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

[matqkks]

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?

 

[scottdave]

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.

 

[phinds]

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

"Class, today I am going to introduce you to your nemesis."

 

[jack action]

• LAPLACE TRANSFORMS AND ITS APPLICATIONS
• APPLICATIONS OF LAPLACE TRANSFORM IN ENGINEERING FIELDS
• Examples and applications

 

[DaveE]

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.

 

[DaveE]

Just to elaborate a bit about Laplace in the EE world. Consider an RLC low pass filter.

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.

 

[Dr Transport]

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.

 

[DaveE]

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.