SIMULATE USING LTSPICE
If you plan to simulate Chua’s Circuit using LTspice, I strongly urge you to again read the LTspice manuals listed on page 4 of this website and perform the suggested exercises.
Two simulations of Chua’s Circuit which each use a different form of the nonlinear resistor are presented below. The resulting strange attractors that are produced are similar, yet a little different. Even if you are new to using LTspice, you will find it easy to do the simulations.
I first recommend that you visit Juho-Eric’s fine website where, among other things, he has simulated a version of Chua’s Circuit. The url is:
http://juho-eric.blogspot.com/2011/12/ltspice-simulation-of-chuas-circuit.html
A direct link to his website is provided below.
Juho-Eric has simulated Chua’s Circuit using LTspice and has provided a lot of useful information. He even has the LTspice circuit file available for your download and use. (LTspice circuit files all have an .asc extension even though they are actually .txt files.)
If you chose to use Juho-Eric’s circuit file, download the file into a folder of your choice. (Why not name the folder “Chua’s Circuit”?)
If for some reason you are unable to download Juho-Eric’s .asc file from his website, click on the link below for the copy I downloaded. After you save it, remember that the file name must have an .asc extension and not a .txt extension.
SIMULATING FROM A SAVED .ASC FILE
For the benefit of those who are new to LTspice, I will describe a step-by-step procedure by which you can simulate Chua’s Circuit using Juho-Eric’s circuit file.
1. From the LTspice toolbar, select “File” and then “Open.”
2. Go to the folder containing Juho-Eric’s .asc file and select it. The circuit diagram should now be visible on your monitor.
3. Select “Simulate” on the toolbar and then “Run.”
4. Select “View” from the toolbar and click on “Visible traces.”
5. Choose “V(v2)” and the voltage across C2 will appear as a function of time. (It looks like a bunch of noise.)
What you want is an x-y display with V(v2) plotted on the vertical axis and V(v1) plotted on the horizontal axis.
6. Put your cursor on the horizontal axis of the plot and left click. A box will appear labeled “Horizontal Axis.”
7. In that box is a line labeled “Quantity Plotted.” On that line the word “time” currently appears.
8. Change the word “time” to “V(v1)” and then click “OK.” You now should see your first strange attractor.
Remember that the variables for this system are the voltage (v1) across C1, the voltage (v2) across C2, and the current I(L1) through the inductor. Any one of these quantities plotted versus another gives an interesting view of the strange attractor.
Juho-Eric used a different circuit to create Chua’s Diode and used different values for some of the other circuit components compared with the reference papers listed on the previous page. Nonetheless, what Eric used is still truly “Chua’s Circuit.” You will enjoy simulating it. His circuit is shown below.
(Click on the image to enlarge it.)
Shown below are a few of the many strange attractors I obtained when I simulated Eric’s version of Chua’s Circuit with LTspice. I varied the value of the resistor (R6) connecting C1 and C2 between approximately 1500 Ohms and 1900 Ohms.
(I use the “Snipping Tool” utility that comes with Windows 7 to capture the simulation displays from LTspice. Click on the images to enlarge them.)
SIMULATING FROM A NETLIST
I also am making available the netlist that was generated by LTspice when I used Juho-Eric’s circuit file. A netlist allows you to simulate the circuit but it does not produce a schematic diagram of the circuit.
1. Open the file above by clicking on it.
2. Do a “Save-As” of the file. This will allow you to change the file name if you wish.
3. Save the file in a folder of your choice. (You might want to call the folder “LTspice Netlists.”)
4. LTspice wants netlist files to have an extension of .net (or .cir or .sp ) so now change the .txt extension of the file you just saved to one of these. (I prefer to use the .net file extension on my netlists.)
To simulate Chua’s Circuit from this netlist, do the following:
1. From the LTspice toolbar, select “File” and then “Open.”
2. Go to the folder containing the netlist and select it. The netlist should now be visible on your monitor.
3. Select “Simulate” on the toolbar and then select “Run.”
4. Select “Plot Settings” from the toolbar and click on “Visible traces.”
5. Choose “V(v2)” and the voltage across C2 will appear as a function of time. (It looks like a bunch of noise.)
What you want is an x-y display with V(v2) plotted on the vertical axis and V(v1) plotted on the horizontal axis.
6. Put your cursor on the horizontal axis of the plot and left click. A box will appear labeled “Horizontal Axis.”
7. In that box is a line labeled “Quantity Plotted.” On that line the word “time” currently appears.
8. Change the word “time” to “V(v1)” and then click “OK.” You now should see the strange attractor.
(Note that Step 4. when performing the simulation from a netlist is slightly different from what you do if you are performing the simulation from a saved .asc file.
MY SIMULATION OF CHUA’S CIRCUIT USING TL082 OP-AMPS
I have made my own LTspice simulation of Chua’s circuit using the paper cited earlier by Michael Kennedy as a guide. Kennedy’s circuit uses TL082 op-amps and utilizes a different form of Chua’s Diode than did Juho-Eric. I did and it is the version that I like best. The schematic diagram I made using LTspice is shown below. The 14 Ohm resistor (R8) in series with the inductor represents the resistance of the inductor’s windings.
(Click on the image to enlarge it.)
I have included the .asc file for this circuit.
I also have included the LTspice netlist.
As before, after you save these files remove any .txt or other extensions so that you end up with only CHUA_TL082.asc and CHUA_TL082.net (respectively) as the saved file names.
PLEASE NOTE: I have included all the necessary information for the TL082 op-amps on the schematic (in a form called a “spice directive”) so that it is not necessary for you to have added the TL082 to your LTspice component library. Everything is ready for you to run, either as an .asc file or as a netlist.
Below are several of the strange attractors this circuit produced. You can click on the images to enlarge them.
(R7 = 1.40K Ohms)
(R7 = 1.50K Ohms)
(R7 = 1.60K Ohms)
(R7 = 1.70K Ohms)
NOW TRY A COMPLETE SIMULATION ON YOUR OWN.
I encourage you now to create the schematic diagram for Chua’s Circuit from scratch in LTspice so that you will know how to simulate other chaotic circuits you may come across. You can use the LT1113 op-amps that already are in the LTspice component file, if you wish or you can follow the instructions on page 4 of this website and add the TL082 op-amp to the LTspice component library, if you have not already done so.
DON’T JUST SIMULATE, ACTUALLY BUILD THE CIRCUIT.
We talked about building Chua’s Circuit out of actual components on page 7 of this website. I strongly encourage you to do so, if you have the inclination. It’s easy and fun!
WE’RE ALMOST READY TO CREATE THE LORENZ BUTTERFLY.
Before we get back to the Lorenz butterfly, however, I want to finish discussing the early history of chaos by examining the work of Otto Rössler. He is a German chemist who made significant contributions to chaos theory after the work of Lorenz but before Chua’s work.
