Modeling 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 2nd integrator to be reduced by the gain of the 1st integrator Errors further downstream are even less significant A D DSP EECS 247 Lecture 22 Sigma Delta Nonlinearities 2002 B Boser 1 Capacitor Voltage Coefficient Ideal capacitor Q CV Practical capacitor 1st order model Q C V V with Q V Co 1 V K V Typical voltage coefficients Poly poly capacitors Metal metal capacitors A D DSP EECS 247 Lecture 22 Sigma Delta Nonlinearities 10 ppm V 1 10 ppm V 2002 B Boser 2 Integrator 1 1D vIN CIN 2 2 1 vCM 1 VREF CR1 2 D 2 CIN s voltco causes initial charge to vary nonlinearly with vIN harmonic distortion 1 vCM 1 CR2 2 D 2 CFB v1OUT 1 vCM VREF A D DSP v1N vCM EECS 247 Lecture 22 Sigma Delta Nonlinearities 2002 B Boser 3 Integrator 1 1D vIN CIN 2 2 1 vCM 1 VREF CR1 2 D 2 The voltage coefficients of CR1 and CR2 only generates a small offset negligible 1 vCM 1 CR2 2 D 2 VREF A D DSP v1N v1OUT 1 vCM CFB vCM EECS 247 Lecture 22 Sigma Delta Nonlinearities 2002 B Boser 4 Integrator 1 1D vIN CIN 2 2 1 The effect of CFB s voltco is non obvious so we ll have to analyze it vCM 1 VREF CR1 2 D 2 1 vCM 1 CR2 2 D 2 CFB v1OUT 1 VREF A D DSP v1N vCM vCM EECS 247 Lecture 22 Sigma Delta Nonlinearities 2002 B Boser 5 CIN Voltage Coefficient From charge conservation VCM 0 CR1 CR2 CR V1OUT k V1OUT k 1 14243 integration C C IN VIN k 1 IN V IN2 k 1 C C 14FB4444 4244FB4444 3 converter input C D R VREF C 142 4 FB43 4 1 bit feedback A D DSP EECS 247 Lecture 22 Sigma Delta Nonlinearities 2002 B Boser 6 Output Spectrum dBFS Int Noise dBV CIN Voltage Coefficient Vin VFS 1V Spectrum scaled for VFS 0dB window lowers peak Noise integral excludes DC fundamental 10 ppm V 0 50 100 2nd harmonic at 103dB dominates noise 150 Output Spectrum Integrated Noise 200 0 A D DSP 1 2 3 Frequency Hz 4 5 Let s characterize it 4 x 10 EECS 247 Lecture 22 Sigma Delta Nonlinearities 2002 B Boser 7 CIN Voltage Coefficient Output Spectrum dBFS 0 10 ppm V 1 ppm V 50 2nd harmonic increases 1dB per 1dB increase of 20dB 100 150 200 0 A D DSP 1 2 3 Frequency Hz 4 EECS 247 Lecture 22 Sigma Delta Nonlinearities 5 x 10 4 2002 B Boser 8 CIN Voltage Coefficient 0 Output Spectrum dBFS 20dB A 1V A 0 1V 50 100 2nd harmonic increases 2dB per 1dB increase of the input signal amplitude 40dB 150 200 A D DSP 0 1 2 3 Frequency Hz 4 5 4 x 10 EECS 247 Lecture 22 Sigma Delta Nonlinearities 2002 B Boser 9 CFB Voltage Coefficient Let s look next at the voltage coefficient of the feedback capacitor in the 1st integrator We turn off all other nonidealities CIN voltage coefficients noise etc Evaluating the effect of the CFB voltage coefficient requires solving a quadratic equation see next slide A D DSP Makes it easier to find the effect of CFB on 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 EECS 247 Lecture 22 Sigma Delta Nonlinearities 2002 B Boser 10 CFB Voltage Coefficient VOUT 1 QFB CFB 1 VOUT 1 1 1 4 QFB CFB 1 2 QFB k QFB k 1 C INVIN DC RVREF same as output from linear integrator A D DSP EECS 247 Lecture 22 Sigma Delta Nonlinearities 2002 B Boser 11 CFB Voltage Coefficient 1 512 1 16 1 64 b1 0 1z 1 1 1 z 1 X I1 3 g b2 z 1 sqrt 1 4 alpha u 1 1 2 alpha 0 25z 1 1 z 1 Nonlinear CFB a1 0 125z 1 00 25 1 z 1 I2 1 z 1 I3 1 z 1 I4 I5 2 3 4 5 I 1 I 2 I 3 I 4 1 a2 0 5 a3 0 5 a4 0 25 6 I 5 a5 0 25 7 Q Comparator 1 Y QFB linear VFB A D DSP EECS 247 Lecture 22 Sigma Delta Nonlinearities 2002 B Boser 12 CFB Voltage Coefficient Output Spectrum dBFS 0 Effect less pronounced than for CIN FB1 10 ppm V FB1 1000 ppm V Noise remains zero at DC 50 First order noise for large 100 Nonlinearities operating on shaped noise change the shape of the noise 150 No linear model can predict this 200 0 A D DSP 1 2 3 Frequency Hz 4 5 4 x 10 No harmonics why EECS 247 Lecture 22 Sigma Delta Nonlinearities 2002 B Boser 13 1st Integrator Output 0 The input signal appears much attenuated at the output of the 1st Integrator by its gain Output Spectrum dBFS Modulator Output Output of 1st Integrator 50 100 This signal appears across CFB and since it contains no strong tones it produces no harmonics 150 200 0 A D DSP 1 2 3 Frequency Hz 4 5 x 10 EECS 247 Lecture 22 Sigma Delta Nonlinearities Or does it 4 2002 B Boser 14 DC Input 0 For 1000 ppm V tones produced by CFB are much larger than native tones but move with the same velocity as native tones 1kHz mV 50 100 Where are these tones coming from 150 200 0 1 A D DSP 2 3 Frequency Hz 4 5 4 x 10 EECS 247 Lecture 22 Sigma Delta Nonlinearities 2002 B Boser 15 Output dBFS CFB Voltage Coefficient Tones appear near fs 2 as expected 0 100 Apparently these are folded to the baseband 140dB linearity requirement 200 1 45 1 46 1 47 1 48 1 49 1 5 6 Output dBFS Output Spectrum dBFS 5mV DC 8mV DC x 10 0 How 100 200 A D DSP 0 1 2 3 Frequency Hz 4 EECS 247 Lecture 22 Sigma Delta Nonlinearities 5 10 ppm V 4 x 10 2002 B Boser 16 Quantization Noise Nonlinearity Native tones at a frequency fD close to fs 2 have much higher power than in band tones fD f s f 2 2 When this tone passes a nonlinearity in the modulator loop filter it produces distortion sin 2 2 f D t A D DSP 1 1 cos 2 2 f D t 2 …
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