The Behavior of Ideal Resonant Inductors

Jared Dwarshuis and Lawrence Morris 20-March-2004

Introduction

We developed the following equations to describe the behavior of our resonant transformers . However, these results apply to all wire length dependant resonant inductors.

Scope of Application

We concern ourselves with ideal resonant inductors, whose operating description fits:

L₀f₀ = λ₀f₀ = c

Where:
c	the speed of light
L₀	length of wire for full wavelength
f₀	frequency
&lambda₀	wavelength

Resonant Inductance

Where:
n	number of nodes on inductor: 1/2, 2/2, 3/2, ...
<turns>	number of turns in full-wave inductor
<area>	area of each turn (meters²)
H	height of full-wave inductor (meters)
C	Capacitance (farads)

The capacitance (C) is chosen to satisfy the equation.

Example: quarter wave

If we want a quarter wave, we set n = 1/2, because we are representing 1/2 of a node. Our expression would read:
<<quarter-wave math>>

Where:
l	length of wire in 1/4-wave inductor

Example: full wave

If we want a full wave (like in a Saskia's coil), we set n = 2, because we are representing two whole nodes. Our expression now reads:
<<full-wave math>>

Where:
l	length of wire in full-wave inductor

Interpretation

In a nutshell, this says we must consider only a portion of the full-wavelength solenoid when calculating the resonant inductance and capacitance:

1/4	at one wavelength
1/8	at two wavelengths

This should not be too surprising: resonant systems exhibit periodicity.

Derivation of Correspondence

Now, we are ready to show a correspondence between the resonance model for ideal ropes and the resonance model for ideal resonant inductors.

Results of Rope Resonance

The results of rope resonance can be summarized by the following chain of equalites:
<<rope results>>

Where:
N	number of nodes on rope; 1,2,3...
L	total length of rope
<tension>	tension on rope
μ	linear mass density of rope
K	spring constant of rope
M	total mass of rope
<velocity>	wave velocity along rope

Mapping Rope Resonance to Inductor Resonance

When we map rope resonance to inductor resonance we must account for the fact that the wave velocity is always the speed of light:
<velocity> = c

Substitutions

Now, by definition:
     <velocity> = λf
     ω = 2πf
And, since we are talking about a resonant inductor:
     L = λ

Conclusion

After substituting all of this, and simplifying, we get:
ω_n = nω₀
This is the expected result.

Because this derivation is based on a chain of equalities, the correspondence is completely bidirectional.

Commentary

Our design axiom has always been that there are waves of energy travelling at the speed of light. Current theory describes velocity inhibited waves. We believe the two descriptions can be reconciled with suitable relativistic arguments.

Our resonant transformer designs

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