# 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.

#### Convolution Integral

