TELK
OMNIKA
T
elecommunication,
Computing,
Electr
onics
and
Contr
ol
V
ol.
23,
No.
6,
December
2025,
pp.
1635
∼
1645
ISSN:
1693-6930,
DOI:
10.12928/TELK
OMNIKA.v23i6.27250
❒
1635
No
v
el
fractional
order
sinusoidal
oscillators
using
operational
trans
r
esistance
amplier
Battula
T
irumala
Krishna
1
,
V
anitha
Kak
ollu
2
,
Manchala
Madhusudhan
Prasad
3
1
Department
of
Electronics
and
Communication
Engineering,
Uni
v
ersity
Colle
ge
of
Engineering
Kakinada,
Ja
w
aharlal
Nehru
T
echnological
Uni
v
ersity
Kakinada,
Kakinada,
India
2
Department
of
Computer
Science,
GIT
AM
School
of
Science,
GIT
AM
Uni
v
ersity
,
V
isakhapatnam,
India
3
Department
of
Mechanical
Engineering,
Uni
v
ersity
Colle
ge
of
Engineering
Kakinada,
Ja
w
aharlal
Nehru
T
echnological
Uni
v
ersity
Kakinada,
Kakinada,
India
Article
Inf
o
Article
history:
Recei
v
ed
May
26,
2025
Re
vised
Sep
12,
2025
Accepted
Oct
19,
2025
K
eyw
ords:
Acti
v
e
element
Continued
fraction
e
xpansion
Fractional
order
Frequenc
y
of
oscillation
virtual
ground
Sinusoidal
oscillator
Sinusoidal
signal
ABSTRA
CT
The
design
of
fractional
order
circuits
in
v
ery
lar
ge-scale
inte
gration
(VLSI)
domain
is
g
aining
the
interest
of
man
y
researchers.
At
the
same
time
design
of
fractional
circuits
using
the
current
mode
de
vices
is
attracting
the
research
community
.
In
this
paper
,
se
v
eral
possible
fractional
order
sinusoidal
oscillators
using
operational
trans
resistance
amplie
r
(O
TRA)
as
a
basic
b
uilding
block
is
presented.
The
necessary
condition
for
the
frequenc
y
of
oscillation
and
condi-
tion
for
oscillations
is
deri
v
ed.
Fractional
order
operator
s
α
is
the
mos
t
crucial
one
to
be
approximated.
In
this
paper
,
the
frac
tional
order
element
is
approxi-
mated
by
the
continued
fraction
e
xpansion
(CFE).
The
approximation
is
carried
out
up
to
fth
order
.
The
circuits
are
tested
with
the
simulation
softw
are
named
L
Tspice.
The
results
agree
with
the
theoretical
one.
The
proposed
circuits
of-
fers
a
frequenc
y
of
15
MHz,
20
MHz,
and
25
MHz
which
is
higher
in
v
alue
as
compared
to
the
e
xisting
circuits.
The
proposed
circuits
nds
applications
in
bio
medical,
communication
circuits.
This
is
an
open
access
article
under
the
CC
BY
-SA
license
.
Corresponding
A
uthor:
Battula
T
irumala
Krishna
Department
of
Electronics
and
Communication
Engineering,
Uni
v
ersity
Colle
ge
of
Engineering
Kakinada
Ja
w
aharlal
Nehru
T
echnological
Uni
v
ersity
Kakinada
Kakinada,
India
533003
Email:
tkbattula@gmail.com
1.
INTR
ODUCTION
The
circuit
which
is
capable
of
producing
oscillations
without
an
y
e
xternal
input
is
called
an
oscilla
tor
[1].
This
can
be
sinusoidal
or
relaxation
type
oscillator
.
The
sinusoidal
oscillators
nd
wide
applications
in
the
areas
of
control
systems,
signal
processing,
communications
and
other
elds.
Con
v
entionally
operational
amplier
is
used
to
be
the
acti
v
e
element.
Man
y
of
the
lter
circuits,
oscillators
(sinusoidal
as
well
as
relaxation)
are
proposed
in
the
literature
with
operational
amplier
as
a
basic
element.
Fractional-order
oscillators
possess
se
v
eral
uses
in
engineering
and
ph
ysics,
pro
viding
more
preci
se
models
of
real-w
orld
systems
by
ef
fecti
v
ely
capturing
memory
and
h
ysteresis
phenomena.
Principal
applica-
tions
encompass
vibrati
on
control
and
structural
healt
h
monitoring,
sophisticated
ltering
and
signal
processing
within
communication
systems,
the
generation
of
intricate
signals
for
chaotic
systems
and
security
applications
such
as
v
oice
encryption,
as
well
as
the
modeling
of
complicated
phenomena
in
bio
engineering.
J
ournal
homepage:
http://journal.uad.ac.id/inde
x.php/TELK
OMNIKA
Evaluation Warning : The document was created with Spire.PDF for Python.
1636
❒
ISSN:
1693-6930
There
are
se
v
eral
number
of
acti
v
e
elements
present
in
the
literature
such
as
current
con
v
e
yor
,
op-
erational
transconductance
amplier
,
current
feedback
operati
onal
amplier
,
and
operational
trans
resistance
amplier
.
Fe
w
of
them
are
a
v
ailable
commercially
also.
These
de
vices
ha
v
e
their
o
wn
adv
antages
and
disad-
v
antages
[2].
In
this
article,
operational
trans
resistance
amplier
(O
TRA)
is
considered
as
an
acti
v
e
de
vice
as
in
[3],
[4].
Realization
of
fractional
oscillator
is
not
an
old
topic
dates
back
to
2000
[5].
Similarly
non
sinusoidal
oscillators
are
also
designed
in
[6].
No
v
el
topologie
s
of
sinusoidal
oscillators
are
proposed
in
[7]
and
are
simulated
using
Orcad
PSpice
.
Fractional-order
oscillators
emplo
y
fracti
onal-order
capacitors
or
inductors,
enhancing
the
design
e
xibility
and
tunability
of
oscillat
o
r
s
relati
v
e
t
o
con
v
entional
inte
ger
-order
oscillators
[8].
These
oscillators
are
characterized
by
non-inte
ger
-order
dif
ferential
equations,
pro
viding
control
o
v
er
oscillation
frequenc
y
and
phase.
T
w
o
oscillator
topologies
with
tw
o
impedances
for
a
uni
v
ersal
fractional
order
osci
llator
based
on
a
tw
o
port
netw
ork
are
co
v
ered
in
paper
[7],
[8].
The
tw
o
fractional
impedances
can
be
combined
in
four
w
ays
for
each
topology
.
A
thorough
analysis
and
design
of
fractional-or
d
e
r
oscillators
using
double
Op-Amps
are
presented
in
the
paper
in
[9].
T
o
v
erify
ho
w
the
fractional
order
af
fects
the
oscillation
parameters,
MA
TLAB
simulations
are
analyzed.
The
numerical
solution
in
[9]
is
v
alidated
by
PSpice
simul
ations
and
e
xperimental
results.
Ahmad
et
al.
[10],
a
fractional
order
W
ien
bridge
oscillator
based
on
state
space
equations
is
proposed.
Carlson
approximation,
mastudas
approximation,
po
wer
series
e
xpansion,
and
continued
fraction
e
x-
pansion
can
all
be
used
to
realize
fractional
order
transfer
functions.
Continued
fraction
e
xpansion
has
yielded
better
r
esults
as
compared
to
other
s
[11].
A
fractional
order
sinusoidal
os
cillator
(FSO)
with
tw
o
f
ractional
ca-
pacitors,
four
resistors,
and
one
O
TRA
is
pro
vided
by
the
w
ork
in
[12].
Nodal
analysis
w
as
used
to
determine
the
frequenc
y
,
condition,
and
phase
dif
ference
between
the
output
v
oltages
of
the
proposed
fractional
order
oscillator
and
simulations
are
carried
out
using
PSpice.
A
dif
ferential
v
oltage
current
con
v
e
yo
r
(D
VCC)
based
fractional
order
Butterw
orth
lo
wpass
lter
is
realized
in
[13].
A
tw
ofold
comple
xity
,
fractional-order
dual
coupling
duf
ng
system
is
proposed
in
the
study
[14]
to
detect
ship-radiated
noise.
Jacobi
collocation
and
nested
picard
iteration
(NPI)
are
used
in
[15]
to
solv
e
the
fractional
order
oscillatory
che
mical
reaction
model.
The
outcomes
of
the
NPI
and
Jacobi
collocation
approaches
are
strikingly
similar
.
K
ubanek
et
al.
[16],
the
transfer
functions
of
order
(1
+
α
)
are
proposed.
Tsirimok
ou
et
al.
[17]
discusses
the
use
of
fractional
order
analog
circuits
in
medicine.
In
study
[18],
[19],
fractional
lters
with
v
oltage
dif
ferencing
transconductance
amplier
(VDT
A)
are
suggested
and
modeled.
Sac
¸
u
and
Alc
¸
ı
[20]
designs
and
simulates
a
lo
w-v
oltage,
lo
w-po
wer
,
operational
transconductance
amplier
(O
T
A)-based
fractional
order
lo
w-pass
and
high-pass
lter
of
order
(
n
+
α
)
using
Cadence-PSpice,
where
0
<
α
<
1
and
n
<
1
.
Sac
¸
u
and
Alc
¸
ı
[21]
describe
a
uni
v
ersal
lter
topology
with
lo
w-v
oltage
acti
v
e
elements
that
of
fer
all
types
of
responses
in
a
single
circuit.
Tsirimok
ou
et
al.
[22]
proposed
no
v
el
fractional-order
generalized
lter
topologies.
multiple-feedback
topologies
and
operational
transconductance
ampliers
were
used
to
achie
v
e
the
abo
v
e.
The
proposed
designs
are
v
alidated
by
simulations
using
Cadence
inte
grated
circuit
(IC)
design
suite
and
the
Austrian
micro
systems
design
kit.
Ne
w
fractional
order
inte
grator
implementation
methods
are
pro
vided
in
[23].
After
analyzi
ng
inte
ger
order
approximation
methods,
fractional
order
inte
grators
with
acti
v
e
circuit
elements
are
des
igned
utilizing
successfully
implemented
approximate
methods
from
the
literature.
Historical
perspecti
v
e
of
the
fractional
order
elements
is
presented
by
Kartci
et
al.
[24],
the
mathe-
matical
e
xpressions
for
the
fractional
order
circuits
using
tw
o
ladder
elements
is
de
v
eloped
in
[25].
Current
mode
fractional
order
dif
ferentiators
and
inte
grators
is
proposed
in
[26]
which
are
recongurable.
The
objec-
ti
v
e
of
the
study
in
[27]
is
to
use
commercially
a
v
ailable
L
T1228
ICs
with
three
inputs
and
one
output
(TISO)
to
de
v
elop
an
electronically
tunable
v
oltage-mode
(VM)
uni
v
ersal
lter
.
T
w
o
L
T1228s,
four
resistors,
and
tw
o
grounded
capacitors
mak
e
up
the
recommended
lter
.
PSpice
simulation
and
hardw
are
implementation
sup-
port
the
suggested
lter’
s
performance.
Arora
[28]
in
v
estig
ates
the
potential
uses
of
v
oltage
dif
ferencing
current
con
v
e
yors
(VDCC)
as
sinusoidal
oscillators
and
current
mode
uni
v
ersal
lters.
Only
grounded
passi
v
e
compo-
nents
and
a
smal
l
number
of
acti
v
e
elements
were
used
in
the
resultant
netw
ork.
The
digitally
programmable
complementary
metal–oxide–semiconductor
(CMOS)
is
used
to
de
v
elop
a
digital
modulation
system
in
[29].
PSpice
softw
are
is
used
to
sim
ulate
and
analyze
this
ne
w
communication
approach.
The
study
suggests
further
research
on
noise
resistance
in
digitally
programmable
(DP)
modulation
techniques
and
e
xploring
digitally
programmable
CMOS
technology-enabled
modulation
methods
lik
e
quadrature
amplitude
modulation
(QAM)
or
orthogonal
frequenc
y
di
vision
multiple
xing
(OFDM).
TELK
OMNIKA
T
elecommun
Comput
El
Control,
V
ol.
23,
No.
6,
December
2025:
1635–1645
Evaluation Warning : The document was created with Spire.PDF for Python.
TELK
OMNIKA
T
elecommun
Comput
El
Control
❒
1637
A
current-controlled
current
con
v
e
yor
second
generation
(CCCII)-bas
ed
shado
w
lter
and
oscill
ator
is
presented
in
the
study
in
[30].
Natural
frequenc
y
and
lter
quality
f
actor
are
determined
by
CCCII
current
g
ains.
It
is
simpler
to
combine
the
resistorless
lter
with
grounded
capacitors.
In
a
single
topology
,
the
lter
utilizes
input
signals
to
generate
all
types
of
transfer
functions.
T
o
conrm
the
unique
architecture,
simulation
program
with
inte
grated
circuit
emphasis
(SPICE)
simulates
the
suggested
circ
u
i
t
using
A
T&T
ALA400-CBIC-
R
bipolar
transistor
arrays.
In
this
paper
,
the
con
v
entional
capacitor
s
are
replaced
with
fractional
order
one.
The
necessary
e
x-
pression
for
frequenc
y
of
oscillation,
condition
for
sustained
oscillations
are
deri
v
ed.
By
v
arying
the
v
alues
of
fractional
order
,
the
output
v
ariat
ion
is
studied.
The
simulations
are
carried
out
using
L
Tspice.
The
results
agree
with
the
theoretical
one.
The
paper
is
structured
as
follo
ws.
Section
2
deals
with
the
rational
approximations.
Section
3
d
e
als
with
the
deri
v
ation
for
frequenc
y
of
oscillation
and
condition
for
sustained
oscillations.
Implem
entation
of
the
circuits
and
results
are
presented
in
section
4.
Section
5
deals
with
the
conclusions.
2.
PR
OPOSED
MA
THEMA
TICAL
METHODS
2.1.
Rational
appr
oximations
T
o
calculate
the
inte
ger
-order
rational
fractance
approximations,
a
number
of
frequenc
y-domain
ap-
proximation
techniques
ha
v
e
been
documented
in
the
literature
[11].
This
te
chnique
is
popular
for
function
e
v
aluation
because
it
e
xpands
the
comple
x
plane’
s
domain
more
quickly
than
po
wer
series
e
xpansions.
Con-
tinued
fraction
e
xpansion
is
described
by
considering
the
continued
fraction
e
xpansion
of
(1
+
x
)
α
as:
(1
+
x
)
α
=
1
1
−
αx
1+
(1+
α
)
x
2+
(1
−
α
)
x
3+
(2+
α
)
x
2+
(2
−
α
)
x
5+
.....
(
N
+
α
)
x
2+
(
N
−
α
)
x
2
N
+1
(1)
Substituting
x
=
s
−
1
,
and
considering
the
rst
ten
terms
the
rational
function
is
gi
v
en
by
[11].
s
α
=
P
0
s
5
+
P
1
s
4
+
P
2
s
3
+
P
3
s
2
+
P
4
s
+
P
5
Q
0
s
5
+
Q
1
s
4
+
Q
2
s
3
+
Q
3
s
2
+
Q
4
s
+
Q
5
(2)
The
co-ef
cients
P
0
,
P
1
,
....P
5
,
Q
0
,
Q
1
,
......Q
5
are
gi
v
en
by:
P
0
=
Q
5
=
−
α
5
−
15
α
4
−
85
α
3
−
225
α
2
−
274
α
−
120
P
1
=
Q
4
=
5
α
5
+
45
α
4
+
5
α
3
−
1005
α
2
−
3250
α
−
3000
P
2
=
Q
3
=
−
10
α
5
−
30
α
4
+
410
α
3
+
1230
α
2
−
4000
α
−
12000
P
3
=
Q
2
=
10
α
5
−
30
α
4
−
410
α
3
+
1230
α
2
+
4000
α
−
12000
P
4
=
Q
1
=
−
5
α
5
+
45
α
4
−
5
α
3
−
1005
α
2
+
3250
α
−
3000
P
5
=
Q
0
=
α
5
−
15
α
4
+
85
α
3
−
225
α
2
+
274
α
−
120
(3)
The
magnitude,
phase
and
err
or
responses
of
the
ideal
and
approximated
functions
for
α
=
0
.
5
and
α
=
0
.
25
are
as
sho
wn
in
Figure
1.
From
the
Fi
gure
1
it
is
e
vident
that
the
proposed
response
matches
the
ideal
one
for
lar
ger
range
of
frequencies
in
both
magnitude
and
Phase.
The
error
is
minimum
in
the
range
of
frequencies
from
[10
−
2
]
to
[10
2
]
Hz.
The
v
alues
of
the
5
th
order
coef
cients
obtained
for
v
arious
α
is
tab
ulated
in
T
able
1.
The
synthesized
resistor–capacitor
(RC)
netw
ork
v
alues
using
partial
fraction
e
xpansion
method
are
as
sho
wn
in
T
able
2.
One
of
the
possible
synthesized
RC
netw
ork
is
as
sho
wn
in
Figure
2.
In
the
ne
xt
section
details
of
the
current
mode
de
vice
named
operational
transresistance
amplier
is
e
xplained.
No
vel
fr
actional
or
der
sinusoidal
oscillator
s
using
oper
ational
tr
ans
...
(Battula
T
irumala
Krishna)
Evaluation Warning : The document was created with Spire.PDF for Python.
1638
❒
ISSN:
1693-6930
Figure
1.
Magnitude,
phase,
and
error
plots
T
able
1.
5
th
order
approximated
coef
cents
for
v
arious
v
alues
of
α
α
P
0
P
1
P
2
P
3
P
4
P
5
0.1
149.7365
3.3350e+03
1.2387e+04
1.1588e+04
2.6851e+03
94.7665
0.2
184.5043
3.6901e+03
1.2748e+04
1.1154e+04
2.3902e+03
73.5437
0.3
224.8689
4.0649e+03
1.3078e+04
1.0701e+04
2.1152e+03
55.8741
0.4
271.4342
4.4593e+03
1.3378e+04
1.0230e+04
1.8600e+03
41.3338
0.5
324.8438
4.8727e+03
1.3643e+04
9.7453e+03
1.6242e+03
29.5313
0.6
385.7818
5.3045e+03
1.3873e+04
9.2489e+03
1.4074e+03
20.1062
0.7
454.9746
5.7541e+03
1.4066e+04
8.7435e+03
1.2092e+03
12.7284
0.8
533.1917
6.2206e+03
1.4218e+04
8.2317e+03
1.0290e+03
7.0963
0.9
621.2470
6.7029e+03
1.4330e+04
7.7164e+03
866.1229
2.9360
T
able
2.
Resistor
and
capacitor
v
alues
for
5
th
approximation
α
R
a
R
b
C
b
R
c
C
c
R
d
C
d
R
e
C
e
R
f
C
f
0.1
0.6329
0.1849
0.3024
0.1163
2.7561
0.1126
9.3996
0.1535
23.2565
0.3798
61.4173
0.2
0.3986
0.2569
0.2450
0.1970
1.7371
0.2183
5.1342
0.3432
11.1552
1.0947
24.7963
0.3
0.2485
0.2652
0.2652
0.2448
1.4900
0.3100
3.8278
0.5648
7.2878
2.3913
13.4318
0.4
0.1523
0.2394
0.3262
0.2634
1.4742
0.3809
3.3000
0.8080
5.4904
4.7229
8.2238
0.5
0.0909
0.1975
0.4366
0.2569
1.6076
0.4240
3.1414
1.0536
4.5509
8.9771
5.3886
0.6
0.0521
0.1503
0.6302
0.2296
1.9113
0.4325
3.2646
1.2681
4.0999
17.0544
3.6856
0.7
0.0280
0.1043
0.9941
0.1859
2.5063
0.4000
3.7450
1.3956
4.0548
33.6310
2.5950
0.8
0.0133
0.0629
1.7977
0.1302
3.7972
0.3203
4.9649
1.3438
4.6034
73.2658
1.8647
0.9
0.0047
0.0279
4.4048
0.0668
7.8470
0.1883
8.9763
0.9616
7.0677
210.3463
1.3590
R
a
R
b
C
b
R
c
C
c
R
d
C
d
R
e
C
e
R
f
C
f
Figure
2.
Fifth
order
approximated
fractance
TELK
OMNIKA
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elecommun
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Control,
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ol.
23,
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6,
December
2025:
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OMNIKA
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elecommun
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Control
❒
1639
2.2.
Operational
trans
r
esistance
amplier
Analog
IC
designers
ha
v
e
recently
paid
a
lot
of
attention
to
O
TRA.
O
TRA
is
an
acti
v
e
g
adget
wi
th
three
terminals.
The
output
v
oltage
of
O
TRA
is
dependent
on
the
input
current
and
can
be
sho
wn
as
the
transfer
matrix.
V
p
V
n
V
0
=
0
0
0
0
0
0
R
m
−
R
m
0
I
p
I
n
I
0
(4)
where
R
m
is
the
trans-resistance
g
ain
of
the
O
TRA.
The
output
v
oltage
V
0
is
gi
v
en
by
(5).
V
0
=
R
m
(
I
p
−
I
n
)
(5)
The
blockdiagram
of
O
TRA
is
as
sho
wn
in
Figure
3.
Figure
3.
Blockdiagram
of
O
TRA
In
v
erting
and
nonin
v
erting
O
TRA
are
possible.
Both
scenarios
ha
v
e
the
same
g
ain.
O
TRA
w
as
pro-
posed
in
1992
and
is
used
in
analog
signal
processing.
The
follo
wing
subsections
e
xplain
the
e
xisting
O
TRA
circuits
named
Salama
O
TRA
and
CMOS
based
O
TRA
as
sho
wn
in
Figures
4(a)
and
(b).
(a)
(b)
Figure
4.
Dif
ferent
types
of
operational
transconductance
amplier:
(a)
salama
O
TRA
and
(b)
O
TRA
using
CMOS
2.2.1.
Salama
O
TRA
The
modied
dif
ferential
current
con
v
e
yor
(MDCC)
and
common
source
amplier
are
cascaded
to
form
the
CMOS
realization
of
O
TRA,
as
illustrated
in
Figure
4(a).
While
the
common
source
amplier
pro
vides
high
g
ain,
MDCC
performs
current
dif
ferencing
and
ef
fecti
v
ely
grounds
the
tw
o
input
terminals.
The
current
mirror
is
created
by
(
M
1
−
M
4
).
The
output
v
oltage
in
the
O
TRA
design
is
produced
by
biasi
ng
the
output
stage
transistors
with
currents
M
9
and
M
11
.
2.2.2.
CMOS
based
O
TRA
A
viable
CMOS
O
TRA
internal
circuit
construction
is
sho
wn
in
Figure
4(b)
and
consists
of
tw
o
pri-
mary
functional
blocks:
a
v
oltage
b
uf
fer
circuit
and
a
dif
ferential-current-controlled
current
source.
All
P-
channel
metal–oxide–semiconductor
(PMOS)
and
N-channel
metal–oxide–semiconductor
(NMOS)
transistors
No
vel
fr
actional
or
der
sinusoidal
oscillator
s
using
oper
ational
tr
ans
...
(Battula
T
irumala
Krishna)
Evaluation Warning : The document was created with Spire.PDF for Python.
1640
❒
ISSN:
1693-6930
in
this
circuit
ha
v
e
their
substrate
terminals
connected
to
positi
v
e
and
ne
g
ati
v
e
supply
v
oltages,
respecti
v
ely
.
As
current
reectors,
transistors
M
1
,
M
3
,
and
M
5
guarantee
that
the
currents
o
wing
through
transistors
M
9
and
M
10
are
equal.
Similarly
,
equal
currents
are
forced
into
tra
n
s
istors
M
7
and
M
8
by
the
current
mirrors
composed
of
M
2
,
M
4
,
and
M
6
.
T
ransistors
M
1
and
M
2
ha
v
e
grounded
sources.
3.
METHODOLOGY
FOLLO
WED
This
section
pro
vides
the
proposed
fractional
order
oscillators.
Three
of
the
circuits
e
xisting
in
[7]
are
modied
by
replacing
the
capacitors
by
fractional
order
one
and
the
conditi
o
ns
for
sustained
oscillations
and
frequenc
y
of
oscillations
is
deri
v
ed.
3.1.
O
TRA
based
fractional
order
sinusoidal
oscillators
The
generalized
block
diagram
of
O
TRA
based
fractional
order
sinusoidal
oscillators
is
as
sho
wn
in
Figure
5.
Let
V
a
be
the
v
oltage
at
the
interconnection
of
Y
1
,
Y
3
,
Y
4
,
Y
6
and
Y
7
.
The
currents
at
the
in
v
erting
and
non-in
v
erting
terminals
can
be
written
as:
V
0
Y
2
+
V
a
Y
3
=
I
p
(6)
V
0
Y
5
+
V
a
Y
4
=
I
n
(7)
From
the
ideal
beha
vior
od
O
TRA
since
the
current
I
p
,
I
n
to
be
equal,
V
0
=
V
a
Y
4
−
Y
3
Y
2
−
Y
5
(8)
Writing
KCL
at
node
a
,
V
0
(
Y
1
+
Y
6
)
=
V
a
(
Y
1
+
Y
3
+
Y
4
+
Y
6
+
Y
7
)
(9)
Substituting
the
v
alues
of
V
a
and
simplifying
the
characteristic
equation
for
the
oscillator
is
going
to
be:
Y
1
Y
2
+Y
2
Y
3
+Y
2
Y
4
+Y
2
Y
6
+Y
2
Y
7
+Y
1
Y
3
+Y
3
Y
6
−
Y
1
Y
5
−
Y
1
Y
4
−
Y
4
Y
6
−
Y
3
Y
5
−
Y
4
Y
5
−
Y
5
Y
6
−
Y
5
Y
7
=
0
(10)
Dif
ferent
topologies
can
be
obtained
by
changing
the
v
alues
of
Y
’
s.
Some
of
the
possible
topologies
by
replac-
ing
Y
with
fractional
order
capacitor
are
as
e
xplained
in
the
coming
sections.
Figure
5.
Blockdiagram
of
O
TRA
based
sinusoidal
oscillator
3.2.
Case
A
Y
1
=
Y
6
=
0
,
Y
2
=
s
α
C
2
,
Y
4
=
s
β
C
4
,
the
characteristic
equation
becomes,
Y
2
Y
3
+
Y
2
Y
4
+
Y
2
Y
7
−
Y
3
Y
5
−
Y
4
Y
5
−
Y
5
Y
7
=
0
.
On
substituting
the
v
alues
of
Y
’
s
and
simplifying
the
equations
for
the
frequenc
y
of
operation
and
condition
for
oscillations
will
be:
ω
α
+
β
cos
(
α
+
β
)
π
2
c
2
c
4
+
ω
α
cos
απ
2
c
2
(
G
3
+
G
7
)
−
ω
β
cos
β
π
2
G
5
c
4
−
G
5
(
G
3
+
G
7
)
=
0
(11)
ω
α
+
β
sin
(
α
+
β
)
π
2
c
2
c
4
+
ω
α
sin
α
π
2
c
2
(
G
3
+
G
7
)
−
ω
β
sin
β
π
2
G
5
c
4
=
0
(12)
TELK
OMNIKA
T
elecommun
Comput
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Control,
V
ol.
23,
No.
6,
December
2025:
1635–1645
Evaluation Warning : The document was created with Spire.PDF for Python.
TELK
OMNIKA
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elecommun
Comput
El
Control
❒
1641
3.3.
Case
B
Y
2
=
Y
4
=
0
,
Y
5
=
s
α
C
5
,
Y
6
=
s
β
C
6
,
the
characteristic
equation
becomes,
Y
1
Y
3
+
Y
3
Y
6
−
Y
1
Y
5
−
Y
3
Y
5
−
Y
6
Y
5
−
Y
5
Y
7
=
0
.
On
substituting
the
v
alues
of
Y
’
s
and
simplifying
the
equations
for
the
frequenc
y
of
operation
and
condition
for
oscillations
will
be:
ω
α
+
β
cos
(
α
+
β
)
π
2
c
5
c
6
+
ω
α
cos
απ
2
c
5
(
G
1
+
G
3
+
G
7
)
−
ω
β
cos
β
π
2
G
3
c
6
−
G
1
G
3
=
0
(13)
ω
α
+
β
sin
(
α
+
β
)
π
2
c
5
c
6
+
ω
α
sin
α
π
2
c
5
(
G
1
+
G
3
+
G
7
)
−
ω
β
sin
β
π
2
G
3
c
6
=
0
(14)
3.4.
Case
C
Y
1
=
Y
5
=
0
,
Y
3
=
s
α
C
3
,
Y
6
=
s
β
C
6
,
the
characteristic
equation
becomes,
Y
1
Y
3
+
Y
3
Y
6
−
Y
1
Y
5
−
Y
3
Y
5
−
Y
6
Y
5
−
Y
5
Y
7
=
0
.
On
substituting
the
v
alues
of
Y
’
s
and
simplifying
the
equations
for
the
frequenc
y
of
operation
and
condition
for
oscillations
will
be:
ω
α
+
β
cos
(
α
+
β
)
π
2
c
3
c
6
+
ω
α
cos
απ
2
G
2
c
3
+
ω
β
cos
β
π
2
(
G
2
c
6
−
G
4
c
6
)
+
G
2
(
G
4
+
G
7
)
=
0
(15)
ω
α
+
β
sin
(
α
+
β
)
π
2
c
3
c
6
+
ω
α
sin
α
π
2
G
2
c
3
+
ω
β
sin
β
π
2
(
G
2
c
6
−
G
4
c
6
)
=
0
(16)
3.5.
Special
cases
The
follo
wing
analysis
illustrates
the
inuence
of
the
fractional
order
on
the
oscillation
parameters.
−
α
=
β
=
1
s
α
+
β
=
e
j
π
ω
2
=
−
ω
2
s
α
=
e
j
π
2
ω
=
j
ω
(17)
The
frequenc
y
of
oscillation
is
found
to
be:
ω
2
k
1
+
k
4
=
0
⇒
ω
2
=
−
k
4
/k
1
(18)
The
condition
for
the
sustained
oscillations
becomes:
k
2
+
k
3
=
0
(19)
−
α
=
β
̸
=
1
cos
(
α
π
)
≈
cos
απ
2
=
1
,
sin
(
α
π
)
≈
α
π
,
sin
απ
2
≈
απ
2
.
Substituting
the
abo
v
e
conditions
and
simplifying
(20).
ω
2
α
k
1
+
1
2
(
ω
α
k
2
+
ω
α
k
3
)
=
0
(20)
The
(20)
reduces
to:
ω
α
=
−
k
2
+
k
3
2
k
1
(21)
Substituting
the
abo
v
e
condition
in
the
(22).
ω
2
α
k
1
+
ω
α
k
2
+
ω
α
k
3
+
k
4
=
0
(22)
This
(22)
reduces
to:
k
2
+
k
3
=
2
p
k
1
k
4
(23)
No
vel
fr
actional
or
der
sinusoidal
oscillator
s
using
oper
ational
tr
ans
...
(Battula
T
irumala
Krishna)
Evaluation Warning : The document was created with Spire.PDF for Python.
1642
❒
ISSN:
1693-6930
The
deri
v
ed
e
xpressions
for
frequenc
y
of
oscillations
and
condition
for
sustained
oscillations
for
both
the
cases,
α
=
β
=
1
and
α
=
β
̸
=
1
are
tab
ulated
in
T
able
3.
T
able
3.
Conditions
of
oscillators
α
=
β
=
1
α
=
β
̸
=
1
Circuit
ω
Condition
ω
α
Condition
Case
A
q
G
5
(
G
3
+
G
7
)
c
2
c
4
c
2
(
G
3
+
G
7
)
=
G
5
c
4
G
5
c
4
−
c
2
(
G
3
+
G
7
)
2
c
2
c
4
c
2
(
G
3
+
G
7
)
=
G
5
c
4
Case
B
q
G
1
G
3
c
5
c
6
c
5
(
G
1
+
G
3
+
G
7
)
=
G
3
c
6
G
5
c
6
−
c
5
(
G
1
+
G
3
+
G
7
)
2
c
5
c
6
c
5
(
G
1
+
G
3
+
G
7
)
=
G
3
c
6
Case
C
q
G
2
(
G
4
+
G
7
)
c
3
c
6
c
6
G
4
=
G
2
(
c
3
+
c
6
)
G
4
c
6
−
G
2
c
6
−
G
2
c
3
2
c
3
c
6
G
2
c
3
+
G
2
c
6
−
G
4
c
6
=
2
p
G
2
c
3
c
6
(
G
4
+
G
7
)
4.
RESUL
TS
AND
DISCUSSIONS
The
proposed
circuits
in
three
cases
has
been
simulated
using
L
Tspice.
The
fractional
order
of
v
alue
0.5
is
chosen
and
the
simulations
are
carried
out
for
all
the
three
topologies.
The
component
v
alues
used
for
the
simulation
are
Case
A
(
Y
3
=
60Ω
,
Y
5
=
1000Ω
,
Y
7
=
300Ω
),
Case
B
(
Y
3
=
150Ω
,
Y
7
=
15Ω
),
Case
C
(
Y
2
=
1000Ω
,
Y
4
=
60Ω
,
Y
7
=
50Ω
).
The
obtained
results
are
as
sho
wn
in
Figure
6.
The
f
ast
fourier
transform
(FFT)
analysis
is
depicted
in
Figure
7
and
the
corresponding
results
are
tab
ulated
in
T
able
4.
Figure
6.
Simulation
results
for
Case
A
circuit
Figure
7.
FFT
analysis
of
Case
A
circuit
TELK
OMNIKA
T
elecommun
Comput
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Control,
V
ol.
23,
No.
6,
December
2025:
1635–1645
Evaluation Warning : The document was created with Spire.PDF for Python.
TELK
OMNIKA
T
elecommun
Comput
El
Control
❒
1643
T
able
4.
FFT
Analysis
of
proposed
oscillators
S.
no
T
ype
of
circuit
Frequenc
y
Magnitude(dB)
1
Circui
t
A
15MHz
-138
2
Circuit
B
20
MHz
-148
3
Circuit
C
25
MHz
-150
5.
CONCLUSION
The
main
purpose
of
this
paper
is
to
propose
a
no
v
el
fractional
order
oscillator
using
opera
tional
transconductance
amplier
.
The
deri
v
ati
on
s
pertaining
to
the
fractional
order
sense
are
deri
v
ed.
The
basic
idea
of
the
oscillator
is
de
v
eloped
by
replacing
con
v
entional
capacitors
with
fractional-order
elements.
The
frac-
tional
order
elements
are
to
be
approximated
by
a
rational
approximations
to
a
nite
order
.
In
this
paper
,
the
rational
approximation
of
the
fractional
elements
is
obtained
using
a
continued-fraction
e
xpansion
approach.
The
mathematical
equations
for
the
sustained
oscillations
is
deri
v
ed
in
terms
of
fractional
order
and
are
tab-
ulated.
The
special
cases
of
similar
fractional
order
and
dissimilar
fractional
orders
is
also
discussed.
F
or
simulation
purpose
the
fractional
order
α
=
0
.
5
is
chosen.
The
v
alues
chosen
for
the
simulation
are
also
indi-
cated.
The
FFT
analysis
of
the
results
is
carried
out.
It
has
been
observ
ed
that
at
α
=
β
=
1
,
the
e
xpressions
are
similar
to
the
e
xpressions
for
the
con
v
entional
oscillators.
The
theoretical
v
alues
of
oscillation
frequenc
y
is
pro
v
en
to
be
same
as
the
theoretical
one.
The
circuits
proposed
are
simple
and
can
be
used
in
communications,
encryption
and
decryption
applications.
A
CKNO
WLEDGEMENTS
The
authors
are
deeply
indebted
to
administration
of
Ja
w
aharlal
Nehru
T
echnological
Uni
v
ersity
Kak-
inada,
Gitam
Uni
v
ersity
for
the
support.
The
authors
also
wish
the
anon
ymous
re
vie
wers
for
the
suggestions
to
impro
v
e
the
manuscript.
FUNDING
INFORMA
TION
Authors
state
no
funding
in
v
olv
ed.
A
UTHOR
CONTRIB
UTIONS
ST
A
TEMENT
The
corresponding
author
will
be
responsible
for
all
conceptualization,
methodology
,
formal
analys
is,
or
in
v
estig
ation,
as
well
as
at
least
one
aspect
of
writing.
Name
of
A
uthor
C
M
So
V
a
F
o
I
R
D
O
E
V
i
Su
P
Fu
Battula
T
irumala
Krishna
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
✓
V
anitha
Kak
ollu
✓
✓
✓
✓
Manchala
Madhusudhan
Prasad
✓
✓
✓
✓
✓
✓
C
:
C
onceptualization
I
:
I
n
v
estig
ation
V
i
:
V
i
sualization
M
:
M
ethodology
R
:
R
esources
Su
:
Su
pervision
So
:
So
ftw
are
D
:
D
ata
Curation
P
:
P
roject
Administration
V
a
:
V
a
lidation
O
:
Writing
-
O
riginal
Draft
Fu
:
Fu
nding
Acquisition
F
o
:
F
o
rmal
Analysis
E
:
Writing
-
Re
vie
w
&
E
diting
CONFLICT
OF
INTEREST
ST
A
TEMENT
Authors
state
no
conict
of
interest.
D
A
T
A
A
V
AILABILITY
The
data
can
be
a
v
ailable
from
the
author
with
request.
No
vel
fr
actional
or
der
sinusoidal
oscillator
s
using
oper
ational
tr
ans
...
(Battula
T
irumala
Krishna)
Evaluation Warning : The document was created with Spire.PDF for Python.
1644
❒
ISSN:
1693-6930
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.
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ulej,
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TELK
OMNIKA
T
elecommun
Comput
El
Control,
V
ol.
23,
No.
6,
December
2025:
1635–1645
Evaluation Warning : The document was created with Spire.PDF for Python.