Anna.les de la. Fondation Louis de Broglie, VoL 16, nO 1, 1991 23

Tesla 's nonlinear oscillator-shut tle-circuit (OSe) t heory

compared with linear, nonlinear-feedback

and nonlinear-element electrical engineering circui t tbeory'

T. W. BARRETT

13521 S.E. 52nd st . Bellevue, WA 98006 , U.S.A.

ABSTRACT_ Tesla's approach to electrical engineering addresses p rimaril,v the reactive part of elect romagnetîc field-matter illterat­tians, rather than the resist ive part. His approach is more compara­ble with the physics of nonlinear optics and many-body systems than with that of sing!e-body systems. It is fundament ally a nonlinear approach and may be contrasted with the approach of mainstream e!eclrical enginccring, bath linear and nonlinear. The nonlinear as­pects of mainstream electricaI engineering are based on feedback in the resistive field. whereas the nonlinearity in Tesla's approach is based on oscilla tors using to-and-fro shllttling of energy to capacita­tive stores throllgh non-circuit clements attached ta circuits_ These oscillator-shuttle-circnit connections result in adiabatic nonlineari­ties in the complete oscillatar-shuttle-circuit systems (OSCs). Tesla oses are reacti,-e or active rather than resis the, the latter being the mainstream approach, therefore device nonlinear s1J5C€ptibilities are possible using the Tesla approach.

As a development of this approach, 3-wave, 4-wave ... n-wave mixing is prapased here using ose devices, rathêr than laser-matter inter­actions. The interactions of oscillator-shuttles (OS) and circuits (C) ta which they are attached as monopoles forming OSCs are not de­scribable by Kirchhoff 's and Ohm 's laws. It is snggested that in the OSC formulation , floating grounds are fllnctionally independent and do not fnnction as corn mon grounds_ Tesla employed, rather, a concept of multiple gronnds far energy storage and removal by oscillator-shuttles which cannot be fitted in the simple monolithic circuit format, permitting a many-body definitioll of the internai activity of device subsystems which act at different phase relations. This concept is the basislor his polyphase system of energy transfer.

Originally disclosed in document N° 225395 , 1988, L".S_ Patcnt Office.

24 T. W. Barrett

The Tcsla oses are Ilila iogs of quatcrn ionÎc systems. lt is shown t hat more colll plex oses Ilrc analogs of more complex Humber cle­ments (e.g. , Carle}' numhers and "bc}'und Cayley lIumbcrs" ). The ad \"a nt.a ges of [ Tafting cucrgy ill quatcrn ic.mÎc. ()f SU(2) group, and highrr gruu p, sym melry fo rrn, lie in. (1) paramctric pumping with only il one dri ,"e system (powe r rontml) ; (2) cont rol of the E field or .lnu le/c.\·c1e (cflt!cgy cordml); (3) pllase lIlodula t ion at ra tes grcatcr Ihan t he carrier (phase con/rol); (-il rrouct ion of noise in encrgy t ransmission (noue c011lrol) for communicat ions; and (5) rcd uction of power loss in pOwer transmission. Engiut'e ring applicat ions are suggested.

Fï llally. il is shown 1 hal Tesla's ose approach 15 more appropriatcly (suççinct l..- ) describcd in A four polent la! form . than in e, H , Blind D field f UTl D or by Ohm 's law. Tha t 15. t he boundary cond itions are of crucial importance i ll dcfi ning the funct ioning of oses.

IlESUME. La manière dOIl! Tesla approche l "ingénierie rE/cc trique concerne principalement la par!ie réuctive de l"in/erac/ioll champ. matière, plutôt que la partie rbistive" Son approche est plus campa· rable li. [a phYSIque de l ·Qptrque " on linéa ire ct des systèmes à plu" sieur.s corps qu 'Ii. celle de systèmes li un seul corps. L ·apPl·oche de 1b;/a est basée ",ur des cormections Clrctlil-naueile-oscillateur (OSe) permettant des su.!uplibildés nonlinéair"C.! du düpositij. Les ose de Tesla sont de.! analogues des système, de qua leTTIiolls. Des ose plus r.omplezes sont des a1lalogues de lIombre! plu.s complexes (par ex. nombre$ de Caylcy) . lI y a des avantages pratiques à m ettre l 'b,er­gie 501/ 5 fo rme de Quaternions ou d ·éléments de groupes de symétrie de type 5U(2) ou plu.s élevé : par ezemple, dans 1) le colllni le de puissance, 2) le CQntrôle d·énergie, 3) le cou /role de phllJe, .f) le contrôle du bruit. J, 'approche ose d~ Tesla esl décrite d~ mafliere plus appropriée SOlU la. forme de qUlldripotel1liel A plutôt que sou.! r;clle de champ! de forct..

L Introduction

T hcre is almost unin!rsaJ agreement t hat :;-Îcolai Tesla approached eletl rical engineering from a different ,"iewpoint than conwnt ionaJ ci rcui t t hcory. T hcre is, howe,"er, no agreemellt Oll the I,hysical modcl behind his pa n icul aT approach. l n th is paper l hope 10 show that Tesla. 's ap proach look ad\'antage of the mally possib ilities of IIOllli ncar interaction in joined oscillator-shuttles, and his nonlinear oscillator-shuttle-circuit approach, which wc shaH cali T.he OSC approach, ta n be contrastoo sharply wit h linear circuit lheory. Thc particula r nonlinearity of O SCs arises because

Tesla 's non linear oscillator -shutt le-circuit (ose) ... 25

of the use of multiple indcpendent fioaling "grounds" which provide separate energy storage capacitative repositories from which energy is oscillator-shuttled to-and-fro. The use of illdependent and Ilonintcract­ing energy storage "cul-de-sacs" is a trademark of Tesla's work and sets it apart from linear circuit theory as weil as nonlinear feedback theory (cf Tesla, 1956; Ford, 1985).

The ose arrangemeni cannot be adequately described either by Kirchhoff's or Ohm's law. ln section nI, belo\\" . the field relations arc derived for oses within the constrainls of the ose arrangement conBid­ered as boundary conditions.

The OSC arrangement is t reated in section TIl as another method of energy crafting or conditioning similar to that of \Van' guides or other field-matter interactions . . Viewed from this perspect i\·c. Tesla's ose arrangements ofJer methods to achieue. macroscopic or device nonlinear interactions presently only achielJcd, with diffie1.l./ty, in nonlinear opties (cf Bloembergen, 1965, 1982; Bloembergen .s..: Shen. 196-/' ; Shen. 1984 ; Yee & Gustafson, 1978).

A clear distinction can be made bet\\'een the adiabatic nonlinear oscillator-shuttle-circuits addressing the dynamics of the reactive field considered here and nonadiabatic circuits addressing the resistive field . For example, Chua and coworkers (Chua. 1060; Chua ct al. 1086; Î\latsumoto et al. 198-/', 1985, 1986, 1987a.b; Kahlert &:: CllUa, 198"1; Rodriguez-Vasquez et al, 1985; Kennedy &. Chua. 1986: Abidi & Chua, 1979; Pei et al, 1986) have described man}" nonliucarities in physical systems sucll as, e.g., rour linear passive clements (2 capacitors, 1 induc­tor and 1 resistor) and one actiw nonlinear 2-terminal resistor charac­terized by a 3 segment piccewise linear '1.' - i characteristic. Such cir­cuits exh ibit bifurcation phenomena, Hopf bifurcation. period-doubling cascades , Rossler's spiral and sere\\' type attractors, periodic \yindows, Shilnikov phenomenon, double scroU and bauudary crisis. The tunnding current of Josephson-junct.ion circuits can e\'en be modeled br a non lin­car tl.ux-cantrolled inductor (Abid i & Chua, 1979) . HO~'e\"er, in ail these instances, (i) the nonlinear resistive elements require an cllergy source tü the lIonlinear resistor which is cxternal to tllat of the circuit, (ii) the resistive field, not the reactive field , is the operati"e mode, and (iîi) of coursc the physical system is a circuit, not an ose.

Treat ments of eledrical circuits by the orie nted graph approach (In­gram & Cramlet, 19H j Van Valkenburg, 1955: Seshu ..\..; Recd, 1961 ; Bray ton & }':Ioser , 1964a,b; Rez & Secly, 1959; Branin, 1959, 1966 ;

26 T. W. Barrett

Smale, 1972) have all commenced with a one-dimensional œil complex (i .e., a grapb) v.':ith vertices and branches connecting them, as weil as separable loops. Representing the connectivity relations of an oriented linear graph by a branch-vertex matrix A = L a...J> the elements have values of + l, - l , and 0, depending on whether current is flowing inta, out of, or stat ionary, at a particular vertex (i .e., ai} = (+ 1, - 1,0)). This linear graph representation does not, however, take inta account an)' representation (resu lting from modulation) which does noL conform to the \"a!ues fOf aij, e.g., when aij takes on spinor values, that is, abeys the eH!n subalgebra of a Clifford a1gebra.

There are, how€ver, ather approaches to circuit analysis which are com patible with Tesla. OSCs. Kron (1938, 1939, 1944 , 1945a,b, 1948) equated circuits with their l.ensor representations. Kron's methods were supported by Roth's demonstration (1955) that network analysis i3 a pmctical application of algebmîc top%gy. Roth (1955a,b) showed that Kirchhoff's current law is the electrical equivalent of a homology se­quence of a lînear graph, and Kirchhoff 's voltage law corresponds ta

a cohomology sequenœ, these sequences being related by an isomor­phism corresponding ta Ohm's law. The algebraic topology approach was enhanced considerably further by Bolinder (1957a,b, 1958, 1959a,b, 1986, 1987) who introduced th ree-dimensional hyperbolic geometrical transformations to circuit analysis and showed how partially polarizcd electromagnetic or optical waves can be transformed bl' Clifford a1gebra. Tesla oses also cao be described in Clifford a1gebra terms. Below, the OSCs are descrihed in quaternion algebra, which is the even subalgebra of a three-dimensional Clifford algebra with Euclidean metric.

In the immedia.tely following section Il the rea.der is introduced ta Tesla oses shown in Tesla (1956, J986) and Ford (1985), establishing the case of unique use of osciJ1ator-shuttie (OS) arrangements joined in a monopole fashion, i.e., with one connect ion , ta circuits (C) , thereby forming OSCs. Simple OSC models are then related to the Tesla models highlighting the operatîng principles.

Il. Sorne Tesla oses

There are unifying physical themes present in Tesla oses and an­tennas (Figures lA-J). Figure lA is t he prototypical oscillator-shuttle (which we sball calI OS) witb the common ground situated between two inductances one of which is joined ta a capacitive encrgy store indicated by the circle. The OS, imbedded in conventional circuits, (which we shall

Tesla's nonlinear oscillator· shuttle-circuit (ose) ... 27

refer to as Cs), (Figures lB & lC), we shaH cali an ose. The frequency of the OS becomes the signal frequency (13 ) for the pump frequency (0:) of the circuit, C, resulting in an idler freq uency h) for the ose, using the signal, pump, idler nomenclature of the theory of parametric excitation. However, as will be indicated below, this is a unique form of adiabatic invariant pammctric excitation and distinct from the conventional farm which requires energy expenditure in the signal as well as the pump. The ose only requires energy expenditure in the pump.

Figures ID, E & F are further examples of OSCs in which the pri­mary coil (our pump or circuit inductance) is wound around the sec­ondary (our signal or OS inductance). In Figures lG & H in a variation, the signal, or secondary, OS inductance has two energy storage capac­itors for shuttle operation and is couple<! to the circuit br the primary field alone.

Figures 11 & J are pancake antennas which utilize two princip/es: (1) the ose principle already discussed in which t he pancake is an in­d uctance OS for energy storage in the pancake coil, and, in the case of Figure lJ, even with a capacitance store at the vertex of the panca.ke; and (2) E field overlap due ta the in- plane winding of the pancake re­sulting in a rnany·to-one mapping of E fie lds and a strict boundary con­dition constraint. The pancake antenna is also a frequency-independent antenna.

Figures 2A-G are OSC diagrams illust rating in a more simplified fashion the principles exhibited in Figures lA-J. Figure 2:\ represents the design of Figures lB & C; Figure 2B rep resents polarization modulation; Figures 2C & D represents Figures 1 B & C ; Figure 2E represents Figures lG; Figure 2F represents Figure lH ; and Figure 2G represents Figures 11 & J.

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

28 T. 'V. Barrett

A o

B

E

c

- ,il~l ~r:=::r;1

.~ : .

--

Figure 1.

Tesla 's nonlinear oscillator-shuttle-circuit (ose). .. 29

1

,~ G Ir 1 1

F ""l1 ", . ...

.... ... H ,--------,-.. ,

J --------0 , o

F igure I bis.

30

A

B

c

a

l' L:~""" 1 v"

= C l

G2

-r ' '--"=----1'9-"'1:---11' r:c-. 0' , " ~ a

Fig ure 2.

D

E

F

G

T . \V. Barr e tt

, ri~~I' ~ G 0. Gl

"r ' 02

'rr&:.J ~, ~I' G2 l' 03 G'

, ~~:-Jra--I~l l 0-"""-0

G 2 l' G3

G'

G2

II 1. R eactive versus r esistive fie lds : analogy betwee n oscilla tor­shu t tle-circ ui ts a nd cohe re nt coupling betwee n modes in a non­li near optical waveguide

The operation of a many-body Tesla ose system can be descri bed by a model already used in nonlinear optics for describing radiatioll­malter interactions. Specifically, there exis lS an analogy between Tesla ose theory and coherent coupling between modes in a nonlinear waveg­uide.

One can commence \\'ith the .\'la.'i:well 's equations :

'V xE:::: -iwJ1.fJ H

v x H = iwD

(1)

(2)

(3)

Tcsla's nonlinear oscillator-shuttle-circ uit (ose) .. . 31

v· p.olI = 0 ( 4)

and D = f.E , (5)

if no free charge density is present and the medium is isotropie. By set ting

and int roducing this into the first Maxwell equation gives :

vx (E + ù ... '.4.)= O

T hereforc E + iwA is the gradient of a scalar potential tjJ:

E = - i;.;A. - v tjJ.

Introducing this inta the second !\Im.\\·cll equation givcs:

Using the identity fo r curl A, gives:

Using the Lorentz gauge:

v . A + 1i.J-'110f. tjJ = 0

and with no source ter ms, gives:

whicll permit solutions of the form :

A = x>P(x,y)exp [-i;3'],

A = y"IjJ(x, y) exp[-i,&:],

(6)

(7)

(9)

( I l)

(12)

(13)

( 14)

for media uniform along the z-direction. The scalar funct ion then obeys the scalar wave equation :

( 15)

32 T . W. Barrett

where <;Ir ~ xô/ôx + yOjôy.

From equ.s (8) and (Il) v .. e have:

E := -iwA - i'V(\7 · A.)jWl1fJE. ,

and the E and H fields for the x-polarized vector potentials are:

E = - i ... :lx{~ + Ofw2 JJ.of..)éP1/J/8x2) + Y(1/w 2 /i-of. )éPtb/ôxây

- ;(~,/w'"",)ihP/ôxJ

paH = v x A = [- x X VT + iy.8J,pexp(-i,lJz).

T he fie lds fOf the y- polarizcd veetar potcnlials arc similar.

(16)

(1 T)

(18)

( 19)

\Ye no\\' introduce the Impedance changes in a Tesla ose, or the equh"alcm of a device nonlioear second-order susceptibility tcosor X(2)

due to the clcctrical control field a nd the particular waveguide condi­tionings of an ose. (Higher-order devjce nonli nearities are considered below). If the frequency of the control or signal fields is designated: w, a nd tha l of the pump field is wp , then the Impedance change caused by the signal or control field is:

(20A)

(20B)

where Z, the complex impedance is :

Z(i",) ~ R + i(wL - l /wC),

with magnitude:

Z ~ v1R' + (wL - l /wC)'J .

\Vith the signal or control field defined :

(21)

the pump field defined:

E(w" t) ~ E,(t) expJi(w,t - ~, t )J, (22)

Tesla's Ilonlinear oscillator·shuttle-circuit (ose) ... 33

and the two fields cou pied br the ose device-generated nonlinear sus­ceptibility, the follow ing changes in the impedance OCCUI :

/!"Zr = ZOX(2 j E;(t)Ep(t) exp[i(wp - w8 )t - i(/3p - /3s)t] ,

/!"Z j = ZOXf2 IE;(t)E .. (t).

(23A)

(23B)

If X (2) is purcly rcaJ (inducti'-e) , then only phase changes are produced. If X (2j has an imaginary (resistiveJ component, then power transfer, and even gain, can be obtained for one of the inputs.

The idler field (Tesla coUload output) is then:

E,(w" , ) ~ ] E, (') ]' E, (')ex p]i("" - w,)' - i(fip - fi,)']. (24)

where

(fi, - P,)' ~]! """°.4, . 6Z · Ad/III! A, · .-Id/]. (25)

and Es and Ep are defined by equ.s (17). Thus the Ihree-body interactive system of E j , E3 and Ep is defined in terms of the A venor potential:

E,(w"l) ~ lI-i • .4 ,(') - iqv· .4,«))/"'"0< l']

X [-iwAp(t) - ï \ ("\· .4p(t)/wpot]

X exp[i(:....·p - "-' s)t - i(8p - ,O~ )tJ .

The quaternionic impedance for the ose is then :

where the subscrîpts on il. i2 and i3 distinguish the separate field con­ditioning of the waveguîde-like properties of the ose and the suhscrîpts on R, wL and !L'C dist înguîsh circuit. C , elements (1) from oscillator­shuttle, OS, element s (2). As the wa\"cguide properties of circuits, C, (1) arc fundamentally differ€lI t from oscillator· shut lles, OS, (2), distinguish­ing the i" ,l/( = A, x, y = 1, 2,3 . . . ,i>:i~ = -ivi", i" anticommutes \\'ith i,,), is a nccessary for distinguishing the OS and C dynamic interaction of the ose totaJ arrangement.

The distinguishing characteristics of higher order oses is in analogy to the dimensions of the number system. The dimensionality of the real numbers is 20 ; of complex num bers is 2\ ; of quaternions îs 22 ;

34 T. W . Barre tt

of Cayley numbers is 23 ; of "beyond Cayley lIumbers" is 24 ; etc. Each lI umhcr system has an ose de\i.ce a nalog associated wit h a higher-order non linear susceptibility.

Quaternions arc four-dimensional numbers. The Appendix rev iews quat ernion number interactions and gi ves Equ.(24) in quatc l'Il iouic Conn. Figure 3 shows a mapping of a fo ur-dimclisional quaternionic signal in lhrcc-dilll cnsions (one dimension representing t\\"o). Sim il ar arguments apply 10 ose fo r represen la l ions higher than quatern ions, c.g. , Cay­ley lIumbers and involve highcr a rder dcvice nonlincar susœpt ibilities. oses are shawn in Figure 4A-C. Figure 4:\ i5 a quaternion ose with di­mension 22(SU(2) group sym metry), to whicb I1umbcr associat Î\-ity and unique di \"Îsioll applies, but oommutativity does not apply. Figure 4B is a Carley numbcr ose with 23 dimension (SU(3) grouJl sym metry), ro wh ich unique division applies, but l:\.Ssociati\' ity and commutat ivity tloes Ilot apply. Figure 4C is a nu mbcr of 24 dimension (SU(·I) group s~·mmct ry), to which neit her associativity, nor commutati\"il)" Il Of unique didsion applies.

Figure 3.

Tesla's non li near oscillator-s hutt le--cir cuit (ose)" , 35

0 , 0 (Q:o.ll '~o. )

, @ l'> (II)

y 0 ,

, , ,

"' 0' . ,. (aj'l)y • y(o.jJl

F igure 4 .

@ ,

,

I II (II )

r-a ,

'---

, ,

•

• ;

aj'l • jJo. (o.llly • y(o.jJ)

o ~ • j'l • 0 BUT ~ tKlT UHlQUE

0

'( , , ,

~

IV, The virt ues of oscillator-shuttle-c ircuit (OSe) arrange­ments

Among the many virtues, or beneficial a pplications of the ose ar­rangements, a nonexhaust ive list would îndude the fol1o~'ing:

1. Parametric pumping with a one-drive cnergy source system Ure­quency of energy ftow or power control, i,e" Manley-Rowe (1959) rela­tions without an extemal drive oUieT" than a primary source). In con lrast. parametric pumping (th ree-wave mi:xing) in convcntional circuit theory is restriCted ta two active systems (pump and signal). Even in the case of thre€-wave mixing in nonlinear opties. two beam sources are required (cf Kaup, 1976, 1980 ; Zakharov & }'lanakov, 1976). The ose arrange­ment can also be contrasted with conventional harmonic generation, as the idler frequency generated is controlled by choice of signal and pump inductances which may not be harmonically related, ln cont rast ta the com'entional , parametric pumping using OSCs requires only one active system (the pump) permitti ng "energy bleeding" from that acth'e sys­tem, The total OSC power Bow for the second-order ose s~'stem, W,

3. T. \ V. Barrett

is gi ,",~n by the .\lanlcy-Rowe rela.tion (l\lanley & Rowe, 1959):

1\ ' = {c1 j2ii")[{kpT ,: cos2 a" 1 EpT 12 jwp } + {ksT,z cos1 a , 1 E8T 1

2 Jw,}

+ {k,T. : cos2 Qi 1 EiT 1

2 !w, }], (27)

\\"hcre I.:pT.:, k&T,;. k,T,: arc the phases of the pump, signal and idler.

EpT , E~T , S,r are the tolal transmittcd energies, i.e .. t he pm\'ers, of the pump. signal and idler.

<1,,, a~ . ai arc the angles bctween ET(";.) and Er.dw,). Th at is. whereas the pO\\1~ r , IF , i5 an adiabatic invariallt fo r ose

arrangements , il is Ilot fOf COII\"ent ional pa rame! ric circuits which f(>quire ail cx lcruaJ signaJ power source. In the case of convent iona! paramet ric ci rcuits the :\fanley-Rowe relation applics to a com"cntional ci rcuit and ilS power source, together with an external signal po\\'cr source, Le., t he ap plication is to a llonadiabat ic device.

Higher-order ose arrangements permit more complex dynamics. For cxample, due 10 parametric processes. tllird-order ose dC"ice 11011-

linca r susccpt ibilil ics permit ":phase conjugale mirror" - Iike signal reœp­lion and communi cation wil h ca ncellation of noise-in-medium similar la

that achien'd wit h phase conjugale mirroTS based on rad iat ion-matter interact ion.

2. E field control (e nergy per cycle (Jouleslcycle)controQ . T he ose principles permit E field cont rol as exhibit ed in frozen Hcrtzian \\tl\'e generation (Figure 5) (For frozen Hcrtzian wa\'e generat ioll cf: Crollson , 1975; Zuckcr et a l, 1976 ; Prout! & Norman , 19i8; ~lathur et al. 1982; Chang e t al, 1982, 1984).

3. P ha..<;e mod'Jla.tion al rates grealer than t he carrier (pha..se éontrol) is pcrmitted by ose arrangements because the OS or signal circu it is t;tllll rll ined alwllys 10 he the signal oscillator - the load or id 1er always being a fUllct ion of t he circuit , C . T hereforc even if t he signal frequency of the OS is a highcr frcquency than that of pump signal of the primary, the power Hm .... to the idler will be a modulation of the primar)' or pump frequency, rather than a modulation of secondary or signal frcquency.

4. l\oise reduction in communica tions t ransmission due to condi­t ioni ng of fields in higher order sy rn melry fo nn (noise controQ. As an out put from an ose, Le., a t ransmitted wave, is in higher-order group symmetry form (see section nT alJo\'e) and such higher-order symme­t rics have a low probability of occu rrence naturally, environ mental noise,

Tesla. 's n o nlinear oscillator-s hut tle-cir c ui t (o se). 37

which is of lower symmctry form (usually, U{ l ) and has a high prob­ability of na tural occurrence, will be excluded from a receivcr dcsigned for o s e t ransmitted wavc reception. Thercfore in the case of commu­nications, lcss lloise will be processed statistically at a receiver designed fOf SU(2) or higher group syffi metry operat ion, rcsulting in enhanced signal-to-noise.

5. A similar statistical argument holds for lcss loss in power t ransmission by higher-order sy mmetry energy craft ing or cond ilion ing. Higher-order group sy rn rnetry "receiw rs" will have enhan ced reccption over lower-order reccivers, i.e., Icakage to grouud.

The type of anlcuna to be used I\'ith o se de\'iecs is a small loop antenna which has an induct ive reactance (cf. Smit h, 1988), Such an­tennas are the dual oC a short di pale which !Jas a capacita t ive reaetance. Thus a smallloop a ll lenna can be substituted for the ,8 inductance of the o se of Fig. 4A , for t he f!. or 6 ind uctances of Fig. 4B , an d for the !/J or e ind uctances of Fig. 4C, Such antenn<l.S t ould he used in either the send or receive operat ion modes and the oses \\"ould function as "act il'c media" .

V, AJlpe ndix

Quaternions-oven 'icl\'

The algebra of quaternions is the c\'cn subalgeb ra of a three­dimensional Cliffo rd algebra with Euclideun mel ric, A qua ternion is:

where the scalar multiplicat ion is :

and the SUffi is :

x + y = (xo +yo)l + (Xl + yt}i + (X;l + 'Y'l )j + (X3 + Y3) ~"

T he product is :

xy =(xoYo - XIY . - X2YZ - x3Y3)1 + (xoY\ - X\Yo ~ X'lY3 - x3yz)i

+ (:COY2 + X:l"Yo + X3Y. - X\ Y3)j + (:COY3 + X3YO + IIYZ - X:lY . )k,

38

F igure 5.

and

T . W. Barre tt

~ 1 " o • ::~ -.- . • • •

E.,,'erp(.l U1 . tj . R . ~,

Eperp(lw,t -' !l, tl .. Pnmp • le _ :nj .:>::Ir.. l':.· .. rp(-Io.:.o.t) .. Siguai (RIOU) _ R - ~,

r::.ezp (·! !3 .t) .. S ignal Om.agw ary) · IR - :0:, ~

R g Eo"ezp(-lW.t) .. signal (Real)

C" R ... 1R .. E.,,'e%p(.lw ,tl ... 1Eoezp(-1j3. tl .. Sipal (Real ~ lmagI.nuy)

Q .. c .. le .. c ... l(E,.erp(lw ,t) - 1JI:pe%JI(-l p ,tll ._, WU " Il

:o; · xot • ..:.\ +Q:I'" Z3r. r.o .. E.,,'e"'l'(.lw .t) "1 .. E.u:pHp.tl 1:2 .. YE,erp{lw"t) Q '!' YEpezp(-lPptJ

12 = j2 = k2 = _ [2,

l i = i l = i lj = jl = j,

lk = kl = k

j k = - kj = i

ij = -ji= k,

ki = - i k = j .

T he set of quaternions is a dh-iSÎon ring. It sat Îsfi es ail the ILxioms for a

Tesla's non li D(~ar oscillator-shutt le-cir cuit (ose).

field except the commutative law of mu ltiplication.

Let; x = E I exp[i:.....,It - ,6yt], y = Ey exp[i:"""yt - !3~t], and

then: [x , y) = xy - yx,

( x ,y, ~ y,x, YlX2 - XIY2 XOY2 - YOX2

[x,y] ~ 0 YoX) - XOYI

YOX2 - XOY2 I OYI - YOX I 0

X3Y2 - Y3X2 YII3 - XIY3 XOY3 - YOX3

If X () = E; exp(i ... 'st)

X2 = .,JE; exp(i:.vpO

Xl = E. exp(i/36 t ),

X3 = ,JE; exp(i.Bpt),

then

or Equ.(2-1) of the text .

R efer en ces

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(Manuscrit reçu le 11 janvier 1990)