Norton's theorem

The Norton theorem (after Edward Lawry Norton ; also: Mayer-Norton theorem ) states in the theory of linear electrical networks that every possible combination of linear voltage sources , current sources and resistances with regard to two terminals is electrically equivalent to a parallel connection of one current source and one ohmic resistance is. Equivalence means that with the same external load, the same behavior of voltage and current strength occurs. ${\displaystyle R}$

This equivalent circuit is called the Norton equivalent or equivalent power source . This theorem is used, for example, to simplify circuit analysis.

Calculation of the Norton equivalent

Any electrical circuit that consists entirely of linear voltage sources, current sources, and resistors can be converted to a Norton equivalent.

The Norton equivalent consists of an ohmic resistor and a current source with the short-circuit current . To determine the two unknowns and , two equations are needed. These equations can be set up in a number of ways. ${\displaystyle R_{\text{No}}}$ ${\displaystyle I_{\text{No}}}$ ${\displaystyle R_{\text{No}}}$ ${\displaystyle I_{\text{No}}}$

If the circuit does not behave like an ideal voltage source, then : ${\displaystyle I_{\text{No}}}$

• The output current in the event of a short circuit between A and B is at the same time . ${\displaystyle I_{\text{No}}}$

For there are different methods: ${\displaystyle R_{\text{No}}}$

• In the circuit diagram, all voltage sources are replaced by short circuits and all current sources by interruptions. The internal resistances of the sources remain in the circuit. The equivalent resistance is then calculated. This is equal to the Norton equivalent resistance.
• If an open circuit (no connection from A to B) is permitted and the open circuit voltage is known, Ohm's law is used: ${\displaystyle U_{\text{L}}}$

${\displaystyle R_{\mathrm {No} }={\frac {U_{\text{L}}}{I_{\mathrm {No} }}}}$

• A known resistor is connected to AB and the current is measured. The current divider law can then be used to determine. ${\displaystyle R_{\mathrm {No} }}$

The evaluation is particularly simple if the known resistance can be adjusted so that half the short-circuit current flows through the resistance. Then the set resistance is as great as . ${\displaystyle R_{\text{No}}}$

The proof of Norton's theorem is based on the superposition principle .

Conversion between Norton and Thévenin equivalent

Two equivalent sources

A Norton equivalent (linear current source) and a Thévenin equivalent (linear voltage source) are mutually equivalent sources. Interchangeability is given under the following two conditions:

• This is the same in both circuits shown opposite (where it must be) ${\displaystyle R_{\mathrm {i} }}$ ${\displaystyle 0<R_{\text{i}}<\infty }$
• ${\displaystyle U_{0}=I_{\mathrm {K} }\cdot R_{\mathrm {i} }}$

Question for understanding

question

“A current source with a parallel resistor and a voltage source with a series resistor are hidden in two black boxes, so that the above equations are fulfilled. Can you tell from the outside which black box the Norton circuit is in? "

answer

Yes! The little box with the Norton circuit is warmer because it is constantly consuming power . The Thévenin circuit does not consume any power and therefore does not get warmer. The equivalence only applies to the output terminals. However, if you load both boxes with a short circuit, the box with the Thévenin circuit takes up the power , since current now flows through the Thévenin resistor. The Norton circuit, on the other hand, no longer consumes power because the Norton resistor is short-circuited. The power consumed by the Norton circuit in the open case is the same as the power consumed by the Thévenin circuit in the short-circuited case. ${\displaystyle P_{\mathrm {No} }=I_{\mathrm {No} }^{2}R_{\mathrm {No} }}$ ${\displaystyle P_{\mathrm {Th} }={\frac {U_{\mathrm {Th} }^{2}}{R_{\mathrm {Th} }}}}$

This question is useful in order to clarify the limits of the theory of Norton and Thévenin equivalent.

Linked to this difference is the difference in the efficiency of the voltage source and the current source, see efficiency of the current source . Wherever a high degree of efficiency is important, the equivalents are not interchangeable.

Extension for alternating current

Norton's theorem can also be generalized to harmonic AC systems by using impedances instead of ohmic resistances. When used in the alternating current range, however, there are also sources with frequency-dependent amplitude and phase. Therefore a practical application for alternating current equivalent circuits is rather rare or limited to one frequency.

history

The Norton theorem is an extension of the Thévenin theorem.

It was discovered in 1926 simultaneously and independently by Hans Ferdinand Mayer (1895–1980) (at Siemens & Halske ) and Edward Lawry Norton (1898–1983) (at Bell Labs ). Mayer published his discovery in the journal Telegraphen- und Fernsprech-Technik , Norton published his discovery in an internal report by Bell Labs.

literature

• Karl Küpfmüller, W. Mathis, A. Reibiger: Theoretical electrical engineering . Springer, Berlin, Heidelberg 2006, ISBN 3-540-29290-X .

Individual evidence

1. ↑ Marlene Marinescu, Nicolae Marinescu: Electrical engineering for study and practice: direct, alternating and three-phase currents, switching and non-sinusoidal processes . Springer Vieweg, 2016, p. 61 ff
2. ↑ Heinz Josef Bauckholt: Basics and components of electrical engineering. Hanser, 7th edition, 2013, p. 87 f
3. ↑ ^{a b} Peter Kurzweil (eds.), Bernhard Frenzel, Florian Gebhard: Physics collection of formulas: With explanations and examples from practice for engineers and natural scientists . Vieweg + Teubner, 2nd edition, 2009, p. 223.
4. ^ Wilfried Weißgerber: Electrical engineering for engineers 1: DC technology and electromagnetic field. Springer Vieweg, 11th edition, 2018, p. 47 f