Lecture 16 - Laplace Transforms

Laplace Transforms allow differential equations to be converted to algebraic relationships. These algebraic relationships can be rearranged and solved for the dependent variable. Once they are rearranged, an inverse Laplace Transforms gives the solution in the time-domain.

• Lecture 16 Notes
• Lecture 16 Handout
• Laplace Transform Table

There are a number of proficiencies that are required to transform to and from the Laplace domain. Some of these include integration and partial fraction expansion. These are topics that were covered in prerequisite math courses so much of the material should be review. The new application will be applying these techniques to physical systems for solving process control problems.

Solve ODE in MATLAB in Time and LaPlace Domain

• MATLAB Example Files

Discussion Questions

Test the lecture's ideas before or after class - commit to an answer first, then check it with a classmate or an AI tutor.

1. Why do engineers transform differential equations into the Laplace domain at all? What operation becomes trivial there, and what is the price paid on the way back to the time domain?
2. The transform of a derivative is $$sF(s) - f(0)$$. Why does working in deviation variables around a steady state make the initial-condition terms vanish, and why is that the standard practice in this course?
3. The final value theorem gives $$y(\infty) = \lim_{s \to 0} sY(s)$$. Construct an example where applying it blindly gives a finite "final value" for a response that actually grows without bound. What must you check first?
4. A pure time delay appears as $$e^{-\theta s}$$ - not a polynomial in s. Why can no finite set of poles and zeros represent it exactly, and what does that mean for a delayed process in block-diagram algebra?

Tip: Verify every hand inversion numerically: solve the original ODE in MATLAB or Python and overlay your analytic y(t). Agreement is the strongest evidence your transform algebra is right - and when the curves disagree, the transform table entry you misused is usually easy to find.