Temporal and frequency response Notes | EduRev

: Temporal and frequency response Notes | EduRev

 Page 1


Objectives_template
file:///G|/optical_measurement/lecture10/10_1.htm[5/7/2012 11:56:40 AM]
 Module 2: Review of Probes and Transducers
 Lecture 10: Temporal and frequency response
 
The Lecture Contains:
Transient and Frequency Response
Lumped Analysis
Analysis with Spatial Variations
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Page 2


Objectives_template
file:///G|/optical_measurement/lecture10/10_1.htm[5/7/2012 11:56:40 AM]
 Module 2: Review of Probes and Transducers
 Lecture 10: Temporal and frequency response
 
The Lecture Contains:
Transient and Frequency Response
Lumped Analysis
Analysis with Spatial Variations
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Objectives_template
file:///G|/optical_measurement/lecture10/10_2.htm[5/7/2012 11:56:40 AM]
 Module 2: Review of Probes and Transducers
 Lecture 10: Temporal and frequency response
 
Transient and Frequency Response
There are three fundamental questions that need to be answered with respect to the temporal
response of a probe and a measurement system which are subjected to a non-zero input. These are :
1. If the input is steady, how long will it take for the probe response to become steady?
2. If the input is steady, is the probe response oscillatory?
3. If the input is periodic, what is the critical frequency beyond which the output has a negligible
amplitude?
Question 3 addresses the problem of attenuation of signals as they pass through the probe and the
measurement system. Further attenuation of signals can take place in spatially distributed systems
due to a non-uniform response of different parts of the probe. For example, in a pitot tube the fluid
close to the wall is always at rest while the bulk of the fluid within it will move during a transient. In a
hot-wire anemometer, the temperature may not be uniform along its length and in particular, the
portion of the wire close to the prongs will be at the prong temperature. Hence it would take finite time
to re-establish the temperature profile along the wire. The reciprocal of this time determines the cut-
off frequency beyond which the signal amplitude is unacceptably small.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Page 3


Objectives_template
file:///G|/optical_measurement/lecture10/10_1.htm[5/7/2012 11:56:40 AM]
 Module 2: Review of Probes and Transducers
 Lecture 10: Temporal and frequency response
 
The Lecture Contains:
Transient and Frequency Response
Lumped Analysis
Analysis with Spatial Variations
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Objectives_template
file:///G|/optical_measurement/lecture10/10_2.htm[5/7/2012 11:56:40 AM]
 Module 2: Review of Probes and Transducers
 Lecture 10: Temporal and frequency response
 
Transient and Frequency Response
There are three fundamental questions that need to be answered with respect to the temporal
response of a probe and a measurement system which are subjected to a non-zero input. These are :
1. If the input is steady, how long will it take for the probe response to become steady?
2. If the input is steady, is the probe response oscillatory?
3. If the input is periodic, what is the critical frequency beyond which the output has a negligible
amplitude?
Question 3 addresses the problem of attenuation of signals as they pass through the probe and the
measurement system. Further attenuation of signals can take place in spatially distributed systems
due to a non-uniform response of different parts of the probe. For example, in a pitot tube the fluid
close to the wall is always at rest while the bulk of the fluid within it will move during a transient. In a
hot-wire anemometer, the temperature may not be uniform along its length and in particular, the
portion of the wire close to the prongs will be at the prong temperature. Hence it would take finite time
to re-establish the temperature profile along the wire. The reciprocal of this time determines the cut-
off frequency beyond which the signal amplitude is unacceptably small.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Objectives_template
file:///G|/optical_measurement/lecture10/10_3.htm[5/7/2012 11:56:40 AM]
 Module 2: Review of Probes and Transducers
 Lecture 10: Temporal and frequency response
 
Lumped Analysis
A lumped parameter analysis of probes is given below. The effect of spatial variability is discussed
through specific examples later in this chapter. Let  be the flow input and  the probe output.
The order of a probe, a transducer or a measurement system is determined by the order of the
differential equation relating  and  with time  as the independent variable. Hence we have:
In the above equations  is the static sensitivity of the probe that can be determined once-and-for-all
from a steady state experiment. Consider the response of these systems to a step input , a
constant and a periodic input . Here  is frequency and  the imaginary unit 
. For a step input, we assume the initial conditions to be quiescent, i.e., 
. For a periodic input we assume that the system has reached a dynamic steady state and the output 
 oscillates with the same frequency as the forcing frequency . The second part of this
assumption is strictly true only for linear systems, i.e. coefficients , ,  and  are independent of
,  and . In both laboratory and field experiments the fluctuations in the input will displace the
measurement system only marginally with respect to the operating point and so its performance can
be locally linearized. Hence the linear analysis presented here is not severely restrictive.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Page 4


Objectives_template
file:///G|/optical_measurement/lecture10/10_1.htm[5/7/2012 11:56:40 AM]
 Module 2: Review of Probes and Transducers
 Lecture 10: Temporal and frequency response
 
The Lecture Contains:
Transient and Frequency Response
Lumped Analysis
Analysis with Spatial Variations
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Objectives_template
file:///G|/optical_measurement/lecture10/10_2.htm[5/7/2012 11:56:40 AM]
 Module 2: Review of Probes and Transducers
 Lecture 10: Temporal and frequency response
 
Transient and Frequency Response
There are three fundamental questions that need to be answered with respect to the temporal
response of a probe and a measurement system which are subjected to a non-zero input. These are :
1. If the input is steady, how long will it take for the probe response to become steady?
2. If the input is steady, is the probe response oscillatory?
3. If the input is periodic, what is the critical frequency beyond which the output has a negligible
amplitude?
Question 3 addresses the problem of attenuation of signals as they pass through the probe and the
measurement system. Further attenuation of signals can take place in spatially distributed systems
due to a non-uniform response of different parts of the probe. For example, in a pitot tube the fluid
close to the wall is always at rest while the bulk of the fluid within it will move during a transient. In a
hot-wire anemometer, the temperature may not be uniform along its length and in particular, the
portion of the wire close to the prongs will be at the prong temperature. Hence it would take finite time
to re-establish the temperature profile along the wire. The reciprocal of this time determines the cut-
off frequency beyond which the signal amplitude is unacceptably small.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Objectives_template
file:///G|/optical_measurement/lecture10/10_3.htm[5/7/2012 11:56:40 AM]
 Module 2: Review of Probes and Transducers
 Lecture 10: Temporal and frequency response
 
Lumped Analysis
A lumped parameter analysis of probes is given below. The effect of spatial variability is discussed
through specific examples later in this chapter. Let  be the flow input and  the probe output.
The order of a probe, a transducer or a measurement system is determined by the order of the
differential equation relating  and  with time  as the independent variable. Hence we have:
In the above equations  is the static sensitivity of the probe that can be determined once-and-for-all
from a steady state experiment. Consider the response of these systems to a step input , a
constant and a periodic input . Here  is frequency and  the imaginary unit 
. For a step input, we assume the initial conditions to be quiescent, i.e., 
. For a periodic input we assume that the system has reached a dynamic steady state and the output 
 oscillates with the same frequency as the forcing frequency . The second part of this
assumption is strictly true only for linear systems, i.e. coefficients , ,  and  are independent of
,  and . In both laboratory and field experiments the fluctuations in the input will displace the
measurement system only marginally with respect to the operating point and so its performance can
be locally linearized. Hence the linear analysis presented here is not severely restrictive.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Objectives_template
file:///G|/optical_measurement/lecture10/10_4.htm[5/7/2012 11:56:41 AM]
 Module 2: Review of Probes and Transducers
 Lecture 10: Temporal and frequency response
 
For zeroth order system the output will match the input at every instant of time except for a scale
factor  that is predetermined. There is no attenuation or phase lag for any value of  and . Hence
it represents an ideal probe or an instrument.
The response of a first order system is:
(Figure 2.20)
(Figure 2.21)
 
 
Figure 2.20: Step Response of a First Order System.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Page 5


Objectives_template
file:///G|/optical_measurement/lecture10/10_1.htm[5/7/2012 11:56:40 AM]
 Module 2: Review of Probes and Transducers
 Lecture 10: Temporal and frequency response
 
The Lecture Contains:
Transient and Frequency Response
Lumped Analysis
Analysis with Spatial Variations
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Objectives_template
file:///G|/optical_measurement/lecture10/10_2.htm[5/7/2012 11:56:40 AM]
 Module 2: Review of Probes and Transducers
 Lecture 10: Temporal and frequency response
 
Transient and Frequency Response
There are three fundamental questions that need to be answered with respect to the temporal
response of a probe and a measurement system which are subjected to a non-zero input. These are :
1. If the input is steady, how long will it take for the probe response to become steady?
2. If the input is steady, is the probe response oscillatory?
3. If the input is periodic, what is the critical frequency beyond which the output has a negligible
amplitude?
Question 3 addresses the problem of attenuation of signals as they pass through the probe and the
measurement system. Further attenuation of signals can take place in spatially distributed systems
due to a non-uniform response of different parts of the probe. For example, in a pitot tube the fluid
close to the wall is always at rest while the bulk of the fluid within it will move during a transient. In a
hot-wire anemometer, the temperature may not be uniform along its length and in particular, the
portion of the wire close to the prongs will be at the prong temperature. Hence it would take finite time
to re-establish the temperature profile along the wire. The reciprocal of this time determines the cut-
off frequency beyond which the signal amplitude is unacceptably small.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Objectives_template
file:///G|/optical_measurement/lecture10/10_3.htm[5/7/2012 11:56:40 AM]
 Module 2: Review of Probes and Transducers
 Lecture 10: Temporal and frequency response
 
Lumped Analysis
A lumped parameter analysis of probes is given below. The effect of spatial variability is discussed
through specific examples later in this chapter. Let  be the flow input and  the probe output.
The order of a probe, a transducer or a measurement system is determined by the order of the
differential equation relating  and  with time  as the independent variable. Hence we have:
In the above equations  is the static sensitivity of the probe that can be determined once-and-for-all
from a steady state experiment. Consider the response of these systems to a step input , a
constant and a periodic input . Here  is frequency and  the imaginary unit 
. For a step input, we assume the initial conditions to be quiescent, i.e., 
. For a periodic input we assume that the system has reached a dynamic steady state and the output 
 oscillates with the same frequency as the forcing frequency . The second part of this
assumption is strictly true only for linear systems, i.e. coefficients , ,  and  are independent of
,  and . In both laboratory and field experiments the fluctuations in the input will displace the
measurement system only marginally with respect to the operating point and so its performance can
be locally linearized. Hence the linear analysis presented here is not severely restrictive.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Objectives_template
file:///G|/optical_measurement/lecture10/10_4.htm[5/7/2012 11:56:41 AM]
 Module 2: Review of Probes and Transducers
 Lecture 10: Temporal and frequency response
 
For zeroth order system the output will match the input at every instant of time except for a scale
factor  that is predetermined. There is no attenuation or phase lag for any value of  and . Hence
it represents an ideal probe or an instrument.
The response of a first order system is:
(Figure 2.20)
(Figure 2.21)
 
 
Figure 2.20: Step Response of a First Order System.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Objectives_template
file:///G|/optical_measurement/lecture10/10_5.htm[5/7/2012 11:56:41 AM]
 Module 2: Review of Probes and Transducers
 Lecture 10: Temporal and frequency response
 
Figure 2.21: Periodic Response of a First Order System.
For a step input it takes a time  for  to reach within  of the steady state. Here,  is
called the time constant of the system. For a periodic input, the first order system shows attenuation
for increasing values of . For a time constant of 0.2 second, the attenuation factor  is 0.04,
when Hz, (=125.6 rad/s) and amplitude reduction by a factor of 25.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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