[NewCandle] Resonance and Circulation: an introduction

[Keith Nagel]

Hi All,

The understanding of the concept of resonance is fundamental to understanding
the underpinnings of the physical world. Yet the phenomena itself is often
shrouded in a mystery of mathematical approximations, sufficient to obtain
a result yet providing little insight into true nature of the subject. Because those
approximations are so simple they are often taught first to the student,
who is left with a rather poor and confused understanding.
The situation is analogous to the teaching of the Bohr model of the atom. The
model works, but it takes years of additional teaching to undo the 
inevitable conceptual misunderstandings that arise as a result... 

The approximations I am referring to are known as the lumped constant
approximations in the field of analog electronics. They assume that the
circuit elements are much smaller than a wavelength of the stimulus.
Consequently we can ignore the effects of wave propagation in the
components, substituting the idea of phase for the richer and more
encompassing ideas of time and duration. The relationship between
current and voltage becomes fixed; 90 degrees for reactive components
and 0 degrees for resistive components. This makes it easy to
perform computations at the expense of any real understanding of
the underlying reality.

As the frequency of the stimulus rises, this model becomes less and less
appropriate. All physical components have spatial extent, as a result
capacitors possess some induction and inductors possess some capacity.
It becomes necessary to use the distributed constant model and switch the
basic circuit component from resistor/capacitor/inductor to the
transmission line, which has all of these properties. It is this model
which will give us the proper understanding of wave phenomena in
resonance, and show that the relations between current and voltage are
quite fluid and directly due to wave interference.

Let us consider a demonstrator circuit that will show the relation between
current and voltage in resonance and how that can be manipulated to
produce both the out of phase resonance that all are familiar with and
the in phase resonance that I claim.

The circuit consists of a piece of transmission line formed into a loop.
The outer shield and inner conductor of each end are connected to the
same on the other end forming a continuous circuit. This circuit is a cavity and can
store energy at resonance. Because this is a distributed circuit, the
frequency at which resonance occurs is a function of the circumference
of the loop. The longer the loop, the lower the frequency. Additionally
there are modes of oscillation, related to the number of half wavelengths
that can be fit into the loop. The lowest order mode is a single half wavelength.

To achieve resonance we need to feed energy into this loop at the correct
frequency. Due to the fact that we are using linear circuit elements,
energy will want to come back out of the circuit through the same element that we
use to inject it into the circuit. This coupling element is critical
to understanding the limits of magnification in a physical circuit. When
the element is matched to the circuit impedance, all the energy of the
stimulus is transferred into the loop, and it all comes back out after
one half of a cycle. When the element is mismatched, less energy enters
the resonant cavity, but less still is returned to the driver. As a
consequence energy can build up over time, until again the point is
reached when the amount of energy coming out equals the amount coming in.

Our coupling element to demonstrate ordinary resonance will be a small
capacitor that connects the sine wave generator to the inner conductor
of the loop. Now consider what happens when we turn on the generator.
A wave is injected into the loop, and immediately begins to travel
in both a clockwise and counterclockwise direction around the loop. At
this point, the current and voltage in each wave are in phase. This
condition is called a traveling wave, and is the fundamental element
from which all other wave phenomena arise.

Half way around the loop the two waves will collide and travel over each other.
When they do, something remarkable happens. While the voltage component
of the waves are the same sign, the current components are oppositely signed
due to their different direction of travel. This results in the voltage
adding, and the currents canceling. As the waves progress and the system
reaches a new equilibrium, we notice two important facts.

1) Energy is no longer traveling in this circuit because the energy
in the clockwise wave is equal to the energy in the counterclockwise wave.
Thus the net energy flow is zero. The combination of two oppositely traveling
waves is a standing wave.

2) Current and voltage are no longer in phase. Using a current and voltage
probe, one can measure the standard 90 degree phase difference noted in
all lumped circuit resonators all around the loop.

One can see in this circuit that the underlying reality of the standing
wave is a pair of oppositely directed traveling waves. It is this that
produces the peculiar 90 degree phase relation in a resonant circuit.
This same phenomena can be seen in a straight length of transmission line
with an open or closed circuit at the end. The termination will result
in a reflection of the forward moving wave and a mixing of it with the
reflected wave resulting in the 90 degree phase shift.

Now we come to my claim. What if, instead of using a coupling element
which injects both a clockwise and counterclockwise wave into the loop,
we somehow could just inject a single wave moving in one direction or
the other. Now, when the head of the wave completes the loop, it will
be traveling in the same direction as the injected wave. Voltage, and
more crucially current, will be of the same sign. I assert that the
result will be very much like our first resonance, with the important
difference that everywhere in that loop when we make current and voltage
measurements we will see them both to be in phase. I call this form
of resonance a circulation to distinguish it from the 90 degree form
that we are all familiar with. In addition we note that rather than
creating a big standing wave, we have created a big traveling wave,
and as a result the energy is not stationary but moving around the loop.
Hence my calling this a circulation.

Fortunately for me, a magical single wave injector is close to hand. Most
ham buffs are familiar with the quarter/half wave coupler. This circuit
consists of a terminated length of transmission line in parallel with
a second unterminated line. When the length is correct and we drive
a signal into the first line, signal will emerge from one end of
the second line and _not_ from the other. You can find out more about
these couplers by doing some web searching, but for now just accept the fact
that such a device exists and has been used for decades in radio work.

Now in order to make my circulator, we break the loop and insert the
coupler in the place of the capacitor. When the generator is started
a traveling wave will be injected into the loop and begin to build
in strength as it circles the loop again and again and more cycles
are injected. Our loop length is matched to the wavelength so that the
waves add constructively, just like in the ordinary resonance.
At some point equilibrium will be reached and the circulation will stop
building. However, much to our surprise, when we make current and voltage measurements
around the loop, we see that current and voltage are everywhere in perfect phase.

In this new circuit we have two additional modes to consider, clockwise
and counterclockwise, along with the length related modes we saw
with the ordinary resonance. And if we were to mix both of those
modes in the same physical circuit, we get back to our old friend
the standing wave resonance.

I developed this idea in the late 80's when I was learning about electronics.
I started my study backwards, from transmission line theory to lumped
elements, and so developed quite a different understand of this material
than the mainstream teachings. Not that there is anything wrong with
those teachings, other than they produce in the student the attitude
that the circuit I describe above is "impossible" and that I am therefore
an uneducated kook. 

I mention this circuit publicly in an attempt to better understand the dynamics
of the educational process. The circuit is impossible, as any EE will tell you.
Now, what does it take to convince such a person that it is possible?
A description of the theory (as above)? A simulation of the circuit using
pspice? A physical model? I can and have done all of these things. Yet
people often have ready answers for all of these things. 

I'm sure we can all think of a few other choice "impossible" things that we
may one day have to prove to others. Perhaps from discussion of this we
can come to some sort of protocol concerning how to introduce a new
idea to those poor souls that have yet to discover the nature of science
and it's practice. The uni's seem to be producing more and more such
people every day, so it's worth a little thought here.

K.