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.Read More

Offer running on EduRev: __Apply code STAYHOME200__ to get INR 200 off on our premium plan EduRev Infinity!