Time response of RCR circuit to single sinusoidal pulse (2: piecewise)

Antonio Sala, UPV

Difficulty: *** ,       Relevance: ,      Duration: 10:59

*Enlace a Spanish version

Summary:

This video calculates the time response of a circuit with 2 resistors and 1 capacitor to a sinusoidal pulse (semi-period) and non-zero initial conditions. The same problem was addressed in the video [sinpulRCREN], where the idea of “linearity” and “temporal invariance” was used to be able to express the output by superposition of two sinusoid responses, one of them delayed. The statement of the problem and the summary of the aforementioned video are outlined in the first 2 and a half minutes of this material.

Here, the same problem is solved in a “piecewise” way, dispensing with the concepts of delay and superposition from the video referred to above. Two main concepts are used:

• causality: if the pulse and the full sine coincide during the first 0.01 seconds, their outputs will also coincide in that interval (the output does not depend on future changes in the input waveform).

• The concept of state: everything that happened in the past is summarized in the state (capacitor charge) at the moment when the input is no longer sinusoidal.

These two ideas allow us to solve two differential equation problems, one for the interval $[0,0.01]$ and another for $t\ge 0.01$, with the initial condition of the second interval equal to the “final” condition of the first .

Repetition of this idea at many intervals will be used to calculate the response to a train of sinusoidal pulses in the video [trenpulRCREN].

*Link to my [ whole collection] of videos in English. Link to larger [ Colección completa] in Spanish.