Large Signal Analysis of Low-Voltage BiMOS Analog Multipliers Using Fourier-Series Approximations

Transcription

1 Proc. atl. Sci. Couc. ROC(A) Vol. 4, o. 6, (Short Commuicatio) Large Sigal Aalysis of Low-Voltage BiOS Aalog ultiliers Usig Fourier-Series Aroximatios UHAAD TAHER ABUELA ATTI ig Fahd Uiversity of Petroleum ad ierals Dhahra, Saudi Arabia (Received Setember 30, 999; Acceted Jauary 7, 000) ABSTRACT This aer discusses the large sigal erformace of BiOS low-voltage aalog multiliers. Fourier-series aroximatios are obtaied for the trasfer characteristics of the basic buildig blocks, such as the BiOS folded Gilbert multilier cell, the squarig multilier made from two cross-couled emitter-couled airs ad drive by a OS quadritail, ad the triler made from cross-couled emitter-couled airs ad drive by a OS quartersquare multilier. Usig the Fourier-series aroximatios of the trasfer fuctios of these basic buildig blocks, closed-form exressios are obtaied for the amlitudes of the harmoics ad itermodulatio roducts at the outut of these aalog multiliers whe excited by multisiusoidal iut sigals. Usig these exressios, comariso betwee the large sigal erformace of these aalog multiliers ca be made ad the arameters required for a redetermied erformace ca be determied. ey Words: aalog multiliers, BiOS circuits, Fourier-series aroximatios, cross-couled emitter-couled airs I. Itroductio At reset there is growig iterest i desigig low-voltage aalog fuctioal elemets usig the BiOS techology. Because of their wide sread use i aalog alicatios, multiliers are amog the may fuctioal circuits which ca be realized usig this techology. With all the biolar trasistors assumed idetical ad their basewidth modulatio igored, ad with all the OS trasistors assumed idetical ad oeratig i the saturatio regio ad igorig their body effect ad chaellegth modulatio, the use of the square-law model for OS trasistors yields the exressio of Eq. () for the differetial outut curret of the BiOS folded Gilbert multilier cell, show i Fig. (imura, 994). y f(z) tah(x), (a) where f() z z z ; z, (b) Fig.. BiOS folded Gilbert multilier cell. f() z sg z; z, (c) where y I out I o α F, x V V T, z C@@@@@?@@@@@g@@@@@??@@@@@?f@@@@@??B@@@@@@@@@@@? I o /β, α F is the dc commo-base curret gai factor for a trasistor, V T kt/q is the thermal voltage, k is the Boltzma s costat, T is the absolute temerature i degrees elvi, q is the charge of a electro, β µc ox (W/L) is the trasco- 480

2 Large Sigal Aalysis of BiOS ultiliers 3 ( z + ) + 6( z )sg( z))); 5 z + + z z + z 3 (3c) 5 hz (, ) sg ( z)tah( x); z + z, 3 (3d) where V C@@@@@?@@@@@g@@@@@??@@@@@?f@@@@@??B@@@@@@@@@@@? I o /β. Uder small sigal coditios, with the three ormalized iut voltages x, z ad restricted, resectively, to x < /, z < / ad < /, Eqs. () (3) ca be aroximated, usig the first term of the Taylorseries exasio of the tah ad square-root fuctios, by C@@@@@?@@@@@g@@@@@??@@@@@?f@@@@@??B@@@@@@@@@@@? xz, (4) Fig.. Cross-couled, emitter-couled airs drive by a OS quadritail cell. y xz (5) ductace arameter of the OS trasistor with effective surface mobility µ ad gate caacitace er uit area C ox, W is the gate width, ad L is the gate legth. Also, the differetial outut curret of the cross-couled emitter-couled airs drive by a OS quadritail cell, as show i Fig, ca be exressed as follows (imura, 994): where y g(z) tah(x), gz () z ; z, 3 ( ) gz () z 3+ z ( 6 z ); z, 9 3 (a) (b) (c) ad y xz. (6) Equatios (4) (6) show that, uder small sigal coditios, the three multiliers i Figs. 3 will erform the required multilicatio. However, i their reset forms, Eqs. () (3) ca ot be used to redict the erformace of the three multiliers uder large sigal coditios. This ca be attributed, largely, to the lack of a sigle cotiuous fuctio describig the trasfer fuctio of the multilier ad stretchig over the useful rage of its oeratio, i additio to the ivolvemet of the tah ad square-root fuctios. Cosequectly, the tolerable oliearity ad the dyamic rage of the three multiliers ca ot be aalytically defied. gz () ; z, (d) where y I out (α F I o ), α F is the dc commo-base curret gai factor for a trasistor. Fially, the differetial outut curret of the cross-couled emitter-couled airs drive by a OS quarter-square cell, as show i Fig. 3, ca be exressed as follows (imura, 995): y h(z,) tah(x), hz (, ) z; z + + z<, (3a) (3b) hz (, ) ( z ( + ( z+ ) ( z ) Fig. 3. Cross-couled, emitter-couled airs drive by a OS quartersquare multilier. 48

3 .T. Abuelma atti The mai objective of this aer is, therefore, to reset a geeral aalysis for redictig the oliear erformace of BiOS multiliers uder large sigal coditios. Usig accurate models for the differetial outut curret trasfer fuctios, closed-form exressios are obtaied for the amlitudes of the outut roducts from a BiOS multilier with large-amlitude siusoidal iut sigals. Usig these exressios, a comariso betwee the differet BiOS multiliers ca be made, ad the otimum arameters required to meet a redetermied level of circuit erformace ca be determied. II. Fourier-Series Aroximatio Here we roose to aroximate the fuctios tah(x), f(z), g(z) ad h(z,w) of the three multiliers i Figs. 3 usig the Fourier-series model of Eq. (7): π π f( φ) ao + acos φ bsi φ + B B φ. Equatio (7) imlies that the fuctio f(φ) ca be rereseted by a Fourier-series i the variable φ with eriodicity, equal to B, chose such that the workig rage of the variable φ is a aroriate segmet aroud zero. The arameters a o, a ad b,,,,, ca be obtaied usig the rocedure described by Abuelma atti (993). Sice, tah(x), f(z) ad h(z,) are odd fuctios, a 0, 0,,,,. Tables ad show tyical values of B, Table. Values of the Parameters B ad b for the Fuctios tah(x) ad f(z) b + tah(x) f(z) B 8.0 B 8.0 b b b b b b b b b b b b b b b b b b ote: a o a b 0,,,,. (7) Table. Values of the Parameters B ad b for the Fuctio h(z,) b + h(z,) h(z,) h(z,) h(z,) B 8.0 B 8.0 B 8.0 B 8.0 b b b b b b b b b b b b b b b b b b ote: a o a b 0,,,,. a, b for the fuctios tah(x), f(z) ad h(z,) i Eqs. () ad (3). Sice, g(z) is a eve fuctio, b 0,,,,. Table 3 shows tyical values of the arameters a o, a,,,,, for the fuctio g(z). From Table, it aears that the arameters b,, 3, 5, of the fuctio h(z,) are, themselves, fuctios of. Variatios of the arameters b,, 3, 5, 7, with are almost liear ad ca, therefore, be aroximated by b () δ, (8) where δ.779, δ , δ ad δ The variatios of the arameters b, > 7 ca, if required, be fitted to olyomials as fuctios of. Usig the arameters listed i Tables 3, calculatios were erformed, ad the results are show i Figs. 4 7 together with the trasfer fuctios of Eqs. () (3). From Figs. 4 7, it aears that the aroximatio of Eq. (7) accurately reresets the trasfer fuctios i Eqs. () (3) with a relative root-mea-square (RRS) error of for the trasfer fuctio tah(x); for the trasfer fuctio f(z) i Eq. (); 0.07 for the trasfer fuctio g(z) i Eq. (); ad for the trasfer fuctio f(z,) i Eq. (3) with 0., with 0., 0.009, with 0.3 ad with 0.4. III. Large Sigal Aalysis of the ultiliers If a oliear device, with a trasfer characteristic modelled by Eq. (7), is excited by a multisiusoidal iut sigal of the form 48

4 Large Sigal Aalysis of BiOS ultiliers.0 tah(x).0 f(z) :Exact :Aroximated usig Eq. (3) :Exact :Aroximated usig Eq. (3) 3 4 ormalized iut, x Fig. 4. The trasfer fuctio tah(x). tah(x) is a odd fuctio. 3 4 ormalized iut, z Fig. 6. The trasfer fuctio g(z) i Eq. (). g(z) is a eve fuctio..0 g(z) 0.6 h(z,) :Exact :Aroximated usig Eq. (3) 3 4 ormalized iut, z :Exact :Aroximated usig Eq. (3) 3 4 ormalized iut, z Fig. 7. The trasfer fuctio h(z,) i Eq. (3). Fig. 5. The trasfer fuctio f(z) i Eq. (). f(z) is a odd fuctio. B φ( t) o + ksi ωkt, o + k, (9) where o may rereset a exterally alied dc bias voltage, the by combiig Eq. (7) with Eq. (9), the ormalized outut curret ca be obtaied as π f( t) ao + acos o + ksiω kt π + bsi o + ksi ω kt. Usig the trigoometric idetities: (0) si( θsi ζ) Jl+ ( θ)si( l+ ) ζ, cos( θsi ζ) Jo( θ) + Jl ( θ)cos lζ, it is easy to show that the amlitude of a outut comoet of frequecy l 0 λ k ω k, where λ k is a ositive or egative iteger or zero, ca be give by π π F( λ, λ,, λ a b ) c cos o si L + o l 0 π Π J λ k k for λk eve (a) 483

5 .T. Abuelma atti ad or where J λk (θ) is the Bessel fuctio of order λ k ad λ k is the order of the ormalized outut curret comoet. Therefore, the amlitude of a ormalized outut curret comoet of frequecy ω ca be give by () the amlitude of a ormalized outut mth harmoic curret comoet of frequecy mω ca be give by ad π π F( λ, λ,, λ b a ) s cos o si L o F b π o a o π cos si F F (3a) (3b) ad the amlitude of a itermodulatio roduct of frequecy rω qω s ad order r + q ca be give by F π a b π a π cos cos π a m o o m π π cos J π Π o k for m odd, k b + m o o m π Π J λ k k for λk, odd (b) Π J k J π Π o k for m eve k π k, b + π si π si si π (,) rq o o o π r r J π q q J π Π o k k r, q for r+ q eve (4a) for r+ q add. (4b) I a similar way, the amlitude of a itermodulatio roduct of ay order ca be obtaied usig Eq. (). From Eqs. (3) ad (4), oe ca see that the amlitudes of odd-order roducts ca be miimized through roer selectio of the arameter o. Thus, roer selectio of the dc bias voltage ca yield miimum values of the amlitudes of the odd-order roducts. ow if the iut to the multilier show i Fig. is comosed of the two ormalized sigals ad F x(t) X siω t (5) z(t) Z siω t, (6) the, usig Eq. (4b), the amlitude of the ormalized outut curret of frequecy ω ± ω ca be exressed as (7) where the suscrit f meas the arameters b m of the fuctio f(z). For sufficietly small values of X ad Z such that (π/b)x << ad (π/b)z <<, the Bessel fuctios J ((π/b)x) ad J ((mπ/b)z) ca be aroximated by the first term of their Taylor series, that is, J (φ) φ/, ad Eq. (7) reduces to ω ± ω b b J B X π b J mπ mf Z, ω± ω π cos π b mbf XZ. (8) From Table, we get b.549 ad b mf ; thus, Eq. (8) ca be reduced to a si π (,) rq o o π r r J π q q J π Π o k k r, q ω ±ω 0.699XZ. (9) This is i excellet agreemet with 0.707XZ, obtaiable from Eq. (4). I a similar way, the amlitude of the ormalized 484

6 Large Sigal Aalysis of BiOS ultiliers outut curret comoet of frequecy ω ±ω at the outut of the multilier show i Fig. ca be exressed as (0) where the subscrit g meas the arameters a m of the fuctio g(z). For sufficietly small values of X ad Z such that (π/b)z <<, the Bessel fuctio J ((mπ/b)z) ca be aroximated by the first term of its Taylor series, that is, J (φ) φ /8, ad Eq. (0) reduces to ω ± ω π b J B X a J mπ mg Z, ω± ω π 8 From Tables ad 3, we get b m amg XZ. b.549 ad m a m 6.09; thus, Eq. () reduces to 3 () ω ±ω 0.7XZ. () This is i excellet agreemet with 0.5XZ, obtaiable from Eq. (5). Fially, the amlitude of the ormalized outut curret comoet of frequecy ω ± ω at the outut of the multilier show i Fig. 3 ca be exressed as where the subscrit h meas the arameters b m of the fuctio h(z,). I this case, the arameters b mh are themselves fuctios of the third iut (t). Thus, if the third iut (t) ca be exressed as (t) P siω 3 t, (4) the, combiig Eqs. (8), (3) ad (4), the ormalized outut curret comoet of frequecy ω ± ω ± ω 3 at the outut of the multilier show i Fig. 3 ca be exressed as π P b J B X J mπ ω ω ω δ m B Z ± ± 3. (5) For sufficietly small values of X, Z, Eq. (5) reduces to ω ± ω ± ω 3 π 4 b mδ m XZP. From Tables ad, we get b.549 ad mδ m 5.36; thus, Eq. (6) reduces to (6) ω ±ω ±ω XZP. (7) b J B X π b J mπ mh Z, ω± ω (3) This is i excellet agreemet with 0.5XZP, obtaiable from Eq. (6). Table 3. Values of the Parameters B ad a o, a for the Fuctio g(z) g(z) a B 8.0 a o a a a a a a a a a a a a a a a a a a ote: a + b 0, 0,,,,. IV. Simulatio Results The multilier circuits show i Figs. ad were (i(r)-i(r))/i(i) Fig. 8. ormalized outut waveform obtaied from Fig. with V 70 mv, V.4 V, f. khz ad f.0 khz. 485

7 .T. Abuelma atti (i(r)-i(r))/i(i) Fig. 9. Frequecy sectrum of the waveform show i Fig. 8. (i(r)-i(r))/i(i) Fig.. Frequecy sectrum of the waveform show i Fig. 0. (i(r)-i(r))/i(i) Fig. 0. ormalized outut waveform obtaied from Fig. with V 00 mv, V.0 V, f. khz ad f.0 khz. Fig.. Calculated ad simulated results obtaied usig the multilier show i Fig.. simulated usig the evaluatio versio of the PSPICE circuit-simulatio rogram. The arameters used were I s 4 fa, β f 00 ad V Af 00 V for the ad the trasistors, ad V To V, ma/v ad γ 0.3 V / for the OS trasistors. The suly voltage V CC 5 V, the curret I o ma, ad the idividual load resistor R L l kω. The results obtaied are show i Figs Figure 8 shows a tyical outut waveform obtaied from the multilier circuit show i Fig. with ω 00 πrad/sec, ω 000 πrad/sec ad X Z.4, which corresods to V l 70 mv ad V.4 V. Figure 9 shows the frequecy sectrum of the outut waveform show i Fig. 8. Figure 0 shows a tyical outut waveform obtaied from the multilier circuit show i Fig. with ω 00 πrad/sec, ω 000 πrad/sec ad X Z.0, which corresods to V 00 mv ad V.0 V. Figure shows the frequecy sectrum of the outut waveform show i Fig. 0. It aears from Figs. 8 that the multilier circuits show i Figs. ad ca successfully erform multilicatio uder relatively large sigals, ad that their oeratio is ot limited to small sigals as metioed by imura (994). Figures 4 show the variatio of the ormalized outut amlitudes with the ormalized iuts. Also show are the calculated results obtaied usig Eqs. (7) ad (9) for the multilier circuit show i Fig., usig Eqs. (0) ad () for the multilier circuit show i Fig. ad usig Eqs. (5) ad (7) for the multilier circuit show i 486

8 Large Sigal Aalysis of BiOS ultiliers Fig. 3. Calculated ad simulated results obtaied usig the multilier show i Fig.. Fig. 3. From Figs. ad 3 it aears that the simulatio results are i fairly good agreemet with the calculated results obtaied usig the theory reseted here. Also, it aears from Figs. 4 that Eqs. (9), () ad (7) ca be used to redict the multilier erformace accurately for values of X Z P < 0.4. V. Coclusio Fig. 4. Calculated results obtaied usig the multilier show i Fig. 3. This aer has reseted large sigal aalysis for the BiOS low-voltage aalog multilier circuits. By aroximatig the oliear trasfer characteristics of the basic buildig blocks, such as the folded Gilbert multilier cell, the squarig multilier made from two cross-couled emitter-couled airs ad drive by a OS quadritail, ad the triler made from cross-couled emitter-couled airs ad drive by a OS quarter-square multilier, the outut sectrum of the aalog multiliers ca be obtaied. I geeral, the sectrum comutatio erformed usig these exressios requires the use of the ordiary Bessel fuctios. I cotrast with the commets made by imura (994), the results obtaied i this aer clearly show that these BiOS aalog multiliers ca be successfully used for large sigal as well as small sigal multilicatio. It is worth metioig here that the results reseted i this aer were obtaied based o the assumtio that the basic buildig blocks are memoryless. This may be a reasoable assumtio at relatively low frequecies. At high frequecies, however, the arasitic effects of the trasistors must be take ito cosideratio. This will result i frequecy-deedet oliear trasfer characteristics. Refereces Abuelma atti,. T. (993) A simle algorithm for fittig measured data to Fourier-series models. Iteratioal Joural of athematical Educatio i Sciece ad Techology, 4, 07-. imura,. (994) Some circuit desig techiques usig two cross-couled, emitter-couled airs. IEEE Trasactios o Circuits ad Systems-I: Fudametal Theory ad Alicatios, 4, imura,. (995) Circuit desig techiques for very low-voltage aalog fuctioal blocks usig trile-tail cells. IEEE Trasactios o Circuits ad Systems-I: Fudametal Theory ad Alicatios, 4,

9 .T. Abuelma atti UHAAD TAHER ABUELA ATTI ig Fahd Uiversity of Petroleum ad ierals Dhahra, Saudi Arabia BiOS BiOS Gilbert OS OS 488