Toleranceanalysis实用教案

1、Lecture #4Richard Li, 20041 1. Importance of tolerance analysiso Tolerance analysis is not important for a prototype design in the Laboratory but, it is very important for a product in the massive production line.o The existence of a product in massive production line depends mainly on the yield rat

2、e of the producto Yield rate of a product is mostly determined by the tolerance of all the parts.o The tool of the tolerance analysis in the simulation stage of design is “Monte Carlo”.o In some circuit designs, the performance are mainly limited by the tolerance. The designer must do the tolerance

3、analysis first.The following example, a tunable filter design, illustrates the first priority of the tolerance analysis.第1页/共40页第一页，共41页。Lecture #4Richard Li, 20042o Plot of relative number of resistors versus value of resistor 2.Fundamentals of tolerance Relative number of resistorsFigure 4.1 Histo

4、gram of relative number of resistors versus the value of resistor 0.800k 0.850k 0.900k 0.950k 1.000k 1.050k 1.100k 1.105k 1.120k R, ohm第2页/共40页第二页，共41页。Lecture #4Richard Li, 20043 ohms , 1000m%52RemmRToliilativeiiR2 ohms .mRz2222)(xexdxdxxzfzezx020222)()(o Gaussian probability function , * Average v

5、alue :* Relative tolerance :* Variance or standard deviation Ri , * Sample value :where* Gaussion distribution :第3页/共40页第三页，共41页。Lecture #4Richard Li, 20044 -4 -3 -2 -1 0 +1 +2 +3 +4 z =(R-m)/m-4 m-3 m-2 m-1 m m+1 m+2 m+3 m+4 R, ohm(z)Figure 4.2 Distribution of the random variable, R, is a Normal pr

6、obability function or a Gaussian distribution0.39890.24200.05400.00440.0001第4页/共40页第四页，共41页。Lecture #4Richard Li, 20045o Tolerance and normal distribution (z)Figure 4.3 Integral of (z) from 0 to z or from m to m+z . -4 -3 -2 -1 0 +1 z +2 +3 +4 z =(R-m)/m-4 m-3 m-2 m-1 m m+1 m+z m+2 m+3 m+4 R, ohm第5页

7、/共40页第五页，共41页。Lecture #4Richard Li, 20046o Six sigma and 100% yield rateGaussian distribution :dxzfzemx022222)(where m = average of the variable x ; = square root of the variable x.Normal probability function is a Gaussian distribution function when :m = 0 , and = 1 .)()(2210222zzeerfdxzfxdyexerfxy0

8、22)(第6页/共40页第六页，共41页。Lecture #4Richard Li, 20047Figure 4.4 At each interval, z, the appearing percentage of a random variable with a Normal distribution when the interval is-1 z +1 ,then the area isf(z) = 68.26%,when the interval is-2 z +2 ,then the area isf(z) = 95.44%,when the interval is-3 z +3 ,

9、then the area isf(z) = 99.74%,when the interval isz +3,then the area isf(z) = 0.26%, (z)0.13%0.13%34.13%34.13%13.59%13.59%2.15%2.15% -4 -3 -2 -1 0 +1 +2 +3 +4 z =(R-m)/m-4 m-3 m-2 m-1 m m+1 m+2 m+3 m+4 R, ohm第7页/共40页第七页，共41页。Lecture #4Richard Li, 20048o 6, Cp, and CpkFigure 4.5 The various terminolo

10、gies of a process with normal distribution mUSLLSLDesign tolerance Process capability = 6 DefectsDefects -4 -3 -2 -1 0 +1 +2 +3 +4 z =(R-m)/m-4 m-3 m-2 m-1 m m+1 m+2 m+3 m+4 R, ohmUSL = Upper specification limit, , LSL = Lower specification limit, .第8页/共40页第八页，共41页。Lecture #4Richard Li, 20049LSLUSLT

11、olerance)()(1LSLzfUSLzfDefects6_Pr_LSLUSLCapabilityocessToleranceDesignCpCapability Index, Cp , is defined as)1(KCCppk2/LSLUSLmmKwhere m = Nominal process mean, , m = Actual process mean, .Adjusted Capability Index, Cpk, is defined aswhere USL = Upper specification limit, , LSL = Lower specification

12、 limit, .第9页/共40页第九页，共41页。Lecture #4Richard Li, 200410Example #102/LSLUSLmmK%56. 5%72.47*21)2(21zfDefects6667. 0646LSLUSLCp6667. 0)01 (6667. 01KCCppkFigure 4.6 A normal distribution with m=10, =0.01, USL-LSL=4, m=m .Specification mean, m Process mean, m USLLSL2.28%2.28% -4 -3 -2 -1 0 +1 +2 +3 +4 z =

13、(R-m)/m-4 m-3 m-2 m-1 m m+1 m+2 m+3 m+4 R, ohm第10页/共40页第十页，共41页。Lecture #4Richard Li, 200411Example #202/LSLUSLmmK%26. 0%87.49*21)3(21zfDefects0000.1666LSLUSLCp0000. 1)01 (0000. 11KCCppkFigure 4.7 A normal distribution with m=10, =0.01, USL-LSL=6, m=m .USLLSL0.13%0.13%Specification mean, m Process m

14、ean, m -4 -3 -2 -1 0 +1 +2 +3 +4 z =(R-m)/m-4 m-3 m-2 m-1 m m+1 m+2 m+3 m+4 R, ohm第11页/共40页第十一页，共41页。Lecture #4Richard Li, 200412Example #3 5000. 02/412/LSLUSLmmK%16%87.49%13.341)()3(1zfzfDefects6667.0646LSLUSLCp3334. 0)5 . 01 (6667. 01KCCppkFigure 4.8 A normal distribution with m=10, =0.01, USL-LSL

15、=4, m=9.99 .Specification mean, m Process mean, m USLLSL0.13%15.87% -4 -3 -2 -1 0 +1 +2 +3 +4 z =(R-m)/m-4 m-3 m-2 m-1 m m+1 m+2 m+3 m+4 R, ohm第12页/共40页第十二页，共41页。Lecture #4Richard Li, 200413Example #412/422/LSLUSLmmK%50%50%01)4()0(1zfzfDefects6667.0646LSLUSLCp0) 11 (0000. 11KCCppkFigure 4.9 A normal

16、 distribution with m=10, =0.01, USL-LSL=4, m=10.02 .USLLSL50%Specification mean, m Process mean, m -4 -3 -2 -1 0 +1 +2 +3 +4 z =(R-m)/m-4 m-3 m-2 m-1 m m+1 m+2 m+3 m+4 R, ohm第13页/共40页第十三页，共41页。Lecture #4Richard Li, 200414o Yield rate and DPU pdNNDPUwhere Nd = Number of defects found at all acceptanc

17、e points,Np = Number of units processed.第14页/共40页第十四页，共41页。Lecture #4Richard Li, 200415 !xexPxo Poisson distributionPoisson distribution, the occurrence of random defects in manufactured product can be statistically predicted. Poisson distribution is a mathematic model to describe the probability di

18、stribution of the arrived entities. For instance, in one hour lunch time, say, statistically from 12.00 to 13.00 oclock, the number of customers getting into a restaurant for lunch. Assuming that the average number of customers is , and the random number of arrivals at the same time interval is x, t

19、hen the probability of the random arrival number, Px, is第15页/共40页第十五页，共41页。Lecture #4Richard Li, 200416 DPUDPUeeDPUP!000DPUeFTY !xeDPUxPDPUx And, in generalthis is the relationship between the defect-free probability and the average DPU. The first time yield, FTY, can be approximated by the formula:

20、第16页/共40页第十六页，共41页。Lecture #4Richard Li, 200417CountPartsDPUpartPPMrateDefectpartPPM_/_/ NFTYDFTYCFTYBFTYAFTYFTYrolled*.*Regardless of process flow or order, the rolled yield can be calculated from the summation of the DPU values in all the steps.If a process with N steps, 1,2,3,4.N, and the value o

21、f DPU in each step is a, b, c, d, n respectively. Then, in terms of the formula, the first time yield, FTY, is e-a, e-b, e-c, e-de-n correspondingly. The fist time rolled yield for the process isndcbarolledeFTY.第17页/共40页第十七页，共41页。Lecture #4Richard Li, 200418Example : Rolled yield for a printed circu

22、it board. Assuming that there are totally 3 steps : Parts placed, parts soldered, and parts assembly. a) Part placement = 300 PPM, it means 300 parts failed in 1million parts.b) Part assembly defect rate = 800 PPM, it means 800 assembly-components failed in 1 million parts.c) Solder defect rate = 20

23、0 PPM, it means 200 connected points failed in 1million parts.*) On average, there are 2.3 connections for each part. Table 5.1 Calculated FTY for a printed circuit board assembly by 3 process stepsParts Parts Parts # of parts placementassemblysoldered TotalRolled FTYabc a+b+ce-(a+b+c) 1000.0300.080

24、0.046 0.15685.6%5000.1500.4000.230 0.78045.8%10000.3000.8000.460 1.56021.0% 第18页/共40页第十八页，共41页。Lecture #4Richard Li, 200419 o Approach to 6 design and production Figure 4.10 Difference of defects when the value of is 50% decreased mUSLLSLDesign tolerance Process capability = 6Defects = 6.68%Defects

25、= 6.68%Design tolerance USLLSLDefects = 0.13%Defects = 0.13%Process capability = 6m -4 -3 -2 -1 0 +1 +2 +3 +4 z =(R-m)/m-4 m-3 m-2 m-1 m m+1 m+2 m+3 m+4 R, ohm-4 -3 -2 -1 0 +1 +2 +3 +4 z =(R-m)/m-4 m-3m-2 m-1 m m+1m+2m+3m+4 R, ohm第19页/共40页第十九页，共41页。Lecture #4Richard Li, 2004203.Approach to 6 design

26、and productionkRRRRR521k 10R%20100RRRo Usually, component with higher values has lower relative tolerance. For instance, with the same processing,o Resistor has highest tolerance in IC design For instance,第20页/共40页第二十页，共41页。Lecture #4Richard Li, 200421kRRRRRtt1022o Resistors in parallelkRRRRRRtR1022

27、kRtRR51kRRRRRtt511To build a 5k resistor by 5k resistorby 10k resistors112221ttttRRRRkRRRkRRR1052IfThen,R=5kR=10kR=10kFigure 4.11 A 5 k resistor is replaced by two 10 k resistors in parallel第21页/共40页第二十一页，共41页。Lecture #4Richard Li, 200422pFCCCCCtt1022o Capacitors in seriespFCCCCCCtC1022pFCtCC51pFCCC

28、CCtt511To build a 5pF capacitor by 5pF capacitorby 10pF capacitors112221ttttCCCCpFCCCpFCCC1052IfThen,C=5pFC=10pF C=10pFFigure 4.12 A 5 pF capacitor is replaced by two 10 pF capacitors in series第22页/共40页第二十二页，共41页。Lecture #4Richard Li, 200423nHLLLLLtt1022o Inductors in parallelnHLLLLLLtL1022nHLtLL51n

29、HLLLLLtt511To build a 5nH inductortor by 5nH inductortorby 10nH inductors112221ttttLLLLnHLLLnHLLL1052IfThen,Usually it does not reduce the tolerance by the way of inductors in parallel because it is too expensive !L=5nHL=10nHL=10nHFigure 4.13 A 5 nH inductor is replaced by two 10 nH inductors in ser

30、ies第23页/共40页第二十三页，共41页。Lecture #4Richard Li, 200424R=5kR=10kR=10k(a) A 5 k resistor is replaced by two 10 k resistors in parallelm = 5k = 400 = 8% of mm = 10 k/2 = 5 k = 200 = 4% of mC=5pFC=10pFC=10pF(b) A 5 pF capacitor is replaced by two 10 pF capacitors in seriesm = 5pF = 0.4 pF = 8% of mm = 10 p

31、F/2 = 5 pF = 0.2 pF = 4% of mL=5nHL=10nHL=10nH(c) A 5 nH inductor is replaced by two 10 nH inductors in seriesm = 5nH = 0.4 nH = 8% of mm = 10 nH/2 = 5 nH = 0.2 nH = 4% of mFigure 4.14 Variation of in the various replacements第24页/共40页第二十四页，共41页。Lecture #4Richard Li, 2004254. An example : Tunable fil

32、ter designo Schematic of a tunable filterCVR1R1C2L3L2L1L1InL1L1OutL2C3C4VR1(C1)VR1(C1)VR1(C1)VR1(C1)L1 : Inductor 4.5 nH / 27 k / 0.2 pFL2 : Inductor 8.8 nH / 27 k / 0.2 pFL3 : Inductor 40.5 nH / 27 k / 0.2 pFC1 : Capacitance of varactor 13.75 pF when CV=5vC2 : Capacitor 2.4 pF C3 : Capacitor 100 pF

33、C4 : Capacitor 10000 pFR1 : Resistor 20 k CV : 5 vFigure 4.15 Schematic of a UHF tunable filter 第25页/共40页第二十五页，共41页。Lecture #4Richard Li, 200426o Main coupling : Inductive coupling * An improper coupling unable a tunable filter tuned over a wide frequency tuning range . * In the case of capacitor co

34、upling, Q = o /(BW) = 1/(2RC o ) BW = 2RC o2 (BW)/ ( o) = 4RC o * In the case of inductor coupling,)2/()/(RLBWQooLRBW/2where Q = Quality factor of filter ; o = Tuned central frequency ;BW = Bandwidth o ;L = Inductance of coupling inductor ;R = Equivalent resonant resistance of tank circuit o . (BW)/

35、 (o) = 0第26页/共40页第二十六页，共41页。Lecture #4Richard Li, 200427From the above expressions , it is concluded that1)In the case of inductor-coupling, the bandwidth of the filter is a constant or independent of the tuning frequency, while 2) In the case of capacitor-coupling, the bandwidth of the filter is a

36、function or dependent of the tuning frequency. It is increased as the tuning frequency is increased. Consequently,The bandwidth could be kept in an almost constant for a wide tuning frequency range if the coupling element is only an inductor. It is therefore decided to use an inductor as the primary

37、 coupling element between the 2 tank circuits, which is L3 as shown above.第27页/共40页第二十七页，共41页。Lecture #4Richard Li, 200428o Second coupling : Capacitive coupling * The capacitor, C2, is the second coupling component between the two tank circuits; * It forms a “zero” at the imaginary frequency; * Thi

38、s “zero” traces the central frequency, o , over a wide frequency tuning range. 第28页/共40页第二十八页，共41页。Lecture #4Richard Li, 200429o Monte Carlo AnalysisFigure 7.15 Simulation page for Monte-Carlo analysis (Strictly speaking, it is not a Gaussian but a normal distribution here.)FLTPort 1Port 250 Termina

39、tor50 TerminatorC1 = CaC2 = CbL1 = LaL2 = LbL3 = LcEquation Ca = Randvar Gaussian, 13.75 pF 5%Equation Cb = Randvar Gaussian, 2.4 pF 5%Equation La = Randvar Gaussian, 4.5 nH 7%Equation Lb = Randvar Gaussian, 8.8 nH 7%Equation Lc = Randvar Gaussian, 40.5 nH 7%Figure 4.16 Simulation page for Monte-Car

40、lo analysis (Strictly speaking, it is not a Gaussian but a normal distribution here.) 第29页/共40页第二十九页，共41页。Lecture #4Richard Li, 200430* Frequency response without tolerance200 300 400 fo 500 600 700 -10S21, dB0-20-40-60 10-50-70-80-90-30f , MHzIL=1.76 dBImag. Rej. =83.1 dBf o =435.43 MHzFigure 4.17

41、Typical frequency response of tunable filter without tolerance 第30页/共40页第三十页，共41页。Lecture #4Richard Li, 200431* Case #3 : Bandwidth is too narrowS21, dB 200 300 400 fo 500 600 700 -100-20-40-60 10-50-70-80-90-30f , MHzIL=1.76 dBImag. Rej. =80.1 dBf o =435.43 MHzFigure 4.18 Frequency response of tuna

42、ble filter with tolerance - Bandwidth is too narrowIL=5.10 dB第31页/共40页第三十一页，共41页。Lecture #4Richard Li, 200432* Case #2 : Bandwidth is appropriate 200 300 400 fo 500 600 700 -10S21, dB0-20-40 10-50-70-80-90-30f , MHzIL=1.76 dBImag. Rej. =80.1 dBf o =435.43 MHz-60Figure 4.19 Frequency response of tuna

43、ble filter with tolerance -Bandwidth is appropriate第32页/共40页第三十二页，共41页。Lecture #4Richard Li, 200433* Case #1 : Bandwidth is too wide 200 300 400 fo 500 600 700 -10S21, dB0-20-40 10-50-90-30 f , MHzIL=1.76 dBImag. Rej. =80.1 dBf o =435.43 MHzFigure 4.20 Frequency response of tunable filter with toler

44、ance - Bandwidth is too wide-70-80-60第33页/共40页第三十三页，共41页。Lecture #4Richard Li, 200434* Yield rate & its histogram of IL less than 2.5 dB ( For case #2 )Yield rate, % 100.00-2.16 -1.71IL, dB350%Figure 4.21 Display of insertion loss histogram and yield rate for IL60 dB 第36页/共40页第三十六页，共41页。Lecture

45、#4Richard Li, 200437* Image Rejection performance of parts with tolerance ( For case #2 )-64-8413.2 142Imag.Rej., dBC1 , pFL1 , nH-64-842.3 2.5C2 , pFImag.Rej., dB-64-848.3 9.3L2 , nHImag.Rej., dB-64-844.2 4.8Imag.Rej., dB-64-8438.1 42.6L3 , nHImag.Rej., dBNote : Capacitors are more sensitive to the

46、 image rejection than inductors !Figure 4.24 The effect of individual parts value on the image rejection 第37页/共40页第三十七页，共41页。Lecture #4Richard Li, 2004385. Appendix: Table of the normal distriution Z 0 1 234 5 6 7 8 90.0.0000.0040.0080.0120.0160.0199.0239.0279.0319.03590.1.0398.0438.0478.0517.0557.0

47、596.0636.0675.0714.07540.2.0793.0832.0871.0910.0948.0987.1026.1064.1103.11410.3.1179.1217.1255.1293.1331.1368.1406.1443.1480.15170.4.1554.1591.1628.1664.1700.1736.1772.1808.1844.18790.5.1915.1950.1985.2019.2054.2088.2123.2157.2190.22240.6.2258.2291.2324.2357.2389.2422.2454.2486.2518.25490.7.2580.261

48、2.2642.2673.2704.2734.2764.2794.2823.28520.8.2881.2910.2939.2967.2996.3023.3051.3078.3016.3133 0.9.3135.3186.3212.3238.3264.3289.3315.3340.3365.3389 1.0.3413.3438.3461.3485.3508.3531.3554.3577.3599.36211.1.3643.3665.3686.3708.3729.3749.3770.3790.3810.38301.2.3849.3869.3888.3907.3925.3944.3962.3980.3997.40151.3.4032.4049.4066.4082.4099.4115.4131.4147.4162.41771.4.4192.4207.4222.4236.4251.4265.4279.4292.4306.43191.5.4332.4345.4357.4370.4382.4394.4406.4418.4429.4441