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Two-stage chaotic Colpitts oscillator
Tamasevicius, Arunas; Mykolaitis, G.; Bumeliene, S.; Cenys, Antanas; Anagnostopoulos, A. N.;
Lindberg, Erik
Published in:
Electronics Letters
Publication date:
2001
Document Version
Publisher's PDF, also known as Version of record
Link back to DTU Orbit
Citation (APA):
Tamasevicius, A., Mykolaitis, G., Bumeliene, S., Cenys, A., Anagnostopoulos, A. N., & Lindberg, E. (2001). Two-
stage chaotic Colpitts oscillator. Electronics Letters, 37(9), 549-551.
http://ieeexplore.ieee.org/iel5/2220/19919/00920906.pdf
straints on the feasible set of the objectives. An initial population
size of 5&100 and up to 200 generations were tested. Various
parameter values such as crossover and mutation probabilities
were also examined. Results showed that convergence of the
model
was
steady and fast. Deterministic results were also exam-
ined and achieved.
0-
-5
-10
-15
-20
%
-25
-30
-35
-40
-50
I
I
I
I
I
I
30
32
34
36
30
40
frequency,
GHz
1915/31
Fig.
3
Measured frequency response of isolation
-
-
-
-
-
-
-
-
-45
I
I
I
I
I
-1
00
-50
0 50
100
1915/41
angle, deg
Fig.
4
Patterns
of
difference-beam at
37.5GHz
measured results
____
computed results
I
-35
t
-40
I
I
I I
I
100
50
0
50
100
angle, deg
Fig.
5
Patterns
of
sum-beam at
37.5GHz
__
measured results
____
computed results
Results:
The optimised results were
a
population of chromosomes
with matching objective values. However, because of the computa-
tional complexity and dependence on the feeding system, the fre-
quency response of the input
VSWR
was not considered as an
objective, Thus, the results were also determined according to the
travelling-wave mechanism
of
the
TSA
[3]. The optimised and
measured sample
is
shown in Fig.
1.
The measured frequency
responses of the VSWR at the suddifference ports and the isola-
tion between the two ports, which are the accumulated results
of
the reflection and cross-coupling of the antenna, the printed
duplexer, and the microstrip-waveguide transitions, are shown in
Figs.
2
and 3, respectively. The frequency bands for a VSWR
<
2:l
were 34.54O+GHz at the sum-port and 334+GHz at the differ-
ence-port. The band for an isolation
<
36dB was 33.%O+GHz.
The computed and measured radiation patterns of the difference-
beam and the sum-beam at the central frequency 37.5GHz are
shown in Figs. 4 and 5, respectively. The null-depth of the differ-
ence-beam is better than -40dB with a null-slant of +21". The -3dB
beamwidth of the sum-beam was
10
x
15" (for the
E-
x
H-
plane).
In
addition, the measured gain of the sum-beam was 14dB. Fat-
terns were also measured at 35 and 4OGHz. The results
of
the dif-
ference-beams were a null-depth of -28dB with 26" null-slant and
-27dB with
48",
respectively; steep slopes at the beam-centre were
also obtained. The results of the sum-beams were -3dB
beamwidth of 12
x
15" and 14
x
26", and a gain of 12 and lOdB,
respectively. The measured sidelobe levels were satisfactory. The
measured cross-polarisation levels were sufficiently low and are
not presented here for the sake
of
brevity.
Conclusions:
Although the linear-edged CTSA samples in our pre-
vious research work produced good VSWR and isolation proper-
ties, they were
not
capable
of
maximising the slope of the
difference beam in a wide band. Although the optimised CTSA
configuration appears to be counterintuitive, it performs better in
terms of the desired aspects. With the fast development of com-
puters, more refined objectives can be taken into consideration.
Therefore, further improvements in terms of both the "/differ-
ence beams and impedance matching are still possible.
0
IEE
2001
Electronics Letters Online
No:
20010315
DOI:
10.1049/el:20010315
K.
Kang,
W.X.
Zhang and
J.J.
Li
(State Key Laboratory of Millimeter
Waves, Southeast University, Nanjing,
210096,
People's Republic of
China)
14 March
2001
References
1
YIN,
M.,
KANG,
K.,
and
ZHANG,
w.x.:
'Suddifference beam duplexer
for coupled tapered slot-line antenna',
Electron. Lett.,
2000,
36,
(1
I),
pp. 935-936
2
BARRETT, R., BERRY, M., CHAN,
T.F.,
DEMMEL, J., DONATO,
J.,
DONGARRA, J., EIJKHOUT,
v.,
POZO,
R., ROMINE,
c.,
and
VAN
DER
VORST,
H.:
'Templates for the solution of linear systems; building
blocks for iterative methods'
(SIAM
Press, Philadelphia,
PA,
1994)
3
YNGVESSON,
K.s.,
KORZENIOWSKI,
T.L.,
KIM,
Y.,
KOLLBERG,
E.L.,
and
JOHNANSSON, J.F.:
'The tapered slot antenna
-
a new integrated
element for millimeter-wave applications',
IEEE Trans. Microw.
Theory Tech.,
1989,
37,
(2), pp. 365-374
Two-stage chaotic Colpitts oscillator
A.
TamaSeviEius,
G.
Mykolaitis,
S.
Bumeliene,
A.
Cenys,
A.N.
Anagnostopoulos and
E.
Lindberg
A
novel version
of
the chaotic Colpitts oscillator is proposed. It
contains two bipolar junction transistors coupled in series. The
resonance loop consists of an inductor and three capacitors. The
two-stage oscillator, compared with the classical circuit, enables
the fundamental frequency of chaotic oscillations to
be
increased
by a factor of three. The PSpice simulations performed with
9
GHz
threshold frequency transistors demonstrate that the highest
fundamental frequencies of chaotic behaviour are
1
and 3GHz
for
the classical and the two-stage Colpitts oscillator, respectively.
Introduction:
Classical oscillators such as the Colpitts, the Hartley
and the Wien-bridge, are commonly used to generate periodic
waveforms. However, with special sets
of
circuit parameters and
No.
9
5
49
circuitry modifications these oscillators can exhibit chaotic behav-
iour. For example, the classical Colpitts oscillator (Fig.
1)
with
a
bipolar junction transistor (BJT) has been demonstrated to gener-
ate chaos at the fundamental frequencyf
=
IOOkHz
[1].
Various
features of the chaotic Colpitts oscillator have been considered in
[2, 31. Chaos in the Colpitts oscillator has
also
been studied in the
HF range. Using the general purpose 2N2222A type BJT with
threshold frequency
fr
=
300MHz, chaotic oscillations have been
observed atf
=
25MHz
[4].
The PSpice simulations indicate that
for chaotic oscillations the highest
f
achievable with the 2N2222A
type transistor is -30MHz, i.e. ten times less than the fp Mean-
while periodic oscillations can be generated at frequencies very
close to thefP In the UHF range chaos has been predicted in the
Colpitts oscillator with the Avantek
AT41486
type BJT
(fT
=
3GHz) by means
of
PSpice simulations atf
=
SOOMHz
[4].
Very
recently, the Colpitts oscillator has been shown experimentally to
generate chaos in the UHF range
[5].
Employing a 2T938A-2 type
transistor (Russian
n-p-n
silicon BJT with fr
c-
5GHz) and a
microstrip line based distributed resonator, much more compli-
cated than the L-C1-C2 in Fig.
I,
the authors of
[5]
have demon-
strated narrowband
(Af
=
l0Y0)
chaos in the UHF range at
f
=
900
to 1000MHz.
In this Letter we propose a novel version of the chaotic Colpitts
oscillator, specifically a two-stage circuit. The modified circuit,
compared with the classical Colpitts oscillator, is shown to move
the highest fundamental frequency
J*
from approximately
0.1
to
0,3fT,
i.e. essentially closer to the threshold frequency
fT.
H
lo?
i-
m
Fig.
1
Circuit diagram
of
classical Colpitts oscillator
'OQ
tc2
Fig.
2
Circuit diagram
of
novel two-stage Colpitts oscillator
The fundamental frequency can be estimated as
r
I
I
0
0.2
0.4
0.6
0.8
1
.o
a
0
3
frequency,
GHz
C
Fig.
3
Power spectra
of
VK
from classical Colpitts oscillator
VO
=
15V,
IO
=
20mA
af
=
300MHz,
R
=
28l2,
L
=
25nH,
C1
=
C2
=
20pF
b
f
=
LGHz,
R
=
25Q
L
=
SnH,
C1
=
C2
=
5pF
c
f
=
3GHz,
R
=
25l2,
L
=
2.5nH, C1
=
IpF,
C2
=
0.5pF
I
I
0
0.2
0.4
0.6
0.8
1
.o
a
-20
m
-40
-60
1
2
b
A
3
-20
-
m
-40
-
v)
I
-60
Modijied
circuit:
The circuit diagram of the novel chaotic oscilla-
(Fig.
1)
it contains
an
extra transistor, Q2, coupled in series with
tion), and an extra capacitor, C3. The lower stage based on the Q2
acts
as
a current source for the upper stage based on the Q1, of
both the DC and the AC signals. The emitter of the Q1 is supplied
with the feedback signals
V,
and
Va+c3
from the capacitor C2
via the 42 and directly from the capacitor C3.
tor is presented in Fig.
2.
Compared with the classical oscillator
0
2
4
6
8
10
the transistor Q1 (both transistors are in "mon base configura-
C
1588/41
frequency,
GHz
Fig.
4
Power spectra
of
V,,from two-stage Cobitts oscillator
v1
=
3ov,
v2
=
15v,
Io
=
30mA
bf
=
IGHz,
R
=
34l2,
L
=
8nH, C1
=
4pF,
C2
=
C3
=
15pF
cf
=
3GHz,
R
=
20Q
L
=
2nH,
C1
=
IpF,
C2
=
C3
=
3pF
=
~OOMH~,
R
=
25~
L
=
25n~,
c1
=
a
=
c3
=
47pF
550
ELECTRONICS LETTERS
26th
April2001
Vol.
37
No.
9
Results:
To compare the features of the classical Colpitts oscillator
and of the novel chaotic oscillator, simulations of the circuits in
Figs.
1
and 2 were performed by means of the Electronics Work-
bench Professional simulator, based on the PSpice software. The
BFG520 type BJTs with
fr
=
9GHz were used in both circuits.
The simulation results are shown in Figs. 3 and 4. The fundamen-
tal frequencyf of the chaotic oscillations for the classical Colpitts
oscillator was not higher than
1
GHz.
Meanwhile, the
f
as high as
3GHz
can be achieved with the same type of BJT in the novel
oscillator, i.e.
f
could be increased by a factor of three.
This result was confirmed experimentally with the 2N3904 type
BJTs having similar characteristics to the above-mentioned
2N2222A transistors. The
f
could be shifted from 25MHz in the
classical chaotic Colpitts oscillator to 90MHz in the novel one.
Conclusion:
The suggested modification of the chaotic Colpitts
oscillator allows shifting of the fundamental frequencies to higher
ones, closer to the threshold frequencies of the transistors.
Acknowledgment:
The work of A. Tamaievicius,
G.
Mykolaitis,
A.
Cenys and A.N. Anagnostopoulos was supported in part by
NATO under Contract No. PSTCLG 977018.
0
IEE 2001
Electronics Letters Online
No:
20010398
DOI: lO.l049/el:20010398
A. TamaSeviEius, G. Mykolaitis,
S.
Bumelienk and A. Cenys
(Semiconductor Physics Institute, A. Goitauto 1
I,
LT-2600 Vilnius,
Lithuania)
E-mail: tamasev@uj.pfi.lt
A.N.
Anagnostopoulos
(Department
of
Physics, Aristotle University of
Thessaloniki, GR-54006 Thessaloniki, Greece)
E. Lindberg
(Department
of
Information Technology, 344, Technical
University
of
Denmark, DK-2800 Lyngby, Denmark)
G.
Mykolaitis: Also with the Vilnius Gediminas Technical University,
Sauletkkio 7, LT-2054 Vilnius, Lithuania
29 January
2001
References
1
KENNEDY,
M.P.: ‘Chaos in the Colpitts oscillator’,
IEEE Trans.
Circuits Syst.
I,
1994,
CAD-41,
(ll), pp. 771-774
2
LINDBERG,
E.:
‘Colpitts, eigenvalues and chaos’. Proc. 5th Int.
Workshop NDES’97, Moscow, Russia, June 1997, pp. 262-267
the Colpitts oscillator and applications to design’,
IEEE
Trans.
Circuits Syst.
I,
1999,
CAD-46,
(9), pp. 11
18-1
130
4
WEGENER,
c.,
and
KENNEDY,
M.P.: ‘RF chaotic Colpitts oscillator’.
Proc. 3rd Int. Workshop NDES’95, Dublin, Ireland, July 1995, pp.
5
PANAS,
A.,
KYARGINSKY,
B.,
and
MAXIMOV,
N.: ‘Single-transistor
microwave chaotic oscillator’. Proc. Int. Symp. NOLTA’2000,
Dresden, Germany, Sept.
2000,
pp. 445448
3
MAGGIO,
G.N.,
FEO, O.DE.,
and
KENNEDY,
M.P.:
‘Nonlinear analysis Of
255-258
Fault tolerant computation
of
large inner
products
L.
Imbert
and
G.A.
Jullien
A new technique for applying fault tolerance to modulus
replication RNS computations by adding redundancy
to
the
independent computational channels is introduced. This technique
provides a low-overhead solution to fault tolerant large inner
product computations.
Introduction:
In the MRRNS system, numbers are represented as
polynomials of indeterminates which are powers of 2. As an exam-
ple, if we use the indeterminate x
=
8,
we have
79
=x2
+Z
f7
=
x2
+22
-
1
=
2%’
-62
-
1
=
...
(1)
The MRRNS system makes use of the fact that every polynomial
of degree
n
can be uniquely defined by its values at
n
+
1
distinct
points, and that closed arithmetic operations can be performed
over completely independent channels [I]. Another advantage of
using the reduced dynamic range channels is the ability to imple-
ment the arithmetic over finite fields
GF(P)
[2]. If we choose
p
as a
ELECTRONICS LETTERS
26th
April
2007
Vol.
37
Fermat prime, i.e. a number of the form
22n
+
1
,
modular multi-
plication can be much simplified [3].
As in [4] we express
our
algorithm in term of matrix transfor-
mations and denote
Mp
and
Mil
as the matrices used for the
evaluation and the interpolation steps, respectively. In the follow-
ing, we shall use
mk
to denote the vector composed of the kth row
of
Mi1,
and
S
for the set of
N
distinct points. It is important that
N
must be large enough, not only to represent the input poly-
nomials, but more importantly the result of the computation(s).
The technique is very efficient for computing many additions
and multiplications in between the mapping and recovery stages,
particularly where we can restrict the number of cascaded multi-
plications to one in any signal flow path. Fortunately, inner prod-
uct computations, which are heavily used in
DSP
algorithms,
allow this restriction for an arbitrary inner product length.
The advantages of implementing VLSI include reducing the
interconnect span in the computational data path; this leads to
easier testing and lower power [3]. Since the computations are
independent we can also purposely skew the clocks between the
independent data paths, thus reducing the clock current spike.
One of the potential advantages of independent computations is
the ability to perform fault detection and correction at a much
redudd complexity compared to more classical computational
fault tolerant techniques. Although this has been previously
explored at the circuit level [5], no one has yet taken advantage of
the algebraic structure of the MRRNS. In this Letter we open the
exploration of
a
new technique for fault tolerance using the inde-
pendent computational structure of the MRRNS system.
Error
detection:
The detection technique we propose is based on
being able to easily compute the constant term of the final polyno-
mial since it depends only on the constant term of each of the ini-
tial polynomials involved in the computation,
Definition
1:
If
U,
v
are two vectors in
Gay,
we denote
d(u,
v)
the distance between
U
and
v
as the number of coordinates in
which
U
and
v
differ.
Thus, if
U
is the correct result and
v
is the vector obtained after
the inner product,
d(u,
v)
gives the number of errors. This defini-
tion can be seen as an extension of the Hamming distance used in
classical coding theory. The following theorem can be used to
detect a single error.
Theorem
I:
Let
w,
z
be the correct and computed vectors,
respectively, and let Q(x)
=
qo
+
qlx
+
q2x2
+...+
qnx“
be the final
polynomial. Let us assume
0
E
ml
(this is the case if
0
E
S),
and
d(w,
z)
S
1
,
i.e. that at most one error occurred. Then
d(w,
z)
=
1
iff
ml
.
zf
qo.
Prooj
See [6].
Error
correction:
Let us consider the following example. We com-
pute the inner product 79
x
47
+
121
x
25
=
6738 with correction
for one channel in error. We first define four polynomials that
correspond to the values 79, 47, 121, 25; let the indeterminate x
=
8,
and assign the following polynomial representations: 79
-+
-1
+
2x
+
xz,
47
+-1
+
6x, 121
-+
1
+
7x
+
x2
and 25
-+
1
+
3x. Since
the final polynomial is third-order, four distinct points are suffi-
cient for its representation, but to correct for one error, we add
a
redundant channel; i.e. we compute over five points. We will Iet
S
=
{-1,
1,
-2,
2,
3).
If we express the evaluation step in terms
of
.
matrix operations, the vectors composed of the polynomial coeffi-
cients now have five co-ordinates, the last one clearly being equal
to
0.
We also assume that the coefficients of the final polynomial
belong to the set 1-128,
...,
128}, which allows us to compute over
Gq257). Evaluating the polynomials at these five points gives the
vectors
u1
=
(-2, 2, -1, 7, 14),
v1
=
(-7,
5,
-13,
11,
17),
u2
=
(-5,
9,
-9, 19, 31) and
v2
=
(-2, 4, -5, 7, 10). The component-wise opera-
tions then give:
w1
=
u1
0
v1
=
(14,
10,
13, 77, -19),
w2
=
u2
0
v2
=
(10,
36, 45, -124, 53) and
w
=
w1
0
w2
=
(24, 46, 58, -47, 34).
The final result is recovered by computing
q=M,-,:xw
-128
1
77 128 -77 24
85
-85
-107 107
0
=
I
75 -22 -107 22 321
x
[[!j
43
-43
107 -107
0
-75 -107 30
107
45
=
(2,2,33,9,0) (2)
No.
9
55
1