Norton's theorem

Norton's theorem for linear electrical networks, known in Europe as the Mayer–Norton theorem, states that any collection of voltage sources, current sources, and resistors 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 impedances, not just resistors. The Norton equivalent is used to represent any network of linear sources and impedances, at a given 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 and was introduced in 1926 separately by two people: Siemens & Halske researcher Hans Ferdinand Mayer (1895–1980) and Bell Labs engineer Edward Lawry Norton (1898–1983). The Norton equivalent circuit is a current source with current I_No in parallel with a resistance R_No. To find the equivalent,

1. Find the Norton current I_No. Calculate the output current, I_AB, with a short circuit as the load (meaning 0 resistance between A and B). This is I_No.
2. Find the Norton resistance R_No. When there are no dependent sources (i.e., all current and voltage sources are independent), there are two methods of determining the Norton impedance R_No.

• Calculate the output voltage, V_AB, when in open circuit condition (i.e., no load resistor — meaning infinite load resistance). R_No equals this V_AB divided by I_No.

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 R_No.

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 of 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

Step 0: The original circuit

Step 1: Calculating the equivalent output current

Step 2: Calculating the equivalent resistance

Step 3: The equivalent circuit

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

$I_\mathrm{total} = {15 \mathrm{V} \over 2\,\mathrm{k}\Omega + 1\,\mathrm{k}\Omega \| (1\,\mathrm{k}\Omega + 1\,\mathrm{k}\Omega)} = 5.625 \mathrm{mA}.$

The current through the load is then, using the current divider rule:

$I = {1\,\mathrm{k}\Omega + 1\,\mathrm{k}\Omega \over (2\,\mathrm{k}\Omega + 1\,\mathrm{k}\Omega)} \cdot I_\mathrm{total}$

$= 2/3 \cdot 5.625 \mathrm{mA} = 3.75 \mathrm{mA}.$

And the equivalent resistance looking back into the circuit is:

$R_\mathrm{eq} = 1\,\mathrm{k}\Omega + 2\,\mathrm{k}\Omega \| (1\,\mathrm{k}\Omega + 1\,\mathrm{k}\Omega) = 2\,\mathrm{k}\Omega.$

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:

$R_{Th} = R_{No} \!$

$V_{Th} = I_{No} R_{No} \!$

$V_{Th} / R_{Th} = I_{No}.\!$

Queueing theory

The term "Norton equivalent" is also used in queueing theory 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.^[1]

References

1. ^ KM Chandy, U Herzog, L Woo, "Parametric analysis of queuing networks", IBM Journal of Research and Development, 1975

External links

• Origins of the equivalent circuit concept
• Norton's theorem at allaboutcircuits.com