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Lecture 9 Sensors, A/D, sampling noise and jitter

Lecture 9 Sensors, A/D, sampling noise and jitter. Forrest Brewer. Light Sensors - Photoresistor. voltage divider V signal = (+5V) R R /(R + R R ) Choose R=R R at median of intended measured range Cadmium Sulfide (CdS) Cheap, relatively slow (low current) t RC = C l *(R+R R )

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Lecture 9 Sensors, A/D, sampling noise and jitter

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  1. Lecture 9Sensors, A/D, sampling noise and jitter Forrest Brewer

  2. Light Sensors - Photoresistor voltage divider Vsignal = (+5V) RR/(R + RR) • Choose R=RR at median of intended measured range • Cadmium Sulfide (CdS) • Cheap, relatively slow (low current) • tRC = Cl*(R+RR) Typically R~50-200kW C~20pF so tRC~20-80uS => 10-50kHz

  3. Light Sensors - Phototransistor • Much higher sensitivity • Relatively slow response (~1-5uS due to collector capacitance)

  4. Light Sensors - Pyroelectric Sensors • lithium tantalate crystal is heated by thermal radiation • tuned to 8-10 m radiation – maximize response to human IR signature • motion detecting burglar alarm • E.g. Eltec 442-3 sensor - two elements, Fresnel optics, output proportional to the difference between the charge on the left crystal and the charge on the right crystal.

  5. Force strain gauges - foil, conductive ink conductive rubber rheostatic fluids Piezorestive (needs bridge) piezoelectric films capacitive force Charge source Sound Microphones Both current and charge versions Sonar Usually Piezoelectric Position microswitches shaft encoders gyros Acceleration MEMS Pendulum Monitoring Battery-level voltage Motor current Stall/velocity Temperature Voltage/Current Source Field Antenna Magnetic Hall effect Flux Gate Location Permittivity Dielectric Other Common Sensors

  6. Incidence -- photoreflectors

  7. Rotational Position Sensors • Optical Encoders • Relative position • Absolute position • Other Sensors • Resolver • Potentiometer Jizhong Xiao

  8. Optical Encoders mask/diffuser • Relative position light sensor decode circuitry light emitter grating Jizhong Xiao

  9. Optical Encoders • Relative position - direction light sensor - resolution decode circuitry light emitter Phase lag between A and B is 90 degrees (Quadrature Encoder) Ronchi grating A B A leads B Jizhong Xiao

  10. Optical Encoders • Detecting absolute position • Typically 4k-8k/2p • Higher Resolution Available – Laser/Hologram (0.1-0.3” resolution) Jizhong Xiao

  11. Gray Code 0000 0001 0011 0010 0110 0111 0101 0100 1100 .. 1001 • Almost universally used encoding • One transition per adjacent number • Eliminates alignment issue of multiple bits • Simplified Logic • Eliminates position jitter issues • Recursive Generalization of 2-bit quadrature code • Each 2n-1 segment in reverse order as next bit is added • Preserves unambiguous absolute position and direction Jizhong Xiao

  12. Other Motor Sensors • Resolver • Selsyn pairs (1930-1960) • High speed • Potentiometer • High resolution • Monotone but poor linearity • Noise! • Deadzone! Jizhong Xiao

  13. Draper Tuning Fork Gyro • The rotation of tines causes the Coriolis Force • Forces detected through either electrostatic, electromagnetic or piezoelectric. • Displacements are measured in the Comb drive

  14. Improvement in MEMS Gyros • Improvement of drift • Little drift improvement in last decade • Controls/Fabrication issue • Improvement of resolution

  15. Piezoelectric Gyroscopes • Basic Principles • Piezoelectric plate with vibrating thickness • Coriolis effect causes a voltage form the material • Very simple design and geometry

  16. Piezoelectric Gyroscope • Advantages • Lower input voltage than vibrating mass • Measures rotation in two directions with a single device • More Robust • Disadvantages • (much) Less sensitive • Output is large when Ω = 0 • Drift compensation

  17. Absolute Angle Measurement • Bias errors cause a drift while integrating • Angle is measured with respect to the casing • The mass is rotated with an initial θ • When the gyroscopes rotates the mass continues to rotate in the same direction • Angular rate is measured by adding a driving frequency ωd

  18. Design consideration • Damping needs to be compensated • Irregularities in manufacturing • Angular rate measurement For angular rate measurement Compensation force

  19. Measurement Accuracy vs. Precision • Expectation of deviation of a given measurement from a known standard • Often written as a percentage of the possible values for an instrument • Precision is the expectation of deviation of a set of measurements • “standard deviation” in the case of normally distributed measurements • Few instruments have normally distributed errors

  20. Deviations • Systematic errors • Portion of errors that is constant over data gathering experiment • Beware timescales and conditions of experiment– if one can identify a measurable input parameter which correlates to an error – the error is systematic • Calibration is the process of reducing systematic errors • Both means and medians provide estimates of the systematic portion of a set of measurements

  21. Random Errors • The portion of deviations of a set of measurements which cannot be reduced by knowledge of measurement parameters • E.g. the temperature of an experiment might correlate to the variance, but the measurement deviations cannot be reduced unless it is known that temperature noise was the sole source of error • Error analysis is based on estimating the magnitude of all noise sources in a system on a given measurement • Stability is the relative freedom from errors that can be reduced by calibration– not freedom from random errors

  22. Model based Calibration • Given a set of accurate references and a model of the measurement error process • Estimate a correction to the measurement which minimizes the modeled systematic error • E.g. given two references and measurements, the linear model:

  23. Noise Reduction: Filtering • Noise is specified as a spectral density (V/Hz1/2) or W/Hz • RMS noise is proportional to the bandwidth of the signal: • Noise density is the square of the transfer function • Net (RMS) noise after filtering is:

  24. Filter Noise Example • RC filtering of a noisy signal • Assume uniform input noise, 1st order filter • The resulting output noise density is: • We can invert this relation to get the equivalent input noise:

  25. Averaging (filter analysis) • Simple processing to reduce noise – running average of data samples • The frequency transfer function for an N-pt average is: • To find the RMS voltage noise, use the previous technique: • So input noise is reduced by 1/N1/2

  26. ‘Normal’ Gaussian Statistics • Mean • Standard Deviation • Note that this is not an estimate for a total sample set (issue if N<<100), use 1/(N-1) • For large set of data with independent noise sources the distribution is: • Probability

  27. Issues with Normal statistics • Assumptions: • Noise sources are all uncorrelated • All Noise sources are accounted for • Enough time has elapsed to cover events • In many practical cases, data has ‘outliers’ where non-normal assumptions prevail • Cannot Claim small probability of error unless sample set contains all possible failure modes • Mean may be poor estimator given sporadic noise • Median (middle value in sorted order of data samples) often is better behaved • Not used often since analysis of expectations are difficult

  28. Characteristic of ADC and DAC • DAC • Monotonic and nonmonotonic • Offset , gain error , DNL and INL • Glitch • Sampling-time uncertainty • ADC • missing code • Offset , gain error , DNL and INL • Quantization Noise • Sampling-time uncertainty

  29. Monotonic and missing code If DNL < - 1 LSB => missing code. (A/D)

  30. Offset and Gain Error D/A A/D

  31. D/A nonlinearity (D/A) Differential nonlinearity (DNL): Maximum deviation of the analog output step from the ideal value of 1 LSB . Integral nonlinearity (INL): Maximum deviation of the analog output from the ideal value.

  32. D/A nonlinearity (A/D) • Differential nonlinearity (DNL): Maximum deviation in step width (width between transitions) from the ideal value of 1 LSB • Integral nonlinearity (INL): Maximum deviation of the step midpoints from the ideal step midpoints. Or the maximum deviation of the transition points from ideal.

  33. Glitch (D/A) • I1 represents the MSB current • I2 represents the N-1 LSB current • ex:0111…1 to 1000…0

  34. Sampling Theorem • Perfect Reconstruction of a continuous-time signal with Band limit f requires samples no longer than 1/2f • Band limit is not Bandwidth – but limit of maximum frequency • Any signal beyond f aliases the samples

  35. Aliasing (Sinusoids)

  36. Aliased Reconstruction • Reconstruction assumes values on principle branch – usually lower frequency • Nyquist Theorem assumes infinite history is available • Aliasing issues are worse for finite length samples • Don’t crowd Nyquist limits!

  37. Alaising • For Sinusoid signals (natural band limit): • For Cos(wn), w=2pk+w0 • Samples for all k are the same! • Unambiguous if 0<w<p • Thus One-half cycle per sample • So if sampling at T, frequencies of f=e+1/2T will map to frequency e

  38. Quantization Effects • Samples are digitized into finite digital resolution • Shows up as uniform random noise • Zero bias (for ideal A/D)

  39. Quantization Error • Deviations produced by digitization of analog measurements • For white, random signal with uniform quantization of xlsb: +lsb/2 x -lsb/2

  40. Quantization Noise (A/D)

  41. Quantization Noise • Uniform Random Value • Bounded range: –VLSB/2, +VLSB/2 • Zero Mean

  42. Sampling Jitter (Timing Error) • Practical Sampling is performed at uncertain time • Sampling interval noise – measured as value error • Sampling timing noise – also measured as value error

  43. Sampling-Time Uncertainty • (Aperture Jitter) • Assume a full-scale sinusoidal input, • want • then

  44. Jitter Noise Analysis • Assume that samples are skewed by random amount tj: • Expanding v(t) into a Taylor Series: • Assuming tj to be small:

  45. Sampling Jitter Bounds • Error signal is proportional to the derivative • Bounding the bandwidth bounds the derivative • For tRMS, the RMS noise is: • If we limit vRMS to LSB – we can bound the jitter • So for a 1MHz bandwidth, and 12 bit A/D we need less than 100pS of RMS jitter

  46. DAC Timing Jitter • DAC output is convolution of unit steps • Jitter RMS error depends on both timing error and sample period Dv tj

  47. DAC Timing Jitter • Error is: • Energy error: • RMS jitter error: • Relating to continuous time:

  48. DAC Jitter Bounds • We can use the same band limit argument as for sampling to find the jitter bound for a D-bit DAC: • So a 10MHz, 5-bit DAC can have at most 85pS of jitter.

  49. Decoder-Based D/A converters • Inherently monotonic. • DNL depend on local matching of neighboring R's. • INL depends on global matching of the R-string.

  50. Decoder-Based D/A converters • 4-bit folded R-string D/A converter

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