Norton's Theorem
Norton's Theorem states that any linear electrical network can be replaced by an equivalent circuit consisting of a single current source (I_N) in parallel with a single resistance (R_N). It is the dual of Thevenin's Theorem and is particularly convenient when analysing circuits where current distribution is of primary interest. Norton and Thevenin equivalents are interconvertible, and choosing between them depends on whether the circuit is better suited to series or parallel analysis.
Key Formula
I_N = short-circuit current; R_N = R_th = V_oc / I_sc
LaTeX: I_N = I_{sc}, \quad R_N = R_{th} = \frac{V_{oc}}{I_{sc}}
| Symbol | Meaning | Unit |
|---|---|---|
| I_N | Norton equivalent current (short-circuit current at terminals) | Ampere (A) |
| R_N | Norton equivalent resistance (same as Thevenin resistance) | Ohm (Ω) |
| I_{sc} | Short-circuit current measured at the terminals | Ampere (A) |
| V_{oc} | Open-circuit voltage at the terminals | Volt (V) |
Worked Example
Problem
A circuit has a 12 V source in series with 3 Ω. Find the Norton equivalent and the current through a 6 Ω load.
Solution
Step 1: Find I_N (short-circuit current): short the terminals A–B. I_sc = 12 / 3 = 4 A (short circuit across output, all current flows through 3 Ω). Step 2: Find R_N: deactivate 12 V source (short it). R_N = 3 Ω (only the series 3 Ω remains between terminals). Step 3: Norton equivalent: 4 A source in parallel with 3 Ω. Step 4: Current through 6 Ω load (current divider): I_load = I_N × R_N / (R_N + R_load) = 4 × 3 / (3 + 6) = 12/9 = 1.33 A.
Answer
I_N = 4 A, R_N = 3 Ω, I_load = 1.33 A
Comparison of Norton and Thevenin Equivalent Circuits
| Property | Thevenin Equivalent | Norton Equivalent | Relationship |
|---|---|---|---|
| Source type | Voltage source (V_th) | Current source (I_N) | Dual of each other |
| Source connection | Series with R_th | Parallel with R_N | Different topology |
| Equivalent resistance | R_th | R_N | R_th = R_N always |
| Conversion formula | V_th = I_N × R_N | I_N = V_th / R_th | Source transformation |
| Best used when | Series loads | Parallel loads | Context-dependent |
| Analysis method | Open-circuit voltage | Short-circuit current | Complementary |
Interactive Tools
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Related Terms
Thevenin's Theorem
Thevenin's Theorem states that any linear electrical network with voltage sources, current sources, and resistances can be replaced by an equivalent circuit consisting of a single voltage source (V_th) in series with a single resistance (R_th). This simplification makes it much easier to analyse the behaviour of a load connected to a complex network, as only the terminal behaviour matters. It is widely used in circuit design, power systems, and electronics to simplify analysis without solving the full network repeatedly.
Superposition Theorem
The Superposition Theorem states that in any linear circuit with multiple independent sources, the response (voltage or current) at any element equals the algebraic sum of the responses caused by each independent source acting alone, with all other independent sources deactivated. Voltage sources are deactivated by replacing them with short circuits, while current sources are deactivated by replacing them with open circuits. This theorem greatly simplifies the analysis of circuits with multiple sources and applies only to linear systems.
Kirchhoff's Current Law
Kirchhoff's Current Law (KCL) states that the algebraic sum of all currents entering and leaving any node (junction) in an electrical circuit equals zero. This law is a consequence of conservation of electric charge — charge cannot accumulate at a node under steady-state conditions. KCL is the basis for nodal analysis, a powerful technique for solving complex parallel and combined circuits.
Named after Edward Lawry Norton (1898–1983), an American engineer at Bell Labs who described the theorem in an internal memo in 1926. The theorem was independently stated by Hans Ferdinand Mayer in Germany the same year. "Norton" is an English surname meaning "north town".