Problem Solving

Linear algebra can be used to solve large ohmic circuits. This page explains how to define the circuit problem in terms of a linear system that can be solved with computational methods like matlab or python.

The Main Idea

We can describe an electrical circuit as a graph where the nodes represent specific points and the edges represent the components connecting them. E.g:

Incidence Matrix

The incidence matrix, A, is a matrix of dimensions m x n, where m is the number of edges and n is the number of nodes. If x is the vector that contains all the nodes in order, A satisfies: [math]\displaystyle{ A\vec{x}_{edge} = x_{end} - x_{start} }[/math] Thus, the resulting matrix will have zeros in every element of each row, except for the start and the end nodes, which will be -1 and 1 respectively:

Here, [math]\displaystyle{ x_{1}, x_{2}, x_{3} }[/math] and [math]\displaystyle{ x_{4} }[/math] represent the voltages at the nodes. Then [math]\displaystyle{ A\vec{x}_{edge} = x_{end} - x_{start} }[/math] gives the voltage differences [math]\displaystyle{ ∆V }[/math].

Loop Rule (Kirchhoff's voltage law)

Using this creative way of transforming a circuit to a graph and getting the incidence matrix from it allows us to do some interesting things. Kirchhoff's loop rule is a principle of conservation of energy that implies that the directed sum of the voltage differences around any closed network is zero.

This means that the components of [math]\displaystyle{ A\vec{x} = b }[/math] add to zero around every loop. Moreover, the voltage law can decide whether b is in the column space of A.

We know from Ohm's law that [math]\displaystyle{ V = I * R }[/math]. Then, in the previous graph for example, we would have that [math]\displaystyle{ V_{edge1} = I * R = (direction) * R_{1} = 1 * R_{1} }[/math].

Kirchhoff's Current Law

A Mathematical Model

What are the mathematical equations that allow us to model this topic. For example [math]\displaystyle{ {\frac{d\vec{p}}{dt}}_{system} = \vec{F}_{net} }[/math] where p is the momentum of the system and F is the net force from the surroundings.

A Computational Model

How do we visualize or predict using this topic. Consider embedding some vpython code here Teach hands-on with GlowScript

Examples

Be sure to show all steps in your solution and include diagrams whenever possible

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Connectedness

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History

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Problem Solving

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