E1C03 09/14/2010
15:24:53 Page 95
The functions M(v) and F v
ð Þ represent the frequency response of the measurement system to
periodic inputs. These equations and universal curves provide guidance in selecting measurement
systems and system components.
Determination of Frequency Response The frequency response of a measurement system is
found by a dynamic calibration. In this case, the calibration would entail applying a simple periodic
waveform of known amplitude and frequency to the system sensor stage and measuring the
corresponding output stage amplitude and phase shift. In practice, developing a method to produce a
periodic input signal in the form of a physical variable may demand considerable ingenuity and
effort. Hence, in many situations an engineer elects to rely on modeling to infer system frequency
response behavior. We can predict the dynamic behavior if the time constant and static sensitivity of
the system and the range of input frequencies are all known.
Example 3.6
A temperature sensor is to be selected to measure temperature within a reaction vessel. It is suspected
that the temperature will behave as a simple periodic waveform with a frequency somewhere between
1 and 5 Hz. Sensors of several sizes are available, each with a known time constant. Based on time
constant, select a suitable sensor, assuming that a dynamic error of Æ2% is acceptable.
KNOWN
1 f 5 Hz
dðvÞj 0:02
j
ASSUMPTIONS First-order system
F t
ð Þ ¼ A sin vt
FIND Time constant, t
SOLUTION With d v
ð Þj 0:02
j
, we would set the magnitude ratio between 0.98 M 1.02.
From Figure 3.12, we see that first-order systems never exceed M ¼ 1. So the constraint becomes
0.98 M 1. Then,
0:98 M v
ð Þ ¼
1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 þ vt
ð Þ
2
q
1
From Figure 3.12, this constraint is maintained over the range 0 vt 0:2. We can also see in this
figure that for a system of fixed time constant, the smallest value of M(v) will occur at the largest
frequency. So with v ¼ 2pf ¼ 2pð5Þrad/s and solving for M(v) ¼ 0.98 yields, t 6:4 ms. Accordingly, a sensor having a time constant of 6.4 ms or less will work.
Second-Order Systems
Systems possessing inertia contain a second-derivative term in their model equation (e.g.,
see Ex. 3.1). A system modeled by a second-order differential equation is called a second-order
3.3 Special Cases of the General System Model 95
15:24:53 Page 95
The functions M(v) and F v
ð Þ represent the frequency response of the measurement system to
periodic inputs. These equations and universal curves provide guidance in selecting measurement
systems and system components.
Determination of Frequency Response The frequency response of a measurement system is
found by a dynamic calibration. In this case, the calibration would entail applying a simple periodic
waveform of known amplitude and frequency to the system sensor stage and measuring the
corresponding output stage amplitude and phase shift. In practice, developing a method to produce a
periodic input signal in the form of a physical variable may demand considerable ingenuity and
effort. Hence, in many situations an engineer elects to rely on modeling to infer system frequency
response behavior. We can predict the dynamic behavior if the time constant and static sensitivity of
the system and the range of input frequencies are all known.
Example 3.6
A temperature sensor is to be selected to measure temperature within a reaction vessel. It is suspected
that the temperature will behave as a simple periodic waveform with a frequency somewhere between
1 and 5 Hz. Sensors of several sizes are available, each with a known time constant. Based on time
constant, select a suitable sensor, assuming that a dynamic error of Æ2% is acceptable.
KNOWN
1 f 5 Hz
dðvÞj 0:02
j
ASSUMPTIONS First-order system
F t
ð Þ ¼ A sin vt
FIND Time constant, t
SOLUTION With d v
ð Þj 0:02
j
, we would set the magnitude ratio between 0.98 M 1.02.
From Figure 3.12, we see that first-order systems never exceed M ¼ 1. So the constraint becomes
0.98 M 1. Then,
0:98 M v
ð Þ ¼
1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 þ vt
ð Þ
2
q
1
From Figure 3.12, this constraint is maintained over the range 0 vt 0:2. We can also see in this
figure that for a system of fixed time constant, the smallest value of M(v) will occur at the largest
frequency. So with v ¼ 2pf ¼ 2pð5Þrad/s and solving for M(v) ¼ 0.98 yields, t 6:4 ms. Accordingly, a sensor having a time constant of 6.4 ms or less will work.
Second-Order Systems
Systems possessing inertia contain a second-derivative term in their model equation (e.g.,
see Ex. 3.1). A system modeled by a second-order differential equation is called a second-order
3.3 Special Cases of the General System Model 95