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3.5 Suppose a thermometer similar to that in Example 3.3 is known to have a time constant of 30 s in a
particular application. Plot its time response to a step change from 32
to 120
F. Determine its 90%
rise time.
3.6 Referring back to Example 3.3, a student establishes the time constant of a temperature sensor by first
holding it immersed in hot water and then suddenly removing it and holding it immersed in cold
water. Several other students perform the same test with similar sensors. Overall, their results are
inconsistent, with estimated time constants differing by as much as a factor of 1.2. Offer suggestions
as to why this might happen. Hint: Try this yourself and think about control of test conditions.
3.7 A thermocouple, which responds as a first-order instrument, has a time constant of 20 ms. Determine
its 90% rise time.
3.8 During a step function calibration, a first-order instrument is exposed to a step change of 100 units. If
after 1.2 s the instrument indicates 80 units, estimate the instrument time constant. Estimate the error
in the indicated value after 1.5 s. y(0) ¼ 0 units; K ¼ 1 unit/unit.
3.9 Estimate any dynamic error that could result from measuring a 2-Hz periodic waveform using a firstorder system having a time constant of 0.7 s.
3.10 A signal expected to be of the form F(t) ¼ 10 sin 15.7t is to be measured with a first-order instrument
having a time constant of 50 ms. Write the expected indicated steady response output signal. Is this
instrument a good choice for this measurement? What is the expected time lag between input and
output signal? Plot the output amplitude spectrum; y(0) ¼ 0 and K ¼ 1V/V.
3.11 A first-order instrument with a time constant of 2 s is to be used to measure a periodic input. If a
dynamic error of Æ2% can be tolerated, determine the maximum frequency of periodic input that can
be measured. What is the associated time lag (in seconds) at this frequency?
3.12 Determine the frequency response [M(v) and f(v)] for an instrument having a time constant of
10 ms. Estimate the instrument’s usable frequency range to keep its dynamic error within 10%.
3.13 A temperature measuring device with a time constant of 0.15 s outputs a voltage that is linearly
proportional to temperature. The device is used to measure an input signal of the form T(t) ¼
115 þ 12 sin 2t
C. Plot the input signal and the predicted output signal with time assuming firstorder behavior and a static sensitivity of 5 mV/
C. Determine the dynamic error and time lag in the
steady response. T(0) ¼ 115
C.
3.14 A first-order sensor is to be installed into a reactor vessel to monitor temperature. If a sudden rise in
temperature greater than 100
C should occur, shutdown of the reactor will need to begin within
5 s after reaching 100
C. Determine the maximum allowable time constant for the sensor.
3.15 A single-loop LR circuit having a resistance of 1 MV is to be used as a low-pass filter between an
input signal and a voltage measurement device. To attenuate undesirable frequencies above 1000 Hz
by at least 50%, select a suitable inductor size. The time constant for this circuit is given by L /R.
3.16 A measuring system has a natural frequency of 0.5 rad/s, a damping ratio of 0.5, and a static
sensitivity of 0.5 m/V. Estimate its 90% rise time and settling time if F(t) ¼ 2 U(t) and the initial
condition is zero. Plot the response y(t) and indicate its transient and steady responses.
3.17 Plot the frequency response, based on Equations 3.20 and 3.22, for an instrument having a damping
ratio of 0.6. Determine the frequency range over which the dynamic error remains within 5%. Repeat
for a damping ratio of 0.9 and 2.0.
3.18 The output from a temperature system indicates a steady, time-varying signal having an amplitude
that varies between 30
and 40
C with a single frequency of 10 Hz. Express the output signal as a
Problems 113
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3.5 Suppose a thermometer similar to that in Example 3.3 is known to have a time constant of 30 s in a
particular application. Plot its time response to a step change from 32
to 120
F. Determine its 90%
rise time.
3.6 Referring back to Example 3.3, a student establishes the time constant of a temperature sensor by first
holding it immersed in hot water and then suddenly removing it and holding it immersed in cold
water. Several other students perform the same test with similar sensors. Overall, their results are
inconsistent, with estimated time constants differing by as much as a factor of 1.2. Offer suggestions
as to why this might happen. Hint: Try this yourself and think about control of test conditions.
3.7 A thermocouple, which responds as a first-order instrument, has a time constant of 20 ms. Determine
its 90% rise time.
3.8 During a step function calibration, a first-order instrument is exposed to a step change of 100 units. If
after 1.2 s the instrument indicates 80 units, estimate the instrument time constant. Estimate the error
in the indicated value after 1.5 s. y(0) ¼ 0 units; K ¼ 1 unit/unit.
3.9 Estimate any dynamic error that could result from measuring a 2-Hz periodic waveform using a firstorder system having a time constant of 0.7 s.
3.10 A signal expected to be of the form F(t) ¼ 10 sin 15.7t is to be measured with a first-order instrument
having a time constant of 50 ms. Write the expected indicated steady response output signal. Is this
instrument a good choice for this measurement? What is the expected time lag between input and
output signal? Plot the output amplitude spectrum; y(0) ¼ 0 and K ¼ 1V/V.
3.11 A first-order instrument with a time constant of 2 s is to be used to measure a periodic input. If a
dynamic error of Æ2% can be tolerated, determine the maximum frequency of periodic input that can
be measured. What is the associated time lag (in seconds) at this frequency?
3.12 Determine the frequency response [M(v) and f(v)] for an instrument having a time constant of
10 ms. Estimate the instrument’s usable frequency range to keep its dynamic error within 10%.
3.13 A temperature measuring device with a time constant of 0.15 s outputs a voltage that is linearly
proportional to temperature. The device is used to measure an input signal of the form T(t) ¼
115 þ 12 sin 2t
C. Plot the input signal and the predicted output signal with time assuming firstorder behavior and a static sensitivity of 5 mV/
C. Determine the dynamic error and time lag in the
steady response. T(0) ¼ 115
C.
3.14 A first-order sensor is to be installed into a reactor vessel to monitor temperature. If a sudden rise in
temperature greater than 100
C should occur, shutdown of the reactor will need to begin within
5 s after reaching 100
C. Determine the maximum allowable time constant for the sensor.
3.15 A single-loop LR circuit having a resistance of 1 MV is to be used as a low-pass filter between an
input signal and a voltage measurement device. To attenuate undesirable frequencies above 1000 Hz
by at least 50%, select a suitable inductor size. The time constant for this circuit is given by L /R.
3.16 A measuring system has a natural frequency of 0.5 rad/s, a damping ratio of 0.5, and a static
sensitivity of 0.5 m/V. Estimate its 90% rise time and settling time if F(t) ¼ 2 U(t) and the initial
condition is zero. Plot the response y(t) and indicate its transient and steady responses.
3.17 Plot the frequency response, based on Equations 3.20 and 3.22, for an instrument having a damping
ratio of 0.6. Determine the frequency range over which the dynamic error remains within 5%. Repeat
for a damping ratio of 0.9 and 2.0.
3.18 The output from a temperature system indicates a steady, time-varying signal having an amplitude
that varies between 30
and 40
C with a single frequency of 10 Hz. Express the output signal as a
Problems 113