Circuit Theory/Transients

Transient analysis produces an equation of voltage or current.

Energy Imbalance

The temporary conditions are caused by an energy imbalance. The energy imbalance occurs between power sources and capacitors or inductors. Power sources may charge capacitors or inductors. The capacitors or inductors may dump previously stored energy into resistors, each other or charge power sources.

Transients occur while energy is being balanced in the circuit.

Transient Origins

The events that can cause transients are:

switching on power
changing a component's value in a functioning circuit
discharging a component after the circuit has been turned off

Turning a knob can change a value, but so can temperature, light, acceleration, stretching, compressing, humidity, etc. Electrical transducers sense non-electrical energy and cause resistance, capacitance or inductance values to change or change the energy stored in a capacitor/inductor.

First & Second Order Analysis

First order circuits have either a capacitor or inductor.

Second order circuits have two energy storage elements and require a different analysis technique.

First order transients voltages and currents are typically one sinusoidal riding one exponential.

Second order transients are typically described as one of the following:

overdamped
critically damped
underdamped
undamped

The word "damped" refers how two different types of energy storage elements (capacitors and inductors) interact as energy is dissipated. The math in general shows the combination of two sinusoidals riding on two exponentials.

Math Summary

There are two ways to solve transient problems.

Differential Equations

Something causes the energy imbalance. The goal is to find the final value of a charging circuit, not the equation describing how energy was added to the circuit.

The goal is to find an equation for the discharge. The steps are to form a differential equation (no integrals at any step) and evaluate constants. Differential equation analysis requires computing a constant C. Unfortunately, C = 0 in most of the examples below. This is because the equation describes time from 0 to ∞. At t=∞, most designs want all energy eliminated from a circuit.