Re: Parallel and Series LCR Circuit Qs

["Antonio Carlos M. de Queiroz" <acmq-at-compuland-dot-com.br> (by way of Terry Fritz <twftesla-at-uswest-dot-net>)]

Tesla list wrote:
> 
> Original poster: "Gavin Dingley" <gavin.dingley-at-astra.ukf-dot-net>

> I have a question regarding basic ac theory, something I thought I had
> sorted, but this does not appear to be the case. I was looking at
> building a simple crystal set and so had a look at what the ideal design
> involves. Apparently you should use more capacity and less inductance,
> that is, a coil wound from fairly stout wire and a large tuning
> capacitor.

Consider the following: 
A pure LC circuit, irrelevant if series or parallel, resonates when 
the inductive and capacitive reactances cancel each other. This occurs 
at the frequency w=1/sqrt(LC) rads/s.
With just one resistor in the circuit, that can be in series with the
LC elements or in parallel with them, Q is defined as:
Q = |inductive or capacitive reactance at resonance|/series resistance =
  = sqrt(L/C)/Rseries
Q = parallel resistance/|inductive or capacitive reactance at resonance|
=
  = sqrt(C/L)*Rparallel
So, the definition that you use depends on where are the most
significant
losses in the circuit.

A crystal set is usually made with the antenna and the detector in
parallel with the LC tank (maybe connected at taps of the inductor, what
allows some control of the Q). The second definition appears to be
more adequate. And you don't want a series resistance in the inductor
to worsen the loading problem.

> Now, am I right in thinking that the Q of a parallel tuned circuit is
> relevant to current magnification, that is, you want as much current to
> flow through the coil and capacitor so as to have a large Q. If this is
> so, then this explains this design for crystal sets.
> However, I am no left wondering about tesla coil secondaries. I was
> under the opinion that a secondary resonator needed to have a fairly
> high Q; I also thought that the secondary resonator was a series tuned
> circuit. If this is the case, then there should be more inductance than
> capacitance to get a high Q-factor. However, from what I have read (an
> understand) the design dictates a large top capacity and relatively
> small inductance; this resulting in white streamers showing a relatively
> substantial current.
> I once built a TC that had a large secondary inductance and small
> capacity, as per a high Q series resonant LC circuit. This TC produced
> nice violet streamers, indicating very little current but high voltage;
> was this poor design?

The reason to have large terminal capacitance is to have more powerful
streamers/sparks, as they are fed from the energy available there.
With small top load you can obtain very high voltages (but you need a
large top load to avoid premature breakout...), but no powerful
sparks because there is no charge immediately available for them.
A large top load also helps to keep the Q of the resonator high when
there are streamers attached to it (load in parallel with the LC tank),
after the primary gap opens (no sparks to ground, as they discharge the
terminal immediately). The series resistance of the coil is only
an important factor in the determination of the Q if there is no
breakout at the terminal, as while the primary-seciondary energy 
transfer is occuring, before breakout.
 
> I guess what I'm asking is what makes a high Q in a parallel and series
> tuned LCR circuit. Also what physical parameters does this dictate, i.e.
> number of turns and thickness of wire in the inductance.

The equations above answer these questions. Note that the difference
between both forms, if the tanks are closed and not connected to other
circuits, is just where the loss is. 
By the way, with both series (Rs) and parallel (Rp) resistance in an 
closed LC tank, Q results as:
Q=sqrt(1+Rs/Rp)/(sqrt(C/L)*Rs+sqrt(L/C)/Rp),
or approximately Qseries//Qparallel if Rp>>Rs. The lowest dominates.

> Another question is regarding primary and secondary TC circuits. As far
> as I understand it, the primary is a parallel tuned circuit, while the
> secondary resonator is a series tuned circuit.

As both LC tanks are closed, I would say that the primary works as a
series circuit, with the main loss in the spark gap, that acts
as a series resistance, and so high Q requires large inductance and
small capacitance. For the secondary, if you want efficient energy
transfer before breakout, you need low series resistance, and so it
behaves also as a series circuit. After breakout, the heavy load
is in parallel (streamers), and so it behaves as a parallel circuit.
 
> These are pretty basic questions (I think?) but I am not sure of their
> answers.

Note that a Tesla coil is actually a 4th-order system, with two
different
resonances, and so discussions about primary and secondary Qs don't
have a so direct relation with the actual behavior of the circuit.
The magnetic coupling mixes both in a complicated way.

Antonio Carlos M. de Queiroz