![TheveninEquivalent-2](https://upload.wikimedia.org/wikipedia/commons/d/dc/TheveninEquivalent-2.png)
As originally stated in terms of direct-current
resistive
The electrical resistance of an object is a measure of its opposition to the flow of electric current. Its reciprocal quantity is , measuring the ease with which an electric current passes. Electrical resistance shares some conceptual parallels ...
circuits only, Thévenin's theorem states that ''"For any
linear
Linearity is the property of a mathematical relationship (''function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear r ...
electrical network
An electrical network is an interconnection of electrical components (e.g., batteries, resistors, inductors, capacitors, switches, transistors) or a model of such an interconnection, consisting of electrical elements (e.g., voltage sources, c ...
containing only
voltage sources,
current sources
A current source is an electronic circuit that delivers or absorbs an electric current which is independent of the voltage across it.
A current source is the dual of a voltage source. The term ''current sink'' is sometimes used for sources fed ...
and
resistances can be replaced at terminals A–B by an equivalent combination of a voltage source V
th in a
series
Series may refer to:
People with the name
* Caroline Series (born 1951), English mathematician, daughter of George Series
* George Series (1920–1995), English physicist
Arts, entertainment, and media
Music
* Series, the ordered sets used in ...
connection with a resistance R
th."''
* The equivalent voltage ''V''
th is the voltage obtained at terminals A–B of the network with terminals A–B
open circuited.
* The equivalent resistance ''R''
th is the resistance that the circuit between terminals A and B would have if all ideal voltage sources in the circuit were replaced by a short circuit and all ideal current sources were replaced by an open circuit.
* If terminals A and B are connected to one another, the current flowing from A to B will be ''V''
th/''R''
th. This means that ''R''
th could alternatively be calculated as ''V''
th divided by the short-circuit current between A and B when they are connected together.
In
circuit theory
Circuit may refer to:
Science and technology
Electrical engineering
* Electrical circuit, a complete electrical network with a closed-loop giving a return path for current
** Analog circuit, uses continuous signal levels
** Balanced circui ...
terms, the theorem allows any
one-port
In electrical circuit theory, a port is a pair of terminals connecting an electrical network or circuit to an external circuit, as a point of entry or exit for electrical energy. A port consists of two nodes (terminals) connected to an outside ...
network to be reduced to a single
voltage source
A voltage source is a two-terminal device which can maintain a fixed voltage. An ideal voltage source can maintain the fixed voltage independent of the load resistance or the output current. However, a real-world voltage source cannot supply unli ...
and a single impedance.
The theorem also applies to frequency domain AC circuits consisting of
reactive
Reactive may refer to:
*Generally, capable of having a reaction (disambiguation)
*An adjective abbreviation denoting a bowling ball coverstock made of reactive resin
*Reactivity (chemistry)
*Reactive mind
*Reactive programming
See also
*Reactanc ...
(inductive and capacitive) and
resistive
The electrical resistance of an object is a measure of its opposition to the flow of electric current. Its reciprocal quantity is , measuring the ease with which an electric current passes. Electrical resistance shares some conceptual parallels ...
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
Hermann Ludwig Ferdinand von Helmholtz (31 August 1821 – 8 September 1894) was a German physicist and physician who made significant contributions in several scientific fields, particularly hydrodynamic stability. The Helmholtz Association, ...
and in 1883 by
Léon Charles Thévenin
Léon Charles Thévenin (; 30 March 1857, Meaux, Seine-et-Marne – 21 September 1926, Paris) was a French telegraph engineer who extended Ohm's law to the analysis of complex electrical network, electrical circuits.
Biography
Born in Meaux, F ...
(1857–1926), an
electrical engineer with France's national
Postes et Télégraphes telecommunications organization.
Thévenin's theorem and its dual,
Norton's theorem
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 ...
, are widely used to make circuit analysis simpler and to study a circuit's initial-condition and steady-state response.
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
Kirchhoff's circuit laws are two equalities that deal with the current and potential difference (commonly known as voltage) in the lumped element model of electrical circuits. They were first described in 1845 by German physicist Gustav Kirc ...
.
Calculating the Thévenin equivalent
The equivalent circuit is a voltage source with voltage ''V''
Th in series with a resistance ''R''
Th.
The Thévenin-equivalent voltage ''V''
Th is the open-circuit voltage at the output terminals of the original circuit. When calculating a Thévenin-equivalent voltage, the
voltage divider
In electronics, a voltage divider (also known as a potential divider) is a passive linear circuit that produces an output voltage (''V''out) that is a fraction of its input voltage (''V''in). Voltage division is the result of distributing the in ...
principle is often useful, by declaring one terminal to be ''V''
out and the other terminal to be at the ground point.
The Thévenin-equivalent resistance ''R''
Th 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
Two-terminal components and electrical networks can be connected in series or parallel. The resulting electrical network will have two terminals, and itself can participate in a series or parallel topology. Whether a two-terminal "object" is an ...
. This method is valid only for circuits with independent sources. If there are
dependent source
In the theory of electrical networks, a dependent source is a voltage source or a current source whose value depends on a voltage or current elsewhere in the network.I. D. Mayergoyz, Wes Lawson ''Basic electric circuit theory: a one-semester tex ...
s 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 the sources' values (meaning their voltage or current) are set 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 the source is replaced with a short circuit. Similarly, a zero valued current source and an ''open'' circuit both pass zero current.
Example
In the example, calculating the equivalent voltage:
(Notice that ''R''
1 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 R
1 and therefore no voltage drop along this part.)
Calculating equivalent resistance (
is the total resistance of two
parallel resistors
A resistor is a passive two-terminal electrical component that implements electrical resistance as a circuit element. In electronic circuits, resistors are used to reduce current flow, adjust signal levels, to divide voltages, bias active el ...
):
Conversion to a Norton equivalent
![Norton-to-thevenin](https://upload.wikimedia.org/wikipedia/commons/b/bd/Norton-to-thevenin.png)
A
Norton equivalent circuit
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 term ...
is related to the Thévenin equivalent by
Practical limitations
* Many circuits are only linear over a certain range of values, thus the Thévenin equivalent is valid only within this linear range.
* The Thévenin equivalent has an equivalent I–V characteristic only from the point of view of the load.
* The power dissipation of the Thévenin equivalent is not necessarily identical to the power dissipation of the real system. However, the power dissipated by an external resistor between the two output terminals is the same regardless of how the internal circuit is implemented.
A proof of the theorem
The proof involves two steps. The first step is to use
superposition theorem
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 (notably including the subcategory of time-invariant l ...
to construct a solution. Then,
uniqueness theorem In mathematics, a uniqueness theorem, also called a unicity theorem, is a theorem asserting the uniqueness of an object satisfying certain conditions, or the equivalence of all objects satisfying the said conditions. Examples of uniqueness theorems ...
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'',
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
*
Extra element theorem The Extra Element Theorem (EET) is an analytic technique developed by R. D. Middlebrook for simplifying the process of deriving driving point and transfer functions for linear electronic circuits.
Much like Thévenin's theorem, the extra element t ...
*
Maximum power transfer theorem 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 ...
*
Millman's theorem In electrical engineering, Millman's theorem (or the parallel generator theorem) is a method to simplify the solution of a circuit. Specifically, Millman's theorem is used to compute the voltage at the ends of a circuit made up of only branches i ...
*
Source transformation
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.CPP. https://ww ...
References
Further reading
*
First-Order Filters: Shortcut via Thévenin Equivalent Source— showing on p. 4 complex circuit's Thévenin's theorem simplication to first-order
low-pass filter
A low-pass filter is a filter that passes signals with a frequency lower than a selected cutoff frequency and attenuates signals with frequencies higher than the cutoff frequency. The exact frequency response of the filter depends on the filter des ...
and associated
voltage divider
In electronics, a voltage divider (also known as a potential divider) is a passive linear circuit that produces an output voltage (''V''out) that is a fraction of its input voltage (''V''in). Voltage division is the result of distributing the in ...
,
time constant In physics and engineering, the time constant, usually denoted by the Greek letter (tau), is the parameter characterizing the response to a step input of a first-order, linear time-invariant (LTI) system.Concretely, a first-order LTI system is a sy ...
and
gain
Gain or GAIN may refer to:
Science and technology
* Gain (electronics), an electronics and signal processing term
* Antenna gain
* Gain (laser), the amplification involved in laser emission
* Gain (projection screens)
* Information gain in de ...
.
External links
*
{{DEFAULTSORT:Thevenin's Theorem
Circuit theorems
Linear electronic circuits