Norton's theorem

Edward Lawry Norton

Known in Europe as the Mayer–Norton theorem, Norton's theorem holds, to illustrate in DC circuit theory terms, that (see image):

• Any linear electrical network with voltage and current sources and only resistances can be replaced at terminals A-B by an equivalent current source I_NO in parallel connection with an equivalent resistance R_NO.
• This equivalent current I_NO is the current obtained at terminals A-B of the network with terminals A-B short circuited.
• This equivalent resistance R_NO is the resistance obtained at terminals A-B of the network with all its voltage sources short circuited and all its current sources open circuited.

For AC systems the theorem can be applied to reactive impedances as well as resistances.

The Norton equivalent circuit is used to represent any network of linear sources and impedances at a given frequency.

Any black box containing resistances only and voltage and current sources can be replaced by an equivalent circuit consisting of an equivalent current source in parallel connection with an equivalent resistance.

Norton's theorem and its dual, Thévenin's theorem, are widely used for circuit analysis simplification and to study circuit's initial-condition and steady-state response.

Norton's theorem was independently derived in 1926 by Siemens & Halske researcher Hans Ferdinand Mayer (1895–1980) and Bell Labs engineer Edward Lawry Norton (1898–1983).^{[1][2][3][4][5]}

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 (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.

This is equivalent to calculating the Thevenin resistance.

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. 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

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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:

$I_\mathrm{No} = {1\,\mathrm{k}\Omega + 1\,\mathrm{k}\Omega \over (1\,\mathrm{k}\Omega + 1\,\mathrm{k}\Omega + 1\,\mathrm{k}\Omega)} \cdot I_\mathrm{total}$

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

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:

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

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

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

Queueing theory

The passive circuit equivalent of "Norton's theorem" in queuing theory is called the Chandy Herzog Woo theorem.^[6][7] 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.^[8]

See also

• Ohm's Law
• Millman's theorem
• Source transformation
• Superposition theorem
• Thévenin's theorem
• Maximum power transfer theorem
• Extra element theorem

References

1. Mayer
2. Norton
3. Johnson (2003b)
4. Brittain
5. Dorf
6. Johnson (2003a)
7. Gunther
8. Chandy et al.

Bibliography

• Brittain, J.E. (March 1990). "Thevenin's theorem". IEEE Spectrum 27 (3): 42. doi:10.1109/6.48845. Retrieved 1 February 2013.
• Chandy, K. M.; Herzog, U.; Woo, L. (Jan 1975). "Parametric Analysis of Queuing Networks". IBM Journal of Research and Development 19 (1): 36–42. doi:10.1147/rd.191.0036.
• Dorf, Richard C.; Svoboda, James A. (2010). "Chapter 5 – Circuit Theorems". Introduction to Electric Circuits (8th ed.). Hoboken, NJ: John Wiley & Sons. pp. 162–207. ISBN 978-0-470-52157-1.
• Gunther, N.J. (2004). Analyzing computer systems performance : with PERL::PDQ (Online-Ausg. ed.). Berlin: Springer. p. 281. ISBN 3-540-20865-8.
• Johnson, D.H. (2003). "Origins of the equivalent circuit concept: the voltage-source equivalent" (PDF). Proceedings of the IEEE 91 (4): 636–640. doi:10.1109/JPROC.2003.811716.
• Johnson, D.H. (2003). "Origins of the equivalent circuit concept: the current-source equivalent" (PDF). Proceedings of the IEEE 91 (5): 817–821. doi:10.1109/JPROC.2003.811795.
• Mayer, H. F. (1926). "Ueber das Ersatzschema der Verstärkerröhre (On equivalent circuits for electronic amplifiers]". Telegraphen- und Fernsprech-Technik 15: 335–337.
• Norton, E. L. (1926). "Technical Report TM26–0–1860 – Design of finite networks for uniform frequency characteristic". Bell Laboratories.

External links

• Norton's theorem at allaboutcircuits.com