E1C03 09/14/2010
15:24:54 Page 110
device, y s (t), will be a second-order response appropriate for input F 2 (t),
y s t
ð Þ ¼ y h t
ð Þ þ
K t K s Asin vt þ F t þ F s
½
1 þ vt
ð Þ
2
h
i 1=2
1 À v=v n
ð
Þ
2
h
i 2 þ 2zv=v n
½
2
&
' 1=2
F s ¼ Àtan
À1
2zv=v n
1 À v=v n
ð
Þ
2
ð3:38Þ
where subscript s refers to the chart recorder and y h (t) is the transient response. The output signal
displayed on the recorder, y s (t), is the measurement system response to the original input signal to
the transducer, F 1 (t) ¼ A sin vt. The steady amplitude of the recorder output signal is the product
of the static sensitivities and magnitude ratios of the first- and second-order systems. The phase shift
is the sum of the phase shifts of the two systems.
Based on the concept behind Equation 3.38 we can make a general observation. Consider the
schematic representation in Figure 3.24, which depicts a measurement system consisting of H
interconnected devices, j ¼ 1, 2, . . . , H, each device described by a linear system model. The
overall transfer function of the combined system, G(s), is the product of the transfer functions of
each of the individual devices, G j (s), such that
KG s
ð Þ ¼ K 1 G 1 s
ð ÞK 2 G 2 s
ð Þ . . . K H G H s
ð Þ
ð3:39Þ
At s ¼ iv, Equation 3.37 becomes
KG iv
ð Þ ¼ K 1 K 2 . . . K H
ð
ÞÂM 1 v
ð ÞM 2 v
ð Þ . . . M H v
ð Þ
½
e
i F 1 v
ð ÞþF 2 v
ð ÞþÁÁÁþF H v
ð Þ
½
ð3:40Þ
According to Equation 3.40, given an input signal to device 1, the system steady output signal at
device H will be described by the system frequency response G iv
ð Þ ¼ M v
ð Þe
iF v
ð Þ , with an overall
system static sensitivity described by
K ¼ K 1 K 2 . . . K H
ð3:41Þ
K 1 G 1 (s)
F(t)
K 2 G 2 (s)
y 1 (t)
y 2 (t)
y 1 (0), ...
KG(s) = K 1 G 1 (s) K 2 G 2 (s) ... K H G H (s)
F(t)
y(t)
y(0), ...
y 2 (0), ...
K j G j (s)
y 1 (t)
y j (0), ...
K H G H (s)
y(t)
y H (0), ...
(a) H-coupled transfer functions
(b) Equivalent system transfer function
Figure 3.24 Coupled systems: describing the system transfer function.
110 Chapter 3 Measurement System Behavior
15:24:54 Page 110
device, y s (t), will be a second-order response appropriate for input F 2 (t),
y s t
ð Þ ¼ y h t
ð Þ þ
K t K s Asin vt þ F t þ F s
½
1 þ vt
ð Þ
2
h
i 1=2
1 À v=v n
ð
Þ
2
h
i 2 þ 2zv=v n
½
2
&
' 1=2
F s ¼ Àtan
À1
2zv=v n
1 À v=v n
ð
Þ
2
ð3:38Þ
where subscript s refers to the chart recorder and y h (t) is the transient response. The output signal
displayed on the recorder, y s (t), is the measurement system response to the original input signal to
the transducer, F 1 (t) ¼ A sin vt. The steady amplitude of the recorder output signal is the product
of the static sensitivities and magnitude ratios of the first- and second-order systems. The phase shift
is the sum of the phase shifts of the two systems.
Based on the concept behind Equation 3.38 we can make a general observation. Consider the
schematic representation in Figure 3.24, which depicts a measurement system consisting of H
interconnected devices, j ¼ 1, 2, . . . , H, each device described by a linear system model. The
overall transfer function of the combined system, G(s), is the product of the transfer functions of
each of the individual devices, G j (s), such that
KG s
ð Þ ¼ K 1 G 1 s
ð ÞK 2 G 2 s
ð Þ . . . K H G H s
ð Þ
ð3:39Þ
At s ¼ iv, Equation 3.37 becomes
KG iv
ð Þ ¼ K 1 K 2 . . . K H
ð
ÞÂM 1 v
ð ÞM 2 v
ð Þ . . . M H v
ð Þ
½
e
i F 1 v
ð ÞþF 2 v
ð ÞþÁÁÁþF H v
ð Þ
½
ð3:40Þ
According to Equation 3.40, given an input signal to device 1, the system steady output signal at
device H will be described by the system frequency response G iv
ð Þ ¼ M v
ð Þe
iF v
ð Þ , with an overall
system static sensitivity described by
K ¼ K 1 K 2 . . . K H
ð3:41Þ
K 1 G 1 (s)
F(t)
K 2 G 2 (s)
y 1 (t)
y 2 (t)
y 1 (0), ...
KG(s) = K 1 G 1 (s) K 2 G 2 (s) ... K H G H (s)
F(t)
y(t)
y(0), ...
y 2 (0), ...
K j G j (s)
y 1 (t)
y j (0), ...
K H G H (s)
y(t)
y H (0), ...
(a) H-coupled transfer functions
(b) Equivalent system transfer function
Figure 3.24 Coupled systems: describing the system transfer function.
110 Chapter 3 Measurement System Behavior