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

- Forrest Brewer

Light Sensors - Photoresistor

- voltage divider Vsignal (5V) RR/(R RR)
- Choose RRR at median of intended measured range
- Cadmium Sulfide (CdS)
- Cheap, relatively slow (low current)
- tRC Cl(RRR)
- Typically R50-200kW C20pF so tRC20-80uS gt

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 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.

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

mask/diffuser

- Relative position

light sensor

decode circuitry

light emitter

grating

Jizhong Xiao

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

Optical Encoders

- Detecting absolute position
- Typically 4k-8k/2p
- Higher Resolution Available Laser/Hologram

(0.1-0.3 resolution)

Jizhong Xiao

Gray Code

0000 0001 0011 0010 0110 0111 0101 0100 1100 .. 10

01

- 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

Other Motor Sensors

- Resolver
- Selsyn pairs (1930-1960)
- 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 O 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 ?
- When the gyroscopes rotates the mass continues to

rotate in the same direction - Angular rate is measured by adding a driving

frequency ?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/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

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

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

Normal Gaussian Statistics

- Mean
- Standard Deviation
- Note that this is not an estimate for a total

sample set (issue if Nltlt100), 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 lt - 1 LSB gt 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
- ex01111 to 10000

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 - Dont crowd Nyquist limits!

Alaising

- For Sinusoid signals (natural band limit)
- For Cos(wn), w2pkw0
- Samples for all k are the same!
- Unambiguous if 0ltwltp
- Thus One-half cycle per sample
- So if sampling at T, frequencies of fe1/2T will

map to frequency e

Quantization Effects

- Samples are digitized into finite digital

resolution - Shows up as uniform random noise
- Zero bias (for ideal A/D)

Quantization Error

lsb/2

x

-lsb/2

- Deviations produced by digitization of analog

measurements - For white, random signal with uniform

quantization of xlsb

Quantization Noise (A/D)

Quantization Noise

- Uniform Random Value
- Bounded range VLSB/2, VLSB/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

tj - Expanding v(t) into a Taylor Series
- Assuming tj to be small

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

DAC Timing Jitter

- DAC output is convolution of unit steps
- Jitter RMS error depends on both timing error and

sample period

Dv

tj

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

Binary-array charge-redistribution D/A converter

- 4 bit R-2R based D/A converter
- No wide-range scaling of resistors.

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

en

Decimation Filter

fs

fs

2 fo

Sampler

Modulator

x(t)

xn

yn

16 bits

Bandlimited to fo

Digital

Analog

Over Sampling Ratio 2fo is Nyquist

frequency Transfer function for an Lth order

modulator given by

Modulator Characteristics

- Highpass character for noise transfer function
- In-band noise power is given by

- no falls by 3(2L1) for doubling of Over Sampling

Ratio - L0.5 bits of resolution for doubling of Over

Sampling Ratio - no essentially is uncorrelated for
- Dithering is used to decorrelate quantization

noise