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. 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 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.
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:
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∞(s) is the transfer function with the extra element open-circuited. H0(s) is the transfer function with the extra element short-circuited. Z(s) is the impedance of the extra element. 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.
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