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Supernode Analysis

In the previous post about node analysis, we learned about nodal analysis without voltage sources to make it easier to understand. We now consider how voltage sources can make nodal analysis harder to analyze.

Supernode Analysis Theory

See Figure.(1) for illustration. There are two possibilities,
supernode analysis electric circuit
Figure 1. Supenode electric circuit
CASE 1 - If a voltage source is connected between the reference node and a nonreference node, we place the nonreference equal to the voltage source. For example the voltage source in the top left branch in Figure.(1),
supernode analysis electric circuit
(1)
Hence, our job in analysis is simplified by this knowledge of voltage at this node.

CASE 2 - If the voltage source (independent or dependent) is connected between two nonreference nodes, the two nonreference nodes form a generalized node or supernode; we use both KCL and KVL to determine the node voltages.
A supernode is formed by enclosing a (independent or dependent) voltage source connected between two nonreference nodes and any elements connected in parallel with it.
In Figure.(1), nodes 2 and 3 form a supernode. (We could have more than two nodes forming a single supernode. We can still use the three steps from previous post about nodal analysis without voltage source, but we will modify it a little bit.

The main idea about nodal analysis is KCL, which we need to find the current flowing in each element. And it is impossible to find how much current is flowing through the voltage source in advance. However, KCL must be satisfied at a supernode like any other node.
Thus, at the supernode in Figure.(1),
supernode analysis electric circuit
(2)

To apply Kirchhoff's voltage law to the supernode in Figure.(1),we redraw the circuit in Figure.(1) to Figure.(2).
supernode analysis electric circuit
Figure 2. Loop in the supernode
Going around the loop in the clockwise direction gives
supernode analysis electric circuit
(3)
From Equations.(1) and (2) we obtain the node voltages.
Give attention to 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.

Supernode Analysis Examples

For better understanding let us review the examples below.
1. For the circuit shown in Figure.(3), find the node voltages.
supernode analysis electric circuit
Figure 3
Solution :
The supernode contains the 2 V source, nodes 1 and 2, and 10 Ω resistor.
supernode analysis electric circuit
Figure 4a
Applying KCL to the supernode in Figure.(4a) gives
supernode analysis electric circuit
Expressing i1 and i2 in terms of the node voltages,
supernode analysis electric circuit
(1.1)
To get the relationship between v1 and v2, we apply KVL to the circuit in Figure.(4b),
supernode analysis electric circuit
Figure 4b
Going around the loop, we obtain
supernode analysis electric circuit
(1.2)
From (1.1) and (1.2) we get
supernode analysis electric circuit
and v2 = v1 + 2 = -5.333 V. Note that the 10 Ω resistor does not make any difference because it is connected across the supernode.

2. Find the node voltages in the circuit of Figure.(5)
supernode analysis electric circuit
Figure 5
Solution :
Nodes 1 and 2 form a supernode, also do nodes 3 and 4. We apply KCL to the two supernodes as in Figure.(6a).
supernode analysis electric circuit
Figure 6a
At supernode 1-2,
supernode analysis electric circuit
Expressing this in terms of the node voltages,
supernode analysis electric circuit
(2.1)
At supernode 3-4,
supernode analysis electric circuit
(2.2)
We now apply KVL to the branches involving the voltage sources as shown in Figure.(6b).
supernode analysis electric circuit
Figure 6b
For loop 1,
supernode analysis electric circuit
(2.3)
For loop 2,
supernode analysis electric circuit
But vx = v1 - v4 so that
supernode analysis electric circuit
(2.4)
For loop 3,
supernode analysis electric circuit
But 6i3 = v3 - v2 and vx = v1 - v4 . Thus,
supernode analysis electric circuit
(2.5)
We need four node voltages, v1v2v3, and v4, and it requires only four out of the five (2.1) to (2.5) to find them. Although the fifth equation is redundant, it can be used to check results.
Substituting (2.3) to both (2.1) and (2.2) respectively, gives
supernode analysis electric circuit
(2.6)
and
supernode analysis electric circuit
(2.7)
(2.4) , (2.6) and (2.7) can be replaced in matrix form as
supernode analysis electric circuit
Using Cramer's rule gives
supernode analysis electric circuit
Thus, we arrive at the node voltage as
supernode analysis electric circuit
and v2 = v1 - 20 = 6.667 V.

Untuk Bahasa Indonesia baca Analisis Supernode.
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