Thévenin's theorem

From Wikipedia, the free encyclopedia

In electrical circuit theory, Thévenin's theorem for 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, 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 EMF). The circuit consists of an ideal voltage source in series with an ideal resistor.

Image:Thevenin equivalent.png

Any black box containing only voltage sources, current sources, and resistors can be converted to a Thévenin equivalent circuit.

1 Calculating the Thévenin equivalent
2 Conversion to a Norton equivalent
3 Example of a Thévenin equivalent circuit
4 In popular culture
5 See also
6 External links

[edit] Calculating the Thévenin equivalent

To calculate the equivalent circuit, one needs a resistance and a voltage - two unknowns. And so, one needs two equations. These two equations are usually obtained by using the following steps, but any conditions one places on the terminals of the circuit should also work:

Calculate the output voltage, V_AB, when in open circuit condition (no load resistor - meaning infinite resistance). This is V_Th.
Calculate the output current, I_AB, when those leads are short circuited (load resistance is 0). R_Th equals V_Th divided by this I_AB.

The equivalent circuit is a voltage source with voltage V_Th in series with a resistance R_Th.

Case 2 could also be thought of like this:

2a. Now replace voltage sources with short circuits and current sources with open circuits.

2b. Replace the load circuit with an imaginary ohm meter and measure the total resistance, R, "looking back" into the circuit. This is R_Th.

The Thévenin-equivalent voltage is the voltage at the output terminals of the original circuit. When calculating a Thevenin-equivalent voltage, the voltage divider 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 is the resistance measured across points A and B "looking back" into the circuit. It is important to first replace all voltage- and current-sources with their internal resistances. For an ideal voltage source, this means replace the voltage source with a short circuit. For an ideal current source, this means replace the current source with an open circuit. Resistance can then be calculated across the terminals using the formulae for series and parallel circuits.

[edit] Conversion to a Norton equivalent

Image:Thevenin to Norton2.PNG

To convert to a Norton equivalent circuit, one can follow the following equations:

[edit] Example of a Thévenin equivalent circuit

Image:Thevenin and norton step 1.png

Step 0: The original circuit

Image:Thevenin step 2.png

Step 1: Calculating the equivalent output voltage

Image:Thevenin and norton step 3.png

Step 2: Calculating the equivalent resistance

Image:Thevenin step 4.png

Step 3: The equivalent circuit

In the example, calculating equivalent voltage:

<math>

V_\mathrm{AB} = {R_2 + R_3 \over (R_2 + R_3) + R_4} \cdot V_\mathrm{1} </math>

<math>

= {1\,\mathrm{k}\Omega + 1\,\mathrm{k}\Omega \over (1\,\mathrm{k}\Omega + 1\,\mathrm{k}\Omega) + 2\,\mathrm{k}\Omega} \cdot 15 \mathrm{V} </math>

<math>

= {1 \over 2} \cdot 15 \mathrm{V} = 7.5 \mathrm{V} </math>

Calculating equivalent resistance:

<math>

R_\mathrm{AB} = R_1 + \left ( \left ( R_2 + R_3 \right ) \| R_4 \right ) </math>

<math>

= 1\,\mathrm{k}\Omega + \left ( \left ( 1\,\mathrm{k}\Omega + 1\,\mathrm{k}\Omega \right ) \| 2\,\mathrm{k}\Omega \right ) </math>

<math>

= 1\,\mathrm{k}\Omega + \left({1 \over ( 1\,\mathrm{k}\Omega + 1\,\mathrm{k}\Omega )} + {1 \over (2\,\mathrm{k}\Omega ) }\right)^{-1} = 2\,\mathrm{k}\Omega </math>

[edit] In popular culture

While one might doubt that there is any popular culture around electrical theorems, both Thévenin's theorem and Norton's theorem feature in the 4th and 10th of May 2006 Doonesbury comic strip panels [1], [2].