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The Behavior of Ideal Resonant Inductors

Jared Dwarshuis and Lawrence Morris


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
cthe speed of light
L0length of wire for full wavelength

Resonant Inductance

     <<resonant inductance formula>>
nnumber of nodes on inductor: 1/2, 2/2, 3/2, ...
<turns>number of turns in full-wave inductor
<area>area of each turn (meters2)
Hheight of full-wave inductor (meters)
CCapacitance (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>>
llength 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>>
llength of wire in full-wave inductor


In a nutshell, this says we must consider only a portion of the full-wavelength solenoid when calculating the resonant inductance and capacitance:
     1/4at one wavelength
     1/8at 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>>
Nnumber of nodes on rope; 1,2,3...
Ltotal length of rope
<tension>tension on rope
μlinear mass density of rope
Kspring constant of rope
Mtotal 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


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


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

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


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