Effect of Non-Linear Settling Error on The Harmonic Distortion of Fully-Differential Switched-Current BandPass Sigma-Delta Modulators

EFFECT OF NON-LINEAR SETTLING ERROR ON THE HARMONIC DISTORTION OF FULLY-DIFFERENTIAL SWITCHED-CURRENT BANDPASS EA MODULATORS Jose' M. de la Rosa, Bele'n Pe'rez-Verdk Fernando Medeiro, Rocio del Rio and Angel Rodriguez-Vhzquez Instituto de Microelectronica de Sevilla, IMSE (CNM-CSIC) Edif. CNM-CICA, Avda. Reina Mercedes sln, 41012 Sevilla, SPAIN Phone: +34 95056666, FAX: +34 95056686, E-mail: [email protected] ABSTRACT This paper presents a detailed study of the effect of the non-linear settling on the harmonic distortion of Bandeass ZA Modulators (BP-UMs) realized using Fully Differential (FD) &vlrched-current (SI) circuits. Based on the analysis of building blocks, closed-form expressions are derived for the third-order intermodulation distortion of BP-XAMs due to defective settling, on the one hand, and to the non-linexities of the sampling process, on the other. Time-domain simulations and measurements taken from a 0.8pm CMOS 4th-order BPSAh4 silicon prototype validate our approach.(*) 1. INTRODUCTION Up to now, the potential of Sw&c/ted-currerit (SI) circuits has been barely demonstrated through actual, practical circuits. Thus, in the case of XA Modulators (CAMS), performances featured by reported SI silicon prototypes are well below than those of - Switched-Capacitor (SC) counterparts, even if the latter are realized in standard technologies without good passive capacitors. Such poorer performances are partly due to the larger influence of SI non-idealities, as well as to the incomplete modeling of their influence. Particularly, for &md&ss CaMs (BP-XAMs), and due to the necessity to cope with the frequency specifications required for modem digital wireless communication systems [I], harmonic distortion due to non-linear settling becomes one of the dominant limiting factors. There have been several attempts to model the non-linear transient response of SI memory cells [2][3][4][5][6]. Regarding harmonic distortion, a precise study of the isolated SI memory cell was presented in [5], but its mathematical complexity precludes to extend its usage to circuits containing heavily coupled memory cells, as it happens for ZAMs. The simplified model for Fully Differential (FD) memory cells presented in this paper enables hierarchical systematic analysis of SI circuits composed of memory cells, such as integrators and resonators. This analysis provides closed-form expressions for the third-order intermodulation distortion of BP-Us caused by two non-linearities: the incomplete settling and the sampling process at the modulator front-end. The latter causes large harmonic distortion levels even for a low settling error, as confirmed by measurements from a 0.8pm CMOS 4th-order BP-XAM [7]. 2. FD MEMORY CELLS WITH NON-LINEAR SETTLING ERROR .' + - '3 I. "1s Figure 1. FD memory cell with non-linear settling error. a) Schematic. b) Equivalent circuit during the sampling phase. sistance is much smaller than that due to the gate-source capacitance, C,, , and the small-signal transcbnductance, gHIQ. In such a case, the cell can be modelled by the equivalent circuit in Fig.l(b) during the sampling phase, $,. In this circuit, the large.- signal behaviour is modelled by g,,,, and g,,,-, which re resent the transconductances of M+,-. given by g,,,-= gf,lQ,/k-, where mi+,. = ij+.-/Ibia,, ii+.- = fii/2, and ij is the input cur.- rent [3]. Assuming that ii keeps stationary during the sampling phase, and that the switch becomes OFF at (n - 1/2)T, (T, is the sampling period) the differential drain current, i, = id+-id- , can be calculated by solving the circuit in Fig.l(b) for the initial condition I' = 19 +I, giving: s+.. SI ... 11 - 1 'd. 11 - I/' = 'j, 11 - 1 /Z - y(nlj. 11 - 1 /') 1 + (1:) + 'd. it - lY("'i, ti - I/?) where with T = C /x,,,~ and mi,fl = ii~lI/(21bfas). current, io = io+ -io-, is given by: 3, At the end of the ri-th hold phase, Q2, the differential outpul (3) 'a I1 = -'d. I1 - 1/1 From (1)-(3) and considering id. ,I - I = i, - 3/2 , one obtains: To calculate the harmonic distortion, the function Y(.) must be approximated by a polynomial inside a given interval. For that purpose, we have combined Taylor series expansion for mi (( l and numerical fitting for -0.5 5 inj 5 0.5 , 0.01 % <E,~ < 10% , to obtain the following approximation: Fig.l(a) shows a FD second-generation memory cell. In what (5) follo& it will be assumed that the incomplete settling is the dominant non-ideality. Therefore, the effect of the charge injection error and the finite output conductance, analysed elsewhere [SI, will where E, = exp[-k,l is the line,. settling error, k, = T,,(2t) , = a,EJks[( I + k,v(321tfa,\1, and a, = 3/2 is a fitting not be considered. Besides, in most practical cases the time constant formed by the drain-source capacitance and the switch-on re- '*'This work has been supported by the Spanish CICYT Project parameter. .I 1, The notation is used to represent ,tR(ji~,,) . TIC 97-0580 0-7803-6685-9/01/$10.0002001 IEEE 1-340 SubstiNting (5) into (4), yields: where Thus the analysis of a FD memory cell with non-linear settling error can be accomplished considering a memory cell with linear error, E,, which has an input current equal to (7). If it is a shewave of amplitude Zi and frequency fi , io will contain harmonics of fi . In FD circuits, the Total &r"nnic Distortion (THD ) is approximately equal to the third-order harmonic distortion, HD3 . The analysis of HD3 can be simplified if io is approximated by its first-order harmonic, such that io, E -Zjsin(2sr fpT,) . Performing a Fourier series expansion of (7) it can be shown that the amplitude of the third-order harmonic is approximately given by: where f, = 1/T, is the sampling frequency, and HD3 is: with Mi = z;/(2Zbjo,). Fig.' compares the theoretical model with HSPICE by plotting HD3 vs. f ;/ f, , using the same example as in [3] [5] with level47 MOS models of a 0.8pm standard CMOS technology. In that example, gnIQ = 82.8pAN, C,, = 22.lpF, Zbias = 20pA, Mi = 0.5 and f, = 5 I2kHz. Note that predictions given by (9) agree with HSPICE and with those made by the model in [5]. However, as a difference to this latter model, the new one can also be used for predicting the harmonic distortion of higher-level SI blocks, such as integrators and resonators, and finally, complete BP-ZAMs. 3. HARMONIC DISTORTION IN SI RESONATORS Resonators are the basic building blocks of BP-UMs, playing the same role as integrators in LP-CAMS. Most of BP-XAMs reported in the literature obtain their architecture by applying the transformation : + -z to the corresponding LP-ZAMs [I]. As a consequence of this transformation, the original integrators become resonators with a transfer function Fa/( 1 + I-') , where 0 < a 5 2 . This function can be realized by several filter structures [I]. Fig.3(a) shows the block diagram of one based on LD Integrators (LDI's). This structure is advantageous as compared to the -1 -2 HD3 Figure models. others because it remains stable under changes in the loop coefficients. Let us consider that the integrators are realized as shown in Fig.3(b) and analyse their isolate operation, assuming that memory cells are described by (4). After clock phase , the differential drain current of cell 2, is: L1 - y(i.r, n)lix, n + 'ds, ,,,- Iv(ix, TI) (lo) (idsl+,,, - 'ds, .,,,) repreidsz.,, E ids2+,,, - where 'x, ir = 'i, I, - 'ds ,,,,- and 'ds ],,, sents the differential drain current of cell 1 .After clock phase $? , i dsl,,,+1,2 -y(-ids2,,,)]id~2.,, + idsl~~,~ly(-ids2~,J) (I1) Assuming that the output stage (represented in Fig.3(b) as a simple current mirror) is ideal, the output current of the integrator is: From (5) and i, =. with Thus, the analysis of a SI FD integrator formed by memory cells with non-linear settling error can be accomplished considering an integrator with linear settling error whose input is equal to (14). Let us consider the resonator of Fig.3(a). Assuming that the integrators can be modelled by (13) and (14), the finite-difference equations that govern the behaviour of the resonator are: (15) (16) .. ll,,l = l;,,,-'o,,, ;2,,l=-(l -&,)jl.,,_1/2+&,i,.,,-3/2+i2 ,,,- 1 +i?H.,, ~O.,l~-(~-~,~~2,,~-I/2+~~~2.,,-3/2+~0,,1-1 +iOH,I, (17) where i, and i2 are respectively the input and the output of the first integrator in the loop and iZH and io, are non-linear terms, respectively given by: (19) 2,012 N 1 , solving for i?.,, in (17) and '. 'OH, . ,I = -E sz c2 '0. ,112. . ,I - I/? + i,. ,I - 1'2. ,I - 3/2) Assuming that E,, substituting it in (16), obtains: io. ,, (1 - 2&,Vj, ,, + 4&,jo, ,, - I - (1 - 4&,)iO, ,I - (20) Figure 3. LDI-loop Resonator. a) Block diagram. b) SI FD LDI. 1-341 Figure 4. HD, at the output of a LDI-loop resonator, with f; : a) 0.001 f ,/4 . b) 0.002 f ,/4 . c) 0.003 f ,/4 . where yi, ,I = ;;,,I - I + (ioH, ,I - ;,ff, ,I - 1 - &ff, ,, - ,,2) ' The harmonic distortion referred to the resonator input can be calculated by analysing the harmonic content of the above expression. Therefore, assuming that ii is a sinewave of amplitude I; and frequency f , the output of the resonator will be a quasi-sinusoidal signal, with an amplitude approximately given by: Performing a Fourier series expnnsion of ri, ,I , it can be shown that the amplitude of the third-order harmonic at the resonator input is given by: (22) where, f ,' = f , - f,/4 and f ,'T, (( 1 has been assumed. Multiplying A, by the resonator gain (evaluated at 3f, ) and dividing the result by I,, obtains HD3 at the resonator output. This analysis has been validated by time-domain simulation using the behavioural simulator for SI circuits reported in [9]. Figd shows HD, at the output of the resonator as a function of E, , for different values of f,'. This simulation was done by changing glIrQ, for Cs5 = IpF, I, = 0.5yA and Ibios = 200yA, when clocked at f, = lOMHz . Note that HD, does not increase with E, because the open loop gain of the resonator is also attenuated by the linear error. Although simulated behaviour is well predicted by theory, their differences become larger when the condition f,'Ts <( I is not satisfied as assumed in theory. 4. HARMONIC DISTORTION IN SI BANDPASS XA MODULATORS Let us consider the 4th-order BP-ZAM shown in Fig.5, where A,,,? = 2A,,,,A,,, , and assume that LDI's are realized as in Fig.3(b). For the analysis of the harmonic distortion, the following considerations have been taken into account: *The harmonic distortion referred to the modulator input is equal to the harmonic distortion referred to the modulator output because the signal transfer function is unity in the signal band. Figure 5. Block diagram of the 4th-order BP-ZAM. *The contribution of the second resonator will not be considesed because it is attenuated by the gain of the first resonator in the signal band. *The quantization error, modelled as an additive white noise source, does not contribute to the harmonic distortion. The analysis of the modulator in Fig.5 reveals that the amplitude of the first resonator output is given by: (23) where A, is the input amplitude. Substituting (23) into (22), assuming that AD,,, = -1 and dividing by A,, obtains that HD, at the output of the modulator is: HD, G 4..&A:( 1 + 5n f ,'T,) E 4&,,A: (24) Note that due to the oversampling, it is f,'T, (< I , and hence HL), practically does not depend on f , . A more appropriate parameter for characterizing the harmonic distortion in BP-ZAMs is the third-order intermodulation distortion, given by: where I,,, is the DAC output current and nb .= Ibios/tDAC. This expression has been validated by time-domrun simulation as shown in Fig.6(a) by representing IM, vs. E, t2 for different values of I,,, with A ~ = IDAC/2, Ibias = 200pA, C,, = 1 pF and f, = IOh4Hz. The input signal consisted on two tones of amplitude A 'I/ 4 and frequencies f , I s 0.247 f , and fl? = 0.248fS. As an illustration, Fig.G(b) shows the output spectrum corresponding to E, = 1 % and I,,, = 50pA. Note that, other intermodulation products appears - not critical since they are outside the signal band. - 'Iheory .Simulation Frequency (MHz) Figure 6. a) IM, vs. E, for different values of I,,, . b) Outpllt spectrum for E, = 1 % and IDAC = 50pA. t2. O.I%<E,<l%. The sin~ulation was canied out by varying gnrp such that 1-342 5. EFFECT OF S/H PROCESS AT THE FRONT-END OF SI BANDPASS ZA MODULATORS In the previous analysis it has been assumed that the input current is constant during the sampling phase. However this assumption does not apply to the memory cell connected at the input node of a BP-EAM. In this case, the ratio between fi and f, is close to unity (fi/fs = 1/4 in the case of the modulator in Fig.5) and hence, there will be large variations of the input current during the sampling phase. As a consequence, additional harmonic distortion will appear even if E, (< I , which can not be explained by (9). The harmonic distortion in memory cells connected to continuous-time sinewave currents was analysed in [6]. Here, we will extend that analysis to BP-XAMs. Let us consider first the cell in Fig.I(a) and assume a sinewave input current of frequency fi and amplitude Ii. In this case, id can not be solved as an step-response. In order to find an explicit solution, the circuit in Fig.l(b) was solved for gm+,- = gnIp. with the initial condition 17 = vg+ pkced by g,,p JKi, yielding to t3: Once the solution was found, gmQ was reio* ll z - ‘i,r?-l/~~l~~l/~~‘i.rl-l/~~ + + [‘i.r7-1~~~-,(’i,r1-1/~)y(’i,11-1,~) +‘~,i~-l~(’i,i~-~,~)l (26) being, 1 - 2xfizcot(2xfinT,) 6nfizcot(2xfi~1Ts) ,2 qll(ii) E 1; (27) 1 + (2x f 32I;;,, where ii/Ihias <( I has been assumed. Following the same procedure as in Section 2, it can be shown that, for xfiz (< 1 , 371 f iT 16( 21,,i0,)- AH. 3 71; (28) In BP-XAMs, only the first memory cell connected to the input signal will contain the above harmonic. To calculate IM, at the output of the modulator, it is necessary to express Ii as a function of A., . The analysis of Fig.5 gives Ii 3 2&A,v. Substituting this expression in (28), and dividing by A., it can be shown that: (29) where fi zf,/4 has been assumed. Fig.7(a) compares (29) with time-domain behavioural simulation by plotting IM, vs. z for IM3(dBi fc = SO MHz -60 -70 AA Figure 7. IM, due to the S/H process at the front-end. a) IM3 vs. z . b) Comparison with IM, due to non-linear E,. ~ ~~ t3. A more rigorous analysis can be done by using the Volterra series method as we demonstrated in [SI. yielding to similar results. O ,IM3 = -54m do f =2MHz -en . -%23 023 02s Figure 8. Measured output spectra for different values off,. . Relative frequency (to&, Relative frequency (tof,) different values off,, nb = 4 and AX/IDAg = 1/2. The theoretical model accurately predicts the simulatlon results except for some cases where a maximum error of 4dB occurs. In these cases a more exact analysis using the Volterra series method should be used. To conclude this study, Fig.7(b) compares ZM, caused by the non-linear settling error and the S/H process for fs = loMHz and A,/IDA, = 1/2. Note that, for E, > 3% , both expressions approximately converge. However, for practical designs, i.e, for < 0.1 % , IM, due to the S/H process dominates, limiting the performance of SI BP-UMs unless a S/H circuit will be used at the front-end. This fact has been confirmed by experimental results from a 0.8pm CMOS 4th-order BP-CAM [7]. Fig.8 shows two measured output spectra for A,/l,,, = 0.42 when clocked at f, = 2MHz and f, = IOMHz, obtaining IM, = -61dB and -54dB respectively. In this case, g, - 360pAN and Cgs = 2.8pF (E, = 0.16% at f, = 10MI-f~; which according to (29) gives IM, = -64dB and IM, = -55dB respectively. 6. CONCLUSIONS The effect of non-linear dynamic SI errors on the harmonic distortion of FD BP-ZAMs has been analysed in detail. Closed-form expressions, validated through time-domain simulation, have been derived for IM, due to the non-linear settling and the sampling process. The latter constitutes the main source of harmonic distortion in practical designs as demonstrated by experimental results. REFERENCES S.R. Norsworthy, R. Schreier. G.C. Temes: “Delta-Sigmo Cornwrters. Theory, Design arid Sirnulatioil”. New York, IEEE Press. 1997. D.B. Nairn: “Analytic Response of MOS Current Copiers”. IEEE Trarzsactioizs on Circuits arid System 11, pp. 133-135, February 1993. P.J. Crawley and G.W. Roberts: “Predicting Harmonic Distortion in Switched-Current Memory Circuits”, IEEE trans. Cirruits ond Systems II, pp. 73-86. February 1994. N.Moeneclaey and A. Kaiser: “Accurate Modelling of the Non-Linear Settling Behaviour of Current Memory Circuits”. Pmr. 1994 IEEE hit. Synzp. Circuits arid Systems (ISCAS), pp. 339-342. M.Helfenstein and G. Moschytz: “Distortion Analysis of Switched-Current Circuits”, Proc. 1998 IEEE hit. Synzp. Cirruits arui Systerizs (ISCAS), pp. 29-32. J.M. Martins and V.F. Dias: “Harmonic Distortion in SI Cells: Settling and Clock Fall-Tmie Effects”, Proc. of 1998 Design of Integrated Circuits arid Systems Conference (DCIS). pp. 292-297. J.M. de la Rosa. B. P&ez-Verdu, R. del Rio and A. Rodriguez-Vizquez: “A CMOS 0.8pm Transistor-Only 1.63MHz Switched-Current Bandpass EA Modulator for AM Signal A/D Conversion”, IEEE Joumal of Solid-state Cirruits, pp. 12201226. August 2OOO. J.M. de la Rosa: “Modelling and Design of Switched-Current Bandpass ZA Modulators for Digital Communication Systems”, Ph.D. Dissertation. Univ. of Seville. December 2000. J.M. de la Rosa. A. Kaiser and B. Ptrez-Verdu: “Interactive Verification of Switched-Current Sigma-Delta Modulators”. Proc. of 1998 IEEE lrztcniotioiinl Coifererice 017 Electrorzics. Circuits arid Systems (ICECS), pp. 2.157-2.160. 1-343