Saturday, February 19, 2011

Would Kron Recognize The Idea Behind The Relativistic Machine?

When one realizes the enormity of his work, one might wonder why Gabriel Kron's name isn't more commonplace than it is. An Electrical Engineer by education and apprenticeship, he formulated an equation that applies to all types of machinery ever designed and ever will be designed in the future. He then used this tensor equation to analyze and systematically design induction motors. He is also responsible for a method called 'Diakoptics' which he used to design the automatic electrical load flow distribution system of the state of New York, the first such system anywhere in the world. However, one might wonder endlessly about a great many things that have come to pass, like for example, the strange fact that Kennelly's & Steinmetz's introduction of the concept of impedance didn't give rise to a wider scientific discussion of the nature of induction in electrical coils and a search for its mechanical analogue beyond the simple spring-and-block arrangements of the early mechanical physicists.

Throughout the later half of the 19th and the early half of the 20th century, so many important scientific discoveries were happening in so many different fields that one may be forgiven for pleading ignorance in certain esoteric areas. If the field is too vast however, one must seize aspects of the science that help us to focus on understanding the general plan of the field before plunging into the details as this leads to the common syndrome of being unable to see the forest for the trees. In this endeavor, Kron's Tensor Equation will be your everlasting friend, willing to venture with you in your searches into the most exotic designs for machines of your choice, no matter what your field or your time - always helping you design better machines by teaching you how to classify the components that form the machine. By neatly classifying the underlying components of the machine in question into purely inductive, resistive and capacitive components and using his trademark analytical method, Kron is able to set up equations and analyze everything from say, a steam turbine governing system (divided into the governor, the linkage, the pilot valve, and the turbine) to the electric speed drive/control (divided into the synchronous motor, the induction motor, and the stationary network).

Although his field of work was circumscribed by his assignments at GE (General Electric, where he worked), Kron sensed that he was playing with something much bigger than 'just' Electro-Magnetism. In reading Kron's work, I am convinced that he would approve of the relativistic machine's theory and design and see that it meshes with his own thinking. The following is an attempt to cite evidence for this supposition.

Inductive Angular Momentum And Kron's Tensor Equation

Gabriel Kron is the author of a method of analysis of rotating electrical machinery, in which one and the same tensor equation applies to ever conceivable type of machinery. - P. le corbellier in the preface to his book on the analysis of rotating electrical machinery.

IEEE Transactions On Circuit Theory, September 1968 noting the passing away of Gabriel Kron summarize that "Dr. Kron was the author of the classic paper entitled, “Non-Riemannian Dynamics of Rotating Electrical Machinery,” that became the basis of his theories covering all types of rotating machines and power systems. The pioneering work of Gabriel Kron demonstrated convincingly the superior organizational powers of the matrix-tensor notation in network theory."

All types of electrical machinery can be analyzed using the one tensor equation that Kron discovered in his intense theoretical and experimental forays. When one first come's across this promise in his work, one might wonder what kind of an equation can accomplish this mammoth task. There is however, more than just an equation involved in the analysis of a given machine. The equation is only relevant if you first break the machine down into components according to the rules that Kron gives, and in addition you set up the tensors for the various components. Kron found that he could reduce any conceivable machine (even a completely mechanical one)into a network of components which could be solved for the behavior of the specific machine using a single tensor equation. Kron himself went on to make clear that to him, "It is surprising how few ultimate types of elements there are that form the building blocks of the great variety of engineering structures. Most stationary networks consist of a collection of one-dimensional "coils" only; all rotating machines consist only of a collection of two dimensional "windings." The great variety of structures differ only by the manner of interconnections of these ultimate coils and windings, and the variety of theories differ only by the type of hypothetical reference frame assumed. It is only the study of the ultimate building blocks that requires analytical work. The interconnection of these units into a given system is a routine procedure."

The main tensor equation itself is analogous to the statement for Kirchhoff's Voltage Law in a circuit with tensors replacing vectors. As Banesh Hoffman notes in his paper (Kron's Non-Riemannian Dynamics, Hoffman, Banesh, Reviews of Modern Physics, July 1949) "The work of Gabriel Kron constitutes a significant enlargement of the domain of application of tensor analysis." Hoffman also wrote in the introduction to the book Tensors for circuits (authored by Kron) that he (Kron) uses tensors to unify great classes of physical systems. "With him, a tensor transformation changes, the equations of one electrical machine to those of another eletrical machine of a different type. He constructs (the equations of) a prototype machine -the primitive machines - from whose equations he obtains those of all other electrical machines by applying appropriate tensor transformations."

It turns out that Kirchhof's 2 circuit laws - the voltage law and the current law - can help us accomplish an unbelievable amount of modeling of the physical world. Nor is their usefulness restricted to electrical circuits. Kirchhoff's circuit laws laid the foundations also for the field of Topology. (Topology is highly relevant to General Relativity).

The Voltage Law turns out to also be at the heart of analyzing rotating electrical machinery (infact, all machinery ever made or to be made) and it requires a tensorial statement and methods of analysis. This is not a coincidence. Kron himself comments that in his research,he had found that it is interesting that Kirchhof laid the foundations of topology even while working simply with circuits. "It is not a coincidence but a consequence of some hitherto hidden relation between the properties of space and those of electricity that the science of electrical engineering and that of topology meet again on a common ground when both are viewed from an invariant point of view."

It was Kirchhof's laws that laid the foundations for Topology. Topology had hitherto used tensors and now Kron was using a tensorial version of Kirchhof's law to analyze rotating electrical machinery. Tensors used in topology were now also in the very heart of the analysis of rotating electrical networks;

It is worth our while to ponder what we are to make of the fact that nature has designed the phenomenon of electricity such that one equation can serve to model all manner of electrical machinery? Is it something innate to the phenomenon of electricity? Or is the breathtakingly sophisticated mathematical edifice describing all electromagnetic phenomena with the aid of just a few parameters such as impedance (inductive and capacitive), voltage and current and rate of change of current something that derives its seeming perfection from other, even more fundamental?

Kron himself is clear on this subject. He commented that "Although the method of reasoning will be employed [by him in his book Tensors for Circuits] only for stationary and rotating electrical networks, exactly the same reasoning applies also to mechanical and other physical systems. That is, all reasonings and all symbolic formulas to be studied are independent of electrical engineering. The electrical applications are only illustrations."

Kron emphasized that mechanical systems behave in ways that are identical to electrical systems and that they can therefore be studied analogously to electrical systems, using infact the same methods and tensor equation he discovered for (all )electrical machinery.

Impedance is one of that small set of basic building blocks making up the wide variety of electrical structures. It represents the opposition from the medium and is mathematically represented as a combination of three different kinds kinds of opposition possible - purely resistive, pure capacitive and purely inductive.

The one dimensional coils and two dimensional windings mentioned by Kron are in fact inductances. Now, while the purely resistive and purely capacitive opposition of the medium are known and extensively used in both electrical and mechanical machinery, the third kind of opposition possible, the inductance makes a prominent appearance hitherto only in electrical machinery - there it is everywhere. But there has not been the recognition that the inductive suspension of angular momentum is the mechanical equivalent of the one-dimensional coils in electrical machinery. The incorporation of this new type of opposition into mechanical machinery will bring the possibility of designing a new class of mechatronic machines that will revolutionize manufacturing systems in the way induction coils transformed electrical engineering into the modern electronics engineering. Putting aside the more arcane aspects of Gabriel Kron's work, looking directly at the main equation governing electrical machinery are presented by Kron in his book, we cannot but wonder what the shape such mechanical systems might take.

A passage from Kron's book Tensors for Circuits is reproduced below to illustrate not only Kron's mastery of the subject of the analysis of machines but also an inkling of how electromagnetic devices and mechanical devices are different manifestations of the same fundamental geometric and physical phenomenon. (Click on the images to open a larger version)

[Experimental Update: All parts are ready. Assembly starts now. Prototype unveiling in 3 wks. Testing in 4.]