Friday, July 1, 2011

Tensor Model of Impedances

So what's so special about a flywheel suspended inductively? The key is that if infact gyroscopes show inductive behavior that has certain implications in tensor mathematics. This mathematics might be used to understand how the machine may achieve flight analogous to how certain electrical circuits resonate EM waves at certain frequencies or transform voltages and currents up and down etc. In other words tensor mathematics can be used to analyze the machine even as we are still building it and help us in better designing it.

We must assess the nature of impedance (inductance and capacitance are the two kinds of impedance found in nature) in a mathematical way in relation to other physical entities. Tensors offer a way of understanding impedances that sets them naturally apart from other electrical parameters. As we move forward, I have quoted so extensively from Kron in his book Tensors For Circuits that I have found it expedient to italicize his words to keep my train of thought comprehensible to myself.

Wherever in this entry there is reference to voltage, the reader should be aware that it represents also velocity in the analysis of purely mechanical or electro-mechanical systems and current to angular acceleration. See my entry on the analogy between mechanical and electrical behavioral parameters.

Further, as my entry on Grabriel Kron makes it clear, he himself saw all his analysis as being fully applicable to mechanical machines, indeed all and any machine. Thus, although we are flying through Kron's tensor model of electrical rotating machinery, we are also able to trace in the same model, information regarding our own machine. The flywheels suspended the way they are in the new prototype are the live inductors with a pulsating torque going through them at a certain frequency and the rest of the machine that is spinning under the influence of the Y-axis torque is the capacitor being charged up and their combined impedance is related to the velocity and the acceleration of the overal system as velocity = Z*acceleration (analogously to voltage, e = Z*i in the electrical scenario), where the impedance is a term with time units in the mechanical scenario. A single closed network in the electrical model can be envisioned as the mechanically invariant closed system of components capable of expending energy through torque or other methods in a different kind of machine.

Impedances are mathematically reprensentable as tensors of valence 2. The tensor of valence 2 is a collection of 2-way matrices describing a physical entity. A tensor of valence 1 like voltage e or of current i is called a vector.

Impedances thus form their own class of devices of valence 2 whose tensor laws of transformation require two transformation matrices C1 and C2.

Energy is a tensor of valence 0, i.e. a scalar. A vector of valence 1 is a vector such as a voltage or current and they require only one transformation matrix C, power or energy requires no C's. Because of this "chemical" property of a tensor of attracting a different number of C's the expression "tensor of valence n" originated.

Impedances, being of valence 2, would have transformation tensors with 2 C's, C= C1*C2.

Now, if the variables describing a specific impedance have been changed from i to i' by C1, then from i' to i" by C2 , then again from i" to i"' by C3., the successive transformations may be performed in one step with the aid of one transformation tensor C= C1. C2.C3... This important property of C is called a "group property". (Kron, Tensors for circuits, "WHy tensors")

With the help of a group transformation tensor C, one may model an inductor's behavior as follows.

The fact that power, P is a valence 0 tensor means that it does not undergo transformation and doesn't need a C to transform it from the reference frame of one machine to another. Kron uses this property of tensors to first extract information about the behavior of voltage in such a system.

The law of transformation of the voltage vector may be found from the physical fact that in going from one reference frame to another the instantaneous power input e*i (a linear form) remains unchanged or "invariant". That is P= P' or e*i = e'*i'. This relation is the physical link that connects all networks together (for any given machine)
Now let the current change from i to i' by i = C*i'
Substituting, e*C*i' = e'*i'
Cancelling i' e*C = e'
Hence e' = Ct*e
and e = Ct(inverse)*e'
It should be noted that though both e and i are vectors, they are transformed to a new reference frame in a different manner. But both being tensors for valence 1, they require C once only.

Kron then follows this up with a brilliant analysis of the impedance tensor itself.

Tensor analysis requires that if the equation of a system in one reference frame is e= Z*i, it should have the same form in every other frame. This property will give the law of transformation of Z. In the old reference frame let e = Z*i
Express i and e along the new reference frame. That is, replace i by C*i' and e by Ct(inverse)*e'.
Ct(inverse)*e' = Z*C*i'
Multiplying both sides by Ct
e' = Ct*Z*C*i'
If the following definition is introduced as the law of transformation of Z
Ct*Z*C= Z'
then the equation in the new reference frame becomes e' = Z'*i'
The equation of the new system is of the same form s that of the old system...

In summary, Kron proved that, if there is at all an impedance in the system, then an analysis of the system at any and all levels must involve an analysis of the impedance, with the invariant form of the law of impedance, e = Z*i (where e is voltage or velocity depending on the machine and similarly, i is current or angular acceleration depending on the machine).

We can see darkly from this statement, for example the answer to the questions: Why could this prototype fly? or Why does a gyroscope precess?

The answer, Kron's discoveries seem to say, is that tensor theory mandates that e = z*i is a permanent invariant law in any system involving an inductor or a capacitor (sources of impedance). So then we may further propose that
a) the purely capacitive impedance condition is addressed in traditional newtonian rotational dynamics,
b) the purely inductive impedance condition is met by gyroscopes. (This was something Eric Laithwaite partly recognized when he stated that the gyroscopes are like inductors. He also said that gyroscopes do not do any work. They rotate without being able to accomplish any energy transfer on a meaning level. It is clearer now that, that is because the system is purely inductive without any significant capacitance. The gyro rotates in a set of gimbals that isolate the flywheel's spin so that there are no levers connecting it to the external frame.)

c) the LC resonance condition is what is being targeted in the new prototype's tests.

Kron also gives certain guidelines for machine analysis. For example when coils, beams, wheels etc are connected into an engineering structure, the constrained reference axes are ignored.

Using his methods and his modeling techniques, I will attempt an analysis of the prototype in the next few entries. One of Kron's models is a good starting point for that analysis and is given below.

Experiment Update: Repairs complete. Testing begins tomorrow.

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