E1C03 09/14/2010
15:24:54 Page 105
system to the input signal. The second term describes its transient response. Included in both terms,
the transfer function G(s) plays a role in the complete time response of the measurement system. As
indicated in Figure 3.19, the transfer function defines the mathematical operation that the
measurement system performs on the input signal F(t) to yield the time response (output signal)
of the system. As a review, Appendix C provides common Laplace transforms with their application
to the solution of ordinary differential equations.
The system frequency response, which has been shown to be given by M(v) and F(v), can be
found by finding the value of G(s) at s ¼ iv. This yields the complex number
G s ¼ iv
ð
Þ¼G iv
ð Þ ¼
1
tiv þ 1
¼ M v
ð Þe
iF v
ð Þ
ð3:26Þ
where G(iv) is a vector on the real–imaginary plane having a magnitude, M(v), and inclined at an
angle, F(v), relative to the real axis as indicated in Figure 3.20. For the first-order system, the
magnitude of G(iv) is simply that given by M(v) from Equation 3.10 and the phase shift angle by
F(v) from Equation 3.9.
For a second-order or higher system, the approach is the same. The governing equation for a
second-order system is defined by Equation 3.13, with initial conditions y(0) ¼ y 0 and _
y 0
ð Þ ¼ _
y 0 .
The Laplace transform yields
Y s
ð Þ ¼
1
1=v 2
n
À
Á s 2 þ 2z=v n
ð
Þs þ 1
KF s
ð Þ þ
s_ y 0 þ y 0
1=v 2
n
À
Á
s 2 þ 2z=v n
ð
Þs þ 1
ð3:27Þ
which can again be represented by
Y s
ð Þ ¼ G s
ð ÞKF s
ð Þ þ G s
ð ÞQ s
ð Þ
ð3:28Þ
By inspection, the transfer function is given by
G s
ð Þ ¼
1
1=v 2
n
À
Á s 2 þ 2z=v n
ð
Þs þ 1
ð3:29Þ
Solving for G(s) at s ¼ iv, we obtain for a second-order system
G s ¼ iv
ð
Þ¼
1
iv
ð Þ
2 =v 2
n þ 2ziv=v n þ 1
¼ M v
ð Þe
iF v
ð Þ
ð3:30Þ
which gives exactly the same magnitude ratio and phase shift relations as given by Equations 3.20
and 3.22.
Re[G(i )]
Im[G(i )]
Real
Imaginary
M( )
( )
Figure 3.20 Complex plane approach to describing
frequency response.
3.4 Transfer Functions 105
15:24:54 Page 105
system to the input signal. The second term describes its transient response. Included in both terms,
the transfer function G(s) plays a role in the complete time response of the measurement system. As
indicated in Figure 3.19, the transfer function defines the mathematical operation that the
measurement system performs on the input signal F(t) to yield the time response (output signal)
of the system. As a review, Appendix C provides common Laplace transforms with their application
to the solution of ordinary differential equations.
The system frequency response, which has been shown to be given by M(v) and F(v), can be
found by finding the value of G(s) at s ¼ iv. This yields the complex number
G s ¼ iv
ð
Þ¼G iv
ð Þ ¼
1
tiv þ 1
¼ M v
ð Þe
iF v
ð Þ
ð3:26Þ
where G(iv) is a vector on the real–imaginary plane having a magnitude, M(v), and inclined at an
angle, F(v), relative to the real axis as indicated in Figure 3.20. For the first-order system, the
magnitude of G(iv) is simply that given by M(v) from Equation 3.10 and the phase shift angle by
F(v) from Equation 3.9.
For a second-order or higher system, the approach is the same. The governing equation for a
second-order system is defined by Equation 3.13, with initial conditions y(0) ¼ y 0 and _
y 0
ð Þ ¼ _
y 0 .
The Laplace transform yields
Y s
ð Þ ¼
1
1=v 2
n
À
Á s 2 þ 2z=v n
ð
Þs þ 1
KF s
ð Þ þ
s_ y 0 þ y 0
1=v 2
n
À
Á
s 2 þ 2z=v n
ð
Þs þ 1
ð3:27Þ
which can again be represented by
Y s
ð Þ ¼ G s
ð ÞKF s
ð Þ þ G s
ð ÞQ s
ð Þ
ð3:28Þ
By inspection, the transfer function is given by
G s
ð Þ ¼
1
1=v 2
n
À
Á s 2 þ 2z=v n
ð
Þs þ 1
ð3:29Þ
Solving for G(s) at s ¼ iv, we obtain for a second-order system
G s ¼ iv
ð
Þ¼
1
iv
ð Þ
2 =v 2
n þ 2ziv=v n þ 1
¼ M v
ð Þe
iF v
ð Þ
ð3:30Þ
which gives exactly the same magnitude ratio and phase shift relations as given by Equations 3.20
and 3.22.
Re[G(i )]
Im[G(i )]
Real
Imaginary
M( )
( )
Figure 3.20 Complex plane approach to describing
frequency response.
3.4 Transfer Functions 105