Abstract:

Fourier transform and Laplace transform are essential tools in electrical and electronic engineering, yet students are often confused about, among others, their possible interrelation as well as their applicability to electrical and electronic systems. Based on students’ answers to a relevant short questionnaire, most students argued that Fourier transform might be a special case of Laplace transform (this misconception steming from the fact that the “jωt” exponent of former is the imaginary part of the “s” exponent of the latter) while almost no student could comment on the integration limits of the two transforms, more specifically on the possible reflection of the lower integration limits being minus infinity and zero for Fourier and Laplace transform, respectively. Finally, students did not seem to realize why Fourier transform (and Fourier analysis, in general) is the preferable tool in telecommunications unlike, e.g., control theory where Laplace transform dominates. Another point of inadequate understanding was the physical meaning of the “s” exponent of Laplace transform, particularly its real part.

To deal with the above issues, Fourier and Laplace transforms were presented in a comparative manner in a two teaching-hour lecture in the framework of the “Telecommunication Systems” course at the Electrical & Electronic Engineering Educators Dept. of the School for Pedagogical & Technological Education (ASPETE), Athens, Greece. This lecture took place towards the end of the course in order students to have already encountered both, Fourier analysis and Laplace transform.

Since the mathematical formula that relates Laplace and Fourier transforms does not easily lend itself for a comparative presentation, a problem-solving approach was applied focusing on the rationale and applicability of the two transforms. Due to the teaching time restrictions and with the aim to enhance students’ active participation, prior to the lecture the students were given a set of examples (that were actually variations of a simple series RL resistance-inductance circuit powered by a voltage source) and were asked to apply Fourier and Laplace analysis. In the first case, the source was a sinusoidal one, in the second case the source provided the sum of two sinusoidal voltages, in the third case (that would serve as a counter-example for the Laplace transform) the voltage was a constant (DC) one, and in the fourth (and last) case the power source provided a constant voltage after the closing of a switch. The students were asked to solve the first two cases by using phasors while for the third and fourth variation were asked to just calculate the current in the frequency (f or s) domain with the aim to provide the time-domain solution during the lecture. The lecture also included commenting on the calculation of Fourier and Laplace transforms of the DC signal x(t) = 1 and the unit-step signal (that are suitable for illustrating similarities and differences between the two transforms), a presentation of the mathematical relation between them as well as a comparison regarding the applicability and versatility of the two transforms as mathematical tools.

Keywords:

Engineering Education, Electrical Engineering Education, Electronic Engineering Education, Laplace transform, Fourier transform.