Chapter 7 Motion and Dimensional Measurement Instruments

23/4/21 [email protected] 1

Chapter 7 Motion and Dimensional Measurement Instruments

Liner Motion Angular MotionDisplaceme

nt

Velocity

Acceleration

Jerk( 冲击量 )

Repetitive displacement-time relationships are called vibration, while a single event may be called a shock. These two fundamentally different events may be measured with the same diagnostics, although the measurement technique may be quite different.


Conversion factors for distance

1 inch (in) = 2.540000 centimeter (cm)1 foot (ft) = 12 inches (in)1 yard (yd) = 3 feet (ft)1 mile (mi) = 5280 feet (ft)


Conversion factors for velocity

1 mile per hour (mi/h) = 88 feet per minute (ft/m) = 1.46667 feet per second (ft/s) = 1.60934kilometer per hour (km/h) = 0.44704 meter per second (m/s) = 0.868976 knot (knot – international)


Chapter 7 Motion and Dimensional Measurement Instruments

A. Motion and Displacement( 运动量和位移量 )

Resistive Potentiometers

a. Ideal Characteristics

Resistive potentiometers are used for two basic functions:

i). Voltage Control (Wheatstone Bridge – variable resistor)

ii. Motion Measurement (linear or angular)

Range ~ 102 – 105 Ohms/inch


Resistive PotentiometersDeviations for Ideality

i. Finite Resolution

~ 500 – 1,000 “turns”/inch (20-40 /mm) for typical wire wrapped devices


Resistive Potentiometers Deviations from Ideality (cont.)

ii. Loading

Any “reading” device constitutes a “load” on the system which is being read. More specifically, the device draws current, which causes a drop in voltage of the “source” (eo in this case).

In effect, an additional resistance, Rm, is inserted in parallel with the device.

For a voltage divider, the result is:

Note: As RP/Rm 0, eo/eex xi/xt

(“High” input impedence Low i draw)

Note: “Max %” Error ~ 15 RP/Rm

(Text: p. 233)


R2





Resistive Potentiometers Deviations from Ideality (cont.)

iii. Power Constraints

Since RP is limited (since Rm is typically 106 Ohms), sensitivity is maximized by increasing eex (in order to maximize eo).

However, there are power constraints due to “Joule Heating” (i2R = iV)

PMAX ~ 5 Watts (typically)

which gives

Example

P = 2 WattsRP = 104 Ohmseex, max ~ 150 V (total)


Some Pictures of Potentionmetric Displacement Sensors

Linear Motion Rotational Motion


B. Resistance Strain Gauge( 电阻应变规 )

Consider a conductor having a uniform cross-sectional area, Ac, and a length, L,

made of a material having a resistivity, . For this electrical conductor,the resistance,R,is given by:

8


If the conductor is subjected to a normal stress along the axis of the wire,the cross-sectional area and the length will change,resulting in a change in the total electrical resistance,R. The total change in R is due to several effects,as illustrated in the total differential:Which may be expressed in terms of Poisson’s ratio νp as:


The dependence of resistivity on mechanical strain is called piezoresistance. And may be expressed in terms of a piezoresistance coefficient, ( 纵向压阻效应系数 )defined by:

With this definition,the change in resistance may be expressed:


Gauge Factor

The change in resistance of a strain gauge is normally expressed in terms of an empirically determined parameter called the gauge factor,GF. It can be expressed as:

The gauge factor is dependent on the Poisson ratio for the gauge material and its piezoresistivity.


Semiconductor Strain Gauges

Silicon crystals are the basic material for semiconductor strain gauges; the crystals are sliced into very thin sections to form strain gauges.

＊ In general,materials exhibit a change in resistivity with strain,characterized by the piezoresistance coefficient,


R1 R2

R3 R4

E1

E0

A simple strain gauge Wheatstone bridge circuit is shown in right FIGURE.

Consider the case all the resistors are equal,and the bridge balanced,if the gauge experiences a change in resistance ,then


EXAMPLE

A strain gauge,having a gauge factor of 2,is mounted on a rectangular steel bar( ),as shown in Figure. The bar is 3cm wide and 1cm high,and is subjected to a tensile force of 30kN. Determine the resistance change of the strain gauge. If the resistance of the gauge was 120 in the absence of the axial load.


Gaugeaxi s Tensile

loading

E1

E0


KNOWN: GF=2 ; FN=30kN ;

FIND: The resistance change of the strain gauge for a tensile force of 30kN.

SOLUTION:The stress in the bar under this loading condition is:

And the resulting strain is


For strain along the axial of the strain gauge,the change in resistance is:


Linear Variable Differential Transformer(LVDT)

Basic Principlesa. AC current flows through “primary” coil, due to excitation voltage eex.b. Current is “induced” through a pair of secondary coils (eo1, eo2).c. The frequency of the induced AC current is the same as the excitation

frequency.d. The amplitude of the induced current in each secondary coil depends upon

the location of the movable “core”.


Ferromagnetic Core

Core

x

vo

(Measurement)

vref

Primary Coil

Insulating Form

Secondary Coil Segment

Secondary Coil Segment

An LVDT transducer shown in FIG comprises a coil former on to which three coils are wound.

The primary coil is excited with an AC current, the secondary coils are wound such that when a ferrite core is in the central linear position, an equal voltage is induced in to each coil.

The secondary are connected in opposite so that in the central position the outputs of the secondary cancels each other out.


LVDT Basic Principle (cont).If core is located in “null” position then secondary voltages are equal, as illustrated below.


LVDT Basic Principles (cont.)If the two secondary coils are connected in anti-series (+ + and - -) then the resulting output is the difference between the outputs of the individual seconary coils. The amplitude depends upon the position of the rod.

(There is also a Phase shift between eex and eo as we will show later)


LVDT As Displacement Sensor

The input, xi, is the MOTION of the rod to which the core is connected.

The output, eo, is the voltage difference between the induced voltages in the two secondary loops.

Note: The output is inherently AM modulated

(The “carrier” is AC excitation of the primary loop).

Note: Phase Shift occurs as xi crosses “null” point


LVDT – Simulated Output (lvdtsim01.dsb)

Generator00

FFT00Formula00

Y/t Chart00

Generator01

Y/t Chart01

ms25 50 75 100 125 150 175

1.000.750.500.250.00

-0.25-0.50-0.75-1.00

5.0

2.5

0.0

-2.5

-5.0

Hz250 500 750 1000 1250 1500 1750 2000 2250

0.90.80.70.60.50.40.30.20.10.0

2.252.001.751.501.251.000.750.500.250.00

xi

eo


More Accurate LDVD Simulation (lvdtsim02)

xi

eo,1

eo,2

eo

ms25 50 75 100 125 150 175

1.00

0.00

-1.00

1.00

0.00

-1.00

1.00

0.00

-1.00

1.00

-0.75


A Closer Look at the LVDT Output

ms5 10 15 20 25 30 35 40 45 50 55 60 65 70

Y/t Chart 0

1.00

0.75

0.50

0.25

0.00

-0.25

-0.50

-0.75

(Note Phase Shift at Null Point)


How Do We Recover the LVDT Signal?

l(vdtsim03.dsb)

ms25 50 75 100 125 150 175

1.00

0.00

-1.00

1.00

-0.75

4

0

-4

1.5

0.0

-1.5

Motion

LVDT output

Demod LVDT

Recovered Motion


LVDT Signal Recover – Frequency Domain

Motion

LVDT output

Demod LVDT

Recovered Motion

Hz250 500 750 1000 1250

Y/t Chart 0 Y/t Chart 1

Y/t Chart 2 Y/t Chart 3

0.8

0.4

0.0

0.35

0.00

1.25

0.00

1.25

0.00


The DasyLab ProgramGenerator00

Generator01

1Formula00 Y/t Chart00

Formula00

Y/t Chart01

Generator02Formula01

Filter00

FFT00 Y/t Chart02


LVDTs – Some Math(Where does the Phase shift come from?)


Bandwidth of LVDT (Demodulation)

The “transfer function” derived previously describes the sensitivity of the output signal to the Excitation frequency of the Primary Loop

(NOT the frequency response to input motion!!)

The frequency response to input motion is dictated by the requirement to Demodulate the signal.

Let’s look at this in more detail.


Modulation/Demodulation

SignalS

Input Signal s (t)

Carrier c (t)

Demodulated Output

2 C + S

S

2C - S

S

Carrier c (t)

Amplitude Modulation Process

Amplitude Demodulation Process

Output

Primary Loop Excitation

C

LVDT Transducer AM OutputC + S

C - S


Frequency Domain Picture(Fill in in Class)

We need to filter demodulated output such that we transmit at s and attenuate at 2c z


Some Examples (Worked in Class)

Example 1: Single Stage RC filter

c = 10 KHz (a typical value)

What is Maximum frequency of motion that can be detected with LVDT?

(Lets verifty our conclusion with DaisyLab)


Piezoelectric Materials - Intro Piezoelectricity describes the phenomenon of generating an electric

charge in a material when subjecting it to a mechanical stress (direct effect) and conversely generating a mechanical strain in response to an applied electric field.

Discovered in 1880 by Pierre and Jacques Curie Types

Natural and Synthetic Crystals ( 单晶压电晶体 ): Quartz, Rochelle Salt (Natural)( 石英、罗歇尔盐（四水酒石酸钾

钠） ) Lithium Sulfate, Ammonium Dihydrogen Phosphate (Synthetic) ( 硫酸锂、磷酸二氢铵 )

Piezoceramic elements( 多晶压电陶瓷 ) Lead Zirconate Titanate (PZT)( 锆钛酸铅 ) Barium Titanate ( 极化的铁电陶瓷（钛酸钡） ) , Cadmium Sulfide

Piezoelectric Polymer ( 高分子压电薄膜 ) Polyvinylidene Fluoride (PVDF)


Piezoelectric Materials Piezoelectric materials belong to a class of materials called

Ferroelectrics. Piezoelectric Crystals exhibit the piezoelectric effect naturally, without any processing.

Piezoelectric Ceramics must be polarized by applying a strong electric field to the material while it is simultaneously heated. They are (isotropic) before poling and after poling exhibit tetragonal symmetry (anisotropic structure) below the Curie temperature. Above this temperature they lose the piezoelectric properties.

On a microscopic level the materials are made of ions which is the reason for electric dipole behavior. Groups of dipoles with parallel orientation are called Weiss domains. The Weiss domains are randomly oriented in the raw ceramic material, before the poling treatment has been finished. For this purpose an electric field (> 2000 V/mm) is applied to the (heated) piezo ceramics.

When an electric voltage is applied to a poled piezoelectric material, the Weiss domains increase their alignment proportional to the voltage. The result is a change of the dimensions (expansion, contraction) of the PZT material.


Piezoelectric Materials (PZT)

Unpolarized Crystal

Polarized Crystal

After poling the zirconate-titanate atoms are off center. The molecule becomes elongated and polarized

(1) Unpolarized- Random Weiss Domains

(2) During Polarization(3) After polarization; Remnant

Polarization exist


石英晶体是常用的压电材料之一。其中纵轴 Z—Z 称为光轴， X—X 轴称为电轴，而垂直于 X—X轴和 Z—Z 轴的 Y—Y 轴称为机轴。沿电轴 X—X 方向作用的力所产生的压电效应称为纵向压电效应，而将沿机轴 Y—Y 方向作用的力所产生的压电效应称为横向压电效应。当沿光轴 Z—Z 方向作用有力时则并不产生压电效应。

石英晶体

（ a ）左旋石英晶体的外形（ b ）坐标系（ c ）切片


Piezoelectric Materials - Intro

Applications: Mechanical to Electrical

Force, Pressure, and acceleration sensors Smart Sensors for Side Impact Diagnostics High Voltage - Low Current Generators: Spark Igniters for Gas

grills, small engines, etc. Yaw Rate( 偏航角速度 )Sensors Platform Stabilization Sensors

Electrical to Mechanical: Ultrasonic motors, Small Vibration Shakers Microactuators (High Precision Macro actuators) Sonar( 水声测位仪 ) array arrays for collision avoidance Pumps for Inkjet Printers


Actuator Types

Actuators are sometimes called motors Sensors are called generators

Bimorph (bending) Bimorph (extension)

Longitudinal Wafer

Transverse Wafer

Stack Actuator


Actuator Types Longitudinal and Transverse Wafers: When an electrical field having

the same polarity and orientation as the original polarization field is placed across the thickness of a single sheet of piezoceramic, the piece expands in the thickness or "longitudinal" direction (i.e., along the axis of polarization) and contracts in the transverse direction (perpendicular to the axis of polarization). Reversing the field reverses the effect.

Unimorphs: A unimorph is a single -layer piezoelectric element bonded to shim stock. They can be made to elongate, bend, or twist depending on the polarization. Electrode pattern and wiring configuration of the layers. The shim laminated between the two piezo layers adds mechanical strength and stiffness and amplifies motion in bending.

Bimorphs: Two -layer elements can be made to elongate, bend, or twist depending on the polarization and wiring configuration of the layers. A center shim laminated between the two piezo layers adds mechanical strength and stiffness, but reduces motion.

Stack Actuators: Stack actuators can be formed when a large number of piezo layers (wafers) are combined into one monolithic structure. The tiny motions of each layer contribute to the overall displacement.


Piezoelectric Coordinate System

The behavior of the materials are defined by the g and d constants. For the piezoelectric constants gij and dij, the first value (i) in the

subscript represents the axis of initial polarization. This is usually the axis that the electrodes are parallel to. The second value (j) relates to the mechanical axis or the axis or applied stress or strain.

g33 or d33 (which we will now define as g)

g31 or d31


Piezoelectric Coefficient g(Piezoelectric Pressure Transducers)

t = thicknessw = widthL = length

eo = Potential Difference Developed (V)F = Force (normal to LW plane)


g coefficient (cont.)

Example: (worked in class)

Quartz pressure transducer

t = 1 mmF/Area = 106 Pascals (N/m2)

eo =


Piezoelectric d Coefficient

Piezoelectrics are fundamentally charge-based devices

(Application of stress results in charge separation)


A Digression on Parallel Plate Capacitors

In General,

lCl = an appropriate length scaleC = capacitance = permittivity of medium (Farads/m)

o

dielectric constant (dimensionless)

For a parallel plate capacitor:

εwL

t

CQ

e

C

Qe

100

(text calls the “dielectric constant”)


Back to Piezos

For quartz, ~ 4 x 10-11 (farads/m) (very small!!)

(This means that small charge separation results in large voltage!)

We will show in class that:d = g

So for quartz:

(Note: farads x volts = Coulombs)

(2 picocoulombs / Newton


Displacement to Voltage (displacement transducer / actuator)

The Charge generated by a deformed crystal is given by:

Kq = Constant, C/m xi = deflection, cm

(4.63)

(Note: text notations changes from Q to q)

e0 = q/C so that eoC = q = Kqxi

Where Kq/C is defined as K, the static sensitivity (Volts/cm)

K ~ 10 – 100 V per 10-6 meters (or ~ 10 nm / Volt!!)


A Few Summary Notes

Piezoelectric transducers are inherently charge devices. Input displacement results in charge separation between

faces.

iqxKq

• Charge is related to output voltage by means of:

(K =Kq/C is the “static” sensitivity)

• Charge eventually “leaks” off due to internal resistance


A Few More Additional Summary Notes

Because the piezo is based on charge, capacitance is of fundamental importance.

Ccr ~ 10-9 farads

(This is rather high, but not so high that we can ignore capacitance of other system components)

• The internal resistance is very high (Rleak ~ 1011 ohms). For this reason resistance of other system elements (cables, amplifiers, meters, etc.) is of fundamental importance.

Let’s look at this a bit more??


Displacement to Voltage Measurement

The voltage generated by a deformed crystal is

Fig 4.40

But we MUST consider the capacitance of the ENTIRE MEASUREMENT SYSTEM!

(We will also need to consider the effective impedance of the entire system, as we will see shortly)


Example – Sensitivity (K)

A piezoelectric transducer has a Kq value of 10-5 C/inch

and an internal capacitance of 10-9 F.

What is the “static” sensitivity, K, of the transducer alone?

If the transducer is part of a measurement system in which

i. Ccable = 300 pFii.Camplifier = 50 pF

What is K for the system?


Piezoelectrics – Frequency Response

The Charge generated by a deformed crystal is

Kq = Constant, C/m

xi = deflection, cm

Fig 4.40

From figure 4.40 e

4.63

4.66

4.67

4.68


Displacement to Voltage Measurement – Dynamic Response

Taking the derivative of (68) and subbing in (66) gives (after a bit of manipulation, which we will do in class)

cm/V ,C

KysensitivitK q s ,RC

4.70

4.69

(or in “s” notation)


Time Domain – Step / Pulse Inputs

Step Input

At t = 0, eo = AK (Input “steps” to a finite value, xi, resulting in voltage output)

Pulse Input

At t = 0, eo = AK (Input “steps” to a finite value)

At t = T eo (Input steps to –xi)


The Result isResponse of Piezo Transducer to Pulse

Input

Fig. 4.41


Piezo Dynamic Response – Frequency Regime

Lets Return to eqn. (4.70) 1

s

sKs

x

e

i

21

K

)(x

e

i

o 190 tan


Sensitivity – Bandwidth Trade off

We mentioned previously that increasing the bandwidth (frequency range) of an instrument often requires that the sensitivity be decreased. Consider the magnitude of the frequency response.

where

To Increase Bandwidth we need to increase which generally means we need to increase C. (Why not increase R?)

But

So Increasing C decreases K!!


An Example – Static/Dynamic Sensitivity

Problem

A piezoelectric transducer has a capacitance of 1,000 pF and Kq of 10-5 C/in. The connecting cable has a capacitance of 300 pF while the oscilloscope used for readout has in input impedance of 1 M paralleled with 50 pF.

a. What is the sensitivity (V/in) of the transducer alone?b. What is the high-frequency sensitivity of the total system.c. What is the lowest frequency that can be measured with 5%

amplitude error.d. What value of C must be connected in parallel to extend the

range of 5% error down to 10 Hz.e. If the value of C in part d is used, what will the system high-

frequency sensitivity be?


Ccr= 1,000 pF , Kq =10-5 C/in ,Ccable= 300 pF , Rample=1 M Cample= 50 pFC=Ccr+Ccable+Cample=1000+300+50=1350pF ,R=Rample=1M


Seismic Transducers

A seismic transducer consists of two basic components:i. Spring – Mass – Damper Elementii. Displacement Transducer

(MS Fig 4.77)

(Note: xo = xi – xM)


Seismic Transducers – Acceleration Sensor

Let’s explore the dynamic response of the spring-mass-damper element alone.

Noting the sign conventions in Fig. 4.77, we have, from Newton’s second law for the motion of mass M, xM:

oioosM xxMxBxKxMF (Where xM = xi – xo)

(A classic 2nd Order System)

We define (again)

With the result


Seismic Transducer – Acceleration (cont).

Let’s associate the following:

ii xq

oo xq

21

n

K

Input is object accelerationOutput is relative displacement of M and objectStatic Sensitivity (sec2)

Classic 2nd Order System


Seismic Accelerometer – Freq. Responce


Seismic Accelerometer – Freq Response (cont)

Question:Over what range of frequencies can we

actually use a seismic accelerometer?

Answer: To be most useful we desire a “flat”

frequency response and a “linear” phase shift. In other words, we need

i. SIG << n

ii. ~ 0.4 – 0.6

But recall that


Seismic Accelerator – “Readout”

The previous discussion ignored the response of the displacement sensor used to measure xo!!

We need to consider this!RECALL

System 1

qi,1 System 2

qi,2 qo,1

qo,2


Seismic Accelerometer

i. Resistive Potentiometer Readout


Acceleration-measuring transducers (accelerometer)

10-2

10-1

100

10110

-3

10-2

10-1

100

101

n /

0

20

un

Directly proportional region between ü0 and 0

Best value of for accelerometer = 0.65

Accelerometer gives accurate result when forcing frequency is 60% less than natural frequency.

Accelerometer must have relatively very large natural frequency


Weakness of high natural frequency

10-2

10-1

100

10110

-3

10-2

10-1

100

101

n /

0

20

un

For a given value of u0, the relative displ. is directly proportional to 1/n2 .

So, high natural freq. makes electric signal very small.

Large amplification is needed.

System will be very sensitive.

We must compromise between high sensitivity and the highest attainable natural frequency.


Accelerometer requirement for shock Accelerometer must have relatively low natural period:

The response of accelerometer follows up the pulse most faithfully when the natural period of the accelerometer is smallest relative to the period of the pulse.

Accelerometer must have sufficient damping: Damping in the transducer reduces the response of the

transducer.

Accelerometer must have low zero shift as possible: Zero shift is the displacement of zero reference line due to

intense shock involving zero frequency.


Important Characteristics of Accelerometer Sensitivity

Ratio of its electrical output to its mechanical input Parameters used in checking sensitivity:

Average, rms, peak

Resolution Smallest change in mechanical input for which a change in the

electrical output is discernible Resolution can be limited by noise levels in the instrument.


Transverse sensitivity Coordinate transformation can be applied in sensitivity.

where e is output voltage

Amplitude linearity and limits A transducer is linear only over a certain range of

amplitude values. The lower end of this range is determined by the electrical

noise. The upper end of linearity imposed by the electrical

characteristics of the transducing element.

cosmaxee


Frequency range

Lower frequency limit

frequency

ampl

itude

peak-to-peak displ.ac

celer

ation

Upper frequency limit

Maximum acceleration

Minimum acceleration

Maximum displacement

Operating range


Environmental effect

Temperature The sensitivity, natural frequency and

damping can be affected.

Humidity A transducer which operated at a high

electrical impedance is affected by humidity.


Acoustic noise High-intensity sound wave often

accompany high-amplitude vibration.

Strain sensitivity Accelerometer may generate a

spurious output when its case is strained or distorted.

Typically, this occurs when the transducer mounting is not flat against the surface.


Physical properties

Size and weight of transducer are very important considerations.

A large instrument may require a mounting structure. it changes local vibration, so it can be treated as added mass.

The smaller is the transducer, the higher is its sensitivity.


Piezoelectric Accelerometer

Principle of operation

mass

Electrical output

Piezoelectric element

Mechanical input (Vibration)


Typical response curve

frequency

Ou

tpu

t vo

ltag

e

Low frequency

limit

High frequency

limit

Usable frequency

range


Determination of acceleration

we can determine acceleration in low frequency region before resonance frequency.

10-2 10-1 100 10110-3

10-2

10-1

100

101

n /

0

20

un

frequency

Ou

tpu

t vo

ltag

e


Types of piezoelectric accelerometers

Compression type

mass


baseoutput


Shear type

mass


baseoutput


Beam type

+

-

Tension part

Compression part

Two piezoelectric plates which are rigidly bonded together


Piezoresistive Accelerometers

Principle of operation

mass

beam

Piezoresistive element

• semiconductor material

• change its resistance in proportion to applied stress or strain

• connected electrically in a Wheatstone-bridge circuit


Seismic Accelerometer

ii. Piezo Readout

“Usable” range

depends upon

damping


reflecting surface

displacement to check

check intensity

fibercheck intensity

Fiber-optic displacement sensor


Electrodynamic(velocity coil) pickups

uses Lenz’s law. Blve

NS S

edirection of motion


Eddy Current TransducersPrinciple of Eddy current:

An eddy current is caused by a moving magnetic field intersecting a conductor or vice-versa.

The relative motion causes a circulating flow of electrons, or current, within the conductor.

These circulating eddies of current create electromagnets with magnetic fields that oppose the change in the external magnetic field.

The stronger the magnetic field, or greater the electrical conductivity of the conductor, the greater the currents developed and the greater the opposing force.

This principle is used in eddy current proximity sensor

FIG illustrates concept of Eddy current FIG


Eddy current proximity sensor The Eddy Current Transducer

uses the effect of eddy (circular) currents to sense the proximity of non-magnetic but conductive materials.

A typical eddy current transducer contains two coils: an active coil (main coil) and a balance coil as shown in FIG

The active coil senses the presence of a nearby conductive object, and balance coil is used to balance the output bridge circuit and for temperature compensation.

FIG


Schematic diagram of eddy current proximity sensor

•Active coil and compensating coil forms arms of inductance bridge.

•When a measurand brought to near to active coil, due to eddy current which produces eddy current magnetic field that opposes active coil field causes change in inductance and thus creates imbalance in inductance bridge.

•This change is noted in calibrated unit.


Accelerometer Calibration

1) Static

a) Plus or minus 1 g turnover method

b) Centrifuge method

2) Steady-state periodic

a) Rotation in a gravitational field

b) Using a sinusoidal shaker or exciter

3) Pulsed

a) One-g step, using free fall

b) Multiple Spring-mass device

c) High-g methods.


1. Static Calibration

Plus of minus 1g

You can calibrate an accelerometer through ±1g by simply putting it upright, and then rotating it exactly 180º.

Centrifugal Method

In a rotating situation, the normal acceleration is

Axis of rotation is vertical

an r2 r 2f 2


Steady-State Periodic CalibrationRotation in a gravitational field

Same as centrifugal, but axis of rotation is horizontal

Sinusoidal Vibrational Exciter


Pulsed CalibrationFree-Fall Method

If suspend the accelerometer and suddenly drop it, it experiences a step change in acceleration of 1g.

High-g Methods


Seismic Displacement

How about a seismic displacement transducer?

(We’ll let you do this one as homework).


Capacitance Transducers

Consider a basic parallel plate capacitor, with

C = Capacitance (pF)A = Plate Area (in2)x = Plate Separation (in)If either x or A are changed, then C will change!!


Basic Capacitance Transducer Geometries

Linear Motion Rotational Motion

Typical Capacitance Values


Capacitance Transducers – Signal Conversion

Capacitance is not easy to measure directly. We need to convert “signal” to current or voltage.a. AC Voltage ApproachWe apply a constant amplitude AC voltage, Vex, at = ex

This will result in a variable amplitude AC current at ex

Let’s Work This Out!


Capacitance Transducers – C to I ConversionV(t) = q(t)/C(t)q(t) = C(t)V(t)

IF we apply an AC Voltage with 1/ex which is short as compared to time scales for the change in displacement to be measured, then C can be considered approximately constant, so that:

dt

dVC

dt

dqI ex Or (taking LaPlace

transform)

Note: This approach is most useful for transducers in which xi modifiers A (the plate Area) Why??


Capacitance Transducers – Signal Conversion

b. AC Current Approach

We apply a constant amplitude AC current, Iex, at = ex

This will result in a variable amplitude AC Voltage at ex, eo

Let’s Work This Out!

Note: This approach is most useful for transducers in which xi modifiers the gap (x).

Why??


Signal Conversion – AM Modulation

Of course in either of the last two cases, the actual signal is AM modulated (Carrier Frequency = ex

MS Fig 4-37c


Signal Recovery – Current Measurement

A good approach to convert current to voltage is to use an Operational Amplifier as shown below

iai ~ 0

if + ix = 0


Op Amp Current – Voltage Conversion

xfxf iiii 0

(Output voltage 1/Cx)


One Final Note

This configuration is best for transducers in which gap is varied. (because Cx 1/x so that eo varies linearly with xi)

Alternatively, we can exchange the positions of Cf and Cx, giving Best for transducers in

which Area is varied. (because Cx A so that eo again varies linearly with xi)


Elastic element methods

Sensors that are used for measurement of force, torque or pressure often contain an elastic element that converts the mechanical quantity into a deflection or strain which can then be transformed using another sensor into an electrical signal. Electrical resistance strain gauges are widely used in this capacity.


Diagram of strain gauge


Various forms of elastic members are used. The simplest is just a spring to make a device called spring balance. The extension of the spring represents the force applied.

Load cells, i.e. elastic members which transform force into displacement or strains, can take many forms .


The structure of typical elastic element and its designing calculation as follows:

Columnar load cell


Proving ring

测力环

Bending beam

Applied force

4


The relative extension of strain gauge in elastic element :

l F

l AE E

Δl-------Total extension of strain gauge

l--------the original length of strain gauge

F------applied force

A------working area of elastic element

E------Yang model of elastic element

σ------stress of elastic element


Sensitivity : / /

/

R R R Rk

l l

31

1 3

2 4

2 4

RR Fk k

R R AE

R R Fk k

R R AE

----poison constant of elastic element

Total strain of elastic element:

0 1 2 3 4

2(1 )F

AE


Output of electrical bridge: 0

00

(1 )

2(1 )

2 4

i

i

U kU F

AEU k k

FU AE

Voltage sensitivity(mv/v):

0 1 2 3 4

2(1 )2(1 )F

AE

•In general, select k=2


An Introduction to Optical Detectors and CCD Cameras

I. The Photoelectric Effect

In 1887, Heinrich Hertz discovered that illuminating a metal surface with ultra-violet (UV) light ( < ~ 350 nm) caused electrons to be emitted from surface.

This is termed the photoelectric effect and was puzzling for two reasons.

i. The kinetic energy of ejected electrons was independent of light intensity.

ii. The effect ONLY occurred if the wavelength of light was LESS than a threshold value, which was different for different metals. (It also did NOT depend upon light intensity)


Einstein’s Explanation(for which he won the Nobel Prize)

hcvmEnergyKineticElectron e

2

2

1

(where is termed the “work function” of the metal)

Einstein, of course, was aware of the work of Planck in the area of Blackbody Radiation

Planck proposed (in late 19th century) that the energy of a light “photon” was equal to:

where h is now known as “Planck’s Constant”

(Note: Planck supposedly did not actually believe his theory, but it was the only way he could “fit” the data)

Einstein applied this idea to Hertz’s observation by stating that:


Illustration of Photoelectric Effect

(and Einstein’s Explanation)

hcKE


Photoelectric Effect – A Simple Example

When a lithium surface is irradiated with light the kinetic energy of the ejected electrons is found to be:

i) 2.935 x 10-19 Joules if = 300 nm

ii) 1.280 x 10-19 Joules if = 400 nm

Calculate

a. Planck’s Constant

b. The Threshold Frequency (and wavelength) for electron ejection

c. The work function


II Photodiode Detectors

Photodiodes (such as the one used in Lab 6) are semiconductors (usually silicon) in which absorption of a photon with h exceeding the work function results in the creation of an “electron – hole” pair.

If a potential difference (voltage bias) is applied across the interface of “p” and “n” type material, then the electrons will drift across the junction, creating a current.

This current is detected by the voltage drop across a “load” resistor.

The PE effect is basis for most optical detectors(Some thermal - passive solar hot water heater)


Photodiodes – Some Circuit Details

Photodiode Equivalent Circuit - “Conductive” Mode


Photodiode Sensitivity(Quantum Efficiency)

Silicon

Indium Gallium Arsenide(“InGas”)


Sensitivity – Bandwidth Trade-off

Photodiodes are fundamentally current devices, so that the static sensitivity is often given in units of:

Example: Typical laser pointer has output power of ~ 3 mWatts ( ~ 700 nm)Output current detected on photodiode is ~ 1 mamp (0.33 mamp/mWatt)

The detector output Voltage depends upon the load resistor.

i. 103 ohms 1 Voltii. 106 ohms ?? Volt (Detector “saturates” at ~ 5 Volts)

What about Bandwidth??


An Example: Laser Doppler Velocimetry

sindp

2

If two laser beams are crossed, an interference fringe pattern (of low and high intensity) are formed. (This occurs because laser beams are coherent).

The fringe spacing, dp, is given by:

If individual particles, seeded into the flow, traverse through the fringe pattern, they scatter light with an intensity which is amplitude modulated by the fringe pattern. The modulation frequency is:

sin

dp

vd

2 v = velocity


III. CCD Cameras*

(* Material taken from “Charge Coupled Devices and Their Applications,” J. Beynon and D. Lamb, McGraw-Hill, 1980)

CCD stands for “Charge Coupled Device”

It consists of a two-dimensional array of “pixels” which can be though of as micro scale (~ 5-10 microns) capacitors which store electrons in a “potential well”.

Each individual pixel has an optically sensitive surface, which is similar to a small silicon photodiode. Electrons are “ejected” at a rate proportional to the incident flux of photons.

However, “ejected” photo-electrons are temporarily trapped beneath the surface in a spatially localized area, and then read out using a set of shift registers.


Sensitivity vs Bandwidth (Again)

In class we have always described “bandwidth” in terms of temporal behavior

Freq = 1/time (sec-1)

However

In imaging applications we often consider “spatial frequency”

Spatial Frequency = 1/spatial resolution (m-1).

High spatial resolution means that we can resolve high spatial frequencies or “sharp” edges.

CCD cameras increase number of “Mpixels”) by decreasing their physical size!!

Higher spatial frequency comes at the cost of lower sensitivity because smaller pixels cannot store as much light!


CCD – How is Charge Stored?

CCD “Potential Wells” Formed by Application of “Gate” Voltage to Array of Pixels Each With Discrete Electrode.

• + Voltage repels “holes” near surface forming “depletion” layer• Ejected photoelectrons fill top portion, termed “Inversion Layer”


CCD – How is Charge “Coupled” Out?

• With all “gates” up, charge fills potential wells as surface irradiated.• After suitable “exposure” time, charge sequentially shifted through a series of

clocking pulses.• Depending upon chip architecture, charge is shifted over, up, and out (output

port)• A/D conversion performed on charge sequence.


More Realistic Picture of Charge Coupling


Interline Transfer Architecture


Frame Readout by “Frame” Transfer

Documents

Chapter 7 Motion and Dimensional Measurement Instruments