A Resonant Band-Pass Filter-Based Facile Recipe for the Estimation of Number Pi

Abstract

In this article, we report an interesting exploration of the basic principles of the Fourier series by using a simple electrical resonant band-pass filter (BPF), driven by a periodic square wave input, and a dual-trace cathode ray oscilloscope (CRO). Here, the BPF circuit is presented as a remarkable Fourier analyzer tool, used to isolate individual frequency components (i.e., the odd harmonics) of the applied square wave. At a fixed resonant frequency (f₀), one can probe the higher-order Fourier components of the square wave, corresponding to lower frequencies of f₀/n. The amplitudes of these odd harmonics can be measured directly from the CRO display or from screenshots captured with a camera. These amplitudes and the corresponding square wave frequencies can then be easily compared with the theoretical prediction of 1/n behavior for n = odd, enabling an engaging and illustrative experiment for students in the laboratory. We also propose the possibility of estimating the value of the famous mathematical constant denoted by the Greek letter π, not only from experimentally measured amplitude values but also from the slope of linearly fitted experimental data. The experimentally determined mean value of π (Pi) was 3.15873, with a total uncertainty of 0.128430. The relative percentage error compared to the accepted value of π (3.14159) was 0.546%, and the percent uncertainty was approximately 0.0003%. Thus, the BPF-based approach yields results consistent with the true value of π within experimental uncertainty. To the best of our knowledge, this paper is unique in presenting such an interesting exploration of the BPF circuit using a CRO while incorporating Fourier components. Finally, in this contemporary age of rapid computation, the suggested experiments provide a valuable addition to physics laboratory experiments involving concepts of Fourier analysis, electrical resonance, BPF circuits, and harmonics.