Frequency-Domain Steady-State Analysis of Circuits with Mem-Elements

Transcription

1 ELECO 211 7h Inernaional Conference on Elecrical and Elecronics Engineering, 1-4 December, Bursa, TURKEY Frequency-Domain Seady-Sae Analysis of Circuis wih em-elemens Zdene Kola 1, Dalibor Biole 2, Viera Biolova 1 1 Brno Universiy of Technology, Technica 358/1, 616 Brno, Czech Republic ola@feec.vubr.cz, biolova@feec.vubr.cz 2 Universiy of Defence Brno, Kounicova 65, Brno, Czech Republic dalibor.biole@unob.cz Absrac The paper deals wih frequency-domain models of memrisor, memcapacior, and meminducor, and heir use in he seady-sae analysis by means of he harmonicbalance mehod. The models are based on a polynomial approximaion of consiuive relaions, allowing analyical formulaion of relaions beween he specral componens of simulus and response, for boh periodic and quasi-periodic seady-sae condiions. I is no necessary o ransform signals beween he ime and he frequency domains o obain he mem-elemen response. Example analyses demonsrae he model use. 1. Inroducion The seady-sae analysis represens an imporan ool for he characerizaion of elecrical circuis. In he case of linear imeinvarian circuis, he harmonic seady sae can be obained using he Heaviside operaor mehod, which is well-nown as he AC analysis. The purpose of he paper is o presen simple analyical frequency-domain models of mem-elemens, which are suiable for seady-sae analysis by means of he Harmonic Balance mehod [1]. The fabricaion of memrisor in 28 in he form of a nanoscale device by HP laboraories [2] riggered a wave of ineres in mem-sysems. As he HP memrisor is sill no available for researchers, any verificaion of proposed circuis is based on simulaions or emulaions [3], [4]. A number of models of memrisor have been proposed so far; see [5], [6], [7] and he references herein. The models are more or less based on he HP memrisor, which is no an ideal elemen bu a generalized memrisive sysem [8]. In addiion o memrisive sysems, here are memcapaciaive and meminducive sysems as well as heir special subsysems, namely memcapaciors and meminducors [9], [1]. An enirely new approach o modeling was proposed in [11] and [12]. The proposed models are based on explici consiuive relaions of ideal mem-elemens. For example, he wo possible consiuive relaions of ideal memrisor have he form ( and q q( ), (1a,b) ) d, q ) d (2a,b) are he flux and he charge, respecively, while ( and q() are he nonlinear consiuive funcions for charge- and fluxconrolled memrisors, respecively. emrisor erminal quaniies are hen [11] d( d( R (, (3a) d d ) ) d G ( ), (3b) d d d d( R ( and ) G ( ) are he d memresisance and he memconducance, respecively. The mehod of harmonic balance (HB) [1] is suiable for cases he waveforms of all circui quaniies can be approximaed by a relaively small number of harmonic componens X (), i.e. by N x( X e, (4) N he se of frequencies is chosen before he analysis. The por volage-curren relaions of individual newor elemens have o be ransformed ino he operaor domain, assigning a vecor of specral componens of response quaniy o a given vecor of simulus. I is a rivial as for linear circui elemens. However, i requires a raher complicaed processing in he case of nonlinear componens including ransformaions beween he frequency and he ime domains [13]. Alhough he memrisor is a nonlinear elemen, he model (1)-(3) allows an explici calculaion of specral componens of he response variable in case consiuive relaion (1) is expressed as a polynomial. Secion 2 of he paper deals wih deriving frequency-domain models of mem-elemens, and secion 3 gives a numerical example. 2. Frequency-Domain odels of em-elemens 2.1. emrisor In he case of he charge-conrolled memrisor, erminal volage v can be calculaed from he independen erminal curren i (simulus) as follows q(r (q = R (, (5) and vice versa for he flux-conrolled memrisor j 37

2 ELECO 211 7h Inernaional Conference on Elecrical and Elecronics Engineering, 1-4 December, Bursa, TURKEY (G ( = G (). (6) Le us suppose ha he consiuive relaion (1a) of a chargeconrolled memrisor is expressed in he polynomial form as ( r q, (7) 1 r are he polynomial coefficiens. Le us furher suppose ha he sysem under analysis is in he periodic seady sae wih one fundamenal frequency, and ha he ime domain waveforms are approximaed by N harmonics. Then he independen erminal curren i can be expressed as N N ( ) I, (8) e j I () are he complex specral componens. The charge is hen N I j () ) d e. (9) j N q( Q As can be seen, (9) imposes a resricion on he DC componen of curren, I () =, and inroduces he DC componen of charge Q () as anoher unnown variable. Thus he se of equaions of he harmonic balance mehod for he whole circui will be exended by one equaion and one unnown for each memrisor. The condiion I () = can be seen as a necessary condiion for he exisence of seady sae. The nex sep consiss in he calculaion of memresisance (3a). Wih respec o consiuive relaion (7), R will be again a polynomial of q Le us consider he square of charge R 1 ( r q. (1) 1 2 N 2N min( N, N ) 2 j j j j q Q e ( ) ( ) ( ) Q Q e N 2N jmax( N, N) (11) Q () are he specral componens. Afer performing (11), he number of harmonics is doubled. The operaion is equivalen o he (full) convoluion of vecors of specral componens Q. Formally, he operaor of muliplicaion in (1) will be replaced by he operaor of convoluion. The specrum of R will be 1 R r ( ) Q. (12) 1 The number of harmonics of R depends on he degree of polynomial (1). Finally, he vecor of harmonic componens of he erminal volage will be V I R. (13) Since he number of specral componens of he erminal volage should be he same as he number of componens of he erminal curren, vecor V should be runcaed ino he inerval N,..., +N. The model of a flux-conrolled memrisor can be derived in he same way. In he muli-one case, frequencies in (4) are no ineger muliples of he fundamenal frequency, he procedure for he calculaion of frequency-domain response will be similar. Wih respec o he mapping of specral componens ino a one-dimensional array, (12) and (13) will no be expressed in erms of he convoluion operaor emcapacior The consiuive relaion of ideal volage-conrolled memcapacior is (), (14) q( ) d (15) is he inegral of charge [1]. The memcapacior charge and curren are hen d ( ) d ( ) d q( C ( ) (16) d d d d C ( ), (17) d d ( ) C ( ) is he memcapaciance. d The erminal curren i can be calculaed from he independen erminal volage v (simulus) as follows (C ( =d/d ( C ()). (18) The specrum of flux can be calculaed similarly o (9), which imposes he resricion V () =, and inroduces he DC componen of flux () as anoher unnown variable. Similarly o (7) and (1), consiuive relaion (14) will be assumed in he polynomial form wih coefficiens c. Then he specrum of memcapaciance will be ( C c 1 ). (19) 1 Finally, he vecor of harmonic componens of he erminal curren will be I ( V C ), (2) of differeniaion in he frequency domain, i.e. each specral componen is muliplied by. The consiuive relaion of ideal charge-conrolled memcapacior is [1] 38

3 ELECO 211 7h Inernaional Conference on Elecrical and Elecronics Engineering, 1-4 December, Bursa, TURKEY The memcapacior volage is hen ( ). (21) L 1 l 1 ( ) Q. (31) d( ) d( ) d D ( ) q(, (22) d d d d( ) D ( ) is he inverse memcapaciance. d The erminal volage v can be calculaed from he independen erminal curren i (simulus) as follows q((d ( = q( D (). (23) The specra of charge and is inegral can be calculaed similarly o (9). There will be wo resricive condiions, I () = and Q () =, and he DC componen of he inegral of charge S () will be inroduced as anoher unnown variable. Similarly o (7) and (1), consiuive relaion (21) will be assumed in he polynomial form wih coefficiens d. Then he specrum of inverse memcapaciance will be 1 D d ( ) S. (24) 1 Finally, he vecor of harmonic componens of he erminal volage will be 2.3. eminducor V Q * D. (25) The consiuive relaion of ideal curren-conrolled meminducor is (, (26) ( ) d (27) is he inegral of flux [1]. The meminducor flux and volage are hen d( d( ( L ( (28) d d d L (, (29) d d( L ( is he meminducance. The erminal volage v can be calculaed from he independen erminal curren i (simulus) as follows q(l (q =d/d ( L (). (3) Similarly o (19), he specrum of meminducance will be The vecor of harmonic componens of he erminal curren will be V ( IL ). (32) The consiuive relaion of ideal flux-conrolled meminducor is [1] The meminducor curren is hen q q(). (33) ) ) d ( ) ( (34) d d d ) ( ) is he inverse meminducance. d The erminal curren i can be calculaed from he independen erminal volage v (simulus) as follows (( ( = ( (). (35) Similarly o (9), here will be wo resricive condiions, V () = and () =, and he DC componen of he inegral of flux F () will be inroduced as anoher unnown variable. Then he specrum of inverse meminducance will be 1 ( ) F (36) 1 and he vecor of harmonic componens of he erminal curren will be I *. (37) 3. Linear Circui wih em-elemen Le us consider a connecion of a linear circui, which may include independen sources, wih one mem-elemen, Fig. 1. For curren/charge-conrolled mem-elemens, he linear par can be replaced by he Thévenin equivalen circui represened by (2N+1) specral componens of equivalen volage V () and series impedances Z (). KVL for he loop in Fig. 1a can be wrien as V Z I V ( I ), = -N,..., N, (38) V ( I ) is he -h specral componen of mem-elemen volage, which is a funcion of all specral componens of curren I. For volage/flux-conrolled mem-elemens, he Noron equivalen circui in Fig. 1b gives I Y V I ( V ), = -N,..., N. (39) 39

4 ELECO 211 7h Inernaional Conference on Elecrical and Elecronics Engineering, 1-4 December, Bursa, TURKEY If he memrisor is conneced o a linear circui, here is no means how o deermine he mem-elemen inernal sae by a linear observer in order o se or sabilize he DC componens of inegrals (see Secion II). Thus he necessary condiion for he () () exisence of he seady sae, I or V, should be fulfilled by design. Then he DC componens can be arbirary, i.e. he sysem has an infinie number of seady saes. i I (1), ma 1 2 Q () = Q () =5 mc Q () =1 mc linear newor v Z () I () I () f, Hz V () V () I () Y () V () Fig. 3. Resonance curves of I (1) for differen values of Q (). a) b) Fig. 1. Linear circui wih mem-elemen and is Thévenin (a) and Noron (b) equivalen circui for individual specral componens. v (, V.1 f = 2 Hz Wih respec o he symmery of specral componens, (38) and (39) represen N complex equaions for N complex unnowns (or equivalenly 2N real equaions for 2N real unnowns) in he implici form F( ) or F ( ). (4) I The soluion of (4) represens an approximaion of he seady sae by an a priori chosen number N of harmonics. The iniial guess may be given by he AC soluion for I (1) or V (1). Le us consider he single-one circui in Fig. 2 wih currenconrolled meminducor. The meminducor consiuive relaion (26), (31) is characerized by wo polynomial coefficiens, l 1 =.1 H and l 3 = 1 3 HC -2 (). The necessary condiion I is fulfilled by design. Parameers were chosen such ha he resonance occurs a 15.9 Hz, considering only he linear erm l 1 of (26). v = 1 2 V 1mF Fig. 2. Example of linear circui wih curren-conrolled meminducor. The mehod of harmonic balance was experimenally implemened in alab. Equaion (4) was solved using he fsolve funcion. Figure 3 shows he seady-sae ampliude of I (1) for differen values of Q (). I was calculaed for he number of harmonics N = 1. I is eviden ha he resonance frequency depends on he DC componen of charge. Figure 4 shows he seady-sae waveforms of meminducor volage for Q () = 1 mc for frequencies of 2 Hz and 8 Hz. i v v (, V ime, s 2 f = 8 Hz ime, s Fig. 4. Time-domain waveforms of v ( for Q () = 1mC. 4. Conclusions The frequency-domain models presened in he paper allow formulaing analyically he relaions beween he specral componens of simulus and response, boh for periodic and quasi-periodic seady-sae condiions. I is no necessary o ransform signals beween he ime and he frequency domains o obain he mem-elemen response. Acnowledgmen This wor has been suppored by he Czech Science Foundaion under gran No P12/1/1614, he Czech inisry of Educaion under research projec No S , by he projec CZ.1.7/2.3./2.7 WICOT of he operaional programme Educaion for compeiiveness, and Projec for he developmen of K217 Deparmen, UD Brno odern elecrical elemens and sysems. The research leading o hese resuls has received funding from he European Communiy's Sevenh Framewor Programme (FP7/27-213) under gran agreemen No

5 ELECO 211 7h Inernaional Conference on Elecrical and Elecronics Engineering, 1-4 December, Bursa, TURKEY 7. References [1] R. J. Gilmore,. B. Seer Nonlinear circui analysis using he mehod of harmonic balance-a review of he ar, In. Journal of icrowave and illimeer-wave Compuer- Aided Engineering, 1991, vol. 1, no. 1, pp [2] D. B. Sruov, G. S. Snider, D. R. Sewar, R. S. Williams The missing memrisor found, Naure, 28, vol. 453, pp [3] Y. V. Pershin,. di Venra Pracical approach o programmable analog circuis wih memrisors, IEEE Transacions on Circui and Sysems I, 21, vol. 57, pp [4] J. Valsa, D. Biole, Z. Biole An analogue model of he memrisor, In. Journal of Numerical odelling: Elecronic Newors, Devices and Fields, 21, published online in Wiley Online Library. DOI: 1.12/jnm.786. [5] Z. Biole, D. Biole, V. Biolova SPICE model of memrisor wih nonlinear dopan drif, Radioengineering, 29, vol. 18, no. 2, pp [6] A. Rá, G. Cserey acromodeling of he emrisor in SPICE, IEEE Transacions on Compuer-Aided Design of Inegraed Circuis and Sysems, 21, vol. 29, no. 4, pp [7] O. Kavehei, A. Iqbal, Y. S. Kim, K. Eshraghian, S. F. al- Sarawi, D. Abbo The fourh elemen: characerisics, modelling, and elecromagneic heory of he memrisor, Proceedings of he Royal Sociey Aahemaical Physical and Engineering Sciences, 21, vol. 466, no. 212, pp [8] L. O. Chua and S.. Kang, emrisive devices and sysems, Proc. of he IEEE, vol. 64, no. 2, pp , [9]. di Venura, Y.V. Pershin and L.O. Chua, Circui elemens wih memory: memrisors, memcapaciors and meminducors, arxiv: v1 [cond-ma.mes-hall] 23 Jan 29. [1] Biole, D., Biole, Z., Biolová, V. SPICE odeling of emrisive, emcapaciaive and eminducive Sysems, Proceedings of he 19h European Conference on Circui Theory and Design, ECCTD '9, 29, Analya, Turey, pp [11] V. Biolova, Z. Kola, D. Biole, Z. Biole emrisor modeling based on is consiuive relaion, Proceedings of he European Conference of Circuis Technology and Devices 211, Tenerife, Spain: NAUN, 21, pp [12] D. Biole, Z. Biole, V. Biolová PSPICE modeling of meminducor, Analog Inegraed Circuis and Signal Processing 211; 66(1): [13] A. Brambilla ulione signal harmonic balance mehod, Elecronics Leers, 1999, vol. 35, no. 21, pp