Alciatore D.G., Histand M.B. Introduction to Mechatronics and Measurement Systems

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Figure 9.39 Law of intermediate junction metals.

+−

Figure 9.40 Law of intermediate temperatures.

+−

1/2

1/3

2/3

3 . Law of intermediate junction metals. As illustrated in Figure 9.39 , if a

third metal is introduced within a junction creating two new junctions (A-C

and C-B), the measured voltage will not be affected as long as the two new

junctions are at the same temperature ( T

⫽ T

). Therefore, although soldered

or brazed joints introduce thermojunctions, they have no resulting effect on the

measured voltage. If T

⫽ T

, the effective temperature at C is the average of

the two temperatures (( T

⫹ T

)/2).

4 . Law of intermediate temperatures. Junction pairs at T

and T

produce the

same voltage as two sets of junction pairs spanning the same temperature range

( T

to T

and T

to T

); therefore, as illustrated in Figure 9.40 ,

1/3

= V

1/2

+ V

2/3

(9.49)

This equation can be read: The voltage resulting from measuring temperature

relative to T

is the same as the sum of the voltages resulting from T

relative

to T

and T

relative to T

. This result supports the use of a reference junction

to allow accurate measurement of an unknown temperature based on a fixed

reference temperature (described below).

5 . Law of intermediate metals. As illustrated in Figure 9.41 , the voltage pro-

duced by two metals A and B is the same as the sum of the voltages produced

by each metal (A and B) relative to a third metal C:

A/B

= V

A/C

+ V

B/C

(9.50)

This result supports the use of a standard reference metal (e.g., platinum) to be

used as a basis to calibrate all other metals.

A standard configuration for thermocouple measurements is shown in Figure

9.42 . It consists of wires of two metals, A and B, attached to a voltage-measuring

9.4 Temperature Measurement 411

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Figure 9.41 Law of intermediate metals.

+−

−

+−

A/C

A/B

B/C

Figure 9.42 Standard thermocouple configuration.

voltage-

measuring

device

ice bath

reference

junction

ref

= 0°C)

measuring

junction

(T)

412 CHAPTER 9 Sensors

device with terminals made of metal C. The reference junction is used to estab-

lish a temperature reference for one of the junctions to ensure accurate temperature

measurements at the other junction relative to the reference. A convenient reference

temperature is 0 ⬚ C, because this temperature can be accurately established and main-

tained with a distilled water ice bath (i.e., an ice-water mixture). If the terminals of

the voltage measuring device are at the same temperature, the law of intermediate

leadwire metals ensures that the measuring device terminal metal C has no effect

on the measurement. For a given pair of thermocouple metals and a reference tem-

perature, a standard reference table can be compiled for converting voltage measure-

ments to temperatures.

An important alternative to using an ice bath is a semiconductor reference (e.g.,

a thermistor), which electrically establishes the reference temperature based on solid

state physics principles. These reference devices are usually included in thermo-

couple instrumentation to eliminate the need for an external reference temperature.

Video Demo 9.12 describes a commercially available digital thermometer, which

can be used for accurate thermocouple temperature measurements.

Figure 9.43 illustrates a two-reference junction configuration, which allows

independent choice of the leadwire metal. Copper is a good choice, because copper

leadwires are inexpensive and no junctions are introduced at the voltmeter connec-

tions, which are usually copper.

Figure 9.44 illustrates the configuration for a thermopile, which combines N

pairs of junctions, resulting in a voltage N times that of a single pair. In the example

shown in the figure, the multiplication factor would be 3. If the measuring junctions

(at T ) are at different temperatures, the output would represent the average of these

temperatures.

Video Demo

9.12Thermo-

couple with a

digital ther-

mometer

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Figure 9.43 Attaching leadwires of selected metal.

copperconstantan

iron

ice bath

voltage

measuremen

device

copper

Figure 9.44 Thermopile.

ref

voltage

measurement

device

9.4 Temperature Measurement 413

A standard two-junction thermocouple configuration is being used to measure the tempera-

ture in a wind tunnel. The reference junction is held at a constant temperature of 10 ⬚ C. We

have only a thermocouple table referenced to 0 ⬚ C. A portion of the table follows. We want to

determine the output voltage when the measuring junction is exposed to an air temperature

of 100 ⬚ C.

Junction Temperature (°C) Output Voltage (mV)

10 0.507

20 1.019

30 1.536

40 2.058

50 2.585

60 3.115

70 3.649

80 4.186

90 4.725

100 5.268

By applying the law of intermediate temperature for this example, we can write

100/0

100/10

10/0

Thermocouple Con guration with Nonstandard Reference

EXAMPLE 9.2

(continued )

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414 CHAPTER 9 Sensors

We wish to find V

100/10

, the voltage measured for a temperature of 100 ⬚ C relative to a refer-

ence junction at 10 ⬚ C. We can get the other voltages in the equation, V

100/0

and V

10/0

, from the

table because both are referenced to 0 ⬚ C. Therefore,

100/10

100/0

10/0

– 5.268 0.507–() mV 4.761 mV== =

(continued )

The six most commonly used thermocouple metal pairs are denoted by the let-

ters E, J, K, R, S, and T. The 0 ⬚ C reference junction calibration for each of the types

is nonlinear and can be approximated with a polynomial. The metals in the junc-

tion pair, the thermoelectric polarity, the commonly used color code, the operating

range, the accuracy, and the polynomial order and coefficients are shown for each

type in Table 9.2 . The general form for the polynomial using the coefficients in the

table is

i=0

∑

= c

+++++++++

(9.51)

where V is the thermoelectric voltage measured in volts and T is the measuring junc-

tion temperature in ⬚ C, assuming a 0 ⬚ C reference junction. Figure 9.45 shows the

sensitivity curves for some commercially available thermocouple pairs. Even though

we use a ninth-order polynomial to represent the temperature voltage relation, pro-

viding an extremely close fit, the relationship is close to linear as predicted by the

Seebeck effect. MathCAD Example 9.1 shows examples of how the polynomial

coefficients in Table 9.2 can be used to make calculations for thermocouple voltage

measurements.

9.5 VIBRATION AND ACCELERATION

MEASUREMENT

An accelerometer is a sensor designed to measure acceleration, or rate of change of

speed, due to motion (e.g., in a video game controller), vibration (e.g., from rotating

equipment), and impact events (e.g., to deploy an automobile airbag). Accelerom-

eters are normally mechanically attached or bonded to an object or structure for

which acceleration is to be measured. The accelerometer detects acceleration along

one axis and is insensitive to motion in orthogonal directions. Strain gages or piezo-

electric elements (described in Section 9.5.1 ) constitute the sensing element of an

accelerometer, converting acceleration into a voltage signal. As a simple example of

acceleration sensing, Video Demo 9.13 shows a toy ball with blinking lights that can

sense acceleration associated with a bounce. Its sensing element is simply a spring

surrounding a metal post.

MathCAD Example

9.1Thermocouple

calculations

Video Demo

9.13Bouncing

ball accelerometer

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9.5 Vibration and Acceleration Measurement 415

Table 9.2 Thermocouple data

Type E Type J Type K Type R Type S Type T

Metal pair Chromel (⫹)

and

constantan (⫺)

Iron (⫹)

and

constantan (⫺)

Chromel (⫹)

and

alumel (⫺)

87% platinum,

13% rhodium (⫹)

and

platinum (⫺)

90% platinum,

10% rhodium (⫹)

and

platinum (⫺)

Copper (⫹)

and

constantan (⫺)

Color code Purple Black Yellow Green Green Blue

Operating

range

⫺100⬚C to

1000⬚C

0⬚C to

760⬚C

0⬚C to

1370⬚C

0⬚C to

1000⬚C

0⬚C to

1750⬚C

⫺160⬚C to

400⬚C

Accuracy ⫾0.5⬚C ⫾0.1⬚C ⫾0.7⬚C

⫾0.5⬚C ⫾0.1⬚C ⫾0.5⬚C

pproximate

sensitivity

(mV/⬚C)

0.079 0.054 0.042 0.012 0.011 0.049

Polynomial

order

958897

0.104967 ⫺0.0488683 0.226585 0.263633 0.927763 0.100861

17,189.5 19,873.1 24,152.1 179,075. 169,527. 25,727.9

⫺282,639. ⫺218,615. 67,233.4 ⫺4.88403 ⫻ 10

⫺3.15684 ⫻ 10

⫺767,346.

1.26953 ⫻ 10

1.15692 ⫻ 10

2.21034 ⫻ 10

1.90002 ⫻ 10

8.99073 ⫻ 10

7.80256 ⫻ 10

⫺4.48703 ⫻ 10

⫺2.64918 ⫻ 10

⫺8.60964 ⫻ 10

⫺4.82704 ⫻ 10

⫺1.63565 ⫻ 10

⫺9.24749 ⫻ 10

1.10866 ⫻ 10

2.01844 ⫻ 10

4.83506 ⫻ 10

7.62091 ⫻ 10

1.88027 ⫻ 10

6.97688 ⫻ 10

⫺1.76807 ⫻ 10

— ⫺1.18452 ⫻ 10

⫺7.20026 ⫻ 10

⫺1.37241 ⫻ 10

⫺2.66192 ⫻ 10

1.71842 ⫻ 10

— 1.38690 ⫻ 10

3.71496 ⫻ 10

6.17501 ⫻ 10

3.94078 ⫻ 10

⫺9.19278 ⫻ 10

— ⫺6.33708 ⫻ 10

⫺8.03104 ⫻ 10

⫺1.56105 ⫻ 10

—

2.06132 ⫻ 10

— — — 1.69535 ⫻ 10

—

Source: G. Burns, M. Scroger, and G. Strouse, “Temperature-Electromotive Force Reference Functions and Tables for the Letter-Designated Thermocouple Types

Based on the ITS-90,” NIST Monograph 175, April 1993.

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Figure 9.45 Thermocouple types and characteristics. (Courtesy of OMEGA

Engineering Inc., Stamford, CT)

416 CHAPTER 9 Sensors

The design of an accelerometer is based on the inertial effects associated with

a mass connected to a moving object through a spring, damper, and displacement

sensor. Figure 9.46 illustrates the components of an accelerometer along with the

displacement references, terminology, and free-body diagram. When the object

accelerates, there is relative motion between the housing and the seismic mass. A

displacement transducer senses the relative motion. Through a frequency response

analysis of the second-order system modeling the accelerometer, we can relate the

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Figure 9.46 Accelerometer displacement references and free-body diagram.

vibrating

object

free-body

diagram

displacement

transducer

accelerometer

housing

seismic

mass

(m)

9.5 Vibration and Acceleration Measurement 417

displacement transducer output to either the absolute position or acceleration of the

object.

To determine the frequency response of the accelerometer, we first express the

forces shown in the free-body diagram. To do this, we define the relative displace-

ment x

between the seismic mass and the object as

= x

− x

(9.52)

It is measured by a position transducer between the seismic mass and the housing.

Therefore, the spring force can be expressed as

= k(x

− x

) = kx

(9.53)

and the damper force can be expressed as

–()bx

(9.54)

where the overdots represent time derivatives. Applying Newton’s second law, the

equation of motion for the seismic mass is

ext

∑

··

(9.55)

– F

– mx

··

(9.56)

The forces have negative signs in this equation because they are in the opposite direc-

tion from the reference direction x

in the free-body diagram. Substituting Equations

9.53 and 9.54 , the result is

– bx

– mx

··

(9.57)

Because the relative displacement x

= x

− x

(9.58)

we can replace x

with

··

(9.59)

Therefore, we can write Equation 9.57 as

– bx

– mx

··

+()=

(9.60)

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418 CHAPTER 9 Sensors

which can be rearranged as

··

++ m– x

··

(9.61)

This second-order differential equation relates the measured relative displacement x

to the input displacement x

. As in the analysis of a second-order system presented in

Section 4.10, we can rewrite this equation as

··

r n

++ x

··

–=

(9.62)

where the natural frequency is

----=

(9.63)

and the damping ratio is

2 km

--------------

(9.64)

■ CLASS DISCUSSION ITEM 9.13

Effects of Gravity on an Accelerometer

The free-body diagram and resulting equation of motion for the accelerometer do

not show a gravitational force explicitly. Explain why.

For a frequency response analysis, the input displacement is assumed to be com-

posed of sinusoidal terms of the form:

t() X

ωt()sin=

(9.65)

Because the system is linear, the resulting relative output displacement is also sinu-

soidal of the same frequency but different phase:

t() X

ωt φ+()sin=

(9.66)

Using the procedure presented in Section 4.10.2, the frequency response analysis

results in the amplitude ratio (Question 9.27)

-----

ωω

⁄()

------

⎝⎠

⎛⎞

–

4ζ

------

⎝⎠

⎛⎞

⎝⎠

⎛⎞

⁄

------------------------------------------------- ----------------------

(9.67)

and the phase angle

φ tan

1–

2ζω ω

⁄()

------

⎝⎠

⎛⎞

–

-------------------------

⎝⎠

⎜⎟

⎛⎞

–=

(9.68)

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Figure 9.47 Ideal accelerometer amplitude response.

0.01

0.1

)/(X

)

ω/ω

0.01 0.1 1

ζ = 0.01

0.3

0.707

9.5 Vibration and Acceleration Measurement 419

To relate the relative output displacement signal x

to the input acceleration

, we differentiate Equation 9.65 , resulting in

··

t() X–

ωt()sin=

(9.69)

Note that the amplitude of the input acceleration is

(9.70)

Rearranging Equation 9.67 , we have

ω()

−−−−−

−−−

–

4ζ

−−−

⁄

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

( )

(9.71)

where H

( ␻ ) is used to represent the ratio (X

␻

)/ (X

␻

) as a function of frequency ␻ .

Figures 9.47 and 9.48 illustrate this term and the phase angle relationship ( Equation

9.68 ) graphically for different values of damping ratio.

The denominator of H

( ␻ ) is the input acceleration amplitude X

␻

, and the

numerator is the product of the output displacement amplitude X

and the square of

the natural frequency ␻

. Therefore, we can relate the measured output displacement

amplitude to the input acceleration amplitude as

--------

⎝⎠

⎛⎞

ω()X

()=

(9.72)

so the input acceleration amplitude can be expressed as

()

ω()

---------------=

(9.73)

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Figure 9.48 Ideal accelerometer phase response.

420 CHAPTER 9 Sensors

If we design the accelerometer so that H

( ␻ ) is nearly 1 over a large frequency

range, then the input acceleration amplitude is given directly in terms of the relative

displacement amplitude scaled by a constant factor ␻

()ω

()X

(9.74)

As can be seen in Figure 9.47 , the largest frequency range resulting in a unity ampli-

tude ratio occurs when the damping ratio ␨ is 0.707 and the natural frequency ␻

as large as possible. Also, as is clear in Figure 9.48 , a ␨ of 0.707 results in the best

phase linearity for the system. We can make the natural frequency large by choosing

a small seismic mass and a large spring constant. This is easy to accomplish in the

very small package common to commercial accelerometers.

Equation 9.74 applies to every frequency component ␻ lying within the band-

width of the sensor. If we have an arbitrary input composed of a number of frequen-

cies that lie within the bandwidth, each frequency contributes to the signal according

to Equation 9.74. Therefore, the total acceleration, due to all frequency components,

is also directly related to the total measured relative displacement:

··

t() ω

t()=

(9.75)

The same spring-mass-damper configuration used to measure acceleration can

be designed to measure displacement instead. This type of device is called a vibro-

meter. The amplitude ratio given in Equation 9.67 provides us with the necessary

relationship between the input and output displacements. As we did in the acceler-

ometer analysis, we can now define a displacement ratio as

ω()

-----

(9.76)

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