EECS 247 Lecture 16: Data Converters © 2005 H.K. Page 1EE247Lecture 16D/A converters continued:• Current based DACs-unit element versus binary weighted• Static performance– Component matching-systematic & random errors• Practical aspects of current-switched DACs• Segmented current-switched DACs• DAC self calibration techniques– Current copiers– Dynamic element matchingADC Converters• Sampling– Sampling switch induced distortion– Sampling switch charge injectionEECS 247 Lecture 16: Data Converters © 2005 H.K. Page 2Current Source DACUnit Element• “Unit elements ”• 2B-1 current sources & switches • Monotonicity does not depend on element matching• Suited for both MOS and BJT technologies• Output resistance of current source causes gain error IrefIrefIoutIrefIref…………………………EECS 247 Lecture 16: Data Converters © 2005 H.K. Page 3Current Source DACUnit Element • Output resistance of current source à gain error problemà Use transresistance amplifier- Current source output held @ virtual ground - Error due to current source output resistance eliminated- New issues: offset & speed of the amplifierIrefIrefIrefIref…………………………VoutR-+EECS 247 Lecture 16: Data Converters © 2005 H.K. Page 4Current Source DACBinary Weighted• “Binary weighted”• B current sources & switches (2B-1 unit current sources but less # of switches)• Monotonicity depends on element matching4 IrefIrefIout2Iref2B-1Iref…………………………EECS 247 Lecture 16: Data Converters © 2005 H.K. Page 5Static DAC INL / DNL Errors• Component matching• Systematic errors– Finite current source output resistance – Contact resistance– Edge effects in capacitor arrays– Process gradient• Random errors– Lithography– Often Gaussian distribution (central limit theorem)*Ref: C. Conroy et al, “Statistical Design Techniques for D/A Converters,” JSSC Aug. 1989, pp. 1118-28.EECS 247 Lecture 16: Data Converters © 2005 H.K. Page 6Gaussian Distribution-3 -2 -1 0 1 2 300.050.10.150.20.250.30.350.4x /σProbability density p(x)( )22x2221p(x)e2wherestandarddeviation:E(x)µσπσσµ−−==−EECS 247 Lecture 16: Data Converters © 2005 H.K. Page 7Yield()2xX2XPXxX1edx2Xerf2π+−−−≤≤+===∫00.10.20.30.4Probability density p(x)0 0.5 1 1.5 2 2.5 300.20.40.60.81X38.368.395.4P(-X ≤ x ≤ +X)EECS 247 Lecture 16: Data Converters © 2005 H.K. Page 8YieldX/σ P(-X ≤ x ≤ X) [%]0.2000 15.85190.4000 31.08430.6000 45.14940.8000 57.62891.0000 68.26891.2000 76.98611.4000 83.84871.6000 89.04011.8000 92.81392.0000 95.4500X/σ P(-X ≤ x ≤ X) [%]2.2000 97.21932.4000 98.36052.6000 99.06782.8000 99.48903.0000 99.73003.2000 99.86263.4000 99.93263.6000 99.96823.8000 99.98554.0000 99.9937EECS 247 Lecture 16: Data Converters © 2005 H.K. Page 9Example• Measurements show that the offset voltage of a batch of operational amplifiers follows a Gaussian distribution with σ = 2mV and µ = 0.• Fraction of opamps with |Vos| < 6mV:– X/σ = 3 à 99.73 % yield• Fraction of opamps with |Vos| < 400µV:– X/σ = 0.2 à 15.85 % yieldEECS 247 Lecture 16: Data Converters © 2005 H.K. Page 10Component MismatchRR∆10000100200300400No. of resistors100410081012996992988R[]ΩExample: Side-by-side resistorsE.g. Let us assume in this example large # of Rs with average of 1000OHM measured: 68.5% within +-4OHM or +-0.4% of averageà 1σ for resistorsà 0.4%Large # of devices measured & curved àtypically if sample size is large shape is Gaussian…….…….EECS 247 Lecture 16: Data Converters © 2005 H.K. Page 11Component Mismatch12122dRRRRR2dRRR1Areaσ+==−∝RR∆000.050.10.150.20.250.30.350.4Probability density p(x)σ2σ3σ−σ−2σ−3σdRRTwo side-by-sideResistorsFor typical technologies & geometries1σ for resistorsà 0.02 το 5%In the case of resistors σ is a function of areaEECS 247 Lecture 16: Data Converters © 2005 H.K. Page 12DNL Unit Element DACiirefRI∆=DNL of unit element DAC is independent of resolution!E.g. Resistor string DAC:IrefiimedianrefiirefmedianiimedianimedianimedianmedianDNL dRRRIRIDNLRRdRdRRRRσσ∆=∆=∆−∆=∆−==≈=VrefEECS 247 Lecture 16: Data Converters © 2005 H.K. Page 13DNL Unit Element DACExample:If σdR/R= 0.4%, what DNL spec goes into the datasheet so that 99.9% of all converters meet the spec? DNL of unit element DAC is independent of resolution!Note similar results for all unit-element based DACsE.g. Resistor string DAC:iiDNLdRRσσ=EECS 247 Lecture 16: Data Converters © 2005 H.K. Page 14YieldX/σ P(-X ≤ x ≤ X) [%]0.2000 15.85190.4000 31.08430.6000 45.14940.8000 57.62891.0000 68.26891.2000 76.98611.4000 83.84871.6000 89.04011.8000 92.81392.0000 95.4500X/σ P(-X ≤ x ≤ X) [%]2.2000 97.21932.4000 98.36052.6000 99.06782.8000 99.48903.0000 99.73003.2000 99.86263.4000 99.93263.6000 99.96823.8000 99.98554.0000 99.9937EECS 247 Lecture 16: Data Converters © 2005 H.K. Page 15DNL Unit Element DACExample:If σdR/R= 0.4%, what DNL spec goes into the datasheet so that 99.9% of all converters meet the spec? Answer:From table: for 99.9% à X/σ = 3.3σDNL= σdR/R= 0.4%3.3 σDNL= 1.3%àDNL= +/- 0.013 LSBE.g. Resistor string DAC:iiDNLdRRσσ=EECS 247 Lecture 16: Data Converters © 2005 H.K. Page 16DAC INL AnalysisBAN=2B-1nnNOutput [LSB]Input [LSB]EIdeal VarianceA=n+E n n.σε2B=N-n-E N-n (N-n).σε2E = A-n r =n/N N=A+B= A-r(A+B)= A (1-r) -B.rà Variance of E:σE2=(1-r)2.σΑ2+ r 2.σB2=N.r .(1-r).σε2EECS 247 Lecture 16: Data Converters © 2005 H.K. Page 17DAC INL• Error is maximum at mid-scale (N/2):• INL depends on DAC resolution and element matching σε • While σDNL= σεRef: Kuboki et al, TCAS, 6/198222E2EBINLBn1nNdTofindmax.variance:0dnnN/2121 2 with N21εεσσσσσ−=×=→==−=−EECS 247 Lecture 16: Data Converters © 2005 H.K. Page 18Untrimmed DAC INLExample:Assume the following requirement:σINL= 0.1 LSBThen:σε= 1% à B = 8.6σε= 0.5% à B = 10.6σε= 0.2% à B = 13.3σε= 0.1% à B = 15.3+≅−≅εεσσσσINLBINLB2log22 1221EECS 247 Lecture 16: Data Converters © 2005 H.K. Page 19Simulation Exampleσε= 1%B = 12Random # generator used in MatLabComputed INL:σDNL= 0.01 LSBσINL= 0.3 LSB(midscale)500
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