AC Analysis of Idealized Switched-Capacitor Circuits in Spice-Compatible Programs

Transcription

1 A Analyi of Idealized Switched-apacitor ircuit in Spice-ompatible Program DALIBOR BIOLEK, VIERA BIOLKOVA, ZDENEK KOLKA Dept. of Microelectronic and Radioelectronic, Brno Unierity of echnology Dept. of EE, Unierity of Defence, Brno, zech Republic Abtract: Direct A analyi of idealized witched-capacitor circuit in Spice-compatible program i decribed in the paper. he capacitor i decribed by a pecial macro-circuit, whoe mathematical model i baed on modified z- domain charge euation and on the theory of euialent ignal. Keyword: - witched capacitor, analyi, SPIE. Introduction heir inability of direct mall-ignal A analyi of witched-capacitor (S) and witched-current (SI) circuit belong to well-known limitation of Spicecompatible program. he only unieral method, which i ery laboriou and time conuming indeed, conit in the teady tate tranient analyi with the input ignal wept by a harmonic ignal of a gien freuency, the election of an A component of output ignal, and it comparion with the input. hi procedure mut be repeated for more freuencie. A certain degree of automation of uch action in the WinSpice (SPIE3) program i decribed in []. hi method ue a multitone excitation of the witched circuit. he correponding complex tranfer are then determined ia pectral analyi. Another method i decribed in []. he z-domain euialent circuit i modeled in the confinuou-time domain uing the lole tranmiion line element to implement the reuired one-port torage element (toritor). he method i retricted to thoe S network where the effect of nonideal OpAmp can be neglected []. he utilization of direct Spice-like A analyi cannot be performed here becaue it tart from the circuit linearization around the D operating point. Howeer, the operating point of witched circuit i periodically wept by irtue of the witching ignal. A number of algorithm for gaining the aboe freuency repone directly and with much le computational effort are decribed in the literature [3], [4]. Howeer, thee procedure are not compatible with mathematical algorithm, implemented in the SPIEfamily program. hat i why they were ued on the platform of pecial analyi program. Such program can ole witched circuit, coniting of ideal capacitor, ideal witche with zero on-reitance and zero off-conductance, and ideal controlled ource. Alo ome real propertie of circuit element can be modeled. Mot of uch method are baed on o-called charge euation. After applying the z-tranform, they lead to dicrete-time euialent model of witched circuit. he freuency repone can be obtained after the well-known ubtitution z = exp(jω S ), where S i the witching period. Other pecial method, which are appropriate for modeling the A behaior of general linear circuit with periodically controlled witche, e.g. the method of generalized tranfer function (GF) [5], can be applied in SPIE only for unified imple block, for which the GF i determined analytically and then it i modeled uing the mean of behaioral modeling [6]. A direct A analyi of idealized two-phae S circuit in Spice-compatible program i decribed below. he freuency repone are not obtained ia repeated tranient analyi or by the method of multitone excitation, but by the conentional Spice-like A analyi of a pecially made-up model of witched circuit. he uer mut create a double circuit, taking into account the configuration of witche in both witching phae. hi procedure i time-aing thank to pecial model of capacitor which reflect the interlacing of their behaior between the witching phae. he method can be eaily extended to multi-phae witching. Howeer, making-up uch model would be rather laboriou in the SPIE enironment. It i alo determined by limited SPIE feature in generating more complicated netlit ia conentional PSpice template [7]. he method can be alo partially extended to Spice modeling of real propertie of circuit element, e.g. witch reitance or OpAmp bandwidth. More detail can be found in [8].

2 Behaioral SPIE model of capacitor in ideal S circuit onider the witched capacitor in Fig. (a). he number or at the witch denote the witching phae in which thi witch i on. he witching diagram together with the ketch of the capacitor oltage waeform i in Fig. (b). Note that the o-called inconitent initial condition [9] can appear in the circuit due to ideal witch model, which reult in the dicontinuou recharging of capacitor by current in the form of Dirac impule. hat i why one hould expect dicontinuitie in capacitor oltage. It i neceary to dicriminate between the left-ide and the right-ide limit at intance when the witche change their tate. c (a) c D S k k +D k + k + +D S S S S S S S S (b) D' S Fig. : (a) Switched capacitor, (b) witching diagram with marked limit alue of capacitor oltage at the end of witching phae (o) and (x). D (0,) i the duty ratio, D =-D. he following charge euation are true for the circuit in Fig. : ( k + D ) = [ ( k + D ) ( k ( k ) = [ ( k + ) ( k + D )]. + Here ( ) are the difference of electric charge on the capacitor within the time interal from the end of witching phae to the end of witching phae (from the end of witching phae to the end of witching phae ). he upercript denote the leftide limit of the correponding circuit ariable. t he charge euation can be tranferred into the current euation, which can be already implemented in Spice: < i > = [ ( k + D ) ( k () < i > = [ ( k + ) ( k + D where and are the aerage alue of capacitor current in witching phae and, aeraged oer the entire witching period: < i > < i > ( k ( k + D + ) = ) = k + D k k + k + D i( t) dt = i( t) dt = ( k + D ) ( k + ). For the purpoe of the A analyi of the witched circuit whoe part i the witched capacitor in Fig., conider the periodic teady tate in the circuit due to harmonic input ignal e t, = jω. It can be hown that each ignal in the witched circuit can be approximated by the o-called euialent ignal, which i alo of harmonic nature with the identical repeating freuency ω [5]. More concretely, the euialent ignal e t = V ˆ e, where ˆ V i the complex phaor of thi ignal, can be interleaed with the o point in Fig. (b), which correpond to the end of phae in the ene of the left-ide limit. Similarly, the euialent e t harmonic ignal = V ˆ e i interleaed with the x point. hen E. () can be rewritten ia thee euialent ignal. Simple arrangement yield: ˆ [ ˆ = V V z ], ˆ [ ˆ ˆ D I = V V z ], () ˆ D I where the phaor of euialent current in witching phae and are on the left ide of euation, and z i the well-known operator of the z tranform, with z = exp( ). he behaioral model of the capacitor in Fig. for witching phae and i made up on the bai of E. (). hi model can be directly implemented a a Spice ubcircuit. he controlled ource in the model may be implemented ia Laplace ource, utilizing the aboe ubtitution z = exp( ). he model i alid both for the witched and the fixed capacitor.

3 phae phae 400kHz with a uality factor of 0. he S-H circuit at the input ample the ignal in phae and hold them during the whole witching period. IˆR Vˆ ˆ I R z D IˆR Vˆ Iˆ R z D in S-H 0pF 0.47pF 8.50pF HP pf 0.47pF 50.7pF Fig. : A model of the capacitor in S circuit. 3 Modeling of other circuit element in S circuit apacitor, ideal witche, and ideal oltage-controlled oltage ource (VVS) for the modeling of oltage amplifier including OpAmp are typical circuit element in idealized S circuit. he VVS repreent only the non-inertial tranformation of gate oltage. hat i why the mathematical model of VVS can be ued independently in both witching phae. Modeling of other circuit element beyond the frame of idealized S circuit can be problematic in Spice-compatible program. For example, reitor for modeling nonzero on-reitance of witche introduce coniderable time contant of tranient phenomena. hen the oltage drop on them at the end of witching phae cannot be neglected a i the cae in the circuit in Fig. (a). Howeer, thee oltage do not depend on aerage current in E. () and (), but on the current at the end of witching phae.. he relation between aeraged and intantaneou current depend on the type of tranient phenomenon and thu on the total circuit configuration. A a reult, the witch euation would depend not only on thi witch, but alo on the remaining part of the circuit. hi inolement can be atifactorily oled in Spice only by trade-off modeling. A imilar problem appear, for example, when modeling the freuency-dependent gain of the operational amplifier. Detail can be found in [9]. 4 Demontration of A analyi onider the witched-capacitor biuad in Fig. 3 according to [0]. hi filter i deigned for a witching freuency of 6MHz. It center freuency i 0pF Fig. 3: Switched-capacitor biuad [0]. A= 0p in 0.47p 0p V(in)=V(in)*exp(-*d/f) in.param f=6meg d=0.5 HP HP 8.50p p 0.47p BP 50.7p BP phae phae BP Fig. 4: Spice modeling of filter from Fig. 3 for ubeuent A analyi. A demontration of the filter model in the chematic editor of Spice-compatible program Micro-ap 9 i hown in Fig. 4. he chematic ymbol of capacitor

4 are duplicated for both witching phae, repreenting Spice ubcircuit which model euation () on the bai of the ubtitution diagram in Fig.. In order to make the circuit creation more comfortable, the chematic ymbol of the ideal OpAmp i alo duplicated. he Opamp i modeled by an ideal VVS with adeuately large oltage gain. he influence of S- H circuit i modeled by the oltage ource, connected to the node in. he A analyi reult in Fig. 5 fully correpond to the reult in [0] oncluion he paper decribe a modification of the method of charge euation for modeling idealized two-phae witched-capacitor circuit. he aboe modification i baed on the theory of euialent ignal and on the tranformation of charge euation into current euation. he witched capacitor i then decribed by gate oltage and aeraged gate current, which i the firt aumption of ucceful implementation in SPIE-compatible program. Each capacitor in the circuit i repreented by a pair of model for witching phae and. he graphic repreentation of uch model, the double chematic ymbol, facilitate making up the model of witched-capacitor circuit in the chematic editor. In principle, an extenion of thi method to circuit with multi-phae witching i poible. Howeer, it would poe ome practical problem in the enironment of mot Spice-compatible program. A trade-off Spice analyi hould be alo applied for real witched-capacitor circuit [8] K 00K M 3M db((hp)) db((bp)) db((hp)) db((bp)) F (Hz) 0 Acknowledgment hi work i upported by the Grant Agency of the zech Republic under grant No. 0/05/077 and 0/05/077, and by the reearch programme of BU MSM , MSM , and UD Brno MO FV Reference K 00K M 3M phae((hp)) (Degree) phae((bp)) (Degree) phae((bp)) (Degree) phae((hp)) (Degree) F (Hz) Fig. 5: A analyi of filter from Fig. 3 by Micro-ap program. [] Bičák, J., Hopodka, J. Freuency repone of witched circuit in SPIE. Proceeding of ED 03, Krakow, IEEE, 003, pp. I [] Nelin, B.D. Analyi of Switched-apacitor Network Uing General-Purpoe ircuit Simulation Program, IEEE ran. On AS, Vol. 30, No., January 983, pp [3] Unbehauen, R., ichocki, A. MOS witchedcapacitor and continuou-time integrated circuit and ytem. Springer-Verlag, 989. [4] Vlach, J., Singhal, K. omputer Method for ircuit Analyi and Deign. Van Notrand Reinhold ompany, New York, 987.

5 [5] Biolek, D. Modeling of Periodically Switched Network by Mixed -z Decription, IEEE ran. on AS-I, Vol. 44, No. 8, Augut 997, pp [6] Biolek, D., Biolkoá, V., Dobeš, J. Modeling of witched D-D conerter by mixed -z decription. IEEE Int. Symp. on ircuit and Sytem, ISAS006, Greece, Ko, 006, pp [7] PSpice A/D Reference Guide. Product erion 5.7. adence, July 006. [8] Biolek, D., Biolkoá, V. SPIE analyi of real witched-capacitor network. Submitted to the 4 th IEEE Int. onference on Electronic, ircuit and Sytem (IES 007), Dec. -4, 007, Marrakech, Morocco. [9] Wojcziechowki, J., Vlach, J., Opal, A. Analyi of Nonlinear Network with Inconitent Initial ondition. IEEE ranaction on AS-I, 995, ol. 4, no. 4, p [0] Dolíka, L., Hopodka, J. Switched-apacitor Filter Optimization with Repect to Switch On- State Reitance and Feature of Real Operational Amplifier. Radioengineering, Vol. 6, No., Jule 007, pp