Re: Energy Equations For LC Standing Wave Resonance

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Original poster: Jared E Dwarshuis <jdwarshui@xxxxxxxxx>


What is inductance?

L = u Nsqrd Area/ Solenoid length

Or:

L = u Nsqrd pi (r)sqrd / Solenoid length

Multiply numerator and denominator by 4pi

L = u [Nsqrd 2sqrd (pi)sqrd (r)sqrd]/ 4pi  Solenoid length

L = u [ 2pi r N]sqrd/ 4pi Solenoid length

But [2pi r N] is the wire length.

Then:

L = u [wire length]sqrd /4pi  Solenoid length


Now:

Let the total sum of capacitance for a series LC system be represented
by a spherical capacitor with the second plate at infinity.

Capacitance = 4pi e R

Then we can write:

w = 1/ sqrt ( u (wire)sqrd / 4pi Solenoid length x 4pi e R )

We cancel 4pi and remove the constants u and e from the radical as C
(the speed of light.)
We remove the (wire)sqrd from the radical.

We get:

w = C/wire length  x sqrt (solenoid length/ R)

>From classic description we can see that our units for frequency 1/s
comes from C/wire length.

But this is still an incomplete description, it does not describe
multiple node formation observed in inductors at high frequency under
conditions of standing wave resonance.

We will elaborate?

LC resonance was modeled after mass spring resonance, they share the
same differential equation.  For low frequency applications the
solutions for LC resonance are essentially correct, they match the
lumped analysis of a mass and spring. But at high frequencies with
inductors whose wire length is some multiple of a quarter wavelength
of the LC resonant frequency, the rules change considerably.

We examined rope resonance as it paralleled mass spring resonance,
both sharing the characteristic of a clear delineation between kinetic
and potential energy.

  w rope = N pi sqrt(K/M) = N pi  Velocity / length

Normally with a rope N = 1,2,3? but we argue that with a sliding
mechanism at one end of the rope we could get N = 1/2, 2/2, 3/2?..


With a standing wave in a rope N sqrt(K/M) and N Velocity/length are
already equal, as they must be. But for standing wave resonance in an
inductor we must create this  equality by ensuring that the length of
wire in the inductor matches the wave length frequency of LC.

We must realize that current and voltage under standing wave resonance
partitions the inductor using (n) as follows:


 w = n/2  2pi C/ wire length = 1/ sqrt( u (Turns/2n)sqrd Area
2n/Length  [Cap] )
 where n = 1/2, 2/2, 3/2 ?.

Example:

With a quarter wave where n = 1/2  (1/2 a node represented) we get:

2pi C/4wire = 1/ sqrt LC   or C/ 4 wire = 1/ 2pi sqrtLC

This is Tesla?s classic quarter wave formulae. The rope resonance
model has allowed us to extend Tesla?s formulae to the general case.

Example:

With n = 2 we can take an inductor of a given length and make it a
full wave.

w = 2 pi C/ wire = 1/ sqrt ( u (Turns/4)sqrd Area 4/Length [cap] )

When we make an inductor a full wave, our inductance is now based on
only one quarter of the solenoids length and one quarter of the number
of turns.

 Justification:

We can look at inductance as:       V (a to b) = -L (di/dt  a to b ),
we always examine the same region (a to b) where the voltage drops
and changing currents are found.
  .

We argue that  under standing wave resonance, the changing currents
(including both conduction and displacement currents) within the
inductor and changes in potential that define inductance are always
found in the quarter wave region of a uniform inductor, a direct
consequence of the mathematics of standing waves.

Self capacitance is also found in the same quarter wave region, if you
like you can think of self capacitance as being analogous to the
stretching of a rope between the nodes.
 ( rope stretch is usually ignored)

Perhaps it helps to think of inductance as having a density, much like
a ropes Mass/ length.

You can take a fixed length of wire and wind it with a constant pitch
around cylinders with different radius. Each cylinder will have a
distinct L and a distinct L/H,  only L x H remain a constant.


Question? Experts claim that the wave velocity is less then C down a
wire. How can we reconcile this?

Answer: We believe that the phase velocity is always C but group
velocity does not have to be C unless we arrange for it. Phase
velocity is not measured by a scope, only the group speed is
measurable.

Question? How is rope resonance different from LC standing wave
resonance?

Answer (1): With a rope a node is a place where we find no motion. We
call the bump part of the rope an antinode With an inductor we call
both positions nodes. The current node is equivalent to a ropes
stationary node but the voltage node is equivalent to the ropes anti
node.   (word differences)

Answer(2): We lose a degree of freedom in the LC world. We cannot
change tension, our velocity is fixed at C

Modeling Resonant transformers after the energy equations of rope
resonance has allowed us to develop an array of new coils, all of them
working directly from the drawing board.

Respectfully:
Jared Dwarshuis and Larry Morris
July 9, 2005