E1C03 09/14/2010
15:24:53 Page 97
Step Function Input
Again, the step function input is applied to determine the general behavior and speed at which the
system will respond to a change in input. The response of a second-order measurement system to a
step function input is found from the solution of Equation 3.13, with F(t) ¼ AU(t), to be
y t
ð Þ ¼ KA À KAe
Àzv n t
z
ffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 À z
2
p
sin v n t
ffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 À z
2
q
þ cos v n t
ffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 À z
2
q
"
#
0 z < 1 ð3:15aÞ
y t
ð Þ ¼ KA À KA 1 þ v n t
ð
Þ e
Àv n t
z ¼ 1 ð3:15bÞ
y t
ð Þ ¼ KA À KA
z þ
ffiffiffiffiffiffiffiffiffiffiffiffi ffi
z
2
À 1
p
2
ffiffiffiffiffiffiffiffiffiffiffiffi ffi
z
2
À 1
p
e
Àzþ
ffiffiffiffiffiffiffi ffi
z
2 À1
p
À
Á
v n t þ
z À
ffiffiffiffiffiffiffiffiffiffiffiffi ffi
z
2
À 1
p
2
ffiffiffiffiffiffiffiffiffiffiffiffi ffi
z
2
À 1
p
e
ÀzÀ
ffiffiffiffiffiffiffi ffi
z
2 À1
p
À
Á
v n t
"
#
z > 1 ð3:15cÞ
where we set the initial conditions, y(0) ¼ _
y 0
ð Þ ¼ 0 for convenience.
Equations 3.15a–c are plotted in Figure 3.14 for several values of z. The interesting feature is
the transient response. For under-damped systems, the transient response is oscillatory about the
steady value and occurs with a period
T d ¼
2p
v d
¼
1
f d
ð3:16Þ
v d ¼ v n
ffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 À z
2
q
ð3:17Þ
where v d is called the ringing frequency. In instruments, this oscillatory behavior is called
‘‘ringing.’’ The ringing phenomenon and the associated ringing frequency are properties of the
measurement system and are independent of the input signal. It is the free oscillation frequency of a
system displaced from its equilibrium.
The duration of the transient response is controlled by the zv n term. In fact, its influence is
similar to that of a time constant in a first-order system, such that we could define a second-order
time constant as t ¼ 1=zv n . The system settles to KA more quickly when it is designed with a larger
2
4
= 0
0.25
0.50
1.0
2.0
6
8
1 0
Output signal
y(t)
n t
y(0)
KA
Figure 3.14 Second-order system time response to a step function input.
3.3 Special Cases of the General System Model 97
15:24:53 Page 97
Step Function Input
Again, the step function input is applied to determine the general behavior and speed at which the
system will respond to a change in input. The response of a second-order measurement system to a
step function input is found from the solution of Equation 3.13, with F(t) ¼ AU(t), to be
y t
ð Þ ¼ KA À KAe
Àzv n t
z
ffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 À z
2
p
sin v n t
ffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 À z
2
q
þ cos v n t
ffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 À z
2
q
"
#
0 z < 1 ð3:15aÞ
y t
ð Þ ¼ KA À KA 1 þ v n t
ð
Þ e
Àv n t
z ¼ 1 ð3:15bÞ
y t
ð Þ ¼ KA À KA
z þ
ffiffiffiffiffiffiffiffiffiffiffiffi ffi
z
2
À 1
p
2
ffiffiffiffiffiffiffiffiffiffiffiffi ffi
z
2
À 1
p
e
Àzþ
ffiffiffiffiffiffiffi ffi
z
2 À1
p
À
Á
v n t þ
z À
ffiffiffiffiffiffiffiffiffiffiffiffi ffi
z
2
À 1
p
2
ffiffiffiffiffiffiffiffiffiffiffiffi ffi
z
2
À 1
p
e
ÀzÀ
ffiffiffiffiffiffiffi ffi
z
2 À1
p
À
Á
v n t
"
#
z > 1 ð3:15cÞ
where we set the initial conditions, y(0) ¼ _
y 0
ð Þ ¼ 0 for convenience.
Equations 3.15a–c are plotted in Figure 3.14 for several values of z. The interesting feature is
the transient response. For under-damped systems, the transient response is oscillatory about the
steady value and occurs with a period
T d ¼
2p
v d
¼
1
f d
ð3:16Þ
v d ¼ v n
ffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 À z
2
q
ð3:17Þ
where v d is called the ringing frequency. In instruments, this oscillatory behavior is called
‘‘ringing.’’ The ringing phenomenon and the associated ringing frequency are properties of the
measurement system and are independent of the input signal. It is the free oscillation frequency of a
system displaced from its equilibrium.
The duration of the transient response is controlled by the zv n term. In fact, its influence is
similar to that of a time constant in a first-order system, such that we could define a second-order
time constant as t ¼ 1=zv n . The system settles to KA more quickly when it is designed with a larger
2
4
= 0
0.25
0.50
1.0
2.0
6
8
1 0
Output signal
y(t)
n t
y(0)
KA
Figure 3.14 Second-order system time response to a step function input.
3.3 Special Cases of the General System Model 97