Norton's Theorem

Norton's theorem provides an alternative but equally powerful way to simplify linear networks at two terminals. It states that any linear two-terminal network can be replaced by an equivalent circuit consisting of a single Norton current source In in parallel with a Norton resistance Rn. Norton's theorem is the dual of Thevenin's theorem, and both equivalents carry exactly the same information about the terminal behavior of the original network.

Core Concept of Norton's Theorem

The Norton equivalent is the current-source dual of the Thevenin equivalent. While Thevenin uses a voltage source in series with a resistance, Norton uses a current source in parallel with a resistance. Both representations are mathematically equivalent and can be converted into each other using source transformation. The choice between them depends on the circuit structure and what is most convenient for further analysis.

The Norton current In is defined as the short-circuit current that flows between terminals A and B when they are connected together (shorted). When terminal A is shorted to terminal B, the current flowing through this short from A to B is In. The Norton resistance Rn is identical to the Thevenin resistance Rth: it is the resistance seen at the terminals after all independent sources are deactivated.

The relationship between the Thevenin and Norton equivalents is given by source transformation: Vth = In * Rn and Rn = Rth. This means knowing any two of Vth, In, and Rth/Rn allows finding the third. This is often used as a quick calculation shortcut in GATE problems.

Mathematical Expression

For a Norton equivalent circuit, when a load RL is connected across terminals A and B, the load voltage is found using the current divider principle. The Norton current In splits between Rn and RL in parallel:

Load current IL equals In multiplied by Rn divided by (Rn plus RL). Load voltage VL equals IL multiplied by RL. This can also be written as VL equals In multiplied by (Rn parallel with RL). The three defining equations are: In = Isc, Rn = Voc / Isc = Rth, and Vth = In * Rn.

Practical Understanding

Norton's theorem is particularly natural for circuits driven by current sources, because the equivalent directly mirrors the source structure. In transistor small-signal models, the transistor is modeled as a dependent current source (gmVbe or gmVgs) in parallel with an output resistance. This is precisely a Norton form, and the circuit analysis proceeds directly without needing to convert to Thevenin.

In digital circuits, logic gate outputs are characterized by their drive strength, which is essentially the Norton equivalent current available to charge load capacitances. The Norton representation also appears in antenna theory, where a receiving antenna is modeled as a Norton equivalent with an induced current source and radiation resistance.

Solved Numerical Example

A circuit has a 24 V source in series with a 6 ohm resistor, feeding two terminals A and B. A 12 ohm resistor is also connected directly between A and B. Find the Norton equivalent as seen from A-B and calculate the voltage across a 4 ohm load connected at A-B.

Given:
Vs = 24 V, R1 = 6 Ω (series with Vs), R2 = 12 Ω (shunt across A-B terminals)
Load RL = 4 Ω

Why this formula applies:
Norton theorem: find Isc (short A-B) for In, find Rn from deactivated network.

Step 1 - Find In = Isc (short A to B):
With A-B shorted, R2 is short-circuited (bypassed), so only R1 carries current from Vs.
Isc = Vs / R1 = 24 / 6 = 4 A
In = 4 A

Step 2 - Find Rn (kill Vs = short circuit, A-B open):
Rn = R1 parallel R2 = (6 * 12) / (6 + 12) = 72 / 18 = 4 Ω

Verification: Vth = In * Rn = 4 * 4 = 16 V
Check: Voc = Vs * R2/(R1+R2) = 24*12/18 = 16 V ✓

Formula:
IL = In * Rn / (Rn + RL)

Substitution:
IL = 4 * 4 / (4 + 4) = 16 / 8 = 2 A

Calculation:
VL = IL * RL = 2 * 4 = 8 V

Final Answer:
In = 4 A, Rn = 4 Ω, Load voltage VL = 8 V

Exam Tip: The fastest way to solve Norton and Thevenin problems in GATE is to find any two of the three quantities: Voc, Isc, and Rth/Rn. The third follows from the relation Rth = Voc / Isc. Avoid computing Rth by source deactivation when dependent sources are involved; use the test source method or the Voc/Isc ratio instead.

Mechanism: Thevenin to Norton Conversion

• In equals the short-circuit current at terminals A-B. Short the terminals and measure or calculate the current flowing through the short.
• Rn equals Rth, the resistance seen at A-B after killing all independent sources. The two equivalents share the same resistance value.
• Source transformation from Thevenin to Norton: In = Vth / Rth, Rn = Rth. From Norton to Thevenin: Vth = In * Rn, Rth = Rn.
• With a load RL connected, current through RL uses current divider: IL = In * Rn / (Rn + RL).
• The Norton form is natural for transistor small-signal models where the active device is inherently modeled as a dependent current source.

Quick Revision

• Norton theorem: any linear two-terminal network equals In (short-circuit current) in parallel with Rn (equivalent resistance).
• In = Isc: current flowing when terminals A and B are shorted together.
• Rn = Rth: found by killing independent sources. Test source method needed if dependent sources exist.
• Relationship: Vth = In * Rn = In * Rth. All three quantities are linked.
• Load current: IL = In * Rn / (Rn + RL) by current divider.
• Norton is preferable for current-source dominated circuits; Thevenin for voltage-source dominated circuits.
• GATE trap: applying Norton to nonlinear circuits (e.g., with diodes), or treating Rn differently from Rth.