5/13/2020
12. A SIMPLER CHAOTIC OSCILLATOR | CHAOTIC CIRCUITS
CHAOTIC CIRCUITS
Electronic Circuits that Exhibit Chaotic Responses
12. A SIMPLER CHAOTIC OSCILLATOR
REMEMBERING CHUA’S
CIRCUIT
In our previous work, we learned
that Chua’s Circuit is a chaotic
oscillator. In its fundamental form,
Chua’s Circuit requires an inductor,
two capacitors, plus a linear resistor
and a non-linear resistor. Two opamps and several more linear
resistors are required to implement
the nonlinear resistor.
When we eliminated the real
inductor by using a synthesized one,
we had to use even more op-amps
and linear resistors. The result was a
somewhat complex circuit.
https://www.chaotic-circuits.com/chaotic-oscillators/
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There are simpler chaotic oscillator
circuits that have been developed
that do not require op-amps. Some
do not require inductors. The circuit
that we will discuss here does not
require either.
“Build a chaos
generator in 5
minutes!”
This is the title of an “Instructables”
article by a person named “Kajnjaps”
which was published in May of 2016.
The article describes a chaotic
oscillator that does not require an
inductor nor any op-amps. This
article can be read by clicking on the
following link.
Build a chaos generator in 5 minutes!
https://www.chaotic-circuits.com/chaotic-oscillators/
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Alternatively, you can go to the
original article on the
“Instructables” web site by clicking
on
http://www.instructables.com/id/A-SimpleChaos-Generator/
I also have included the source paper
for Kajnjaps “Instructable” article.
This source paper is entitled
“Simple Two-Transistor SingleSupply Resistor-Capacitor Chaotic
Oscillator” and was written by Lars
Keuninckx, Guy Van der Sande and
Jan Danckaert. This paper appeared
in the December 2015 issue of
the IEEE TRANSACTIONS ON
CIRCUITS AND SYSTEMS II.
The IEEE article can be read by
clicking on the link below. You will
see that this article is, in many ways,
virtually identical with the
“Instructables” article. The IEEE
https://www.chaotic-circuits.com/chaotic-oscillators/
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article, however, does include a
discussion of the mathematical
model for this circuit together with a
very good bibliography. It is very
much worth reading.
Simple Two-Transistor Single-Supply RC Chaotic Oscillator
It turns out that “Kajnjaps” is a
pseudonym for one of the three
authors of this IEEE article.
In any event, the schematic diagram
for this circuit is shown below. It
was taken from the “Instructables”
document but is identical with that
in the IEEE paper.
https://www.chaotic-circuits.com/chaotic-oscillators/
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(Click on the image to enlarge it.)
The circuit is that of a resistorcapacitor (R-C) phase shift oscillator.
All oscillator circuits require that a
portion of the output signal be fedback to the input, in phase with the
input. This is called “positive
feedback” and results in the output
being sustained.
In the circuit above, the output at the
collector of transistor Q1 is fed-back
to its base (input) through a threesection R-C network. Each section of
that network produces a phase shift
of 60 degrees. The transistor, itself,
produces a phase shift of 180 degrees
between the signal at its base and
that at its collector.
The total phase shift due to the R-C
network is, therefore, 360 degrees. A
360 degree phase shift is equivalent
to a zero degree phase shift. Thus,
https://www.chaotic-circuits.com/chaotic-oscillators/
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12. A SIMPLER CHAOTIC OSCILLATOR | CHAOTIC CIRCUITS
the signal fed-back is in phase with
that already at the input of Q1 and
oscillations are sustained.
The frequency of the oscillations is
determined by the values of the
resistors and capacitors in the
feedback network. For the values of
the resistors and capacitors shown,
the frequency of oscillation is
approximately 45 kHz.
As long as the value of resistor R3 is
large, transistor Q2 does not conduct
and the components in the dotted
box have no effect on the operation
of the oscillator. When the value of
R3 is reduced, however, chaotic
effects can be observed.
Let’s Try It Ourselves
https://www.chaotic-circuits.com/chaotic-oscillators/
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The circuit provided in the
“Instructables” article is especially
easy to simulate in LTspice. The
schematic diagram I created in
LTspice for simulating this simple
chaotic oscillator is shown below.
Click on the image to enlarge it.
As you can see, this schematic is
identical with that shown in the
“Instructables” article. There is one
important fact to keep in mind.
I used two BC547C transistors in my
LTspice simulation, as did the author
of the “Instructables” article in the
physical circuit he built. These
transistors are contained in the
LTspice component library.
https://www.chaotic-circuits.com/chaotic-oscillators/
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However, any other common, easily
obtained and low-cost general
purpose NPN transistors such as the
2N3904, the 2N2222, or the 2N2222A
can be used in this circuit. These,
too, are contained in the LTspice
component library.
Also, while the values of the resistors
and capacitors in this circuit are not
especially critical, you may have to
“tweak” their values a bit to get your
strange attractor images to look
exactly like those in the
“Instructables” article. This is
because the author of that article
showed what his actual circuit
produced on an oscilloscope.
We need to remember that none of
the component models in any
computer simulation program are
perfectly precise representations of
“real” components. In addition, all
real circuit components have
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tolerances, however small, and
therefore are not identical from one
to another even though they may
have identical markings.
The following is the LTspice .asc
file for this circuit.
SIMPLE CHAOTIC OSCILLATOR.asc
RESULTS
The strange attractors I obtained
from my LTspice simulations are
shown below. In each case, the value
of R3 was equal to 31k Ohms.
x-axis: vce1, y-axis: vce2
Click on the image to enlarge it.
https://www.chaotic-circuits.com/chaotic-oscillators/
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x-axis: v1, y-axis: v2
Click on the image to enlarge it.
x-axis: vce1, y-axis: v1
Click on the image to enlarge it.
The strange attractors that I got are
quite similar (but not identical) to
those shown in the “Instructables”
paper. The reasons for the slight
differences were discussed earlier.
https://www.chaotic-circuits.com/chaotic-oscillators/
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I then built a physical circuit using
2N3904 transistors and, again, my
results were very similar to my
LTspice simulations and to those
shown in the “Instructables” paper
(both of which used BC547C
transistors). It’s always amazing to
me how well LTspice can simulate
real circuits.
https://www.chaotic-circuits.com/chaotic-oscillators/
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