e x a m p l e 13 .5 b o d e p l o t f o r s e r i e s r l c i r c u i t Let
us sketch the Bode plot for the RL circuit of Figure 13.11. From Equation 13.47, the
system function here is a voltage ratio:
H( jω) =
Vo
=
Vi
R/L
jω + R/L
.
(13.97)
To make the example specific, let us assume that the time constant L/R has a value of
50 msec. Thus the break frequency is at
a=
R
L
= 20 rad/s
and the system function becomes
H( jω) =
20
jω + 20
.
(13.98)
The system function has two terms: a constant term and a term of the form (s + a).
Figures 13.26 and 13.27 show the construction of the magnitude and phase plots,
respectively. The dashed lines in Figures 13.26c and 13.27c form the composite Bode
plot, and are obtained by simple subtraction of Figures 13.26b from Figure 13.27a. For
reference, the solid lines show the true magnitude and phase functions. Note that at the
break frequency, the true magnitude is given by:
|H( jω)| = 1/1.41
(13.99)
= 0.707.
The principal advantage of the Bode plot is that the composite magnitude asymptotes
for system functions that can be written in the form of Equation 13.94 are always
lines of integer slope in log space. Further, any system function that can be written as
a ratio of polynomials in ω (regardless of whether the roots are real or complex) must
approach at both low and high frequencies ( jω)n , where n is some integer. Hence the
magnitude asymptotes on Bode plots for both small and large ω must be straight lines
of integer slope in log space, and the phase must approach a multiple of 90◦ .
742a
3
100.00
2
10.00
1
1.00
0
0.10
-1
log |R/L
|R/L| log scale
1000.00
0.01
10-1
100
101
102
103
ω
1000.00
3
100.00
2
10.00
1
1.00
0
0.10
-1
log |jω + R/
|jω + R/L| log scale
(a)
0.01 -1
10
F I G U R E 13.26 Magnitude curve
of the Bode plot for RL circuit:
(a) the magnitude curve for R/L;
(b) the magnitude curve for
jω + R/L; (c) the composite
magnitude curve obtained by
subtracting (b) from (a).
100
101
102
103
|
|||| | | | |
100.00
|
|||| | | | |
10.00
|
|||| | | | |
1.00
|
|||| | | | |
0.10
3
ω
log |H|
1000.00
|||| | | | |
|H| log scale
(b)
2
1
0
-1
|
|
0.01 |
10-1
|
|
| | | | | ||
|
|
| | | | | ||
100
101
(c)
|
|
| | | | | ||
102
|
|
| | | | | ||
103
ω
742b
<R/L
<R/L + jω
<H
F I G U R E 13.27 Phase curve of
the Bode plot for RL circuit: (a) the
phase curve curve for R/L; (b) the
phase curve for jω + R/L; (c) the
composite phase curve obtained by
subtracting (b) from (a).
742c
90
70
50
30
10
-10 -1
10
-30
-50
-70
-90
90
70
50
30
10
-10 -1
10
-30
-50
-70
-90
90
70
50
30
10 vv
-10 -1
10
-30
-50
-70
-90
100
101
102
103
ω
(a)
100
101
102
103
ω
(b)
100
101
102
103
ω
(c)