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.

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 is found.

Next, two correctional factors must be found and combined with the previously derived function to find the exact expression.

[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.

H∞(s) is the transfer function with the extra element open-circuited.

H0(s) is the transfer function with the extra element short-circuited.

Zd(s) is the single-injection driving point impedance "seen" by the extra element.

Zn(s) is the double-null-injection driving point impedance "seen" by the extra element.

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.

Zd(s) is found by making the input to the system's transfer function zero (short circuit a voltage source or open circuit a current source) and determining the impedance across the terminals to which the extra element will be connected with the extra element absent.

Then, Zn(s) is defined as the ratio of voltage across the terminals of this second test source to the current leaving its positive terminal when the output of the system's transfer function is nulled for any value of the primary input to the system's transfer function.

In practice, Zn(s) can be found from working backward from the facts that the output of the transfer function is made zero and that the primary input to the transfer function is unknown.

Then using conventional circuit analysis techniques to express both the voltage across the extra element test source's terminals, vn(s), and the current leaving the extra element test source's positive terminals, in(s), and calculating

Although the computation of Zn(s) is an unfamiliar process for many engineers, its expressions are often much simpler than those for Zd(s) because the nulling of the transfer function's output often leads to other voltages/currents in the circuit being zero, which may allow exclusion of certain components from analysis.

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, Zd is the same as the impedance of the input test current source signal made zero or equivalently with the input open circuited.

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

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

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

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.