**for**

**Background**

Solution of linear systems can be applied to
resistor network circuits. Kirchoff's voltage law says
that the sum of the voltage drops around any closed loop in the
network must equal zero. A closed loop has the obvious
definition: starting at a node, trace a path through the circuit that
returns you to the original starting node.

**Network #1**

Consider the network consisting of six
resistors and two battery, shown in the figure
below.

There are two closed loops. When Kirchoff's voltage law is
applied, we obtain the following linear system of
equations.

**Proof ****Kirchoff's
Law** **Kirchoff's
Law**

**Computer
Programs ****Kirchoff's
Law** **Kirchoff's
Law**

**Example
1.** Solve the network #1 for the
currents given
the following value for the resistors and battery:

.

**Solution
1.**

**Example
2.** Solve the network #1 for the
currents given
the following value for the resistors and battery:

.

**Solution
2.**

**Network #2**

Consider the network consisting of nine
resistors and one battery, shown in the figure below.

There are three loops. When Kirchoff's voltage law is
applied, we obtain the following linear system of
equations.

**Example
3.** Solve the network #2 for the
currents given
the following value for the resistors and battery:

.

**Solution
3.**

**Example
4.** Solve the network #2 for the
currents given
the following value for the resistors and battery:

.

**Solution
4.**

**Network #3**

Consider the network consisting of six
resistors and two batteries, shown in the figure below.

There are three loops. When Kirchoff's voltage law is
applied, we obtain the following linear system of
equations.

**Example
5.** Solve the network #3 for the
currents given
the following value for the resistors and batteries:

.

**Solution
5.**

**Example
6.** Solve the network #3 for the
currents given
the following value for the resistors and batteries:

.

**Solution
6.**

**Research Experience for
Undergraduates**

**Kirchoff's
Law** **Kirchoff's
Law** Internet hyperlinks to web sites and a
bibliography of articles.

**Download this
Mathematica Notebook****
****Kirchoff's
Law**

**Return
to Numerical Methods - Numerical Analysis**

(c) John H. Mathews 2004