Simulation of a non-linear, time-variant circuit using the Haar wavelet transform

Wavelet theory has disentangled numerous complexities, including those pertinent to transient and steady-state responses of systems, when Laplace and Fourier transforms face insoluble obstacles. Reactive linear components (e.g. inductors and capacitors) are typically handled in the frequency plane. Non-linear (e.g. diodes) or time-variant components (e.g. switches) are conventionally simulated in the time plane (e.g. a diode via its I-V characteristic) and are considered open or short circuits in AC analyses (e.g. in circuit simulation software). Although translating circuits in an alternative plane, such as the Haar wavelet plane, significantly simplifies the process, a wide integration of wavelets into instruments and education is not yet realised; an underlying reason is the considerate complexity of applying wavelet theory to circuits and signals. The aim of this paper is to bridge this gap, providing a new user-friendly, Laplace-alike approach, utilising measurement-based models and the Haar wavelet. The Haar wavelet transform and a numerical method for the inverse Laplace transform which uses the Haar operational matrix are applied, to calculate the total current of a non-linear, time-variant system, that is the total current of a voltage source which powers a non-linear, time-variant load.