Patrick F. Dunn, Measurement, Data Analysis, and Sensor Fundamentals for Engineering and Science, 2nd Edition

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66 Measurement and Data Analysis for Engineering and Science

FIGURE 3.3

A strain gage with a typical sensing area of 5 mm × 10 mm.

For a wire, the resistance R can be written as

R = ρ

, (3.3)

where ρ is the resistivity, L the length, and A the cross-sectional area. Taking

the total derivative of Equation 3.3 yields

dR =

dL +

dρ −

ρL

dA. (3.4)

Equation 3.4 can be divided by Equation 3.3 to give the relative change in

resistance,

= (1 + 2ν)

dρ

. (3.5)

Equation 3.5 shows that the relative resistance change in a wire depends on

the strain of the wire and the resistivity change.

A local gage factor, G

, can be deﬁned as the ratio of the relative

resistance change to the relative length change,

dR/R

dL/L

. (3.6)

This expression relates diﬀerential changes in resistance and length and

describes a factor that is valid only over a very small range of strain.

An engineering gage factor, can be deﬁned as

∆R/R

∆L/L

. (3.7)

Measurement Systems 67

FIGURE 3.4

Schematic of an integrated-silicon pressure sensor.

This expression is based on small, ﬁnite changes in resistance and length.

This gage factor is the slope based on the total resistance change throughout

the region of strain investigated. The local gage factor is the instantaneous

slope of a plot of ∆R/R versus ∆L/L. Because it is very diﬃcult to measure

local changes in length and resistance, the engineering gage factor typically

is used more frequently. Equation 3.5 can be rewritten in terms of the engi-

neering gage factor as

= 1 + 2ν + [

∆ρ



]. (3.8)

For most metals, ν ≈ 0.3. The last term in brackets represents the strain-

induced changes in the resistivity, which is a piezoresistive eﬀect. This term

is constant for typical strain gages and equals approximately 0.4. Thus, the

value of the engineering gage factor is approximately 2 for most metallic

strain gages.

An alternative expression for the relative change in resistance can be

derived using statistical mechanics where

= 2

−

dλ

−

. (3.9)

Here v

is the average number of electrons in the material in motion between

ions, λ is the average distance travelled by an electron between collisions, and

is the total number of conduction electrons. Equation 3.9 implies that the

diﬀerential resistance change and, thus, the gage factor, is independent of

the material properties of the conductor. This also implies that the change

in resistance only will be proportional to the strain when the sum of the

changes on the right hand side of Equation 3.9 is either zero or directly

proportional to the strain. Fortunately, most strain gage materials have this

behavior. So, when a strain gage is used in a circuit such as a Wheatstone

bridge, strain can be converted into a voltage. This system can be used as

a displacement sensor.

68 Measurement and Data Analysis for Engineering and Science

Strain gages also can be mounted on a number of diﬀerent ﬂexures to

yield various types of sensor systems. One example is four strain gages

mounted on a beam to determine its deﬂection, as described in Chapter 2.

As force is applied to the beam, it deﬂects, producing a strain. This strain

is converted into a change in the resistance of a strain gage mounted on the

beam. This is called a force transducer. Another example involves one or

more strain gages mounted on the surface of a diaphragm that separates

two chambers exposed to diﬀerent pressures. As the diaphragm is deﬂected

because of a pressure diﬀerence between the two chambers, a strain is pro-

duced. The resultant resistance usually is converted into a voltage using a

Wheatstone bridge. This system is called a pressure transducer, although

it actually contains both a sensor (the strain gage) and a transducer (the

Wheatstone bridge). A schematic of a miniature, integrated-silicon pressure

sensor is shown in Figure 3.4. The calibration and use of this type of pres-

sure sensor in a model rocket’s on-board measurement system is presented

in Section 3.7.

An accelerometer uses a strain gage ﬂexure arrangement. An ac-

celerometer in the 1970’s typically contained a small mass that was moved

against a spring as the device containing them was accelerated. The displace-

ment of the mass was calibrated against a known force. This information

then was used to determine the acceleration from the displacement using

Newton’s second law. Accelerometers then used strain gages or piezoelec-

tric transducers instead of a spring, although the size did not change much.

Now micro-accelerometers are available [10]. These contain a very small mass

attached to a silicon cantilever beam that is instrumented with a piezore-

sistor. As the device is accelerated, the beam deﬂects, the piezoresistor is

deformed, and its resistance changes. The piezoresistor is incorporated into

an on-board Wheatstone bridge circuit which provides a voltage output that

is linearly proportional to acceleration. The entire micro-accelerometer and

associated circuitry is several millimeters in dimension. The calibration and

use of this type of accelerometer in a model rocket’s on-board measurement

system are presented in Section 3.7.

Capacitive Sensor

A capacitive sensor consists of two small conducting plates, each of

area, A, separated by a distance, d, with a dielectric material in between.

The capacitance between the two plates is

C = 

A/d, (3.10)

where 

is the permittivity of free space and  the relative permittivity.

When used, for example, to measure pressure, the dielectric is air and one

plate is held ﬁxed. As the other plate moves because of the forces acting on

it, the capacitance of the sensor changes. The change in capacitance is pro-

portional to the diﬀerence in pressure from the reference pressure measured

Measurement Systems 69

at zero plate deﬂection. When used in a capacitive Wheatstone bridge cir-

cuit, the pressure diﬀerence is converted into a voltage. This system forms a

capacitive pressure transducer. A central plate ﬁxed to a small mass can

be used instead of air between the two capacitor plates. As the mass and its

attached central plate are accelerated, the change in capacitance with re-

spect to time is sensed. This is converted into a voltage that is proportional

to the acceleration. This system constitutes an accelerometer. Another

use of this type of sensor is to expose the dielectric material to moist air

while keeping the distance between the two plates ﬁxed. The permittivity

of the dielectric material changes with relative humidity, which leads to a

change in the sensor’s capacitance. This type of sensor can be used as a

relative humidity sensor.

Optical-based Sensor

An optically based measurement system can be designed to measure

non-invasively the velocity and velocity ﬂuctuations of a transparent ﬂuid

over the velocity range from ∼1 cm/s to ∼500 m/s with ∼1 % accuracy.

This system is known as the laser Doppler velocimeter (LDV) and op-

erates on the principle of the Doppler eﬀect [9]. A coherent beam of laser

light of a given frequency is directed into the moving ﬂuid containing mi-

croparticles (∼1 µm diameter), which ideally follow the ﬂow. Because these

microparticles are moving with respect to the beam, the frequency of light

as received by the microparticles is Doppler shifted. A photodetector in the

same reference frame as the laser receives the light that is scattered from

the microparticles. This scattered light is frequency shifted once again at

the receiver. The frequency of the scattered light, however, is too high to be

detected using conventional detectors.

This limitation can be overcome by using two beams of equal fre-

quency, intensity, and diameter, and crossing the beams inside the ﬂow.

This produces an ellipsoidal measurement volume with sub-millimeter di-

mensions. This method is called the dual-beam or Doppler frequency dif-

ference method. The crossed beams produce an ellipsoidal measurement

volume, on the order of 0.5 to 1 mm in length and 0.1 to 0.3 mm in diam-

eter. The velocity component, U , of the ﬂow perpendicular to the bisector

of the incident beams separated by an angle, θ, is related to the Doppler

frequency diﬀerence, f

, by

U =

λf

2 sin(θ/2)

, (3.11)

where λ is the wavelength of the incident laser light. The Doppler frequency

diﬀerence is the diﬀerence between the frequencies of the two scattered light

beams, as received in the laboratory reference frame. Further modiﬁcations

can be made by adding other beams of diﬀerent frequencies in diﬀerent

Cartesian coordinate directions to yield all three components of the velocity.

Also, frequency shifting usually is employed to compensate for insensitivity

70 Measurement and Data Analysis for Engineering and Science

of Equation 3.11 to ﬂow direction. If two additional, equal-spaced detectors

are added, the phase lag between the signals of the three detectors is re-

lated to the diameter of the microparticle passing through the measurement

volume. This system is called a phase Doppler anemometer.

3.3.3 *Sensor Scaling

Sensors have evolved considerably since the beginning of scientiﬁc instru-

ments. Marked changes have occurred in the last 300 years. The tempera-

ture sensor serves as a good example. Daniel Gabriel Fahrenheit (1686-1736)

produced the ﬁrst mercury-in-glass thermometer in 1714 with a calibrated

scale based upon the freezing point of a certain ice/salt mixture, the freezing

point of water, and body temperature. This device was accurate to within

several degrees and was approximately the length scale of 10 cm. In 1821,

Thomas Johann Seebeck (1770-1831) found that by joining two dissimilar

metals at both ends to form a circuit, with each of the two junctions held

at a diﬀerent temperature, a magnetic ﬁeld was present around the circuit.

This eventually led to the development of the thermocouple. Until very re-

cently, the typical thermocouple circuit consisted of two dissimilar metals

joined at each end, with one junction held at a ﬁxed temperature (usually

the freezing point of distilled water contained within a thermally insulated

ﬂask) and the other at the unknown temperature. A potentiometer was used

to measure the mV-level emf. Presently, because of the advance in micro-

circuit design, the entire reference temperature junction is replaced by an

electronic one and contained with an ampliﬁer and linearizer on one small

chip. Such chips even are being integrated with other micro-electronics and

thermocouples such that they can be located in a remote environment and

have the temperature signal transmitted digitally with very low noise to a

receiving station. The simple temperature sensor has come a long way since

1700.

Sensor development has advanced rapidly since 1990 because of MEMS

(microelectromechanical system) sensor technology [2]. The basic nature of

sensors has not changed, although their size and applications have changed.

Sensors, however, simply cannot be scaled down in size and still operate ef-

fectively. Scaling laws for micro-devices, such as those proposed by W.S.N.

Trimmer in 1987, must be followed in their design [10]. As sensor sizes

are reduced to millimeter and micrometer dimensions, their sensitivities to

physical parameters can change. This is because some eﬀects scale with the

sensor’s physical dimension. For example, the surface-to-volume ratio of a

transducer with a characteristic dimension, L, scales as L

−1

. So, surface

area-active micro-sensors become more advantageous to use as their size is

decreased. On the other hand, the power loss-to-onboard power scales as

−2

. So, as an actuator that carries its own power supply becomes smaller,

power losses dominate, and the actuator becomes ineﬀective. Further, as

sensors are made with smaller and smaller amounts of material, the prop-

Measurement Systems 71

erties of the material may not be isotropic. A sensor having an output that

is related to its property values may be less accurate as its size is reduced.

For example, the temperature determined from the change in resistance of

a miniature resistive element is related to the coeﬃcients of thermal expan-

sion of the material. If property values change with size reduction, further

error will be introduced if macro-scale coeﬃcient values are used.

The scaling of most sensor design variables with length is summarized

in Table 3.5. This can be used to examine the scaling of some conventional

sensors. Consider the laminar ﬂow element, which is used to determine a liq-

uid ﬂow rate. The element basically consists of many parallel tubes through

which the bulk ﬂow is subdivided to achieve laminar ﬂow through each tube.

The ﬂow rate, Q, is related to the pressure diﬀerence, ∆p, measured between

two stations separated by a distance, L, as

Q = C

πD

∆p

128µL

, (3.12)

where D is the internal diameter of the pipe containing the ﬂow tubes, µ

the absolute viscosity of the ﬂuid, and C

the ﬂow coeﬃcient of the element.

What happens if this device is reduced in size by a factor of 10 in both length

and diameter? According to Equation 3.12, assuming C

is constant, for the

same Q, a ∆p 1000 times greater is required! Likewise, to maintain the same

∆p, Q must be reduced by a factor of 1000. The latter is most likely the

case. Thus, a MEMs-scale laminar ﬂow element is limited to operating with

ﬂow rates that are much smaller than a conventional laminar ﬂow element.

Example Problem 3.2

Statement: Equation 3.12 is valid for a single tube when C

= 1, where it reduces

to the Hagen-Poiseuille law. How does the pressure gradient scale with a reduction in

the tube’s diameter if the same velocity is maintained?

Solution: The velocity, U , is the ﬂow rate divided by the tube’s cross-sectional area,

U = 4Q/(πD

), where D is the tube diameter. Thus, Equation 3.12 can be written

∆p/L = 32µU D

−2

. This implies that the pressure gradient increases by a factor of 100

as the tube diameter is reduced by a factor of 10. Clearly, this presents a problem in

sensors using micro-capillaries under these conditions. This situation necessitates the

development of other means to move liquids in micro-scale sensors, such as piezoelectric

and electrophoretic methods.

Decisions on the choice of a micro-sensor or micro-actuator are not based

exclusively on length-scaling arguments. Other factors may be more appro-

priate. This is illustrated by the following example.

Example Problem 3.3

Statement: Most conventional actuators use electromagnetic forces. Are either elec-

tromagnetic or electrostatic actuators better for micro-actuators based upon force-

scaling arguments?

72 Measurement and Data Analysis for Engineering and Science

Solution: Using Table 3.5, the electrostatic force scales as L

and the electromag-

netic force as L

. So, a reduction in L by a factor of 100 leads to a reduction in the

electrostatic force by a factor of 1 × 10

and in the electromagnetic force by a factor of

1 × 10

! If these forces are comparable at the conventional scale, then the electrostatic

force is 10 000 times larger than the electromagnetic force at this reduced scale.

The ﬁnal choice of which type of micro-actuator to use, however, may be based

upon other considerations. For example, Madou [11] argues that energy density also

could be the factor upon which to scale. Energy densities several orders of magnitude

higher can be achieved using electromagnetics as compared to electrostatics, primarily

because of limitations in electrostatic energy density. This could yield higher forces

using electromagnetics as compared to electrostatics for comparable micro-volumes.

3.4 Ampliﬁers

An ampliﬁer is an electronic component that scales the magnitude of an

input analog signal, E

(t), producing an output analog signal, E

(t). In

general, E

(t) = f {E

(t)}. For a linear ampliﬁer f {E

(t)} = GE

(t); for

a logarithmic ampliﬁer f {E

(t)} = G log

(t)], where G is the gain of

the ampliﬁer. Ampliﬁers are often used to increase the output signal of a

transducer to a level that utilizes the full-scale range of an A/D converter

that is between the transducer and the board. This minimizes errors that

arise when converting a signal from analog to digital format.

The common-mode rejection ratio (CMRR) is another characteristic

of ampliﬁers. It is deﬁned as

CMRR = 20 log

, (3.13)

in which G

is the gain when diﬀerent voltages are applied across the am-

pliﬁer’s positive and negative input terminals, and G

is the gain when the

same voltages are applied. Ideally, when two signals of the same voltage

containing similar levels of noise are applied to the inputs of an ampliﬁer,

its output should be zero. Realistically, however, the ampliﬁer’s output for

this case is not zero, but rather it is some ﬁnite value. This implies that the

ampliﬁer eﬀectively has gained the signal diﬀerence by a factor of G

, when,

ideally, it should have been zero. Thus, the lower G

is, and, consequently,

the higher the CMRR is, the better it is. Typically, CMRR values greater

than 100 are considered high and desirable for most applications.

Today, almost all ampliﬁers used in common measurement systems are

operational ampliﬁers. An op amp is comprised of many transistors, resis-

tors, and capacitors in the form of an integrated circuit. For example, the

LM124 series op amp, whose schematic diagram is shown in Figure 3.6,

consists of 13 transistors, 2 resistors, 1 capacitor, and 4 current sources.

Variable

Equivalent

displacement

distance

strain

length change/length

strain rate or shear rate

strain change/time

velocity

distance/time

surface

width × length

volume

width × length × height

force

mass × acceleration

line force

force/length

surface force

force/area

body force

force/volume

work, energy

force × distance

power

energy/time

power density

power/volume

electric current

charge/time

electric resistance

resistivity × length/cross-sectional area

electric ﬁeld potential

voltage

electric ﬁeld strength

voltage/length

electric ﬁeld energy

permittivity × electric ﬁeld strength

resistive power loss

voltage

/resistance

electric capacitance

permittivity × plate area/plate spacing

electric inductance

voltage/change of current in time

electric potential energy

capacitance × voltage

electrostatic potential energy

capacitance × voltage

with V ∼ L

electrostatic force

electrostatic potential energy change/distance

electromagnetic force

electromagnetic potential energy change/distance

ﬂow rate

velocity × cross-sectional area

pressure gradient

surface force/area/length

FIGURE 3.5

Variable scaling with length, L.

74 Measurement and Data Analysis for Engineering and Science

FIGURE 3.6

Internal layout of a low cost FET operational ampliﬁer (National Semicon-

ductor Corporation LM124 series).

FIGURE 3.7

An operational ampliﬁer in an open-loop conﬁguration.

When used in an open-loop conﬁguration, as shown in Figure 3.7, the

output is not connected externally to the input. It is, of course, connected

through the internal components of the op amp. For the open-loop conﬁg-

uration, E

(t) = A [E

(t) − E

(t) − V

], where V

is the op amp’s oﬀset

voltage, which typically is zero. E

is called the inverting input and E

the non-inverting input. Because A is so large, this conﬁguration is used

primarily in situations to measure very small diﬀerences between the two

inputs, when E

(t)

∼

(t).

The op amp’s major attributes are as follows:

• Very high input impedance (> 10

Ω)

• Very low output impedance (< 100 Ω)

• High internal open-loop gain (∼ 10

to 10

)

These attributes make the op amp an ideal ampliﬁer. Because the input

impedance is very high, very little current is drawn from the input circuits.

Also, negligible current ﬂows between the inputs. The high internal open-

loop gain assures that the voltage diﬀerence between the inputs is zero. The

Measurement Systems 75

FIGURE 3.8

An operational ampliﬁer in a closed-loop conﬁguration.

very low output impedance implies that the output voltage is independent

of the output current.

When used in the closed-loop conﬁguration, as depicted in Figure 3.8,

the output is connected externally to the input. That is, a feedback loop is

established between the output and the input. The exact relation between

(t) and E

(t) depends upon the speciﬁc feedback conﬁgura-

tion.

Op amps typically can be treated as black boxes when incorporating

them into a measurement system. Many circuit design handbooks provide

equations relating an op amp’s output to its input for a speciﬁed task. This

can be a simple task such as inverting and gaining the input signal (the

inverting conﬁguration), not inverting but gaining the input signal (the non-

inverting conﬁguration), or simply passing the signal through it with unity

gain (the voltage-follower conﬁguration). An op amp used in the voltage-

follower conﬁguration serves as an impedance converter. When connected

to the output of a device, the op amp eﬀectively provides a very low out-

put impedance to the device-op amp system. This approach minimizes the

loading errors introduced by impedance mismatching that are described in

Chapter 2. Op amps also can be used to add or subtract two inputs or to

integrate or diﬀerentiate an input with respect to time, as well as many

more complex tasks. The six most common op amp conﬁgurations and their

input-output relations are presented in Figure 3.9.

Example Problem 3.4

Statement: Derive the expression given for the input-output relation of the diﬀer-

ential ampliﬁer shown in Figure 3.9.

Solution: Let node A denote that which connects R

and R

at the op amp’s

positive input and node B that which connects R

and R

at the op amp’s negative

input. Essentially no current passes through the op amp because of its very high input

impedance. Application of Kirchhoﬀ’s ﬁrst law at node A gives

− E

− 0