Supernode analysis is a technique used in nodal analysis when a voltage source appears between two non-reference nodes in a circuit. In this scenario, the two nodes and any elements connected in parallel with the voltage source are combined into a single entity called a supernode.

### Supernode Analysis

Supernode analysis is a technique used in nodal analysis when a voltage source appears between two non-reference nodes in a circuit. In this scenario, the two nodes and any elements connected in parallel with the voltage source are combined into a single entity called a supernode. This approach simplifies the analysis by allowing the voltage source’s constraint to be incorporated directly into the node voltage equations.

### Steps for Supernode Analysis

**Step 1: Identify the Supernode:**

Locate the voltage source between two non-reference nodes. Combine these two nodes and any elements connected in parallel with the voltage source into a supernode.

**Step 2: Write KCL Equations for the Supernode:**

Apply Kirchhoff’s Current Law (KCL) to the supernode, summing the currents entering and leaving the supernode.

**Step 3: Incorporate the Voltage Source Constraint:**

Write an equation that represents the voltage source constraint between the two nodes in the supernode. This equation directly relates the voltages of the two nodes.

**Step 4: Solve the System of Equations:**

Use the KCL equations and the voltage source constraint to form a system of linear equations. Solve these equations to find the node voltages.

### Supernode Circuit Analysis Example.

Find the I_{x} in below circuit by using supernode concept.

#### Step-by-Step Solution:

**Identify the Supernode:**

- Nodes ( A ) and ( B ) with voltage source ( V_s ) form a supernode.

**Write KCL Equations for the Supernode:**

Given nodes 1 and 2 with a voltage source between them, the supernode includes both nodes and any elements connected in parallel. We write the KCL equation for the supernode.

2A = (V_{1} /0.5) + (V_{2} / 0.5) + (V_{2} – V_{3 } /0.5) Equation-1

For Node 3, the equation of KCL is

4A = (V_{3} – V_{2 } /0.5) + 2. Equation-2

and the 3rd equation is

V_{2} – V_{1} = 6 Equation-3

By combining all these 3 equations and solving

V_{1} = −2 V, V_{2} = 4 V, V_{3} = 5 V.

So finally current is

Ix = V_{2} / 0.5 = 4 / 0.5 = 8 A.

so Ix is 8 A.

By solving these equations, you can determine the voltages at nodes ( A ) and ( B ), and subsequently find the currents through the resistors.

### Summary

When a voltage source appears between two non-reference nodes, combining these nodes into a supernode and using supernode analysis simplifies the circuit analysis. This method allows the incorporation of voltage source constraints directly into the nodal equations, making it easier to solve for unknown node voltages and currents.