non-oscillatory exponential responses (overdamped or critically damped).
• Complex Conjugate Poles: Always result in an oscillatory response (underdamped). The further the poles are from the real axis, the higher the oscillation frequency.
• Poles at the Origin (s=0): A pole at the origin indicates an integration effect, leading to a response that approaches a constant non-zero value or grows linearly with time.
• Zeros (Numerator Roots): The zeros of the transfer function (roots of the numerator) affect the amplitude and phase of the response components but not the fundamental nature (e.g., oscillation frequency, decay rate) which is set by the poles.

Frequently Asked Questions (FAQ)

Why is partial fraction expansion necessary?

Partial fraction expansion breaks a complex function into simpler parts whose inverse Laplace transforms are listed in standard tables. It’s a “divide and conquer” strategy for complex fractions.

What does this calculator handle?

This calculator is specifically designed for rational functions with a second-order denominator and a first-order (or constant) numerator. It correctly handles distinct real, repeated real, and complex conjugate roots.

What if the numerator’s degree is higher than the denominator’s?

If the degree of the numerator is greater than or equal to the degree of the denominator, you must first perform polynomial long division until the remainder fraction is “proper” (numerator degree is smaller). This calculator assumes the function is already proper.

What do the ‘poles’ of a system represent physically?

The poles represent the natural modes of the system. They are the inherent frequencies and decay rates at which the system will respond when “disturbed.” Their location in the complex plane tells you everything about the system’s stability and transient response. For more insight, research Control System Stability.

Are the input coefficients unitless?

Yes, for this general mathematical calculator, the coefficients A, B, C, and D are treated as unitless real numbers. In a physical application (like an RLC circuit), these coefficients would be derived from physical quantities like resistance, inductance, and capacitance.

How does this relate to solving differential equations?

The Laplace transform converts a linear ordinary differential equation (ODE) in the time domain into an algebraic equation in the s-domain. After solving for the output variable algebraically, you use the inverse Laplace transform (the process this calculator performs) to get the solution to the original ODE.

What is an underdamped vs. overdamped response?

An underdamped response, caused by complex poles, oscillates as it settles to its final value. An overdamped response, from distinct real poles, approaches the final value slowly without any oscillation.

Can this calculator handle repeated roots?

Yes. If you enter coefficients that result in a determinant (C² – 4D) of zero, the calculator will apply the correct partial fraction expansion for repeated roots, leading to a `t*e^(pt)` term in the solution.