Thevenin's Theorem
 

Thevenin's Theorem

In circuit theory, Thévenin's theorem for linear electrical networks states that any combination of voltage sources, current sources, and resistors with two terminals is electrically equivalent to a single voltage source V and a single series resistor R. For single frequency AC systems the theorem can also be applied to general impedances, not just resistors. The theorem was first discovered by German scientist Hermann von Helmholtz in 1853,[1] but was then rediscovered in 1883 by French telegraph engineer Léon Charles Thévenin (1857–1926).


This theorem states that a circuit of voltage sources and resistors can be converted into a Thévenin equivalent, which is a simplification technique used in circuit analysis. The Thévenin equivalent can be used as a good model for a power supply or battery (with the resistor representing the internal impedance and the source representing the electromotive force). The circuit consists of an ideal voltage source in series with an ideal resistor.



Thevenin's Theorem: Detial

In principle, all currents and voltages of an arbitrary network of linear components and voltage/current sources can be found by any of the three methods discussed previous, namely, the branch current method, the loop current method and the node voltage method.
However, if only the current and/or voltage associated with one component are of interest, it is unnecessary to find voltages and currents elsewhere in the circuit. The methods considered below can be used in such situations.
Any one-port (two-terminal) network of resistance elements and energy sources is equivalent to an ideal voltage source http://fourier.eng.hmc.edu/e84/lectures/ch2/img112.png in series with a resistor http://fourier.eng.hmc.edu/e84/lectures/ch2/img113.png, where

• http://fourier.eng.hmc.edu/e84/lectures/ch2/img112.png is the open-circuit voltage of the network, and
• http://fourier.eng.hmc.edu/e84/lectures/ch2/img113.png is the equivalent resistance when all energy sources are turned off (short-circuit for voltage sources, open-circuit for current sources).

http://fourier.eng.hmc.edu/e84/lectu...rtonfigure.gif
If we are only interested in finding the voltage http://fourier.eng.hmc.edu/e84/lectures/ch2/img29.png across and current http://fourier.eng.hmc.edu/e84/lectures/ch2/img100.png through one particular resistor in a complex circuit containing a large number of resistors, voltage and current sources, we can ``pull'' the resistor out and treat the rest of the circuit as a Thevenin voltage source http://fourier.eng.hmc.edu/e84/lectures/ch2/img114.png, and use Thevenin's theorem to find http://fourier.eng.hmc.edu/e84/lectures/ch2/img29.png and http://fourier.eng.hmc.edu/e84/lectures/ch2/img100.png.
Proof:
http://fourier.eng.hmc.edu/e84/lectu...veninProof.gif
Assume with the load the network's terminal voltage and current are http://fourier.eng.hmc.edu/e84/lectures/ch2/img29.png and http://fourier.eng.hmc.edu/e84/lectures/ch2/img100.png respectively.

• Replace the load by an ideal current source http://fourier.eng.hmc.edu/e84/lectures/ch2/img100.png while keeping the terminal voltage http://fourier.eng.hmc.edu/e84/lectures/ch2/img29.png the same (b), the voltage or current anywhere in the system should not be affected.
• Find the terminal voltage http://fourier.eng.hmc.edu/e84/lectures/ch2/img29.png in terms of the internal energy sources inside the network and the external current source by superposition principle:
  • When the external current source is off (open circuit), the terminal voltage http://fourier.eng.hmc.edu/e84/lectures/ch2/img115.png is due only to the sources internal to the network (c).
  • When all internal sources are turned off (short-circuit for voltage sources, open-circuit for current sources), the terminal voltage http://fourier.eng.hmc.edu/e84/lectures/ch2/img116.png where http://fourier.eng.hmc.edu/e84/lectures/ch2/img117.png is the equivalent resistance of the network with all energy sources off (d).
  The overall terminal voltage is http://fourier.eng.hmc.edu/e84/lectures/ch2/img118.png.

As far as the port voltage http://fourier.eng.hmc.edu/e84/lectures/ch2/img29.png and current http://fourier.eng.hmc.edu/e84/lectures/ch2/img100.png are concerned, a one-port network is equivalent to an ideal voltage source http://fourier.eng.hmc.edu/e84/lectures/ch2/img119.png equal to the open-circuit voltage across the port, in series with an internal resistance http://fourier.eng.hmc.edu/e84/lectures/ch2/img120.png, which can be obtained as the ratio of the open-circuit voltage and the short-circuit current.