John Popelish wrote:
Roy Lewallen wrote:
John Popelish wrote:

. . .
I think I agree with just about every conclusion you are making about
treating coils as slow wave transmission lines. . .


A coil itself isn't a slow wave transmission line.

Not at all? It seems to me that any real, physical inductor must have
some lumped properties and some transmission line properties, and it is
the balance of these that must be considered in any particular case to
decide which analysis is the more accurate way to deal with it in a
circuit. Solenoidal air core inductors have a lot of transmission line
properties if the frequency is high enough. If this were not so, they
would look exactly like fixed capacitors above self resonance, instead
of having multiple impedance peaks and valleys.

In conjunction with shunt C, it can be analyzed as a transmission
line, but only in conjunction with shunt C.

But any real, physical inductor has shunt capacitance to its
surroundings. So if you neglect this without considering whether or not
this is reasonable, you are going to be blindsided by its effects,
eventually.

I don't disagree with anything you've said. The point I was trying to
make was that the resemblance of a coil to a transmission line depends
not only on the coil but also its capacitance to other objects -- and
not to its relationship to traveling current waves. One thing I've seen
done on this thread is to use the C across the inductor in transmission
line formulas, appearing to give the coil a transmission line property
all by itself and without any external C. This is incorrect.


Remove the shunt C and it ceases looking like a transmission line.

How do I remove the shunt C of an inductor? With an active guarding
scheme?

Actually, you can reduce it to a negligible value by a number of means.
One I've done is to wind it as a physically small toroid. In the example
discussed in the next paragraph, removing ground from the model reduces
the external C to a small enough value that the current at the coil ends
become nearly equal. That of course isn't an option in a real mobile
coil environment, but it illustrates that the current drop from one end
to the other, which in some ways mimics a transmission line, is due to
external C rather than reaction with traveling waves as Cecil claims. In
my modification to Cecil's EZNEC file I showed how the coil behaves the
same with no antenna at all, just a lumped load impedance. As long as
the load impedance and external C stay the same, the coil behavior stays
the same. This isn't, however, to discount the possibility of the coil
interacting with the antenna's field. It just wasn't significant in that
case.

The earlier example of the modification to Cecil's EZNEC model
illustrated this -- when the ground (the other side of the shunt
capacitor) was removed, the current drop across the coil disappeared.

So whether or not this coil is acting as a slow wave transmission line
in addition to being inductive depends on the surrounding fields and
connections? I have no trouble with that.

Well, not a "slow wave" transmission line. We shouldn't confuse an
ordinary lumped LC transmission line approximation with a true slow wave
structure such as a helical waveguide (next item). The propagation
velocity of the equivalent transmission line is omega/sqrt(LC), so the
speed depends equally on the series L and the shunt C.

And let's talk for a minute about the coil "acting like" a transmission
line. A transmission line is of course a distributed circuit. But you
can make a single pi or tee section with lumped series L and shunt C
which has all the characteristics of a transmission line at one
frequency(*), including time delay, phase shift, characteristic
impedance, impedance transformation, and everything else. If put into a
black box, you wouldn't be able to tell the difference among the pi,
tee, or transmission line -- at one frequency. You could even sample the
voltage and current with a Bird wattmeter and conclude that there are
traveling voltage and current waves in both cases, and calculate the
values of the standing waves on either "transmission line". And this is
with a pure inductance and capacitance, smaller than the tiniest
components you can really make. With a single section, you can mimic any
transmission line Z0 and any length from 0 to a half wavelength. (The
limiting cases, however, require some components to be zero or
infinite.) So you can say if you wish that the inductor in this network
"acts like" a transmission line -- or you can equally correctly say that
the capacitor does, because it's actually the combination which mimics a
transmission line. But only over a narrow range of frequencies, beyond
which it begins deviating more and more from true transmission line
behavior. To mimic longer lines or mimic lines over a wider frequency
range requires more sections.

So what can we conclude about inductors from this similar behavior?
Certainly not that there's anything special about inductors interacting
with traveling waves or that inductors comprise some kind of "slow wave
structure". The duality comes simply from the fundamental equations
which describe the nature of transmission lines, inductances, and
capacitances.

Because the LC section's properties are identical to a transmission
line's at one frequency, we have our choice in analyzing the circuit. We
can pretend it's a transmission line, or we can view it as a lumped LC
network. If we go back to the fundamental equations of each circuit
element, we'll find that the equations end up exactly the same in either
case. And the results from analyzing using each method are identical --
if not, we've made an error.

The coil in the EZNEC model on Cecil's web page acts just like we'd
expect an inductor to act. With ground present constituting a C, the
circuit acts like an L network made of lumped L and C which behaves
similarly to a transmission line. With ground, hence external C, absent,
it acts like a lumped L. (There are actually some minor differences, due
to imperfect coupling between turns and to coupling to the finite sized
external circuit.) The combination of L and C "act like" a transmission
line, just like any lumped L and C. And it doesn't care whether the load
is a whip or just lumped components.

(*) It actually acts like a transmission line at many frequencies, but a
different length and Z0 of line at each frequency. To mimic a single
line over a wide frequency range requires additional sections.

As far as considering a coil itself as a "slow wave structure", Ramo
and Whinnery treat this subject. It's in the chapter on waveguides,
and they explain how a helix can operate as a slow wave waveguide
structure. To operate in this fashion requires that TM and TE modes be
supported inside the structure which in turn requires a coil diameter
which is a large part of a wavelength. Axial mode helix antennas, for
example, operate in this mode. Coils of the dimensions of loading
coils in mobile antennas are orders of magnitude too small to support
the TM and TE modes required for slow wave propagation.

I'll have to take your word for this limitation. But it seems to me
that the length of the coil in relation to the wavelength and even the
length of the conductor the coils is made of are important, also.

Important for what? No matter how long the coil or how many turns of the
wire, a small (in terms of wavelength) inductor won't act like a slow
wave structure or an axial mode helical antenna. This is for the same
reason that a two inch diameter pipe won't perform as a waveguide at 80
meters -- there's not enough room inside to fit the field distribution
required for that mode of signal propagation. There will of course be
some point at which it'll no longer act as a lumped inductor but would
have to be modeled as a transmission line. But this is when it becomes a
significant fraction of a wavelength long. If the turns are very loosely
coupled to each other, the wire length becomes more of a determining
factor. As I mentioned in earlier postings, there's a continuum between
a straight wire and that same wire wound into an inductor. As the
straight wire is wound more and more tightly, the behavior transitions
from that of a wire to that of an inductance. There's no abrupt point
where a sudden change occurs.

Roy Lewallen, W7EL