NODAL ANALYSIS 

In this topic we will tackle about nodal analysis ..

Nodal Analysis provides a general procedure for analyzing circuits using node voltage as the circuits variables. Choosing node voltages instead of voltage elements as circuit variables is convenient and reduces the number of equations one must solve simultaneously.
To simplify matters, we shall assume in this section that circuits do not contain voltage sources. Circuits that contain voltage sources will be analyzed in the next section.
In nodal analysis, we are interested in finding the node voltages given a circuit with n nodes without voltage sources, the nodal analysis of the circuit involves taking the following three steps.

Step to Determine Node Voltages:
1. Select a node as the reference node or a ground. Assign voltages v1, v2, ... Vn -1 to the remaining n-1 nodes. The voltages are referenced with respect to the reference node.

2 Apply KCL to each of the n-1 non reference nodes. Use Ohms law to express the branch currents in terms of node voltages.

For example, for the node to the right KCL yields the equation:
Ia + Ib + Ic = 0


Express the current in each branch in terms of the nodal voltages at each end of the branch using Ohm's Law (I = V / R). Here are some examples:



The current downward out of node 1 depends on the voltage difference V1 - V3 and the resistance in the branch.





In this case the voltage difference across the resistance is V1 - V2 minus the voltage across the voltage source. Thus the downward current is as shown.






In this case the voltage difference across the resistance must be 100 volts greater than the difference V1 - V2. Thus the downward current is as shown.




The result, after simplification, is a system of m linear equations in the m unknown nodal voltages (where m is one less than the number of nodes; m = n - 1). The equations are of this form:

where G₁₁, G₁₂, . . . , G_mm and I₁, I₂, . . . , I_m are constants.


NODAL ANALYSIS WITH VOLTAGE SOURCES

Case !

If a voltage souve is connected between the reference node and a nonreference node, we simply set the voltage at the non reference node equal to the voltage of the voltage source. In fig. 3.7 for example 

V1=10v

Thus our analysis is somewhat simplified by this knowledge of the voltage at this node.

Case 1

If the voltage source( dependent of independent ) is connected between two nonreference nodes, the two nonreference nodes form generalized node or supernode, we apply both KCL and KVL to determine the node voltages.

A supernode is formed by enclosing a (dependent or independent) voltage source connected between two nonreference nodes and any elements connected in parallel with it.

In Fig. 3.7, nodes 2 and 3 form a supernode. (We could have more than two nodes forming a single supernode. For example, see the circuit in Fig. 3.14.) We analyze a circuit with supernodes using the same three steps mentioned in the previous section except that the supernodes are treated differently. Why? Because an essential component of nodal analysis is applying KCL, which requires knowing the current through each element. There is no way of knowing the current through a voltage source in advance. However, KCL must be satisfied at a supernode like any other node. Hence, at the supernode in Fig. 3.7,

i1 + i4 = i2 + i3                                                                                   (3.11a)
or
v1 − v2 / 2 + v1 − v3  /4 = v2 − 0 / 8 + v3 − 0 / 6                              (3.11b)

To apply Kirchhoff’s voltage law to the supernode in Fig. 3.7, we redraw the circuit as shown in Fig. 3.8. Going around the loop in the clockwise direction gives

−v2 + 5 + v3 = 0 ⇒ v2 − v3 = 5                                                          (3.12)

Note the following properties of a supernode:

1. The voltage source inside the supernode provides a constraint equation needed to solve for the node voltages.

2. A supernode has no voltage of its own.

3. A supernode requires the application of both KCL and KVL.