What is the best way to introduce Laplace transforms in an Engineering Mathematics course?

In summary: If I showed it to someone and then didn't tell them what it was for, they would be lost. If I introduce it earlier on in the course, when they are more motivated, then they will remember it better and be more likely to use it.In summary, Laplace transforms are a way to turn a differential equation into an algebraic equation. This can be helpful when trying to solve problems in the ##s## domain.
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Are there any practical applications of Laplace transform? I would not use Laplace transforms to solve first, second-order ordinary differential equations as it is much easier by other methods even if it has a pulse forcing function.

How can Laplace transforms be introduced so that students are motivated to learn? It needs to have an impact.

What are the applications of Laplace transforms?
 
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In one of my Electrical Engineering courses (I think Signal Processing), the component reactance values are represented with Laplace.

For example, a resistor is just a number (the resistance in Ohms).

An inductor is L*s, where L is the inductance in Henries, and Capacitor is 1/(C*s). From there it's just Algebra to get the impedance of the entire circuit, then can use Unit Step functions or Sine functions as the source to drive them and get the response at each component.

There are other types of problems that are easier to work with in the 's' domain, rather than time domain.
 
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matqkks said:

What is the best way to introduce Laplace transforms ...​

"Class, today I am going to introduce you to your nemesis."
 
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matqkks said:
Are there any practical applications of Laplace transform?
OMG. Ask an analog/controls EE, like me. It is the go to method for transient response solutions. It is the language used for transfer functions. In my world ##j \omega## and Fourier transforms hardly exist, everything is done in the ##s## domain. Having said that, it's more looking things up in tables and applying theorems like shifting and superposition, not really brute force mathematical solutions.
 
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Just to elaborate a bit about Laplace in the EE world. Consider an RLC low pass filter.

PXL_20230227_194703676~2.jpg


A physics student would write out something like
##v_i = Ri + L \frac{di}{dt} + \frac{1}{C} \int{i}{dt} ##
##v_o = \frac{1}{C} \int{i}{dt} ##
and proceed to solve the DEs

An analog EE would use the familiar voltage divider formula with complex impedances and just write out the solution in the ##s## domain immediately.
##\frac{v_o}{v_i} = \frac{ \frac{1}{sC} }{R+sL+\frac{1}{sC}} = \frac{1}{1 + sRC + s^2LC}##

Chances are we'd never invert it into the ##t## domain but just look at Bode plots and such. Our spectrum analyzers and FRAs give us the frequency domain data anyway. Many of us work in the frequency domain most of the time. If we needed to invert the transform, we'd first look in a table of common transform pairs.

IMO, Laplace transforms are the basis for phasors and complex impedances, even though they aren't taught that way initially. You can always set ##s=j \omega## and do that complex arithmetic when needed.
 
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I introduce it as a method to turn a differential equation into an algebraic equation. They only need to so a transform and an inverse transform to get the homogeneous and inhomogeneous solutions.
 
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Dr Transport said:
I introduce it as a method to turn a differential equation into an algebraic equation. They only need to so a transform and an inverse transform to get the homogeneous and inhomogeneous solutions.
I like this; first explain why students should pay attention. Explain that this is something that can be really useful IRL. The method I saw first looked like just another math theorem, another transform that looked confusing.
 
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1. What is a Laplace transform?

A Laplace transform is a mathematical operation that is used to convert a function of time into a function of complex frequency. It is commonly used in engineering and physics to solve differential equations and analyze systems in the frequency domain.

2. Why is it important to introduce Laplace transforms in an Engineering Mathematics course?

Laplace transforms are important in engineering because they can simplify complex differential equations into algebraic equations, making it easier to model and analyze systems. They are also commonly used in control systems, signal processing, and circuit analysis.

3. What are the key concepts students should understand before learning Laplace transforms?

Before learning Laplace transforms, students should have a solid understanding of calculus, including differentiation and integration. They should also be familiar with complex numbers and the concept of functions of complex variables.

4. How should Laplace transforms be taught in an Engineering Mathematics course?

Laplace transforms should be taught with a combination of theory and practical examples. Students should first be introduced to the definition and properties of Laplace transforms, and then given the opportunity to practice solving problems using them. Visual aids, such as diagrams and graphs, can also be helpful in understanding the concept.

5. Are there any helpful resources for students to learn Laplace transforms?

Yes, there are many helpful resources for students to learn Laplace transforms, including textbooks, online tutorials, and practice problems. Many universities also offer tutoring or study sessions specifically for engineering mathematics courses. Additionally, students can seek help from their professors or teaching assistants if they have any questions or difficulties understanding the material.

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