WHY IS THERE AN INDUCTOR IN CHUA’S CIRCUIT AND WHY MIGHT WE WANT TO “ELIMINATE” IT?
Chua’s Circuit is basically an oscillator connected to a nonlinear resistor. An inductor is used in the oscillator portion of the Chua’s Circuit we discussed previously.
Electrical oscillations are produced when energy moves back and forth between the inductor and capacitor, C2, in Chua’s Circuit. That’s the way oscillators work. An electrical oscillator must have both an inductor and a capacitor.
You can trick the oscillator circuit into thinking that an inductor is present by using a synthetic inductor (aka “impedance converter” or “gyrator”). OK, but are there any reasons why would we would not want to use a real inductor in Chua’s Circuit?
Well, no, there aren’t any really compelling reasons. The inductor needed for Chua’s Circuit is not all that expensive nor is it difficult to obtain.
Nonetheless, we are going to talk about how we can build a version of Chua’s Circuit using a synthetic inductor just because we will learn some really interesting new things in the process. Learning something new is always fun.
Before we get to our “inductorless” Chua’s Circuit, however, let’s talk about what inductors are and what their drawbacks are. Then we will understand what motivated people to figure out interesting ways to eliminate them. (It’s more accurate to say “synthesize” them.)
WHAT’S AN INDUCTOR?
An inductor is basically an electrical circuit component that stores energy in a magnetic field. (Similarly, a capacitor is an electrical circuit component that stores energy in an electrical field.)
Whenever an electrical current flows through a wire, a magnetic field is created that surrounds that wire. Electrical energy is stored in that field.
If we wind the wire into a multi-turn coil, the strength of the magnetic field produced is increased. So, too, is the amount of energy stored in that field. You also can increase the strength of the magnetic field produced by an inductor by winding the wire on an iron core.
The unit that measures the ability of an inductor to create a magnetic field and store energy in that field is the “Henry” (H). One Henry is a fairly large amount of inductance so often you will see inductors with values measured in millihenries (mH) or microhenries (µH).
ARE THERE DOWNSIDES ASSOCIATED WITH USING INDUCTORS?
In general, yes. Inductors with any substantial amount of inductance tend to be physically large, heavy, and expensive. Equally importantly, if there is more than one inductor in a circuit, their magnetic fields can interact with each other. This can cause serious problems unless the inductors are magnetically shielded from one another.
FILTER CIRCUITS SPARKED THE NEED TO ELIMINATE INDUCTORS.
There are many occasions when it is desirable to have an electrical circuit pass (or alternatively, block) only a certain range of frequencies. Filter circuits are used to accomplish this. Worldwide, communications systems use many millions of filter circuits.
Prior to the advent of op-amps, the only way to build filter circuits was to use capacitors and inductors. A really good filter circuit, however, requires many capacitors and many inductors. We’ve already discussed the drawbacks of inductors: namely their size, weight, cost, and the fact they have to be magnetically shielded when they are located near one another. A way to eliminate the inductors in filter circuits was needed.
THE OP-AMP COMES TO THE RESCUE (AGAIN).
Previously, we talked about op-amps and some of the amazing things that they can do. One thing that wasn’t mentioned is that an op-amp can act as an “impedance converter.”
For our purposes, the term “impedance converter” means an op-amp circuit that can make a capacitor act as an inductor or make an inductor act as a capacitor. Since there is seldom any reason to want to make an inductor act as a capacitor, we will focus our discussion on using op-amps to make a capacitor act as an inductor.
Several different impedance converter circuits have been devised to eliminate inductors. We will discuss just one of these circuits. It’s called the “Antonio Inductance-Simulation Circuit.” If you want to read a detailed but easy to understand discussion of this circuit, you can click on the title below.
AN INDUCTORLESS CHUA’S CIRCUIT
The Antoniou inductance-simulation circuit is used in the inductorless version of Chua’s Circuit described in the paper “INDUCTORLESS CHUA CIRCUIT” listed below which I put together using information from Valentin Siderskiy’s website
You can click on the link below to read the paper.
The schematic diagram and the component list for the inductorless Chua’s Circuit we will discuss are shown below.
INDUCTORLESS CHUA’S CIRCUIT
COMPONENTS
R=2.5 kΩ (pot.) C=100 nF
R1=220 Ω C1=10 nF
R2=220 Ω C2=100 nF
R3=2.2 kΩ
R4=22.0 kΩ
R5=22.0 kΩ
R6=3.3 kΩ
R7=100 Ω
R8=1.0 kΩ
R9=1.0 kΩ
R10=2.5 kΩ (pot.)
All op-amps are TL082 or equivalent.
The variable resistor, R10, allows you to adjust the value of the simulated inductance precisely.
As you can see in this schematic, the inductor in the original Chua’s Circuit we discussed previously (on Page 5) has been replaced by the Antoniou inductance simulation circuit located in the box to the left of C2. The value of the simulated inductance is given by
L = (R7 x R9 x R10 x C)/R8
L is in Henries when the resistor values are in Ohms and C is in Farads.
Set R10 to 1.8 kOhms and use the values given for R7, R8, R9, and C.
(Remember, 1 nF = 1 x 10^(-09) F thus 100 nF = 1 x 10^(-07) F )
With these values, L = .018 H. This is the value of L used in the original Chua’s Circuit. Alternatively, you can replace R10 with a 1.8 kOhm fixed resistor.
SIMULATING THE INDUCTORLESS CHUA CIRCUIT
I simulated the inductorless Chua Circuit shown in the schematic above with one slight modification. I added a 14 Ohm resistor between R7 and the C2-R node (junction).
I did this to include the 14 Ohm resistance of the real 18 mH inductor used in the version of Chua’s Circuit we discussed on pages 5 and 8. The Antoniou inductance-simulation circuit, by itself, yields an inductance with zero internal resistance.
The LTspice schematic diagram for the inductorless Chua Circuit is shown below. The value of R between C1 and C2 is 1.50k Ohms.
Click on image to enlarge
The strange attractor produced by this circuit is shown below.
Click on image to enlarge
The strange attractor for this inductorless Chua Circuit is virtually identical with that for the Chau Circuit which used a real inductor and TL082 op-amps (with R7 = 1.50k Ohms) on page 8.
The LTspice .asc file for this circuit can be obtained by clicking on the following link.
Remember to drop the .txt follower when you download and save this file so that you end up with a file named INDUCTORLESS_CHUA_TL082.asc
I strongly encourage you to build this circuit using real components. Then compare the strange attractor you get from this circuit with the one obtained from the Chua Circuit you built previously that uses an actual inductor.
