Improved Model for a Primary Charging CKT

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

Original poster: "Gerry Reynolds" <gerryreynolds-at-earthlink-dot-net> 

Hi,

One of the items on my wish list for TC designer tools is to be able to 
handle (at least to some extent) the effects charging resonance has on 
charging time, charge levels, and BPS calculations.  To me, the ability to 
take charging resonance's into account becomes more important for LTR 
designs.  The current tools (at least the ones I use), calculate the 
transformer's impedance by computing Vs/Is (usually name plate data).  This 
impedance is assumed to be all inductive for purposes of calculating Cres 
and is used as a resistance (as in computing RC charging time constants) to 
determine the time to charge Cp to the sparkgap breakdown voltage (like for 
static gaps) and from this, to compute the BPS.

This model has always bothered me because the charging is not an RC circuit 
and the equation Vc = Vs(1-e^(-t/RC)) doesn't seem appropriate for this 
purpose.  Also, the excitation is AC and not DC.  Maybe for STR designs, 
this works OK.  But for us NST users, this approach seems to break down.

The following proposal models the charging circuit as an LRC circuit, with 
the purpose of introducing the charging resonance into the circuit's 
behavior.  The L and R come from the NST and the C is the Cp of the tank.

The transformer model is based on a four element model instead of a more 
accurate six element model because, I believe, the measurements required 
are more within reach of the novice.  Using this model, primary and 
secondary resistance's will need to be measured and added to the name plate 
data.  More about this later.

The LRC circuit will have both forced (50 or 60Hz response) and natural 
(freq associated with the LRC circuit) responses.  The following approach 
is based on the steady state forced response only as adding the natural 
response stimulated by the spark gap firing may require use of computer 
simulations. Notably, this may be a weakness of this approach and I do 
solicit the opinions of ANTONIO and PAUL as well as other experts.  My hope 
is not for perfect accuracy, but for a model that will better reflect 
reality when using a Cres or LTR value of Cp.


Transformer Model:

With the four element model, Rp and Lp will be series elements on the 
primary side, and Rs and Ls will be series elements on the secondary 
side.  Primary elements can be transformed to the secondary side by using 
the turns ratio (n) squared (see richieburnett's web site):

L = Lp * n^2  +  Ls
R = Rp * n^2  +  Rs

The four element model can then be distilled to a two element thevenin 
model that has a voltage source (Vs) driving the L, R, and C of the 
charging circuit.  Rp and Rs can be measured to compute R.  L can be 
deduced from the total transformer impedance (that I believe represents the 
current limiting affects of Rp, Rs, Lp, and Ls) using the following:

Z = Vs / Is  (as is currently done)   [name plate or measured data]

Z = sqrt (XL^2  + R^2)  therefore,

XL =  sqrt (Z^2 - R^2)

L = XL / (2*pi*freq_line)


Charging Circuit Analysis:

This part requires EE circuit analysis skills and utilizes the Laplace 
transforms,  in particular  ZL = sL   and Zc = [1/sC]  where ZL is the 
impedance of the inductor, Zc is the impedance of the capacitor, s = jw (a 
complex variable), j=sqrt(-1), and
w = 2*pi*f.

Given this, the steady state capacitor voltage is:

Vc = Vs * Zc / (ZL + R + Zc)

substituting the "Laplace impedance",

Vc = Vs * [1/sC] / (sL + R + [1/sC])

multiply both numerator and denominator by s/L and you get

Vc = Vs * [1/LC] / (s^2 + sR/L + [1/LC])

substitute s=jw, realizing s^2 = - w^2 because of the j, collecting real 
and imaginary terms and you get

Vc = Vs * [1/LC] / ([1/LC] - w^2 + jwR/L)

The magnitude of Vc is then:

|Vc| = |Vs| * [1/LC] / sqrt { ([1/LC] - w^2)^2 + (wR/L)^2 }

If Vs = Vs_peak, then Vc = Vc_peak.

Plot:

for w<<[1/LC], Vc = Vs

for w = [1/LC], R will limit the peak voltage to Vs * L/R  [note if R=0 
then Vc = infinity and L/R is the Q of the circuit]

as w > [1/LC] the voltage starts decreasing from peak at resonance and 
eventually becomes smaller than Vs.

With LTR values, it is easier to see that we can still have a fully charged 
Cp at 120 BPS if chosen correctly  (see Terry Fritz's derivation).  I have 
measured Vcp for various LTR values with the TC secondary and sparkgap 
removed (to eliminate introducing natural responses) and the results agree 
closely with this equation.

Please comment (especially ANTONIO),

Gerry R
Ft Collins, CO