O the affie liearity i circuit thery Emauel Gluski The Kieret Cllege the Sea f Galilee; ad Ort Braude Cllege (Carmiel), Israel. gluski@ee.bgu.ac.il; http://www.ee.bgu.ac.il/~gluski/
E. Gluski, O the affie liearity NDES 202 [my first lecture there] ABSTRACT (SUMMARY) fr the vixra pstig Accrdig t the defiiti f the liear peratr, as accepted i system thery, a affie depedece is a liear e. This implies the liearity f Thevei's -prt, while the battery itself is a strgly liear elemet that i the -prt's "passive mde" (whe the -prt is fed by a "strger" circuit) ca be replaced by a hardlimiter. Fr the thery, t the actual creati f the equivalet -prt, but the selecti f e f the prts f a (liear) may-prt fr iterpretig the circuit as a -prt, is imprtat. A practical example f the affie liearity is give als i terms f wavefrms f time fuctis. Emphasizig the imprtace f the affie liearity, it is argued that eve whe straighteig the curved characteristic f the slar cell, we retai the mai part f the liearity. Fially, the "fractal-ptetial" ad "f-cecti-aalysis" f - prts, which are missed i classical thery, are metied. 2 Ad w, the rigial slides start:
E. Gluski, O the affie liearity, The questi is NDES 202 N cfrtati is expected here regardig the defiiti, rigiatig frm aalytical gemetry, f the affie depedece. The questi is whether we have affie liearity, r affie liearity. Thugh the latter pssibility is smewhat paiful, because it readily meas that Thevei (Nrt, Helmhltz) equivalet is a liear circuit, the aswer may deped ly which psiti we are stayig: that f system thery, r that f the gemetry, ad e wishig t stay the psitis f system thery shuld accept this ews. 3
E. Gluski, affie liearity, The affie depedece,.. NDES 202 Plaimetry itrduces affie depedece as the straight lie: y = ax b; () a, b -- cstats Fr x > 0, the particular case f y a= 0, ie.. y b which is als see as liear depedece: y b 0 x is a direct aalgy t the fllwig circuit: 4 0 Liear depedece i plaimetry. x x(t) V CC (y saturated) x( t) > 0 y( t) Vcc The saturated (liear) amplifier
E. Gluski, O the affie, Liearity i System Thery NDES 202 I system thery (r fuctial aalysis, r thery f peratrs) the fllwig defiiti f liearity is used: 5 = Fr this is the liear scalig As well, we have p p = p p p= p= y ( k x ) k y ( x ),, {{ k },{ x }}. p i.e. ay liear map satisfies 0 0. Hece, affie map that des t satisfy 0 0, is a liear map. That is, i system thery affie depedece is affie liearity ( ANN ). p y ( k x ) = k y ( x ). y ( x x ) = y ( x ) y ( x ), (2) superpsiti!
E. Gluski, O the affie liearity, A remark NDES 202 Substitutig it we btai Thus, if y = ax b y( k p x p ) = k p y( x p ) p= p= b k =, p= = m= p k b p. [ () ] [ (2) ] ( * ) -- as if {k p } are sme prbabilities, -- the () satisfies (2), ad we have a filtrati f the liear effect. { } Thugh, as the pit f priciple, cstraits are permitted fr liearity f a system, let us give als a circuit realizati fr this specific case, bservig hw a structure ca realize the prbability cditi ( * ): k p 6
E. Gluski, O the affie liearity, A circuit realizati f the specific case NDES 202 E R Z ip is ifiite axb p p = p p p= p= y( k x ) k y( x ) E 2 R 2 E 0 axb Here, " x " E : p p 7 E E E 2 E R 3 axb axb axb equivalet t: k k 2 () k Σ E p= = E p R p Kirchhff s laws p= prvide R ( * ). p p= p p p m= p R = k E, k = ; p= R k p =. This is ( * ). m
E. Gluski, O the affie liearity, Oe-prt, equivalet t () NDES 202 Returig t the affie characteristic y = ax b by itself, we rewrite it i EE tis: v= Ri E This is the prt-characteristic f may circuits f which the simplest e is the Thevei s equivalet: R E (E Th ) (R Th ) i a v b v= Ri E Ntice the directi f i(t). 8
E. Gluski, O the affie, Our -prt fed by a active circuit NDES 202 r: E Th RTh i a v Ntice the directi f i(t). v= Ri E b This case f i eterig the circuit (ad sig[i(t)] = cst) is equivalet t the fllwig bviusly liear circuit: R Th i v z = E Th a v This is the partial (t fr ay sig[i]), passive mde equivalece that is imprtat fr creati f liear resistrs v(i), see Slide. 9 b
E. Gluski, O the affie liearity, A mre geeral view ANN NDES 202 E ANN E 2 The chse utput The same utput E Th E 3 4-prt (a) -prt (b)! (a) is a liear 4-prt if we accept (recgize) all f the prts. If we recgize ly e prt, i.e. start t see this whle circuit as a -prt, the the circuit becmes ANN, i the sese f (). 0 (b) is the (liear) Thevei equivalet f (a), after (a) is recgized as a -prt. Hwever, actual realizati f the equivalet versi is NOT ecessary fr the pit: whe a liear N-prt (N >) (a) is apprached as a -prt, it becmes a liear (ANN) circuit.
E. Gluski, O the affie liearity, The aximatic aspect NDES 202 That such reducti f the umber f the defied prts is the reas fr liearity, is csidered als i E. Gluski, A exteded frame CASS Newsletter, Dec. 20; ad with mre stress the aximatic side i: E. Gluski, A Applicati f Physical Uits (Dimesial) Aalysis t the Csiderati f Nliearity i Electrical Switched Circuits its, Circuits Syst.. Sigal Prcess vl. 3, 737 752 752 (Apr. 202). where als LTV systems are cmpared with NL systems.! Carefully defie yur system! Where are the prpsed surces, iside r utside? If the iputs are defied s that the system appears t be active, as i case (b) (mre geerally, at least e f the prts is rejected as iput), -- it is NL. E 2 E 3 E 4-prt (a) The chse utput E Th -prt (b) The same utput
E. Gluski. O the pedaggical side NDES 202 The very sigificat (ad uique whe cmpared t ther umerus cmm textbks basic circuit thery) atteti t the liear resistive -prts i Deser&Kuh (ad i the kw bk by Prfessr L.O. Chua) shuld have, histrically, the ANN f a -prt as sme backgrud. Ideed, it is reasable t start the tpic f creati f the liear resistive characteristics v(i), frm the simple ANN characteristic: Our -prt: v(i) v i A "strger" circuit 2 A -prt is used fr creati f a liear resistr
E. Gluski, O the affie liearity, A lk at ZIR ZSR NDES 202 A dyamic versi y( t) = y(0) f ( t) ( Tx ˆ )( t) ZIR ZSR A utlk the iitial cditis: Sice y(0) is give by us, it is als a iput Is it the case (a) f: T [ y(0), x( t)] y( t) y(0) x(t) LTI (dyamic) y(t) x(t) The same system y(0) y(t) i.e. the liear e (superpsiti), (a) (b) r the case (b) f: 3 x( t) y( t) (with zer ZIR spilig superpsiti) i.e. the NL e? Ntice that i the dmai f the Laplace variable, the iterpretati f the iitial values f state variables as iputs is eve a stadard e. Hwever, the ZIR ZSR sluti s structure exists als fr LTV circuits.
E. Gluski, O the affie li, Watch the ANN cmpets! NDES 202 Let us csider the slar cell characteristic: i(v) i = v/r L Writig i( v) = i(0) ( i( v) i(0)) = i(0) f ( v), ( where f (0) = 0), 0 A pwer-supply uit v e ca see the affie kerel i(0) as the mai liear term, defiig the pwer supply t the lad.! E Nte that E -3 ca be physical iputs (here su radiati), t ecessarily batteries. 4 E 2 E 3 4-prt (a) The chse utput E Th -prt (b) The same utput
E. Gluski. ANN i terms f a steady-state NDES 202 Csider w x y give i the terms f time fuctis as Lˆ y( t) = Lˆ x( t) ψ ( t) 2 (a) with sme liear peratrs ad ψ kw. Obviusly, it is the same ANN as (). Example fr (a) Csider: di L Asig[ i( t)] i( t) dt = v( t) = Uξ( t) dt C (3) where ξ is a T-peridic (usually, sie) give fucti r wave-frm, rmed i sme way. Equati (3) plays a imprtat rle i a liear thery f flurescet lamp circuits very imprtat pwer csumers [see, e.g., [6] E. Gluski, O the thery f a itegral equati, Advaces i Applied Mathematics, 5(3), 994 (305-335), ad als: IEEE Tras. CAS, Pt., May 999]. As well, a mechaical versi f (3) is kw i the thery f systems with Culmb fricti. (The, eergy csumpti is t the mai tpic.) 5!
! E. Gluski, With equati (3): the way t a ANN starts NDES 202 The 50-60 Hz L-C-flurescet lamp circuit:: v(i) The KVL gives i C L v - lamp Uξ(t) (Usiωt) - i U is chaged usig labratry varjak. 6 di L A sig[ i ( t )] i ( t ) dt = U si ω t dt C where L, A, C ad U are psitive cstats. Eq.(3a) ad the circuit are liear, bviusly, ad is the parameter f the liearity. (3a) The The resistive resistive (light-emittig) (light emittig) term term is is physically physically mst mst imprtat; imprtat; LC LC is is the the lamp s lamp ballast ballast. x = A / U
7 E. Gluski, O the affie liearity, The lg way t ANN NDES 202 x= A/ U Asig[ i( t)] Fr prperly limited, i(t) is ([6]) a zercrssig fucti: ad the, is a rectagular wave. Usig als that (similarly t the iput fucti): i( t T / 2) = i( t), we have A sig[i(t)] as the simple square wave 4A si ω ( t t( x)) Asig[ i( t)] =. π,3,5,... This equality has the frm f F( t,{ t }) = Aζ ({ t t ( x)}); t = t (md T), k k k k with the zercrssigs f i(t) as parameters, adζkw. Thus, (3) becmes di L F( t,{ tk}) i( t) dt = Uξ( t). dt C (3b)
E. Gluski, O the affie, The cstacy f the zercrssigs NDES 202 If ω = LC 2ω where ω is the basic frequecy f the peridic iput, the (see [6]) are cstat, i.e. umved with the permitted chage f U. That is, thus ad t ( x) t (0) [ x= A/ U], k is kw befre i(t) is determied. I this case, after rewritig (3) as k ζ ({ t t ( x)}) ζ ({ t t (0)}), k F = Aζ = Asig[ i( t)] ~ A e 8 ca mistakely cclude that this is a liear equati (system). k di L i ( t ) dt = Uξ ( t ) Aζ ({ t tk (0)}) dt C t k This meas that whe U is chaged, t k are t shifted The whle right-had side is cmpletely kw
E. Gluski, O the affie liearity, This is the ANN NDES 202 Hwever, sice the lamp (r the mechaical Culmb-fricti uit) remais i the actual circuit, the liearity must remai, ad i fact is a ANN equati: r di L i ( t ) dt = Uξ ( t ) Aζ ({ t tk (0)}) dt C ( Ti ˆ )( t) F( t,{ t (0)}) = Uξ( t), 2 k Lˆ y( t) = Lˆ x( t) ψ ( t). ˆT is the liear peratr f the L-C sub-circuit Of curse, the liearity als has t be well see via pwer features f the circuit, ad ideed fr ω, as fr ay ther, we / ω ω / ω d t have (see the refereces) fr the average pwer 2 P ~ U, which wuld be ecessary fr ay liear circuit i the peridic steady state. 9
E. Gluski, O the affie liearity, Back t the algebraic ANN NDES 202 Yu see that ANN ca take the duties f a sigular liearity! ------------------------------------------------------------------------------------------ Let us retur, hwever, t the algebraic characteristic, itrducig w a quatitative measure fr ANN. The, we shall a bit cmplete the classical thery f -prts t which the ccept f ANN belg. Observe that we deal with the algebraic -prts, -- t ecessarily resistive, pssibly als magetic ad ferrelectric. (Csider, e.g., the magetic circuits with reluctaces ). 20
E. Gluski, O the affie The quatitative ANN NDES 202 v= Ri E Returig t the simple case f, let us defie, fr sme quatitative estimatis, the measure f affie liearity as ANN e E Ri usig sme stadard. i Thus defied, ANN e ca be chaged, fr istace, by meas f parallel cectis f Thevei s -prts: R Th E Th R Th 2 E Th 2 R Th E Th R 0 E 0 2 (a) (b)
E. Gluski, O the affie Parallel cecti f the -prts NDES 202 Fr Fig. (b)( here: R Th E Th R Th 2 E Th 2 R Th E Th R 0 E 0 we have with (a) ANN e = E R i (b) E = ( ET h k / RT h k ) k= / RT h k k= ad R = / RTh k k= 22
E. Gluski, ANN Circuits ad "What is Life?" The parallel cecti (ct) NDES 202 R Th E Th R Th 2 E Th 2 R Th E Th R 0 E 0 (a) (b) Takig, fr simplicity, all E ad all R i Fig.(a) similar, we have i (b): which gives E = E, ad R = R / e E E ANN = = ~. R i Ri 23!
24 E. Gluski, ANN... Whe, NDES 202 While the aalytical side we btai, as, ( ANN ~ ), the structural side we have R i.e. circuit (b) becmes pure vltage hardlimiter f --------------------------------------------------------- e = R / 0, E = E. T this simple trasfer t the hardlimiter we fid a trivial aalgy i elemets with (geerally very imprtat, see belw) pwer-law i(v) characteristic (4) ad just as it is/was with the slar-cell characteristic i Slide 3, ANN ca be cected with (4), i.e. a affie kerel like E is bserved i (4). R Th E Th (a) R Th 2 E Th 2 R Th E Th A ew far-reachig pit starts here! i ~ v α Nte, here, this, (b) R 0 E 0 is a sigle elemet.
E. Gluski, The affie kerel i i ~ v α NDES 202 Ideed, rewritig the pwer-law cductivity characteristic i ~ as the dimesially mre reasable: with sme give i ad v, we have fr the respective v(i), the limit v α i / i = ( v / v ) α, that is, / α v / v = ( i / i ), v v α as it is fr ANN whe. 25 R ~ 0
E. Gluski, ANN... The similarity ad disticti NDES 202 Cmpare the tw trasfers t hard-limiters: v v E " if " 0 0 v= ( R / ) i E ( ) v 2 v / v = ( i / i ) i ( α ) "α if " i i / α 2 26 Remark: Fr 0< i<, bth trasfers are -uifrm.
E. Gluski, ANN... The affie kerel NDES 202 Usig that fr ε l a<< a ε ε l a, we btai fr v / v = ( i / i ) / α, v v= v l( i / i ), α >> l( i / i ), α i.e. the affie kerel v is separated i the pwer-law characteristic. Thus, fr the mutual limitati α ad i, a circuit mdel f the pwer-law elemet (csumig eergy, i.e. with i directed iside) ca ivlve vltage hardlimiter r battery. Thus, it appears that the kerel feature f ANN ca be istructive als fr this liearity. 27
E. G. A step twards circuit cmplexity (pard!): The pwer-law characteristic ad fractal -prts NDES 202 Dealig with ANN circuits, we deal with the very basic ccept f -prt, ad w we are i psiti t bserve tw remarkable features f -prts (t metied i the classical thery f algebraic circuits) the secd f which is assciated with a ew circuit cecti.. Observe that sice each circuit brach is a - prt, each -prt is a specific ptetial fractal. There is the pssibility f repeatig the whle structure i each brach, r i sme f the braches. This recursive repetiti wrks very well with the pwer-law elemets, i ~ v α, because the the iput cductivity, is f the same type (~ v ipα ), just as it is fr α = (liear resistrs) r ly liear capacitrs r idictrs. Advice: make cmputer simulatis f such recursive prcedure, ad study i = F( v ). 28 ip ip a - b i ~ v α
E. Gluski, ANN... The f-cecti NDES 202 The specific features f the -prts with i ~ v α suggest wrkig with the mre flexible fr applicatis 2 i( v) Dv α α = D2v T make sme cclusi f this mdel, let us itrduce a ew circuit cecti, amed f-cecti, which relates t circuits f the same tplgy ad is a geeralizati f the usual parallel cecti (i geeral, f t ecessarily -prts). G(v i ) F m (v i ) a b f m -circuit F(v i ) F m ct (v i ) a b f m ct -circuit v i - F (v i ) v i - F ct (v i ) a a f -circuit b f ct -circuit b 29 Usual parallel cecti f-cecti
E. Gluski, ANN... f-cecti ad a use f the pwer-law characteristic NDES 202 The btaiig f a circuit with the prescribed tplgy, α 2 2 cmpsed f elemets i( v) = D v D v : α Here, each f-cected circuit has similar elemets i its braches. v i F(v i ) - a b a b - f () a b - f (2) v i F(v i ) α V α 2 i = D v α i = 2 D v α 2 30
E. Gluski, ANN... f-cecti ad the iput curret f the α-circuits NDES 202 Fr the tw basic (idividual) iitial states we have at the de i fcus: {v, r v 2, r v} 3 v i D v : = ; v i D v 2 : ; 2 2 v < v< v 2 2 2 2 2 2 D v D v D v D v ; α α = F(v α i ) Remark: Values f D ad D 2 d t ifluece the iitial vltage distributis, just scale the iput currets, i.e. F ~ D. After f-cecti, we have That is, hypthesis Ff cct ( vip ) F ( vip ) F2 ( vip ) apprximate aalytical superpsiti. v i α 2 V The right-had side relates t the usual parallel cecti Pssible applicatis:. Perclati thery (pwer degrees). 2. Spatial filterig (hmgeeus structures).?
E.Gluski, Numerical Example E. Gluski NDES 2 Errr i the aalytical superpsiti (bserve the tw vertical circuits) F cct F,3 F 3 cct f-cecti: Left side: α = Right side: α =3 v i (v i =, D = D 3 = ). Results: The circuit F F cct Percet chage α =:.4.466 4.7% α = 3:.4.044-8.4% f-cecti 2.5 MatLab simulati: Usual parallel cecti gave 2.54.5% errr versus 2.5, 32 i.e. F = F < F F,3 f cected 2 ~
E. Gluski, ANN... Sme refereces the pwer-law characteristic NDES 202 Sme wrks the circuits cmpsed f the elemets with the pwer-law characteristic : i ~ v α "Oe-prts cmpsed f pwer-law resistrs", IEEE Tras. Circuits ad Systems II: Express Briefs 5(9), 2004 (464-467). O the symmetry features f sme electrical circuits, It'l. J. f Circuit Thery ad Applicatis 34, 2006 (637-644). f - cecti: a ew circuit ccept, IEEE 25th Cveti f Electrical ad Electrics Egieers i Israel ( IEEEI 2008 ), 2008, 3-5 Dec., pp: 056 060. "A estimati f the iput cductivity characteristic f sme resistive (perclati) structures cmpsed f elemets havig a tw-term plymial characteristic, Physica A, 38, 2007 (43-443). A apprximati fr the iput cductivity fucti f the liear resistive grid, It l. J. f Circuit Thery ad Applicatis, 29, 200 (57-526). 33 See als my ArXiv wrks devted t α-circuits ad apprximate aalytical superpsiti.
E. Gluski O the affie liearity (ANN) i circuit thery NDES 202 END f the lecture 34