A designation of Kirchhoff’s equation for a DC electric circuit. The electric circuit, which is examined in the example, is composed of four voltage sources, two current sources and resistors. The number of Kirchhoff’s current law (KCL) and the Kirchhoff’s voltage law (KVL) equations is given by following formula:
The number of nodes is marked as n; the number of the equations for the Kirchhoff’s current law is equal to (n-1).
The number of the Kirchhoff’s voltage law equations depends on the number of branches and the nodes in the electric circuit. The general formula for the number of Kirchhoff’s voltage law equations is given by equation:
where:
m-number of branches
n-number of nodes
The examined electric circuit has got in its topology number of nodes n=3. It means that two Kirchhoff’s current law (KCL) equations are needed to solve the circuit.
On the electrical drawing diagram of the electric circuit, the nodes are marked from “1” to “7”. It should be noted that the nodes marked as "1" and "2" are in reality the same node. The same case is with nodes "3" and "4", "3" and "5", "6" and "7". The current marked as I_6 does not exist at all because nodes "1" and "2" have the same electric potential; an electric current flow is possible only between points with different electric potentials. The same case is with the current I_7 because nodes "3" and "5" have also the same electric potential.
The Kirchhoff’s current law (KCL) equation for node "1" and "2":
The Kirchhoff’s current law (KCL) equation for node "3" and "4":
The next step is marking by arrows the voltages on elements in the electric circuit. The number of the Kirchhoff’s voltage law (KVL) equations is equal to the number of branches in circuit "m" minus the number of the Kirchhoff’s current law (KCL) equations. The number of the branches in the examined circuit is m=5. It has to be remembered that a source current is not a branch.
The Kirchhoff’s voltage equation for mesh 1:
The Kirchhoff’s voltage equation for mesh 2:
The Kirchhoff’s voltage equation for mesh 3:
Three Kirchhoff’s voltage law (KVL) equations have been written down. To solve the examined electric circuit only two voltage equations are needed. One of equations is redundant, however, it has been written down for the reader's convenience as a benchmarking point during their exercises with electrical circuits. At this point electric circuits theory ends and the mathematical calculations are necessary to find selected unknowns for selected parameters.