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Sideband Production
Subject:
Sideband Production
Date:
Fri, 21 Mar 1997 12:46:35 +1200
From:
"Malcolm Watts" <MALCOLM-at-directorate.wnp.ac.nz>
Organization:
Wellington Polytechnic, NZ
To:
tesla-at-pupman-dot-com
Hi all,
Here's a brief note to try and explain this topic a little.
Before starting, I should explain that we have yet to explain the
sinusoidally related energy transfer/cycle. Accept that the
relationship exists. The mathematics says it does and it certainly
looks that way on the scope. An investigation as to why this
relationship is sinusoidal is currently under way.
Assume for the moment that the shape of the beat envelope is a
sinusoid. We have two frequencies present according to what is seen
on the oscilloscope; the resonant frequency and the frequency of
variation of its amplitude as the energy trade proceeds. A sinewave
of constant amplitude may be written as:
f(t) = A * sin(2*PI*Fr*t) where A is the peak amplitude (constant)
and Fr=resonant frequency and t is time
ASIDE: notations like f(t) simply mean "a function of time". END ASIDE
In the energy trading system, A is not constant but varying. The term
for A can be replaced by the second frequency mentioned above (beat
frequency), so let's define it as:
b(t) = B * sin(2*PI*Fb*t) where B is the peak amplitude of the beat
frequency and Fb is that frequency.
Combining the two gives: f(t) = b(t) * sin(2*PI*Fr*t)
= B * sin(2*PI*Fb*t) * sin(2*PI*Fr*t)
NB - this multiplication is exactly what happens in a mixer or
modulator in radio equipment. However, we are not modulating one
signal with another in the coil. The envelope shape arises as a
result of physical processes as mentioned above.
(ASIDE):
But, B is not constant either! The beat envelope itself decays with
each 2-way trade. On checking with various L/C combinations I found
that the shape of B is dictated by the lossiest part of the system.
For a primary where the gap is the main loss, B is a negative-sloped
linear function. If a poor capacitor or resistive primary coil or
both dominate losses, an exponential decay arises and this is also
true if a poor secondary is used or if excessively long interconnects
are used in the tank.
For the linear case, B(t) = -nx where the slope is -n
For the exponential case, B(t) = e^-nt
This really doesn't concern the explanation because we are interested
in what happens in the first beat but it does hint at more
complications if quenching is done away with.
(END OF ASIDE).
Let us say then that we are only interested in Beat Number 1. In this
case, B is constant. Now it so happens that our sin(x)*sin(y) form
above can be rearranged to give: 0.5*cos((x-y)/2) - 0.5*cos((x+y)/2)
This is a trig identity and has a formal proof - i.e. it is standard
fare.
The important thing is that we have two frequencies present,
neither of which is either of the original frequencies. These are:
x+y and x-y. However, it is not so cut and dried because this assumes
that the function is continuous. Things start to happen if we examine
this over short periods of time. To do this, those who have access to
a computer math program might like to pop the following equation in
and see what pops out as different values for t are plugged in:
f(t) = B/2 * [cos(2*PI*t*(Fr-Fb)) - cos(2*PI*t*(Fr+Fb))]
B can be an arbitrary value and therefore one can make B/2 = 1 to
simplify it a bit. Things to look for: what are the amplitudes of the
two new frequencies as time is increased from zero? What is the
amplitude of the two original frequencies as time is increased from
zero? To do this, the equation must be translated from the time
domain where amplitude is a function of time to the frequency domain
where amplitude is a function of frequency.
The upshot of all this is that the waveshape that the physical
processes give rise to is *equivalent* to that you would see if you
summed two disparate frequencies together. So the two side
frequencies are present in the system by virtue of physical act of
transferring energy from one circuit to another over several cycles.
Since the speed that process occurs at is dictated by k, the
resultant sideband frequencies are a function of k.
A final note: As k is increased in the 2-coil system, the
frequency Fr-Fb tends to zero, and Fr+Fb tends to infinity so there
is a difference between our situation and a modulator. This is
because both of the frequencies fed into a modulator are fixed but
the beat frequency in our systems varies with k. We can say that
for a given k, both frequencies are fixed but it is important to note
that in our case, Fb can never exceed Fr - not true of the modulation
case. If this happens in a modulator, it gives rise to a phenomemon
known as aliasing.
What we see on the scope is true to what is happening with
respect to time but tells us nothing about what is happening with
respect to frequency.
Malcolm