Norton’s Theorem in Electric Circuits

An American engineer, E.L. Norton at Bell Telephone Laboratories, proposed a theorem similar to Thevenin’s theorem.

Norton’s theorem states that:

A linear two-terminal network can be replaced by an equivalent circuit consisting of a current source $i_N$ in parallel with a resistor $R_N$, where $i_N$ is the short-circuit current through the terminals and $R_N$ is the equivalent resistance at the terminals when the independent sources are turned off.

The Norton equivalent circuit consists of:

• Norton current source $i_N$
• Norton resistance $R_N$

---

Norton Current

The Norton current is the short-circuit current flowing through the output terminals.

$$i_N = I_{sc}$$

where:

• $I_{sc}$ = short-circuit current

---

Norton Resistance

The Norton resistance is the equivalent resistance seen from the terminals after deactivating all independent sources.

$$R_N = R_{th}$$

Thus, Norton resistance is equal to Thevenin resistance.

---

Relationship Between Thevenin and Norton Equivalents

In fact, source transformation of the Thevenin equivalent circuit leads to Norton’s equivalent circuit.

The relationship between Thevenin and Norton forms is:

$$V_{th} = i_N R_N$$

or

$$i_N = \frac{V_{th}}{R_N}$$

---

Procedure for Finding Norton’s Equivalent Circuit

If the network contains resistors and independent sources, follow the instructions below.

Step 1: Find $R_N$

Deactivate all independent sources and find the equivalent resistance using circuit reduction techniques.

Source Deactivation

• Independent voltage sources are replaced with short circuits.
• Independent current sources are replaced with open circuits.

---

Step 2: Find Norton Current

Find the short-circuit current $i_N$ with all sources activated.

$$i_N = I_{sc}$$

---

Step 3: Draw Norton Equivalent Circuit

Replace the original network with:

• A current source $i_N$
• A parallel resistance $R_N$

Reconnect the load resistance to the equivalent circuit.

---

Load Current Using Norton’s Theorem

If a load resistance $R_L$ is connected across the Norton equivalent circuit, the load current can be determined using current division.

$$I_L = i_N \left( \frac{R_N}{R_N + R_L} \right)$$

---

Advantages of Norton’s Theorem

• Simplifies complex circuits.
• Useful in parallel network analysis.
• Makes load current calculations easier.
• Applicable to both AC and DC circuits.
• Convenient for current source networks.

---

Limitations of Norton’s Theorem

• Applicable only to linear bilateral networks.
• Not directly suitable for non-linear elements.
• Additional calculations may be required for power analysis.

---

Applications of Norton’s Theorem

• Circuit simplification
• Electrical network analysis
• Electronic circuit design
• Communication systems
• Power system studies

---

Summary

• Norton’s theorem converts a complex network into an equivalent current source and parallel resistance.
• Norton current is the short-circuit current across the terminals.
• Norton resistance is the equivalent resistance seen from the terminals.
• Norton resistance is equal to Thevenin resistance.
• Source transformation relates Thevenin and Norton equivalents.
• Norton’s theorem simplifies the analysis of electrical circuits.