Norton's theorem

# Norton's theorem

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Norton's theorem for linear electrical network
Electrical network
An electrical network is an interconnection of electrical elements such as resistors, inductors, capacitors, transmission lines, voltage sources, current sources and switches. An electrical circuit is a special type of network, one that has a closed loop giving a return path for the current...

s, known in Europe as the Mayer–Norton theorem, states that any collection of voltage source
Voltage source
In electric circuit theory, an ideal voltage source is a circuit element where the voltage across it is independent of the current through it. A voltage source is the dual of a current source. In analysis, a voltage source supplies a constant DC or AC potential between its terminals for any current...

s, current source
Current source
A current source is an electrical or electronic device that delivers or absorbs electric current. A current source is the dual of a voltage source. The term constant-current sink is sometimes used for sources fed from a negative voltage supply...

s, and resistor
Resistor
A linear resistor is a linear, passive two-terminal electrical component that implements electrical resistance as a circuit element.The current through a resistor is in direct proportion to the voltage across the resistor's terminals. Thus, the ratio of the voltage applied across a resistor's...

s with two terminals is electrically equivalent to an ideal current source, I, in parallel with a single resistor, R. For single-frequency AC systems the theorem can also be applied to general impedance
Electrical impedance
Electrical impedance, or simply impedance, is the measure of the opposition that an electrical circuit presents to the passage of a current when a voltage is applied. In quantitative terms, it is the complex ratio of the voltage to the current in an alternating current circuit...

s, not just resistors. The Norton equivalent is used to represent any network of linear sources and impedances, at a given frequency
Frequency
Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...

. The circuit consists of an ideal current source in parallel with an ideal impedance (or resistor for non-reactive circuits).

Norton's theorem is an extension of Thévenin's theorem
Thévenin's theorem
In circuit theory, Thévenin's theorem for linear electrical networks states that any combination of voltage sources, current sources, and resistors with two terminals is electrically equivalent to a single voltage source V and a single series resistor R. For single frequency AC systems the theorem...

and was introduced in 1926 separately by two people: Siemens & Halske
Siemens
Siemens may refer toSiemens, a German family name carried by generations of telecommunications industrialists, including:* Werner von Siemens , inventor, founder of Siemens AG...

researcher Hans Ferdinand Mayer
Hans Ferdinand Mayer
Hans Ferdinand Mayer was a German mathematician and physicist and perhaps most notable for the Oslo Report which revealed German technological secrets to the British Government shortly after the start of World War II.-Biography:Hans Ferdinand Mayer studied mathematics, physics and astronomy at...

(1895–1980) and Bell Labs
Bell Labs
Bell Laboratories is the research and development subsidiary of the French-owned Alcatel-Lucent and previously of the American Telephone & Telegraph Company , half-owned through its Western Electric manufacturing subsidiary.Bell Laboratories operates its...

engineer Edward Lawry Norton
Edward Lawry Norton
Edward Lawry Norton was an accomplished Bell Labs engineer and scientist famous for developing the concept of the Norton equivalent circuit. He attended the University of Maine for two years before transferring to M.I.T. and received a S.B. degree in 1922. He received an M.A...

(1898–1983). The Norton equivalent circuit is a current source with current INo in parallel with a resistance RNo. To find the equivalent,
1. Find the Norton current INo. Calculate the output current, IAB, with a short circuit
Short circuit
A short circuit in an electrical circuit that allows a current to travel along an unintended path, often where essentially no electrical impedance is encountered....

If an electric circuit has a well-defined output terminal, the circuit connected to this terminal is the load....

(meaning 0 resistance between A and B). This is INo.
2. Find the Norton resistance RNo. When there are no dependent sources (i.e., all current and voltage sources are independent), there are two methods of determining the Norton impedance RNo.
• Calculate the output voltage, VAB, when in open circuit
Open circuit
The term Open circuit may refer to:*Open-circuit scuba, a type of SCUBA-diving equipment where the user breathes from the set and then exhales to the surroundings without recycling the exhaled air...

condition (i.e., no load resistor — meaning infinite load resistance). RNo equals this VAB divided by INo.
or
• Replace independent voltage sources with short circuits and independent current sources with open circuits. The total resistance across the output port is the Norton impedance RNo.
or
• Use a given Thevenin resistance: as the two are equal.
However, when there are dependent sources, the more general method must be used. This method is not shown below in the diagrams.
• Connect a constant current source at the output terminals of the circuit with a value of 1 Ampere and calculate the voltage at its terminals. The quotient
Quotient
In mathematics, a quotient is the result of division. For example, when dividing 6 by 3, the quotient is 2, while 6 is called the dividend, and 3 the divisor. The quotient further is expressed as the number of times the divisor divides into the dividend e.g. The quotient of 6 and 2 is also 3.A...

of this voltage divided by the 1 A current is the Norton impedance RNo. 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

In the example, the total current Itotal is given by:

The current through the load is then, using the current divider rule
Current divider rule
In electronics, a current divider is a simple linear circuit that produces an output current that is a fraction of its input current . Current division refers to the splitting of current between the branches of the divider...

:

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

A Norton equivalent circuit is related to the Thévenin equivalent by the following equations:

## Queueing theory

The term "Norton equivalent" is also used in queueing theory
Queueing theory
Queueing theory is the mathematical study of waiting lines, or queues. The theory enables mathematical analysis of several related processes, including arriving at the queue, waiting in the queue , and being served at the front of the queue...

for a similar concept. In a reversible queueing system, it is often possible to replace an uninteresting subset of queues by a single (FCFS or PS) queue with an appropriately chosen service rate.