Thévenin's theorem

The Thévenin's theorem (after Léon Charles Thévenin ; also: Helmholtz-Thevenin theorem or Helmholtz set ) states in the theory of linear electrical networks that each possible combination of linear voltage sources , current sources and resistors with respect to two terminals electrically equivalent to a series connection of a voltage source and an ohmic resistor . Equivalence means that with the same external load, the same behavior of voltage and current strength occurs. ${\ displaystyle R}$

This equivalent circuit is called Thevenin equivalent or equivalent voltage source. This theorem is used, for example, to simplify circuit analysis.

Calculation of the Thévenin equivalent

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

The Thévenin equivalent consists of an ohmic resistor and a voltage source with the open circuit voltage . To determine the two unknowns and , two equations are needed. These equations can be set up in a number of ways. ${\ displaystyle R _ {\ text {Th}}}$ ${\ displaystyle U _ {\ text {Th}}}$ ${\ displaystyle R _ {\ text {Th}}}$ ${\ displaystyle U _ {\ text {Th}}}$

If the circuit does not behave like an ideal current source, then : ${\ displaystyle U _ {\ text {Th}}}$

The output voltage with open terminals A – B is the open circuit voltage and at the same time . ${\ displaystyle U _ {\ text {Th}}}$

For there are different methods: ${\ displaystyle R _ {\ text {Th}}}$

In the circuit diagram, all voltage sources are replaced by short circuits and all current sources are replaced by interruptions. The internal resistances of the sources remain in the circuit. The equivalent resistance is then calculated. This is equal to the Thévenin equivalent resistance.

If a short circuit from A to B is permissible and the short circuit current is known, Ohm's law is used ${\ displaystyle I _ {\ mathrm {K}}}$

{\ displaystyle R _ {\ text {Th}} = {\ frac {U _ {\ text {Th}}} {I _ {\ mathrm {K}}}}}

A known resistor is connected to A – B and the voltage is measured. The voltage divider law can then be used to determine. ${\ displaystyle R _ {\ text {Th}}}$

A common variant of this method is that of half-voltage : The resistance at A – B can be varied so that half of the open-circuit voltage drops across A – B. The variable resistance is then the same .

{\ displaystyle R _ {\ text {Th}}}

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

Conversion between Norton and Thévenin equivalent

Two equivalent sources

A Thévenin equivalent (linear voltage source) and Norton equivalent (linear current 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) ${\ displaystyle R _ {\ mathrm {i}}}$ ${\ displaystyle 0 <R _ {\ text {i}} <\ infty}$
${\ displaystyle U_ {0} = I _ {\ mathrm {K}} \ cdot R _ {\ mathrm {i}}}$

Nevertheless, there is a difference in efficiency between the equivalent voltage source and the equivalent power source , see efficiency of the power source . Wherever a high degree of efficiency is important, the equivalents are not interchangeable.

The difference can also be seen in the example of the short circuit: While no current flows through the internal resistance of the current source, the internal resistance of the voltage source is "heated" by the entire short-circuit current , and that at maximum voltage . ${\ displaystyle I _ {\ mathrm {K}}}$ ${\ displaystyle U_ {0}}$

Extension for alternating current

Thévenin'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 Thévenin theorem was first discovered by the German scientist Hermann von Helmholtz in 1853. It was then rediscovered in 1883 by the French engineer Léon Charles Thévenin (1857–1926).

literature

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

Individual evidence

↑ 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
↑ Heinz Josef Bauckholt: Basics and components of electrical engineering. Hanser, 7th ed., 2013, pp. 82-88
↑ ^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
^ Wilfried Weißgerber: Electrical engineering for engineers 1: DC technology and electromagnetic field. Springer Vieweg, 11th edition, 2018, p. 47
^ Johnson, DH (2003). Origins of the equivalent circuit concept: the voltage-source equivalent. Proceedings of the IEEE, 91 (4), 636-640. doi : 10.1109 / JPROC.2003.811716 .

[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 ed., 2013, pp. 82-88

[FG-3] 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

[5] Johnson, DH (2003). Origins of the equivalent circuit concept: the voltage-source equivalent. Proceedings of the IEEE, 91 (4), 636-640. doi : 10.1109 / JPROC.2003.811716 .