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Berkeley ELENG 247A - Lecture Notes

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EECS 247 Lecture 22: Sigma-Delta Nonlinearities © 2002 B. Boser 1A/DDSPModeling Σ∆ Nonlinearities• Many component nonlinearities contribute negligible errors– Don’t waste CPU cycles modeling the voltage coefficient of every capacitor in the loop– Unnecessarily complex models reduce the chance to find relevant problems, and, perhaps, solutions– As with all nonidealities, model one at a time• Expect errors from the 2ndintegrator to be reduced by the gain of the 1stintegrator– Errors further downstream are even less significantEECS 247 Lecture 22: Sigma-Delta Nonlinearities © 2002 B. Boser 2A/DDSPCapacitor Voltage Coefficient• Ideal capacitor• Practical capacitor (1storder model)• Typical voltage coefficients– Poly-poly capacitors 10 ppm/V– Metal-metal capacitors 1 … 10 ppm/V()( )VVCVQVVCQoK++==α1)(h wit CVQ=EECS 247 Lecture 22: Sigma-Delta Nonlinearities © 2002 B. Boser 3A/DDSPIntegrator 1φ1DCINφ2v1OUTCFBvINφ2vCMφ1φ1φ2vCMφ1CR1φ2•DVREFφ1φ2vCMφ1CR2VREFvCMφ2•Dv1NCIN’s voltco causes initialcharge to vary nonlinearlywith vIN⇒ harmonicdistortionEECS 247 Lecture 22: Sigma-Delta Nonlinearities © 2002 B. Boser 4A/DDSPIntegrator 1φ1DCINφ2v1OUTCFBvINφ2vCMφ1φ1φ2vCMφ1CR1φ2•DVREFφ1φ2vCMφ1CR2VREFvCMφ2•Dv1NThe voltage coefficients of CR1and CR2only generates a (small) offset à negligibleEECS 247 Lecture 22: Sigma-Delta Nonlinearities © 2002 B. Boser 5A/DDSPIntegrator 1φ1DCINφ2v1OUTCFBvINφ2vCMφ1φ1φ2vCMφ1CR1φ2•DVREFφ1φ2vCMφ1CR2VREFvCMφ2•Dv1NThe effect of CFB’s voltco is non-obvious, so we’ll have to analyze itEECS 247 Lecture 22: Sigma-Delta Nonlinearities © 2002 B. Boser 6A/DDSPCINVoltage Coefficient• From charge conservation (VCM=0, CR1=CR2=CR):()()( ) ( )4434421444444 3444444 2143421feedbackbit -1inputconverter 2nintegratio11111REFFBRFBININFBINOUTOUTVCCDkVCCkVCCkVkVIN−−+−+−=αEECS 247 Lecture 22: Sigma-Delta Nonlinearities © 2002 B. Boser 7A/DDSPCINVoltage Coefficient0 1 2 3 4 5x 104-200-150-100-500Output Spectrum [dBFS] / Int. Noise [dBV]Frequency [Hz]Output SpectrumIntegrated Noise• Vin= VFS= 1V• Spectrum scaled for VFSà 0dB(window lowers peak)• Noise integral excludes DC, fundamental• α = 10 ppm/V• 2ndharmonic at –103dB dominates noise!• Let’s characterize it …EECS 247 Lecture 22: Sigma-Delta Nonlinearities © 2002 B. Boser 8A/DDSPCINVoltage Coefficient2ndharmonic increases 1dB per 1dB increase of α0 1 2 3 4 5x 104-200-150-100-500Frequency [Hz]Output Spectrum [dBFS]α=10 ppm/Vα=1 ppm/V20dBEECS 247 Lecture 22: Sigma-Delta Nonlinearities © 2002 B. Boser 9A/DDSPCINVoltage Coefficient2ndharmonic increases 2dB per 1dB increase of the input signal amplitude0 1 2 3 4 5x 104-200-150-100-500Frequency [Hz]Output Spectrum [dBFS]A = 1VA = 0.1V20dB40dBEECS 247 Lecture 22: Sigma-Delta Nonlinearities © 2002 B. Boser 10A/DDSPCFBVoltage Coefficient• Let’s look next at the voltage coefficient of the feedback capacitor in the 1stintegrator• We “turn off” all other nonidealities – CINvoltage coefficients, noise, etc.– Makes it easier to find the effect of CFBon the modulator– Downside: we miss potential interactions between nonidealities– Often they are negligible: nonidealities (like voltage coefficients) produce small errors … linear superposition applies– Of course it’s a good idea to run a complete verification at the end– And we’ll get to diagnose the “real thing” soon enough … without the insight gained from such idealized simulations it’s next to impossible to diagnose a complex chip• Evaluating the effect of the CFBvoltage coefficient requires solving a quadratic equation, see next slide …EECS 247 Lecture 22: Sigma-Delta Nonlinearities © 2002 B. Boser 11A/DDSPCFBVoltage Coefficient( )αα24111111FBFBOUTFBFBOUTCQVCQV++−==()())integratorlinear fromoutput as (same 1REFRININFBFBVDCVCkQkQ++−=EECS 247 Lecture 22: Sigma-Delta Nonlinearities © 2002 B. Boser 12A/DDSPCFBVoltage Coefficient7Q6I_55I_44I_33I_22I_11Y3g1/16-1/64b21/512b10.25a50.25a40.5a30.5a21a1(sqrt(1 + 4*alpha*u(1)) - 1) / 2 / alphaNonlinear CFB0.125z -11-z -1I500.25 1-z -1I40.25z -11-z -1I3z -11-z -1I20.1z -11-z -1I1Comparator1XQFB (linear)VFBEECS 247 Lecture 22: Sigma-Delta Nonlinearities © 2002 B. Boser 13A/DDSPCFBVoltage Coefficient• Effect less pronounced than for CIN• Noise remains zero at DC• First order noise for large α• Nonlinearities operating on shaped noise change the shape of the noise …– No linear model can predict this• No harmonics … why?0 1 2 3 4 5x 104-200-150-100-500Frequency [Hz]Output Spectrum [dBFS]αFB1 = 10 ppm/VαFB1 = 1000 ppm/VEECS 247 Lecture 22: Sigma-Delta Nonlinearities © 2002 B. Boser 14A/DDSP1stIntegrator Output• The input signal appears much attenuated at the output of the 1stIntegrator(by its gain)• This signal appears across CFB… and since it contains no strong tones it produces no harmonics• Or does it?0 1 2 3 4 5x 104-200-150-100-500Frequency [Hz]Output Spectrum [dBFS]Modulator OutputOutput of 1st IntegratorEECS 247 Lecture 22: Sigma-Delta Nonlinearities © 2002 B. Boser 15A/DDSPDC Input• For α = 1000 ppm/V, tones produced by CFBare much larger than native tones, but move with the same velocity as native tones (1kHz/mV)• Where are these tones coming from?0 1 2 3 4 5x 104-200-150-100-500Frequency [Hz]Output Spectrum [dBFS]5mV DC8mV DCEECS 247 Lecture 22: Sigma-Delta Nonlinearities © 2002 B. Boser 16A/DDSPCFBVoltage Coefficient• Tones appear near fs/2, as expected• Apparently these are “folded” to the base-band• How?α = 10 ppm/V 1.45 1.46 1.47 1.48 1.49 1.5x 106-200-1000Output [dBFS]0 1 2 3 4 5x 104-200-1000Output [dBFS]Frequency [Hz]140dB linearity requirementEECS 247 Lecture 22: Sigma-Delta Nonlinearities © 2002 B. Boser 17A/DDSPQuantization Noise Nonlinearity• Native tones at a frequency fDclose to fs/2 have much higher power than in-band tones• When this tone passes a nonlinearity in the modulator loop filter, it produces distortion22δfffsD−=( ) ( )[ ]tftfDD22cos21212sin2ππ −=EECS 247 Lecture 22: Sigma-Delta Nonlinearities © 2002 B. Boser 18A/DDSPQuantization Noise Nonlinearity( ) ( )[ ]tftfDD22cos21212sin2ππ −=• In the sampled data system,maps to fδ• Small nonlinearities applied to aggressively shaped quantization noise can produce big tone problems …δfffsD−=2EECS 247 Lecture 22: Sigma-Delta Nonlinearities ©


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Berkeley ELENG 247A - Lecture Notes

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