Electronic Circuits Modeling Using Artificial Neural Networks

Transcription

1 JOUNAL OF AUTOMATIC CONTOL, UNIVEITY OF BELGAE, VOL. 3():3-37, 3 Electronc Crcuts Modelng Usng Artfcal Neural Networks Mona Andrejeć and Vančo Ltosk Abstract - In ths paper artfcal neural networks () are appled to modelng of electronc crcuts. s are used for applcaton of the black-box modelng concept n the tme doman. Modelng process s descrbed, so the topology of the, the testng sgnal used for exctaton, together wth the complexty of are consdered. The procedure s frst exemplfed n modelng of resste crcuts. MO transstor, as a four-termnal dece, s modeled. Then nonlnear negate resste characterstc s modeled n order to be used as a pece-wse lnear resstor n Chua s crcut. Examples of modelng nonlnear dynamc crcuts are gen encompassng a arety of modelng problems. A nonlnear crcut contanng quartz oscllator s consdered for modelng. Verfcaton of the concept s performed by erfyng the ablty of the model to generalze.e. to create acceptable responses to exctatons not used durng tranng. Implementaton of these models wthn a behaoural smulator s exemplfed. Eery model s mplemented n realstc surroundng n order to show ts nteracton, and of course, ts usage and purpose. I. INTOUCTION There are two basc approaches to the modelng of electronc crcuts: the physcal and the black-box approach. When the physcal laws undergong the behaour of the component are known one may create a set of expressons (usually by solng dfferental equatons) relatng the nput and output termnals. The obtaned current-oltage relatons are referred to as physcal model of the component. Man adantage of ths concept may be deoted to the exstence of physcal meanng of the coeffcents arsng n the modelng expressons. There are, howeer, many dffcultes n the mplementaton of such models []. Frstly, one rarely knows the physcs of the component n such a detal that enables establshng the mutual domnance of all physcal and technologcal parameters. The number of such parameters s usually so large leadng to ery complex models []. Further, n most cases t s not possble to descrbe the complete behaour by one equaton only hang n mnd dfferent operatng regmes of the component [3]. The equatons descrbng parts of the model, frequently become ncompatble leadng to nonanalytcal oerall approxmatng functon. When no full knowledge of the physcs of the dece s aalable one uses the so called black-box approach. The behaour s captured by measurng of nput (exctaton sgnal) and output (response) quanttes. After that an approxmaton procedure s performed oer the set of measured data n order to get an analytcal expresson Faculty of Electronc Engneerng, Unersty of Nš, Beogradska 4, 8 Nš, erba and Montenegro e-mal:mona@elfak.n.ac.yu, anco@elfak.n.ac.yu conenent for equaton formulaton n the crcutsmulaton process. The queston of the choce of adequate approxmant s crucal for ths type of modelng. In some cases polynomal nterpolaton s used n between two measured ponts [4]. In other cases the complete measurement s descrbed by lnear segments leadng to pece-wse lnear models [], [5]. To our knowledge there s no general recept for the choce of an analytcal functon for ths approxmaton. Man adantage of the black-box approach s related to the fact that one doesn t need to hae full knowledge on the physcs of the dece beng modeled. In general there are no lmtatons about the choce of the approxmants, most frequently, the man restrcton s that they need to be analytcal functon. From the other sde, man problem encountered durng use of ths approach s modelng smultaneously of the nonlnear and dynamc behaour of the dece. In such cases the exctaton sgnal used actates only part of the nner propertes of the deces meanng that the model generated based on one measurement may be nadequate for other exctatons. In addton, the problem of parameterzaton arses. Namely, the model obtaned by the black-box approach s useful only for one dece wth fxed parameters. If parameterzaton s preferred one should use measurements for many components beng produced by araton of one or more parameters and nclude the parameter alue nto the approxmaton process as f t s an nput sgnal. Artfcal neural networks were shown to be an excellent canddate for the approxmant needed n the black-box modelng. The frst example of applcaton of for modelng an electronc dece was gen n [6]. There the output characterstcs of a MO transstor are approxmated by a feed-forward three layer. The mplementaton of such model s lmted by the need of exstence of behaoural smulator beng able to formulate crcut equatons for system contanng smultaneously component descrbed by electrcal equatons and others descrbed by functons (.e. s) [7, 8]. After publcaton of the frst results n [6], s were successfully appled n electronc crcuts modelng seeral tmes [9]. In all these applcatons feed-forward networks were used meanng that only resste propertes of the deces were captured. The frst attempt of modelng dynamc behaour was descrbed n []. A mcro-electromagneto-mechancal actuator was modeled but the modelng was n fact quas-dynamc. Namely, by ts rtue t was possble to separate the resste and the dynamc part of the model. The was appled for the resste part but strongly connected to the rest of the model. In ths paper we wll frst descrbe our method of usng s for modelng of nonlnear resste and dynamc

2 3 ANEJEVIĆ, M., LITOVKI, V., ELECTONIC CICUIT MOELING UING ATIFICIAL NEUAL NETWOK networks. Then we wll exemplfy the complete modelng procedure and the propertes of the models generated. Fnally, we wll demonstrate the possblty of mplementaton of ths procedure nto a behaoural electronc smulator. II. MOELING UING ATIFICIAL NEUAL NETWOK Among the frst attempts to use s for modelng of dynamc nonlnear deces was the one descrbed n []. Here, howeer, the sgnals were transformed so that the approxmaton was performed n the frequency doman. urng the tranng process the error s mnmzed between the spectral characterstcs of the responses of the orgnal dece and the. Here we propose a method where the approxmaton s performed n a natural-tme doman. We start from the fact that the may be used as an unersal approxmant [, 3]. Followng that we propose soluton beng able to explot the propertes of the s. These solutons are related to the followng: synthess of the waeform of the exctaton sgnal, synthess of the topology of the, gettng the complexty of the. Before proceedng to the descrpton of the solutons to the aboe problems we wll descrbe the bascs of feed-forward neural network used as the core approxmant throughout our work. An example of a feed-forward neural network [] s gen n Fg.. It s a fully connected network wth one hdden layer. n, n, and n' are the number of neurons n the nput, hdden, and output layer, respectely. θ j s the threshold of the -th neuron n the j-th layer, whle w(p, j)(q, ) s the weght of the connecton between the j-th neuron n the p-th layer and the -th neuron n the q-th layer. The neurons belongng to the hdden layer are actated by the followng functon: z = () s e λ For the neurons n the output layer we use: y = λ q Fg.. Feed-forward λ and λ are constants, whle z and y are the responses of the neurons n the hdden and the output layer, respectely. In addton: n s = w(, j)(, ) x j θ j= () (3) n (, )(3, ) j 3 (4) j= q = w j z θ x j are the nput sgnals of the correspondng neurons. ynthess of a sgnal for exctaton s a fundamental ssue. The sgnal used n modelng s supposed to actate the complete behaour of the dece to be modeled. In the same tme t s supposed to be ratonal enough to shorten both the modelng process and the smulaton tme. Obserng the C characterstc, for example, t should hae ampltude large enough to actate eery nonlnearty. From the other sde, n order to capture the dynamc propertes, ts spectre should be broad enough to coer the complete "pass-band" of the component. Both the ampltude and the spectre are to be taken nto account when deces wth nherent dynamc nonlneartes are to be modeled. Chrp sgnal s proposed to be used for ths purpose. It s a frequency modulated snusodal sgnal. uch a sgnal s depcted n Fg.. and gen by: t () = I sn( π ( f k t) t) (5) k s calculated from the smulaton tme ( f hgh hghest frequency of nterest ( ): Current [ma] k = f hgh t f fnal t fnal ) and the Tme [s] Fg. Chrp sgnal used for exctaton Optmal network topology s neer known n adance. The soluton should be based on experence and knowledge related to the propertes of s. Topology sutable for modelng of nonlnear dynamc crcuts presented n ths paper s ecurrent Tme-elay Neural Network, Fg. 3 [4]. The network output, denoted by y n, s a functon of the exctaton n the present and seeral preous tme nstants, and n the same tme, s a functon of ts own alues n some preous nstants. The network complexty was always accommodated to the problem complexty whle recommendatons [5, 6] were accepted. In ths paper, all these consderatons wll be appled to modelng of resste and nonlnear dynamc crcuts. (6)

3 JOUNAL OF AUTOMATIC CONTOL, UNIVEITY OF BELGAE 33 xn t t xn xn- t xn- yn- yn- t Fg. 3. ecurrent Tme-elay Neural Network III. yn MOELING OF THE EITIVE CICUIT Frst model of MO transstor, Leel model, bult n the frst erson of PICE, was ery smple. As technology was mong on, there were many new models deeloped, expected to encompass more dfferent effects, so they were becomng more and more complex. At frst, the dmensons of the transstors were drectly bult n the equatons, consdered as ndependent arables. The mproement was accomplshed wth BIM model, where there was new dependence between parameters and dmensons. Ths model also had ts dsadantages, so ts further correcton s now aalable n BIM33 model. As technology and smulaton tools are n the constant progress, t s necessary to nclude more and more physcal effects and technologcal mproements n the model. Ths leads to permanent deelopment of the model by addng new mathematcal expressons and new parameters. o, the actual model of MO transstor s ery complex wth great number of parameters. It s shown that number of parameters s doubled eery ten years [7]. When we use black-box approach, we are not nterested n the physcs of the component, and we do not een hae to know eery one of ts parameters. The only thng s to measure ts nput-output dependences, and then make an approxmaton of ts functon usng neural network, of course. Parameterzaton s also possble, because we can ntroduce dfferent nput to the neural network for eery parameter, and then present correspondng data durng the learnng process. In that way, temperature or some other ambent parameter can be used. In the followng example, a four termnal dece s modeled (Fg. 4). In ths case, the dran current I d s a functon of three arables: V ds, V gs, V bs. The correspondng neural network s a feed-forward network wth one hdden layer, three nputs (current alues of dran-source oltage, gatesource oltage and bulk-source oltage) and one output (current alue of the dran current). Ten hdden neurons are used. It s necessary to nclude as many dfferent combnatons of three nputs as possble, n order to coer all operatng regmes of the transstor. In ths example, oltage appled to all of three nputs s changed n the range from to 5V. G B G B a) b) Fg. 4. a) MO transstor as a four-termnal component, b) model Implementaton of such model s presented n Fg. 5. MO transstor and ts model are loaded by resstance ( L =kω). The alues of resstors n dran and source are: =kω, =kω. It s shown here that ths model can be used n a complex crcut, because t can be loaded and connected to other crcut elements. Transfer characterstcs are depcted n Fg 6. n n d V a) L V b) Fg. 5. MO transstor a), and ts model b), loaded by resstor Modelng of resste crcuts wll also be presented n an example of modelng nonlnear negate resste characterstcs, and ts mplementaton n Chua s crcut, a standard paradgm for studyng chaotc phenomena. In fact, t s one of the ery few physcal systems n whch a formal proof of the exstence of chaos has been accomplshed and n whch the theoretcal, smulaton and expermental results match precsely [8]. It s a chaotc attractor consstng of only one nonlnear element, pecewse-lnear resstor. Under the acton of external perodc exctaton, ths crcut exhbts a large arety of bfurcaton sequences, ncludng perod doublng and perod addng n chaos regme [8]. d L d

4 34 ANEJEVIĆ, M., LITOVKI, V., ELECTONIC CICUIT MOELING UING ATIFICIAL NEUAL NETWOK Fg. 6. Transfer characterstcs of the orgnal crcut and ts model from Fg. 5 Here, an s used for modelng the nonlnear resstor n the Chua s crcut. Concernng that resstance, we know the current-oltage relaton, meanng that current s a functon of ts controllng oltage. In [9] the smulator for pecewse-lnear crcut analyss was used for the modelng of ths dependence. Ths smulator s not n a common use, and, generally, PICE-lke smulators are preferred. We wll present here an model for ths pecewse-lnear characterstc. The new model beng expressed by analytcal functons wll not ask for PWLPICE any more. Crcutry gen n Fg. 7a s called Chua s crcut. g( ) s the pecewse lnear functon, Fg. 7b, gen by: g( ) = m.5( m m ) B.5( m m ) B p m L G C L C g ( ) -Bp a) = g ( ) Bp m p (7) Equaton (7) wth the specfed parameters has three equlbra. Each equlbrum has one real and two complex egenalues. A typcal trajectory n the attractor rotates around one of the two outer equlbra [], suppose the upper one, n a counterclockwse drecton wth respect to the left handed coordnate system. After each rotaton the trajectory gets further from the equlbrum untl a certan tme after whch there are two possbltes: ) the trajectory goes back to a poston closer to the equlbrum and repeats a smlar process, ) the trajectory does not go back to a pont close to the equlbrum but descends downward n a spral path and lands on the lower part of the attractor. The pont where t lands s close to the lower equlbrum and starts rotatng counterclockwse around the lower hole. After ths, the behaour s smlar to that n the upper part of the attractor except for the fact that t starts ascendng after rotatng around the lower equlbrum seeral tmes. Theoretc nterpretaton of the chaotc behaour s gen n []. The parallel connecton of C and L consttutes a lossless oscllatory mechansm n the (V - I L )-plane, whereas the conductance G prodes nteractons between the (C, L)-oscllatory component and the acte resstor together wth C. The acte resstor s responsble for the crcut s chaotc behaour. nce g( ) s acte, t keeps supplyng power to the external crcut. The attractng nature of the chaotc trajectores s, therefore, due to the power dsspated n the passe element. There exst many problems concernng the realzaton of Chua s attractor, because of the nonlnear, negate resstance shown n Fgure 7b. The man problem s to realze a pecewse-lnear characterstc of the resstor. In fact, n the solutons offered n the lterature, there s a smooth transton of the cure near the breakponts. Though the characterstc s resste, a feed-forward network wth one hdden layer would ft qute well. nce current s the nonlnear functon of the controllng oltage, =g( ), a lnear change of oltage s used, n order to capture behaour of the resstance. The actual parameters for the consttute relaton of the pecewse lnear characterstc are: m =-5* -4, m =-8* -4, B p =. Electrcal smulaton of the resstor was performed frst. The change of oltage was lnear, as mentoned aboe, n the range -3V 3V. For the example used here, one nput, four hdden, and one output neuron were ncorporated. The current-oltage relaton of the model s shown n Fg. 8 together wth the orgnal current-oltage relaton used for tranng. Excellent agreement of the relatons was obtaned. In order to show the qualty of the approxmaton procedure and the generalzaton capabltes of the, the new model s mplemented n a double scroll crcut wth the followng parameter set (Fg. 7a): C =.55µF, L =7.7mH, C =.495µF, G=7m. Projectons onto three dfferent planes are presented n Fg. 9. b) Fg. 7. a) Chua s crcut, b) Consttute relaton of the nonlnear resstor Ths crcut behaes lke an chaotc attractor only f we specfy parameter alues exactly (C, C, L, G, m, m, B p ).

5 JOUNAL OF AUTOMATIC CONTOL, UNIVEITY OF BELGAE.3 orgnal Fg. 8 Consttute relaton of the nonlnear resstor n Fg. 7 descrbed n smulator and model [V] a) [V] MOELING OF NONLINEA YNAMIC CICUIT The motaton for modelng of ths knd of crcuts appeared wth the problem of modelng mplanted hearng ads []. Here, howeer, n order to present reproducble results the nonlnear crcut, Fg., contanng quartz crystal, Fg., wll be consdered for modelng. The schematc symbol for a quartz crystal s shown n Fg. a. The equalent crcut for a quartz crystal near fundamental resonance s shown n Fg. b. The equalent crcut s an electrcal representaton of the quartz crystal s mechancal and electrcal behaor. The components C, L, r, are called the motonal arm that represents the mechancal behaor of the crystal element. C represents the electrcal behaor of the crystal element and holder []. C s motonal arm capactance representng the elastcty of the quartz, the area of the electrodes on the face, thckness and shape of the quartz wafer. Values range n femtofarads. [V] L s motonal arm nductance representng the bratng mechancal mass of the quartz n moton. Low frequency crystals hae thcker and larger quartz wafers and range n a few Henrys. Hgh frequency crystals hae thnner and smaller quartz wafers and range n few mllhenrys..3.. r represents the real resste losses wthn the crystal. C s shunt capactance representng the sum of capactance due to the electrodes of the crystal plate plus stray capactances due to the crystal holder and enclosure c) Fg. 9. ouble scroll attractor wth model performng lke negate resstance. Projectons onto: a) (- ) plane, b) (- )plane, c) (- )-plane IV [V] b) Crystal has two resonant frequences characterzed by a zero phase shft. The frst s the seres resonant, fs frequency. The equaton s: fs = (8) π LC When the crystal s operatng at ts seres resonant frequency the mpedance wll be at a mnmum and current flow wll be at a maxmum. The second resonant frequency s the ant-resonant fa, frequency. The equaton s:

6 36 ANEJEVIĆ, M., LITOVKI, V., ELECTONIC CICUIT MOELING UING ATIFICIAL NEUAL NETWOK f a = π CC L C C When the crystal s operatng at ts ant-resonant frequency the mpedance wll be at a maxmum and current flow wll be at a mnmum. As an example of modelng of nonlnear dynamc crcuts, the electronc crcut depcted n Fg. wll be modeled. The par of branches contanng dodes s ntroduced enablng the nonlnearty of the crcut to be accounted for. Values of the elements n the crcut are: C =.8pF, L =mh, r =3Ω, C =4.5pF, =kω, =kω, E =E =V. esonant frequency of the crystal oscllator s 8MHz, meanng that both f s and f a are close to that alue. C C L r a) b) Fg.. a) Crystal equalent crcut and b) ts symbol E E Fg.. Nonlnear dynamc crcut chosen for modelng o, a chrp (t) sgnal s needed to coer the frequency band around 8MHz. eccurent tme delay neural network wth fe nput, four hdden and one output neuron s used, because the structure from Fg. s hghly nonlnear. The response of ths crcut excted by a chrp sgnal wth the change of frequency from MHz s gen n Fg.. eres resonant frequency can be notced frst, and then, ant-resonant frequency [V] f s Tme[ms] Fg.. esponse of the crcut, Fg., excted by a chrp sgnal The responses of the modeled crcut and the model are shown n Fg. 3. It s obtaned as an enelope of the tme doman response [3]. f a (9) [V] orgnal Tme[ms] Frequency [MHz] Fg. 3. esponses of the orgnal crcut (Fg. ) and the model (only the enelopes of the poste perods are shown) V. CONCLUION Artfcal neural networks are used for applcaton of the black-box concept n the tme doman for modelng of electronc crcuts. The topology, the testng sgnal used for exctaton, and the complexty of the are consdered. Modelng of resste crcuts s shown on two examples: MO transstor modelng, and modelng of negate nonlnear resstance. Nonlnear dynamc crcuts are also modeled and erfed by gettng acceptable responses to exctatons not used durng tranng. Implementaton of all these models wthn a behaoural smulator s exemplfed. EFEENCE [] Chua, L., and Ln, P., Computer-Aded Analyss of Electronc crcuts: Algorthms and Computatonal Technques, Prentce-Hall, New Jersey, 975. [] Cheng, Y., BIM33 Manual (Fnal Verson), Unersty of Calforna at Berkley, EL, 996. [3] Foty,., MOFET Modelng Crcut mulaton, IEEE Crcuts and eces, Vol. 4, No 4, July 998, pp [4] hahram, M., Calbratng mulatons Tools for Nanometer esgns, IEEE pectrum, Vol. 36, No 6, June 999, pp [5] Katzenelson, J., An Algorthm for olng Nonlnear esste Networks, Bell ystem Techncal Journal, Vol. 44, October 965, pp [6] Ltosk, V.B., et all., MO transstor modelng usng neural networks, Electronc Letters, Vol. 8, No. 8, 99, pp [7] Glozć,., et all., Alecss, the mulator, Elektronsk Fakultet, LEA, Nš, Yugoslaa (nternal report) 996. [8] Ltosk, V., Maksmoć,., and Mrčarca, Ž., Mxed-sgnal Modelng wth AleC: pecfc features of the HL, mulaton, Practce and Theory, Elseer, Vol. 8,, pp [9] Wang, F., and Zhang, Q-j., Knowledge-based Neural Models for Mcrowae esgn, IEEE Trans. on MTT, Vol. 45, No., ec. 997, pp [] Ltosk, V., Mrčarca, Ž., Ilć, T., "mulaton of non-lnear magnetc crcuts modeled usng artfcal neural network", mulaton Practce and Theory, Elseer, No. 5, 997, pp [] Cttero, C., Pur,V., occa, L., Functon Approxmaton-A Fast Conergence Neural Approach Based on pectral Analyss, IEEE Trans. on Neural Networks, Vol., No. 4, July 999. [] Hecht-Nelsen,., Neurocomputng, Addson-Wesley Publshng Company, eadng, MA, 989.

7 JOUNAL OF AUTOMATIC CONTOL, UNIVEITY OF BELGAE 37 [3] carsell, F., Tso, A.C., Unersal Approxmaton Usng Feedforward Neural Networks: A urey of ome Exstng Methods, and ome New esults, Neural Networks, Elseer cence, Vol., No., pp 5-37, 998. [4] Ilć, T., Zarkoć, K., Ltosk, V.B., and Mrčarca, Ž., Applcaton n Modelng of ynamc Lnear Crcuts, Proc. of the mall ystems mulaton ymposum, ', Nš, Yugoslaa, eptember, pp [5] Masters, T., Practcal Neural Network ecpes n C, Academc Press, London, 993. [6] Amrkan, B., Nshmura, H., What ze Network Is Good for Generalzaton of a pecfc Task of Interest?, Neural Networks, Vol. 7, No., pp. 3-39, 994. [7] Ltosk V.B.: Projektoanje elektronskh kola: smulacja, optmzacja, testranje, fzčko projektoanje, Noa Jugoslaja, Vranje,. [8] Jose M. Cruz, Leon O. Chua: A CMO IC Nonlnear esstor for Chua s crcut, IEEE Trans. Crcuts yst:-fundamental Theory and Applcatons, Vol. 39, No., pp , ec. 99. [9] Leon O. Chua, Ln Yang, Cellular Neural Networks: Applcatons, IEEE Trans. Crcuts yst., ol. 35, no., pp. 73-9, Oct [] Takash Matsumoto, Leon O. Chua, Motomasa Komuro: The double scroll, IEEE Trans. Crcuts yst., Vol. CA-3, pp , Aug [] Ball, G., and Maxfeld, B., Floatng Mass Transducer for Mddle Ear Applcatons: An Analytcal Model, prate communcaton, 997. [] Crystal Oscllator Bascs and Crystal electon for rfpic and PICmcro eces. [3] Andrejeć, M., Mloanoć,., Petkoć, P., Ltosk, V.: Extracton of Frequency Characterstcs of wtched-capactor Crcuts Usng Tme-oman Analyss, Proceedngs of the Conference MIEL, May, Nš, Yugoslaa, pp