测试技术英文课件

F u n d a m e n t a l s o f
M e a s u re m e n t Te ch n o l o g y P ro b l e m B o o k
Wang Boxiong Luo Xiuzhi
Dept. of Precision Instruments and Mechanology
Tsinghua University
November, 20
P r e f a c e
This problem book is compiled and used for the course “ Fundamentals of Measurement Technology ”. All problems contained in this book are designed for the students with the aim of strengthening their understanding of theories of measurement technology and enhancing their capabilities in analyzing and solving problems relating measurement technology .
The authors
C h a p t e r 2 S i g n a l R e p r e s e n t a t i o n s
2-1 Two time sequences :
(1)()⎪⎭
⎫ ⎝⎛-=π8n j e
n x (2)()n
e n x π10cos
2
=
Are they periodic functions? If so , what are the periods?
2-2 A periodic signal )(t x has its Fourier coefficients n a ，n b ，n c . Prove that a time-delayed signal )(0t t x - has the following expression of Fourier series :
()∑∑∞
+-∞
=∞
='=
⎪
⎭⎫ ⎝⎛'
+'+=-n t jn n n n n e
c t n b t n a a t t x 001
000sin cos ωωω
where ：0000sin cos t n b t n a a n n n
ωω-='
0000cos sin t n b t n a b n n n ωω+='
00t jn n n e c c ω-='
2-3 Determine the Fourier series of the following periodic signals and plot their spectra .
(1)
t
Fig 2-1
(2)
t
Fig 2-2
(3)
sawtooth wave
t T t x •=/2)( )
2/2/T t T ≥≤-
)(t x ()
()
⎩⎨
⎧≤≤≤≤--=2002T/t A t T/A
t
Fig 2-3
(4)
t
Fig 2-4
(5)
t
Fig 2-5
2-4 Determine the Fourier transforms of the following nonperiodic signals , and plot their spectra .
(1) exponentially-decaying function )0,0()(≥>=-t a Ae t x at ； (2) signum function(Fig (a)), and unit-step function (Fig (b)).
full-rectified sinusoidal
t A t x 0sin )(ω=
exponential function
()0)(>=a e t x at
()22/T/t T ≤≤-
rectangular impulse for τ3=T and τ5=T
t
t
(a) (b)
Fig 2-6
Hint ：a signum function is symbolized as )(t Sgn ，Make first the Fourier transform of the function
⎩⎨⎧><->=-00
0)(λλλt e
t e t x t
t
and let λ→0 to obtain the Fourier transform of )(t Sgn . A unit-step function
)(t u can be obtained by shifting a signum function along the ordinate .
(3)
t
Fig 2-7 truncated cosine wave
⎩⎨
⎧><='
'cos )(0T t T t t t x ω
Using graphic method to discuss the effect of different s T 0 on the spectrum for
'0'0'0,,T T T T T T ><=.
(4) e xponentially-decaying t e t x at 0cos )(ω-=
(5) triangular impulse with time shift τ
(Calculate with one method and present two other ideas for solutions)
t
Fig 2-8
(6) Two rectangular impulses , one centered at origin and the other centered at
τ.
t
Fig 2-9
(7) Plot the waveform of time signal k t f A t x ++=)2cos(
)(0ϕ，and its Fourier spectrum .
(8) Find the spectrum of cosine impulse .
t
Fig 2-10
()
()
≥<2/τt 2/τt
(9) Find the spectrum of two cosine impulses .
t
Fig 2-11
(10) Find the spectrum of the periodic modulated signal by use of graphic method :
)0,0(cos )(0≥>=-t a t
e t x at ω
Fig 2-12
2-5 Sig nal )(t x and its frequency spectrum are shown in Fig 2-13. An oscillatory signal )(2cos 00m f f t f >π is multiplied with )(t x . (In this case , signal )(t x is called the modulating signal , and the oscillatory signal t f 02cos π is called carrier .) Find the Fourier transform of the modulated signal t f t x 02cos )(π, and plot its time waveform and frequency spectrum . What will be the situation if
m f f <0.
Fig 2-13
2-6 Two signals , )(1t x and )(2t x are shown in Fig 2-14 )(1t x has the frequency spectrum )(1f X . Determine the frequency spectrum of )
(2t x .
Fig 2-14
2-7 Find the convolution of )(1t x and )(2t x
using graphic method .
t
Fig 2-15
2-8 Find the Fourier transform of the signal ()()bt at e e --*, and plot its spectrum )0,0,0(>>>t b a .
C h a p t e r 3 A n a l y s i s o f M e a s u r i n g s y s t e m
3-1 A piezoelectric transducer with a sensitivity N PC S cp /00.9= is connected with a charge amplifier of a sensitivity PC V S vc /005.0=, which are then connected to a light trace oscilloscope of a sensitivity V mm S xv /20= . Draw the whole system using block diagram , and determine the resultant sensitivity .
3-2 A micro-ammeter with a measurement range of A μ60 is calibrated . The following are the obtained data :
number of measurements 1 2 3 4 5 readings of calibrating meter 10 20 30 40 50 readings of calibrated meter 10 20.5 29.5 39 50.5 Calculate the linearity of the meter (using the least-square regression).
3-3 Find the total sensitivity S for the following systems formed by elements connected in series , in parallel , and with negative feedback loop . Assuming that all elements in the figure are linear ones .
⋯
Fig 3-1
3-4 A first-order device with a transfer function ()s
s H τ+=
11
and a time constant s 35.0=τ is used to measure sinusoidal signals with period of 1s ,
2s and 5s . Determine the relative amplitude error for each case .
3-5 A gear box with a reduction ratio of 1:5 has unbalances on its two shafts . A signal )(t y of the vibration caused by the unbalances is picked up by a sensor on the box case , whose frequency spectrum is ()f Y .
t
)
Fig 3-2
Question ：
(1) What are the input and output of the whole system ？ (2) Is the system a linear one?
(3) What are the two frequency components in the spectrum that are caused by the unbalances of the two gear shafts? Assume that the meter has a rotation speed of rpm 3000.
3-6 A periodic signal )4/300cos(100cos 2)(π-+=t t t x ) passes through a first-order system whose frequency response )105.0/(1)(+=ωωj j H .
Find its steady-state frequency response .
(1) Design a procedure for evaluating the output response ; (2) Use graphic method to determine :
①the synthesized wave forms of the input and the output ; and
②the amplitude spectra and the phase spectra of the input and output respectively .
3-7 Second-order measuring devices usually have a damping ratio
7.0~6.0=ζ. Explain the reason .
3-8 A second-order device has the transfer function
()()()⎪⎪⎭
⎫ ⎝⎛++=n n n j j s H 2222ωωζωωω. Its damping ratio 7.0=ζ and the natural frequency Hz f n 50=. What is its steady-state output )(t y under the excitation )(t x shown in Fig 3-3.
3-9 A measuring device has its amplitude spectrum shown in Fig 3-4. The phase spectrum shows a phase-shift of 75︒ for s rad /5.125=ω, a phase-shift of 90︒ f or s rad /6.150=ω, and a phase-shift of 180︒ for s rad /626≥ω. The device is used to measure the following two signals :
t A t A t x 6.150sin 5.125sin )(211+=
t A t A t x 700sin 626sin )(432+=
Is it possible to obtain a distortionless measurement of )(1t x and )(2t x using the device? Indicate the reason .
A(f)
Fig 3-4
3-10 Find the steady-state response of the device with a transform function
()1
005.01+=s s H when its input signal )4510cos(2.010cos 5.0)( -+=t t t x .
3-11 Insert suddenly a thermometer from the air of 20℃ into the water of 80℃. If the thermometer has a time constant s 5.3=τ，what is the reading indicated by the thermometer after 2 seconds ？
3-12 A meteorological balloon with a thermometer having a time constant τ15 (the thermometer can be considered as a first -order s ystem) is passing through the air at a speed of 5m/s . The temperature decreases 0.15℃ for every m 30 increase in altitude . The balloon transmits the altitude and the temperature data to the ground .By calculation , the temperature at the altitude of m 3000 is -1℃.What is the altitude when a temperature of -0.1℃ is practically reached.？
C h a p t e r 4 T r a n s d u c e r s
4-1 A steel plate with a length of m L 1= and an
elastic modulus 26/101.2cm f kg E •⨯= is pulled
by a force P . The recorded tensile strain by an
HP-3 foil strain gage （Ω=120R ,sensitivity
coefficient 2=K ）is με300.Calculate the elongation L ∆ of the plate , its stress σand R R /∆ Fig 4-1
of the strain gage . If a strain
of με1 must be
measured what is the related R R /∆？
4-2 A resistance strain gage has a sensitivity
2=S and a resistance Ω=120R .If its strain
is με1000 when it operates . What is the
resistance change R ∆?
If the strain gage is connected in a circuit
(see Fig 4-2), determine :
(1) the current indicated when there is no strain ; Fig 4-2
(2) the current indicated when there is a strain ；
(3) the relative variation of the current indication ；
(4) whether it is possible to read out this variation from the ammeter .
4-3 A capacitance micro-displacement measuring
instrument has an initial gap mm d 3.00= between its two plates in an air medium . Each plate has a diameter mm r 4=. Determine :
(1) the capacitance variation when the gap has Fig 4-3
a displacement m d μ1±=∆ from its initial position ；and
(2) the variation in graduations of the instrument .
Assuming that the amplification factor of the measuring circuit PF mV K /12=，the sensitivity mV s graduation K /50=, and the displacement m d μ1±=∆.
4-4 A planar plate-capacitor displacement sensor
with a width cm b 4= and a plate gap
mm 2.0=σis used to measure displace-
ment .What is its sensitivity?
Fig 4-4
4-5 A platinum-resistance thermometer is used to measure temperatures from C 200~0.The resistance-temperature relationship is )1(20T T R R T βα++=. For Ω=1000R ，Ω=5.138100R and R 200=175.83Ω, determine :
(1) values of α and β；and
(2) the nonlinearity the thermometer displays at C 100.
4-6 A capacitance liquid level gage has two metal cylinders located in a liquid tank . The outer cylinder has an inner diameter of R 2 and the inner cylinder has an outer diameter of r 2. If the liquid
dielectric constant is 1ε，the air dielectric
constant is 0ε，the total length of the plate is L and the level height is 1L , determine :
(1) the relationship between the level height 1L and the capacitance C ；and Fig 4-5
(2) the sensitivity S . Are 1L and C linearly related ?
4-7 Use the following two self-inductance transducers to show the sensibilities for
Fig 4-6
differential and nondifferential transducers .
4-8 An inertial velocity pickup of a natural frequency Hz f n 15=，and a damping ratio ζ=0.7 is used to measure the following two vibrations :
)2sin()(),2sin(5)(2211t f a t x t f a t x ππ==，where Hz f 101=，Hz f 1002=
Calculate the pickup’s output . If a piezoelectric accelerometer is used for the measurement, what is its output?
4-9 A velocity transducer is designed for measuring vibrations of frequencies higher than Hz 30. The maximum amplitude error should not be larger than ±5%. For a damping ratio ζ=0.6，find the natural frequency of the transducer .
2
4-10 What are the working frequency ranges of velocity transducers and accelerometers ? Describe the influence of m , k and ζ on frequency range .
4-11 A pressure transducer (considered as a second-order vibration system) has a natural frequency Hz f 8000=and a damping ζ=0.14. It is used to measure a sinusoidal force of Hz f 400=. Find its amplitude ratio )(ωA and the phase difference )(ωϕ. If ζis changed to 0.7，determine )(ωA and )(ωϕ again .
4-12 A Hall-element ，whose sensitivity is KGS mA mV K H •=/2.1，is placed in a magnetic field with a gradient of mm KGS /5. If the rated control current is mA 20 and the element vibrates about its equilibrium position with vibration amplitude of mm 01.0±，calculate its output voltage .
4-13 Fig 4-7 shows the equivalent circuit of the combination of a piezoelectric transducer and a change amplifier, where C is the sum of the inherent capacitance of the transducer, the stray capacitance of the cable ,and the input capacitance of the amplifier . The transducer has a sensitivity g PC S q /100=，and a feedback capacitance pF C f 1000=. Find the amplitude voltage of the amplifier for a measured acceleration of g 5.0.
q C f
Fig 4-7
C h a p t e r 5 S i g n a l C o n d i t i o n i n g
5-1 A bridge consists of a resistance strain gage having a resistance Ω=120R ，a sensitivity 2=S and a fixed resistor of Ω120 .The bridge is powered with an excitation of V 3. Assume that the load resistance is infinite . For a strain of με2 and a strain of με2000，determine the output voltage of a single-arm bridge and a double-arm bridge respectively , and compare their sensitivities .
5-2 Someone found in using a strain gage that the sensitivity of the gage is insufficient . To raise the sensitivity ，he increased the number of strain gages . Can the sensitivity be raised under the following conditions when
（1）a strain gage is connected in series with each of the two arms of the
half bridge ；and when
（2）a strain gage is connected in parallel with each of the two arms of the
half bridge?
5-3 An a . c . bridge is shown in Fig. 5-1.
For Ω==50011R Z ,
Ω==100022R Z ,
ωω2.0/13•-=j Z , and the power supply V e 100=of Hz f 1000=, Fig 5-1
calculate :
(1) Z 4 when the bridge is balanced , and tell whether the reactance Z 4 is a
capacitive one or an inductive one ; and
(2) Z 4 when 2Z and 3Z are interchanged .
5-4 A Wheatstone bridge composed of resistive strain gages is used to measure a
structure’s strain . The strain t B t A t 100cos 10cos )(+=ε. The bridge has an excitation voltage t E t e 100000sin )(0=. Find the spectrum of the output . 5-5 An amplitude-modulated signal
))(cos 3cos 20cos 30100()(t t t t x c a ωΩ+Ω+=,
where KHz f c 10= and Hz f 500=Ω. Determine :
(1) the frequencies and amplitudes of the frequency components contained in )(t x a ；
(2) plot the frequency spectra of the modulating signal, the carrier and the modulated signal respectively .
5-6 Two strain gages are connected in the opposite arms of a bridge (see Fig 5-2). The excitation voltage t f e 002cos π=，where Hz f 10000=. Strain gage sensitivity 2=K and R R =1 when the bridge is initially balanced .
Find （1）the bridge output and plot its time waveform ;
（2）plot the amplitude spectrum of the output .
Fig 5-2
5-7 A signal )0,0()(≥>=-t a e t x at is used to modulate a carrier t 0cos ω，The modulated carrier is again used as a reference signal for synchronous demodulotion and phase-sensitive detection . Determine the time waveform and the amplitude spectrum of the signal after synchronous demodulation and phase-sensitive detection. What is the cut-off frequency of a filter if it is used to recover the original waveform .
t
5-8 Determine the output voltage L e when the load L R is connected to the following bridge , and the relation between L e and the output for an open circuit y e .
5-9 Determine the amplitude spectrum of the signal
t f ft m A t f 02sin
)2cos 1()(ππ+=.
5-10 The following graph shows the strain curve measured by an a .c . bridge , determine :
(1) the modulated time waveform ；
(2) the waveform after a phase-sensitive detection ；
(3) the amplitude spectrum of the signal after being modulated and phase -sensitive detected .
Fig 5-4
5-11 A cosine wave signal of frequency f is used to modulate a sine wave carrier of frequency 0f . The modulated signal t f ft m A t x 02sin )2cos 1()(ππ•+= where m is the modulating factor, and f f >>0
t
Determine (1) the spectrum of )(t x ; and
(2) prove that the average power of )(t x is ⎪⎪⎭
⎫ ⎝⎛+=2120m P P where 0P is the average power of the unmodulated carrier .
5-12 T wo band-pass filters , one is an octave filter and the other is a 1/3-octave filter, have the same lower cut-off frequency . How many times is the center frequency of the former filter larger than that of the latter one?
5-13 A 1/3-octave band-pass filter has a center frequency Hz f 800=. Calculate its lower and upper cut-off frequencies 1c f and 2c f .
5-14 A band-pass filter has a lower cut-off frequency 1c f and upper cut-off frequency 2c f ，and a center frequency 0f . Are the following statements correct ? (1) for an octave filter, c1c22f f =
(2) ()c2c1021f f f +=; (3) c2c10f f f -=;
(4) the cut-off frequency of a filter is the frequency at which the -3dB magnitude of the passband is located ; and
(5) the center frequency of an octave filter is 32 times that of a 1/3-octave filter , when the two filters have the same lower cut-off frequency .
5-15 Two RC filters are shown in Fig 5-5. If the upper cut-off frequency of the low-pass filter is to be Hz f c 4402=，and the lower cut-off frequency of the high-pass filter Hz f c 3601=, select the resistor values, when the following
capacitors are available ：4700PF ，0.15F μ, 0.01F μ，0.022F μ，0.033F μ, 0.039F μ，0.047F μ，0.082F μ.
（calculate only one pair of data for each filter ）
R C
low-pass filter high-pass filter
Fig 5-5
5-16 For a filter with multi-channel negative feedback(shown in Fig 5-6)，
(1)determine its frequency response ; and (2) identify what kind of filter it is , and calculate its cut-off frequency .
V out
Fig 5-6
5-17 For a filter with limited voltage amplification , determine
(1) its frequency response ; and
(2) identify what kind of filter it is , and calculate its cut-off frequency .
V out
Fig 5-7
5-18 A low-pass filter )(t x has its amplitude shown in the following figure and its phase angle characteristic 0)(=ωϕ. The input signal to the filter is a periodic square wave of magnitude 1 and period ms T 1=. Determine its output )(t y and the related frequency spectrum )(f Y , sketch the
waveform of )(
t y and its spectrum .
t
f(KHz )
1 2 3
Fig 5-8
5-19 A measurement system is shown in block diagram . The first stage has a form of an exponential signal ax e and the second one is an amplifier . Determine the characteristic of the third linear correction stage f . If f is now placed in a feedback
loop, what is its characteristic ?
Fig 5-9
5-20 If a high-pass filter and a low-pass filter are connected in series (shown in Fig 5-10), is it possible to form a band-pass filter? Derive the frequency response function of the total network . Analyze its amplitude phase spectra
.
Fig 5-10
5-21 How to distinguish between the two signals )(
t x and )(t y (see Fig 5-11) by both their time and frequency waveforms after they have been first amplitude-
modulated and then demodulated .
Fig 5-11
5-22 Calculate the output of the network (see Fig 5-12) for the input of t 10sin .
Sin10t
e y
1 f
Fig 5-12
5-23 The difference between two neighboring quantized levels of an A/D converter is equal to the quantization unit q . Prove that the quantization unit q for an N-bit A/D converter is
N m
V q 2
=
where m V is the maximum converted analog voltage of the converter .
5-24 Fig 5-13 shows a cantilever beam made of steel with a Young’s modulus
Pa E 101020⨯=. Four resistive strain gages are cemented on the beam to form a
full bridge. Each strain gage has a rated resistance of Ω200 and a sensitivity factor of 3.5. The bridge is powered by a dc power supply of V 6.5. Find the output of the bridge when the load is N 100.
Fig 5-13
C h a p t e r6R e c o r d i n g I n s t r u m e n t
6-1 The vibrator of an SC16 Light Trace Oscilloscope has a unit step input )(t i
θ(see Fig 6-1). What are the damping ratio ζand the and gives an output )(t
ωof the vibrator? Deduce its transfer function.
natural frequency
n
t
a
Fig 6-1
6-2 Several vibrators are used to record sine wave signals of different frequencies and analyze the relationship between these different signals.How should one choose the vibrators?
6-3 An FC6-1200 Vibrator is used to record signals from a velocity transducer. Determine its working frequency range. If signals from piezoelectric accelerometer are to be recorded, what is the working frequency range of the accelerometer.
6-4An FC6-30 vibrator is used to record a Hz
100sine wave signal of a magnitude V2. The desired deflection on the recording sheet should be mm
±.
50 Determine the values of the resistors to be connected in series and in parallel. (Assume the inner resistance of the power supply is Ω
200)
6-5A vibrator with a natural frequency of Hz
1200is used to record a square- wave signal with a fundamental frequency of Hz
600. Calculate and analyze the recorded results , and sketch the recorded waveform.
C h a p t e r 7 S i g n a l P r o c e s s i n g a n d A n a l y s i s
7-1 Design a spectrum analyzer with multi-channel filters for a signal frequency range of KHZ 8~0. Ten octave band-pass filters are employed to cover the whole frequency range .
Determine ：(1) the center frequency of each filter, 0f ；
(2) the band width of each filter .
7-2 A tracking filter spectrum analyzer is used to analyze a square wave signal of a period of s 1.0. Calculate the band width of the band-pass filter .
7-3 Ideal sampling with a sampling rate Hz f s 4= is performed to three sinusoidal signal
t t x π2cos )(1=,t
t x π6cos )(2=
and t
t x π10cos )(3=
respectively .
Determine and compare the sampled output sequences of the three signals , sketch the time waveforms and the sample positions , and explain the frequency aliasing phenomenon by use of frequency spectra .
7-4 Calculate the DFTs of the following discrete sequence .
（21,1,21,0,21,1,21,0---）
7-5 Calculate the frequency spectrum of the sequence in Problem 4 using FFT algorithm . Compare and explain the two results .
7-6 Calculate the probability density function of signal t A t x ωsin )(=，sketch the result graph .
7-7 Calculate the autocorrelation and the power spectrum of a periodic cosine wave t A t x 0cos )(ω=. Sketch their graphs .
7-8 Calculate the cross-correlation of a square wave signal and a sine wave signal (see Fig 7-1). Compare the result with that of the autocorrelation of a sine wave signal and explain the reason .
t t
Fig 7-1
7-9 For 2x
y=and the sequence =
x0, 1, 2, 3, 4,5,calculate the
cross-correlation coefficient
xy
ρof x and y. Explain the result.
7-10 Deduce the DFT of the exponential function
t
e
t
x-
=
)(（t≥0）
The waveform of )(t
x and its amplitude spectrum ()f
X are shown in Fig 7-2.
Fig 7-2
7-11 Digital Fourier transform is to be made with a stationary random signal which has been low-pass filtered before hand. If the components with frequencies
lower than Hz 500 are to be analyzed , and the spectrum resolution must be Hz 5.0, determine
(1) the sampling frequency s f ;
(2) the number of samples N ; and
(3) the bandwidth of the window function , T .
C o m p r e h e n s i v e P r a c t i c a l P r o b l e m s
Use your knowledge to solve the following practical problems . You are required to (1) put forward plans for measuring systems , their fundamental arrangements and the necessary explanations ; (2) put forward solutions to the problems these systems might encounter with in practice, such as temperature , variation , and vibration , etc .
1．Truck deadweight gage
The truck body is fixed on the chasis through springs . Indications of the gage must be seen by the driver in driver’s cab . Work out two possible solutions .
2．Rail-breadth measuring instrument ：
The inner width of rails is defined to be within A ±δ where A is the width , and δ is the tolerance . Width larger than the value will cause abnormal operation . Put forward two possible schemes for measuring width using patrol train .
3．Automatic thickness measurement of rolled steel ：
The thickness of rolled steel-plate must be ensured within δ±∆δ. Propose a solution for automatic measurement of plate thickness during rolling process
.
4．Steel cable is made by twisting several strands of wire. How to detect the wire breakage on-line.
5. It is found that a workpiece has vibration trace on its machined surface after being processed by a grinding machine. To find the possible source for the vibration, transducers and spectrum analysis are employed. Design a scheme for the vibration source detection. (Assume that the grinding machine has 3 motors: one for the spindle with a rotation speed of rpm
1500, one for the grinding wheel with a rotation speed of rpm
3000, and one for the cooling oil pump with a rotation speed of rpm
750. No other vibration source is present.)。