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,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),

(1) |

**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.

Asupernodeis 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),

(2) |

To apply Kirchhoff's voltage law to the supernode in Figure.(1),we redraw the circuit in Figure.(1) to Figure.(2).

Figure 2. Loop in the supernode |

Going around the loop in the clockwise direction gives

(3) |

From Equations.(1) and (2) we obtain the node voltages.

Give attention to the following properties of a supernode :

- The voltage source inside the supernode provides a constraint equation needed to solve for the node voltages.
- A supernode has no voltage of its own.
- 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.**

Figure 3 |

__Solution :__
The supernode contains the 2 V source, nodes 1 and 2, and 10 Ω resistor.

Figure 4a |

Applying KCL to the supernode in Figure.(4a) gives

Expressing

*i*and_{1}*i*in terms of the node voltages,_{2}(1.1) |

To get the relationship between

*v*and_{1}*v*, we apply KVL to the circuit in Figure.(4b),_{2}Figure 4b |

Going around the loop, we obtain

and

*v*=_{2}*v*+ 2 = -5.333 V. Note that the 10 Ω resistor does not make any difference because it is connected across the supernode._{1}**2. Find the node voltages in the circuit of Figure.(5)**

Nodes 1 and 2 form a supernode, also do nodes 3 and 4. We apply KCL to the two supernodes as in Figure.(6a).

(2.5) |

*v*,

_{1}*v*,

_{2}*v*, and

_{3}*v*, 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.

_{4}
Substituting (2.3) to both (2.1) and (2.2) respectively, gives

Untuk Bahasa Indonesia baca Analisis Supernode.

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