![NortonEquivalentCircuits](https://upload.wikimedia.org/wikipedia/commons/7/7c/NortonEquivalentCircuits.png)
In
direct-current
Direct current (DC) is one-directional flow of electric charge. An electrochemical cell is a prime example of DC power. Direct current may flow through a conductor such as a wire, but can also flow through semiconductors, insulators, or even ...
circuit theory
Circuit may refer to:
Science and technology
Electrical engineering
* Electrical circuit, a complete electrical network with a closed-loop giving a return path for current
** Analog circuit, uses continuous signal levels
** Balanced circui ...
, Norton's theorem, also called the Mayer–Norton theorem, is a simplification that can be applied to
networks
Network, networking and networked may refer to:
Science and technology
* Network theory, the study of graphs as a representation of relations between discrete objects
* Network science, an academic field that studies complex networks
Mathematics
...
made of
linear time-invariant
In system analysis, among other fields of study, a linear time-invariant (LTI) system is a system that produces an output signal from any input signal subject to the constraints of linearity and time-invariance; these terms are briefly defin ...
resistances,
voltage sources, and
current sources
A current source is an electronic circuit that delivers or absorbs an electric current which is independent of the voltage across it.
A current source is the dual of a voltage source. The term ''current sink'' is sometimes used for sources fed ...
. At a pair of terminals of the network, it can be replaced by a current source and a single resistor in parallel.
For
alternating current
Alternating current (AC) is an electric current which periodically reverses direction and changes its magnitude continuously with time in contrast to direct current (DC) which flows only in one direction. Alternating current is the form in whic ...
(AC) systems the theorem can be applied to
reactive
Reactive may refer to:
*Generally, capable of having a reaction (disambiguation)
*An adjective abbreviation denoting a bowling ball coverstock made of reactive resin
*Reactivity (chemistry)
*Reactive mind
*Reactive programming
See also
*Reactanc ...
impedances as well as resistances.
The Norton equivalent circuit is used to represent any network of linear sources and impedances at a given
frequency
Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
.
Norton's theorem and its dual,
Thévenin's theorem
As originally stated in terms of direct-current resistive circuits only, Thévenin's theorem states that ''"For any linear electrical network containing only voltage sources, current sources and resistances can be replaced at terminals A–B b ...
, are widely used for circuit analysis simplification and to study circuit's
initial-condition and
steady-state
In systems theory, a system or a process is in a steady state if the variables (called state variables) which define the behavior of the system or the process are unchanging in time. In continuous time, this means that for those properties ''p'' ...
response.
Norton's theorem was independently derived in 1926 by
Siemens & Halske
Siemens & Halske AG (or Siemens-Halske) was a German electrical engineering company that later became part of Siemens.
It was founded on 12 October 1847 as ''Telegraphen-Bauanstalt von Siemens & Halske'' by Werner von Siemens and Johann Ge ...
researcher
Hans Ferdinand Mayer
Hans Ferdinand Mayer (born 23 October 1895 in Pforzheim, Germany; died 18 October 1980 in Munich, West Germany) was a German mathematician and physicist. He was the author of the " Oslo Report", a major military intelligence leak which reveale ...
(1895–1980) and
Bell Labs
Nokia Bell Labs, originally named Bell Telephone Laboratories (1925–1984),
then AT&T Bell Laboratories (1984–1996)
and Bell Labs Innovations (1996–2007),
is an American industrial research and scientific development company owned by mult ...
engineer
Edward Lawry Norton
Edward Lawry Norton (July 28, 1898 – January 28, 1983) was an accomplished engineer and scientist. He worked at Bell Labs and is known for Norton's theorem.
His areas of active research included network theory, acoustical systems, electro ...
(1898–1983).
To find the equivalent, the Norton current ''I''
no is calculated as the current flowing at the terminals into a
short circuit
A short circuit (sometimes abbreviated to short or s/c) is an electrical circuit that allows a current to travel along an unintended path with no or very low electrical impedance. This results in an excessive current flowing through the circuit ...
(zero resistance between ''A'' and ''B''). This is ''I''
no. The Norton resistance ''R''
no is found by calculating the output voltage produced with no resistance connected at the terminals; equivalently, this is the resistance between the terminals with all (independent) voltage sources short-circuited and independent current sources
open-circuited. This is equivalent to calculating the Thevenin resistance.
:When there are dependent sources, the more general method must be used. The voltage at the terminals is calculated for an injection of a 1 Amp test current at the terminals. This voltage divided by the 1 A current is the Norton impedance ''R''
no. This method must be used if the circuit contains dependent sources, but it can be used in all cases even when there are no dependent sources.
Example of a Norton equivalent circuit
![Norton-example](https://upload.wikimedia.org/wikipedia/commons/9/9f/Norton-example.png)
In the example, the total current ''I''
total is given by:
:
The current through the load is then, using the
current divider rule
In electronics, a current divider is a simple linear circuit that produces an output current (''I''X) that is a fraction of its input current (''I''T). Current division refers to the splitting of current between the branches of the divider. The cu ...
:
:
And the equivalent resistance looking back into the circuit is:
:
So the equivalent circuit is a 3.75 mA current source in parallel with a 2 kΩ resistor.
Conversion to a Thévenin equivalent
![norton-to-thevenin](https://upload.wikimedia.org/wikipedia/commons/b/bd/Norton-to-thevenin.png)
A Norton equivalent circuit is related to the
Thévenin equivalent by the equations:
:
Queueing theory
The passive circuit equivalent of "Norton's theorem" in
queuing theory
Queueing theory is the mathematical study of waiting lines, or queues. A queueing model is constructed so that queue lengths and waiting time can be predicted. Queueing theory is generally considered a branch of operations research because the ...
is called the
Chandy Herzog Woo theorem.
In a
reversible queueing system
Queueing theory is the mathematical study of waiting lines, or queues. A queueing model is constructed so that queue lengths and waiting time can be predicted. Queueing theory is generally considered a branch of operations research because the ...
, it is often possible to replace an uninteresting subset of queues by a single (
FCFS
Queueing theory is the mathematical study of waiting lines, or queues. A queueing model is constructed so that queue lengths and waiting time can be predicted. Queueing theory is generally considered a branch of operations research because the ...
or
PS) queue with an appropriately chosen service rate.
See also
*
Ohm's Law
Ohm's law states that the current through a conductor between two points is directly proportional to the voltage across the two points. Introducing the constant of proportionality, the resistance, one arrives at the usual mathematical equat ...
*
Millman's theorem In electrical engineering, Millman's theorem (or the parallel generator theorem) is a method to simplify the solution of a circuit. Specifically, Millman's theorem is used to compute the voltage at the ends of a circuit made up of only branches i ...
*
Source transformation
Source transformation is the process of simplifying a circuit solution, especially with mixed sources, by transforming voltage sources into current sources, and vice versa, using Thévenin's theorem and Norton's theorem respectively.CPP. https://ww ...
*
Superposition theorem
The superposition theorem is a derived result of the superposition principle suited to the network analysis of electrical circuits. The superposition theorem states that for a linear system (notably including the subcategory of time-invariant l ...
*
Thévenin's theorem
As originally stated in terms of direct-current resistive circuits only, Thévenin's theorem states that ''"For any linear electrical network containing only voltage sources, current sources and resistances can be replaced at terminals A–B b ...
*
Maximum power transfer theorem In electrical engineering, the maximum power transfer theorem states that, to obtain ''maximum'' external power from a power source with internal resistance, the resistance of the load must equal the resistance of the source as viewed from its ...
*
Extra element theorem The Extra Element Theorem (EET) is an analytic technique developed by R. D. Middlebrook for simplifying the process of deriving driving point and transfer functions for linear electronic circuits.
Much like Thévenin's theorem, the extra element t ...
References
External links
*{{Commons category inline
Norton's theorem at allaboutcircuits.com
Circuit theorems
Linear electronic circuits