User manual

page 36
Analog System Lab Kit PRO
experiment 5
The goal of this experiment is to learn the concept of tuning a lter. The idea
is to adjust the RC time constants of the lter so that in phase response of
a lowpass lter, the output phase w.r.t. input is exactly 90
ø
at the incoming
frequency. This principle is utilized in distortion analyzers and spectrum
analyzers, such self tuned lters are used to lock on to the fundamental
frequency and harmonics of the input.
Inordertodesignself-tunedltersand
other analog systems in subsequent
experiments, we need to introduce
one more building block, the Analog
multiplier. The reader will benet
from viewing the recorded lecture at
[21]. In ASLK PRO, we have used to
MPY634 analog multiplier from Texas
Instruments. Refer to Figure 5.1, which shows the symbol of an analog multiplier.
5.1 Brief theory and motivation
5.1.1 Multiplier as a Phase Detector
Goal of the experiment
2
90
0
1
cos
tan
RC
V V K V K V K V V
V
V
V V
V V
V
V VVV
V
V
VV
K
d
dV
K
V
VRC
V
dV
d
VRC
V
dV
d
dV
d
d
d
dV
d
V
V
Q
s s
H
Q
d
d
Q
dV
d
QV
V
kHz
Q
H QV
1
1
1
2
2
0
0
2
2
0
2
0
0
offset x x y y x y
x
y
r x
y r
r
x yr
r
p p
pd
av
pd
av
r
c
c
r
c
c
c c
i
r
r
c
c
av
i
0 0
0
0
0
0 0
0
0
0
0
1
0
0
0
# # # #
#
$
$
$ $
p
z
z
~
~
~ ~
z
z
~
z
~
~
~
z
~
~
~
~
~
z
~
z
#
#
= + + + +
=
=
=
=
=
= =
=
=
=
=
=
=
p
z
+ +
+
-
-
-
-
l
c
b
b
d
b
l
l
n
l
2
90
0
1
cos
tan
RC
V V K V K V K V V
V
V
V V
V V
V
V VVV
V
V
VV
K
d
dV
K
V
VRC
V
dV
d
VRC
V
dV
d
dV
d
d
d
dV
d
V
V
Q
s s
H
Q
d
d
Q
dV
d
QV
V
kHz
Q
H QV
1
1
1
2
2
0
0
2
2
0
2
0
0
offset x x y y x y
x
y
r x
y r
r
x yr
r
p p
pd
av
pd
av
r
c
c
r
c
c
c c
i
r
r
c
c
av
i
0 0
0
0
0
0 0
0
0
0
0
1
0
0
0
# # # #
#
$
$
$ $
p
z
z
~
~
~ ~
z
z
~
z
~
~
~
z
~
~
~
~
~
z
~
z
#
#
= + + + +
=
=
=
=
=
= =
=
=
=
=
=
=
p
z
+ +
+
-
-
-
-
l
c
b
b
d
b
l
l
n
l
2
90
0
1
cos
tan
RC
V V K V K V K V V
V
V
V V
V V
V
V VVV
V
V
VV
K
d
dV
K
V
VRC
V
dV
d
VRC
V
dV
d
dV
d
d
d
dV
d
V
V
Q
s s
H
Q
d
d
Q
dV
d
QV
V
kHz
Q
H QV
1
1
1
2
2
0
0
2
2
0
2
0
0
offset x x y y x y
x
y
r x
y r
r
x yr
r
p p
pd
av
pd
av
r
c
c
r
c
c
c c
i
r
r
c
c
av
i
0 0
0
0
0
0 0
0
0
0
0
1
0
0
0
# # # #
#
$
$
$ $
p
z
z
~
~
~ ~
z
z
~
z
~
~
~
z
~
~
~
~
~
z
~
z
#
#
= + + + +
=
=
=
=
=
= =
=
=
=
=
=
=
p
z
+ +
+
-
-
-
-
l
c
b
b
d
b
l
l
n
l
2
90
0
1
cos
tan
RC
V V K V K V K V V
V
V
V V
V V
V
V VVV
V
V
VV
K
d
dV
K
V
VRC
V
dV
d
VRC
V
dV
d
dV
d
d
d
dV
d
V
V
Q
s s
H
Q
d
d
Q
dV
d
QV
V
kHz
Q
H QV
1
1
1
2
2
0
0
2
2
0
2
0
0
offset x x y y x y
x
y
r x
y r
r
x yr
r
p p
pd
av
pd
av
r
c
c
r
c
c
c c
i
r
r
c
c
av
i
0 0
0
0
0
0 0
0
0
0
0
1
0
0
0
# # # #
#
$
$
$ $
p
z
z
~
~
~ ~
z
z
~
z
~
~
~
z
~
~
~
~
~
z
~
z
#
#
= + + + +
=
=
=
=
=
= =
=
=
=
=
=
=
p
z
+ +
+
-
-
-
-
l
c
b
b
d
b
l
l
n
l
2
90
0
1
cos
tan
RC
V V K V K V K V V
V
V
V V
V V
V
V VVV
V
V
VV
K
d
dV
K
V
VRC
V
dV
d
VRC
V
dV
d
dV
d
d
d
dV
d
V
V
Q
s s
H
Q
d
d
Q
dV
d
QV
V
kHz
Q
H QV
1
1
1
2
2
0
0
2
2
0
2
0
0
offset x x y y x y
x
y
r x
y r
r
x yr
r
p p
pd
av
pd
av
r
c
c
r
c
c
c c
i
r
r
c
c
av
i
0 0
0
0
0
0 0
0
0
0
0
1
0
0
0
# # # #
#
$
$
$ $
p
z
z
~
~
~ ~
z
z
~
z
~
~
~
z
~
~
~
~
~
z
~
z
#
#
= + + + +
=
=
=
=
=
= =
=
=
=
=
=
=
p
z
+ +
+
-
-
-
-
l
c
b
b
d
b
l
l
n
l
2
90
0
1
cos
tan
RC
V V K V K V K V V
V
V
V V
V V
V
V VVV
V
V
VV
K
d
dV
K
V
VRC
V
dV
d
VRC
V
dV
d
dV
d
d
d
dV
d
V
V
Q
s s
H
Q
d
d
Q
dV
d
QV
V
kHz
Q
H QV
1
1
1
2
2
0
0
2
2
0
2
0
0
offset x x y y x y
x
y
r x
y r
r
x yr
r
p p
pd
av
pd
av
r
c
c
r
c
c
c c
i
r
r
c
c
av
i
0 0
0
0
0
0 0
0
0
0
0
1
0
0
0
# # # #
#
$
$
$ $
p
z
z
~
~
~ ~
z
z
~
z
~
~
~
z
~
~
~
~
~
z
~
z
#
#
= + + + +
=
=
=
=
=
= =
=
=
=
=
=
=
p
z
+ +
+
-
-
-
-
l
c
b
b
d
b
l
l
n
l
(5.1)
(5.2)
(5.3)
(5.4)
Figure 5.1: Analog Multiplier
where
2
90
0
1
cos
tan
RC
VV KVKVKVV
V
V
VV
VV
V
VVVV
V
V
VV
K
d
dV
K
V
VRC
V
dV
d
VRC
V
dV
d
dV
d
d
d
dV
d
V
V
Q
ss
H
Q
d
d
Q
dV
d
QV
V
kHz
Q
HQV
1
1
1
2
2
0
0
2
2
0
2
0
0
offset xxyy xy
x
y
rx
yr
r
xyr
r
pp
pd
av
pd
av
r
c
c
r
c
c
cc
i
r
r
c
c
av
i
00
0
0
0
00
0
0
0
0
1
0
0
0
####
#
$
$
$$
p
z
z
~
~
~~
z
z
~
z
~
~
~
z
~
~
~
~
~
z
~
z
#
#
=+ ++ +
=
=
=
=
=
==
=
=
=
=
=
=
p
z
++
+
-
-
-
-
l
c
b
b
d
b
l
l
n
l
is a non-linear term in
2
90
0
1
cos
tan
RC
VV KVKVKVV
V
V
VV
VV
V
VVVV
V
V
VV
K
d
dV
K
V
VRC
V
dV
d
VRC
V
dV
d
dV
d
d
d
dV
d
V
V
Q
ss
H
Q
d
d
Q
dV
d
QV
V
kHz
Q
HQV
1
1
1
2
2
0
0
2
2
0
2
0
0
offset xxyy xy
x
y
rx
yr
r
xyr
r
pp
pd
av
pd
av
r
c
c
r
c
c
cc
i
r
r
c
c
av
i
00
0
0
0
00
0
0
0
0
1
0
0
0
####
#
$
$
$$
p
z
z
~
~
~~
z
z
~
z
~
~
~
z
~
~
~
~
~
z
~
z
#
#
=+ ++ +
=
=
=
=
=
==
=
=
=
=
=
=
p
z
++
+
-
-
-
-
l
c
b
b
d
b
l
l
n
l
and
2
90
0
1
cos
tan
RC
VV KVKVKVV
V
V
VV
VV
V
VVVV
V
V
VV
K
d
dV
K
V
VRC
V
dV
d
VRC
V
dV
d
dV
d
d
d
dV
d
V
V
Q
ss
H
Q
d
d
Q
dV
d
QV
V
kHz
Q
HQV
1
1
1
2
2
0
0
2
2
0
2
0
0
offset xxyy xy
x
y
rx
yr
r
xyr
r
pp
pd
av
pd
av
r
c
c
r
c
c
cc
i
r
r
c
c
av
i
00
0
0
0
00
0
0
0
0
1
0
0
0
####
#
$
$
$$
p
z
z
~
~
~~
z
z
~
z
~
~
~
z
~
~
~
~
~
z
~
z
#
#
=+ ++ +
=
=
=
=
=
==
=
=
=
=
=
=
p
z
++
+
-
-
-
-
l
c
b
b
d
b
l
l
n
l
. For a precision multiplier,
2
90
0
1
cos
tan
RC
VV KVKVKVV
V
V
VV
VV
V
VVVV
V
V
VV
K
d
dV
K
V
VRC
V
dV
d
VRC
V
dV
d
dV
d
d
d
dV
d
V
V
Q
ss
H
Q
d
d
Q
dV
d
QV
V
kHz
Q
HQV
1
1
1
2
2
0
0
2
2
0
2
0
0
offset xxyy xy
x
y
rx
yr
r
xyr
r
pp
pd
av
pd
av
r
c
c
r
c
c
cc
i
r
r
c
c
av
i
00
0
0
0
00
0
0
0
0
1
0
0
0
####
#
$
$
$$
p
z
z
~
~
~~
z
z
~
z
~
~
~
z
~
~
~
~
~
z
~
z
#
#
=+ ++ +
=
=
=
=
=
==
=
=
=
=
=
=
p
z
++
+
-
-
-
-
l
c
b
b
d
b
l
l
n
l
and
2
90
0
1
cos
tan
RC
VV KVKVKVV
V
V
VV
VV
V
VVVV
V
V
VV
K
d
dV
K
V
VRC
V
dV
d
VRC
V
dV
d
dV
d
d
d
dV
d
V
V
Q
ss
H
Q
d
d
Q
dV
d
QV
V
kHz
Q
HQV
1
1
1
2
2
0
0
2
2
0
2
0
0
offset xxyy xy
x
y
rx
yr
r
xyr
r
pp
pd
av
pd
av
r
c
c
r
c
c
cc
i
r
r
c
c
av
i
00
0
0
0
00
0
0
0
0
1
0
0
0
####
#
$
$
$$
p
z
z
~
~
~~
z
z
~
z
~
~
~
z
~
~
~
~
~
z
~
z
#
#
=+ ++ +
=
=
=
=
=
==
=
=
=
=
=
=
p
z
++
+
-
-
-
-
l
c
b
b
d
b
l
l
n
l
, where
2
90
0
1
cos
tan
RC
VV KVKVKVV
V
V
VV
VV
V
VVVV
V
V
VV
K
d
dV
K
V
VRC
V
dV
d
VRC
V
dV
d
dV
d
d
d
dV
d
V
V
Q
ss
H
Q
d
d
Q
dV
d
QV
V
kHz
Q
HQV
1
1
1
2
2
0
0
2
2
0
2
0
0
offset xxyy xy
x
y
rx
yr
r
xyr
r
pp
pd
av
pd
av
r
c
c
r
c
c
cc
i
r
r
c
c
av
i
00
0
0
0
00
0
0
0
0
1
0
0
0
####
#
$
$
$$
p
z
z
~
~
~~
z
z
~
z
~
~
~
z
~
~
~
~
~
z
~
z
#
#
=+ ++ +
=
=
=
=
=
==
=
=
=
=
=
=
p
z
++
+
-
-
-
-
l
c
b
b
d
b
l
l
n
l
is the reference voltage of the multiplier. Hence, for precision
ampliers,
2
90
0
1
cos
tan
RC
VV KVKVKVV
V
V
VV
VV
V
VVVV
V
V
VV
K
d
dV
K
V
VRC
V
dV
d
VRC
V
dV
d
dV
d
d
d
dV
d
V
V
Q
ss
H
Q
d
d
Q
dV
d
QV
V
kHz
Q
HQV
1
1
1
2
2
0
0
2
2
0
2
0
0
offset xxyy xy
x
y
rx
yr
r
xyr
r
pp
pd
av
pd
av
r
c
c
r
c
c
cc
i
r
r
c
c
av
i
00
0
0
0
00
0
0
0
0
1
0
0
0
####
#
$
$
$$
p
z
z
~
~
~~
z
z
~
z
~
~
~
z
~
~
~
~
~
z
~
z
#
#
=+ ++ +
=
=
=
=
=
==
=
=
=
=
=
=
p
z
++
+
-
-
-
-
l
c
b
b
d
b
l
l
n
l
.
In Experiment 4, if we replace the integrator with a multiplier followed by integrator,
then the circuit becomes a Voltage Controlled Filter (or a Voltage Controlled Phase
Generator). This formsthe basic circuit for self-tuned lter. See Figure 5.2. The
outputoftheself-tunedlterforasquare-waveinput,includingthecontrolvoltage
waveform,is showninFigure5.3.Thegurebringsouttheaspectofautomatic
control and self-tuning.
In the circuit of Figure 5.1, the output of the multiplier is
cos cosV
V
VV
t
2
r
p p
0
$ z ~z
= +-
l
_ i
8
B
After passing through the low-pass lter, the high frequency component gets
lteredoutandonlytheaveragevalueofoutput
2
90
0
1
cos
tan
RC
VV KVKVKVV
V
V
VV
VV
V
VVVV
V
V
VV
K
d
dV
K
V
VRC
V
dV
d
VRC
V
dV
d
dV
d
d
d
dV
d
V
V
Q
ss
H
Q
d
d
Q
dV
d
QV
V
kHz
Q
HQV
1
1
1
2
2
0
0
2
2
0
2
0
0
offset xxyy xy
x
y
rx
yr
r
xyr
r
pp
pd
av
pd
av
r
c
c
r
c
c
cc
i
r
r
c
c
av
i
00
0
0
0
00
0
0
0
0
1
0
0
0
####
#
$
$
$$
p
z
z
~
~
~~
z
z
~
z
~
~
~
z
~
~
~
~
~
z
~
z
#
#
=+ ++ +
=
=
=
=
=
==
=
=
=
=
=
=
p
z
++
+
-
-
-
-
l
c
b
b
d
b
l
l
n
l
remains.
2
90
0
1
cos
tan
RC
VV KVKVKVV
V
V
VV
VV
V
VVVV
V
V
VV
K
d
dV
K
V
VRC
V
dV
d
VRC
V
dV
d
dV
d
d
d
dV
d
V
V
Q
ss
H
Q
d
d
Q
dV
d
QV
V
kHz
Q
HQV
1
1
1
2
2
0
0
2
2
0
2
0
0
offset xxyy xy
x
y
rx
yr
r
xyr
r
pp
pd
av
pd
av
r
c
c
r
c
c
cc
i
r
r
c
c
av
i
00
0
0
0
00
0
0
0
0
1
0
0
0
####
#
$
$
$$
p
z
z
~
~
~~
z
z
~
z
~
~
~
z
~
~
~
~
~
z
~
z
#
#
=+ ++ +
=
=
=
=
=
==
=
=
=
=
=
=
p
z
++
+
-
-
-
-
l
c
b
b
d
b
l
l
n
l
is called the sensitivity of the phase detector and is measured in Volts/radians.
For
2
90
0
1
cos
tan
RC
VV KVKVKVV
V
V
VV
VV
V
VVVV
V
V
VV
K
d
dV
K
V
VRC
V
dV
d
VRC
V
dV
d
dV
d
d
d
dV
d
V
V
Q
ss
H
Q
d
d
Q
dV
d
QV
V
kHz
Q
HQV
1
1
1
2
2
0
0
2
2
0
2
0
0
offset xxyy xy
x
y
rx
yr
r
xyr
r
pp
pd
av
pd
av
r
c
c
r
c
c
cc
i
r
r
c
c
av
i
00
0
0
0
00
0
0
0
0
1
0
0
0
####
#
$
$
$$
p
z
z
~
~
~~
z
z
~
z
~
~
~
z
~
~
~
~
~
z
~
z
#
#
=+ ++ +
=
=
=
=
=
==
=
=
=
=
=
=
p
z
++
+
-
-
-
-
l
c
b
b
d
b
l
l
n
l
,
2
90
0
1
cos
tan
RC
VV KVKVKVV
V
V
VV
VV
V
VVVV
V
V
VV
K
d
dV
K
V
VRC
V
dV
d
VRC
V
dV
d
dV
d
d
d
dV
d
V
V
Q
ss
H
Q
d
d
Q
dV
d
QV
V
kHz
Q
HQV
1
1
1
2
2
0
0
2
2
0
2
0
0
offset xxyy xy
x
y
rx
yr
r
xyr
r
pp
pd
av
pd
av
r
c
c
r
c
c
cc
i
r
r
c
c
av
i
00
0
0
0
00
0
0
0
0
1
0
0
0
####
#
$
$
$$
p
z
z
~
~
~~
z
z
~
z
~
~
~
z
~
~
~
~
~
z
~
z
#
#
=+ ++ +
=
=
=
=
=
==
=
=
=
=
=
=
p
z
++
+
-
-
-
-
l
c
b
b
d
b
l
l
n
l
becomes 0. This information is used to tune the voltage controlled
lter(VCF)automatically.Thevoltage-controlledlter,alongwithphasedetector,is
calledaself-tunedlter.SeeFigure5.2.
2
90
0
1
cos
tan
RC
VV KVKVKVV
V
V
VV
VV
V
VVVV
V
V
VV
K
d
dV
K
V
VRC
V
dV
d
VRC
V
dV
d
dV
d
d
d
dV
d
V
V
Q
ss
H
Q
d
d
Q
dV
d
QV
V
kHz
Q
HQV
1
1
1
2
2
0
0
2
2
0
2
0
0
offset xxyy xy
x
y
rx
yr
r
xyr
r
pp
pd
av
pd
av
r
c
c
r
c
c
cc
i
r
r
c
c
av
i
00
0
0
0
00
0
0
0
0
1
0
0
0
####
#
$
$
$$
p
z
z
~
~
~~
z
z
~
z
~
~
~
z
~
~
~
~
~
z
~
z
#
#
=+ ++ +
=
=
=
=
=
==
=
=
=
=
=
=
p
z
++
+
-
-
-
-
l
c
b
b
d
b
l
l
n
l
of the VCF is given by
Therefore,
The sensitivity of VCF is
2
90
0
1
cos
tan
RC
VV KVKVKVV
V
V
VV
VV
V
VVVV
V
V
VV
K
d
dV
K
V
VRC
V
dV
d
VRC
V
dV
d
dV
d
d
d
dV
d
V
V
Q
ss
H
Q
d
d
Q
dV
d
QV
V
kHz
Q
HQV
1
1
1
2
2
0
0
2
2
0
2
0
0
offset xxyy xy
x
y
rx
yr
r
xyr
r
pp
pd
av
pd
av
r
c
c
r
c
c
cc
i
r
r
c
c
av
i
00
0
0
0
00
0
0
0
0
1
0
0
0
####
#
$
$
$$
p
z
z
~
~
~~
z
z
~
z
~
~
~
z
~
~
~
~
~
z
~
z
#
#
=+ ++ +
=
=
=
=
=
==
=
=
=
=
=
=
p
z
++
+
-
-
-
-
l
c
b
b
d
b
l
l
n
l
radians/sec/Volts. Now
U1
C2
R1
U1•U2
R2
U2
C1
U1•U2
VF1
R8 R6
R3
U3
R9
U4
R7
+
VG1
U5
R5
U1•U2
R4
C3
R11
R10
V3
+
VF3
VF2
Figure 5.2: A Self-Tuned Filter based on a Voltage Controlled Filter
or Voltage Controlled Phase Generator