Fundamentals of
Measurement Technology
(7)
Prof,Wang Boxiong
Both the transfer function and the frequency response
function describe the response of a measuring
instrument or system to sinusoidal excitation,
But the frequency response describes only the transfer
characteristics of a system with steady-state input and
output,
A transient output will reduce gradually to zero,and the
system will then reach the steady-state stage,For
describing the whole process of the two stages,the
transfer function must be employed,and the frequency
response is only a special case of the transfer function,
3.4.3 Responses of measuring system to typical excitations
The dynamic response of a measuring system
can be also obtained through applying other
excitations to the system,
The most commonly used excitation signals
are,unit impulse,unit step,and ramp signals,
3.4.3 Responses of measuring system to typical excitations
1,Unit impulse response
For a unit impulse function δ(t),its
Fourier transform Δ(jω)=1 and the Laplace
transform of δ(t),Δ(s)=L[δ(t)]=1,The output
of a measuring instrument with δ(t) as its
excitation,Y(s)=H(s)X(s)=H(s)Δ(s)=H(s),
Making inverse Laplace transform of Y(s),
then
h(t) is referred to as the impulse response
function or weighting function of a
measuring system.
3.4.3 Responses of measuring system to typical excitations
thsYLty 1 (3.44)
The first-order system
its impulse response h(t)
where time constant,
3.4.3 Responses of measuring system to typical excitations
11 ssH?
teth 1
(3.45)
Fig,3.18 Impulse response of first-order system
A second-order system
(assuming its static sensitivity K=1)
3.4.3 Responses of measuring system to typical excitations
12
1
2
2
nn
sssH
1) d,( o v e r d a m p e
1
1) d a m p e d,y ( c r i t i c a l l
)1 e d,( u n d e r d a m p1s i n
1
11
2
2
2
2
22
tt
n
t
n
n
tn
nn
n
n
eeth
teth
teth
(3.46)
3.4.3 Responses of measuring system to typical excitations
Fig,3.19 Impulse responses for second-order system with different dampimgs
The unit impulse does not exist in reality,Often
in engineering,an approximation is made by
use of a pulse signal with very short time
duration for the impulse signal,
Example,a shock to a system,if the shock duration
is shorter than τ/10,where τ is the system’s time
constant,then the shock can be considered as a unit
impulse,
3.4.3 Responses of measuring system to typical excitations
3.4.3 Responses of measuring system to typical excitations
Fig,3.21 Exact and approximate impulse response
2,Step response
The unit impulse function
The step function is
The response of first-order system to a unit
step input
The related Laplace transform is
3.4.3 Responses of measuring system to typical excitations
dt tdt (3.63)
't dttt (3.64)
tety 1 (3.65)
11 sssY? (3.66)
3.4.3 Responses of measuring system to typical excitations
Fig,3.22 Nondimensional step-function response of first-order system
Fig,3.22 (b) displays the curve of error em
versus time constant τ,Obviously y(t)=0.982
when t=4τ,The difference between the output
and the response in steady state is less than 2%,
Since the step excitation is easy to carry out,
it is often employed to measure dynamic
performances of measuring systems.
3.4.3 Responses of measuring system to typical excitations
A second-order system,
Its unit step response under zero initial conditions,
in nondimensional form,are
where
3.4.3 Responses of measuring system to typical excitations
12
1
2
2
nn
sssH
e d )( u n d e r d a m p1s i n11 22 tety ntn
da m pe d)y ( c r i t i c a l l11 tn netty
d)( o v e r d a m p e
12
1
12
11 1
2
21
2
2 22 tt
nn eety
(3.67)
(3.68)
(3.69)
21a r c ta n
3.4.3 Responses of measuring system to typical excitations
Fig,3.23 Nondimentional step-function response of second-order instrument
All the terms of error in the equations of step-
function response contains the factor e-At,thus
the dynamic error is zero when t→ ∞,
The response of system depends largely on its
damping ratio? and the natural frequency ωn,the
higher the ωn,the faster the system’s response.
3.4.3 Responses of measuring system to typical excitations
If? is chosen between 0.6 and 0.8,then the
maximum overshoot will be within 2.5%~10%,
For a setting time of 5%~2%,the curve for
=0.6~0.8gives a setting time of about
(3~4)/?ωn,and this is optimum,since either
larger or smaller? gives a longer setting time.
Many commercial instruments use?=0.6 to
0.7 because this range of? gives good
frequency response over the widest
frequency range,
3.4.3 Responses of measuring system to typical excitations
3,Ramp-function response
A ramp function can be considered as the
integral of a step function,The unit ramp
response of a system can be obtained by
integrating a step-function response,
A unit ramp function is
3.4.3 Responses of measuring system to typical excitations
0
00
tt
tt? (3.70)
Fig,3.24 Unit ramp function
The unit ramp response of
first-order system is
and the transfer function
3.4.3 Responses of measuring system to typical excitations
tetty 1
(3.71)
112 sssY? (3.72)
Fig,3.25 Ramp response of
first-order system
In steady state,the horizontal (time) displacement
between input and output curves is seen to be τ.
Therefore the instrument inherently has a steady-
state error which is directly proportional to τ.
The unit-ramp response for a second-order
system is
where
3.4.3 Responses of measuring system to typical excitations
e d )( u n d e r d a m p1s i n12 22 tetty n
n
t
n
n
d a m p e d )y ( c r i t i c a l l2122 tn
nn
nettty
o v e r d a m p e d
12
2121
12
21212
1
2
22
1
2
22
2
2
t
n
t
nn
n
n
e
etty
(3.73)
(3.74)
(3.75)
12
12a r c t a n
2
2
Its transform function is
3.4.3 Responses of measuring system to typical excitations
222 22
nn
n
ssssY
(3.76)
Fig,3.26 Unit ramp response of second-order system
The input signal x(t),which can be regarded
as a train of equally spaced rectangular functions
(or impulses) of varying magnitude,
3.4.4 Responses to arbitrary excitations
Fig,3.27 Approximation of an arbitrary function x(t) by use of
a train of impulses
The contribution of the impulse x(kΔτ) to the
total response at time t is [x(kΔτ)Δτ]h(t-kΔτ).
The total response
Letting Δτ→0 and taking limit of Eq,(3.77),
Conclusion:
The response to any arbitrary excitation is a
convolution integral of the excitation function
and the impulse response of system.
3.4.4 Responses to arbitrary excitations
0
)()()(
k
kthkxty
(3.77)
0
00
)()(
)()(lim)(
dthx
kthkxty
k (3.78)
)(*)()( thtxty? (3.79)
The frequency representation of Eq,(3.79) is
Similarly,
3.4.4 Responses to arbitrary excitations
)()()( sHsXsY? (3.80)
)()()( jHjXjY? (3.81)
The parameters of a measuring system
characterize its overall performance,To acquire
correct measurements,it is necessary to know
exactly all parameters of the measuring system
used,
Determination of static parameters of measuring
systems is relatively simple,
Determination of dynamic parameters is
relatively complicated and specific,
3.4.5 Experimental determination of parameters of measuring system
1,Determination of dynamic parameters of first-
order system
For first-order instruments,the static sensitivity K is
found by static calibration,
Determination of τ:
From Eq,(3.65) we can get
and
Define
and then
3.4.5 Experimental determination of parameters of measuring system
tety 1)( (3.84)
tety )(1 (3.85)
)](1ln [ tyZ (3.86)
tZ (3.87)
Further we have
Plot Z versus t,to get a
straight line whose slope
is numerically –1/τ.
3.4.5 Experimental determination of parameters of measuring system
1
dt
dZ (3.88)
Fig,3.28 Step-function test of first-
order system
The ultimate objective of measurement is to
response accurately,by use of measuring
instruments or systems,the measured quantities
or parameters,
Theoretically,a perfect measuring system
reproduces signal waveforms with no time delay
between input and output
In frequency domain,H(jω)=K∠ 0° over the entire
frequency range,a flat amplitude ratio (amplification)
and a zero phase angle.
3.5 Requirements on measuring instrument to ensure accurate
measurement
Many measuring systems can meet the
requirement on amplitude ratio,but cannot meet
the requirement on zero phase angle over the
same frequency (second-order instruments with
small? and large ωn,such as piezoelectric
devices,are exceptions),
A relaxed criterion which can be met by most
practical systems allows the phase angle to be
nonzero,but requires that it very linearly with
frequency over the range of flat amplitude ratio,
3.5 Requirements on measuring instrument to ensure accurate measurement
Thus the original requirement for an accurate
measurement may be changed to
where both K and t0 are constants,
In frequency domain
The system frequency response is
with the amplitude and phase angle being
3.5 Requirements on measuring instrument to ensure accurate measurement
)()( 0ttKxty (3.95)
0)()( tjejKXjY
(3.96)
00)(
)()( tKKe
jX
jYjH tj?
(3.97)
0)(
)(
t
KjA
(3.98)
A measuring system meets the two requirements
are called an accurate or a nondistortional
measuring system,
3.5 Requirements on measuring instrument to ensure accurate measurement
The actual measurements are all carried out in a
certain frequency range,then the actual frequency
response will be like the curves in Fig,3.33,
3.5 Requirements on measuring instrument to ensure accurate measurement
Fig,3.33 Requirements for accurate measurement
For (ω/ωn)<0.3
The amplitude ratio of a second-order system is
nearly a straight line,Its amplitude variation is not
larger than 10%.
The phase shift,however,changes drastically with
different dampings.
→0,φ=0;
→(0.6~0.8),the phase shift curve becomes approximately
an oblique line originating from the coordinate origin.
Many measuring instruments select a value range of
(0.6~0.8) for damping?,ensuring a better phase
response.
3.5 Requirements on measuring instrument to ensure accurate measurement
For (ω/ωn)>3
Phase response curves for all?’s approach -180°,
Using an inverter or subtracting a fixed phase angle
of 180° to get a result with no phase shift.
Disadvantage,the magnitudes for higher-frequency
components are very small and are not favorable to
successive processing.
3.5 Requirements on measuring instrument to ensure accurate measurement
1,Loading effect in measurement
A measuring system can be considered as a
combination of measurand and measuring
instrument,
H0(s),the transfer function of measured object
Hm(s),the transfer function of measuring instrument,
3.6 Loading effects of measuring system
Fig,3.35 Interconnection of measured object and measuring instrument
Due to the influences of intermediate elements
such as sensing and displaying elements,there
occurs an energy exchange between the
preceding and the latter elements of the system,
Thus the output z(t) of the measuring instrument
won’t be equal to the output y(t) of the measured
object any more,
3.6 Loading effects of measuring system
When two systems are interconnected and have
energy exchanges between them,the physical
parameters at connection points will change
correspondingly,
Thus the two systems won’t maintain simply
their original transfer functions,but rather they
give a new transfer function for the combined
system,
3.6 Loading effects of measuring system
3.6 Loading effects of measuring system
Fig,3.36 Examples of loading effects
2,Interconnection of two first-order
systems
Two first-order elements
3.6 Loading effects of measuring system
Fig,3.37 Interconnection of two first-order elements
111
1
1 ;1
1 CR
ssH
222
2
2 ;1
1 CR
ssH
(3.100)
(3.101)
When connected in series directly without
any isolation between them
Assume V2(t) to be the voltage at the connection
point,
The impedance of the right-hand side of the
connection point is
3.6 Loading effects of measuring system
ssV
sV y
22 1
1
(3.102)
sC
s
sC
sCR
sCRZ 2
2
2
22
2
22
111
Letting z represent the impedance of the right-
hand side of R1
Thus
3.6 Loading effects of measuring system
21221
2
2
2
1
2
2
1
2
1
1
11
11
//
1
sCsCC
s
sC
s
sC
sC
s
sC
Z
sC
Z
2212121
2
2
2
112211
2
1
2
1
1
1
1
ssCR
s
ssCRsCCR
s
ZR
Z
sV
sV
x
(3.103)
The transfer function after the connection
becomes
Conclusion,H(s)≠H1(s)·H2(s)
Reason,there are energy exchanges between
the two elements.
3.6 Loading effects of measuring system
2212121
2
2
1
1
ssCR
sV
sV
sV
sV
sV
sV
sH
y
xx
y
(3.104)
2
212121
21 1
1
1
1
1
1
sssssHsH
(3.105)
To avoid loading effect
1,insert a,follower amplifier” between the two
stages
2,let H(s)≈H1(s)
Two alternatives can be adopted in selecting a
measuring instrument,
1,τ2<<τ1;
2,the storage component C2 of the measuring
instrument must be of small capacitance,
3.6 Loading effects of measuring system
Even if reached H(s)=H1(s)·H2(s),to ensure the
measured data to reflect accurately the dynamic
performance of measured object and to remove
the influence of measuring instrument,still H2(s)
must be approximately equal to unity,that is,
H2(s)≈1,Therefore τ2<0.3τ1.
3.6 Loading effects of measuring system
Measurement Technology
(7)
Prof,Wang Boxiong
Both the transfer function and the frequency response
function describe the response of a measuring
instrument or system to sinusoidal excitation,
But the frequency response describes only the transfer
characteristics of a system with steady-state input and
output,
A transient output will reduce gradually to zero,and the
system will then reach the steady-state stage,For
describing the whole process of the two stages,the
transfer function must be employed,and the frequency
response is only a special case of the transfer function,
3.4.3 Responses of measuring system to typical excitations
The dynamic response of a measuring system
can be also obtained through applying other
excitations to the system,
The most commonly used excitation signals
are,unit impulse,unit step,and ramp signals,
3.4.3 Responses of measuring system to typical excitations
1,Unit impulse response
For a unit impulse function δ(t),its
Fourier transform Δ(jω)=1 and the Laplace
transform of δ(t),Δ(s)=L[δ(t)]=1,The output
of a measuring instrument with δ(t) as its
excitation,Y(s)=H(s)X(s)=H(s)Δ(s)=H(s),
Making inverse Laplace transform of Y(s),
then
h(t) is referred to as the impulse response
function or weighting function of a
measuring system.
3.4.3 Responses of measuring system to typical excitations
thsYLty 1 (3.44)
The first-order system
its impulse response h(t)
where time constant,
3.4.3 Responses of measuring system to typical excitations
11 ssH?
teth 1
(3.45)
Fig,3.18 Impulse response of first-order system
A second-order system
(assuming its static sensitivity K=1)
3.4.3 Responses of measuring system to typical excitations
12
1
2
2
nn
sssH
1) d,( o v e r d a m p e
1
1) d a m p e d,y ( c r i t i c a l l
)1 e d,( u n d e r d a m p1s i n
1
11
2
2
2
2
22
tt
n
t
n
n
tn
nn
n
n
eeth
teth
teth
(3.46)
3.4.3 Responses of measuring system to typical excitations
Fig,3.19 Impulse responses for second-order system with different dampimgs
The unit impulse does not exist in reality,Often
in engineering,an approximation is made by
use of a pulse signal with very short time
duration for the impulse signal,
Example,a shock to a system,if the shock duration
is shorter than τ/10,where τ is the system’s time
constant,then the shock can be considered as a unit
impulse,
3.4.3 Responses of measuring system to typical excitations
3.4.3 Responses of measuring system to typical excitations
Fig,3.21 Exact and approximate impulse response
2,Step response
The unit impulse function
The step function is
The response of first-order system to a unit
step input
The related Laplace transform is
3.4.3 Responses of measuring system to typical excitations
dt tdt (3.63)
't dttt (3.64)
tety 1 (3.65)
11 sssY? (3.66)
3.4.3 Responses of measuring system to typical excitations
Fig,3.22 Nondimensional step-function response of first-order system
Fig,3.22 (b) displays the curve of error em
versus time constant τ,Obviously y(t)=0.982
when t=4τ,The difference between the output
and the response in steady state is less than 2%,
Since the step excitation is easy to carry out,
it is often employed to measure dynamic
performances of measuring systems.
3.4.3 Responses of measuring system to typical excitations
A second-order system,
Its unit step response under zero initial conditions,
in nondimensional form,are
where
3.4.3 Responses of measuring system to typical excitations
12
1
2
2
nn
sssH
e d )( u n d e r d a m p1s i n11 22 tety ntn
da m pe d)y ( c r i t i c a l l11 tn netty
d)( o v e r d a m p e
12
1
12
11 1
2
21
2
2 22 tt
nn eety
(3.67)
(3.68)
(3.69)
21a r c ta n
3.4.3 Responses of measuring system to typical excitations
Fig,3.23 Nondimentional step-function response of second-order instrument
All the terms of error in the equations of step-
function response contains the factor e-At,thus
the dynamic error is zero when t→ ∞,
The response of system depends largely on its
damping ratio? and the natural frequency ωn,the
higher the ωn,the faster the system’s response.
3.4.3 Responses of measuring system to typical excitations
If? is chosen between 0.6 and 0.8,then the
maximum overshoot will be within 2.5%~10%,
For a setting time of 5%~2%,the curve for
=0.6~0.8gives a setting time of about
(3~4)/?ωn,and this is optimum,since either
larger or smaller? gives a longer setting time.
Many commercial instruments use?=0.6 to
0.7 because this range of? gives good
frequency response over the widest
frequency range,
3.4.3 Responses of measuring system to typical excitations
3,Ramp-function response
A ramp function can be considered as the
integral of a step function,The unit ramp
response of a system can be obtained by
integrating a step-function response,
A unit ramp function is
3.4.3 Responses of measuring system to typical excitations
0
00
tt
tt? (3.70)
Fig,3.24 Unit ramp function
The unit ramp response of
first-order system is
and the transfer function
3.4.3 Responses of measuring system to typical excitations
tetty 1
(3.71)
112 sssY? (3.72)
Fig,3.25 Ramp response of
first-order system
In steady state,the horizontal (time) displacement
between input and output curves is seen to be τ.
Therefore the instrument inherently has a steady-
state error which is directly proportional to τ.
The unit-ramp response for a second-order
system is
where
3.4.3 Responses of measuring system to typical excitations
e d )( u n d e r d a m p1s i n12 22 tetty n
n
t
n
n
d a m p e d )y ( c r i t i c a l l2122 tn
nn
nettty
o v e r d a m p e d
12
2121
12
21212
1
2
22
1
2
22
2
2
t
n
t
nn
n
n
e
etty
(3.73)
(3.74)
(3.75)
12
12a r c t a n
2
2
Its transform function is
3.4.3 Responses of measuring system to typical excitations
222 22
nn
n
ssssY
(3.76)
Fig,3.26 Unit ramp response of second-order system
The input signal x(t),which can be regarded
as a train of equally spaced rectangular functions
(or impulses) of varying magnitude,
3.4.4 Responses to arbitrary excitations
Fig,3.27 Approximation of an arbitrary function x(t) by use of
a train of impulses
The contribution of the impulse x(kΔτ) to the
total response at time t is [x(kΔτ)Δτ]h(t-kΔτ).
The total response
Letting Δτ→0 and taking limit of Eq,(3.77),
Conclusion:
The response to any arbitrary excitation is a
convolution integral of the excitation function
and the impulse response of system.
3.4.4 Responses to arbitrary excitations
0
)()()(
k
kthkxty
(3.77)
0
00
)()(
)()(lim)(
dthx
kthkxty
k (3.78)
)(*)()( thtxty? (3.79)
The frequency representation of Eq,(3.79) is
Similarly,
3.4.4 Responses to arbitrary excitations
)()()( sHsXsY? (3.80)
)()()( jHjXjY? (3.81)
The parameters of a measuring system
characterize its overall performance,To acquire
correct measurements,it is necessary to know
exactly all parameters of the measuring system
used,
Determination of static parameters of measuring
systems is relatively simple,
Determination of dynamic parameters is
relatively complicated and specific,
3.4.5 Experimental determination of parameters of measuring system
1,Determination of dynamic parameters of first-
order system
For first-order instruments,the static sensitivity K is
found by static calibration,
Determination of τ:
From Eq,(3.65) we can get
and
Define
and then
3.4.5 Experimental determination of parameters of measuring system
tety 1)( (3.84)
tety )(1 (3.85)
)](1ln [ tyZ (3.86)
tZ (3.87)
Further we have
Plot Z versus t,to get a
straight line whose slope
is numerically –1/τ.
3.4.5 Experimental determination of parameters of measuring system
1
dt
dZ (3.88)
Fig,3.28 Step-function test of first-
order system
The ultimate objective of measurement is to
response accurately,by use of measuring
instruments or systems,the measured quantities
or parameters,
Theoretically,a perfect measuring system
reproduces signal waveforms with no time delay
between input and output
In frequency domain,H(jω)=K∠ 0° over the entire
frequency range,a flat amplitude ratio (amplification)
and a zero phase angle.
3.5 Requirements on measuring instrument to ensure accurate
measurement
Many measuring systems can meet the
requirement on amplitude ratio,but cannot meet
the requirement on zero phase angle over the
same frequency (second-order instruments with
small? and large ωn,such as piezoelectric
devices,are exceptions),
A relaxed criterion which can be met by most
practical systems allows the phase angle to be
nonzero,but requires that it very linearly with
frequency over the range of flat amplitude ratio,
3.5 Requirements on measuring instrument to ensure accurate measurement
Thus the original requirement for an accurate
measurement may be changed to
where both K and t0 are constants,
In frequency domain
The system frequency response is
with the amplitude and phase angle being
3.5 Requirements on measuring instrument to ensure accurate measurement
)()( 0ttKxty (3.95)
0)()( tjejKXjY
(3.96)
00)(
)()( tKKe
jX
jYjH tj?
(3.97)
0)(
)(
t
KjA
(3.98)
A measuring system meets the two requirements
are called an accurate or a nondistortional
measuring system,
3.5 Requirements on measuring instrument to ensure accurate measurement
The actual measurements are all carried out in a
certain frequency range,then the actual frequency
response will be like the curves in Fig,3.33,
3.5 Requirements on measuring instrument to ensure accurate measurement
Fig,3.33 Requirements for accurate measurement
For (ω/ωn)<0.3
The amplitude ratio of a second-order system is
nearly a straight line,Its amplitude variation is not
larger than 10%.
The phase shift,however,changes drastically with
different dampings.
→0,φ=0;
→(0.6~0.8),the phase shift curve becomes approximately
an oblique line originating from the coordinate origin.
Many measuring instruments select a value range of
(0.6~0.8) for damping?,ensuring a better phase
response.
3.5 Requirements on measuring instrument to ensure accurate measurement
For (ω/ωn)>3
Phase response curves for all?’s approach -180°,
Using an inverter or subtracting a fixed phase angle
of 180° to get a result with no phase shift.
Disadvantage,the magnitudes for higher-frequency
components are very small and are not favorable to
successive processing.
3.5 Requirements on measuring instrument to ensure accurate measurement
1,Loading effect in measurement
A measuring system can be considered as a
combination of measurand and measuring
instrument,
H0(s),the transfer function of measured object
Hm(s),the transfer function of measuring instrument,
3.6 Loading effects of measuring system
Fig,3.35 Interconnection of measured object and measuring instrument
Due to the influences of intermediate elements
such as sensing and displaying elements,there
occurs an energy exchange between the
preceding and the latter elements of the system,
Thus the output z(t) of the measuring instrument
won’t be equal to the output y(t) of the measured
object any more,
3.6 Loading effects of measuring system
When two systems are interconnected and have
energy exchanges between them,the physical
parameters at connection points will change
correspondingly,
Thus the two systems won’t maintain simply
their original transfer functions,but rather they
give a new transfer function for the combined
system,
3.6 Loading effects of measuring system
3.6 Loading effects of measuring system
Fig,3.36 Examples of loading effects
2,Interconnection of two first-order
systems
Two first-order elements
3.6 Loading effects of measuring system
Fig,3.37 Interconnection of two first-order elements
111
1
1 ;1
1 CR
ssH
222
2
2 ;1
1 CR
ssH
(3.100)
(3.101)
When connected in series directly without
any isolation between them
Assume V2(t) to be the voltage at the connection
point,
The impedance of the right-hand side of the
connection point is
3.6 Loading effects of measuring system
ssV
sV y
22 1
1
(3.102)
sC
s
sC
sCR
sCRZ 2
2
2
22
2
22
111
Letting z represent the impedance of the right-
hand side of R1
Thus
3.6 Loading effects of measuring system
21221
2
2
2
1
2
2
1
2
1
1
11
11
//
1
sCsCC
s
sC
s
sC
sC
s
sC
Z
sC
Z
2212121
2
2
2
112211
2
1
2
1
1
1
1
ssCR
s
ssCRsCCR
s
ZR
Z
sV
sV
x
(3.103)
The transfer function after the connection
becomes
Conclusion,H(s)≠H1(s)·H2(s)
Reason,there are energy exchanges between
the two elements.
3.6 Loading effects of measuring system
2212121
2
2
1
1
ssCR
sV
sV
sV
sV
sV
sV
sH
y
xx
y
(3.104)
2
212121
21 1
1
1
1
1
1
sssssHsH
(3.105)
To avoid loading effect
1,insert a,follower amplifier” between the two
stages
2,let H(s)≈H1(s)
Two alternatives can be adopted in selecting a
measuring instrument,
1,τ2<<τ1;
2,the storage component C2 of the measuring
instrument must be of small capacitance,
3.6 Loading effects of measuring system
Even if reached H(s)=H1(s)·H2(s),to ensure the
measured data to reflect accurately the dynamic
performance of measured object and to remove
the influence of measuring instrument,still H2(s)
must be approximately equal to unity,that is,
H2(s)≈1,Therefore τ2<0.3τ1.
3.6 Loading effects of measuring system
