Extra Element Theorem (EET) and Null-Double Injection

Hello Friends,

Went back and looked at Middlebrook's EET (as described by Vorpérian in Fast Analytical Techniques for Electrical and Electronic Circuits). Read through the derivation, and I can see how he arrived at the answer,

\[H=H_o\frac{1+\frac{Z_N}{Z}}{1+\frac{Z_D}{Z}}\]

Where Z is the impedance of the added element; Ho is the original transfer function with the element removed and open-circuited; Zn is the output nulling impedance; and Zd is the driving point impedance.

And, like anything with electricity, there's a dual:

\[H=H_o^'\frac{1+\frac{Z}{Z_N}}{1+\frac{Z}{Z_D}}\]

where Ho' is a different version of the original transfer function with the element short-circuited instead of left open.

There's lots of math, a bunch of equations, and some derivation performed. And the math seems to work. However, I'm left unsatisfied... I'd like to have an intuitive explanation, and I don't have one yet.

For example, with KVL and KCL the explanation is pretty obvious: sum the voltages around a loop and you'll get zero (because energy is conserved), and whatever currents go into a node must come out (because charge is conserved). Nodal and loop analysis are pretty easy to grasp too: solve equations of KCL using variables of voltage, or solve equations of KVL with variables of current. It doesn't take more than a sentence to explain the procedure or why it's done that way, and the explanation is in plain English, not math.

For the EET, however, the explanation I can make (so far) is not in plain English; even if we add nulling impedance and driving point impedance to our vocabulary, the best explanation I can give is: "Multiply the original transfer function [with the extra element open-circuited] by the ratio of the sum of the added impedance and the nulling impedance over the sum of the added impedance and the driving point impedance." In other words, the explanation is an equation, which means it's not really an explanation. Oh, and if the original transfer function has the element short-circuited, then the equation changes. This is definitely not plain English, and it's easy to forget what goes where in the equation—and therefore, it's not very handy when you need it in a pinch.

So my question is this: Is there an intuitive and easy explanation in a spoken language for the EET? If you use the EET, are there any jingos or simplifications that you use to help you remember, understand, or apply it? How many forum members find the EET useful?

Hi ZekeR,

I don`t know if it will be possible to find a kind of interpretation or explanation of the expressions for H. These formulas are "simply" the result of some intelligent manipulations of the original circuit. However, perhaps the following is helpful:

The EET states that the transfer function H of a linear network N can be calculated in symbolic form for a simpler network N1 by removing an element Z from the original network N. Then the resulting function H1 is to be multiplied with a correction factor which involves the element Z and two driving point impedances (DPI) seen at the node pair where Z was removed.
This procedure proves advantageous if the calculation of the DPI`s is easier than the direct calculation of H. This is true, in particular, if the linear network contains controlled sources.
Instead of finding an expression for the transfer function H, the EET can also be applied for calculating the input impedance of a network.

I took this information from an article written by V. Vorperian - published in EDN (August 1995). However, as you have Vorperian`s book, you have some more information I suppose.
Regards
LvW

I haven't read through the whole of it yet; perhaps more reading will be revealing. But it seems that most of the book is devoted to giving examples of applying the theory rather than explaining it further. Has anybody found the EET a useful approach? Useful enough to keep in your mental toolkit?

ZekeR, I think the usefulness of the method can be revealed only if you compare the mathematics that is necessary to solve a circuit analysis problem for the two cases
(a) direct (conventional) method
(b) using the EET.

The mentioned EDN article contains such an example: A resistive 4-element bridge with another path, which connects two opposite nodes and consists of a controlled source in parallel to another resistor.
The source is controlled by a voltage developed across one of the bridge resistors.
However, if such a circuit has any practical meaning is another question.
LvW