Re: Improved Model for a Primary Charging CKT

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

Original poster: "Antonio Carlos M. de Queiroz" <acmq-at-compuland-dot-com.br> 

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

 > 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.

Is someone using reactance as resistance?

 > 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.

Surely. But things are somewhat more complex. See below.

 >...
 > 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.

This is steady state sinusoidal analysis. Valid only if you assume that
the spark gap never fires and wait several cycles until all the
transients decay.

 > 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

w<<1/sqrt(L*C)

 > 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]

The actual maximum is a bit larger than Vs*Q = Vs*sqrt(L/C)/R, at a
frequency slightly smaller than 1/sqrt(L*C) rad/s.

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

w>1/sqrt(L*C)

 > 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).

 > Please comment (especially ANTONIO),

Your derivation is essentially correct, unless for the errors pointed.
Some problems are:
R would have little effect in a practical transformer, but it's not a
big problem to include it in the calculations.
The presence of a spark gap complicates the analysis substantially.
The circuit would never reach the steady state assumed in the
calculation
between "bangs".

A more precise analysis must consider a transient response caused by
the AC source Vs and the current in the transformer inductance at
the end of the last gap firing. This solution would include the forced
response due to Vs as calculated above, added to a decaying oscillatory
waveform caused by both IL and Vs. The obtained solution would be valid
until the next gap firing, where the capacitor voltage would return to
zero (if you don't complicate considering the fast energy transfer
transient to the secondary circuit too) instantaneously, and the
process would repeat. The gap firings can be (almost) periodical in
a rotary spark gap, that can be syncronous with the power line or
not, or can be determined by Vc in a static gap.
In the general case, the obtained output voltage Vc(t) would be not
periodical, and very difficult to predict exactly without a simulation.
It's not difficult, however, to write a specific simulator for this
problem. (If you are interested in details, contact-me directly.)

Antonio Carlos M. de Queiroz