Nodal Analysis With Sources

The node-equation method is based directly on Kirchhoffs current law unlike loop-current
method which is based on Kirchhoff s voltage law. However, like loop current method, nodal method also has the advantage that a minimum numt
2 2 3 ber of equations need be written to determine
I I"A-/2 fBI-h the unknown quantities. Moreover, it is par-
+ ~ + ticularly suited for networks having many
15 parallel circuits with common ground con-
£2 T nected such as electronic circuits.
For the application of this method, every
juncion in the network where three or more
branches meet is regarded a node. One of
these is regarded as the reference node or
datum node or zero-potential node. Hence
the number of simultaneous equations to be solved becomes (n - 1) where n is the number of independent
nodes. These node equations often become simplified if all voltage sources are converted
into current sources First Case
Consider the circuit of Fig. 2.60 which has three nodes. One of these i.e. node 3 has been take
in as the reference node. VArepresents the potential of node 1 with reference to the datum node 3.
Similarly, VBis the potential difference between node 2 and node 3. Let the current directions which
have been chosen arbitarily be as shown Though the above nodal equations (ii) and (iii) seem to be complicated, they employ a very
simple and systematic arrangement of terms which can be written simply by inspection. Eq. (ii) at
node I is represented by
1. The product of node potential VAand (lIRI + lIR2 + lIR4) i.e. the sum of the reciprocals of
the branch resistance connected to this node. .
2. Minus the ratio of adjacent potential VBand the interconnecting resistance R2.
3. Minus ratio of adjacent battery (or generator) voltage-EI and interconnecting resistacne RI'
4. All the above set to zero.
Second Case
Now, consider the case when a third battery of
e.m.f. E3 is connected between nodes 1 and 2 as
shown in Fig. 2.62.
It must be noted that as we travel from node 1to --£
node 2, we go from the -ve terminal of E3to its +ve T 1
terminal. Hence, according to the sign convention
given in Art. 2.3, E3must be taken aspositive. However,
if we travel from node 2 to node 1, we go from
the +ve to the -ve terminal of E3' Hence, when
viewedfrom node 2, E3 is taken negative.It is exactly the same expression as given under the First Case discussed above except for the
additional tenn involving E3' This additional tenn is taken as +EiR2 (and not as - EiR2) because
this third battery is so connected that when viewed from mode I, it represents a rise in voltage. Had
it been connected the other way around, the additional tenn would have been taken as -EiR2'
~s seen, the additional tenns is -EiR2 (and not + EiR2) because as viewed from this node, E3
represents afall in potential.
It is worth repeating that the additional tenn in the above Eq. (i) and (ii) can be either +EiR2 or
-EiR2 depending on whether it represents a rise or fall of potential when viewed from the node
under consideration.