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

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46 Measurement and Data Analysis for Engineering and Science

Quantity

Famous Person

current

James Joule

charge

Charles Coulomb

electric ﬁeld work

Georg Ohm

electric potential

James Watt

resistance

Andre Ampere

power

Michael Faraday

inductance

Joseph Henry

capacitance

Alessandro Volta

TABLE 2.3

Famous people and electric quantities.

4. The ends of a wire 1.17 m in length are suspended securely between two

insulating plates. The diameter of the wire is 0.000 05 m. Given that

the electric resistivity of the wire is 1.673 × 10

−6

Ω·m at 20.00

◦

C and

that its coeﬃcient of thermal expansion is 56.56 × 10

−5

◦

C, compute

the internal resistance in the wire at 24.8

◦

C to the nearest whole ohm.

5. A wire with the same material properties given in the previous problem

is used as the R

arm of a Wheatstone bridge. The bridge is designed to

be used in deﬂection method mode and to act as a transducer in a system

used to determine the ambient temperature in the laboratory. The length

of the copper wire is ﬁxed at 1.00 m and the diameter is 0.0500 mm.

= R

= 154 Ω and E

= 10.0 V. For a temperature of 25.8

◦

compute the output voltage, E

, in volts to the nearest hundredth.

6. Which of the following eﬀects would most likely not result from routing

an AC signal across an inductor? (a) A change in the frequency of the

output alternating current, (b) a back electromagnetic force on the input

current, (c) a phase lag in the output AC signal, (d) a reduction in the

amplitude of the AC signal.

7. Match each of the following quantities given in Table 2.3 with the famous

person for whom the quantity’s unit is named.

8. Given the electrical circuit in Figure 7, where R

= 37 Ω, R

= 65 Ω,

= 147 Ω, R

= 126 Ω, and R

= 25 Ω, ﬁnd the total current drawn

by all of the resistors to the nearest tenth A.

Questions 9 through 13 pertain to the electrical circuit diagram given

in Figure 2.18.

9. A Wheatstone bridge is used as a transducer for a resistance temperature

device (RTD), which forms the R

leg of the bridge. The coeﬃcient of

thermal expansion for the RTD is 0.005/

◦

C. The reference resistance of

Electronics 47

FIGURE 2.17

Resistor circuit.

FIGURE 2.18

Temperature measurement system.

the device is 25 Ω at a reference temperature of 20

◦

C. Compute the

resistance of the RTD at 67

◦

C to the nearest tenth of an ohm. Use this

procedure to arrive at the answer in the next problem.

10. For the Wheatstone bridge shown in Figure 2.18, R

= R

= 25 Ω

and E

= 5 V. The maximum temperature to be sensed by the RTD is

◦

C. Find the maximum output voltage from the Wheatstone bridge

to the nearest thousandth volt. The answer to this question will be used

in the following problem. (Hint: The answer should be between 0.034 V

and 0.049 V.)

11. A constant gain ampliﬁer, with gain factor G, conditions the output

voltage from the Wheatstone bridge shown in Figure 2.18. The multi-

meter used to process the output voltage from the ampliﬁer, E

, has a

full-scale output of 10 V. Determine the maximum gain factor possible

to the nearest hundred. The answer to this question will be used in the

following problem.

48 Measurement and Data Analysis for Engineering and Science

12. The RTD shown in Figure 2.18 senses a temperature of 60

◦

C. Compute

the voltage output to the multimeter, E

, to the nearest hundredth

volt.

13. What bridge method is used for the RTD measurement system shown

in Figure 2.18? (a) Deﬂection method, (b) null method, (c) strain gage

method, (d) resistance-temperature method.

14. Which of a following is a consequence of the conservation of energy?

(a) Ohm’s law, (b) Kirchhoﬀ’s ﬁrst law, (c) potential diﬀerences around

a closed loop sum to zero, (d) reciprocals of parallel resistances add.

15. Consider the cantilever-beam Wheatstone bridge system that has four

strain gages (two in compression and two in tension). Which of the fol-

lowing statements is not true: (a) The change in resistance in each gage

is proportional to the applied force, (b) temperature and torsional eﬀects

are automatically compensated for by the bridge, (c) the longitudinal

(axial) strain in the beam is proportional to the output voltage of the

bridge, (d) a downward force on the beam causes an increase in the

resistance of a strain gage placed on its lower (under) side.

16. An initially balanced Wheatstone bridge has R

= R

= 120

Ω. If R

increases by 20 Ω, what is the ratio of the bridge’s output voltage

to its excitation voltage?

FIGURE 2.19

Wheatstone bridge circuit.

17. A Wheatstone bridge may be used to determine unknown resistances

using the null method. The electrical circuit shown in Figure 2.19 (with

no applied potential) forms the R

arm of the Wheatstone bridge. If

= R

= 31 Ω and R

= 259 Ω, ﬁnd the necessary resistance of arm

to balance the bridge. Resistances R

, R

, and R

refer to the

resistances in the standard Wheatstone bridge conﬁguration. Use the

standard Wheatstone bridge. Round oﬀ the answer to the nearest ohm.

Electronics 49

18. A Wheatstone bridge has resistances R

=10 Ω, R

=14 Ω, and R

Ω. Determine the value of R

in Ω when the bridge is used in the null

method. Round oﬀ the answer to the nearest ohm.

19. Calculate the power absorbed by each resistor in Figure 2.20.

5 A

3 A

10 V

11 V

(a) (b)

FIGURE 2.20

Two resistors.

20. A 2 mH inductor has a voltage v(t) = 2 cos(1000t) V, with i(t = 0) = 1.5

A. Find the energy stored in the inductor at t = π/6 ms.

21. Determine the coeﬃcient of thermal expansion (in Ω/

◦

R) of a 1 mm-

diameter wire whose resistance increases by 10 % when its temperature

increases by 1.8 K.

22. Determine the current, in A, through a capacitor that is discharging at

a rate of 10 C in 2.5 s.

23. The typical output impedance of an operational ampliﬁer, in ohms, is

(a) 0, (b) < 100, (c) ∼1000, or (d) > 10

24. What is the unit of resistance (Ω) in the base units (kg, m, s, and C)?

2.13 Homework Problems

1. Consider the pressure measurement system shown in Figure 2.21. The

Wheatstone bridge of the pressure transducer is initially balanced at

p = p

atm

. Determine (a) the value of R

(in Ω) required to achieve this

balanced condition and (b) E

(in V) at this balanced condition. Finally,

determine (c) the value of E

(in V) required to achieve E

= 50.5 mV

50 Measurement and Data Analysis for Engineering and Science

when the pressure is changed to p = 111.3 kPa. Note that R

(Ω) =

100[1 + 0.2(p − p

atm

)], with p in kPa.

FIGURE 2.21

An example pressure measurement system conﬁguration.

2. Consider the temperature measurement system shown in Figure 2.22.

At station B determine (a) E

(in V) when T = T

, (b) E

(in V) when

T = 72

◦

F, and (c) the bridge’s output impedance (in Ω) at T = 72

◦

Note that the sensor resistance is given by R

= R

[1 + α(T −T

)], with

α = 0.004/

◦

F , and R

= 25 Ω at T

= 32

◦

F. Also E

= 5 V.

FIGURE 2.22

An example temperature measurement system conﬁguration.

3. Consider the Wheatstone bridge that is shown in Figure 2.10. Assume

that the resistor R

is actually a thermistor whose resistance, R, varies

with the temperature, T , according to the equation

R = R

exp



β(

−

)



Electronics 51

where R

= 1000 Ω at T

= 26.85

◦

C= 300 K (absolute) and β = 3500.

Both T and T

must be expressed in absolute temperatures in Equa-

tion 3. (Recall that the absolute temperature scales are either K or

◦

R.)

Assume that R

= R

. (a) Determine the normalized bridge

output, E

, when T = 400

◦

C. (b) Write a program to compute and

plot the normalized bridge output from T = T

to T = 500

◦

linear? (d) Over what temperature range is the normalized output very

insensitive to temperature change?

FIGURE 2.23

Test circuit.

4. For the test circuit shown in Figure 2.23, derive an expression for the

output voltage, E

, as a function of the input voltage, E

, and the re-

sistances shown for (a) the ideal case of the perfect voltmeter having

= ∞ and (b) the non-ideal voltmeter case when R

is ﬁnite. Show

mathematically that the solution for case (b) becomes that for case (a)

when R

→ ∞.

5. An inexpensive voltmeter is used to measure the voltage to within 1 %

across the power terminals of a stereo system. Such a system typically

has an output impedance of 500 Ω and a voltage of 120 V at its power

terminals. Assuming that the voltmeter is 100 % accurate such that

the instrument and zero-order uncertainties are negligible, determine

the minimum input impedance (in Ω) that this voltmeter must have to

meet the 1 % criterion.

6. A voltage divider circuit is shown in Figure 2.24. The common circuit is

used to supply an output voltage E

that is less than a source voltage

. (a) Derive the expression for the output voltage, E

, measured by

the meter, as a function of E

, R

, and R

, assuming that R

is not negligible with respect to R

and R

. Then, (b) show that the

expression derived in part (a) reduces to E

= E

) when R

becomes inﬁnite.

52 Measurement and Data Analysis for Engineering and Science

FIGURE 2.24

The voltage divider circuit.

7. Figure 2.25 presents the circuit used for a ﬂash in a camera. The capac-

itor charges toward the source voltage through the resistor. The ﬂash

turns on when the capacitor voltage reaches 8 V. If C = 10 µF, ﬁnd R

such that the ﬂash will turn on once per second.

FIGURE 2.25

Camera ﬂash circuit.

8. Find the diﬀerential equation for the current in the circuit shown in

Figure 2.26.

9. Between what pair of points (A, B, C, D) shown in Figure 2.27 should

one link up the power supply to charge all six capacitors to an equal

capacitance?

10. A capacitor consists of two round plates, each of radius r = 5 cm. The

gap between the plates is d = 5 mm. The capacity is given by C =

Electronics 53

−

FIGURE 2.26

RLC circuit.



S/d where S is the surface area, d is the gap between plates, 

is the

permittivity of free space, and  = 1 for air. (a) Determine the maximum

charge q

max

of the capacitor in coulombs if the breakdown potential of

the air is V

max

= 10 kV. (b) Find the capacitor energy in both the

International (SI) and the English Engineering (EE) systems.

11. Consider the ﬂash circuit shown in Figure 2.25 for a camera supplied

with a 9.0 V battery. The capacitor is used to modulate the ﬂash of the

camera by charging toward the battery through the resistor. When the

capacitor voltage reaches 8.0 V, the ﬂash discharges at a designed rate

of once per 5 seconds. The resistor in this circuit is 25 kΩ. What is the

capacitance of the capacitor for this design?

A

B

C

D

FIGURE 2.27

Six-capacitor circuit.

12. A researcher is attempting to decipher the lab notebook of a prior em-

ployee. The prior employee diagrammed the circuit shown in Figure 2.28

but gave no speciﬁcation about the input voltage. Through advanced

forensics you were able to ﬁnd places where he recorded the measured

current through the inductor I

, at time t, the capacitor voltage V

54 Measurement and Data Analysis for Engineering and Science

FIGURE 2.28

Notebook circuit.

and the capacitor capacitance C. Your boss has asked you to make sure

you are using the right resistors, but the lab notebook does not specify

the resistance. Formulate an expression to determine the resistance of

the resistor. (Note: Assume that the time under consideration is small

and that current through the inductor is constant when solving the dif-

ferential equation. Also assume that the capacitor voltage was known at

the beginning of the experiment when no current was ﬂowing.)

13. Design an op-amp circuit with two input voltages, E

i,1

and E

i,2

, such

that the output voltage, E

, is the sum of the two input voltages.

14. Consider the operational ampliﬁer circuit shown in Figure 2.29 and the

information in Table 2.1, in which R is resistance, C is capacitance, I

is current, and t is time. The transfer function of the circuit can be

written in the form where the output voltage, E

, equals a function

of the input voltage, E

, and other variables. (a) List all of the other

variables that would be in the transfer function expression. (b) Using

Kirchhoﬀ’s laws, derive the actual transfer function expression. Identify

any loops or nodes considered when applying Kirchhoﬀ’s laws.

FIGURE 2.29

Op-amp circuit.

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