# Tx line vs lumped parameter

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From:  R M Craven [SMTP:craven-at-globalnet.co.uk]
Sent:  Saturday, June 06, 1998 5:01 AM
To:  W Y Liu; Tesla List
Subject:  Tx line vs lumped parameter

Mr Liu and all,

Thankyou for your contributions to the Tesla Coil list. They are worthwhile
and valid so please do not feel uncomfortable about writing in!

I have a comment here about why the tx line theory falls down as measured by
Terry Fritz, but appears to be technically valid nonetheless. Everyone,
please skip to the end to read it, if you don't want to look at all of the
maths!

I agree with your transmission line comments to a point, but we need to
remember antennas such as helical antenna, and also UHF resonator
structures. These are both cases where the e-m wavelength is much longer
than the biggest dimension of the helix in question. In these cases, tx line
theory is applicable (see work by McAlpine & Schildnecht, ITT Radio Eng.
Reference Book, and also Schelkunoff, Adler Cu and fano, Ramo Whinnery and
Van Duzer etc etc).

(Admittedly, other slow wave structures such as TWT helices are operated
whereby the em wavelength becomes comparable with the structure size and
then tx line analysis becomes wholly dominant).

The point which many people miss is that tx line analysis makes several
assumptions, one of which I argue is dominant: we consider a line to have
some series inductance and a small series resistance, and these are shunted
by a parallel capacitance and a  parallel conductance. In the case of a
Tesla coil secondary, the series R is NOT small enough to ignore, but even
more significantly, the shunt conductance rises to significant values when
sparks occur.

Those who aren't interested in maths should skip this bit, but the
conclusions I suggest are worthwhile
(I hope  :')  )
Thus the voltage "dV" along a  bit of line of length "dx" drops by an amount
"dV":

dV = L dI/dt dx + IR dx by Kirchoff's law. Thus, dV/dx = -LdI/dt + IR

The shunt currents through C and G are considered: dI = -(C dV/dt +GV)dx

therefore dI/dx = -C dv/dt + GV

Now, V = V+ exp (jwt - (a+jb)x): missing out several steps we get Zo = root
(jwL + R/jwC + G)

But we always assume R and G are insignificant, hence the usual Zo = root
(L/C)

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>From this point on, Schelkunoff and others model their tx lines and come up
with notions of average characteristic impedance and so on. Interestingly,
work that I carried out some time ago compared the Schelkunoff
characteristic (Zschel) with the calculated rootL/C value Zo: the
measurements and calcs casrried out on four different coils showed Zschel
differed from Zo by about 25%, but at the moment I can't find my notebook.
Perhaps malcolm Watts has the results that I sent him a couple of years ago!

So, what is the point of all of this? Possibly none, if we pay attention to
the excellent work that Terry Fritz has carried out. I think the important
point to remember, actually stems from comments about determining the
self-capacitance of coils. I commented on this a year or so ago, by quoting
some bits from Terman.

Basically, if you MEASURE the self-capacitance of a coil by determining its
resonance by sweeping it, the results you get are different depending on
what approach you take: I refer to Terman, first second and fourth editions,
and I quote now from Terman and Pettit, "Electronic measurements", McGraw
Hill 1952.
(BTW the info in Radiotron Designers handbook   by Langford-Smith is a
regurgitation of Terman but with more info on coilwinding charts).

Terman and Pettit say on page 102 of the British reissue of the 2nd ed: (my
emphasis in bold, my comments in brackets)

"A method sometimes proposed for determining distributed capacitance
consists in measuring the resonant frequency of the coil when tuned only by
the distributed capcitance. This can be done by loosely coupling the coil to
an oscillator and observing the frequency at which the coil reacts upon the
oscilator to cause a sudden change in the grid or plate current (i.e. the
base impedance of the coil has reached a sharp null: prallel resonance!). A
knowledge of this self-resonant frequency and the true inductance of the
coil will permit a determination of an apparent distributed capacitance.
However, the capacitance obtained in this way will always be smaller than
when the same capacitance is measured by ...(tuning with external
capacitances and plotting 1/f squared or using C1-4C2/3). THIS IS BECAUSE
WHEN THE DRIVING VOLTAGE IS A DISTRIBUTED INDUCED VOLTAGE, THE VOLTAGE AND
CURRENT DISTRIBUTION IN THE COIL WITH NO EXTERNAL TUNING CONDENSER IS QUITE
DIFFERENT FROM THE DISTRIBUTION WHEN AN APPRECIABLE CAPACITANCE IS SHUNTED
ACROSS THE COIL TERMINALS.  For this reason the distributed capacitance of a
coil normally should not be determined by the self-resonant-frequency
method".

Earlier texts by Terman (Radio Engineers' Handbook, 1943 etc.) mention as
well that the effective distributed capcitance will also depend to some
extent on current distribution in the coil and will generaly be bigger when
the coil is tuned by an external tuning capacity.

So what?

1. Measuring self-capacitance of coils generates different results depending
on the method of excitation of the coil.

2. I reckon that a direct injected coil (extra coil in a magnifier) is being
"illuminated" from a point source: the electric field and magnetic field
coupling will be small or zero, and the conditions for tx line response will
be met (whereby you have a wavefront propagating into an uncharged area of
tx line as per the model described above).

3. I reckon that an inductively coupled coil will not show distributed (tx
line) parameters or at least, the dominant response under sparking
conditions will not be tx line response. Terry has shown that the dominant
response under non-sparking conditions is also NOT tx line. This is likely
to be because the area of "tx line" in front is already storing energy due
to the coil living inside the induction field of the primary.

The only way I can see to lay this argument to rest is to carry out Terrys
measurements on an isolated resonator: you could inject a CW signal from a
sig gen and carry out current measuremnts at the topload and at the base of
a resonator (secondary coil, I mean). If Terman and Pettit's discussions
above hold true, we should see that an "extra coil" or direct-injected
resonator behaves like a transmission line (under no-load, no-spark
conditions).

These comments are in support of the ideas originally suggested by Terry
Fritz and Dave Sharpe in last weeks' long post. If a magnetic field is
coupling into lots of the secondary instead of only into the bottom turn
(let's say), then the secondary won't look particularly "distributed" to the
primary. The e and H field coupling will store energy in lots of parts of
the secondary at any one instant: the concept of a wavefront has vanished
and with it, the concept of a distributed tx line. However, if that
secondary was direct injected, the wavefront propagation would hold true and
VSWR effects would live!

Richard Craven, Malvern, England.

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