Re: Series resonance/Was: Waveguide TC

["Tesla list" <tesla-at-pupman-dot-com>]

Original poster: "Paul Nicholson by way of Terry Fritz <twftesla-at-qwest-dot-net>" <paul-at-abelian.demon.co.uk>

Jim wrote:

 > a number of folks have made measurements of the current
 > at top and bottom and find [current phase between top and bottom]
 > are only a few degrees apart.

Yes, this is correct.  There is little or no observed phase change
along the coil, even if the coil is many electrical degrees long [*].

But it is wrong to conclude, as Terry did in
  http://hot-streamer-dot-com/TeslaCoils/MyPapers/topsync/topsync.html

that this invalidates the transmission line model. In this paper,
Terry wrote

: The top and bottom currents in the secondary inductor are almost
: perfectly in phase.

A good measurement...

: If the 1/4 wave model of Tesla coil secondary inductors were true,
: these currents should be 90 degrees out of phase.

...but an incorrect conclusion.  Both the lumped and transmission
line models predict a uniform phase for standing waves.  That's why
they are called 'standing' waves.  This is a common enough error, as
Jim demonstrates,

 > ...the phase shift between current at the top and bottom of the TC
 > is nowhere near 90 degrees, as it would be for a transmission line.

This is a common source of confusion and has led to many futile
debates in the past. Whichever model is used, the current phase
is almost uniform (although of course the amplitude must vary to
satisfy charge conservation).  As long as the reflection coefficient
at each end of the coil has a magnitude close to unity, the waves
along the coil will be 'standing'.  Terminate the coil with its
characteristic impedance in order to suppress the reflected wave and
reveal the underlying phase change of each travelling component
caused by the electrical length of the coil.

The Corum's in their notorious Class Notes incorrectly suggest that
the coiler has the option to run their coil in a lumped or
transmission line mode:

: How can you tell whether your Tesla coil is operating in Tesla's
: pre-1894 tuned lumped element regime or in his post-1894
: distributed resonator regime?

The short answer is you can't!  It is not an alternative regime of
operation, just an alternative mathematical model.

: If lumped analysis describes your coil, cheer up - modify its
: operation to an open resonator and you'll see what Tesla called,
: on July 11, 1899, "a beautiful advance in the art"!)

The authors seem to think that adding top C to the coil (thus making
the current amplitude more uniform along the coil) is necessary to
allow a lumped model to be applied.  This is wrong for two reasons;

a) No matter how much top C you add, you still get a spectrum of
multiple resonant modes;
b) No matter how little top C is applied, you can still represent
the resonator by a lumped LC model.

[The Corums go on to suggest that you can improve the performance of
the TC by making it work like a transmission line rather than a
lumped circuit, which is also incorrect.  The two models predict
essentially the same performance for any given coil].

Of course, the lumped LC model only gives correct answers at one or
a few resonant modes of the coil.  Move to a different resonance and
you must use different values of L and C to make the LC model fit.
The transmission line model allows you to describe the behaviour of
the solenoid over a continuous range of frequencies, and allows the
actual distributed currents and voltages to be calculated.

Typically, the extra computational effort of the transmission line
model is not necessary for TC design, and is used only to calculate
the effective L and C values from which the design can proceed with
the lumped model.  Until recently, this has been a stumbling block
for coilers.  But now, thanks to Bart Anderson, you can use the
transmission line model to compute accurate effective LC values for
use in lumped models:

  http://www.classictesla-dot-com/fantc/fantc.html

The electrical length of a resonator is always some integer multiple
of 90 degrees, which is what defines the resonances in the first
place.  The coiler has a choice over how much of that electrical
length is taken up by the topload reactance and how much is due to
the coil.  With an unloaded coil, the coil itself will span almost
the full 90 degrees (for 1/4 wave, or 180 for 1/2 wave, 270 for 3/4
wave, etc).  Add some top C and the coil will now resonate at a lower
frequency, at which its electrical length is only say 60 degrees - the
top C provides the other 30 degrees.  Addition of still more top C
further reduces the proportion of the electrical length taken up by
the coil. The coil retains the same signal delay throughout, but this
constant delay represents fewer and fewer degrees of phase as the
frequency is lowered by the top C.  But it is important to appreciate
that no matter whether the electrical length of the coil is 90, or
60 or just a few degrees, the observed phase change will be around
zero because all you can measure is the sum of the forward and
reflected waves, which cancel their respective phase changes to give
the familiar standing wave.  The 'zero phase change' is the reason
the waves appear to 'stand'.  Terminate the coil with a matched
impedance in order to see the real underlying phase change of the
'travelling' wave - the phase change that Jim was thinking of.

Bear in mind that so-called lumped L and C components are in fact
transmission lines.  Inductors have a very high characteristic
impedance (large distributed L/C ratio), while capacitors have a
very small characteristic impedance (small distributed L/C). Thus
in the former, L dominates and in the latter, C is dominant. Both
components are normally operated well below their lowest self-
resonant modes, so that they approximate the ideal of a lumped
component.  The transmission line behaviour of ordinary L and C
'lumped' components is however exposed as soon as you put them on a
network analyser to observe their spectrum of self-resonances.

The transmission line model itself starts to break down when the
distances over which the distributed L and C are operating becomes
a noticeable fraction of a free space wavelength.  Then, the rather
artificial concepts of capacitance and inductance begin to break down,
and at the same time a noticeable amount of energy escapes into the
far field.  In this case, both the lumped and transmission line
models, which up to now have both given the right answers, begin to
deliver wrong answers, and the full EM equations are needed in order
to describe the behaviour of the structure.

It may be helpful to think of the lumped LC model as a transmission
line model simplified to the point where it is only valid at one
spot frequency. In this respect the two models are not vastly
different: they both use the same approximations that allow the
use of 'inductance' and 'capacitance' as simplifications of, and
approximations to, Maxwell's full set of equations.

[*] This statement needs a slight caveat.  If you're driving the
secondary by feeding RF current into the base, then you *will* see
something like 90 degrees of phase shift between the base voltage
and the top voltage (but not so the current).   Feed current into
the top of the coil and the reverse is true - uniform phase of the
voltage and 90 degree phase change of the current.  This is a
consequence of the necessity to force the coil into resonance
against the damping due to its intrinsic loss resistance.  At the
driven end, current and voltage are in phase with each other, whereas
at the free end, the two are in quadrature.  The voltage (base drive)
or the current (top drive) undergoes a rapid phase change over a small
fraction of the length of the coil at the driven end, this fraction
becoming smaller as the Q factor of the resonator is increased.

If we vote for none of the above and force the secondary into
resonance by say inductive or capacitive coupling to the coil as a
whole, we find some small phase change of both the voltage and the
current along the coil, occuring because a small in-phase component
is induced across the (distributed) loss resistance of the coil, which
mixes with the dominant resonant quadrature voltage and current of the
resonator.
--
Paul Nicholson,
Manchester, UK
--