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

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(c) John H. Mathews 2004