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 ) Typically R~50-200k  C~20pF so t RC ~20-80uS => 10-50kHz

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

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 sensor - two elements, Fresnel optics, output proportional to the difference between the charge on the left crystal and the charge on the right crystal.

Other Common Sensors 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

Incidence -- photoreflectors

Rotational Position Sensors Optical Encoders –Relative position –Absolute position Other Sensors –Resolver –Potentiometer Jizhong Xiao

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

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

Optical Encoders Detecting absolute position Typically 4k-8k/2  Higher Resolution Available – Laser/Hologram ( ” resolution) Jizhong Xiao

Jizhong Xiao Gray Code 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 2 n-1 segment in reverse order as next bit is added –Preserves unambiguous absolute position and direction

Other Motor Sensors Resolver Selsyn pairs ( ) High speed Potentiometer High resolution Monotone but poor linearity Noise! Deadzone! Jizhong Xiao

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

Improvement in MEMS Gyros Improvement of drift –Little drift improvement in last decade –Controls/Fabrication issue Improvement of resolution

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

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

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 ω d

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

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

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

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

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:

Noise Reduction: Filtering Noise is specified as a spectral density (V/Hz 1/2 ) 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:

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

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/N 1/2

‘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

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

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

Monotonic and missing code If DNL missing code. (A/D)

Offset and Gain Error D/A A/D

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.

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.

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

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

Aliasing (Sinusoids)

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!

Alaising For Sinusoid signals (natural band limit): For Cos(  n),  =2  k+  0 –Samples for all k are the same! –Unambiguous if 0<  <  –Thus One-half cycle per sample So if sampling at T, frequencies of f=  +1/2T will map to frequency 

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

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

Quantization Noise (A/D)

Quantization Noise Uniform Random Value Bounded range: –V LSB /2, +V LSB /2 Zero Mean

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

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

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

Sampling Jitter Bounds Error signal is proportional to the derivative Bounding the bandwidth bounds the derivative For t RMS, the RMS noise is: If we limit v RMS 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

DAC Timing Jitter DAC output is convolution of unit steps –Jitter RMS error depends on both timing error and sample period tjtj vv

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

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.

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.

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

Decoder-Based D/A converters Multiple R- string 6-bit D/A converter interpolating

Decoder-Based D/A converters R-string DACs with binary-tree decoding. Speed is limited by the delay through the resistor string as well as the delay through the switch network.

Binary-Scaled D/A Converters Monotonicity is not guaranteed. Potentially large glitches due to timing skews. Current-mode converter

Binary-Scaled D/A Converters 4 bit R-2R based D/A converter No wide-range scaling of resistors. Binary-array charge-redistribution D/A converter

Thermometer-Code Converter

Flash (Parallel) Converters High speed. Requires only one comparison cycle per conversion. Large size and power dissipation for large N.

Feedback in Sensing/Conversion High Resolution and Linearity Converters –Very expensive to build open-loop (precision components) –Aging, Drift, Temperature Compensation Closed-Loop Converters –Much higher possible resolution –Greatly improved linearity –Can use inexpensive components by substituting amplifier gain for component precision But –Higher Measurement Latency –Decreased Bandwidth –Eg. Successive Approx, Sigma-Delta

Nyquist-Rate A/D converters

Integrating converters Low conversion rate.

Successive-Approximation Converters Binary search

Successive-Approximation Converters DAC-based successive-approximation converter. –Requires a high-speed DAC with precision on the order of the converter itself. –Excellent trade-off between accuracy and speed. Most widely used architecture for monolithic A/D.

Sigma Delta A/D Converter SamplerModulator Decimation Filter x(t) x[n]y[n] Analog Digital fsfs fsfs 2 f o 16 bits e[n] Over Sampling Ratio = 2f o is Nyquist frequency Transfer function for an Lth order modulator given by Bandlimited to f o

Modulator Characteristics Highpass character for noise transfer function: In-band noise power is given by n o falls by 3(2L+1) for doubling of Over Sampling Ratio L+0.5 bits of resolution for doubling of Over Sampling Ratio n o essentially is uncorrelated for Dithering is used to decorrelate quantization noise