E1C03 09/14/2010
15:24:53 Page 100
ASSUMPTIONS Second-order system behavior
FIND Rise and settling times; v d
SOLUTION The ringing behavior of the system noted on the trace supports the assumption that
the transducer can be described as having a second-order behavior. From the given information,
E 0
ð Þ ¼ Kp 0
ð Þ ¼ 1 V
E 1 ¼ Kp 1 ¼ 2 V
so that the step change observed on the trace should appear as a magnitude of 1 V. The 90% rise time
occurs when the output first achieves a value of 1.9 V. The 90% settling time will occurs when the
output settles between 1.9 < E(t) < 2.1 V. From Figure 3.15, the rise occurs in about 4 ms and the
settling time is about 9 ms. The period of the ringing behavior, T d , is judged to be about 13 ms for an
v d % 485 rad/s.
Simple Periodic Function Input
The response of a second-order system to a simple periodic function input of the form F(t) ¼
A sin vt is given by
y t
ð Þ ¼ y h þ
KAsin vt þ F v
ð Þ
½
1 À v=v n
ð
Þ
2
h
i 2 þ 2zv=v n
½
2
&
' 1=2
ð3:19Þ
with frequency-dependent phase shift
F v
ð Þ ¼ tan
À1
À
2zv=v n
1 À v=v n
ð
Þ
2
!
ð3:20Þ
The exact form for y h is found from Equations 3.14a–c and depends on the value of z. The steady
response, the second term on the right side, has the general form
y steady t
ð Þ ¼ y t ! 1
ð
Þ¼B v
ð Þsin vt þ F v
ð Þ
½
ð 3:21Þ
with amplitude B(v). Comparing Equations 3.19 and 3.21 shows that the amplitude of the steady
response of a second-order system subjected to a sinusoidal input is also dependent on v. So the
amplitude of the output signal is frequency dependent. In general, we can define the magnitude ratio,
M(v), for a second-order system as
M v
ð Þ ¼
B v
ð Þ
KA
¼
1
1 À v=v n
ð
Þ
2
h
i 2 þ 2zv=v n
½
2
&
' 1=2
ð3:22Þ
The magnitude ratio-frequency dependence for a second-order system is plotted in Figure 3.16 for
several values of damping ratio. A corresponding plot of the phase-shift dependency on input
frequency and damping ratio is shown in Figure 3.17. For an ideal measurement system, M(v)
would equal unity and F(v) would equal zero for all values of measured frequency. Instead, M(v)
approaches zero and F(v) approaches Àp as v/v n becomes large. Keep in mind that v n is a property
of the measurement system, while v is a property of the input signal.
100 Chapter 3 Measurement System Behavior
15:24:53 Page 100
ASSUMPTIONS Second-order system behavior
FIND Rise and settling times; v d
SOLUTION The ringing behavior of the system noted on the trace supports the assumption that
the transducer can be described as having a second-order behavior. From the given information,
E 0
ð Þ ¼ Kp 0
ð Þ ¼ 1 V
E 1 ¼ Kp 1 ¼ 2 V
so that the step change observed on the trace should appear as a magnitude of 1 V. The 90% rise time
occurs when the output first achieves a value of 1.9 V. The 90% settling time will occurs when the
output settles between 1.9 < E(t) < 2.1 V. From Figure 3.15, the rise occurs in about 4 ms and the
settling time is about 9 ms. The period of the ringing behavior, T d , is judged to be about 13 ms for an
v d % 485 rad/s.
Simple Periodic Function Input
The response of a second-order system to a simple periodic function input of the form F(t) ¼
A sin vt is given by
y t
ð Þ ¼ y h þ
KAsin vt þ F v
ð Þ
½
1 À v=v n
ð
Þ
2
h
i 2 þ 2zv=v n
½
2
&
' 1=2
ð3:19Þ
with frequency-dependent phase shift
F v
ð Þ ¼ tan
À1
À
2zv=v n
1 À v=v n
ð
Þ
2
!
ð3:20Þ
The exact form for y h is found from Equations 3.14a–c and depends on the value of z. The steady
response, the second term on the right side, has the general form
y steady t
ð Þ ¼ y t ! 1
ð
Þ¼B v
ð Þsin vt þ F v
ð Þ
½
ð 3:21Þ
with amplitude B(v). Comparing Equations 3.19 and 3.21 shows that the amplitude of the steady
response of a second-order system subjected to a sinusoidal input is also dependent on v. So the
amplitude of the output signal is frequency dependent. In general, we can define the magnitude ratio,
M(v), for a second-order system as
M v
ð Þ ¼
B v
ð Þ
KA
¼
1
1 À v=v n
ð
Þ
2
h
i 2 þ 2zv=v n
½
2
&
' 1=2
ð3:22Þ
The magnitude ratio-frequency dependence for a second-order system is plotted in Figure 3.16 for
several values of damping ratio. A corresponding plot of the phase-shift dependency on input
frequency and damping ratio is shown in Figure 3.17. For an ideal measurement system, M(v)
would equal unity and F(v) would equal zero for all values of measured frequency. Instead, M(v)
approaches zero and F(v) approaches Àp as v/v n becomes large. Keep in mind that v n is a property
of the measurement system, while v is a property of the input signal.
100 Chapter 3 Measurement System Behavior