Mathematics-Online course: Linear Algebra-Linear Systems of Equations-Classification and General Structure-Electric Circuit

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	Mathematics-Online course: Linear Algebra - Linear Systems of Equations - Classification and General Structure
	Electric Circuit

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Let

denote the counterclockwise oriented loop currents in an electric circuit. Further let $R_{i,j}$ denote the common resistance of the

-th and the

-th loop and let

be the voltage impressed on the

-th loop. Then, by Ohm's and Kirchhoff's laws, we obtain the linear system

$\displaystyle \sum_{i\sim 0} x_i R_{i,0}\, +\, \sum_{i\sim j} (x_i-x_j)R_{i,j} = U_i\,,$

where $i\sim j$ means that the

-th and the

-th loop have a common resistor and

denotes the current through this resistor. The resistances belonging only to the

-th loop are denoted by $R_{i,0}$ , $i\sim 0$ .

$\includegraphics[width=\moimagesize]{b_elektrischer_schaltkreis}$

For example, for the electric circuit pictured above we obtain:

$\displaystyle \left(\begin{array}{rrrrr} 150 & -70 & -80 & 0 & 0 \\ -70 & 12... ...) = \left(\begin{array}{c} 110 \\ 0 \\ 0 \\ 0 \\ -220 \end{array}\right).$

The

-th diagonal entry of the coefficient matrix is the sum of the resistances belonging only to the

-th loop, and the common resistance of loop

and loop

is the entry at position

. The solution for this example is

$\begin{displaymath} x \approx \left( \begin{array}{r} 1.0157\\ 0.5641\\ 0.0358\\ -0.0940\\ -1.1595 \end{array} \right)\,. \end{displaymath}$

automatically generated 4/21/2005