EE 505 Lecture 9. Statistical Circuit Modeling

Transcription

1 EE 505 Lecture 9 Statistical Circuit Modelig

2 eview from previous lecture: Statistical Aalysis Strategy Will first focus o statistical characterizatio of resistors, the exted to capacitors ad trasistors Every resistor ca be expressed as = + P + W + D + GAD + L where is the omial value of the resistor ad the remaiig terms are all radom variables P : adom process variatios W : adom wafer variatios D : adom die variatios GAD : adom gradiet variatios L : Local adom Variatios

3 eview from previous lecture: Summary of esults Structure omial esistace Stadard Deviatio ormalized Stadard Deviatio = Ser 1 Par ote icreasig or decreasig the resistace by a factor of decreased the ormalized stadard deviatio by ote icreasig the area by a factor of decreased the ormalized stadard deviatio by What is the relatioship betwee resistace, area, ad stadard deviatio?

4 eview from previous lecture: Cosider the ormalized resistace 1 W L 1 EF EF 3 L W WL Defie the parameter A by the expressio A EF A is also a process parameter but more coveiet to use tha both ad σ EF WL A A A ote the ormalized variace is idepedet of the resistor value!

5 eview from previous lecture: B 1 How ca A be obtaied? ecall: A A where A EF 1µ B 1. Obtai A from a PDK. Build a test structure to obtai A

6 eview from previous lecture:

7 eview from previous lecture: Measuremet of A 1,,... - calculate variace of these samples ˆ

8 eview from previous lecture: Measuremet of A 1 1 ˆ A A ˆ A A

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10 atio Matchig Effects i Data Coverters atio matchig is ofte critical i ADCs ad DACs Accuracy ad matchig of gais is also critical i some data coverters

11 Amplifier Gai Accuracy V 1 VOUT A CL = - 1 Does the ratio matchig accuracy (A) deped upo the magitude of the gai: Cosider: : 1 1 : 11 Assume ideally 1 = = = = 11 ad the areas of the resistors are also ideally the same. Defie A CL0 to be the omial gai. Defie θ to be the gai error OM A CL0 = - 1OM

12 Amplifier Yield Assume the closed-loop gai A CL is a Gaussia V with mea A CL0 ad stadard deviatio σ ACL where A CL0 is the omial gai. Assume yield is defied by amplifiers with a gai that satisfies the expressio A 1 A A 1 CL0 X CL CL0 X Y P{A (1 ) A A 1 } CL0 X CL CL0 X CL0 CL0 xa 1 X xa (1 ) X ACL Y f x dx A 1 A z CL0 X CL0 ACL A (1 ) A z CL0 X CL0 ACL XA z CL0 ACL XA z ACL CL0 (0,1) (0,1) Y f z dz Y f z dz

13 Amplifier Yield Assume the closed-loop gai A CL is a Gaussia V with mea A CL0 ad stadard deviatio σ ACL where A CL0 is the omial gai Assume yield is defied by amplifiers with a gai that satisfies the expressio A 1 A A 1 CL0 X CL CL0 X XA z CL0 ACL XA z ACL CL0 (0,1) Y f z dz A X CL0 Y F(0.1) 1 ACL X Y F(0.1) 1 ACL A Thus to obtai yield eed to obtai σ ACL or CL0 ACL A CL0

14 Amplifier Gai Accuracy Gai error t follows that A CL0 A CL Thus eed to obtai ACL V 1 VOUT 1 OM 1 i1 11 i ACT i 0 i i i1 11 i i i 11 1 i i 0 i1 0

15 Amplifier Gai Accuracy 11 i 0 i1 0 V 1 VOUT i1 11 i i 0 0 i 0 f =1 f =10 i i 0 0 i i 0 ecall: A WL A A A X CL0 Y F(0.1) 1 ACL

16 Amplifier Gai Accuracy V 16 VOUT V 4 4 V OUT Optio 1 Optio Which will have the lowest σ? ote: TOT = 17 for Optio 1 10 for Optio

17 Amplifier Gai Accuracy May differet ways to achieve a give gai with a give resistor area V V OUT V V OUT V V OUT Which will have the best yield?

18 V EF Strig DAC Statistical Performace -1 V O, V O L is of cosiderable iterest L=Max( L ), 0<<-1 1 V O3 V O V O1 L is difficult to characterize aalytically so will focus o L Assume resistors are ucorrelated Vs but idetically distributed, typically zero mea Gaussia Cosider L = V OUT () V FT () V OUT j1 0 0 j1 j j V 1 1 EF 1 VFT V -1 j j1 EF j1 j 0-1

19 Strig DAC Statistical Performace 1 j j j1 1 j VEF 1 j j j1 j1 L 1 1 VEF 1 j j j1 1 j1 L 1 1 j1 j1 j 1 j j j j1 1 j1 1 j1 L 1 1 j 1 j1 j j1 1 1 j1 L 1 1 j1 j Let j = OM + j

20 Strig DAC Statistical Performace 1 1 OM 1 OM j 1 j j1 1 1 j1 j1 1 1 j1 L 1 1 OM j1 j1 j1 f we do a Taylor s series expasio of the reciprocal of the deomiator ad elimiate secod-order ad higher terms it follows that ote that K is a zero-mea multivariate Gaussia distributio j 1 OM j j j1 1 1 j1 L 1 1 OM 1 1 L 1 1 j j 1 j1 1 1 j1 1 OM 1 j OM j L j j j 1 1 OM j1 1 1 j1 OM j1 j 1 1 L j 1 j 1 1 OM j1 1 1 j1

21 Strig DAC Statistical Performace 1 1 L j 1 j 1 1 OM j1 1 1 j1 Sice the resistors are idetically distributed ad the coefficiets are ot a fuctio of the idex i, it follows that j1 1 j1 1 OM L Sice the idex i the sum does ot appear i the argumets, this simplifies to L OM ote there is a ice closed-form expressio for the L for a strig DAC!!

22 Strig DAC Statistical Performace L assumes a maximum variace at mid-code L max OM

23 Strig DAC Statistical Performace How about statistics for the L? L is ot zero-mea ad ot Gaussia ad closed form solutios do ot exist

24 Curret Steerig DAC Statistical Characterizatio Uary weighted V FF 0 =0 1-1 X Biary to Thermometer Decoder S 0 S 1 S S -1 V OUT Assume uary curret source array ad defie 0 =0 1 VOUT j 1 j= OM +j j0 For otatioal coveiece will ormalize by to obtai 1 OUTX 1 i i 0 Assume curret sources are radom variables with idetical distributios j 0,σ

25 Curret Steerig DAC Statistical Characterizatio Uary weighted L 1 j FT j0 OM 1 X VFF Biary to Thermometer Decoder 0=0 S0 1 S1 S -1 S-1 VOUT 1 1 FT j 1 j1 1 L 1 j 1 1 j j1 1 j1 OM L i i 1 i OM i

26 Curret Steerig DAC Statistical Characterizatio Uary weighted VFF 0=0 1-1 L i i 1 i OM i X Biary to Thermometer Decoder S0 S1 S S-1 VOUT Model the curret sources as j= OM +j L OM OM i 1 i OM t ca be show that the omial part cacels, thus L OM 1 1 i 1 OM i This is a sum of ucorrelated radom variables

27 Curret Steerig DAC Statistical Characterizatio VFF 0=0 1-1 The variace of K ca be readily calculated X Biary to Thermometer Decoder S0 S1 S S-1 VOUT L i i OM OM j= +j 1 L OM This simplifies to 1 1 L OM

28 Curret Steerig DAC Statistical Characterizatio Uary weighted VFF 0=0 1-1 As for the strig DAC, the maximum L occurs ear mid-code at about =/ thus X Biary to Thermometer Decoder S0 S1 S S-1 VOUT L MAX OM j= +j Ad, as for the strig DAC, the L is a order statistic ad thus a closed-form solutio does ot exist

29 Curret Steerig DAC Statistical Characterizatio Biary Weighted V FF 1 3 j deally 1 j OM X S 1 S S 3 S 1 j -1 V OUT The structure loos about the same as for the uary structure but ow the curret sources are biary weighted V b b OUT i j j0 Defie the decimal equivalet of b, b, by b b j 1 j j 1 b b, b 1... b1 For otatioal coveiece will ormalize by to obtai OUTX b bii for <0,0, 0> b <1,1, 1> i 1

30 Curret Steerig DAC Statistical Characterizatio Biary Weighted FT b b 1 i 1 i 0 b -1 VFF X 1 S1 S 3 S3 S VOUT Thus L b OUTX b LSBX FT b for <0,0, 0> b <1,1, 1> or equivaletly for 0 b -1 L b i i b 1 i1 i1 b LSBX i

31 Curret Steerig DAC Statistical Characterizatio Biary Weighted VFF 1 3 Budled curret sources assume comprised of uary curret sources X S1 S S3 S VOUT m m 1 m1 GK G OM G Thus L b i 1 1 bi GK b Gi i1 i1 i 1 1 LSBX Substitutig the values for GK, it ca be show that the omial parts cacel thus L b i 1 1 bi GK b Gi i1 i 1 1 i1 LSBX

32 Curret Steerig DAC Statistical Characterizatio Biary Weighted This ca be expressed as VFF 1 3 L b i 1 b G bi i 1 i 1 1 LSBX X S1 S S3 S VOUT This is ow a sum of ucorrelated radom variables, thus i 1 b L bi b i 1 i 1 1 G LSBX This reduces to i 1 b L bi b i 1 1 G LSBX

33 Curret Steerig DAC Statistical Characterizatio Biary Weighted t ca be show that this assumes a maximum L b occurs at b=< > or b=< > VFF 1 3 X S1 S S3 S VOUT Substitutig b=< > L b=< > 1 1 i1 1 G 1 1 i 1 LSBX This simplifies to L b=< > This ca be expressed as 1 1 i1 1 G 1 1 i 1 LSBX Lb =< > 1 1 G 1 1 LSBX

34 Curret Steerig DAC Statistical Characterizatio Biary Weighted L b=< > 1 1 G 1 1 LSBX VFF X 1 S1 S 3 S3 S VOUT LMAX Lb=<1,0,...0> G LSBX ote this is the same result as obtaied for the uary DAC But closed form expressios do ot exist for the L of this DAC sice the L is a order statistic

35 Ed of Lecture 9