Biomedical Engineering Reference
In-Depth Information
tan
1
(ot
p
) is drawn with a straight line from 0
at
Phase:
an asymptotic approximation to
1
t
p
1
t
p
90
at 1 decade above o
one decade below o
¼
to
¼
.
1
t
p
The pole is located at
.
Zero on the real axis
8
<
:
1
t
z
0
for o
<
Gain:
20 log 1
j
þ
j
ot
z
j ¼
1
t
z
20 log otðÞ
for o
an asymptotic approximation to tan
1
(ot
z
) is drawn with a straight line from 0
at
Phase:
1
t
z
1
t
z
90
at 1 decade above o
one decade below o
¼
to
þ
¼
.
1
t
z
The zero is located at
.
Complex poles
<
0
for o
<
o
n
r
¼
0
1
j
2
2z
r
o
n
r
o
o
n
r
j
o
o
n
r
Gain:
20 log 1
þ
o
þ
@
A
40 log
for o
o
n
r
:
A graph of the actual magnitude-frequency is shown in Figure 13.61, with o
n
¼
1.0 and z
ranging from 0.05 to 1.0. Notice that as z decreases from 1.0, the magnitude peaks at
correspondingly larger values. As z approaches zero, the magnitude approaches infinity
at o
¼
o
n
r
. For values of z
>
0.707 there is no resonance.
Phase:
depending on the value of z
r
, the shape of the curve is quite variable but in general
is 0
at one decade below o
180
at 1 decade above o
o
n
r
.
A graph of the actual phase-frequency is shown in Figure 13.61 with o
n
¼
¼
o
n
r
and
¼
1.0 and z
ranging from 0.05 to 1.0. Notice that as z decreases from 1.0, the phase changes more
quickly from 0
to 180
over a smaller frequency interval.
q
1
z
2
r
The poles are located at
z
r
o
n
r
j
o
.
n
r
Complex zeros
8
<
¼
0
for o
<
o
n
s
j
2
40 log
o
o
n
s
2z
s
o
n
s
o
o
n
s
j
Gain:
20 log 1
þ
o
þ
for o
o
n
s
:
Phase:
depending on the value of z
s
, the shape of the curve is quite variable but in general
is 0
at one decade below o
180
at 1 decade above o
o
n
s
.
Both the magnitude and phase follow the two previous graphs and discussion with
regard to the complex poles with the exception that the slope is
¼
o
n
s
and
þ
¼
þ
40 dB/decade rather
than -40 dB/decade.
The zeros are located at
q
1
z
2
s
z
o
n
s
j
o
.
s
n
s
