_{L}= -X

_{C}. This "cancels out" the reactance (resistance to change of frequency), leaving only the DC resistance. All tuned circuits work this way.

However, you cannot just "cancel out" things like reactance or resistance--if I have 20 ohms of inductive reactance, and 20 ohms of capacitive reactance, I should have 40 ohms of reactance--NOT zero. Where does it go?

I noticed a similarity to birotation, where two oppositely-spinning, parallel rotations cancel each other out, leaving a cosine wave. This is calculated by Euler's formula, (e

^{iθ}+ e

^{-iθ})/2 = cos(θ). This is basically the same concept--two "opposites" that cancel each other out, leaving a lower-dimensional structure (two rotations become a wavy line, or the complex quantity of reactance gets reduced to a single, real component).

Wondered... what if this is the SAME THING? That would infer that reactance is

*angular*(a twist of an axis), not

*linear*(a translation along the impedance axis).

Resonance can only occur with AC (alternating current), not DC (direct current). That means the current is cosine wave (or sine; in the RS, things start with "1" like a cosine, rather than "0" with sine, so cosine is more accurate). That cosine wave can be the result of two "reactance" rotations--which ARE expressed as "imaginary numbers," which we now know to be "rotational operators" expressing an angular velocity (not "make believe" numbers).

Typically, resonance is plotted as an axial displacement, like this: When X

_{L}= -X

_{C}, the reactance is canceled out, leaving only the resistive component. But what if it ISN'T axial, but ANGULAR, as shown on the X

_{LC}arc?

Consider: if I am running on a treadmill, I am running forward at 8mph as the treadmill is sliding backwards at 8mph. If I use the LC formula, 8 + -8 = 0 ... I should not be expending any calories, since I'm not moving! (And I KNOW that isn't the case!) The net motion to the gym is zero, as I don't go flying off the treadmill, but the energy involved is still going, full blast.

That means in an LC circuit, at the resonant point, that "impedance" must be going somewhere else where it is fully active. What if this "2D problem" on graph paper, is actually working in 3D? Once can get the SAME net effect of zero reactance, simply by rotating the resistance axis (DC), to move the projection of the motion on the reactance axis (AC) to zero? You now have the full displacement on the IC axis, zero on the AC axis, and the regular resistance on the DC axis. It

*appears*the reactance has been "canceled," but has just rotated into another dimension.

The same may be occurring with birotation, omitted by Euler because he was only considering a 2D problem--the birotation may be transforming into a quaternion rotation, where the rotational axis is moving into a 2nd, scalar dimension that IS NOT coincident with the reference system--and hence, fully functional but unobservable.