Circuit Theory/State Variables
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[edit] State Variables
A more modern approach to circuit analysis is known as the state variable method, which we will attempt to describe here. We use variables called state variables to describe the current state of the energy storage elements (capacitors and inductors). Here, we are using the word "state" to mean "condition" or "status" of the elements.
[edit] Aim of State Variables
What is the goal of using state variables? We have all sorts of other methods for solving and describing circuits, so why would we introduce another method for dealing with circuits?
An answer to both these questions lay in the theory of the state variable method: the state variable approach attempts to describe a circuit using a system of first-order differential equations instead of a single, higher-order equation. A system of first-order equations can then be easily manipulated using linear algebra techniques, and can be solved through brute-force methods such as cramers rule.
[edit] State Variable Selection
We choose state variables to be either the voltage across a capacitor, or the current through an inductor. If our circuit has multiple irreducible capacitors or inductors, we assign a state variable to each one.
[edit] System of Equations
Once we have assigned our state variables, we are tasked with finding a first-order differential equation that describes each one individually. Once we have all of these equations, we can set them up in matrix form, and use Cramers rule, or another analysis method to solve the system.
