Norton TheoremEdit
Norton Theorem is a cornerstone of practical circuit analysis, providing a reliable way to simplify complex networks into a simple equivalent as seen from two terminals. Named after Edward Lawry Norton, the theorem states that any linear, bilateral network with two output terminals can be replaced by a current source in parallel with an impedance. This Norton equivalent behaves identically to the original network for all loading conditions at the terminals, making it easier to analyze currents and voltages in the rest of the circuit. The Norton form is the dual of Thevenin's theorem and, in many engineering workflows, offers a flexible way to reason about how a network will interact with different loads and sources.
From a practical engineering standpoint, Norton equivalents are especially valuable because they let designers treat complex subsystems as modular parts. This fits well with how products are developed in a competitive market: different subsystems, components, and suppliers can be swapped or upgraded without reworking the entire circuit model. The method applies to both dc and ac situations, and it extends to networks that include dependent sources, provided the right procedures are followed. In many classrooms and industries, Norton analysis is taught alongside its theoretical twin to give students and practitioners a robust toolkit for analyzing power delivery, signal integrity, and impedance matching.
Statement
If a linear two-terminal network consists of independent sources, dependent sources, and impedances, there exists an equivalent Norton circuit: a current source in parallel with an impedance, as seen from the two terminals. The Norton impedance is the same as the Thevenin impedance of the network, and the Norton current is the short-circuit current that flows when the output terminals are directly connected.
The Norton current, I_N, and the Norton impedance, R_N, satisfy I_N = V_th / R_th, where V_th is the open-circuit voltage across the terminals and R_th is the equivalent resistance seen with all independent sources turned off (for networks without dependent sources, or with a straightforward dependent-source configuration, this reduces to the usual resistance seen by the load). In other words, the Norton model is the two-terminal dual of the Thevenin model: I_N and R_N describe the same external behavior as V_th and R_th.
For networks that include dependent sources, R_N is determined by keeping the dependent sources active and using a test source to measure the resulting voltage/current response, rather than simply turning off all sources.
The theorem is valid for both direct current and alternating current analysis, with complex impedances in the ac case. In phasor form, I_N is a complex current source and R_N is a complex impedance that reflects the network’s response at the frequency of interest.
Relationship to Thevenin's theorem
Norton and Thevenin theorems are two ways of describing the same external behavior of a network. If you have the Thevenin equivalent, you can convert to the Norton form via I_N = V_th / R_th and R_N = R_th. Conversely, from a Norton model you can obtain the Thevenin equivalent by V_th = I_N × R_N. This duality makes the pair a versatile choice depending on what is more convenient for a given loading problem, whether you’re calculating current through a particular load or the voltage at a specific node.
Procedure and practical steps
Identify the two terminals where the load connects to the rest of the circuit.
Determine the Norton current, I_N. This is the short-circuit current that flows if the two terminals are connected directly with a wire (i.e., the load is replaced by a short).
Determine the Norton impedance, R_N. With all independent sources deactivated (zeroed), find the resistance “seen” from the terminals. If dependent sources are present, use a test source to characterize the response rather than simply turning everything off.
Form the Norton equivalent: a current source I_N in parallel with R_N, connected to the same two terminals as the original network.
To verify, you can compare with the Thevenin form: V_th = I_N × R_N. For any load, the current and voltage at the terminals should be identical under both representations.
In ac analysis with complex impedances, treat I_N as a phasor current source and R_N as a complex impedance; the same relationships hold, and you can use impedance algebra to evaluate currents and voltages in the load.
Variants, extensions, and practical notes
Dependent sources: When dependent sources are present, you cannot simply “turn off” all sources to find R_N. Instead, leave dependent sources active and use a test source to determine the terminal impedance.
Nonlinear elements: The Norton theorem applies to linear networks. If the circuit contains nonlinear devices (e.g., diodes, transistors outside small-signal models, saturating elements), the exact Norton equivalent may not exist for all operating points; linearization around a chosen bias point is commonly used in such cases.
Educational and design contexts: In teaching and in practice, engineers often switch between Thevenin and Norton viewpoints to simplify multiterminal problems, power-transfer calculations, and impedance matching tasks. The choice can reflect which form yields the easiest algebra or the most transparent intuition for a given loading condition.
Modeling in modern tools: Circuit simulators like SPICE and other electronic design automation packages routinely implement both Thevenin and Norton equivalents as convenient abstractions when analyzing subcircuits, aiding rapid iteration in design and verification.
Example
Consider a simple dc network: a 12 V source in series with a 4 Ω resistor, connected to two terminals A and B where a load can be attached.
Open-circuit condition at A–B gives V_th = 12 V and R_th = 4 Ω.
Short-circuit condition across A–B gives I_sc = 12 V / 4 Ω = 3 A, so I_N = 3 A.
The Norton equivalent is a 3 A current source in parallel with a 4 Ω impedance (R_N = 4 Ω).
If a load R_L = 8 Ω is connected across A–B, the load current can be found either from the Thevenin form (I_L = V_th / (R_th + R_L) = 12 V / (4 Ω + 8 Ω) = 1 A) or from the Norton form (I_L = I_N × R_N / (R_N + R_L) = 3 A × 4 Ω / (4 Ω + 8 Ω) = 1 A). The two methods yield the same result, illustrating the practical interchangeability of the representations.
This kind of modular thinking aligns with how engineers approach systems in real-world settings: you model a network once, then reuse that model across multiple loading scenarios, tests, and product iterations.