Sensor Performance Metrics

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Engineering Institute

Sensor Performance Metrics

Michael Todd

Professor and Vice Chair Dept. of Structural Engineering University of California, San Diego

[email protected]

Email me if you want a copy.

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Outline

•  Sensors as dynamic systems

•  Zero-order sensors

•  First-order sensors

•  Second-order sensors

•  Fundamental sensor performance metrics

•  sensitivity

•  bandwidth/transmission band(s)

•  cross-axis sensitivity

•  sensitivity to extraneous measurands

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The Process of Getting An Acceleration Measurement

plate

vibration epoxy

viscoelastic effects temperature

housing

self-dynamics (e.g., resonances)

boundary conditions

seismic mass

seating

binding post

manufacturing tolerances

self-dynamics (e.g., cross-axis sensitivity)

boundary conditions

observer piezoelectric

element

coupling

stray capacitance

contact wire

contact leads

cabling

EMI

environment

signal conditioning

environment human error

(e.g., filter settings)

DAQ board

quantization error

DSP

human error software glitches (e.g., round-off)

data presentation

human error Microsoft

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Input/Output Representation of Sensors

What we WANT to measure is some kind of response of the structure (e.g., acceleration at point q, ), to some kind of excitation of the plate (e.g., impact hammer strike at point p, );

we usually describe the response in terms of an FRF (frequency response function):

H

_pq

( ! ⁾ = Y

_q

( ! ^{) / X}

_p

⁽ ! ⁾

Just as the plate transfers an excitation into an acceleration, the accelerometer subsequently transfers this acceleration into a measured representation of itself. Thus, what we ACTUALLY measure, , is NOT necessarily equal to what we WANT to measure, :

Y

_q,observed

( ! ) = H (

_epoxy

( ! )H

_{hou sin g}

( ! )H

_post

( ! )H

_mass

( ! )H

_piezo

( ! )H

_wires

( ! ) )

H_sensor(! )

! ########### " ########### $ Y

_q

( ! )

And this doesn’t include all data acquisition and signal conditioning response functions!

We need to understand the behavior of the sensor transfer function in order to choose the right sensor characteristics for an application and calibrate it.

Y_q

( )

! ^X^p

( )

^!

Y_q,obesrved

( )

! ^Yq

( )

!

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A “Failure” Example...Measuring Pendulation Loads

Sea state can lead to dangerous load pendulation during LOTS exercises

Design fiber optic accelerometer array to monitor load and ship motions

Also asked to monitor

centripetal

acceleration (has DC component!)

Actual signal

What we measured

accelerometers + electronics had low-frequency

attenuation!

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General Input/Output Sensor Model: Accelerometer

Newton’s 2nd Law on m:

md²y

dt² + cdy

dt + ky

"output"

! # # " # # $ = cdx

dt + kx

"input"

! " # $ #

housing binding post

inertial mass/

damping block piezoelectric

material contact plate

input motion

housing (base) mass

inertial mass

effective damping/

stiffness

input motion k c

m

x

y

And, in general,

a_n dⁿy

dtⁿ + a_n!1d^n!1y

dt^n!1 +…+ a₁ dy

dt + a₀y = F(t)

Sensors are themselves dynamical systems!

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Zero-Order Sensors

These sensors don’t behave as dynamic systems and can only measure static signals sensor model is ^a0y(t)= F(t) so the output is easily obtained

y(t)= (1/ a₀)

K! static sensitivity! " # F(t)

•  The sensor’s static sensitivity K determines the simple proportional input/output relationship.

•  The sensor has no relevant dynamics, so it responds instantly to changes in F(t), provided the changes aren’t “too fast”

•  Sensor ideally contains no storage or inertial elements (or, more accurately, the behavior of these elements is negligible)

•  There are no sensors that are truly zero-order, but in some applications, sensors act like zero-order

intake valve

plenum

force due to pressure difference

spring

∆y ^piston

P_atm P

P! P_atm = !kA_piston"y, or

"P = !kA_piston

static sensitivity, K! " # $ # "y

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First-Order Sensors

•  These sensors contain storage elements which cannot respond immediately to inputs

•  Modeled as first-order dynamical system ^a1

dy

dt + a₀y= F(t)

! ^dy

dt + y = KF(t) time constant

static sensitivity

•  The zero-state response (with no input) of this system shows the influence of :

y(t)

t

The larger the sensor’s time constant, the longer it takes to respond to input changes.

=0.5

=5

=20

!

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First-Order Sensors (Cont’d)

•  We know from dynamic systems theory that we can utilize the powerful kernel method based on zero-state impulse-response to get the formula for the response to ANY input. The impulse response for an impulse-loaded first-order system

h

₁

(t) = Ke

^{!t /}^"

/ "

! ^dy

dt + y = K " (t)

is

For any general input F(t), then, we know the sensor’s response to it:

y(t) = y

₀

e

^{!t /"}

+ F(T )h

₁

(t !T )dT

T=0 t

#

= y

₀

e

^{!t /"}

transient response

! " # + F(T )(K / " ^)e

!(t!T )/"

dT

T=0 t

#

steady!state response

! $ $ $ $ " $ $ $ $ #

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Example: Bulb Thermometer

Q ˙

E ˙

control volume

The change in energy stored in the bulb, !E, must balance the exchange of heat between sensor and environment:

E = ˙ ˙ Q mc dT

dt = hA(T

_!

"T ) mc

hA

!

#

dT

dt +T = T

_!

# ^dT

dt +T = F(t)

m = liquid mass in the sensor c = specific heat of liquid

h = convective heat transfer coefficient A = sensor surface area

•  Standard from for first-order sensor

•  Note that the static sensitivity, K, is 1

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First-Order Sensors: Step Input

•  Consider the sensor response to a step change in the input, say 0 to A, so F(t)=A:

y(t) = y

₀

e

^{!t /"}

+ A(K / " )e

!(t!T )/"

T=0

t

# dT = KA + (KA ! y

₀

)e

^{!t /"}

•  Our steady response (what we want to measure) is KA, which is the actual measurement A modified by the constant static sensitivity of the sensor K

•  Our current measurement (initial condition) is y₀

•  We can define an error fraction function as

!(t) = y(t) " y

_ss

y

₀

" y

_ss

= e

^{"t /#}

t /! 0.37

When t= , the sensor has responded to 63% of the real measurement;

When t=2.3 , the sensor has responded to 90% of the real measurement;

When t=5 , the sensor has responded to 99% of the real measurement One may experimentally find by plotting the ln of error function during test data collection, since ,

and finding the slope of the best-fit line.

ln! = "(1/#^)t

! !

!

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First-Order Sensors: Periodic Input

•  Consider an input that is periodic in time, so

y(t) = y

₀

e

^{!t /}^"

+ Asin # ^{T (K /} " ^)e

^{!(t!T )/}^"

T=0

t

$ dT = Ce

^{!t /}^"

+ KA

1 + "

²

#

²

^sin( # ^t + % ⁾

where

C = y

₀

+ KA !"

1 + "

²

!

²

^{, tan} # = $ !"

•  Once the transient portion dies out, we have the first-order sensor frequency response to a periodic input

•  The steady-state sensor response matches the input in frequency, but it lags the input by a time delay and gets attenuated:

input output

F(t)= Asin!t

! /" ! /"

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First-Order Sensors: Periodic Input (Cont’d)

•  This attenuation and lag, as a function of parameters are shown here

M phase shift (deg)

Attenuation factor M is the ratio of the dynamic amplitude to the static amplitude:

M =

KA 1+!²"²

KA

= 1

1+!²"²

Phase shift was defined previously,

•  to measure high frequencies, need very small or get too much attenuation (filtering)

•  to measure low-frequency (towards static) signals, can use systems with large

!,"

!" !"

! tan^!1"#

!

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Second-Order Sensors

•  These sensors contain both storage elements (in the form of damping) as well as inertia

•  Modeled as second-order dynamic systems:

a₂ d²y

dt² + a₁dy

dt + a₀y= F(t) 1

!_n² d²y

dt² + 2"

!_n dy

dt + y = KF(t)

!_n = a₀ / a₂ # natural frequency

" = a₁/ 2 a₀a₂ # damping ratio

•  The natural frequency corresponds to a time scale that the sensor “wants” to respond at for any inputs it sees

•  The damping ratio determines the characteristic manner in which the sensor response approaches the steady-state measurement:

•  (underdamped) sensor transient responds will oscillate (at )

•  sensor transient response will monotonically (not oscillate) approach steady-state;

, called critical damping, represents the quickest monotonic approach (optimal)

0 <! < 1 !_n

! > 1

! = 1

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Second-Order Sensors (Cont’d)

•  For any general input F(t), then, we know the sensor’s response to it:

y(t)= y_trans(t)+ F(T )h₂(t!T )dT

T=0 t"

= y_trans(t)

transient response

! " # $ # +_T₌₀^t^" F(T )(K#_n / 1!$²)e^!$#ⁿ^{(t!T )}sin

[

#_n 1!$²(t!T )

]

^dT

steady!state response

! # # # # # # # # # # " # # # # # # # # # # $ h₂(t)= K!_n

1"#² ^e

"#!_nt sin!_n 1"#²t 1

!_n² d²y

dt² + 2"

!_n dy

dt + y = K#(t) results in

•  We can again use the kernel method to get the general sensor response:

•  The vast majority of sensor designs result in sub-critical damping (oscillations), so we will proceed with our discussion with that assumption. The transient response of a second-order sensor system with sub-critical damping is

y_trans(t) = e^!^!"ⁿ^t y₀cos"_n ¹!!²^t + !y₀ + y₀!_n"

!_n 1!"²

"

#$ %

&

' cos!_n ¹!"²^t (

)

**

+ , -- sensor initial conditions

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Second-Order Sensors: Step Input

•  Consider the sensor response to a step change in the input, say 0 to A, so F(t)=A:

y(t)= y_trans(t)+_T₌₀^t^$ A(K!_n / 1"#²)e^"^#!ⁿ^(t^{"T )} sin

[

!_n 1"#²(t "T )

]

^dT

= KA " KAe^"^#!ⁿ^t cos!_dt + #

1"#² sin!_dt

%

&

' (

) * , where !_d =!_n 1"#²

•  if , sensor “rings” at

•  these oscillations die out and the expected measurement, KA, is eventually achieved

•  system performance can be characterized by rise time and settling time

KA

= 0

= 0.25

= 2 = 0.5

= 1

!

! ! < 1 ^!^d

!_nt

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Second-Order Sensors: More Step Input

•  As with the first-order sensor, we can compute an error function :

!(t) = e^"#$ⁿ^t cos$_dt + #

1"#² sin$_dt

%

&

' (

)*

•  rise time: time it takes to first achieve 90% of steady-state (or, error is 10%)

•  settling time: time it takes for oscillations to settle within ±10% of steady-state rise time

settling time •  rise time decreases with decreased , but the trade-off is that settling time increases with decreased

•  the region near the intersection of these curves is an optimal design point for the sensor

!_nt

!(t)

!_nt

!

!(t)

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Second-Order Sensors: Periodic Input

•  Consider an input that is periodic in time, so

y(t)= y_trans+ Asin!T (K!_n / 1"#²)e^"^$!ⁿ^(t^{"T )}sin!_d(t "T )

T=0

t% dT

= y_trans+ KAsin(!t +&) 1"(! /!_n)²

( )

² ^{+ 2#! /!}⁽ ⁿ⁾² , where tan& = "2#! /!ⁿ 1"(! /!_n)²

•  After the transient portion (which oscillates at sensor time scale ) dies out, the sensor response oscillates at the same frequency as the input, , but with a phase shift and an amplitude modification, both of which depend on damping and the ratio

!/!_n = 0.5 !/!_n =1.5

output lags the input and is magnified for <1

output is attenuated for >1, and the lag continues to increase

F(t)= Asin!t

!_n

!

! ! ^/!n

! ^/!n

! /!_n

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Second-Order Sensors: Periodic Input (Cont’d)

•  This amplitude effect M and phase shift are shown here; define

M = 1

1! "²

( )

²^{+ 2#"}⁽ ⁾²

M phase shift (deg) ^{= 0.01}

= 0.25 = 0.5 = 1

= 2

transmission attenuation amplification (resonance)

= 0.01 = 0.25 = 0.5 = 1

= 2

•  resonance occurs if input frequency approaches the sensor’s natural frequency (large attenuation, especially for small )

•  M is near unity for small and goes to zero for large (regardless of )

•  phase shift increasingly lags as increases, with the most abrupt changes occurring at resonance (especially for small )

•  output is in phase with input for small and out of phase for large (regardless of )

! =" /"_n

! !

!

! !

!

! !

!

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Transmission Band: Unity Gain and Linear Phase Regime

•  = 0.7 corresponds to the widest band for retaining a unity transfer function

•  this is also the regime where the phase is most closely linearly related to the frequency

•  nonlinear phase shifts cause a non-uniform phase lead/lag in the original signal and the measured signal (signal distortion)

•  As shown previously, there is a frequency regime for second-order sensors where transmission is near unity and phase shifts are near zero.

!" = 2#+#² 1+#

error tolerance

M

!

! !

!"

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Signal Distortion

•  Suppose have original signal to be measured:

x(t) = sin2!t + sin 4!t

•  A linear phase shift means that the measured signal may be represented by:

y(t) = sin 2!t +" ( ) ^{+ sin 4} ( !t + 2" )

= sin# + sin2#

which by inspection is the same as the original signal but time-shifted

•  Consider a nonlinearly phase shifted measured signal:

y(t) ! = sin 2 ( ! t + 0.1 ) ^{+ sin 4} ( ^! ^t ^{+ 2} )

original

linear phase measured nonlinear phase measured Near sensor

resonance, phase distortion is

maximized!

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Accelerometers: Case of Second-Order Sensor

Newton’s 2nd Law on m: md²y

dt² + cdy

dt + ky = cdx

dt + kx + F(t)

housing binding post

inertial mass/

damping block piezoelectric

material contact plate

input motion direct force

y

housing (base) mass

inertial mass

effective damping/

stiffness

input motion

k c m

x

direct force, F(t)

Accelerometers are designed so that the mass is protected from direct forces, so F(t)=0; if we define z=y-x, we can re-write this equation and normalize as before for second-order systems:

1

!_n²

d²z

dt² + 2"

!_n dz

dt + z = # 1

!_n²

d²x dt²

input x

output z

accelerometers measured acceleration of input by relative motion response of the inertial mass

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Let’s Consider a Periodic Input to this Accelerometer...

M = !²

1" !²

( )

²^{+ 2}⁽ ^#^!⁾²

We can define M for this system depending on what we consider the input: displacement, velocity, or acceleration

Displacement input amplitude: A

Velocity input amplitude:

Acceleration input amplitude:

M = !

1" !²

( )

²^{+ 2#!}⁽ ⁾² ^M ⁼

(

¹^{! "}²

)

¹²^{+ 2#"}⁽ ⁾²

good trans.!

pretty much stinks!

z(t)= KA!² sin(!T "#⁾ 1" !²

( )

² ^{+ 2}

⁽

^$^!

⁾

² ^{, tan}^# ^{= "}

2$! 1" !²

Very response to previous second-order response, but with an extra in

the numerator; phase shift is the same If x = Asin!t, then !!x= "A!² sin!t, and

!²

A! A!²

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Demonstration of Accelerometer in Acceleration Mode

•  Consider a periodic acceleration input labeled in black) and the corresponding measured output (labeled in red) for three different damping ratios over a swept frequency range (0< <8)

! = 0.1 ! = 0.7 ! = 2.0

•  In acceleration mode, output signal is in phase and in near-unity transmission for low and then enters attenuated, out-of-phase behavior for post-resonant

•  Very low or very high damping shrink the useful transmission band ( = 0.7 is optimal)

! !

!

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Measuring Wideband Signals Beyond Transmission Band

We will assume that the sensor behaves as a linear dynamic system, so multiple inputs may be treated by superposition; suppose a sensor’s resonance is at 3 Hz, and we try to measure a signal composed of 1 Hz and 5 Hz components:

x(t)= sin2!t + sin10!t Input components are in black, outputs in red:

1 Hz component + 5 Hz component = total input and response

Not very good signal representation due to distortions induced by resonance

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An Accelerometer Cut-Away

Let’s look at a cut-away view of a seismic accelerometer:

seismic mass

elastic beams (springs) fiber Bragg grating

(fiber optic transducer, attached to beam)

fiber optic cable housing

•  contains seismic mass, as expected

•  utilizes elastic beams as springs

•  in this example, uses fiber Bragg gratings to transduce bending of beams into displacement/acceleration

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Sensor Performance Properties

We will utilize this example of an accelerometer to explore several sensor performance properties:

•  sensitivity: What is the response of the sensor to inputs, usually over a range of time scales? (Usually given in terms of the transfer function)

•  resolution: What is the minimum detectable value of the intended input? (Usually given in terms of power or amplitude spectral density)

•  cross-axis sensitivity: How much does the sensor respond to inputs not aligned with the primary sensing direction? (Usually expressed as a fraction of the main sensitivity)

•  multiple resonances: Does the sensor have multiple nonlinear (resonant) areas that affect sensitivity and response? (The answer is typically, “yes”).

•  sensitivity to extraneous measurands: Does the sensor respond to unintended inputs?

(Does an accelerometer, for example, also respond to strain or temperature inputs yielding “false” signals?)

Sensor Performance Metrics