Don,

How's this for an elegant proof of the extra element theorem. Let me know what you think.

How's this for an elegant proof of the extra element theorem. Let me know what you think.

Add an extra element ** Z**
between any two nodes in a linear system:

Treat
** i** as a
second input, and

_{}
Eqn
1

_{}_{}
Eqn
2

Eliminate *i, v***.**
Solve for_{}:

_{}

_{}

_{}
Eqn
3

This
gives *A(Z)* in
terms of the original transfer
function *A _{1}*,
and three coefficients that can be determined from analysis of the
system
without Z connected. We can reduce these

_{}

_{}
(bilinear)
Eqn 4

where by definition:

_{}
(denominator)
Eqn 5

_{} (numerator)
Eqn 6

These two unknowns __look__
more complex
than the previous three, but they have a simple interpretation, and
they are
often more easily determined. Also, the
bilinear form has a nice interpretation when the original network has
resistors
only, and we are adding one reactive element ** Z**.
Adding this element multiplies the original transfer function
with one
pole and one zero.

The interpretation of the two new parameters is as follows:

_{}
_{}
Eqn
7

_{}
Eqn
8

*Z _{d}*

As will be seen in the
examples,
the computation of ** Z_{n}**
using
eqn 8 in a real circuit is often

I think our students get too bogged down in the math, and miss the
simple insights they need to become good desingers.