Laplace Transform: Solving ODEs, Circuits & Signals

TAKEAWAYS:

• Laplace transforms convert ODEs/integral equations into algebraic equations for easier solving.
• Transfer functions (ratio of output/input in Laplace domain) model systems like RLC circuits or mechanical vibrations.
• Inverse Laplace recovers time-domain solutions from s-domain expressions.
• Step/impulse responses analyze system behavior to inputs like sudden voltage changes.
• Convolution theorem simplifies product integrals in signal processing.
• Final Value Theorem predicts steady-state outputs without solving full ODEs.

1. Core Concepts: Definition & Properties

The Laplace transform of a function $f(t)$ is defined as: $$\mathcal{L}\{f(t)\}=F(s)={\int }_0^{\infty }{e}^{-st}f(t)dt$$ where $s=\sigma +j\omega$ is a complex variable.

Key Properties (Visualized)

Worked Example 1: Solving an ODE Solve $y^{\prime \prime }+4y^{\prime }+3y=e^{-2t}$ with $y(0)=1$, $y^{\prime }(0)=0$.

Steps:

1. Take Laplace transform of both sides: $$s^{2}Y(s)-sy(0)-y^{\prime }(0)+4[sY(s)-y(0)]+3Y(s)=\frac{1}{s+2}$$
2. Substitute initial conditions: $$s^{2}Y(s)-s+4sY(s)-4+3Y(s)=\frac{1}{s+2}$$
3. Solve for $Y(s)$: $$Y(s)=\frac{1}{(s+1)(s+3)(s+2)}+\frac{s+4}{s^2+4s+3}$$
4. Partial fraction decomposition and inverse Laplace yield: $$y(t)=\frac{1}{4}{e}^{-2t}+\frac{1}{2}{e}^{-t}+\frac{1}{4}{e}^{-3t}$$

2. Transfer Functions & System Analysis

A transfer function $H(s)=\frac{Y(s)}{X(s)}$ describes a system’s input-output relationship in the Laplace domain.

Applications in Engineering

flowchart TD
    A["Transfer Function \( H(s) \)"] --> B["RLC Circuits"]
    A --> C["Mechanical Vibrations"]
    A --> D["Control Systems (PID)"]
    A --> E["Signal Processing"]

Worked Example 2: RLC Circuit Analysis For an RLC circuit with $R=2\Omega$, $L=1H$, $C=0.5F$, and input $x(t)=u(t)$, find the current $i(t)$.

Steps:

1. Write the differential equation: $$L\frac{di}{dt}+Ri+\frac{1}{C}\int idt=x(t)$$
2. Take Laplace transform: $$sI(s)+2I(s)+\frac{2}{s}I(s)=\frac{1}{s}$$
3. Solve for $I(s)$: $$I(s)=\frac{1}{s(s^2+2s+2)}=\frac{1}{s}-\frac{s+2}{s^2+2s+2}$$
4. Inverse Laplace: $$i(t)=1-e^{-t}(\cos t+\sin t)$$

3. Step & Impulse Responses

• Unit step response: ${\mathcal{L}}^{-1}\left\{\frac{1}{s}\right\}=u(t)$.
• Unit impulse response: ${\mathcal{L}}^{-1}\left\{1\right\}=\delta (t)$.

Real-World Link: Traffic Flow Modeling At a Kathmandu intersection, the green light duration can be modeled as a step input to a traffic queue system. The Laplace transform helps predict how quickly vehicles clear the intersection after the light turns green.

4. Convolution Theorem

The convolution of two functions $f(t)\ast g(t)$ in time domain becomes a product $F(s)G(s)$ in Laplace domain.

Worked Example 3: Convolution of Signals Find $f(t)\ast g(t)$ where $f(t)=e^{-t}u(t)$ and $g(t)=\sin tu(t)$.

Steps:

1. Compute Laplace transforms: $$F(s)=\frac{1}{s+1},G(s)=\frac{1}{s^2+1}$$
2. Multiply: $$F(s)G(s)=\frac{1}{(s+1)(s^2+1)}$$
3. Partial fractions and inverse Laplace: $$f(t)\ast g(t)=\frac{1}{2}{e}^{-t}-\frac{1}{2}{e}^{-t}\cos t+\frac{1}{2}\sin t$$

5. Final Value Theorem

Predicts the steady-state value of a system: $$\underset{t\to \infty }{\lim }f(t)=\underset{s\to 0}{\lim }sF(s)$$

Real-World Link: Bank Loan Interest If a loan’s remaining balance $L(t)$ satisfies $\frac{dL}{dt}=-rL$, the Final Value Theorem gives the final balance as ${\lim }_{s\to 0}s\mathcal{L}\{L(t)\}$.

6. Applications in Engineering

## In the Real World

1. eSewa & Khalti Payments: The Laplace transform models transaction delays in payment processing. For example, when you pay a bill via eSewa, the system’s response time to confirm the transaction can be analyzed using transfer functions to optimize server load.

2. NTC Power Grid Stability: Nepal’s power grid uses Laplace-based control systems to stabilize voltage fluctuations. During sudden demand spikes (e.g., monsoon season), the grid’s transfer function predicts how quickly generators must adjust output.

3. Pathao Ride Allocation: Pathao’s algorithm for matching drivers to riders uses convolution integrals (via Laplace transforms) to estimate wait times. The system treats rider requests as impulse inputs and calculates the optimal driver response time.

## Exam Tip

• Always check initial conditions when solving ODEs via Laplace transforms.
• Partial fraction decomposition is critical—practice factoring polynomials quickly.
• Transfer functions are often tested in circuit/control problems. Memorize standard forms (e.g., $\frac{1}{s+a}$ for first-order systems).
• Final Value Theorem is a shortcut—use it to verify steady-state answers.
• Graphs are worth marks: Sketch $F(s)$ poles/zeros and time-domain responses when asked for qualitative behavior.

Visual Summary

mindmap
  root((Laplace Transform Applications))
    ODEs
      Solve via algebraic equations
      Initial conditions critical
    Circuits
      RLC analysis
      Transfer functions
    Signals
      Convolution theorem
      Step/impulse responses
    Engineering
      Control systems
      Traffic modeling
      Economic forecasting