Tx line vs lumped parameter

From:  W Y Liu [SMTP:eenwyl-at-sun.leeds.ac.uk]
Sent:  Saturday, June 06, 1998 2:33 PM
To:  tesla-at-pupman-dot-com
Subject:  Re: Tx line vs lumped parameter

> ----------
> 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 transmission line theory can still be used for low frequency application. 
But the standing wave effect is relatively small. For example, if lamda is
far greater than L, all the tangent terms of the transmission line equation
become close to zero, in which case the input impedance Zin to the medium of 
transmission is just the load impedance, ZL. 

> 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.

In dealing with low frequency application, it is usually the damping effect,
coupled with other reactive elements, that affect the network performance the 
most.  But in high frequency application, the damping effect is relatively small
compared to the overall reactance, if the medium of transmission is not air,
but copper wire.

For example, if  the network impedance is given as

     Z = R + j ( wL) ----------------------(1)
where R is the resistance and 
     ( wL) is the inductive reactance dependent on angular frequency w,   
The second term of equation (1) becomes dominant when frequency, w,
become very high, eg. a few giga radians.  In this case, ( wL ) >> R.   

So, in high frequency application, if the damping element in the network 
is mainly copper wire, some people just ignore the value of R and just equate 
       Z = j( wL )  ------------------------(2)
to work out the approximate impedance at a particular frequency, w.
Otherwise, when w is small, say a few hundred radians, the value of R should be 
taken into account in calculating the impedance Z.  

Yes, you are right. If the chief signals that co-oscillate at resonance
in the secondary of the tesla coil are in mega hertz band, and if the coil
is not inductive enough to compensate the damping effect, R and G should be taken
into account to minimize the effects of the standing waves.   But I am not a 
coiler and I am just answering your question by guessing the answer.

> 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)

As is explained earlier,  R and G are relatively insignificant compared to dI/dt
and dV/dt, only when the frequency is very high, say a few hundred mega radians.

> ****************************************************************************
> **************************************
> >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!

In high frequency measurement, say up to a few hundred megaradians, Zo can just be 
root(L/C).  But in low frequency measurement, R and G become dominant, and
somehow standing wave does not even exist! 

But don't confuse that the linear characteristic impedance is frequency dependence. 
In some cases, like long distance power transmission, the frequency is low but
the standing wave is still there because of lamda <= L.  In this case, R and G
are also dominant in calculating characteristic impedance.

I don't know how this 25% difference was arrived.  Honestly I am not qualified to make any 
conclusion with regards to this Zo measurement. 
Perhaps you could tell me more with
some drawings how you made the measurement. I hope I could share with you my info,
but I am not a practical man !