lecture04 dynamic characteristicsmohamed/ukur inst notes/lecture 04.pdf · 2012. 10. 2. · dynamic...
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Dynamic Characteristics
Lecture04 SME3242 Instrumentation 1
Static transfer function – how the output related to input if the input is constant
Dynamic transfer function – also called time response
Lecture04 SME3242 Instrumentation 2
Lecture04 SME3242 Instrumentation 3
FIGURE 1.27 The dynamic transfer function specifies how asensor output varies when the input changes instantaneously intime (i.e., a step change).
Curtis JohnsonProcess Control Instrumentation Technology, 8e] Copyright ©2006 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458All rights reserved.
Lecture04 SME3242 Instrumentation 4
Input and output relationship of a linear measurement system - ordinary differential equation (ODE):
where,u = input, y = output; u and y varies with tn > ma, b = constant coefficients
2.1.3: Mathematical model structure
d dyy yadt
adtda
dta n
n
nn
n
n 011
1
1
dtdtdtd du ubbdbub m
m
mm
m
m 011
1
1
y
u
Lecture04 SME3242 Instrumentation 5
: Transfer Function of An Accelerometer
Differential equation:
xoxi
mk
c FD
=Fs ma
dxdx o d2
2
)()(dt
xmdtdt
cxxk oioi
kk ccx dx ii
ooo x
mdtdx
mx
mdtmdtd
2
2
Applying 2nd Newton’s Law: F = ma
Lecture04 SME3242 Instrumentation 6
Dynamic characteristic- the output response of the instrument against
time when the input is varied- the relation between any input and output for
nth order system can be written as:
- 3 types of response: zero order response, firstorder response and second order response
y d dy ubyadt
adt
adtda n
n
nn
n
n 0011
1
1
y
Lecture04 SME3242 Instrumentation 7
Dynamic response of zero order instrument (i.e. n = 0)
- the zero order instrument is represented bya0y=b0u or y=Ku
or
(y=output, u=input, K=b0/a0=static sensitivity)
- the output responses linearly to the input
y/u = K
Lecture04 SME3242 Instrumentation 8
Eg: potentiometer
Lecture04 SME3242 Instrumentation 9
Dynamic response of first order instrument (i.e. n = 1)
- dividing the equation by a0, and apply D-operator
- T=a1/a0=time constant, K=b0/a0=static sensitivity
1 KuyTDuabydt
dyaa )1(;
0
0
0
ubyadtdya 001
Lecture04 SME3242 Instrumentation 10
Sensoru(t) y(t)
- the operational transfer function
y)1( TD
Ku
- The time constant; T, represents the time taken forthe output to reach 63% of the final value and itreaches its final value (99%) at around 5T.
Lecture04 SME3242 Instrumentation 11
Eg.: Thermocouple
Lecture04 SME3242 Instrumentation 12
Characteristic first-order time response of a sensor.
t y(t) %T 0.63212 63.2122T 0.86466 86.4663T 0.95021 95.0214T 0.98168 98.1685T 0.99326 99.32610T 0.99995 99.995
Lecture04 SME3242 Instrumentation 13
Characteristic first-order time response of a sensor.
General equation as function of time following a step input is given as:
where,yi = initial output from static transfer
function and initial inputyf = final output from static transfer function
and final inputT = time constant = 63% time
Lecture04 SME3242 Instrumentation 14
]1)[()( /Ttifi eyyyty
Lecture04 SME3242 Instrumentation 15
Characteristic first‐order exponential time response of a sensorto a step change of input.
Curtis JohnsonProcess Control Instrumentation Technology, 8e]
Copyright ©2006 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458
All rights reserved.
yf
yi
y
A sensor measures temperature linearly with a statictransfer function of 33 mV/0C and has a 1.5‐s timeconstant. Find the output 0.75 s after input changesfrom 200C to 410C. Find the error in temperature thisrepresents.
* Time response analysis always applied to the outputof the sensor because it is only the output of thesensor that lagged
Lecture04 SME3242 Instrumentation 16
Given static transfer function:V = (33mV/ºC)T
Hence, initial and final output of the sensor are:yi = (33mV/ºC)(20ºC)
= 660mVyf = (33mV/ºC)(41ºC)
= 1353mV
Lecture04 SME3242 Instrumentation 17
Substitute the value of yi and yf,
Lecture04 SME3242 Instrumentation 18
0.75y 660 )[ ]1 /1.50.75e(1353 - 660=
= 932.7 mV
]1)[()( /Ttifi eyyyty =
Time response of first-order system,
Lecture04 SME3242 Instrumentation 19
The corresponding temperature for sensor output of932.7 mV,
C3.28
mVCmV
T/33
7.932
Since the actual temperature is 41ºC, hence the errorin temperature is:
error = (true value – instrument reading)= (41ºC – 28.3ºC)= 12.7ºC
Lecture04 SME3242 Instrumentation 20
When t = 5T i.e. t = 5(1.5) = 7.5 s,
7.5y 660 )[ ]1 /1.57.5e (1353 - 660=
= 1348.3 mV
The corresponding temperature for sensor output of1348.3 mV is:
C41mVCmV
T/33
1348.3
which is the exact measured temperature
Lecture04 SME3242 Instrumentation 21
Dynamic response of second-order instrument(i.e. n = 2)
Applying D operator
dtdty dy ubyaada 0012
2
2
)( 2210
0
DaDaauby
Applying Laplace Transform (with all initialconditions equal to zero) and rearranging theequation:
where, = damping ration= natural frequency
Lecture04 SME3242 Instrumentation 22
uss
Ky
nn
n22
2
2
Lecture04 SME3242 Instrumentation 23
The time response is given as:q0(t) qe-atsin(nt)
where q=amplitude and a= n is output damping ratio
Eg.: Accelerometer
Lecture04 SME3242 Instrumentation 24
Lecture04 SME3242 Instrumentation 25
FIGURE 1.29 Characteristic second-order oscillatory time responseof a sensor.
Curtis JohnsonProcess Control Instrumentation Technology, 8e]
Copyright ©2006 by Pearson Education, Inc.
Upper Saddle River, New Jersey 07458All rights reserved.
Lecture04 SME3242 Instrumentation 26
undamped (=0)
under damped (1>>0)
over damped (1)
Lecture04 SME3242 Instrumentation 27