Biomedical Engineering Reference
In-Depth Information
where
T
d
are all positive integers. Incorporating the preceding terms, the
transfer function is written as
M, P, Z, R, S,
and
!
!
j
2
K
e
j
o
T
d
Q
z
¼1
1
Q
s
¼1
2z
s
o
n
s
o
o
n
s
j
ð
þ
j
ot
z
Þ
1
þ
o
þ
!
!
Gj
ðÞ¼
o
ð
13
:
67
Þ
j
2
ð
M
Q
p
¼1
Q
r
¼1
2z
r
o
n
r
o
o
n
r
j
j
o
1
þ
j
ot
p
1
þ
o
þ
This equation is used as a template when describing the data with a model. To determine
the value of the unknown parameters in the model, the logarithm and asymptotic approx-
imations to the transfer function are used. In general, the logarithmic gain, in dB, of the
transfer function template is
0
@
1
A
j
0
@
1
A
2
20
Z
z
¼
20
X
S
s
¼
2z
s
o
n
s
o
o
n
s
j
20 log
j
Gj
ðÞ
o
j ¼
20 log
K
þ
log 1
j
þ
j
ot
z
j þ
log 1
þ
o
þ
1
1
0
@
1
A
j
0
@
1
A
2
20
X
P
p
¼
20
X
R
r
¼
2z
r
o
n
r
o
o
n
r
j
ð
M
20 log
j
o
log 1
þ
j
ot
p
log 1
þ
o
þ
1
1
ð
13
:
68
Þ
and the phase, in degrees, is
0
1
X
Z
X
S
2z
s
o
n
s
o
o
2
@
A
M
tan
1
otðÞþ
tan
1
90ðÞ
f
ð
o
Þ¼
o
T
d
þ
n
s
o
2
z
¼
1
s
¼
1
0
1
ð
13
:
69
Þ
X
P
X
R
2z
o
n
r
o
@
r
A
tan
1
ot
p
tan
1
o
2
n
r
o
2
p
¼
1
r
¼
1
where the phase angle of the constant is 0
and the magnitude of the time delay is 1.
Evident from these expressions is that each term can be considered separately and added
together to obtain the complete Bode diagram. The asymptotic approximations to the loga-
rithmic gain for the poles and zeros are given by the following:
Poles at the origin
Gain:
20 log
j
ðÞ
j
o
j ¼
20 log o. The logarithmic gain at o
¼
1 is 0 (i.e., the line passes
through 0 dB at o
¼
1 radian/s).
f
¼
90
. If there is more than one pole, the slope of the gain changes by
Phase:
M
(
20) and the phase by M
(
90
).
Pole on the real axis
8
<
1
t
p
0
for o
<
¼
Gain:
20 log 1
þ
j
ot
p
for o
1
t
p
:
20 log ot
p
