barret - tesla's nonlinear oscillator-shuttle circuit (osc) theory

5/14/2018 Barret - Tesla's Nonlinear Oscillator-shuttle Circuit (OSC) Theory

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Annates de la Fondation Louis de Broglie, VoL 16, n" 1, 1991 23

Tesla's nonlinear oscillator-shuttle-circuit (OSe) theorycompared with linear, nonlinear-feedbackand nonlinear-element electrical engineering circuit theory'

T. W. BARRETT13521 S.E. 52nd st. Bellevue, WA 98006, U.S.A.

ABSTRACT. Tes!a's approach to electrical engineering addressesprimarily the reactive part of electromagnetic field-matter interac-tions, rather than the resistive part. His approach is more compara-ble with the physics of nonlinear optics and many-body systems thanwith that of single-body systems. It is fundamentally a nonlinearapproach and may be contrasted with the approach of mainstreamelectrical engineering, both linear and nonlinear. The nonlinear as-pects of mainstream electrical engineering are based on feedback inthe resistive field, whereas the nonlinearity in Tesla's approach isbased on oscillators using to-and-fro shuttling of energy to capacita-tive stores through non-circuit elements attached to circuits. Theseoscillator-shuttle-circuit connections result in adiabatic nonlineari-ties in the complete oscillator-shuttle-circuit systems (OSCs). TeslaOSCs are reactiy or active rather than resistive, the latter beingthe mainstream approach, therefore device nonlinear susceptibilitiesare possible using the Tesla approach.As a development of this approach, 3-wave, 4-wave ... n-wave mixingis proposed here using OSC deviceS, rather than laser-matter inter-actions. The interactions of oscillator-shuttles (OS) and circuits (C)to which they are attached as monopoles forming OSCs are not de-scribable by Kirchhoff's and Ohm's laws. It is suggested that inthe OSC formulation, floating grounds are functionally independentand do not function as common grounds. Tesla employed, rather,a concept of multiple grounds for energy storage and removal byoscillator-shuttles which cannot be fitted in the simple monolithiccircuit format, permitting a many-body definition of the internalactivity of device subsystems which act at different phase relations.This concept is the basisfor his polyphase system of energy transfer.


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24 T. W. Barrett

The Tesla ases are analogs of quaternionic systems. It is shownthat more complex ases are analogs of more complex number ele-ments (e.g., Cayley numbers and "beyond Cayley numbers"). Theadvantages of crafting energy in quaternionic, or SU(2) group, andhigher group, symmetry form, lie in: (1) parametric pumping withonly a one drive system (power control) ; (2) control of the E field orJoule/cycle (enexgy control) ; (3) phase modulation at rates greaterthan the carrier (p hase c on tro l); (4) reduction of noise in energytransmission (no ise co ntrol) for communications; and (5) reductionof power Joss in power transmission. Engineering applications aresuggested.Finally, it is shown that Tesla.'s asc approach is more appropriately(succinctly) described in A four potential form, than in E, H, BandD field form or by Ohm's law. That is, the boundary conditions areof crucial importance in defining the functioning of OSCs.

RESUME. La maniere dont Testa approche l'ingen ierie electriquec on ce rn e p rin cip alemen t la p artie reactive de !'interaction champ-m atiere , plutot que la partie resistive. 80n approche est plus cotnpa-ruble a la physique de I 'optique non lineaire et des system es a plu-sieurs corps qu'li ceile de syslcmes d un seul corps. L 'approche deTesta est basie sur des connec;tions circuit-rum ette-oscillaieur (OSC)perm etiant des susceptibilites nonlineaires du dispositij. Les asc deTesta son t des analogues des systemes de quat ern ions. Des OSC pluscomplexes sont des analogues de nombres p lus complexes (par ex.tunnbres de Cayley). It y a des cuoniaqes pratiques a meitre Iener-gie SOllS forme de quater'n ions o u d 'e lem ents de groupe;; de symetriede type SU (2) ou plus eleve : po ezemple, dans 1) ie conirole depuissance, 2) le contro/e d 'inergie, 3) le cantr-ole de phase, 4) lecontrole du bruit. L 'approche asc de Testa est dicrite de m aniereplus appropriee sous La Jorm e de quadripatentiel A plutot q'u.e souscelle de cham ps de [orce.

I.IntroductionThere is almost universal agreement that ::\icolai Tesla approached

electrical engineering from a different viewpoint than conventional circuittheory. There is, however, no agreement on the physical model behind hisparticular approach. In this paper I hope to show that Tesla's approachtook advantage of the many possibilities of nonlinear interaction in joinedoscillator-shuttles, and his nonlinear oscillator-shuttle-circuit approach,


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Tesla's nonlinear oscillator-s hut tIe-circuit (OS C) ... 25

of the use of multiple independent floating "grounds" which provideseparate energy storage capacitative repositories from which energy isoscillator-shuttled to-and-fro. The use of independent and noninteract-ing energy storage "cui-de-sacs" is a trademark of Tesla's work and setsit apart from linear circuit theory as well as nonlinear feedback theory(cf Tesla, 1956; Ford, 1985).

The OSC arrangement cannot be adequately described either byKirchhoff's or Ohm's law. In section III, below, the field relations arederived for OSCs within the constraints of the OSC arranqemeni consid-ered as boundary conditions.

The OSC arrangement is treated in section III as another methodof energy crafting or conditioning similar to that of wave guides or otherfield-matter interactions .. Viewed from this perspective, Tesla's OSCarrangements offer methods to achieve macroscopic or device nonlinearinteractions presently only achieved, with difficulty .. in nonlinear optics(cf Bloembergen, 1965, 1982; Bloembergen & Sheri , 1964; Sheri , 1984;Vee & Gustafson, 1978).

A clear distinction can be made between the adiabatic nonlinearoscillator-shuttle-circuits addressing the dynamics of the reactive fieldconsidered here and nonadiabatic circuits addressing the resistive field.For example, Chua and coworkers (Chua, 1969; Chua et al, 1986;Matsumoto et al, 1984, 1985, 1986, 1987a,b;. Kahlert o S . : Chua, 1987;Rodriguez-Vasquez et al, 1985; Kennedy & . Chua, 1986: Abidi & Chua,1979; Pei et al, 1986) have described many nonlinearities in physicalsystems such as, e.g., four linear passive elements (2 capacitors, 1 ind uc-tor and 1 resistor) and one active nonlinear 2-terminal resistor charac-terized by a 3 segment piecewise linear v - i characteristic. Such cir-cuits exhibit bifurcation phenomena, Hopf bifurcation. period-doublingcascades, Rossler's spiral and screw type at tractors, periodic windows,Shilnikov phenomenon, double scroll and boundary crisis. The tunnelingcurrent of Josephson-junction circuits can even be modeled by a nonlin-ear flux-controlled inductor (Abidi & Chua, 1979). However, in all theseinstances, (i) the nonlinear resistive elements require an energy sourceto the nonlinear resistor which is external to that of the circuit, (ii) theresistive field, not the reactive field, is the operative mode, and (iii) ofcourse the physical system is a circuit, not an OSC.

Treatments of electrical circuits by the oriented graph approach (In-


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gram Cramlet, 1944; Van Valkenburg, 1965: Sashu & : Reed, 1961;

26 T. W. Barrett

Smale, 1972) have all commenced with a one-dimensional cell complex[i.e., a graph) with vertices and branches connecting them, as well asseparable loops. Representing the connectivity relations of an orientedlinear graph by a branch-vertex matrix A ::::.I:ij, the elements havevalues of +1, ~ 1, and 0, depending on whether cu rren t is flowing into,ou t of, or stat ionary, at a particular vertex (i. ., (tij = (+ 1, ~ 1, 0)). Thislinear graph representation does not, however, take into account anyrepresentation (resulting from modulation) which doe.'>not conform tothe values for aijJe.g., whenajj takes on spinor values, that is, obeysthe even subalgebra of a Clifford algebra.

There are, however, other approaches to circuit analysis which arecompatible with Tesla OSCs. Kron (1938, 1939, 1944, 1945a,b, 1948)equated circuits with their tensor representations. Kron's methods weresupported by Roth's demonstration (1955) that network analysis is apractical application of algebraic topology. Roth (1955a,b) showed thatKirchhoff's current law is the electrical equivalent of a homology se-quence of a linear graph, and Kirchhoff's voltage law corresponds toa cohomology sequence, these sequences being related by an isomor-phism corresponding to Ohm's law. The algebraic topology approachw a s enhanced considerably further by Bolinder (19S7a,.b, 1958, 19S9a,b,1986, 1987) who introduced three-dimensional hyperbolic geometricaltransformations to circuit analysis and showed how partially polarizedelectromagnetic or optical waves can be transformed by Clifford algebra.TesJa OSCs also can be described in Clifford algebra terms. Below, theOSCs are described in quatemion algebra" which is the even subalgebraof a three-dimensional Clifford algebra with Euclidean metric.

In the immediately following section IIhe reader is introduced toTesla OSCs shown in Tesla (1956, 1986) and Fbrd (1985), establishingthe case of unique use of oscillator-shuttle (OS) arrangements joined ina monopole fashion, i.e., with one connection, to circuits (C), therebyforming OSCs. Simple OSC models are then related to the Tesla modelshighlighting the operating principles.

II. Some Tesla oscsThere are unifying physical themes present in Tesla OSCs and an-

tennas (Figures lA-J). Figure lA is the prototypical oscillator-shuttle(which we shall callOS) with the common ground situated between twoinductances one of which is joined to a capacitive energy store indicatedby the circle. The OS, imbedded in conventional circuits, (which we shall


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Tesla's nonlinear oscillaror-shut tle-circui t (OS C) ... 27

refer to as Cs), (Figures IB & IC), we shall call an asc. The frequencyof the OS becomes the signal frequency u n for the pump frequency (o) ofthe circuit, C, resulting in an idler frequency (y) for the asc, using thesignal, pump, idler nomenclature of the theory of parametric excitation.However, as will be indicated below, this is a unique form of adiabaticinvariant parametric excitation and distinct from the conventional formwhich requires energy expenditure in the signal as well as the pump. TheOSC only requires energy expenditure in the pump.

Figures ID, E & F are further examples of OSCs in which the pri-mary coil (our pump or circuit inductance) is wound around the sec-ondary (our signal or as inductance). In Figures IG & H in a variation,the signal, or secondary, OS inductance has two energy storage capac-itors for shuttle operation and is coupled to the circuit by the primaryfield alone.

Figures 11 & J are pancake antennas which utilize two principles:(1) the asc principle already discussed in which the pancake is an in-ductance OS for energy storage in the pancake coil, and, in the case ofFigure lJ, even with a capacitance store at the vertex of the pancake;and (2 ) E field overlap due to the in-plane winding of the pancake re-sulting in a many-to-one mapping of E fields and a strict boundary con-dition constraint. The pancake antenna is also a frequency-independentantenna.

Figures 2A-G are OSC diagrams illustrating in a more simplifiedfashion the principles exhibited in Figures IA-J. Figure 2A represents thedesign of Figures lB & C; Figure 2B represents polarization modulation;Figures 2C & D represents Figures IB & C ; Figure 2E represents FiguresIG; Figure 2F represents Figure IH; and Figure 2G represents Figures11 & J.

In the following section III, the pump, signal and idler fields ofthe simplest, or Tesla, OSC are derived by a treatment in which theOSC arrangement is treated as another method of energy crafting orconditioning similar to waveguides or other field-matter interactions, i.e.,with network theory subsumed under field theory.


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28

A

B

c

u _ , 1- = J I E > " ~ ,-- - t! ! L - , - - . . .

T. W. Barrett

o~...e

" ' t I"~..." ' = . - L . . . . t f.Y

E


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(OSC) ...ttle-circuttllator-s ui ear OSCIesla's non In 29

I Itt...,.'H

J


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30 T. W. Barrett

o

E

G. G1' " I P

G2's:}t~llG2 P os, I T 1 G , } t~G2 1 3 G3

GlIt" T ~ ,'= Gl G1 -=" F02

" I 1 3 I'a 01 G 01

c

02

Figure 2.

III. Reactive versus resistive fields: analogy between oscillator-shuttle-circuits and coherent coupling between modes in a non-linear optical waveguide

The operation ofa many-body Tesla OSC system can be describedby a model already used in nonlinear optics for describing radiation-matter interactions. Specifically, there exists an analogy between TeslaOSC theory and coherent coupling between modes in a nonlinear waveg-uide ..

One can commence with the Maxwell's equations:v x E : : : : : ~iwJ1-0H (1 )

vD =0(2)(3)

v x H = iwD


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Testa's nonlinear oscillator-shuttle-circuit (aSe) ... 31

' i7 . fJ .oH = 0 (4)and

D =EE, (5)if no free charge density is present and the medium is isotropic. Bysetting

MoH =v x Aand introducing this into the first Maxwell equation gives:

(6)

v x (E + it...'A) = 0 (7)Therefore E + iwA is the gradient of a scalar potential :

E= -iwA. - vcjJ.Introducing this into the second Maxwell equation gives:

' i7 x (v x A) =w 2 M ocA . - iWl1oc'i7 r f ; . (9)Using the identity for curl A, gives:

Using the Lorentz gauge:v .A. + iWlloc@ =0 (11)

and with no source terms, gives:(12)

which permit solutions of the form:A = x 1 P ( x , y ) exp[-i;9z],A = y 1 P ( x , y ) e x p [- ip z ] ,

(13)(14)

for media uniform along the z-direction. The scalar function then obeysthe scalar wave equation:


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32 T. w. Barrettwhere

'V T = xa/ax + ya/ay ..From equ.s (8) and (n) we have:

(16)

E =~ iw .4. - i\7('V .. A)/wf. tot . , (17)and the E and H fields for the z-polarized vector potentials are:

E = ~ i . . . . : l x ( 1 , 1 > + (1/w 2 f . tO)a2 /ax2) + Y(1/w2 f.tof.)a 21/J/8x8y~ i(,) z / w2 fLOE )O w / ax] (18)

J . l o H =v x A = [ - x X \IT + iy,BJ"\I'Jexp(-i,Bz). (19)The fields fo r the y-palarized vector potentials are similar.

We now introduce the impedance changes in a Tesla OSC, or theequivalent of a device nonlinear second-order susceptibility tensor Xt2)due to the electrical control field and the particular waveguide condi-tionings of an OSC. (Higher-order device nonlineanties are consideredbelow). If the frequency of the control or signal fields is designated: Wsand that of the pump field is w p then the impedance change caused bythe signal or control field is:

b,.Zr =ZOX(2) E(wp)E~( -ws),.6.Z i = ZOX(2) E(ws)E~(ws),

(20A)(20B)

where Z, the complex impedance is:Z(iw) = R + i(wL - l/wC),

with magnitude:

With the signa! or control field defined:(21)

the pump field defined:(22)


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Tesla's nonlinear oscillator-shutt.le-circult (OSC) ... 33

and the two fields coupled by the oSC device-generated nonlinear sus-ceptibility, the following changes in the impedance occur:

6Z,,. = ZOX(2 ) E; (t)Ep(t) expIi(wp ~ ws)t ~ i(f3p -!3s)tJ, (23 .1'1.)cz, = ZoX(2)E;( t )E.( . t ) . (23B)

If X(2 ) is purely real (inductive), then only phase changes are produced.If X(2) has an imaginary (resistive) component, then power transfer, andeven gain, can be obtained for one of the inputs.

The idler field (Tesla coil load output) is then:

whereCBp- (3S) = [j w 2 /1 oA o . 6Z . Adll/Ij A ~ . A dl]. (25)

and E, and Ep are defined by equ.s (17). Thus the three-body interactivesystem of Ei, E; and Ep is defined in terms of the A vector potential:E;(wi,t) = n - i : . ;As(t) -i~{v As ( t ) ) jW,uOf 12J

X [-iw_-1p(t) - i\{t"" . A,,( t) jw/1iJf]X exp[i(:..,,, - v . . ' , l t - i C B " - f3s)tJ.

The quatsrnionic impedance for the OSC is then:(24)

where the subscripts on ii,i2 and i3 distinguish the separate field con-ditioning of the waveguide-like properties of the OSC and the subscriptson R,. wL and u..C distinguish circuit, C , elements (1 ) from oscillator-shu ttie, OS, elements (2). As the waveguide properties of circui ts, C, (1)are fundamentally different from oscillator-shuttles, as, (2), distinguish-ing the i",y (=p,x, y = 1,2,3 ... ,i"iy = -iyix, ix anticommutes withill)' is a necessary for distinguishing the OS and C dynamic interactionof the OSC total arrangement.

The distinguishing characteristics of higher order OSCs is in analogyto the dimensions of the number system. The dimensionality of the


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34 T. W. Barrett

of Cayley numbers is 2 3 ; of "beyond Cayley numbers" is 2 4 ; etc. Eachnumber system has an ase device analog associated with a higher-ordernonlinear susceptibility.Quaternions are four-dimensional numbers. The Appendix reviewsquatarnion number interactions and gives Equ.(24) in quaternionic form.Figure 3 shows a mapping of a four-dimensional quaternionic signal inthree-dimensions (one dimension representing t.wo). Similar argumentsapply to asc for representations higher than quaternions, e.g., Cay-ley numbers and involve higher order device nonlinear susceptibilities.ascs are shown in Figure 4A-C. Figure 4A is a quaternion ose with di-mension 22(5U(2) group symmetry), to which number associativity andunique division applies, but commutativity does not apply. Figure 4Bis a Cayley number ose with 23 dimension (5U(3) group symmetry),to which unique division applies, but associativity and commutativitydoes not apply. Figure 4C is a number of 24 dimension (5U(4) groupsymmetry), to which neither associativity, nor commutativity nor uniquedivision applies.

Figure 3.


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Testa's nonlinear oscillator-shuttle-circuit (OSe) .. , 35

y

II) III)

(Il (II)

y

J: O ;p po .(o.l'ly y(o;l)lo .x P 0

BUT ~ NOT UNIQUE

Figure 4.IV. The virtues ofo5cillator-shuttle-circuit (OSe) arrange-ments

Among the many virtues, or beneficial applications of the OSCar-rangements, a nonexhaustive list would include the following :

1. Parametric pumping with a one-drive energy source system (fre-quency of energy flow or power control .. s.e., Manley-Rowe (1959) rela-tions without an external drive other than a primary source). In contrast,parametric pu illping (th fee- wave mixing] in con ven tiona! circuit theoryis restricted to two active systems (pump and signal). Even in the caseof three-wave mixing in nonlinear optics, two beam sources are required(cf Kaup, 1976, 1980; Zakharov & l\Ian akov , 1976). The OSCarrange-ment can also be contrasted with conventional harmonic generation, asthe idler frequency generated is controlled by choice of signal and pumpinductances which may not be harmonically related. In contrast to theconventional, parametric pumping using OSCs requires only one activesystem (the pump) permitting "energy bleeding" from that active sys-


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36 T. \V. Barrett

is giYen by the Manley-Rowe relation (Manley & Rowe, 1959) :H' = { C Z /2,,) [{kpT,z C052 O'p 1 EpT 1 2 jwp} + {ksT,z cos2 {ts 1 EsT 1 2 jw s}

+ {kiT,z cos2 0:; 1 EiT 1 2 /W i } ] ,(27)

where kpT,~, ka,, krr., are the phases of the pump, signal and idler.EpT, EsT, EiT are the total transmitted energies, i.e., the powers,

of the pump, signal and idler.up, (1$, Ct'i are the angles between ET(Wi) and ET _ l ( w ; ) .That is, whereas the power, lV, is all adiabatic invariant for OSCarrangements, it is not for conventional parametric circuits which require

an external signal power source. In the case of conventional parametriccircuits the :"Ianley-Rowe relation applies to a conventional circuit andits power source, together with an external signal power source, i.e., theapplication is to a nonadiabatic device.

Higher-order OSC arrangements permit more complex dynamics.For example. due to parametric processes, third-order OSC device non-linear susceptibilities permit "phase conjugate mirror" -Iike signal recep-tion and communication with cancellation of noise-in-medium similar tothat achieved with phase conjugate mirrors based on radiation-matterinteraction.

2. E field control (energy per cycle (Jou/esjcycle)control). TheOSC principles permit E field control as exhibited in frozen Hertzianwave generation (Figure 5) (For frozen Hertzian wave generation cf:Cronson, 19/'5; Zucker et al, 1976; Proud & Norman, 1978; Xlathur etal 1982; Chang et al, 1982, 1984).

3. Phase modulation at rates greater than the carrier (phase control)is permitted by OSC arrangements because the OS or signal circuit iscontrained always to be the signal oscillator -the load or idler alwaysbeing a function of the circuit, C. Therefore even if the signal frequencyof the as is a higher frequency than that of pump signal of the primary,the power flow to the idler will be a modulation of the primary or pumpfrequency, rather than a modulation of secondary or signal frequency.

4. Noise reduction in communications transmission due to condi-tioning of fields in higher order symmetry form (noise controD. As anoutput from an OSC, i.e., a transmitted wave, is in higher-order groupsymmetry form (see section III above) and such higher-order symme-tries have a low probability of occurrence naturally, environmental noise,


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Tesla's nonlinear oscillator- shuttle- circuit (OSC) ... 37

which is of lower symmetry form (usually, U(l)) and has a high prob-ability of natural occurrence, 'will be excluded from a receiver designedfor OSC transmitted wave reception. Therefore in the case of commu-nications, less noise will be processed statistically at a receiver designedfor SU(2) or higher group symmetry operation, resulting in enhancedsignal- to- noise.

5. A similar statistical argument holds for less loss in powertransmission by higher-order symmetry energy crafting or conditioning.Higher-order group symmetry "receivers" will have enhanced receptionover lower-order receivers, i.e., leakage to ground.

The type of antenna to be used with OSC devices is a small loopantenna which has an inductive reactance (cf, Smith, 1988). Such an-tennas are the dual of a short dipole which has a capacitative reactance.Thus a small loop antenna can be substituted for the f 3 inductance ofthe OSC of Fig. 4A, for the e or {;inductances of Fig. 4B, and for the or f inductances of Fig. 4C. Such antennas could be used in either thesend or receive operation modes and the OSCs would function as "activemedia".V. AppendixQua ternions-overview

The algebra of quaternions is the even subalgebra of a three-dimensional Clifford algebra with Euclidean metric. A quaternion is:

where the scalar multiplication is :

and the sum is :

The product is:xy = (xoY o - X IYI - x2YZ - x3Y3)1 + (XOYI - X lYa - X 2Y3 - X3Y2)i


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38 T. W. Barrett

E,erp(-Iw.t]"R" rot

E,.1Up(iWl't -I1 3p t] " Pmnp"IC=X2 j +X3fo .E. e rp( -Iw tJ " Signal (Real)=R"ZO'E.erp(-I13.t) " Signal (lm.agiDary)~IREI;I ~

R " E,,"erp(-Iw.t) .. Slgnal (Re allC " R + IR " E,,"=p(.lw.tl + lE.=p(-I!3.tl.. Signal (Real & : Imaginary)9" C + Ie = C + Il';.erp(lwptJ -lEperp(-I13ptJl~ PmnpIdler. I:l;=rot+l:li.+>


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Tesla's nonIinear oscillator-shuttle-circuit (OSC) ... 39

field except the commutative law of multiplication.Let: x =Ex exp[iwxt - , B y t ] , Y =B y exp[iu_ .'yt - , B y t ] , and

then:[ x , y ] =x y - y x ,

[ x , y ] =(

0XIYl - YII]YOXl, - IOYlX3Y2 - Y3X2

YII2 - XIY2o IOYl - YOXlYOXl - IDYloIOY3 - YOX3

IOYI - VOIIYIT3 - X1Y3

IfX o = E; exp(i;.cost)I2 = . JE; , exp(iwpt)

Xl =E o e x p ( i ,B o t ) ,X3 =JE; exp(i/1pt),

then

or Equ.(24) of the text.ReferencesAbidi, A.A. & Chua, L.O., IEEE Trans. Circuits & Systems, "01. CAS-.'33,nOlO, Oct 1986, pp. 97,1980.Blcembergen, N., Nonlinear Optics, Benjamin, NY, 1965Bloernbergen, N., Reviews of Modern Physics 54,. 1982, 685_BJoembergen, N. & Shell, V.R., Physical Revl'ew 133, 1964, A37-A49_Bolinder, E.F., Impedance transformations by extension of the isometric cIrclemeth od to the three-dimeIlSil1Ilal hyperbolic space. J _ :>lathema tics &

Physics 36, 19571, 49-61-Bolinder, E.F., Noisy and noise-free two-port networks treated by the isometriccircle method, Proc, IRE voi. 45, 1957b, 1412-3_Bolinder, E.F., Radio engineering use of the Cayley-Klein model of three-dimensional hyperbolic space, Proc. IRE vol. 46, 1958, 1650.Bolinder, E.F.) Theory of Noisy Two-Port Network, J. of the Franklin Insti-tute 267, 1959a, n01, 1-23_Bolinder, E_ F ., Impedance, power, and noise transformations b.,' means of th elvfinkowski model of Lorentz space, Ericsson Technics n02, 1959b, 1-35_


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40 T. W. Barrett

al1d Their A.pplications in Mathematical Physics, 1986, D. Reidel, TheNetherlands.

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n O 10, O ct 1986, pp. 974980.Kron, G., Jnrar/allt form of the l\I.axwell-Lorentz field equa Uons for acceleratedsystems, J . Appl, Phys., 9,1938,196-208.Kron, G ., Tensor analysis of networks, WHey, New York, 1939.Kron, G.,. Equiralent circuit of the field equatiolls of Maxwell, Proc, IRE 32,

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Testa's nonJi near oscillaeor-sh uttle-circui t (OS C) ... 41

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barret - tesla's nonlinear oscillator-shuttle circuit (osc) theory

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circuit approach

teslas approach

nonlinear feedback theory

linear circuit theory

circuit ose

circuit systems oscs

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complete oscillatorshuttle