Solution 14.57
(a) Consider the circuit below.
Z
V
I
s
=
1/sC
R
+
−
Vs
+
Vo
−
1/sC
R
I1
I
(b) Similarly,
sLR2
)sLR(R
sL)sLR(||RsL)s( +
+
+=++=Z
Solution 14.58
The circuit parameters for a series RLC bandstop filter are R = 250 Ω, L = 1 mH,
C = 40 pF. Calculate:
(a) the center frequency
(b) the half-power frequencies
(c) the quality factor.
Solution
(b)
6
1025.0
001.0
250 ×=== L
R
B
rad/s
Solution 14.59
Consider the circuit below.
sC1sLR
)sC1sL(R
sC
1
sL||R)s( ++
+
=
+=Z
ω= js
0)Im( in =Z
implies that
R
sL
1/sC
Ro
+
−
Vi
+
Vo
−
LCRLCRRCRRjR
)LC1(R
2
o
2
oo
2
ω−ω−+ω+
ω−
=H
At
1
ω
and
2
ω
,
mzx
H
2
1
=H
CRRj)RR(LCRR
)LC1(R
)RR(2
R
oo
2
o
2
o
ω++ω−+
ω
−
=
+
Solution 14.60
Obtain the transfer function of a highpass filter with a passband gain of 100 and a cutoff
frequency of 40 rad/s.
Solution
j
RCj
ω
ω
Solution 14.61
(a)
i
Cj1R
Cj1 VV ω+
ω
=
+
,
o
VV =
−
(b)
i
Cj1R
RVV ω+
=
+
,
o
VV =
−
Solution 14.62
This is a highpass filter.
RCj1
1
RCj1
RCj
)( ω−
=
ω+
ω
=ωH
(a)
i
o
5j1
1
)Hz200f( V
V
H=
−
==
Solution 14.63
For an active highpass filter,
fi
RsC
Comparing (1) and (2) leads to:
Solution 14.64
ff
f
f
ff CRj1
R
Cj
1
||RZ ω+
=
ω
=
Solution 14.65
ii
RCj1
RCj
Cj1R
RVVV ω+
ω
=
ω+
=
+
Solution 14.66
Solution 14.67
fi
f
R4R
4
1
R
R
gain DC =→==
Solution 14.68
Design a problem to help other students to better understand the design of active highpass
filters when specifying a high-frequency gain and a corner frequency.
Although there are many ways to solve this problem, this is an example based on the
same kind of problem asked in the third edition.
Problem
Design an active highpass filter with a high-frequency gain of 5 and a corner frequency
of 200 Hz.
Solution
if
i
f
R5R
R
R
5gainfrequency High =→==
Solution 14.69
This is a highpass filter with
2f
c
=
kHz.
1
f2 cc =π=ω
Solution 14.70
(a)
)YYY(YYY
YY
)s(
)s(
)s(
321421
21
i
o
+++
== V
V
H
Solution 14.71
Ω= 50R
,
mH40L =
,
F1C µ=
)1040(
K
K
1L
K
K
L
3–
f
m
f
m
×⋅=→=
′
Substituting (1) into (2),
Solution 14.72
Design a problem to help other students to better understand magnitude and frequency
scaling.
Although there are many ways to solve this problem, this is an example based on the
same kind of problem asked in the third edition.
Problem
What values of Km and Kf will scale a 4-mH inductor and a 20-
µ
F capacitor to 1 H and
2 F respectively?
Solution
CL
LC
K
K
LC
CL
2
f
2
f
′′
=→=
′′
Solution 14.73
=×==
′)10800)(12(RKR 3
m
ΩM6.9