A General Subthreshold MOS Translinear Theorem

A Gene al Sub h eshold MOS T anslinea Theo em Te esa Se ano-Go a edona’, Be nab6 Lina es-Ba anco’, and And eas G. And eou2 Na ional Mic oelec onics Cen e (CNM), Ed. CICA, AV. Reina Me cedes sln, 41012 Se illa, Spain * The Johns Hopkins Uni e si y, 3400 N. Cha les S ee , Ba on Hall, Bal imo e, MD 21218, USA Abs ac This pape ou lines he condi ions unde which he T anslinea P inciple can be ully exploi ed o MOS ansis o s ope a ing in sub h eshold. Due o he exponen ial na u e o sub h eshold MOS ansis o s he T anslinea p inciple applies immedia ely as long as he Sou ce- o-Bulk ol ages a e made equal o ze o (o cons an ). This pape add esses he condi ions unde which sub h eshold MOS ansis o s s ill sa is y a T anslinea p inciple bu wi hou imposing his cons ain . I is ound ha he T anslinea p inciple esul s in a mo e gene al o mula ion han he o iginal o BJTs since now mul iple T anslinea loops can be in ol ed. The cons ain o e en numbe o ansis o s is no longe necessa y. I. In oduc ion The anslinea p inciple, in oduced by Ba y Gilbe in 1975 [I], is one o he mos impo an ci cui heo y con ibu ions in he elec onics e a. In i s o iginal o mula ion, he anslinea p inciple p o ides a simple and e icien way o analyze and syn hesize nonlinea ci cui s based on bipola junc ion ansis o s (BJTs). Due o hei exponen ial cha ac e is ics, he anslinea p inciple can be ex ended o MOS ansis o s ope a ing in weak in e sion [2], [3] wi hou and wi h loa ing-ga e de ices [4]. Fo MOS ansis o s ope a ing abo e h eshold he e has also been ound a simila way o analyze and syn hesize nonlinea ci cui s [5]. Fo bipola ansis o s one p ac ical p oblem ha may equi e some a en ion when applying he anslinea p inciple is he nonze o base cu en [6]. In con as , he anslinea p inciple holds o MOS sub h eshold ansis o s in an exac manne i sou ce and bulk a e sho -ci cui ed. Howe e , i has been ound ha he p inciple holds as well in an exac manne unde di e en ci cums ances [2]-[3], al hough a gene al sub h eshold MOS anslinea heo em has no been de ised un il now. In his pape we p o ide his gene al heo em and ou line he condi ions unde which sub h eshold MOS ansis o s, iewed as ou e minal de ices, sa is y a gene al anslinea p inciple. The ope a ion o a sub h eshold MOS can be desc ibed by he ollowing equa ion [2], [7]-[9] whe e V h = KTIq is he mal ol age, I, is a posi i e cons an cu en , S is he ansis o size ac o (S = WIL , whe e W is ansis o wid h and L is i s leng h), and K is a echnology dependen posi i e pa ame e . This equa ion holds ue as long as whe e @FB is he de ice’s la band ol age [lo]. Vol age VBs can ake ei he posi i e o nega i e alues as long as he pa allel PN diode junc ion is biased below i s o wa d conduc ion h eshold ol age. Pa ame e K is known o ha e a sligh dependency on ol age VBS [2]. Howe e , in his pape we will assume K o be cons an , which is a easonable assump ion i ca e is aken o make he VBs ol ages simila o all ansis o s. Fo ope a ion in sa u a ion eq. (1) can be simpli ied o (i VDs >> V h) (3) and be ew i en as whe e I (a “no malized cu en ”) is ansis o cu en no malized wi h espec o ansis o size ac o S = WIL, and i,, i, a e dimensionless numbe s called pseudo-cu en s and equal o (5) GS‘ Ih . ‘BSI ‘1 h i, = e iB = e Le us use he symbol in Fig. 1 o ep esen a weak- in e sion MOS in sa u a ion. Le us call he pa h ha goes om he ga e e minal G o he sou ce e minal S he G - b unch (o Ga e-b unch), and he pa h ha goes om he bulk e minal B o e minal S he B - b unch (o Bulk-b unch). We a e using a diode- like symbol o ep esen he exponen ial ela ionship be ween he ol age o he b anch and he cu en lowing ou o he de ice and a capaci i e-like e mina ion o each diode symbol o ep esen he capaci i e coupling na u e o he Ga e and Bulk e minals. I Vss = 0 (o cons an ) he e is an exac exponen ial ela ionship be ween VGs and ZDs (see eq. (3)) and he o iginal BJT anslinea o mula ion can be di ec ly and exac ly applied. 11. Gene alized T anslinea Theo em o Sub h eshold MOS T ansis o s In his sec ion we will conside he condi ions unde which anslinea p inciples can be applied o ci cui s PB S Fig. 1: T anslinea Symbol Rep esen a ion o Sub h eshold MOS Tkansis o in Sa u a ion 0-7803-5471 -0/99/$10.0001999 IEEE 11-302 Fig. 2: Example o Coupled G-loops using anslinea symbol ep esen a ion Fig. 3: Illus a ion o he G-o de concep . De ices ‘1-2-3- 6’ o m a G-loop which is coupled o G-loop o med by de ices ‘2-4-5’, because de ices ‘1-2-3-4’ o m a B-loop. De ices ‘1-2-3-4-5-6’ o m a Closed T anslinea Se , and so do de ices 6 and 7. De ice 2 has a G-o de o nc2 = 2 because i s G-b anch belongs o wo G-loops o he same Closed T anslinea Se . All o he de ices ha e G-o de one. wi h sub h eshold MOS ansis o s bu wi hou imposing he cons ain o making Vss = 0. We will in oduce i s some p elimina y heo ems and de ini ions, and hen s a e and p oo he gene alized anslinea heo em o sub h eshold MOS de ices. The i s concep s o be in oduced a e G-loop and B-loop. A G-loop (o Ga e-loop) is a closed loop o G- b anches, and a B-loop (o Bulk-loop) is a closed loop o B-b anches. Fo hese loops we can s a e anslinea heo ems o hei pseudo-cu en s, whose p oo is iden ical as o he o iginal BJT T anslinea Theo em. Theo em: In a G-loop con aining an a bi a y numbe o G-b anches, he p oduc o pseudo- cu en s o b anches cpnnec ed in he Clock-Wis~G(CW) di ec ion is equal o he co esponding p oduc o b anches connec ed in he Coun e -Clock-Wise (CCW) di ec ion. No e ha since pseudo-cu en s a e dimensionless en i ies we can ha e an a bi a y numbe o b anches o ien ed CW and ano he a bi a y numbe o b anches o ien ed CCW (as opposed o he case o eq. (7)). A comple ely equi alen heo em holds di ec ly o B- Up o now hings a e simila o classical anslinea loops, excep ha an a bi a y numbe o b anches a e allowed. Howe e , he p esence o wo exponen ial b anch ol ages in eq. (3) is wha makes sub h eshold MOS anslinea loops mo e gene al and complica ed han he classical ones. A i s consequence o his ac is he ollowing concep o coupled loops: De ini ion: Two G-loops a e said o be coupled i a leas one MOS de ice o he i s G-loop and a leas one o he di e en de ice o he second G-loop sha e hei espec i e B-b anches in a common B-loop. loops. This is illus a ed in Fig. 3. De ices ‘1-3-5’ o m a G- loop and de ices ‘2-4-6’ o m ano he G-loop. Howe e , de ices ‘1-2-3-4’ o m a B-loop, hus making he p e ious wo G-loops o be coupled h ough he B- b anches o de ices 1,2,3 and 4. An equi alen de ini ion applies o coupled B- loops. No e ha wo loops may ha e a common b anch wi hou being necessa ily coupled loops. In he example o Fig. 3, we can w i e o he wo G-loops ( ‘SS4 + ‘LIS6 - ‘BSZ) = Ioe whe e all pseudo-cu en s i, ha e cancelled ou by applying he abo e Theo em. Howe e , due o B-loop ‘1-2-3-4’, ‘BSI + ‘BS4 = ‘LIS2 + ‘BS3 (7) which in oduces a coupling be ween he wo equa ions in (Il), and makes G-loops ‘1-3-5’ and ‘2-4-6’ o be coupled loops. When de ices o m mul iple ouching loops i is no clea which ones o choose o how many o choose. Fo example, in Fig. 4 one can choose G-loops ‘1-2-3-6’, ‘6- 7’, and ‘2-4-5’. Bu why no conside ‘1-3-4-56’, ‘6-7’, and ‘2-4-5’, o ‘1-3-4-5-7’ and ‘1-2-3-6’. One can y all possible op ions as long as one chooses a se o Non- Redundan (NR) loops: De ini ion: A se o loops is said o be Non- Redundan (NR) i he sum o b anch ol ages o any loop canno be exp essed as a linea combina ion o he sum o b anch ol ages o o he loops in he se . Fo example, in Fig. 4, o G-loops ‘1-2-3-6’, ‘2-4- 5’, and ‘1-3-4-5-6’ hei espec i e sum o b anch ol ages is, ‘GS6 + ‘GS, = ‘CS2 + ‘GS3 “GS2 + VGS4 = VGSS VGS6 + VCSl+ VGS4 = VGS3 + VGSS (8) Any o hese h ee equa ions can be exp essed as a linea combina ion o he o he wo. Thus he h ee G- loops do no o m a Non-Redundan se o G-loops. Howe e , any wo o hese h ee G-loops do o m a Non-Redundan se o G-loops. The ac ha sub h eshold MOS ansis o s can o m coupled loops yields na u ally o he ollowing concep o Closed T anslinea Se , De ini ion: Gi en a se o MOS de ices, and once a Non-Redundan se o loops has been chosen, a Closed T anslinea Se (CTS) is a se o de ices such ha all loops hey o m a e only coupled among hem, bu no .o loops 11-303 -I A + L BHCL 8 1V3TH BF l 8T&J T- Fig. 4 Example o Closed 3’5 T anslinea Se s. MOS De ices 7 and 8 o m a Closed T anslinea Se , and so do MOS De ices 1-7. whe e b anches o o he de ices (no belonging o he CTS) a e p esen . This is illus a ed in Fig. 5. Le us selec he NR se o G-loops ‘1-2-3-7’, ‘4-5-6’, and ‘7-8’, and he NR se o B-loops ‘1-2-3-4-56’, ‘7’, and ‘8’. G-loops ‘1-2-3-7’ and ‘4-5-6’ a e coupled because he e is a e B-b anches o de ices o bo h G-loops ha a e sha ed in he common B-loop ‘1-2-3-4-5-6’. The wo G-loops ‘1-2-3- 7’ and ‘4-5-6’, and he wo B-loops ‘1-2-3-4-5-6’ and ‘7’ a e no coupled o o he loops (ei he G-loop ‘7-8’ no B-loop ‘8’), hus ( o he chosen NR se o loops) de ices ‘1-2-3-4-5-6-7’ o m a Closed T anslinea Se . On he o he hand, G-loop ‘7-8’ is no coupled o any o he loop, no a e B-loops ‘7’ and ‘8’. The e o e, de ices ‘7-8’ o m ano he Closed T anslinea Se . When wo king wi h mul iple G-loops and B-loops, wi h some o hem being coupled, i is no e y con enien o classi y each b anch as being CW o CCW o ien ed, as will become appa en la e . Le us ins ead classi y all b anches in o wo o ien a ion g oups, he a-wise o ien ed b anches and he p-wise o ien ed b anches. Two b anches a e classi ied in o he same g oup (ei he he a-wise o he p-wise) i hey appea in he same loop wi h he same o ien a ion. On he con a y, wo b anches a e classi ied each in o a di e en g oup (one in o he a-wise, he o he in o he p- wise) i hey appea in he same loop wi h opposi e o ien a ion. No e ha now a CW b anch in one loop and a CCW b anch in ano he loop can be classi ied in o he same a-wise o p-wise g oup. I a b anch is sho - ci cui ed, i o ms a one b anch loop and can be classi ied as ei he a-wise o p-wise. Ano he concep ha is use ul o s a ing he gene alized anslinea sub h eshold MOS heo em is ha o G-o de and B-o de o a MOS de ice in a Closed T anslinea Se : De ini ion 4: Once a NR se o loops has been chosen, a sub h eshold MOS ansis o which is pa o a Closed T anslinea Se is said o ha e a G-o de o alue nc, i i s G- b anch belongs o nc G-loops o he gi en Closed T anslinea Se . An equi alen de ini ion o B-o de can be s a ed o B- loops. The concep is illus a ed in Fig. 4. Le us choose he NR se o G-loops ‘1-2-3-6’, ‘2-4-5’, and ‘6-7’ and o B-loops ‘1-2-3-4’, ‘5’, ‘6’, and ‘7’. G-loops ‘1-2-3-6’ and ‘2-4-5’ a e coupled because de ices ‘1-2-3-4’ o m a B-loop. The e a e no o he couplings among he chosen loops. Consequen ly, de ices ‘ 1-2-3-4-5-6’ o m a Closed T anslinea Se which consis s o G-loops ‘1- 2-3-6’ and ‘2-4-5’ and B-loops ‘1-2-3-4’, ‘5’, and ‘6’. De ices 6 and 7 o m one G-loop (‘6-7’) and wo B- loops (‘6’ and ‘7’) which a e no coupled o any o he loop. The e o e de ices 6 and 7 o m ano he Closed T anslinea Se . MOS de ice 2 has G-o de nG2 = 2 because i s G-b anch appea s in wo G-loops o he same Closed T anslinea Se . MOS de ice 6 does no ha e G-o de 2 because, al hough i s G-b anch belongs o wo di e en G-loops, hese wo loops do no belong o he same Closed T anslinea Se . All MOS de ices ha e B-o de one because hei B-b anches appea only in one B-loop. When a G-b anch has G-o de g ea e han one i belongs o mo e han one G-loop o he same Closed T anslinea Se . In such cases i is possible ha he b anch be classi ied as a-wise in some G-loops and as p- wise in o he G-loops. Unde hese ci cums ances i is con enien o di ide i s G-o de in o wo pa s (9) nc = na.,+n B, G whe e na (le us call i G-a-o de ) deno es he imes his G-b hnch is classi ied as a-wise in a CTS, and n (le us call i G-p-o de ) deno es he imes i is clhied as p-wise in CTS. Simila ly, o B-b anches, he B-o de can be sepa a ed in o he B-a-o de ( na, ) and he B-p-o de ( nP, B). Using he concep s and p elimina y heo ems in oduced un il now, i is possible o s a e and p oo he gene alized anslinea heo em o sub h eshold MOS ansis o s’: Theo em: Gi en a se o sub h eshold MOS de ices and choosing o hem a se o Non- Redundan G-loops and B-loops, o each Closed T anslinea Se he ollowing can be s a ed: I i is possible o ind an a- and p-wise classi ica ion o hei G-loops and B-loops such ha a) he sum o G-a-o de s equals he sum o G- P-o de s Zna,,, = Cnp,cl (10) J E { a-w se) Is {p-wise} bland, e e y ime a de ice’s G-b anch is classi ied as a-wise in a G-loop i s B-b anch can be classi ied as a-wise in some B-loop, and e e y ime a de ice‘s G-b anch is classi ied as P-wise in a G-loop i s B-b anch can be classi ied as p-wise in some B-loop, hen he p oduc o no malized cu en s aised o he powe o hei G-a-o de o all ansis o s in he CTS whose G-b anches ha e been classi ied a-wise equals he p oduc o no malized cu en s aised o he powe o hei G-P-o de o all ansis o s whose G-b anches ha e been classi ied P-wise. P oo : Fo each G-loop in he Closed T anslinea Se he ollowing holds: 11 iG1 le {p-wise) 1. The heo em will be s a ed using G-b unches as p ima y b anches and making B-b anches o depend on hem. Howe e , because o he symme y be ween G-b unches and B-b unches (due o he symme y be ween V,, and V,, ol ages in eq. (3)), he heo em can be s a ed as well by in e changing G-b unches and B- b unches. 11-304 Since his is ue o e e y single G-loop we can mul iply hese equa ions o he chosen se o Non- Redundan G-loops and hei p oduc will s ill be equal o uni y, Fu he mo e, we can aise i o he powe o K, and i s ill be equal o uni y, No e ha , since he de ices o m a CTS, all b anches will be p esen and no b anch o ano he CTS appea s. Consequen ly, eq. (18) includes all G-b anches o he CTS and only he b anches o his CTS. Equi alen ly, he same can be s a ed o all B-loops, bu aising now o he powe o 1 - IC, o con enience. No e ha , due o s a emen b) in Theo em 3, e e y ime a de ice has i s pseudo-cu en iGj in he nume a o o eq. (1 8), i s pseudo-cu en iBj will also appea in he nume a o o eq. (19), and bo h will appea na, cj imes. And he same holds o pseudo-cu en s in he denomina o s o eqs. (18) and (19). The e o e, le us de ine na, and n such ha 1 Also, since he de ices o m a Closed T anslinea Se , eq. (1 8) includes all de ices o he CTS, and so does eq. (1 9). Consequen ly, we can mul iply eqs. (1 8) and (1 9) and index he iG and i pseudo-cu en s o he same de ice wi h he same su&c ip and use his subsc ip o index he MOS de ice, On he o he hand, due o s a emen a) in he heo em he ollowing is sa is ied By mul iplying eqs. (21) and (22), and using eq. (4) we ob ain which concludes he p oo o he Gene alized Sub h eshold MOS T anslinea Theo em. 0 111. Re e ences B. Gilbe , “T anslinea Ci cui s: A P oposed Classi ica ion,” Elec onics Le e s, ol. 1 I, No. 1, pp. 14-16, 1975; e a a, ol. 11, No. 1. A. G. 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