The great and silly transmission line debate

[paul-at-abelian.demon.co.uk (by way of Terry Fritz <twftesla-at-uswest-dot-net>)]

In order to prevent another of those embarrassing debates
between the 'alternative' lumped and transmission line
theories, I'd like to take the opportunity to point out
the following.

All electronic components, from the largest secondary resonator
through to the smallest surface mount chip, all interconnecting
wires, cables, pcb tracks, power lines, even the tiny insulated
gates of a FET, they are all transmission lines!

Many people find this surprising at first, but it inevitably
follows from Maxwell's equations. The simple fact is that
to move a charge from A to B, ie a current flow, requires a
redistribution of electric and magnetic fields along the
path from A to B. In electronics we make use of the slight
abstractions of inductance and capacitance as summary descriptions
of the geometry of the path, and if the path is anything other
than infinitesimal, we have distributed reactance, energy
storage, and propagation delay to deal with.

Fortunately, most of the time, we use components and wires at
frequencies such that their electrical length
is negligible, which gives rise to the somewhat idealised notion
of a 'lumped' component. So usualy we can talk about a 4.7k 1/4 watt
resistor without having to specify its transmission line
properties. Likewise we can wire up our circuits casualy
with hookup wire without worring about its characteristic
impedance. However, if you are designing say, PCB layouts for
high speed processors, you (hopefully!) will be very
concerned with correctly terminating your pcb tracks.

So, the full story is given by Maxwells equations which are
jolly hard to use practicaly. We can ease up a little by
inventing inductance and capacitance and thus transmission line
theory, which make things a little easier, and at low enough
frequencies, we can pretend that the inductance and capacitance
(and resistance) are 'lumped' into a 'component', thus we
get circuit theory. All components have L, C, and R, and
usualy two of the three can be neglected, so we know which
component tray to put it in!

Maxwells theory -> transmission line theory -> circuit
theory: successive levels of approximation.

There is nothing mysterious about all this, the same thing
happens in mechanics. Apply a bending moment to a one end of
a beam and there is a delay before the effect reaches the other
end. We have distributed inertia to deal with, and distributed
stress and strain. At low enough frequencies (a seesaw) we can
use the 'rigid body' approximation, but given a high enough
frequency (loudspeaker cones) or a big enough beam (skyscrapers
responding to earthquake vibrations) the rigid body approximation
is insufficient. 

How do we know when to apply transmission line theory instead
of circuit theory: when the component in question exhibits
time dependant behaviour. Usualy circuit theory can be rescued
a little by introducing slight qualifications into our component
specifications, eg an L as well as a C for a high frequency
capacitor.

Moving on to tesla secondary solenoids, a transmission line
model will always be successful at describing the behaviour
of the solenoid at any frequency, at least in principle.
Luckily however, for practical purposes, we can continue to
use circuit theory by stating that our solenoid has not only
L but a C as well, if we choose the C correctly. This is
the basis of several effective software calculator programs
which successfully estimate the lowest resonant frequency
of the solenoid by using various methods to determine the
effective C value for a given coil geometry.

Summarising, it is up to us if we want to describe a tesla
secondary in terms of a transmission line, most of the
time we dont. It is definately not a question of which mode
the resonator is oscillating in!

Sorry guys, no new science here! Or is there? 

I will add this: effective methods are available to estimate
the equivalent C to go with the solenoid's L. The same is not
true of the R. There is no currently accepted theory which can
give a value of effective R for a given coil geometry. Some
empirical work has led to tables which estimate R but these only
work up to, but not including the lowest resonant frequency.
Thus it is still not possible to predict the Q factor or input
impedance of a tesla resonator, even after a century of use!

But thats another subject, perhaps the one we should be debating!

Regards All.