Re: Pig Ballast -> description of complex conjugate of load (fwd)
---------- Forwarded message ----------
Date: Sun, 19 Jul 98 10:32:13 EDT
From: Jim Monte <JDM95003-at-UCONNVM.UCONN.EDU>
Subject: Re: Pig Ballast -> description of complex conjugate of load
>Date: Sat, 18 Jul 1998 15:16:34 -0500 (CDT)
>From: Larry Bud Melman <gasman-at-althea.a-line-dot-net>
>To: Tesla List <tesla-at-pupman-dot-com>
>Subject: Re: Pig Ballast Question (fwd)
> I have only a limited background in electronics and physics.
>Would you explain what the 'complex conjugate' of the load is?
>Sounds like you're talking about imaginary numbers... :-)
Yes, complex numbers...
Starting with the facts that voltage across an inductor is
directly proportional to its inductance and the rate that the
current is changing and the current through a capacitor is
directly proportional to its capacitance and the rate that the
voltage is changing, you can eventually show for steady-state
conditions with sinusoldal inputs that an inductor can be
treated as an imaginary number j*w*L and a capacitor as -j/(w*C),
where j is the square root of -1 (usually called i in math areas
but j in engineering because i is already commonly used to denote
current), w is the angular frequency, L is inductance, and C is
capacitance. Using these numbers, steady-state voltages and
currents in circuits with resistors, inductors, and capacitors
that have a sinusoidal input can be determined.
As a simple example, suppose you have a 1nF capacitor in series
with a 1k ohm resistor and you plug the unconnected ends of the
capacitor and resistor into a typical 120 V RMS, 60 Hz wall outlet.
Since w=2*pi*f, w=120*pi. The reactance of the capacitor will
be Z= -j/wC = -j/(120*pi*1E-9). The voltage across the resistor can
be found using a voltage divider VR = 120*(1000/(1000+Z)). Similar
calculations can be done for current. Notice that VR and the others
are complex numbers. The magnitude of the number is the voltage and
the phase is relative to the 120 volt input.
Relating this to Tesla coils, ballasting, and power transfer, notice
that wL is positive and -1/wC. Suppose you are given an L, such as
from a transformer. By choosing C appropriately, you can cancel out
the impedance from L. This cancelling is the key to resonant
charging and resonance in general for that matter. Using the complex
conjugate "removes" the reactive part of the load (either L or C) and
allows more current and hence power to be delivered to that load.