Norton's theorem
Norton's theorem is a fundamental principle in electrical circuit analysis that enables the simplification of any linear electrical network—containing independent voltage and current sources and passive impedances such as resistors, inductors, and capacitors—into an equivalent circuit consisting of a single constant current source connected in parallel with a single impedance, known as the Norton equivalent circuit.[1] This theorem serves as the dual counterpart to Thévenin's theorem, which instead represents the network using a single voltage source in series with an impedance, allowing engineers to choose the most convenient form for specific analyses, such as those involving parallel load connections or current-based calculations.[2] The approach is particularly valuable in direct current (DC) and alternating current (AC) circuit design, where it reduces computational complexity for determining currents and voltages across loads without solving the entire network simultaneously.[3] Named after Edward Lawry Norton, the theorem was independently derived in 1926 by Norton, an engineer at Bell Laboratories in New York City, in an internal technical memorandum titled "Design of Finite Networks for Uniform Frequency Characteristic" dated November 11, 1926, which was never formally published.[4] Concurrently, German engineer Hans Ferdinand Mayer at Siemens & Halske developed the identical concept, leading to its alternative designation as the Mayer–Norton theorem.[5] This method underpins broader applications in network theory, including impedance matching for maximum power transfer and the analysis of transistor circuits, amplifiers, and power distribution systems.[1]History
Origins and Discovery
In the early 20th century, electrical engineers at institutions like Bell Laboratories and Siemens & Halske advanced network analysis techniques to develop simplified equivalent circuits for complex linear systems, addressing the growing demands of telecommunications infrastructure.[6] These efforts built on earlier work, such as Léon Charles Thévenin's 1883 formulation of a voltage-source equivalent, which provided a foundational approach to circuit reduction.[6] The primary motivation for these advancements stemmed from challenges in telephone network design, where engineers needed efficient methods to model intricate linear circuits involving voltage and current sources without exhaustive recalculations for varying loads.[6] At Bell Laboratories, this was particularly relevant for optimizing signal transmission in telephone systems, requiring practical tools to represent network behavior at specific terminals.[6] Similarly, researchers at Siemens & Halske encountered analogous issues in amplifier and communication circuit design, pushing for streamlined analysis.[6] The first documented formulation of what became known as Norton's theorem appeared in internal technical memos around 1926, emphasizing its utility for practical simplification in engineering applications.[6] Edward Lawry Norton, a Bell Laboratories engineer, described the current-source equivalent in a technical memorandum titled "Design of finite networks for uniform frequency characteristic," highlighting its equivalence to voltage-based models for network analysis.[6] Independently, Hans Ferdinand Mayer at Siemens & Halske outlined a similar approach in contemporaneous work, focusing on its application to electronic circuits.[6] These early descriptions underscored the theorem's role in expediting circuit evaluations for real-world telephony problems.[6]Independent Developments
Edward Lawry Norton, an engineer at Bell Telephone Laboratories, derived the current-source equivalent circuit in 1926 while working on network design for uniform frequency characteristics.[7] In his internal technical memorandum dated November 11, 1926, titled "Design of Finite Networks for Uniform Frequency Characteristic," Norton described the utility of representing complex impedance networks with a parallel combination of a current source and impedance, emphasizing its application to linear circuits.[4] This formulation provided a practical tool for simplifying analysis in telephone systems and related electrical engineering tasks at Bell Labs.[8] Independently, in the same year, Hans Ferdinand Mayer, a researcher at Siemens & Halske in Germany, developed an equivalent formulation focused on current sources for linear networks.[5] Mayer's work, published in November 1926 in the German journal Telegraphen- und Fernsprech-Technik under the title "Über das Ersatzschema der Verstärkerröhre," extended the concept to amplifier tubes and communication circuits, highlighting its relevance to AC systems in electrical engineering.[9] His publication provided a detailed theoretical basis, including derivations for the equivalent current and impedance.[8] Norton's contribution remained in an internal memorandum at Bell Labs and was not publicly disseminated until the late 1940s, when colleagues began crediting him for the current-source equivalent in technical discussions.[6] In contrast, Mayer's work appeared in print shortly after its completion, gaining early recognition in German technical literature.[5] The parallel developments received limited cross-recognition initially due to the internal nature of Norton's report and linguistic barriers between English and German publications, though both are now acknowledged as foundational to the theorem's history.[8] This dual origin underscores Norton's theorem as a complement to Thévenin's earlier 1883 voltage-source equivalent in circuit theory.[9]Electrical Engineering Applications
Formal Statement
Norton's theorem, developed in 1926 by Edward Lawry Norton at Bell Laboratories, states that any linear time-invariant electrical network containing independent voltage and current sources along with impedances can be replaced, at a pair of output terminals, by an equivalent circuit consisting of a single current source in parallel with a single impedance.[4] This equivalence holds under the assumptions of linearity, meaning the principle of superposition applies and there are no nonlinear elements such as diodes or transistors; the network is treated as a one-port configuration with clearly defined output terminals A and B. The theorem is applicable to both direct current (DC) circuits, where impedances reduce to resistances, and alternating current (AC) circuits, which require analysis at a specific frequency using complex impedances.[10] The key components of the Norton equivalent are the Norton current I_N and the Norton resistance R_N (or impedance Z_N for AC). The Norton current I_N is defined as the short-circuit current flowing between the output terminals A and B when they are directly connected, with all independent sources active. The Norton resistance R_N (or Z_N) is the equivalent resistance (or impedance) measured across the same terminals after deactivating all independent sources in the network—specifically, by short-circuiting voltage sources and open-circuiting current sources—while leaving any dependent sources intact if present.[10][11] For networks containing dependent sources, the theorem remains valid, but calculating R_N (or Z_N) requires a modified approach since deactivating independent sources alone may not suffice. In such cases, a test signal method is used: apply a known current (e.g., 1 A) across the output terminals with all independent sources deactivated, then measure the resulting voltage V across those terminals; the Norton resistance is then R_N = V / [1](/page/1) A (or equivalently for impedance in AC). This ensures the equivalent accurately represents the network's behavior at the terminals.Norton Equivalent Circuit
The Norton equivalent circuit represents any linear electrical network as a simplified two-terminal model consisting of an ideal current source I_N connected in parallel with a resistor R_N, attached across the designated terminals A and B.[12] In this configuration, the current source delivers a constant current I_N, while the resistor R_N provides the equivalent internal resistance of the network as viewed from those terminals.[11] This parallel arrangement ensures that the Norton equivalent behaves identically to the original network for any passive load connected between A and B, allowing straightforward analysis using current division principles.[12] The current source models the network's short-circuit current at the terminals, and the resistor establishes the Thevenin-like impedance that influences load current distribution.[13] In standard schematic diagrams, the ideal current source is symbolized by a circle with an arrow indicating current direction, drawn in parallel with the resistor symbol, and both spanning the A-B terminals for clarity.[2] For alternating current (AC) applications involving impedances, the model extends to a complex current source \mathbf{I}_N in parallel with an impedance \mathbf{Z}_N, where the Norton admittance is defined as Y_N = 1 / Z_N to account for frequency-dependent behavior.[13] The Norton equivalent is valid solely for the specified terminals A and B and does not directly apply to simplifying multi-port networks without additional analysis.[12] It serves as the current-based dual to the Thévenin equivalent, which employs a voltage source in series with resistance.[11]Calculation Procedures
To determine the Norton current I_N, short-circuit the output terminals A and B of the network, and compute the total current flowing through this short circuit.[14] This calculation treats the short circuit as the load and can employ the superposition principle for networks with multiple independent sources: activate one source at a time while deactivating others (replacing voltage sources with shorts and current sources with opens), sum the resulting short-circuit currents from each configuration, and add their contributions to obtain I_N.[11] To find the Norton resistance R_N, first deactivate all independent sources in the network by replacing independent voltage sources with short circuits and independent current sources with open circuits.[14] Then, compute the equivalent resistance seen looking into terminals A and B, using techniques such as series-parallel reduction for simple networks or mesh and nodal analysis for more complex topologies.[11] In circuits containing dependent sources, the standard deactivation method for R_N does not apply directly, as dependent sources must remain active since their behavior depends on circuit variables. Instead, apply the test source method: connect a test voltage source V_t (e.g., 1 V) across A and B with all independent sources deactivated (voltage sources replaced by short circuits and current sources by open circuits), measure the resulting current I_t drawn from the test source, and calculate R_N = V_t / I_t.[15] Alternatively, inject a test current source I_t (e.g., 1 A) with all independent sources deactivated and compute R_N = V_t / I_t, where V_t is the voltage across A and B.[16] For intricate networks, mesh or nodal analysis facilitates the computations required in both steps, particularly when manual series-parallel simplification is infeasible.[11] Circuit simulation software such as SPICE can verify these results by modeling the network and extracting I_N and R_N parameters.[17] The resulting Norton equivalent consists of I_N in parallel with R_N.[14]Example of Application
Consider a simple DC circuit to illustrate the application of Norton's theorem, consisting of two parallel branches connected across output terminals A and B: the first branch includes a 28 V voltage source in series with a 4 Ω resistor, and the second branch includes a 7 V voltage source in series with a 1 Ω resistor.[1] To determine the Norton equivalent circuit across terminals A and B, first calculate the Norton current I_N, which is the short-circuit current flowing through the terminals when A and B are directly connected. With the terminals shorted, the current in the first branch is I_1 = \frac{28 \, \mathrm{V}}{4 \, \Omega} = 7 \, \mathrm{A}, and the current in the second branch is I_2 = \frac{7 \, \mathrm{V}}{1 \, \Omega} = 7 \, \mathrm{A}. Thus, the total short-circuit current is I_N = I_1 + I_2 = 14 \, \mathrm{A}.[1] Next, compute the Norton resistance R_N by deactivating all independent sources: replace the voltage sources with short circuits. The resulting network across A and B consists of the 4 Ω and 1 Ω resistors in parallel, so R_N = 4 \, \Omega \parallel 1 \, \Omega = \frac{4 \times 1}{4 + 1} = 0.8 \, \Omega. [1] The Norton equivalent circuit is therefore a 14 A current source in parallel with a 0.8 Ω resistor. To verify this equivalent, reconnect a 2 Ω load resistor across terminals A and B. The equivalent resistance seen by the current source is R_{eq} = 0.8 \, \Omega \parallel 2 \, \Omega = \frac{0.8 \times 2}{0.8 + 2} = 0.571 \, \Omega. The voltage across the load is V_L = I_N \times R_{eq} = 14 \, \mathrm{A} \times 0.571 \, \Omega = 8 \, \mathrm{V}, and the current through the load is I_L = \frac{V_L}{2 \, \Omega} = 4 \, \mathrm{A}. This matches the load voltage and current obtained by direct analysis of the original circuit, confirming the equivalence.[1] For comparison, this Norton equivalent can be converted to a Thévenin equivalent using V_{th} = I_N R_N = 14 \, \mathrm{A} \times 0.8 \, \Omega = 11.2 \, \mathrm{V} and R_{th} = R_N = 0.8 \, \Omega.[1]Relation to Thévenin's Theorem
Duality and Conversion
Norton's theorem represents the dual of Thévenin's theorem in linear circuit analysis, where the Norton equivalent employs a current source in parallel with an equivalent resistance, contrasting with the Thévenin equivalent's voltage source in series with an equivalent resistance.[15] This duality arises from fundamental source transformation principles, allowing any linear network to be equivalently represented in either voltage- or current-based forms at a pair of terminals.[18] Historically, Norton's theorem, developed in 1926 by Edward L. Norton at Bell Telephone Laboratories, serves as the current dual to the voltage-oriented approach of Thévenin's 1883 theorem, providing a complementary tool for circuit simplification.[19] The conversion between the two equivalents is straightforward, as they share the same equivalent resistance while relating the source values through Ohm's law. Specifically, the Thévenin voltage V_{th} equals the Norton current I_N multiplied by the equivalent resistance R_N, expressed as: V_{th} = I_N \times R_N The Norton current is the Thévenin voltage divided by the equivalent resistance: I_N = \frac{V_{th}}{R_{th}} and the resistances are identical: R_{th} = R_N [15][18] To obtain the Thévenin equivalent from a Norton circuit, compute the open-circuit voltage across the terminals, which yields V_{th}; conversely, to derive the Norton equivalent from a Thévenin circuit, determine the short-circuit current through the terminals, giving I_N.[15] This bidirectional transformation preserves the circuit's behavior for any connected load.[18]Equivalence Formulas
The equivalence between the Thévenin and Norton representations of a linear electrical network is established through specific mathematical relationships that allow direct conversion between the two forms. The Thévenin equivalent circuit features an open-circuit voltage V_{th} in series with an equivalent resistance R_{th}, whereas the Norton equivalent consists of a short-circuit current I_N in parallel with an equivalent resistance R_N. These are related by the core equations V_{th} = I_N R_N and R_{th} = R_N, ensuring identical terminal behavior for any load connected across the output.[11] Conversely, the Norton current can be obtained from the Thévenin parameters via I_N = \frac{V_{th}}{R_{th}}. This inverse relationship is derived by short-circuiting the output terminals of the Thévenin equivalent, which yields the short-circuit current I_N flowing through the equivalent resistance R_{th}.[11] To convert between equivalents, an algorithm involves first computing one representation from the original network—such as the Thévenin voltage and resistance via open-circuit voltage and deactivated sources—and then applying the above formulas to obtain the other. This approach is particularly useful in mixed analyses, where the voltage form simplifies open-circuit evaluations and the current form aids loaded or short-circuit assessments.[11] In alternating current (AC) circuits, the formulations extend naturally by replacing resistance with complex impedance, yielding Z_{th} = Z_N and V_{th} = I_N Z_N, where V_{th} and I_N are phasor quantities. This duality between voltage and current sources underpins the interconversion, maintaining equivalence across DC and AC domains.[20]Mathematical Derivation
General Proof
Norton's theorem holds for linear electrical networks because the voltage-current relationship at the port is affine, arising from the linearity of the underlying equations governing the circuit. A general linear network with passive impedances and independent sources can be modeled using nodal or mesh analysis, leading to a system of linear equations of the form \mathbf{V} = \mathbf{Z} \mathbf{I} + \mathbf{V}_s, where \mathbf{V} is the vector of node voltages, \mathbf{I} is the vector of branch currents, \mathbf{Z} is the symmetric positive-semidefinite impedance matrix representing the network topology and component values, and \mathbf{V}_s is the affine shift vector due to the independent voltage sources (with analogous formulations for current sources). This linearity ensures that the port behavior, when a load is connected, follows a straight-line characteristic in the v-i plane.[21] To derive the Norton equivalent, decompose the port current i using the superposition principle, separating contributions from internal sources and the port excitation. Consider injecting a test current I_{test} at the port while deactivating internal sources (voltage sources shorted, current sources opened), yielding a port voltage v_B = Z_N I_{test}, where Z_N is the input impedance seen at the port under these conditions (assuming passive sign convention where i enters the positive terminal). Next, with the test source removed (I_{test} = 0) and internal sources activated, the short-circuit current at the port is I_{sc}, representing the current due to sources alone (with I_N = I_{sc} directed appropriately). Superimposing these, the total port current is i = I_{sc} - \frac{v}{Z_N}, or equivalently, i = I_N - \frac{v}{Z_N}, where I_N = I_{sc} is the Norton current source and Z_N is the Norton impedance (identical to the Thévenin impedance). This form captures the affine nature, with the constant term I_N as the shift and the linear term from the passive network.[15] The equivalence is established by showing that this Norton model reproduces the original network's port response for any load Z_L. For a load connected, the voltage across Z_L in the original network is V_L = I_N \frac{Z_N Z_L}{Z_N + Z_L}, derived from the linear v-i relation v = -Z_N (i - I_N). By the linearity of the original system, this matches the response obtained via full analysis, as the straight-line characteristic is uniquely determined by the open-circuit voltage V_{oc} = I_N Z_N and short-circuit current I_{sc} = I_N, with Z_N = V_{oc}/I_{sc}. Thus, the Norton equivalent simplifies the network without altering external behavior.[15]Step-by-Step Reasoning
The mathematical derivation of Norton's theorem relies on the linearity of the circuit, allowing the use of superposition and basic circuit analysis to establish the equivalent current source and impedance that replicate the original network's behavior at the terminals.[15] Step 1: Determining the short-circuit current I_{sc}.Short-circuit the output terminals A-B of the linear network containing independent sources. Apply the superposition principle: deactivate all independent sources except one, compute the current contribution through the short circuit due to that active source, then sum the contributions from all sources. The total short-circuit current I_{sc} represents the Norton current source, as it is the current that would flow through the terminals under short-circuit conditions.[21] Step 2: Determining the equivalent impedance Z_{eq}.
Deactivate all independent sources in the network by replacing voltage sources with short circuits and current sources with open circuits, resulting in a passive network. Apply a test voltage V_{test} across terminals A-B and measure the resulting current I_{test}, or vice versa with a test current. The equivalent impedance is then Z_{eq} = \frac{V_{test}}{I_{test}}, which is the impedance seen looking into the terminals of the deactivated network (dependent sources, if present, remain active).[22] Step 3: Establishing load current equivalence for an arbitrary load Z_L.
Construct the Norton equivalent circuit with current source I_{sc} in parallel with Z_{eq}. Connect load impedance Z_L across the terminals. The voltage across the load is the voltage across the parallel combination: V_L = I_{sc} \cdot \frac{Z_{eq} Z_L}{Z_{eq} + Z_L}. The current through the load is I_L = \frac{V_L}{Z_L} = I_{sc} \cdot \frac{Z_{eq}}{Z_{eq} + Z_L}. This I_L matches the load current in the original network, derived from the linear i-v relationship at the terminals ensured by superposition.[23] Step 4: Verifying identity using Kirchhoff's laws and handling dependent sources.
For the original network and Norton equivalent both connected to Z_L, apply Kirchhoff's current law (KCL) at the terminals and Kirchhoff's voltage law (KVL) around loops involving the load; the resulting equations are identical, confirming matching terminal currents and voltages. In networks with dependent sources, the equivalent impedance is determined without deactivating dependent sources, preserving the linear terminal behavior.[15]