Problem 1
a) The transfer function of this process can be expressed as the product of three first order
lag transfer functions. The AR and phase angles of a general 1
st
order lag are:
2
2
K
AR
1
=
τ ω +
and
1
tan
(
)
−
φ =
−τω
(S1.1)
Thus, applying the principle of superposition we get:
2
2
2
3
1
1
AR
64
1
4
1
1
=
ω +
ω +
ω +
(S1.2)
1
1
1
tan
( 8 )
tan
( 2
)
tan
(
)
−
−
−
φ =
− ω +
− ω +
−ω
(S1.3)
b) Asymptotically as w goes to infinity,
the AR is approximated by
3
1
1
AR
8
2
=
ω
ω ω
(S1.4)
while for
ω
going to zero, AR goes to 3.
Thus, the corner frequency will be
obtained by solving the equations
3
1
1
3
0.397
8
2
=
→
ω =
ω
ω ω
(S1.5)
Taking logarithms in the asymptotic
expression for AR, the asymptote slope is
–3.
c)
The
Bode
plots
are
obtained
computationally (i.e. give an array of
values for
ω
and find the corresponding
phase angle and amplitude ratios from the
above formulae). They are shown in
figure 1.
10
-7
10
-6
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
10
1
0.001
0.01
0.1
1
10
100
AR
w
-300
-250
-200
-150
-100
-50
0
0.001
0.01
0.1
1
10
100
φ
w
Problem 1: Bode plots

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