20-AEEM-329 ENGINEERING MEASUREMENTS Engineering Areas Research - PDF document

20-AEEM-329 ENGINEERING MEASUREMENTS

Engineering Areas Research and Development Design (Product and Process) Manufacturing Service and Maintenance Engineering Methods Theoretical Simulation (Computational and Experimental) Experimental

Part 1 Basic Principles

Feedback-Control System Disturbances Input variable Controlled (energy and/or material) variable Process Control Measuring System Element Controller Desired value of controlled variable

Measuring System Measured medium Measured quantity Primary Sensing element sensor Variable converison element Variable manipulation element signal conditioner Data transmission element Data storage element Data presentation element Presented data Observer

Computer-Based Measurement Measured medium Measured quantity Transducer Signal conditioner Analog-to-digital converter Computer Presented data Observer

Part 2 Measurement Characteristics

Instrument Types active versus passive instruments proportional versus null-type analog versus digital indicating versus signal output smart versus conventional

Instrument Characteristics Static Characteristics • accuracy/inaccuracy (uncertainty) absolute, relative, re full-scale tolerance • precision/repeatability/reproducibility low-precision high-precision high-precision low-accuracy low-accuracy high-accuracy • range/span • linearity/nonlinearity • sensitivity output output reading reading measured measured quantity quantity

Instrument Characteristics Static Characteristics (continued) • threshold (absolute/relative) • resolution (absolute/relative) • sensitivity to disturbance (temperature, pressure, etc.) zero drift/sensitivity drift output output reading reading zero drift sensitivity drift characteristic nominal characteristic nominal measured measured quantity quantity • dead space/backlash/hysteresis output output reading reading measured measured quantity quantity dead space dead space

Instrument Characteristics Dynamic Characteristics q i measured quantity q o output reading • general linear, time-invariant dynamic instrument/general input 2 3 2 3 a d d a d b d b d b d + + + = + + + ... a q dt q a dt q dt q b q dt q dt q dt q ... o o o o i i i i 0 1 2 3 0 1 2 3 2 3 2 3 • general linear, time-invariant dynamic instrument/stepped input 2 3 a d d a d + + + = a q dt q a dt q dt q b q ... o o o o i 0 1 2 3 0 2 3 • zero-order instrument = a q b q o i 0 0 • first-order instrument a d + = a q dt q b q o o i 0 1 0 • second-order instrument 2 a d a d + + = a q dt q dt q b q o o o i 0 1 2 0 2

Instrument Characteristics Dynamic Characteristics (continued) zero-order instrument response measured quantity time output reading time t first-order instrument response measured quantity time output reading ~ 63% ~ time t τ time constant ≈ 5 τ ( t settling within 0 5%) . t

Instrument Characteristics Dynamic Characteristics (continued) second-order instrument response measured quantity time output reading low damping high damping time t • delay time • dead time • transition time • settling time • transient frequency • slew rate

Part 3 Measurement Errors

Types of Errors • intrinsic errors of the measurement process extrinsic errors during data transfer, storage, display, evaluation, etc. systematic errors ( < > ≠ • e 0 ) can be reduced by corrections and calibration random errors ( < > = e 0 ) can be reduced by averaging Sources of systematic errors: disturbance in the measured system by the measurement tolerances of components wear, aging environmental influence, etc. Sources of random errors: • truly random stochastic noise Brownian (thermal) motion of molecules Johnson (thermal) noise of resistors shot (electron) noise of current flow flicker (contact) noise Barkhausen (magnetic domain) noise partition noise generation-recombination noise, etc. • incoherent extraneous signals and disturbances rf (radio-frequency electromagnetic) interference mains (60-Hz power line) interference magnetic interference vibrations, shocks, sound temperature oscillations, etc.

Disturbance by the Measurement Example: loading by a voltmeter unloaded Electrical Circuit Voltmeter R m V 1 V o loaded Electrical Circuit Voltmeter R m ' V 1 V = V o m equivalent circuit Electrical Circuit Voltmeter R o R m V = V ' V o o m R ' = m V V o o + R R o m − V V R R R m o m o o = = − = − ≈ − e 1 + + V R R R R R o o m o m m

Reduction of Systematic Errors • careful instrument design low tolerance low temperature coefficient low aging, etc. • opposing inputs, differential measurements Electrical Circuit Voltmeter R o ' R m V V = V V d o m o V ref = + V V V m ref d R m = − V V V ( ) d o ref + R R o m − V V − V V R o ref m o o = ≈ − e V R V o m o • Feed-back measurements Electrical Circuit Voltmeter R o ' R m V V = V V d o m o V ref feedback

High-Gain Negative Feedback Electrical Circuit Voltmeter Feedback Amplifier Device V V R o + d M R m V = V ' V G K V o o m ref _ = + V V V m ref d R m = − V V V ( ) d o ref + R R o m = V V GK ref d R o 1 + + = V GK V ( ) d o R m = + V V GK ( 1 ) m d + GK 1 = ≈ V V V m o o R o + + GK 1 R m − V V R 1 m o o = ≈ − ≈ e 0 V R GK o m

Random Deviations x i is the result of the i th measurement ( i = 1, 2, ... n ) average value n 1 < > = = ∑ x x x mean i n = i 1 median value ( x , is in increasing number) = , x x 1 2 if n is odd median + n ( )/ 1 = , + , x x x if n is even ( ) median + n n / 2 / 2 1 2 deviation from the mean value = − < > d x x i i variance n 1 = ∑ 2 V d i − n 1 = i 1 standard deviation σ = V

Frequency Distributions histogram of n = 50 measurements < x > = 405.16, σ = 1.91 10 Number of Measurements 9 8 7 6 5 4 3 2 1 0 400 401 402 403 404 405 406 407 408 409 410 Measured Value frequency distribution and probability density 0.25 Frequency Distribution 0.2 0.15 0.1 0.05 0 400 402 404 406 408 410 Measured Value

Probability Distributions + = The probability that a measurement is between x and x dx is dP p x dx ( ) , where p x ( ) is called the probability density distribution. ∞ = p x dx ∫ ( ) 1 0 ∞ < > = = ∫ x x x p x dx ( ) mean 0 The probability that a measurement is smaller than x is x = ∫ P x p x dx ( ) ( ) 0 P x ( ) is the cumulative probability = 1 P x lim ( ) →∞ x = 0 5 P x median ( ) . Normal (Gaussian) distribution − < > x x 2 ( ) − 1 σ 2 = 2 p x e ( ) σ π 2 68.0 % of data points is within ±σ of the mean 95.4 % of data points is within ± 2 σ of the mean 99.7 % of data points is within ± 3 σ of the mean

Error Estimates Estimated range from n measurement (68% confidence level): = < > ± x x e Standard error from the mean: σ = e n Combined effects of m unrelated errors = + + + 2 2 2 2 e e e e m ... 1 2 Error in a sum = ± + ± = + ± S a e b e a b e ( 1 ) ( 1 ) ( )( 1 ) a b + 2 2 2 2 a e b e a b = e + a b Error in a difference = ± − ± = − ± S a e b e a b e ( 1 ) ( 1 ) ( )( 1 ) a b + 2 2 2 2 a e b e a b = e − a b Error in a product/quotient = ± × ± = × ± S a e b e a b e ( 1 ) ( 1 ) ( )( 1 ) a b = 2 + 2 e e e a b

Regression Regression is the process of finding a simple mathematical relationship = y f x ( ) between two variables x and y based on a series of measured quantities x i and y i ( i = 1, 2, ... n ) Fitting with a given functional form m m j = = ∑ = + ∑ e.g., y f x a x or f x a b x c ( ) ( ) sin( ) j j j j = 0 j = j 0 difference: = − d y f x ( ) i i i least-squares (or least-mean-squares difference) n n n n 1 1 = ∑ = − = = − 2 ∑ 2 ∑ 2 ∑ 2 S d y f x )] (or S d y f x [ ( [ ( )] ) i i i i n n = = = = i i i i 1 1 1 1 least-squares regression (or fitting) ⇒ S a a a a a a min{ ( , , ... )} , , ... m m 1 2 1 2 initial guess best-fitting curve 40 40 Displacement [mm] Displacement [mm] 30 30 20 20 10 10 0 0 -10 -10 theory theory -20 -20 experiment experiment -30 -30 -40 -40 0 5 10 0 5 10 Position [m] Position [m]

Part 4 Signal Processing

Signal-to-Noise Ratio Over a given bandwidth B : ⎛ ⎞ ⎛ ⎞ P V S S = = SNR ⎜ ⎟ ⎜ ⎟ 10log 20log P V ⎝ ⎠ ⎝ ⎠ N N P S signal power P N noise power V S signal voltage V N noise voltage Amplitude [a. u.] Amplitude [a. u.] Time [a. u.] Time [a. u.] Spectrum [a. u.] Spectrum [a. u.] Frequency [a. u.] Frequency [a. u.]