Thévenin's theorem

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Any black box containing resistances only and voltage and current sources can be replaced by a Thevenin equivalent circuit consisting of an equivalent voltage source in series connection with an equivalent resistance. TheveninEquivalent-2.png
Any black box containing resistances only and voltage and current sources can be replaced by a Thévenin equivalent circuit consisting of an equivalent voltage source in series connection with an equivalent resistance.

As originally stated in terms of direct-current resistive circuits only, Thévenin's theorem states that "For any linear electrical network containing only voltage sources, current sources and resistances can be replaced at terminals A–B by an equivalent combination of a voltage source Vth in a series connection with a resistance Rth."

Contents

In circuit theory terms, the theorem allows any one-port network to be reduced to a single voltage source and a single impedance.

The theorem also applies to frequency domain AC circuits consisting of reactive and resistive impedances. It means the theorem applies for AC in an exactly same way to DC except that resistances are generalized to impedances.

The theorem was independently derived in 1853 by the German scientist Hermann von Helmholtz and in 1883 by Léon Charles Thévenin (18571926), an electrical engineer with France's national Postes et Télégraphes telecommunications organization. [1] [2] [3] [4] [5] [6] [7]

Thévenin's theorem and its dual, Norton's theorem, are widely used to make circuit analysis simpler and to study a circuit's initial-condition and steady-state response. [8] [9] Thévenin's theorem can be used to convert any circuit's sources and impedances to a Thévenin equivalent; use of the theorem may in some cases be more convenient than use of Kirchhoff's circuit laws. [7] [10]

Calculating the Thévenin equivalent

The equivalent circuit is a voltage source with voltage VTh in series with a resistance RTh.

The Thévenin-equivalent voltage VTh is the open-circuit voltage at the output terminals of the original circuit. When calculating a Thévenin-equivalent voltage, the voltage divider principle is often useful, by declaring one terminal to be Vout and the other terminal to be at the ground point.

The Thévenin-equivalent resistance RTh is the resistance measured across points A and B "looking back" into the circuit. The resistance is measured after replacing all voltage- and current-sources with their internal resistances. That means an ideal voltage source is replaced with a short circuit, and an ideal current source is replaced with an open circuit. Resistance can then be calculated across the terminals using the formulae for series and parallel circuits. This method is valid only for circuits with independent sources. If there are dependent sources in the circuit, another method must be used such as connecting a test source across A and B and calculating the voltage across or current through the test source.

As a mnemonic, the Thevenin replacements for voltage and current sources can be remembered as we turn the sources' values (meaning their voltage or current) to zero. A zero valued voltage source would create a potential difference of zero volts between its terminals, just like an ideal short circuit would do, with two leads touching; therefore we replace the source with a short circuit. Similarly, a zero valued current source and an open circuit both pass zero current.

Example

Original circuit
The equivalent voltage
The equivalent resistance
The equivalent circuit Thevenin theorem example.png
  1. Original circuit
  2. The equivalent voltage
  3. The equivalent resistance
  4. The equivalent circuit

In the example, calculating the equivalent voltage:

(Notice that R1 is not taken into consideration, as above calculations are done in an open-circuit condition between A and B, therefore no current flows through this part, which means there is no current through R1 and therefore no voltage drop along this part.)

Calculating equivalent resistance ( is the total resistance of two parallel resistors):

Conversion to a Norton equivalent

Norton-Thevenin conversion Norton-to-thevenin.png
Norton-Thevenin conversion

A Norton equivalent circuit is related to the Thévenin equivalent by

Practical limitations

A proof of the theorem

The proof involves two steps. The first step is to use superposition theorem to construct a solution. Then, uniqueness theorem is employed to show that the obtained solution is unique. It is noted that the second step is usually implied in literature.

By using superposition of specific configurations, it can be shown that for any linear "black box" circuit which contains voltage sources and resistors, its voltage is a linear function of the corresponding current as follows

Here, the first term reflects the linear summation of contributions from each voltage source, while the second term measures the contributions from all the resistors. The above expression is obtained by using the fact that the voltage of the black box for a given current is identical to the linear superposition of the solutions of the following problems: (1) to leave the black box open circuited but activate individual voltage source one at a time and, (2) to short circuit all the voltage sources but feed the circuit with a certain ideal voltage source so that the resulting current exactly reads (Alternatively, one can use an ideal current source of current ). Moreover, it is straightforward to show that and are the single voltage source and the single series resistor in question.

As a matter of fact, the above relation between and is established by superposition of some particular configurations. Now, the uniqueness theorem guarantees that the result is general. To be specific, there is one and only one value of once the value of is given. In other words, the above relation holds true independent of what the "black box" is plugged to.

In three-phase circuits

In 1933, A. T. Starr published a generalization of Thévenin's theorem in an article of the magazine Institute of Electrical Engineers Journal, titled A New Theorem for Active Networks, [11] which states that any three-terminal active linear network can be substituted by three voltage sources with corresponding impedances, connected in wye or in delta.

See also

Related Research Articles

Electrical network Assemblage of connected electrical elements

An electrical network is an interconnection of electrical components or a model of such an interconnection, consisting of electrical elements. An electrical circuit is a network consisting of a closed loop, giving a return path for the current. Linear electrical networks, a special type consisting only of sources, linear lumped elements, and linear distributed elements, have the property that signals are linearly superimposable. They are thus more easily analyzed, using powerful frequency domain methods such as Laplace transforms, to determine DC response, AC response, and transient response.

Electrical impedance Opposition of a circuit to a current when a voltage is applied

In electrical engineering, impedance is the opposition to alternating current presented by the combined effect of resistance and reactance in a circuit.

In electrical engineering, the maximum power transfer theorem states that, to obtain maximum external power from a power source with internal resistance, the resistance of the load must equal the resistance of the source as viewed from its output terminals. Moritz von Jacobi published the maximum power (transfer) theorem around 1840; it is also referred to as "Jacobi's law".

Johnson–Nyquist noise Electronic noise due to thermal vibration within a conductor

Johnson–Nyquist noise is the electronic noise generated by the thermal agitation of the charge carriers inside an electrical conductor at equilibrium, which happens regardless of any applied voltage. Thermal noise is present in all electrical circuits, and in sensitive electronic equipment can drown out weak signals, and can be the limiting factor on sensitivity of electrical measuring instruments. Thermal noise increases with temperature. Some sensitive electronic equipment such as radio telescope receivers are cooled to cryogenic temperatures to reduce thermal noise in their circuits. The generic, statistical physical derivation of this noise is called the fluctuation-dissipation theorem, where generalized impedance or generalized susceptibility is used to characterize the medium.

Nortons theorem DC circuit analysis technique

In direct-current circuit theory, Norton's theorem, also called the Mayer–Norton theorem, is a simplification that can be applied to networks made of linear time-invariant resistances, voltage sources, and current sources. At a pair of terminals of the network, it can be replaced by a current source and a single resistor in parallel.

Negative resistance Property that an increasing voltage results in a decreasing current

In electronics, negative resistance (NR) is a property of some electrical circuits and devices in which an increase in voltage across the device's terminals results in a decrease in electric current through it.

Voltage divider Linear circuit that produces an output voltage that is a fraction of its input voltage

In electronics, a voltage divider is a passive linear circuit that produces an output voltage (Vout) that is a fraction of its input voltage (Vin). Voltage division is the result of distributing the input voltage among the components of the divider. A simple example of a voltage divider is two resistors connected in series, with the input voltage applied across the resistor pair and the output voltage emerging from the connection between them.

Gyrator Two-port non-reciprocal network element

A gyrator is a passive, linear, lossless, two-port electrical network element proposed in 1948 by Bernard D. H. Tellegen as a hypothetical fifth linear element after the resistor, capacitor, inductor and ideal transformer. Unlike the four conventional elements, the gyrator is non-reciprocal. Gyrators permit network realizations of two-(or-more)-port devices which cannot be realized with just the conventional four elements. In particular, gyrators make possible network realizations of isolators and circulators. Gyrators do not however change the range of one-port devices that can be realized. Although the gyrator was conceived as a fifth linear element, its adoption makes both the ideal transformer and either the capacitor or inductor redundant. Thus the number of necessary linear elements is in fact reduced to three. Circuits that function as gyrators can be built with transistors and op-amps using feedback.

Output impedance

The output impedance of an electrical network is the measure of the opposition to current flow (impedance), both static (resistance) and dynamic (reactance), into the load network being connected that is internal to the electrical source. The output impedance is a measure of the source's propensity to drop in voltage when the load draws current, the source network being the portion of the network that transmits and the load network being the portion of the network that consumes.

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Current source Electronic circuit which delivers or absorbs electric current regardless of voltage

A current source is an electronic circuit that delivers or absorbs an electric current which is independent of the voltage across it.

Internal resistance

A practical electrical power source which is a linear electric circuit may, according to Thévenin's theorem, be represented as an ideal voltage source in series with an impedance. This impedance is termed the internal resistance of the source. When the power source delivers current, the measured voltage output is lower than the no-load voltage; the difference is the voltage drop caused by the internal resistance. The concept of internal resistance applies to all kinds of electrical sources and is useful for analyzing many types of electrical circuits.

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Current divider

In electronics, a current divider is a simple linear circuit that produces an output current (IX) that is a fraction of its input current (IT). Current division refers to the splitting of current between the branches of the divider. The currents in the various branches of such a circuit will always divide in such a way as to minimize the total energy expended.

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Source transformation is the process of simplifying a circuit solution, especially with mixed sources, by transforming voltage sources into current sources, and vice versa, using Thévenin's theorem and Norton's theorem respectively.

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The superposition theorem is a derived result of the superposition principle suited to the network analysis of electrical circuits. The superposition theorem states that for a linear system the response in any branch of a bilateral linear circuit having more than one independent source equals the algebraic sum of the responses caused by each independent source acting alone, where all the other independent sources are replaced by their internal impedances.

An equivalent impedance is an equivalent circuit of an electrical network of impedance elements which presents the same impedance between all pairs of terminals as did the given network. This article describes mathematical transformations between some passive, linear impedance networks commonly found in electronic circuits.

RLC circuit Resistor Inductor Capacitor Circuit

An RLC circuit is an electrical circuit consisting of a resistor (R), an inductor (L), and a capacitor (C), connected in series or in parallel. The name of the circuit is derived from the letters that are used to denote the constituent components of this circuit, where the sequence of the components may vary from RLC.

References

  1. von Helmholtz, Hermann (1853). "Ueber einige Gesetze der Vertheilung elektrischer Ströme in körperlichen Leitern mit Anwendung auf die thierisch-elektrischen Versuche" [Some laws concerning the distribution of electrical currents in conductors with applications to experiments on animal electricity]. Annalen der Physik und Chemie (in German). 89 (6): 211–233. Bibcode:1853AnP...165..211H. doi:10.1002/andp.18531650603.
  2. Thévenin, Léon Charles (1883). "Extension de la loi d'Ohm aux circuits électromoteurs complexes" [Extension of Ohm's law to complex electromotive circuits]. Annales Télégraphiques. 3e series (in French). 10: 222–224.
  3. Thévenin, Léon Charles (1883). "Sur un nouveau théorème d'électricité dynamique" [On a new theorem of dynamic electricity]. Comptes rendus hebdomadaires des séances de l'Académie des Sciences (in French). 97: 159–161.
  4. Johnson, Don 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. hdl: 1911/19968 .
  5. Johnson, Don 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.
  6. Brittain, James E. (March 1990). "Thevenin's theorem". IEEE Spectrum . 27 (3): 42. doi:10.1109/6.48845. S2CID   2279777 . Retrieved 2013-02-01.
  7. 1 2 Dorf, Richard C.; Svoboda, James A. (2010). "Chapter 5: Circuit Theorems". Introduction to Electric Circuits (8th ed.). Hoboken, NJ, USA: John Wiley & Sons. pp. 162–207. ISBN   978-0-470-52157-1.
  8. Brenner, Egon; Javid, Mansour (1959). "Chapter 12: Network Functions". Analysis of Electric Circuits. McGraw-Hill. pp. 268–269.
  9. Elgerd, Olle Ingemar (2007). "Chapter 10: Energy System Transients - Surge Phenomena and Symmetrical Fault Analysis". Electric Energy Systems Theory: An Introduction. Tata McGraw-Hill. pp. 402–429. ISBN   978-0-07019230-0.
  10. Dwight, Herbert Bristol (1949). "Section 2: Electric and Magnetic Circuits". In Knowlton, Archer E. (ed.). Standard Handbook for Electrical Engineers (8th ed.). McGraw-Hill. p. 26.
  11. Starr, A. T. (1933). "A new theorem for active networks". Journal of the Institution of Electrical Engineers. 73 (441): 303–308. doi:10.1049/jiee-1.1933.0129.

Further reading