Circuit Theory/Terminology

Basic Terminology
There are a few key terms that need to be understood at the beginning of this book, before we can continue. This is only a partial list of all terms that will be used throughout this book, but these key words are important to know before we begin the main narrative of this text.


 * Time domain : The time domain is described by graphs of power, voltage and current that depend upon time. This is simply another way of saying that our circuits change with time, and that the major variable used to describe the system is time. Another name is "temporal".


 * Frequency domain: The frequency domain are graphs of power, voltage and/or current that depend upon frequency such as Bode plots. Variable frequencies in wireless communication can represent changing channels or data on a channel. Another name is the "Fourier domain". Other domains that an engineer might encounter are the "Laplace domain" (or the "s domain" or "complex frequency domain"), and the "Z domain". When combined with the time, it is called a "spectral" or "waterfall."


 * Circuit response : Circuits generally have inputs and outputs. In fact, it is safe to say that a circuit isn't useful if it doesn't have one or the other (usually both). Circuit response is the relationship between the circuit's input to the circuit's output. The circuit response may be a measure of either current or voltage.


 * Non-homogeneous : Circuits are described by equations that capture the component characteristics and how they are wired together. These equations are non-homogeneous in nature. Solving these equations requires splitting the single problem into two problems: a steady-state solution (particular solution) and a transient solution (homogeneous solution).


 * Steady-state solution : The final value, when all circuit elements have a constant or periodic behavior, is also known as the steady-state value of the circuit. The circuit response at steady-state (when voltages and currents have stopped changing due to a disturbance) is also known as the "steady-state response". The steady-state solution to the particular integral is called the particular solution.


 * Transient : A transient event is a short-lived burst of energy in a system caused by a sudden change of state.


 * Transient means momentary, or a short period of time. Transient means that the energy in a circuit suddenly changes which causes the energy storage elements to react. The circuit's energy state is forced to change. When a car goes over a bump, it can fly apart, feel like a rock, or cushion the impact in a designed manner. The goal of most circuit design is to plan for transients, whether intended or not.


 * Transient solutions are determined by assuming the driving function(s) is zero which creates a homogeneous equation, which has a homogeneous solution.

Summary
When something changes in a circuit, there is a certain transition period before a circuit "settles down", and reaches its final value. The response that a circuit has before settling into its steady-state response is known as the transient response. Using Euler's formula, complex numbers, phasors and the s-plane, a homogeneous solution technique will be developed that captures the transient response by assuming the final state has no energy. In addition, a particular solution technique will be developed that finds the final energy state. Added together, they predict the circuit response.

The related Differential equation development of homogeneous and particular solutions will be avoided.