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Re: non-dissipative circuitry



Regarding hooking a low-impedance high-Q circuit
across the AC mains,

Bernard Cleyet wrote:

I tried this once -- of course the circuit elements were less than
ideal -- fortunately the circuit was fused!! (The 30 A. circuit
breaker)
...
> "A little knowledge is a dangerous thing."

Yes, dangerous indeed.

Acually the less-than-ideal circuit elements
aren't the problem. The dangerous result is
well predicted by theory.

Consider the (R series L series C) circuit as
previously discussed. Let L and C be constant,
and improve the Q by taking the limit as R
becomes small.

Using the usual formula for complex impedances
in series, we find the that impedance of the
(R L C) circuit at resonance is just R, since
the inductive impedance plus the capacitive
impedance add to zero at resonance.

Such a circuit attached to a constant-voltage
source will draw a divergently large amount
of current from the source. The resonant
currents _inside_ the circuit will be even
larger than this, but usually the circuit
breaker (or something else :-) will pop before
you get a chance to observe this.

====

Constructive suggestion: when high-Q circuits
are used in practice, it is traditional to
have some sort of _weak coupling_ arrangement.

As an easy-to-visualize mechanical analog,
imagine a high-impedance resonator such as
a long, massive pendulum: an elephant on
a large swing-set. You can excite this
to large amplitude by pushing. However, you
will have to be clever and/or careful.

Suppose the elephant is already at a moderately
large amplitude, and you want to keep pushing.
If you stand at the midpoint of the arc, you
won't be able to do any useful pushing, because
the elephant will whiz by too quickly. If
you try you'll just get hurt.

*) One fairly good option would be to reposition
yourself to the end of the arc, where the elephant
is nearly stationary. There are disadvangates:
-- You will only be able to push during a tiny
percentage of each cycle, and
-- you will have to repeatedly reposition
yourself as the amplitude changes;
... but you can get the job done this way.

*) Another option would be to stand to one
side and use a long _lever_ to push on the
elephant. You want to have a mechanical
disadvantage scales (roughly) like Q. That
is, if the mass and restoring-force remain
constant, and the dissipative term decreases
by a factor of 2, you want 2x more mechanical
disadvantage. The resulting amplitude will go
up by a factor of 2, but the motion and force
you exert at your place on the lever is unchanged.

The electrical analog of this lever is called
a transformer.

Usually high-Q coils have a tap near one end, so
they can be used as autotransformers. The
mechanical analog of this is to climb up and
push on the pendulum somewhere near the pivot,
so it provides its own leverage.

Another matter: The resonance formula is derived from diff. eq. that
has transient and steady state solutions. Does this have any effect
on this use of the resonant circuit?

Yes.