Norton’s Theorem states that any portion of a linear electrical circuit with independent and/or dependent sources and resistances can be replaced by an equivalent circuit consisting of a current source I_{sc} in parallel with a resistor R_{th}. Here, I_{sc} is the short circuit current at the terminals, and R_{th} is the Thevenin equivalent resistance.

## Steps for Applying Norton’s Theorem.

### 1. Circuit Containing Only Independent Sources:

**a. Find the Short Circuit Current I _{sc} .**

- Short the terminals of interest.
- Calculate the current through the shorted terminals.

**b. Find Thevenin Equivalent Resistance R _{th}.**

- Deactivate all independent sources (replace voltage sources with short circuits and current sources with open circuits).
- Calculate the resistance seen from the terminals.

**c. Construct Norton’s Equivalent Circuit.**

- Place I
_{sc}in parallel with R_{th}. - Reconnect the load resistor to this equivalent circuit.

### 2. Circuit Containing Only Dependent Sources:

**a. Find the Short Circuit Current I _{sc}: **

- Short the terminals of interest.
- Calculate the current through the shorted terminals, considering the dependency relations.

**b. Find Thevenin Equivalent Resistance R _{th}.**

- Apply an external voltage source at the terminals and calculate the resulting current, or apply an external current source and measure the voltage.
- R
_{th}is the ratio of the applied voltage to the resulting current (or vice versa).

**c. Construct Norton’s Equivalent Circuit: **

- Place I
_{sc}in parallel with R_{th}. - Reconnect the load resistor to this equivalent circuit.

### 3. Circuit Containing Both Independent and Dependent Sources:

**a. Find the Short Circuit Current I _{sc}:**

- Short the terminals of interest.
- Calculate the current through the shorted terminals, considering both independent and dependent sources.

**b. Find Thevenin Equivalent Resistance R _{th}:**

- Deactivate all independent sources.
- Use the methods for circuits with dependent sources to determine R
_{th}.

**c. Construct Norton’s Equivalent Circuit: **

- Place I
_{sc}in parallel with R_{th}. - Reconnect the load resistor to this equivalent circuit.

## Circuit Example of Norton’s Theorem.

By Using Norton’s theorem to find V_{0}. We use same example as in Thevenin theorem example.

**Step 1: Find the Short Circuit Current I _{sc} .**

By using KVL around upper loop-1:

12i_{1} + 8 (i_{1} – i_{3} ) + 5 (i_{1} – i_{2} ) = 0

25 i_{1} – 5 i_{2} – 8 i_{3} = 0 Equation-1

By using KVL around upper loop-2.

5 (i_{2} – i_{1}) + 20 (i_{2} – i_{3} ) = 72

-5 i_{1} + 25 i_{2} – 20 i_{3} = 72 Equation-2.

By using KVL around lower right loop-3.

8 (i_{3} – i_{1} ) + 20 (i_{3} – i_{2}) = 0

-8 i_{1} – 20 i_{2} + 28 i_{3} = 0 Equation-3

By solving all these 3 equations by Cramer’s rule.

**i _{1} = 6 A, i_{2} = 12.72 A , i_{3} = 10.8 A**

So finally we got **I _{sc}** =

**10.8 A**

**Step 2: Find Thevenin Equivalent Resistance R _{th}.**

- Deactivate all independent sources.
- Replace Voltage source with a short circuit.
- Calculate the resistance seen from the open terminals.

R_{th} = (12 x 12 ) / (12 + 12) = 144/ 24 = 6Ω

**Step 3: Construct Norton’s Equivalent Circuit:**

Reconnect the load.

As we know that V_{0} = i_{2} x R_{o}

so as shown above calculation V_{0} = 25.92 is the same answer as we prove in Thevenin’s Theorem example.