Accurae Tunable-Gain 1/x Circui Using Capacior Charging Scheme Byung-Do Yang and Seo Weon Heo This paper proposes an accurae unable-gain 1/x circui. The oupu volage of he 1/x circui is generaed by using a capacior charging ime ha is inversely proporional o he inpu volage. The oupu volage is independen of he process parameers, because he oupu volage depends on he raios of he capaciors, resisors, and curren mirrors. The volage gain of he 1/x circui is uned by a 1-bi digial code. The 1/x circui was fabricaed using a.18 μm CMOS process. Is core area is.11 mm (144 μm 78 μm), and i consumes 78 μw a DD = 1.8 and f CLK = 1 MHz. Is error is wihin 1.7% a =.5 o 1. Keywords: 1 over x, analog divider, process independen, capacior charging, unable gain. I. Inroducion The analog 1/x circui is used in analog dividers, filers, fuzzy conrol, neural neworks, and A/D converers [1][6]. Several circuis for he 1/x funcion have been proposed. A volagemode analog divider is accurae, bu i needs a highly accurae wide-range volage conrolled oscillaor [1]. A curren-mode 1/x circui using weak-inversion MOSFETs has low accuracy due o very low reference and inpu currens, which are only under a few nanoamperes []. A volage-mode analog divider [3], volage-mode 1/x circui [4], and curren-mode analog divider [5] have good accuracy; however, heir oupus significanly change due o device parameers and emperaure. In his paper, an accurae unable-gain 1/x circui is proposed. The oupu volage of he 1/x circui is generaed by using a capacior charging ime ha is inversely proporional o he inpu volage. The oupu volage is independen of process parameers, because all variaions from acive and passive devices are canceled ou. In addiion, he volage gain of he 1/x circui is uned by a 1-bi digial inpu code. The res of his paper is organized as follows. Secion II describes he proposed unable-gain 1/x circui. Secion III shows he measuremen resuls of he fabricaed chip. Finally, conclusions are drawn in Secion I. Manuscrip received May 13, 14; revised Dec. 8, 14; acceped Jan., 15. This research was suppored by he Basic Science Research Program hrough he Naional Research Foundaion of Korea (NRF) funded by he Minisry of Educaion, Science and Technology (15R1D1A3A117756). The chip fabricaion was suppored by he IC Design Educaion Cener (IDEC). Byung-Do Yang (bdyang@cbnu.ac.kr) is wih he Deparmen of Elecronics Engineering, Chungbuk Naional Universiy, Cheongju, Rep. of Korea. Seo Weon Heo (corresponding auhor, seoweon.heo@hongik.ac.kr) is wih he School of Elecronic & Elecrical Engineering, Hongik Universiy, Seoul, Rep. of Korea. II. Proposed 1/x Circui The proposed 1/x circui generaes an oupu volage ( ) ha is inversely proporional o he inpu volage ( ). Figures 1 and show he schemaic and operaion of he 1/x circui, respecively. Iniially, he capacior C 1 samples and holds by he rese signal so ha he volage of C 1 ( 1 ) is updaed o. The rese signal also discharges he capaciors 97 Byung-Do Yang and Seo Weon Heo 15 ETRI Journal, olume 37, Number 5, Ocober 15 hp://dx.doi.org/1.418/erij.15.114.577
Curren-adjusing circui DD 1:M DD CNT[1:N] Oupu volage generaor Rese DD BIAS1 1:1 DD BIAS1 I B=M I DD DD 1:K BIAS1 1 + I I C 1 + A + I I B@=1 B + I I R A Rese C A R B Rese C B C R R Fig. 1. Schemaic of proposed 1/x circui. Rese DD DD DD DD 1 I A I B B Q A Q B T CHG T CHG I B I B_MAX B_MAX (1) 1 = 1 () I = = RA RA Fig.. Operaion of proposed 1/x circui. (3) QA I TCHG CA C R C A A A (4) TCHG I (5) IB M I M R B (6) QB IB TCHG CB B_MAX (7) B_MAX I B T C B CHG R C M R C (8) B_MAX M A A B B @ R R, C C A B A B (9) B_MAX R (1) K R (11) M @ R R, C C A B A B R K R C A and C B o he ground. The inpu volage is convered o a curren I by an operaional amplifier, a resisor R A, and a curren mirror. The curren I charges capacior C A during he capacior charging ime (T CHG ) unil he volage of C A ( A ) reaches he reference volage ( ). When A =, he charge sored in C A (Q A = C A ) is equal o he produc of I and T CHG., 1 (1) BIAS1 W=1 I W=1/ W=1/4 W=1/ N CNT[1] CNT[] CNT[N] I B = M I Fig. 3. Curren-adjusing circui. I / R / R, () 1 A A Q C I T. (3) A A CHG From equaions () and (3), he capacior charging ime (T CHG ) is expressed as follows: T C R C A A A CHG. (4) I A reference curren (I = /R B ) is generaed from he reference volage ( ) by an operaional amplifier and a resisor R B. In he curren-adjusing circui in Fig. 3, he reference curren (I ) is muliplied by M for a curren I B (= M I ) using a binary weighed curren mirror and N-bi digial conrol code (CNT[1:N]). 1, where CNT[ ]. (5) N IB M I M M K K RB 1 During T CHG, he swich () urns on, and he curren I B charges capacior C B such ha he charge sored in C B (Q B = C B B_MAX ) becomes he produc of I B and T CHG, where B_MAX is he maximum volage of C B afer he capacior charging operaion. ETRI Journal, olume 37, Number 5, Ocober 15 Byung-Do Yang and Seo Weon Heo 973 hp://dx.doi.org/1.418/erij.15.114.577
Q I T C. (6) B B CHG B B_MAX From equaions (4), (5), and (6), B_MAX is expressed as follows: B_MAX IBTCHG RA CA M, (7) C R C B B B where R A = R B and C A = C B ; B_MAX can be simplified as follows:. (8) B_MAX M Here, B_MAX is inversely proporional o, which is a funcion of 1/x. The unable-gain 1/x circui is achieved by changing he value of M of he curren-adjusing circui. Alhough B_MAX is inversely proporional o, B changes during he capacior charging operaion. A sampling circui composed of a sampling swich and a capacior is used for he coninuous oupu volage of he 1/x funcion. Afer every capacior charging operaion, he capacior C samples and holds B_MAX by he (sample-and-hold) signal. If C is much smaller han C B, hen he volage of C ( ) becomes B_MAX.. B _ MAX (9) An oupu-volage generaor wih an operaional amplifier, a curren mirror, and wo resisors (R and R ) is designed, because he minimum oupu volage of he 1/x circui is. The designed oupu-volage generaor makes easily by reducing he curren going hrough he resisor R. The oupu curren (I ) is K-imes larger han I (= /R ). The oupu volage ( ) is expressed as follows: R. (1) K R From equaions (8), (9), and (1), is simplified as follows: M, where R K R. (11) When a resisive load is conneced o he oupu node, changes in such a way ha an addiional volage buffer is required. However, in many applicaions, he 1/x circui has a high-impedance oupu; hus, no volage buffer is required, because he 1/x circui drives he oher circuis in he same chip. Figure 4 shows he waveforms of he 1/x circui when he inpu volage ( ) changes. The curren I charging he capacior C A is proporional o such ha he charging ime (T CHG ) is inversely proporional o when he volage of C A ( A ) reaches. Therefore, he maximum volage of C B ( B_MAX ) is also inversely proporional o, because C B is I A B Fig. 4. Waveforms of 1/x circui. Table 1. Design and adjusable parameers of 1/x circui. Design parameers Adjusable parameers C A = 1 pf, C B = 1 pf, C =.1 pf R A = 4 kω, R B = 16 kω K = 5, R = 1 kω, R = 5 kω M = 1 for CNT[1:8] =.1 1 Gain 1 charged wih a consan curren I B during T CHG. The oupu volage ( ) becomes a coninuous volage ha is inversely proporional o by sampling B_MAX every clock cycle. From equaions (7), (9), and (1), he oupu volage of he 1/x circui is expressed as follows: R R C. (1) A A M K R R B C B The 1/x circui was implemened wih a.18 μm CMOS process a DD = 1.8. By applying he design parameers in Table 1, (1) can be simplified as follows: Gain M, where Gain. (13) 4 The gain is conrolled by he parameers M and. The parameer M can be changed beween and 1 by he 1-bi digial conrol code (CNT[1:1]), and he reference volage can be changed beween.1 and 1. Therefore, he allowable range of he gain of he 1/x circui is from o 1. 974 Byung-Do Yang and Seo Weon Heo ETRI Journal, olume 37, Number 5, Ocober 15 hp://dx.doi.org/1.418/erij.15.114.577
().8.7.6.5.4.3..1.4 = =.5.8 =.5.8.1..3.4.5.6.7.8 () (a) Table. Simulaion resuls of 1/x circui. Process.18 μm CMOS Supply volage ( DD ) 1.8 Clock frequency (f CLK ) 1 MHz Inpu volage ( ).5.8 Oupu volage ( ) 1 Accuracy of.5% < Error <.5% for =.1.8 1.5% < Error <.5% for =.5.8 Simulaion condiions =.4/ @ M =.5, =.8 Error of (%) 1..5.5 1. 1.5.1..3.4.5.6.7.8 () (b) Table 3. Error sources of 1/x circui. Source Number Sandard deviaion (σ) Offse volage of amplifier 4.7% (1.4 m @ =. ) Mismach of curren mirror 3.18% Resisor mismach.7% Capacior mismach 1.7% Toal error N/A 1.45% () 1..9.8.7.6.5.4.3..1 M =.5 () = M =.5 (1) =.5.8.1..3.4.5.6.7.8 () (c) M (.8) 4 M =.15 (.5) Fig. 5. Simulaion resuls of 1/x circui a M =.5 and =.8 : (a) vs., (b) accuracy of, and (c) vs. M. Figures 5(a) and 5(b) show he simulaion resuls of he 1/x circui a M =.5 and =.8. The 1/x circui has errors less han.5% and 1.5% for =.1 o.8 and =.5 o.8, respecively. The error increases when is under.1, because he offse volages of he amplifiers become he more dominan facors as decreases. Figure 5(c) shows he gain variaions of he 1/x circui according o M a =.8. The simulaion resuls in Table show he high accuracy and wide inpu and oupu ranges of he 1/x circui. In (1), he oupu of he proposed 1/x circui is independen of he process parameers. However, he oupu is sill affeced by he mismaches of he devices. The main error sources are he offse volages of amplifiers (s), he mismaches of curren mirrors, resisor mismaches (R A and R B, R and R ), and capacior mismaches (C A and C B ). In addiion, he oupu is affeced by he reference volage ( ). The reference volage comes from eiher an inernal bandgap reference (BGR) circui or an exernal volage source. In his design, comes from an exernal volage source so as o exhibi he variaions of he 1/x circui exacly. Table 3 shows he Mone Carlo simulaion resuls of he error sources in he implemened 1/x circui wih a.18 μm CMOS process. The sandard deviaion (σ) of he oal error is 1.45%. The larges error comes from he s (σ =.7%). The maximum clock frequency (f CLK ) for he volage conversion of he 1/x circui is deermined by he capacior charging ime (T CHG = R A C A / ) in (4). When R A = 4 kω, C A = 1 pf, and =.8, he maximum capacior charging ime becomes.64 μs for he lowes allowable inpu volage ( =.5 ). In he simulaions and experimens, a clock frequency of 1 MHz was used o cover he.64 μs capacior charging ime. A higher clock frequency can be used by reducing he maximum capacior charging ime wih he parameers R A, C A, and. A lower clock frequency is recommended o save power consumpion, because power consumpion is proporional o he conversion frequency. The ETRI Journal, olume 37, Number 5, Ocober 15 Byung-Do Yang and Seo Weon Heo 975 hp://dx.doi.org/1.418/erij.15.114.577
small capaciors (C A = 1 pf, C B = 1 pf, and C =.1 pf) are implemened o reduce power consumpion and chip area, bu hey are weak from leakage currens and noises, which increase he error of he 1/x circui. The larger capaciors, C A, C B, and C, can improve he accuracy of he 1/x circui. III. Measuremen Resuls The proposed 1/x circui was fabricaed using a.18 μm CMOS process. Figure 6 shows a microphoograph of he chip. The chip occupies an area of.11 mm (144 μm 78 μm) and consumes 78 μw a DD = 1.8 and f CLK = 1 MHz. Figure 7 shows he measured waveforms of according o (= o 1 ) a =.8 and M =.5,.5, and.15. The funcion of he proposed 1/x circui does no have any process parameers heoreically, bu he oupu volage of he proposed 1/x circui is affeced by process variaions, such as he offse volages of he s; he swiching delays of he s; and he mismaches in capaciors, resisors, and curren mirrors. Figure 8 shows he measured disribuion of for unrimmed sample chips a =., M =.5, and =.8. The of each chip is measured a =. o show he disribuion of according o process variaions. Theoreically, =. a =., because he funcion of he 1/x circui is =.4/ a M =.5 and =.8. The mean (μ) and sandard deviaion (σ) of for samples were m and 3.1 m (1.61%), respecively. The measured sample chips showed a maximum and minimum of 5.7 m and 194.7 m, respecively. Therefore, he maximum difference of (Δ ) was 11 m (5.48%). The change in due o he process variaions can be rimmed by he curren-adjusing circui. Figure 9 shows he measured accuracies of for rimmed sample chips a M =.5 and =.8. The rimmed 1/x circuis have errors beween +1.7% and 1.7% for =.5 o.8. arious 1/x circui chips are compared in Table 4. The proposed 1/x circui has a larger normalized inpu/oupu range () () ().9.8.7.6.5.4.3..1.9.8.7.6.5.4.3..1.9.8.7.6.5.4.3..1 =.8/ @M =.5 =.4 @ =. =.83 @ =.83 =. @ =.4 = 1 =.1 @ =.8.1..3.4.5.6.7.8.9 () 1. (a) =.4/ @M =.5 =.4 @ =.1 =. @ =. =.1 @ =.4 = 1 =.5 @ =.8.1..3.4.5.6.7.8.9 () 1. (b) =./ @M =.15 =. @ =.1 =.14 @ =.14 =.1 @ =. = 1 =.5 @ =.4.1..3.4.5.6.7.8.9 () 1. (c) Fig. 7. Measured waveforms of vs. (= o 1 ) a =.8 : (a) M =.5, (b) M =.5, and (c) M =.15. Bias curren circui Resisors CM C Curren adjusing Fig. 6. Microphoograph of chip. Capacior (C A, C B ) Resisors CM of 1 o compared o oher 1/x circuis (from 1 o.67 o 1 o 14). The normalized inpu/oupu range is he lowes value among he inpu and oupu ranges normalized wih heir minimum values. The previous 1/x circuis [3][5] have smaller errors of 1.7%. However, all previous 1/x circuis [3] [6] have addiional lineariy errors due o PT (process parameers, supply volage, emperaure) variaions, as shown in Table 5. The addiional lineariy errors come from he parameers K (= μc OX W/L), g m (= KI D ), and T of he 976 Byung-Do Yang and Seo Weon Heo ETRI Journal, olume 37, Number 5, Ocober 15 hp://dx.doi.org/1.418/erij.15.114.577
Table 4. Performance comparisons of various 1/x circui chips. JSSC 1995 [3] IEICE 3 [4] TCAS-II 5 [5] MWCAS 1 [6] This work Process μm CMOS.5 μm CMOS.5 μm CMOS.5 μm CMOS.18 μm CMOS DD () 5 1.5 1.5 1.5 1.8 SS () 5 1.5 1.5 1.5 Power (μw) N/A 4 8 78 Area (mm ) N/A N/A N/A.3.11 ().3.8.11.5.4.4.51 ().1.3.1.5.11.4.. (sim.).9 I (μa) 17 I (μa) 55 Bandwidh 9 MHz (sim.) N/A 1 MHz 175 khz (sim.).5 MHz* Sampling frequency 1 MHz Normalized inpu/oupu range 1.667 17.5 114 N/A 1 Number of samples 1 1 1 1 Lineariy errors ±1% (sim.) ±1% ±.85% N/A ±1.7% * Maximum bandwidh is equal o half of he sampling frequency, according o Nyquis heorem. Occurrences 6 5 4 3 1 193 195 197 199 1 3 5 7 (m) =. M =.5 & =.8 Sample = µ = m σ = 3.1 m (1.61%) Max = 5.7 m Min = 194.7 m Max Δ = 11 m (5.48%) Fig. 8. Measured disribuion of for unrimmed sample chips a =., M =.5, and =.8. Error of (%). 1.5 1..5.5 1. 1.5. +1.7% Sample = 1.7%.1..3.4.5.6.7.8 () Fig. 9. Measured accuracies of for rimmed sample chips a M =.5 and =.8. JSSC 1995 [3] IEICE 3 [4] TCAS-II 5 [5] Table 5. Lineariy error sources of PT variaions. Funcion g Measuremen condiions m C C =.5 4K A IC K N N KP ( ) I K I DD T N I C = 1 μa μa I N = μa3 μa Lineariy error sources K A (1%) g m (1%) K N (1%) K P (1%) K N (1%) T (%) MWCAS N Δ N =.3 K N (1%) 1 [6] KN M =.5, This work M - =.8 * K (= μc OX W/L) and g m ( KID ) are he ransconducance parameers of MOSFET, and T is he hreshold volage of MOSFET. MOSFET. In he.18 μm CMOS process, he maximum process variaions of K, g m, and T are 1%, 1%, and %, respecively. Bu, heoreically, he proposed 1/x circui is no affeced by he parameers (K, g m, and T ) because all parameers are removed from he equaion of is funcion. Therefore, he addiional lineariy errors wih PT variaions in [3][6] will be much larger han he lineariy errors (±1.7%) of he proposed 1/x circui. The proposed 1/x circui is slower han previous circuis [3] and [5], because is operaion is based on he signal sampling ETRI Journal, olume 37, Number 5, Ocober 15 Byung-Do Yang and Seo Weon Heo 977 hp://dx.doi.org/1.418/erij.15.114.577
and he charge inegraion on capaciors. Is bandwidh (.5 MHz) is considered o be half of he sampling frequency (1 MHz), according o Nyquis heorem. The bandwidh under.5 MHz is adequae for low-speed applicaions using he 1/x funcion. The analog dividers used in mos insrumenaion and conrol applicaions need a bandwidh of under 1 khz [1]. A hearing-aid sysem [] needs a bandwidh of 16 khz. I. Conclusion An accurae unable-gain 1/x circui was proposed. The oupu volage of he 1/x circui is generaed using a capacior charging ime ha is inversely proporional o he inpu volage. The oupu volage is independen of process parameers, because he oupu volage depends on he raios of he capaciors, resisors, and curren mirrors. The 1/x circui achieved low lineariy errors and wide inpu and oupu volages by using low offse amplifiers and by maching he capaciors, resisors, and curren mirrors well. The volage gain of he 1/x circui is uned by a 1-bi digial code. The 1/x circui was fabricaed using a.18 μm CMOS process. Is core area is.11 mm (144 μm 78 μm), and i consumes 78 μw a DD = 1.8 and f CLK = 1 MHz. Is lineariy error is wihin 1.7% a =.5 o 1. References Byung-Do Yang received his BS, MS, and PhD degrees in elecrical engineering and compuer science from he Korea Advanced Insiue of Science and Technology, Daejeon, Rep. of Korea, in 1999, 1, and 5, respecively. He was a senior engineer a he Memory Division, Samsung Elecronics, Hwaseong, Rep. of Korea, in 5, where he was involved in he design of DRAM. Since 6, he has been a Chungbuk Naional Universiy, Cheongju, Rep. of Korea, where he is now an associae professor. His research ineress include analog circuis, memory circuis, and power IC designs. Seo Weon Heo received his BS and MS degrees in elecronic engineering from Seoul Naional Universiy, Rep. of Korea, in 199 and 199, respecively, and his PhD degree in elecrical engineering from Purdue Universiy, Wes Lafayee,, USA, in 1. From 199 o 1998, he was wih he Digial Media Research Laboraory, LG Elecronics Co., Ld., Seoul, Rep. of Korea. From 1 o 6, he worked a he Telecommunicaion R&D Cener, Samsung Elecronics Co., Ld., Hwaseong, Rep. of Korea. Since 6, he has been an associae professor wih he School of Elecronic and Elecrical Engineering, Hongik Universiy, Seoul, Rep. of Korea. His curren research ineress include signal processing, wireless communicaion, and embedded sysem design. [1] T.L. Laopoulos and C.A. Karybakas, A Simple Analog Division Scheme, IEEE Trans. Insrum. Meas., vol. 4, no. 4, Aug. 1991, pp. 77978. [] M. van de Gevel and J.C. Kuenen, Simple Low-olage Weak Inversion MOS 1/x Circui, Elecron. Le., vol. 3, no., Sep. 1994, pp. 1639164. [3] S.-I. Liu and C.-C. Chang, CMOS Analog Divider and Four- Quadran Muliplier Using Pool Circuis, IEEE J. Solid-Sae Circuis, vol. 3, no. 9, Sep. 1995, pp. 1519. [4] W. Liu and S.-I. Liu, CMOS Tunable 1/x Circui and is Applicaions, IEICE Trans. Fundam. Elecron. Commun. Compu. Sci., vol. E86-A, no. 7, July 3, pp. 18961899. [5] W. Liu, S.-I. Liu, and S.-K. Wei, CMOS Curren-Mode Divider and is Applicaions, IEEE Trans. Circuis Sys. II: Exp. Briefs, vol. 5, no. 3, Mar. 5, pp. 145148. [6] I. Padilla-Canoya, Compac Low-olage CMOS Analog Divider Using a Four-Quadran Muliplier and Biasing Conrol Circui, IEEE In. Midwes Symp. Circuis Sys., Boise, ID, USA, Aug. 58, 1, pp. 555. 978 Byung-Do Yang and Seo Weon Heo ETRI Journal, olume 37, Number 5, Ocober 15 hp://dx.doi.org/1.418/erij.15.114.577