Biomedical Engineering Reference
In-Depth Information
Time delay
Gain:
1 for all o
Phase:
o
T
d
Constant
K
Gain:
20 log
K
Phase:
0
The frequency at which the slope changes in a Bode magnitude-frequency plot is called a
break or corner frequency. The first step in estimating the parameters of a model involves
identifying the break frequencies in the magnitude-frequency and/or phase-frequency
responses. This simply involves identifying points at which the magnitude changes slope
in the Bode plot. Poles or zeros at the origin have a constant slope of -20 or
þ
20 dB/decade,
respectively, from
1
to
1
. Real poles or zeros have a change in slope at the break
frequency of -20 or
20 dB/decade, respectively. The value of the pole or zero is the break
frequency. Estimating complex poles or zeros is much more difficult. The first step is
to locate the break frequency o
n
þ
, the point at which the slope changes by 40 dB/decade.
To estimate z, use the actual magnitude-frequency (size of the peak) and phase-frequency
(quickness of changing 180
) curves in Figure 13.61 to closely match the data.
The error between the actual logarithmic gain and straight-line asymptotes at the break
frequency is 3dB for a pole on the real axis (the exact curve equals the asymptote
3dB).
The error drops to 0.3dB one decade below and above the break frequency. The error
between the real zero and the asymptote is similar except the exact curve equals the asymp-
tote
3dB. At the break frequency for the complex poles or zeros, the error between the
actual logarithmic gain and straight-line asymptotes depends on z and can be quite large
as observed from Figure 13.61.
þ
EXAMPLE PROBLEM 13.9
Sinusoids of varying frequencies were applied to an open-loop system, and the results in
Table 13.3 were measured. Construct a Bode diagram and estimate the transfer function.
TABLE 13.3
Data for Example Problem 14.9
Frequency
(radians/s)
0.01
0.02
0.05
0.11
0.24
0.53
1.17
2.6
5.7
12.7
28.1
62
137
304
453
672
1000
20 log
j
G
j
58
51
44
37
30
23
15
6
7
20
34
48
61
75
82
89
96
(dB)
Phase
(degrees)
90
91
91
93
97
105
120
142
161
171
176
178
179
180
180
180
180
Solution
Bode plots of gain and phase versus frequency for the data given are shown in Figure 13.62.
From the phase-frequency graph, it is clear that this system has two more poles than zeros
because the phase angle approaches
. Also note that there is no peaking observed
in the gain-frequency graph or sharp changes in the phase-frequency graph, so there does not
appear to be any lightly damped (z <
180
as
o!1
0.5) complex poles. However, this does not imply that there
are no heavily damped complex poles at this time.
