Extra Element Theorem

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## General formulation

## Driving point impedances

### Single Injection Driving Point Impedance

### Double Null Injection Driving Point Impedance

## Special case with transfer function as a self-impedance

### Example

## Feedback amplifiers

## See also

## Further reading

## References

## External links

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

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.^{[1]} Much like Thévenin'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 Kirchhoff's circuit laws. 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.^{[2]}

The (single) extra element theorem expresses any transfer function as a product of the transfer function with that element removed and a correction factor. The correction factor term consists of the impedance of the extra element and two driving point impedances seen by the extra element: The double null injection driving point impedance and the single injection driving point impedance. Because an extra element can be removed in general by either short-circuiting or open-circuiting the element, there are two equivalent forms of the EET:^{[3]}

or,

- .

Where the Laplace-domain transfer functions and impedances in the above expressions are defined as follows: *H*(*s*) is the transfer function with the extra element present. *H _{∞}*(

The extra element theorem incidentally proves that any electric circuit transfer function can be expressed as no more than a bilinear function of any particular circuit element.

*Z _{d}*(

*Z _{n}*(

In practice, *Z _{n}*(

As a special case, the EET can be used to find the input impedance of a network with the addition of an element designated as "extra". In this case, *Z _{d}* is same as the impedance of the input test current source signal made zero or equivalently with the input open circuited. Likewise, since the transfer function output signal can be considered to be the voltage at the input terminals,

where

- is the impedance chosen as the extra element

- is the input impedance with Z removed (or made infinite)

- is the impedance seen by the extra element Z with the input shorted (or made zero)

- 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.

Consider the problem of finding 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

- .

Removing this capacitor from the circuit,

- .

Calculating the impedance seen by the capacitor with the input shorted,

- .

Calculating the impedance seen by the capacitor with the input open,

- .

Therefore, using the EET,

- .

This problem was solved by calculating three simple driving point impedances by inspection.

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.

- Christophe Basso
*Linear Circuit Transfer Functions: An Introduction to Fast Analytical Techniques*first edition, Wiley, IEEE Press, 2016, 978-1119236375

**^**Vorpérian, Vatché (2002).*Fast analytical techniques for electrical and electronic circuits*. Cambridge UK/NY: Cambridge University Press. pp. 61-106. ISBN 978-0-521-62442-8.**^**Vorpérian, Vatché (2002-05-23).*pp. 137-139*. ISBN 978-0-521-62442-8.**^**Middlebrook R.D. (1989). "Null Double Injection and the Extra Element Theorem" (PDF).*IEEE Transactions on Education*.**32**(3): 167-180. doi:10.1109/13.34149.

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

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