Norton's theorem
 

Norton's theorem


Norton's theorem for linear electrical networks, known in Europe as the Mayer–Norton theorem, states that any collection of voltage sources, current sources, and resistors with two terminals is electrically equivalent to an ideal current source, I, in parallel with a single resistor, R. For single-frequency AC systems the theorem can also be applied to general impedances, not just resistors. The Norton equivalent is used to represent any network of linear sources and impedances, at a given frequency. The circuit consists of an ideal current source in parallel with an ideal impedance (or resistor for non-reactive circuits).


Norton's theorem is an extension of Thévenin's theorem and was introduced in 1926 separately by two people: Siemens & Halske researcher Hans Ferdinand Mayer (1895–1980) and Bell Labs engineer Edward Lawry Norton (1898–1983). Only Mayer actually published on this topic, but Norton made known his finding through an internal technical report at Bell Labs.


Norton's Theorem: Detial


Any one-port (two-terminal) network of resistance elements and energy sources is equivalent to an ideal current source http://fourier.eng.hmc.edu/e84/lectures/ch2/img121.png in parallel with a resistor http://fourier.eng.hmc.edu/e84/lectures/ch2/img122.png, where

• http://fourier.eng.hmc.edu/e84/lectures/ch2/img121.png is the short-circuit current of the network, and
• http://fourier.eng.hmc.edu/e84/lectures/ch2/img122.png is the equivalent resistance when all energy sources are turned off (short-circuit for voltage sources, open-circuit for current sources).

Proof: The proof of this theorem is in parallel with the proof of the Thevenin's theorem. Again, 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 voltage source http://fourier.eng.hmc.edu/e84/lectures/ch2/img29.png while keeping the terminal current http://fourier.eng.hmc.edu/e84/lectures/ch2/img100.png the same, the voltage or current anywhere in the system should not be affected.
• Find the terminal current http://fourier.eng.hmc.edu/e84/lectures/ch2/img100.png in terms of the internal energy sources inside the network and the external voltage source by superposition principle:
  • When the external voltage source is off (short circuit), the terminal current http://fourier.eng.hmc.edu/e84/lectures/ch2/img123.png is due only to the sources internal to the network.
  • When all internal sources are turned off (short-circuit for voltage sources, open-circuit for current sources), the terminal current http://fourier.eng.hmc.edu/e84/lectures/ch2/img124.png where http://fourier.eng.hmc.edu/e84/lectures/ch2/img117.png is the equivalent resistance of the network with all energy sources off.
  The overall terminal current is http://fourier.eng.hmc.edu/e84/lectures/ch2/img125.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 current source http://fourier.eng.hmc.edu/e84/lectures/ch2/img126.png equal to the short-circuit current through the port, in parallel with an internal resistance http://fourier.eng.hmc.edu/e84/lectures/ch2/img127.png, which can be obtained as the ratio of the open-circuit voltage and the short-circuit current. Of course, the Norton's theorem can also be easily proven by converting Thevenin's equivalent circuit of an ideal voltage source http://fourier.eng.hmc.edu/e84/lectures/ch2/img119.png in series with a resistance http://fourier.eng.hmc.edu/e84/lectures/ch2/img113.png to an equivalent circuit of an ideal current source http://fourier.eng.hmc.edu/e84/lectures/ch2/img128.png in parallel with a resistance http://fourier.eng.hmc.edu/e84/lectures/ch2/img129.png.
Load Line and Output Resistance
Due to the Thevenin's and Norton's theorems, any one-port network of resistors and energy sources can be converted into a simple voltage or current source with an internal or output resistance http://fourier.eng.hmc.edu/e84/lectures/ch2/img117.png. Moreover, the relationship between the voltage http://fourier.eng.hmc.edu/e84/lectures/ch2/img29.png across and the current http://fourier.eng.hmc.edu/e84/lectures/ch2/img100.png through the load is a straight line referred to as the load line. The slope http://fourier.eng.hmc.edu/e84/lectures/ch2/img130.png of the load line indicates the internal or output resistance of the network, as shown in the figure below. One the other hand, the input resistance http://fourier.eng.hmc.edu/e84/lectures/ch2/img131.png of the load can also be represented on the graph as a straight line with its slope http://fourier.eng.hmc.edu/e84/lectures/ch2/img132.png. The intersection of these two lines indicates the actual voltage http://fourier.eng.hmc.edu/e84/lectures/ch2/img29.png and current http://fourier.eng.hmc.edu/e84/lectures/ch2/img100.png with the load http://fourier.eng.hmc.edu/e84/lectures/ch2/img131.png.
http://fourier.eng.hmc.edu/e84/lectu...s/loadline.gif
The output resistance http://fourier.eng.hmc.edu/e84/lectures/ch2/img117.png of a network can also be determined experimentally by varying the load http://fourier.eng.hmc.edu/e84/lectures/ch2/img131.png. Assume http://fourier.eng.hmc.edu/e84/lectures/ch2/img133.png are associated with load http://fourier.eng.hmc.edu/e84/lectures/ch2/img134.png and http://fourier.eng.hmc.edu/e84/lectures/ch2/img135.png with load http://fourier.eng.hmc.edu/e84/lectures/ch2/img136.png, then the output resistance of the source network can be found to be:


http://fourier.eng.hmc.edu/e84/lectures/ch2/img137.png

To show this, we assume the source network is converted to a voltage source with http://fourier.eng.hmc.edu/e84/lectures/ch2/img138.png and http://fourier.eng.hmc.edu/e84/lectures/ch2/img117.png, and use two different loads http://fourier.eng.hmc.edu/e84/lectures/ch2/img134.png and http://fourier.eng.hmc.edu/e84/lectures/ch2/img136.png with

http://fourier.eng.hmc.edu/e84/lectures/ch2/img139.png

and
http://fourier.eng.hmc.edu/e84/lectures/ch2/img140.png

The output resistance can therefore be found to be http://fourier.eng.hmc.edu/e84/lectures/ch2/img117.png as expected:
http://fourier.eng.hmc.edu/e84/lectures/ch2/img141.png

Consider two extreme cases for the two loads http://fourier.eng.hmc.edu/e84/lectures/ch2/img134.png and http://fourier.eng.hmc.edu/e84/lectures/ch2/img136.png:

• when http://fourier.eng.hmc.edu/e84/lectures/ch2/img142.png (short circuit), then we get the short-circuit current:
  http://fourier.eng.hmc.edu/e84/lectures/ch2/img143.png
• when http://fourier.eng.hmc.edu/e84/lectures/ch2/img144.png (open circuit), then we get the open-circuit voltage:
  http://fourier.eng.hmc.edu/e84/lectures/ch2/img145.png

and we have http://fourier.eng.hmc.edu/e84/lectures/ch2/img146.png, http://fourier.eng.hmc.edu/e84/lectures/ch2/img147.png, and the output resistance is found to be:
http://fourier.eng.hmc.edu/e84/lectures/ch2/img148.png

Example:
http://fourier.eng.hmc.edu/e84/lectu...eninNorton.gif

• Thevenin's equivalent circuit:
  • Find open-circuit voltage http://fourier.eng.hmc.edu/e84/lectures/ch2/img149.png
  • Find equivalent internal resistance http://fourier.eng.hmc.edu/e84/lectures/ch2/img150.png:
  • The Thevenin's equivalent circuit is shown in (b).
• Norton's equivalent circuit:
  • Find short-circuit current. As there are two sources, superposition principle is used to get http://fourier.eng.hmc.edu/e84/lectures/ch2/img151.png
  • Find equivalent internal resistance http://fourier.eng.hmc.edu/e84/lectures/ch2/img152.png:
  • The Norton's equivalent circuit is shown in (c).

Note that the resistor http://fourier.eng.hmc.edu/e84/lectures/ch2/img136.png does not appear in either of the equivalent circuit. This is because http://fourier.eng.hmc.edu/e84/lectures/ch2/img136.png is in series with an ideal current source which drives a constant current http://fourier.eng.hmc.edu/e84/lectures/ch2/img101.png through the branch, independent of the resistance along the branch. Also note that the Thevenin's voltage source and the Norton's current source can be converted into each other:

• Convert Thevenin's voltage source to Norton's current source:
  http://fourier.eng.hmc.edu/e84/lectures/ch2/img153.png
• Convert Norton's current source to Thevenin's voltage source:
  http://fourier.eng.hmc.edu/e84/lectures/ch2/img154.png