Digital Signal Processing Reference
In-Depth Information
K
1
K
2
+
+
f
1
(
t
)
input
phase
v
(
t
)
output
voltage
G
(
s
)
−
loop filter
q
(
t
)
t
∫
d
t
−∞
integrator
Fig. P8.11. Block diagram
representation of a phase-locked
loop.
8.7
By integrating the impulse response
h
(
t
) of the armature-controlled dc
motor, derive Eq. (8.42) for
ξ
n
=
1.
8.8
Assume that the inductance
L
a
of the induction motor, shown in Fig. 8.8(b),
is zero. Determine the transfer function
H
(
s
) and impulse response
h
(
t
)
for the modified model. Based on the location of the poles, comment on
the stability of the induction motor.
8.9
Repeat Problem 8.7 for Eq. (8.43) with
ξ
n
>
1 and Eq. (8.44) with
ξ
n
<
1.
8.10
Based on Eqs. (8.53)-(8.56), derive the expression for the transfer function
H
(
s
) of the human immune system shown in Eq. (8.57).
8.11
In order to achieve synchronization between the modulating and demod-
ulating carriers, a special circuit referred to as a phase-locked loop (PLL)
is commonly used in communications. The block diagram representing
the PLL is shown in Fig. P8.11.
Show that the transfer function of the PLL is given by
V
(
s
)
φ
(
s
)
K
1
K
2
sG
(
s
)
s
+
K
1
G
(
s
)
,
where
K
1
and
K
2
are gain constants and
G
(
s
) is the transfer function of
a loop filter. Specify the condition under which the PLL acts as an ideal
differentiator. In other words, derive the expression for
G
(
s
) when the
transfer function of the PLL equals
Ks
, with
K
being a constant.
=
8.12
Repeat the simulink simulation for the human immune system for the
following values of the proportionality constants:
α
=
0
.
3
,β=
0
.
1
,γ=
0
.
25
,=
0
.
6
,τ=
1
,λ=
0
.
1
,
σ
=
0.4,
and
η =
0
.
2
Sketch the time evolution of the antigens, plasma cells, and antibodies.
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