# 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 op-amps 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.

## 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!

## Alternatively, you can go to the original article on the “Instructables” web site by clicking on

### http://www.instructables.com/id/A-Simple-Chaos-Generator/

## I also have included the source paper for **Kajnjaps “Instructable” article. ****This source paper is entitled “Simple Two-Transistor Single-Supply 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 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.

##### (Click on the image to enlarge it.)

## The circuit is that of a resistor-capacitor (R-C) phase shift oscillator. All oscillator circuits require that a portion of the output signal be fed-back 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 Q_{1} is fed-back to its base (input) through a three-section 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, the signal fed-back is in phase with that already at the input of Q_{1} 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 R_{3} is large, transistor Q_{2} does not conduct and the components in the dotted box have no effect on the operation of the oscillator. When the value of R_{3} is reduced, however, chaotic effects can be observed.

_{ }

# Let’s Try It Ourselves

## 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.

## 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 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.

#### 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.

## 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.