The circuit of figure A has 4 essential nodes. In order for this circuit to be analyzed, (4-1=3), 3 equations are needed.
figure A : example circuit
Figure B demonstrates the branch currents of figure A.
On figure C, the 4 nodes of the circuit are numbered, and a tree joins all the nodes together creating a path.
A tree is a set of connecting branches that connects every node to every other node, passing only once from each node (does not form a loop).
Some example trees of figure A can be:
Once a tree is defined, the remaining branches are called links or chords. A collection of these links is called a complementary tree or cotree.
A graph is the redrawing of the circuit with each branch represented by a line, like figure B & figure C.
figure B figure C
A cut-set is the smallest set of branches which are removed from the graph in order to divide the graph into two groups of nodes. Some example cut-sets, are shown in figure D(i) & D(ii) below:
figure D(i) figure D(ii)
A fundamental loop is a loop which contains one and only one link. By observing figure E:
fundamental loops: i2-i4-i3, i4-i6-i5, i1-i4-i3,
A fundamental cut-set is a cut-set which contains one and only one tree branch. By observing figure E above, an example of a fundamental cut-set is shown on figure F & G
fundamental cut-set: i1-i2-i3-i7
fundamental cut-set: i1-i2-i4-i5-i7