Biomedical Engineering Reference
In-Depth Information
The experiment to identify model parameters using sinusoidal analysis is simple to carry
out. The input is varied at discrete frequencies over the entire spectrum of interest, and
the output magnitude and phase are recorded for each input. To illustrate this technique,
consider the following analysis. From Figure 13.60, it is clear that the transfer function is
given by
Þ¼
V
o
ð
j
o
Þ
H
ð
j
o
ð
13
:
61
Þ
V
i
ð
j
o
Þ
The Fourier transform of the input is
cos o
x
t
Z
Z
y
o
x
þ
t
j
o
e
j
ol
cos o
x
l
V
i
j
ðÞ¼
FA
o
f
cos o
x
t
þ
ð
y
Þ
g ¼
A
ð
þ
y
Þ
d
l
¼
A
e
ð
Þ
d
t
ð
13
:
62
Þ
y
o
x
by substituting l
¼
t
. Factoring out the terms not involving t, gives
Z
j
oy
o
x
j
oy
o
x
pd o
e
j
ot
cos o
x
t
ðÞ¼
Ae
o
t
¼
Ae
½
ð
o
x
Þþ
pd o
ð
þ
o
x
Þ
ð
13
:
63
Þ
V
i
j
d
Similarly,
j
of
o
x
V
o
j
ðÞ¼
Be
o
½
pd o
ð
o
x
Þþ
pd o
ð
þ
o
x
Þ
ð
13
:
64
Þ
According to Eq. (13.61)
j
of
o
x
j
ofy
ð
Þ
ðÞ¼
V
o
ðÞ
¼
B
ðÞ
j
o
e
¼
B
Hj
o
A
e
o
x
ð
13
:
65
Þ
V
i
j
o
A
j
oy
o
x
e
At steady state with o
¼
o
x
, Eq. (13.65) reduces to
ðÞ¼
B
A
e
j
fy
ð
Þ
Hj
o
x
ð
13
:
66
Þ
Each of these quantities in Eq. (13.66) is known (i.e.,
f,andy), so the magnitude
and phase angle of the transfer function are also known. Thus, o
x
can be varied over the
frequency range of interest to determine the transfer function.
In general, a transfer function,
B, A,
G
ð
s
Þj
s
¼
j
o
¼
G
ð
j
o
Þ
, is composed of the following terms:
1.
Constant term
K
ð
M
poles or zeros at the origin of the form
o
2.
M
j
or Z zeros of the form
Q
z
¼1
1
poles of the form
Q
p
¼1
3.
P
1
þ
j
ot
p
ð
þ
j
ot
z
Þ
. Naturally, the
1
t
.
4.
R complex poles of the form
Q
r
¼1
poles or zeros are located at
or S zeros of the form
j
2
2z
r
o
n
r
j
o
o
n
r
1
þ
o
þ
j
2
Q
s
¼1
2z
s
o
n
s
j
o
o
n
s
1
þ
o
þ
e
j
o
T
d
5.
Pure time delay
