The nodal analysis method is based on the fundamental cut-set technique. Any forming tree (see figure C), should exclude the current sources, and include the voltage sources of the circuit (see figure A), because all the equations derived from the node voltage method involve only the voltages of the circuit. It is worth mentioning that an example tree of the circuit appears on figure C. It joins together the 4 nodes, passing through the (i3-i4-i6) branches. Another example tree could have been the (i2-i4-i5-i7) tree and so on. On this web-page, all the terminologies and the mathematical equations would be based on the tree showed on figure C (i3-i4-i6), because only one tree is needed in order to analyse a circuit.
figure A : the circuit
figure B figure C
By observing figure C above, the fundamental cut-set (i): i7- i3-i2-i1 is formed (see figure D).
Another fundamental cut-set (ii) can be: i7- i5-i4-i2-i1 (see figure E)
A cut-set can be: i6-i4-i2-i1 (see figure F). This cut-set is not a fundamental cut-set because it contains more than one tree branches.
(note that: if a node joins more than one tree branches, it cannot be isolated. That is why node 2 is never isolated and that's why figure F does not represent a fundamental cut-set)
As mentioned above, in order for the circuit of figure A to be analysed, 3 equations are needed. Each fundamental cut-set would provide a single equation. Figure D and figure E above represent 2 fundamental cut-sets, so they represent 2 equations. The last equation is given from figure G below, representing the fundamental cut-set (iii): i5-i6-i7.
By observing figure C and summing the currents in the fundamental cut-sets:
currents i4 and i5 have negative sign because they flow out of the node
Using equations (1), (2) and (4) (neglecting equation (3) because of equation (4)).
Substituting equation (4) into equation (2) =>
Substituting equation (4) into equation (1) =>
V1 = 3.79V
V2 = 6.047V
V3 = 26.04V