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. cite book
author=Vorpérian, Vatché
title=Fast analytical techniques for electrical and electronic circuits
year= 2002
pages=pp. 61-106
publisher=Cambridge University Press
location=Cambridge UK/NY
isbn= 0521624428
url=http://worldcat.org/isbn/0521624428] Much like Thevenin's theorem, the extra element theorem breaks down one complicated problem into several simpler ones.

Driving point and transfer functions can generally be found using KVL and KCL methods, however several complicated equations may result that offer little insight into the circuit's behavior. Using the extra element theorem, a circuit element (such as a resistor) can be removed from a circuit and the desired driving point or transfer function found. By removing the element that most complicates the circuit (such as an element that creates feedback), the desired function can be easier to obtain. Next two correctional factors must be found and combined with the previously derived function to find the exact expression.

The general form of the extra element theorem is called the N-extra element theorem and allows multiple circuit elements to be removed at once. cite book
author=Vorpérian, Vatché
title=pp. 137-139
isbn= 0521624428
url=http://worldcat.org/isbn/0521624428]

Driving point impedances

As a special case, the EET can be used to find the input impedance of a network. For this application the EET can be written as:

:$Z\_\{in\}\; =\; Z^\{infty\}\_\{in\}\; left(\; frac\{1+frac\{Z^0\_\{e\{Z\{1+frac\{Z^\{infty\}\_\{e\{Z\; ight)$

where

:$Z$ is the impedance chosen as the extra element

:$Z^\{infty\}\_\{in\}$ is the input impedance with Z removed (or made infinite)

:$Z^0\_\{e\}$ is the impedance seen by the extra element Z with the input shorted (or made zero)

:$Z^\{infty\}\_\{e\}$ is the impedance seen by the extra element Z with the input open (or made infinite)

Computing these three terms may seem like extra effort, but they are often easier to compute than the overall input impedance.

Example

Consider the problem of finding $Z\_\{in\}$ for the circuit in Figure 1 using the EET (note all component values are unity for simplicity). If the capacitor (gray shading) is denoted the extra element then

:$Z\; =\; frac\{1\}\{s\}$

Removing this capacitor from the circuit we find

:$Z^\{infty\}\_\{in\}\; =\; 2|1\; +1\; =\; frac\{5\}\{3\}$

Calculating the impedance seen by the capacitor with the input shorted we find

:$Z^0\_\{e\}\; =\; 1|(1+1|1)\; =\; frac\{3\}\{5\}$

Calculating the impedance seen by the capacitor with the input open we find

:$Z^\{infty\}\_\{e\}\; =\; 2|1+1\; =\; frac\{5\}\{3\}$

Therefore using the EET, we find

:$Z\_\{in\}\; =\; frac\{5\}\{3\}\; left(frac\{1+frac\{3\}\{5\}s\}\{1+frac\{5\}\{3\}s\}\; ight)$

Note that this problem was solved by calculating three simple driving point impedances by inspection.

Feedback amplifiers

The EET is also useful for analyzing single and multi-loop feedback amplifiers. In this case the EET can take the form of the Asymptotic gain model.

References

External links

* [http://www.edn.com/archives/1995/080395/16df4.htm Examples of applying the EET]
* [http://ece-www.colorado.edu/~ecen5807/course_material/slidesAppC.pdf Derivation and examples]