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What is a Supernode?

 

Circuit Analysis

What Is a Supernode?

A supernode is formed when an ideal voltage source connects two non-reference nodes. It allows us to apply Kirchhoff's Current Law (KCL) to the combined circuit boundary.

1. Understanding a Supernode

In nodal analysis, a voltage source between two unknown-voltage nodes makes the current through that source difficult to express directly using Ohm's law. We combine the two nodes and the voltage source into one extended region called a supernode.

Key idea

Apply KCL around the outer boundary of the supernode, then use the voltage source to write an additional voltage equation.

2. Supernode Circuit Diagram

The voltage source connects nodes VL and VR. The dashed blue boundary surrounds the supernode. Currents through the external resistors are included in the KCL equation.

SUPERNODE V_L V_R R₁ R₂ R₃ R₄ 6 V External circuit Reference / return path
Supernode boundary
Voltage source
Circuit connections

Conceptual diagram. The resistor values and exact circuit connections depend on the specific problem being solved.

3. Why Do We Need a Supernode?

1

An ideal voltage source connects two unknown nodes

The voltage difference is known, but the individual node voltages may be unknown.

2

Current through the source is unknown

We cannot calculate that current directly from the voltage and resistance of an ideal source.

3

Combine the nodes and apply KCL

Treat the source and its two terminals as one supernode, and sum the currents crossing the outer boundary.

4. Example From the Given Circuit

In the original circuit, the 6 V source connects the left midpoint VL and right midpoint VR. These nodes form a supernode.

Equation 1: KCL at the supernode

(V_L − V_T)/1kΩ + V_L/1kΩ

+

(V_R − V_T)/1kΩ + V_R/1kΩ = 0

V_T = V_L + V_R

Equation 2: Voltage-source constraint

V_L − V_R = 6 V

These equations are combined with KCL at the top node to find the unknown node voltages.

5. Steps to Solve a Supernode Problem

  1. Identify an ideal voltage source connecting two non-reference nodes.
  2. Combine the two nodes and the source into a supernode.
  3. Apply KCL to the supernode's outer boundary.
  4. Write a voltage constraint from the source polarity and value.
  5. Solve the resulting equations for the unknown node voltages.

Quick Summary

  • Definition: A combined region formed by two nodes connected by an ideal voltage source.
  • Main law: Apply KCL around the outer boundary.
  • Extra equation: Use the voltage source's polarity and voltage value.
  • Purpose: Simplifies nodal analysis when a voltage source lies between unknown nodes.

Remember

KCL for the supernode plus the voltage-source constraint gives the equations needed to solve the circuit.

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