Operational Calculus without Transforms

Abstract

An operational calculus is outlined that enables the determination of the response of any lumped circuit to a general waveform. It is based on elementary notions of operator algebra (sum, product, and inversion of operators) and is rigorously deducible. All processes are carried out in the time domain, no transform or complex-variable theory being needed. The operators turn out to correspond to superposition integrals of impulse responses. Steady-state theory is derived easily as a special case. In particular, the response to any periodic waveform can be calculated by integrations over a single period and is a distinct improvement over the use of Fourier series or Laplace transforms for this problem. The analog of the calculus in the frequency domain is shown to correspond to the use of the bilateral Laplace transformation.