14 February, 2007

Thevenin's and Norton's Theorem

Thevenin's Theorem :
This 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 any linear bilateral circuit with combination of voltage sources , current sources and resistors with two terminals is electrically equivalent to a single voltage source VTh (Thevenin Voltage) and a single series resistor RTh (Thevenin Resistance).This equivalent is called Thevenin Equivalent.

Calculating the Thevenin 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:
  1. Calculate the output voltage, VAB, when in open circuit condition (no load resistor - meaning infinite resistance). This is VTh.
  2. Calculate the output current, IAB, when those leads are short circuited (load resistance is 0) RTh equals VTh divided by this IAB.
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 RTh.
The Thevenin-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 Vout and the other terminal to be at the ground point.
The Thevenin-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.
A Norton equivalent circuit is related to the Thevenin equivalent by the following equations:
RTh = RNo , VTh = INo RNo
Example of a Thévenin equivalent circuit



In the example, calculating equivalent voltage and resistance:

Norton's Theorem :
Norton's theorem is an extension of Thevenin's theorem and was introduced in 1926 separately by two people: Hause-Siemens researcher Hans Ferdinand Mayer (1895-1980) and Bell Labs engineer Edward Lawry Norton (1898-1983). Mayer was the only one of the two who actually published on this topic, but Norton made known his finding through an internal technical report at Bell Labs.

This theorem states that any linear bilateral circuit with combination of voltage sources , current sources and resistors with two terminals is electrically equivalent to an ideal current soure, INo, in parallel with a single resistor, RNo.This equivalent is called Norton Equivalent.

Calculation of Norton Equivalent
To calculate the equivalent circuit:
  1. Calculate the output current, IAB, when a short circuit is the load (meaning 0 resistance between A and B). This is INo.
  2. Calculate the output voltage, VAB, when in open circuit condition (no load resistor - meaning infinite resistance). RNo equals this VAB divided by INo.
The equivalent circuit is a current source with current INo, in parallel with a resistance RNo.
Case 2 can also be thought of like this:
2a. Now replace independent voltage sources with short circuits and independent current sources with open circuits.
2b. For circuits without dependent sources RNo is the total resistance with the independent sources removed.
* Note: A more general method for determining the Norton Impedance is to connect a current source at the output terminals of the circuit with a value of 1 Ampere and calculate the voltage at its terminals; this voltage is equal to the impedance of the circuit. This method must be used if the circuit contains dependent sources. This method is not shown below in the diagrams.
To convert to a Thevenin equivalent circuit, one can use the following equations:
RTh = RNo
VTh = INo RNo



Example of Norton Equivalent circuit




For more information on Thevenin's theorem visit these sites :
www.eas.asu.edu/~holbert/ece201/ECE201Lect-13.ppt
www.ee.adfa.edu.au/staff/hrp/teaching/docs/superThevininMar96.pdf
www.allaboutcircuits.com/vol_1/chpt_10/7.html
http://www.wisc-online.com/objects/index_tj.asp?objid=DCE5803 -- Online Calculations
http://hyperphysics.phy-astr.gsu.edu/hbase/electric/thevenin.html
http://utwired.engr.utexas.edu/rgd1/lesson09.cfm
http://www.swarthmore.edu/NatSci/echeeve1/Ref/E72WhaKnow/WhaKnow.html
www.ibiblio.org/kuphaldt/socratic/output/millman.pdf --- Some Practice problems
For more information on Norton's theorem visit these sites :
www.eas.asu.edu/~holbert/ece201/ECE201Lect-14.ppt
www.allaboutcircuits.com/vol_1/chpt_10/8.html
http://www.wisc-online.com/objects/index_tj.asp?objid=DCE10004
http://230nsc1.phy-astr.gsu.edu/hbase/electric/nort2l.html
www.bowest.com.au/library/theorems.html
http://utwired.engr.utexas.edu/rgd1/lesson09.cfm

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