so that the phase perturbation is
( ) = 2 tan 1
1
!
tan !
;
;
!
10 ;
5
which is plotted in gure 9.2. The maximum phase shift of 38 9 corresponds to
;
:
about 6dB of gain variation (see the discussion in section 9.1.2).
0
;5
;10
;15
;20
(degrees)
;25
;30
;35
;40
0:1
1
10
100
!
A specic phase shift in std.
Figure
9.2
P
0
Figure 9.3 shows the nominal magnitude of , the actual perturbed magnitude
T
caused by the phase shift in std
0 , and the perturbed magnitude predicted by the
P
rst order analysis,
approx( ) = ( ) exp
( )
1
(9.16)
jT
j
!
j
jT
j
!
j
;
!
=
1 + std
0 ( ) (a)( ) :
P
j
!
K
j
!
9.1.5
Other Log Sensitivities
We have seen that the logarithmic sensitivity of the I/O transfer function is given
T
by another closed-loop transfer function, the sensitivity . Several other important
S
closed-loop transfer functions have logarithmic sensitivities that are also closed-loop
transfer functions. Table 9.1 lists some of these.
From the top line of this table we see that a speci cation such as
log ( )
@
S
j
!
log 0( ) 2 for
bw
(9.17)
!
!
@
P
j
!
9.1 BODE’S LOG SENSITIVITIES
201
10
5
0
;5
dB ;10
pert( )
jT
j
!
j
;15
approx( )
jT
j
!
j
;20
( )
jT
j
!
j
;25
;30
0:1
1
10
!
When the phase factor (10 ) (10 + ) in std
0 is replaced
Figure
9.3
;
s
=
s
P
by (5 ) (5+ ), the magnitude of the I/O transfer function changes from
;
s
=
s
to pert . approx is a rst order approximation of pert computed
jT
j
jT
j
jT
j
jT
j
from (9.16).
log
@
H
H
log 0
@
P
1
0
;P
K
1 + 0
1 + 0
P
K
P
K
0
K
;P
K
1 + 0
1 + 0
P
K
P
K
0
1
P
1 + 0
1 + 0
P
K
P
K
0
1
P
K
1 + 0
1 + 0
P
K
P
K
The logarithmic sensitivity of some important closed-loop trans-
T
able
9.1
fer functions are also closed-loop transfer functions. In the general case,
however, the logarithmic sensitivity of a closed-loop transfer function need
not be another closed-loop transfer function.
202