Year 2 Project

Build and Study electronic circuits that have chaotic behavior

Name Di Peng (ID 200907437) Name Yifan Liu (ID 200780972) Group 41

Supervised by Pro. Steve Hall

March 22, 2013

Abstract This project aims to investigating the chaotic behavior of the electronic circuit. The chaotic is an aperiodic behavior appears in deterministic nonlinear system that is extremely sensitive to initial status. Chaotic circuit is the circuit with nonlinear components, which is diode in this project. The objective of this project is to build chaotic circuit and analyze chaotic behavior in the circuit. RLD is used as the chaotic circuit in the experiment. The experiment is carried out basically successful, important results and conclusions has acquired from the experiment and looking up the papers of predecessors online. In addition, PSpice is used to get simulation results to compare with experimental results. To analyze the chaotic behavior, the relative knowledge such as resonance frequency, diode capacitance, bifurcation phenomenon and Fiegenbaum constant are included. This report will show the method, results, analysis and conclusion in details.

Contents
1 Introduction 1.1 Background information . . . . . . . . 1.2 Theory . . . . . . . . . . . . . . . . . . 1.2.1 RLD circuit . . . . . . . . . . . 1.2.2 Resonance frequency . . . . . . 1.2.3 Diode Capacitance . . . . . . . 1.2.4 Chaotic behavior: bifurcation, harmonic . . . . . . . . . . . . 1.3 Objective . . . . . . . . . . . . . . . . 1.4 Structure . . . . . . . . . . . . . . . . 7 7 8 8 9 9

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. 10 . 10 . 11 12 12 13 13 13

2 Materials and Methods 2.1 Material . . . . . . . . . . . . . . . . . . . . . 2.2 Method . . . . . . . . . . . . . . . . . . . . . 2.2.1 Preparation . . . . . . . . . . . . . . . 2.2.2 Resonance frequency . . . . . . . . . . 2.2.3 The relation between chaotic behavior frequency of input voltage . . . . . . . 2.2.4 Normal behavior in equivalent circuit . 2.2.5 PSpice simulation . . . . . . . . . . . .

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3 Result and analysis 19 3.1 Resonance frequency . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.2 Chaotic behaviour with changing input frequency . . . . . . . . . . 23 3.3 Chaotic behaviour with changing input amplitude . . . . . . . . . . 30 4 Discussion 4.1 Discussion of the RLD circuit . . . . . . . . . . . . . . . . . . . . 4.1.1 The non-linear capacitance of diode . . . . . . . . . . . . . 4.1.2 Chaotic behaviour of RLD circuit . . . . . . . . . . . . . . 4.2 The resonance frequency . . . . . . . . . . . . . . . . . . . . . . . 4.3 Identify the chaotic behaviour . . . . . . . . . . . . . . . . . . . . 4.3.1 Time domain output waveform plots . . . . . . . . . . . . 4.3.2 Output signal spectrum plots . . . . . . . . . . . . . . . . 4.3.3 The X-Y mode plots . . . . . . . . . . . . . . . . . . . . . 4.4 Discussion of bifurcation and verification of Fiegenbaum constant 4.5 Error discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 38 38 41 42 42 42 43 45 45 47

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4.5.1 Equipment and environment 4.5.2 The extra capacitance . . . Assessment and further research . . Applications of chaotic circuit . . .

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5 Conclusion References

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List of Figures
1.1 1.2 1.3 2.1 2.2 2.3 2.4 2.5 2.6 2.7 3.1 RLD circuit used in experiment(taken from [2]) . . . . . . . . . . . 8 Diode forward bias (taken from [2]) . . . . . . . . . . . . . . . . . . 10 Diode reverse bias(taken from [2]) . . . . . . . . . . . . . . . . . . 10 Equivalent circuit with capacitor replacing diode RLD circuit on PSpice . . . . . . . . . . . . . . Parameter of D1N4007 . . . . . . . . . . . . . . Analysis setup . . . . . . . . . . . . . . . . . . . The output voltage waveform on PSpice . . . . Analysis Setup when simulate chaotic behavior . Equivalent circuit on PSpice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 15 16 16 16 17 18

the variation tendency of the output voltage against the input frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 the simulation plot of output voltage against input frequency with resistance lkΩ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 the resonance frequency plot with 5kΩ resistor . . . . . . . . . . . 3.4 the resonance frequency plot with 10kΩ resistor . . . . . . . . . . 3.5 the above groups of plot are the time and frequency domain output plots and X-Y mode plots with different frequency and fixed 5V amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 the above groups of plot are the time and frequency domain output plots and X-Y mode plots with different frequency and fixed 10V amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 the time and frequency domain plots of RLC circuit under 5V input amplitude and variable frequency . . . . . . . . . . . . . . . . . . 3.8 the time and frequency domain plots of RLC circuit under 10V input amplitude and variable frequency . . . . . . . . . . . . . . . 3.9 the Pspice simulation results under the 5V stationary amplitude condition and the frequency was changed . . . . . . . . . . . . . . 3.10 the above groups of plot are the time and frequency domain output plots and X-Y mode plot with different amplitude and 45.09 kHz fixed frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.11 the above groups of plot are the time and frequency domain output plots and X-Y mode plot with different amplitude and the resonance frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.12 the time and frequency domain plots of RLC circuit with variable amplitude and 60 kHz input frequency . . . . . . . . . . . . . . .

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3.13 the time and frequency domain plots of RLC circuit with variable amplitude and 95 kHz input frequency . . . . . . . . . . . . . . . . 36 3.14 the Pspice simulation results with input resonance frequency and the variable amplitude . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.1 4.2 4.3 the time domain plot of complete chaotic status (input frequency was 95 kHz and the amplitude was 4.5V) . . . . . . . . . . . . . . . 43 the spectrum plot of complete chaotic status (input frequency was 95 kHz and the amplitude was 4.5V) . . . . . . . . . . . . . . . . . 43 the X-Y mode plot of complete chaotic status (input frequency was 95 kHz and the amplitude was 4.5V) . . . . . . . . . . . . . . . . . 45

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List of Tables
2.1 2.2 3.1 3.2 3.3 4.1 4.2 Components used in the experiment . . . . . . . . . . . . . . . . . . 12 Material used in the experiment . . . . . . . . . . . . . . . . . . . . 12 the output voltages (voltage across the resistor) collected with variable input frequency . . . . . . . . . . . . . . . . . . . . . . . . . . 20 the frequencies that led to bifurcations and complete chaos under different fixed amplitude . . . . . . . . . . . . . . . . . . . . . . . . 28 the amplitudes that led to bifurcations and complete chaos under different fixed frequency . . . . . . . . . . . . . . . . . . . . . . . . 35 the calculated Fiegenbaum constant with fixed amplitude and changeable frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 the calculated Fiegenbaum constant with fixed frequency and changeable amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

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Chapter 1 Introduction
1.1 Background information

The topic of this mini project is building and studying electronic circuits that have chaotic behavior. The chaotic behavior was firstly found by a meteorologist named Lorentz in 1963 when he was analyzing a forecast model of aerodynamics. The behavior he found could not be explained by linear theory. Then after 12 years research on this weird nonlinear behavior, scientists defined it as chaotic behavior. [1] Electrical circuit with chaotic behavior is also called chaotic circuit. Its an electronic circuit with nonlinear characteristics. A nonlinear system is the system whose output is not proportional to the input. In other words, a system that keeps changing with time is a nonlinear systems, its time evolution equation is nonlinear. The feature of nonlinear system is that the properties are described by dynamical variables. [2] A tiny change in a parameter can cause qualitative and quantitative changes in the system. For electrical circuits, the nonlinear circuits are circuits with nonlinear components and the output not proportional to the input. For the previous study, most of the experiments or projects are focus on the linear system and linear circuits because they are straightforward and easy to be represented by mathematics. But in nature, the nonlinear systems are more important than the linear system, because most of the systems existed in nature are linear systems. And for most of the systems, if the input is big or high enough, it will become nonlinear. For example, if the input light intensity for a dielectric crystal is too big and the output light intensity is not proportional to the input one, it will become a nonlinear dielectric crystal. And if the displacement for a spring is too big, the Hookes law will not work and it will become a nonlinear 7

spring. Chaotic behavior is a complex nonlinear behavior. In addition, chaos implies aperiodicity. So chaotic behavior can be defines as an aperiodic behavior happens in deterministic nonlinear system that is extremely sensitive to initial status. Chaotic behavior in electronic circuit has many useful applications in many areas. For instance, chaotic encryption is an advanced technology to transmit secret signals. Apply chaotic behavior in carrier signal to protect the useful information. The application will be introduced in detail in discussion part.

1.2
1.2.1

Theory
RLD circuit

In this project, RLD circuit is used to investigate the chaotic behavior. RLD circuit is the most simple model of Chuas circuit. Chua’s circuit is a straightforward electronic circuit that has chaotic behavior. It was discovered by Leon O. Chua in 1983 when he was visiting Waseda University in Japan. To show the chaotic behavior in a circuit, there are some conditions must be satisfied. First, the circuit contains at least one nonlinear element. In this project, diode in RLD circuit is the nonlinear element. Secondly, a resistor is necessary to show the chaotic behavior. The RLD circuit used in the project is very straightforward, its a circuits with a diode, a resistor and an inductor in series shown in Figure 1.1.

Figure 1.1: RLD circuit used in experiment(taken from [2])

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1.2.2

Resonance frequency

Resonance frequency is of vital importance for observing chaotic behavior. It is the signal frequency that is relative to the maximum amplitude of the output voltage. It is also the frequency where chaotic behavior may happen. In theory, the resonance frequency of a circuit is determined by the inductance of the inductor and capacitance of the capacitor. In the RLD circuit, the diode has capacitance so it can be regard as a capacitor with changing capacitance. The formula of resonance frequency is: √ f=1 / (2π LC)

1.2.3

Diode Capacitance

The diode used in the RLD circuit can be seen as a parallel combination of diode resistance r, junction capacitance Cj and diffusion capacitance CT. The diode resistance of the diode changes with the change of Q point. In the experiment, the diode resistance can be regard as a constant. The diffusion capacitance only appears in forward bias and is determined by the current at Q point and storage time. Also can be regard as a constant in the experiment. The only variable in diode is the junction capacitance. The junction capacitance of a diode is just like an ordinary capacitor, the capacitance is determined by the space between the positive charge on p side and the negative charge on the n side. But the only difference between this junction capacitance and normal capacitance is that the plate separation changes when the voltage across the diode changes. The details how it changes will be illustrated in Discussion part. The RLD circuit behaves in two modes. The first mode is when the diode is in forward bias, the diode can be seen as a fixed bias in this mode. (Figure 1.2) The second mode is when diode is in reversed bias, the diode can be seen as a capacitor with changing capacitance. (Figure 1.3)

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Figure 1.2: Diode forward bias (taken from [2])

Figure 1.3: Diode reverse bias(taken from [2])

1.2.4

Chaotic behavior: bifurcation, period doubling and sub-harmonic

Chaotic behavior contains many aspects of abnormal phenomenon, in this project, researchers define bifurcation, period doubling and sub-harmonics as the chaotic behavior in electronic circuit. In linear electronic system, one input voltage corresponds to one output voltage. But in chaotic behavior, one input voltage may produce more than one height of peak value of output voltage. If use oscilloscope to observe the x-y plot of the input voltage and output voltage, there will be more than one loop on screen, this is the bifurcation. Each time a bifurcation happens, the period will double and there will be sub-harmonic appear. The details of how these happen will be illustrated in discussion part.

1.3

Objective

There are two main objective of this project. The first one is to design and build the circuit and observe the chaotic behavior in the circuit. The design and build 10

work is quite simple. But the observe work is a little difficult, because getting the chaotic behavior requires certain conditions of amplitude and frequency of input voltage. The second objective is to analyze the chaotic behavior. In this project, researchers analyze chaotic behavior from following aspects: the definition of chaotic behavior, how to distinguish chaotic behavior and normal behavior and how does chaotic behavior happen.

1.4

Structure

In this report, there are mainly four parts: material and method, results, discussion, conclusion. In the material and method part, the material used in the experiment is listed. And the method and procedure of the experiment will be illustrated step by step in details. In addition, the simulation part conducted on computer using PSpice will be shown with screenshot and notes. In the results part, the results of this project are shown in the form of words, pictures and tables, following with appropriate explanation and analysis. In the discussion part, there is integral analysis of the whole project and will answer questions listed in Objective part. In addition, the limitation and how to improve are listed in discussion part. In conclusion part, the conclusion of the whole project will be shown.

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Chapter 2 Materials and Methods
2.1 Material

The material used in the experiment is listed in the table below. Table 2.1: Components used in the experiment Components Resistor Diode (D1N4007) Inductor Theoretical value Experimental value 1kΩ 986 Ω NA 87.6pF 39mH 38.6mH

Table 2.2: Material used in the experiment Components Type Multimeter NA Function Generator FG200 Digital Oscilloscope TDS101213 Bread board SK10

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2.2
2.2.1

Method
Preparation

Before connecting the circuit, the first thing need to do is to measure the experimental value of components, including inductor, resistor and diode. Use multimeter to measure these values and write them down on the logbook. The equipment used in the experiment includes oscilloscope and function generator. The calibration work is also very important to get accurate experimental results, especially for the oscilloscope. Adjust the oscilloscope to make the indication is around zero when there is no input. After the preparation, the next work is to build the circuit. Connect the resistor, inductor and diode in series on the board.

2.2.2

Resonance frequency

The first task in this experiment is to measure the resonance frequency of the circuit. Resonance frequency is the signal frequency that is relative to the maximum amplitude of the output voltage. Usually corresponds to chaotic behavior. Set the frequency of the function generator at a low frequency initially, 50Hz in this experiment. Adjust the peak to peak value of the function generator to 100mV. Because the function generator used in this experiment could not show the input voltage. Oscilloscope is used to read the value of input voltage supplied by function generator. Connect the oscilloscope and function generator. Times 10 probe is used because of the noises existing in the oscilloscope. After set the peak to peak voltage of function generator to 100mv, connect it into the circuit. And connect the oscilloscope to the terminals of resistor to display the output voltage on the screen. Observe the waveform of the output voltage. Then slowly increase the frequency of the input voltage. Watch the change of the output voltage with the change of the frequency. Draw a diagram to show the tendency. When the voltage across the resistor suddenly increased to a very high value and has reached its maximum value, the frequency of the AC power at this time is the resonance frequency. Then with frequency keeps increasing, the voltage across the voltage will decrease to normal value. 13

In this RLD circuit, in theory, the inductor should behave like open circuit and diode should behave like short circuit when conducted under AC condition. So the voltage across the resistor should be very small. When the voltage across the resistor suddenly increases to a high value, this must be something chaotic happen.

2.2.3

The relation between chaotic behavior and amplitude and frequency of input voltage

After measuring the resonance frequency of the input voltage, chaotic can be observed by adjusting the amplitude and frequency. In the experiment, the chaotic behavior is observed in three ways: output voltage waveform, output spectrum, x-y plot of input voltage and output voltage. The next step is to fix the amplitude and change the frequency of the input voltage. The purpose is to find the range of frequency that chaotic behavior appears. First fix the peak to peak voltage of input voltage to 1.2V. Slowly increase the frequency. Observe the x-y plot of the input voltage and output voltage. Record the range of the frequencies where generate two bifurcations, four bifurcations, infinite bifurcations. Then change the amplitude of the peak to peak voltage to 2V, 5V, 10V, 20V one by one. Record the range of the frequencies where generate bifurcation for each group. The next step is to fix the frequency and change the amplitude of the input voltage. The purpose is to find the relation between chaotic behavior and amplitude of input voltage. Fix the frequency of input voltage to 35kHz. Slowly increase the amplitude. Observe the x-y plot of the input voltage and output voltage. Record the range of the amplitudes where generate two bifurcations, four bifurcations, infinite bifurcations. Then change the frequency of the output voltage to 45kHz, 55kHz, 95kHz one by one. Record the range of the frequencies where generate bifurcation for each group.

2.2.4

Normal behavior in equivalent circuit

After investigating the chaotic behavior of the RLD circuit, the next task is to find out the normal behavior of a RLC circuit to compare with the chaotic behavior. The capacitance in the RLD circuit is 89 pF, so researchers use a 22pF capacitor 14

and a 68pF capacitor in parallel to replace the diode. The circuit is shown in Figure 2.1.

Figure 2.1: Equivalent circuit with capacitor replacing diode Fix the peak to peak value of the input voltage constant, change the frequency. Observe the change of the output voltage waveform, output voltage spectrum, x-y plot of input and output voltage.

2.2.5

PSpice simulation

Connect the RLD circuit on PSpice (Figure 2.2). Input the parameter of all components in the circuit. Because there is no D1N4007 in PSpice, researchers has to changes the parameter of D1N4002. Edit the parameter of D1N4002. Details are shown in Figure 2.3.

Figure 2.2: RLD circuit on PSpice After change the parameter of the diode, the analysis setup also needs to be changed which is shown in Figure 2.4. After set all setups of the circuit, the next step is to simulate the circuit. Then add the output voltage as the y axis to get the output voltage which is shown in Figure 2.5. 15

Figure 2.3: Parameter of D1N4007

Figure 2.4: Analysis setup

Figure 2.5: The output voltage waveform on PSpice

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After measuring the resonance frequency of the input voltage, the next step is to simulate the chaotic behavior on PSpice. This time, the input voltage should be Vsin instead of VAC. And the analysis setup needs to be modified which is shown in Figure 2.6.

Figure 2.6: Analysis Setup when simulate chaotic behavior After set all setups of the circuit, the next step is to simulate the circuit. Then add the output voltage as the y axis to get the output voltage waveform. In addition, change the time domain to frequency domain to display the spectrum of the output voltage. Next, add the input voltage as the x axis to get the x-y plot of the input voltage and output voltage. After observing the chaotic behavior using PSpice, the next step is to fix the input voltage amplitude and change the frequency to find out the relation of chaotic behavior and the input voltage amplitude. Set the amplitude to 0.6V, 1V, 2.5V 5V and 10V one by one. Find the range of frequency when generate bifurcation for each group. Then fix the input voltage frequency and change the amplitude to find out the relation of chaotic behavior and the input voltage frequency. Set the frequency to 35kHz, 45kHz, 55kHz, 95kHz one by one. Find the range of amplitude when generate bifurcation for each group. The next step is to simulate the normal behavior in equivalent circuit Draw the equivalent circuit on PSpice shown in Figure 2.7. Simulate the circuit and get the abnormal behavior picture.

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Figure 2.7: Equivalent circuit on PSpice

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Chapter 3 Result and analysis
3.1 Resonance frequency

As the first needing result of this project, the resonance frequency was measured with the function generator and oscilloscope. When the output voltage reached its peak value, the corresponding frequency can be considered as the resonance frequency of the RLD circuit. The added input signals amplitude was 50mV while the frequency was the variable. The output voltages value with changing frequency were collected and displayed in the following table (table 3.1). The table (table 3.1) just showed the one-to-one correspondence relationship between the output voltage and the input frequency. The item Frequency/log is the logarithm format value of the input frequency. The logarithm values were calculated to express the output voltages variation tendency against the frequency more intelligible since the intervals of actual frequency values were too big to show the tendency. In addition, because the amplitude of the inputs accompanied noise signal was 4 mV, then the recorded voltage values were the original values minus 4 mV. Since the table (table 3.1) just displayed the raw data collected from the experiment, the figure (figure 3.1) obtained with Microsoft Excel 2010 was then plotted to demonstrate the relationship and variation tendency clearly. From the plot, the value of output voltage was almost zero when frequency was lower than 10 KHz (in the figure, the point 4 on the abscissa). Then with the increasing frequency, the output voltage was growing rapidly, and at approximate 95 kHz, the value reached to its maximum. After that, the output voltage was decreased rapidly and close 19

Table 3.1: the output voltages (voltage across the resistor) collected with variable input frequency F requency (Hz) F requency (log) Outputvoltage (mV) 0.94 -0.027 0.8 0.690 0.8 4.9 1.004 0.8 10.1 1.692 0.8 49.2 100.6 2.003 0.8 2.698 0.8 499.3 5.054k 3.704 0.8 10.09k 4.004 0.8 19.97k 4.300 1.6 4.478 2.0 30.06k 4.603 3.2 40.12k 49.95k 4.699 4.8 4.779 7.2 60.05k 70.04k 4.845 8.8 80.23k 4.904 10.4 4.954 13.6 89.99k 92.99k 4.968 17.6 93.95k 4.973 40.0 4.978 41.6 95.08k 100.7k 5.003 40.0 119.9k 5.079 28.0 5.117 20.8 130.9k 140.5k 5.148 15.2 150.2k 5.177 10.0 5.204 6.8 159.9k 169.5k 5.230 3.6 179.9k 5.255 2.0 190.6k 5.280 1.6 200.8k 5.303 1.6

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Figure 3.1: the variation tendency of the output voltage against the input frequency to zero again. The resonance frequency should be the frequency that made the RLD circuits output voltage attained the peak value. Therefore, the experimental resonance frequency was around 95 kHz. Then the RLD circuit was simulated with Pspice and the resulting resonance frequency was obtained as 85.399 kHz as the next figure (figure 3.2) displayed. The abscissa represented the input frequency while the ordinate expressed the output voltage. This figure presented the analogous voltage variation tendency with the excel plot. That is, they all demonstrated that the output voltage was nearly zero at the beginning, but then rapidly increased and after reached the apex, the value deduced quickly. This comparison illustrated that the experimental results were collected successful. However, the resonance frequency of simulation was smaller than the experimental value about 10 kHz. Since the inductance of the inductor was known and the capacitance of the diode was measured during week one, the theoretical value of resonance frequency can be calculated. As the inductance was 39mH while the capacitance was 86.7pF, according to the following function (equation 3.1):

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Figure 3.2: the simulation plot of output voltage against input frequency with resistance lkΩ

fresonance =

1 √ 2π LC

(3.1)

Substitute the measured value of inductance and capacitance, the theoretical resonance frequency was 86.6 kHz. Thus, compare these three resonance frequency (experimental, simulated and theoretical), the simulated and theoretical value are quite close while the experimental value is larger than these two results about 10 kHz. There exist many reasons that may cause this discrepancy. The experiment environment was not stable through the measurement sine the oscilloscope had detected the noise signal was exist in both input and output signal. Also, the capacitance of the RLD circuit was not only provided by the diode as the theory expected. The inductor could generate extra capacitance and the circuit may also generate stray capacitance between wires. The measurement and calculation of extra capacitance will be displayed in discussion chapter. Then the Pspice was used to explore the relationship between the resistor and the resonance frequency. The simulation was taken twice and the resistor was 22

changed its value to 5kΩ and 10kΩ.

Figure 3.3: the resonance frequency plot with 5kΩ resistor Observe these two plots (figure 3.3 and figure 3.4) and the first simulation plot (with 1kΩ resistor), the overall voltage variation tendency were all the same. However, since the added resistor has larger value and therefore more voltage was divided to the resistor, the maximum value of the output voltage was larger than the first resonance frequency plot. In addition, with growing resistance, the breadth of the protuberance part was increased. That is, the increase of the voltage was appeared with lower frequency and the rate of change of the output voltage was deduced.

3.2

Chaotic behaviour with changing input frequency

After the measurement of resonance frequency, the next target was to observe the chaotic behaviour of the RLD non-linear system with fixed input signals amplitude and varying frequency. The selected values of stationary amplitude were 0.6V, 1V, 2.5V, 5V and 10V (namely, the peak to peak voltages were 1.2V, 2V, 5V, 10V and 20V). For each selected amplitude situation, the changeable frequency was changed from 17.35 kHz to 205.8 kHz. 23

Figure 3.4: the resonance frequency plot with 10kΩ resistor When the stationary amplitude was set as 0.6V, then with 49.50 kHz frequency, the output voltage signal suddenly became to two different waveforms instead of one stable waveform in time domain. If the oscilloscope showed the output voltage against input signal, the single circle bifurcated to two circles at the frequency. This circuit behaviour indicated that at this frequency, the RLD non-linear system became unstable. However, if the frequency was larger than 52.50 kHz, the bifurcation phenomenon disappeared and there was still a single loop. Therefore, when the fixed amplitude was adjusted to 0.6V, although the bifurcation phenomenon could be observed, the complete chaotic behaviour was not appeared. The outcomes were quite similar with the stationary amplitude was increased to 1V. In this situation, the first bifurcation was also appeared and the complete chaotic behaviour was not been observed. However, the bifurcation frequency was change to 30.41 kHz and when frequency larger than 33.38 kHz, the bifurcation disappeared. For the situation with 2.5V stationary amplitude, both the first and second bifurcation behaviour plots were collected. The first bifurcation region was from 37.6 kHz to 74.5 kHz. When frequency was increased about 74.5 kHz, the second bifurcation began, that is, the oscilloscope displayed four circles on the screen under X-Y mode (display the relationship between input and output signal). Until the frequency increased to 91.5 kHz, the four circles can be observed clearly. After this frequency, the number of circle went back to one. Thus, under this amplitude, the complete chaotic behaviour was not been detected.

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Then the amplitude was enhanced to 5V and 10V. Under these input condition, the complete chaotic behaviour of the RLD circuit arose. Under the amplitude was 5V condition, the first bifurcation was appeared at 30.65 kHz frequency while the second bifurcation can be detected at 51.03 kHz. When frequency was bigger than 54.05 kHz, the first complete chaotic behaviour arose, that is, in the X-Y mode, the number of circles increased to infinite at this frequency. This complete chaotic state was kept until the frequency went to 65.52 kHz. Then the plot displayed three circles until the frequency reached 160.83 kHz. After that, the number of circles was stayed at one. Therefore, the circuit under this situation displayed one complete chaos approaching process and then started the second chaotic behaviour. The finally tested stationary amplitude was 10V. With this condition, the complete chaos phenomenon could be easily collected. The first bifurcation was happened at 27.14 kHz while the second one displayed at 42.53 kHz. Then adjusted the frequency to 44.00 kHz, the infinite circles appeared in the X-Y mode which represent the complete chaotic status of the system. With the growing frequency, at the frequency 51.42 kHz, the number of circles suddenly became to three. Then at 59.88 kHz frequency, the three circles bifurcated to six circles. This status maintained for a long time until the frequency reached 90.33 kHz and made the number of circles reduced to one. The two figures (figure 3.5 and figure 3.6) displayed the collected outcomes from the experiment. For certain fixed amplitude, when the adjustable frequency was reached to certain value and caused the bifurcation or complete chaos, then the output signal waveforms in both time and frequency domain and the X-Y mode plots (output voltage against input voltage) were recorded by photographing. As the above figure showed, the three pictures (in a row) from left to right express the X-Y mode plot, time domain plot and spectrum of output under the labelled input condition. Observed and analysed the displayed plots, the following conclusions can be obtained. When bifurcation happened, the number of circles in X-Y mode was doubled, and the output signal waveform in time domain plot were split to more shapes. As for spectrums, when the bifurcation appeared, the number of the circles in corresponding X-Y mode plot was equal to the number of under harmonic components which is the components with lower frequency than the fundamental or driving frequency (the input frequency) plus one. In addition, the intervals 25

Figure 3.5: the above groups of plot are the time and frequency domain output plots and X-Y mode plots with different frequency and fixed 5V amplitude 26

Figure 3.6: the above groups of plot are the time and frequency domain output plots and X-Y mode plots with different frequency and fixed 10V amplitude 27

between every two adjacent under harmonic components were equal. Furthermore, when the circuit reached the complete chaotic status, the number of circles became infinite as well as the number of output signals in time domain plot. At this moment, the spectrum under the fundamental frequency was continuous which represent infinite components. Table 3.2: the frequencies that led to bifurcations and complete chaos under different fixed amplitude First bifurcation 49.50 kHz 30.41 kHz 37.6 kHz 30.6 kHz 27.14 kHz Three circles Amp=0.6V \ Amp=1.2V \ Amp=2.5V \ Amp=5V 65.5 kHz Amp=10V 51.42 kHz Amp=0.6V Amp=1.2V Amp=2.5V Amp=5V Amp=10V Second bifurcation Complete chaos \ \ \ \ 74.5 kHz \ 51.03 kHz 54.05 kHz 42.53 kHz 44.00 kHz First bifurcation (six circles) Back to one circle \ 52.50 kHz \ 33.38 kHz \ 91.5 kHz \ 160.83 kHz 59.88 kHz 90.33 kHz

The above table (table 3.2) listed the frequencies that led to bifurcation phenomenon under different conditions. This table and the contained data demonstrated one conclusion that the larger stationary frequency made the complete chaotic behaviour easier to detect. In addition, the frequency that can cause bifurcation was tending to be smaller when the amplitude was increasing. The differences between one bifurcation frequency and the frequency that caused next bifurcation were deduced with growing stationary amplitude. Therefore, it can be considered as the RLD system was becoming more sensitive about input frequency with larger fixed amplitude. In order to demonstrate the collected results can represent the chaotic behaviour for a non-linear system correctly, the diode was then replaced by the equivalent capacitors and a linear system (the RLC circuit) was built. Now the input signals amplitude of RLC circuit was fixed at different amplitude and the frequency was still the variable. The outcomes were displayed as below (figure 3.7): Because the number of circle in X-Y mode plot was always one and the spec28

Figure 3.7: the time and frequency domain plots of RLC circuit under 5V input amplitude and variable frequency

Figure 3.8: the time and frequency domain plots of RLC circuit under 10V input amplitude and variable frequency 29

trums showed no under harmonic components when the amplitude was big enough, the above photos (figure 3.7 and 3.8) indicated that the RLC circuit which is a typical linear system has no chaotic behaviour. Therefore, compared to these behaviours, the recorded figures about the RLD circuits outcomes can be identified as canonical chaotic behaviour. In order to further prove the correctness of the experimental chaotic results, the Pspice simulation was used to simulate the RLD circuit behaviour under the same condition. The Pspice results under certain frequency and amplitude condition were also represented by three types of plots. From left to right to each raw, the output waveform, the output spectrum and the X-Y mode plot were displayed in this order. The simulated overall variation tendency of the output voltage with growing frequency was similar with experimental results. That is, the simulation outcomes also showed the first, second bifurcation and the complete chaos with increasing frequency respectively. However, the frequencies that led to the chaotic behaviour were not equal to the experimental values. This discrepancy can be attributed to the unstable experiment environment and the extra capacitance caused by wires and the inductor as the previous analysis about resonance frequency.

3.3

Chaotic behaviour with changing input amplitude

After the explosion of the relationship between the changeable input frequency and the chaotic behaviour, the next target is to research whether the variable amplitude (under same frequency condition) can cause chaotic behaviour. The designed values of stationary frequency were 35 kHz, 45 kHz, 55 kHz and the experimental resonance frequency 95 kHz. The adjustable amplitude was changed from 0V to 11.4V for each fixed frequency. In the situation that the fixed frequency was 35.05 kHz, the first bifurcation appeared at input amplitude 2.42V and the double circles in X-Y mode plot were kept until the amplitude reached the maximum value that the function generator can supply. Then the frequency was adjusted to 45.09 kHz (close to 45 kHz). Under this condition, the first bifurcation was started at 1.32V and the second circle doubling was displayed at 5.8V. At 6.5V, the oscilloscope detected the complete 30

Figure 3.9: the Pspice simulation results under the 5V stationary amplitude condition and the frequency was changed 31

chaotic behaviour. Then the number of circles was suddenly turned to three at 10.7V. For 54.98 kHz situation, the outcomes were quite similar. At 0.92V amplitude, the output bifurcated the first time and then it doubled again at 3.28V. The amplitude region caused complete chaos was from 4.0V to 6.0V. However, the three circle plot was not expressed under this condition, and the circles combined into one loop after 6.0V. The most significant fixed frequency was the resonance frequency. Thus, the frequency was finally adjusted to 95.12 kHz and the results were similar with lower fixed frequency situations. The first and second bifurcation happened at 0.50V and 1.62V respectively. The circuit became completely chaos status at 2.08V. The chaos status was lasted until the amplitude larger than 5.7V. From 5.7V to 6.1V, there were two circles in X-Y mode plot. Finally, the circles combined into one loop. Similar with the results of changeable frequency exploration, the three types outcome plots (figure 3.10 and figure 3.11) of variable amplitude study were recorded and displayed in the order mention by the figure label. The relationship among the three types of plots was the same as previous changeable frequency study. That is, the number of different signal waveforms in time domain plots was always equal to the number of circles in X-Y mode plots. And this value minus one should be the number of under harmonic components (whose frequency was lower than driving frequency) in corresponding spectrum. Furthermore, the intervals of adjacent two components were the same. When complete chaotic behaviour appeared, the number of output waveforms and circles were infinite and for spectrum, the infinite components under the driving frequency made the waveform looks continuous in that frequency region. The following table (table3.3) was made to conclude the relationship between the fixed frequency and the bifurcation and chaos leading amplitude easier and more convenient. As the table showed, the complete chaotic behaviour will be generated only when the driving frequency is large enough. The table also indicated that with the growing fixed frequency, the amplitudes that caused bifurcation and complete chaos were deduced. Another accompanied phenomenon was: the intervals between two contiguous bifurcation leading amplitude were decreased with the increasing input frequency. Thus, the circuit can be considered more sensitive to amplitude with higher driving frequency. In order to ensure the collected resulting plots are able to demonstrate that 32

Figure 3.10: the above groups of plot are the time and frequency domain output plots and X-Y mode plot with different 33 amplitude and 45.09 kHz fixed frequency

Figure 3.11: the above groups of plot are the time and frequency domain output plots and X-Y mode plot with different amplitude and the resonance frequency

34

Table 3.3: the amplitudes that led to bifurcations and complete chaos under different fixed frequency First bifurcation 2.32V 1.32V 0.92V 0.50V Three circles \ 10.7V \ \ Second bifurcation \ 5.8V 3.28V 1.62V Back to one circle \ \ 6.0V 6.1V Complete chaos \ 6.5V 4.0V 2.08V

F=35.05 F=45.09 F=54.98 F=95.12 F=35.05 F=45.09 F=54.98 F=95.12

kHz kHz kHz kHz kHz kHz kHz kHz

RLD circuit has complete chaotic status with variable input amplitude, the equivalent RLC circuit was used. The initial conditions for RLC circuit were kept the same as the RLD circuit and the chaotic behaviour searching process was repeated. The outputs of RLC circuit with variable amplitude under different driving frequencies were displayed below: The resulting plots (fogure 3.12 and figure 3.13) showed that for RLC circuit, the chaotic behaviour cannot be detected with variable amplitude and no matter how large the input frequency was. Namely, the X-Y mode plots only had one circle and there was no under harmonic components appeared in spectrums. This phenomenon proved that chaotic behaviour is the feature of non-linear systems and for linear system (such as RLC circuit), the output signal always can be conjectured with initial conditions. The figure below (figure 3.14) displayed the simulated RLD circuit behaviour under the similar conditions of experiment testing process. The simulation process also detected the bifurcation phenomenon and complete chaos status of the built RLD circuit. The variation tendencies were the same since the first bifurcation slip the one circle into two and the second bifurcation doubled the circles and then the system reached complete chaos status. Then the number of circles deduced to two and finally one. These consistent results ensured the correctness of experimental outcomes. However, the amplitudes that caused chaotic behaviour of simulation results were inconsistent with the experimental values. The difference was calculated smaller than 0.5V. The reason lead to this difference should be the same as the analysed reasons of changeable frequency situation.

35

Figure 3.12: the time and frequency domain plots of RLC circuit with variable amplitude and 60 kHz input frequency

Figure 3.13: the time and frequency domain plots of RLC circuit with variable amplitude and 95 kHz input frequency

36

Figure 3.14: the Pspice simulation results with input resonance frequency and the variable amplitude

37

Chapter 4 Discussion
4.1
4.1.1

Discussion of the RLD circuit
The non-linear capacitance of diode

The RLD circuit can be considered as a non-linear system because it contains one non-linear componentthe diode. A typical diode can be equivalent to a capacitance paralleled with a resistor. In this project, the generation of chaotic behaviour of the RLD circuit was mainly resulted from the non-linear fluctuated equivalent capacitance of the diode. The equivalent capacitance of the diode can be divided into two main parts. One is called as junction or depletion capacitance which dominates reverse bias situation while the other one is recognized as diffusion capacitance which mainly dominates forward bias situation. [3] Therefore, the value of total capacitance of the diode is the sum of junction and diffusion capacitance. When the reverse bias voltage adds to the diode, the capacitance is mainly contributed by junction capacitance while the total capacitance depends on diffusion capacitance in forward bias status. The junction capacitance can be represented by the following equation: Kε . W

Cj = A

(4.1)

The label Cj represents the junction capacitance and its value can be calculated if the values of parameter A (the cross area of diodes P-N junction interface),

38

K (is the silicon dielectric constant), ε (the permittivity of the diode material) and W (the width of depletion layer) were known. The values of A and ε are the characteristic parameters of a diode and they should be treated as constants for a specific diode. The particular value of A and ε can be found in diodes datasheet. The parameter K is a constant and its value should be 11.7. Therefore, the equation 4.1 only contains one variable. The junction capacitance is controlled by the depletion layer width of the diode. The value of junction capacitance is inversely proportion to the depletion layer width. The depletion layer width relates to both the diode itself and the external voltage on the diode. The equation of the depletion layer width is:

W = xp + xn =

2Kε q

Na + Nd (ϕi − V a) N aN d

(4.2)

Inside the equation, xp and xn stand for the depletion layer width in p and n region and the total depletion width is the sum of xp and xn. In the rightmost part of the equation, the value of K and ε are constant for a diode as discussed above. The parameter q is the charge constant for an electron whose value is defined as 1.6 × 10−19 with unit Coulomb. In addition, the parameter Na and Nd denote the doping density of donor in n region and accepter in p region. For a specific diode, the values of these two parameters are constant and they are often displayed in datasheet. The parameter φi which can also be represented by Vbi is the built-in or diffusion potential of a diode in thermal equilibrium condition. [4] N aN d KT ln q n2 i

ϕi = Vbi =

(4.3)

The function 4.3 indicated that the diffusion potential of a diode in thermal equilibrium condition is a constant since the contained parameters are all characteristic parameters of the diode and they only depend on the diode manufacturing material. The parameter Va is the only involved variable inside the equation 4.2 which denotes the external voltage applied to diode. If the voltage is forward bias voltage, then Va is positive. On the other hand, if the applied voltage is reverse bias voltage, the sign of Va is negative sign. Therefore, the value of depletion layer width is only controlled by the external voltage. And according to the function 4.1, the

39

external voltage applied to diode also decides the value of junction capacitance. qKε N aN d 2 (ϕi − V a) N a + N d

Cj =

(4.4)

The function 4.4 indicated the relationship between the external voltage applied to the diode and the junction capacitance is not linear. The second part of the diode capacitance is the diffusion capacitance which only appears under forward bias condition. When the forward bias voltage applied to the diode, the built-in electric field breadth deduced and it allows the majority carriers in both side (holes in p region and electrons in n region) diffuse to the opposite side. Therefore, under the forward bias situation, the diffusion current appeared through the diode. The total diffusion current ID is defined as the sum of holes and electrons current (write as Ip and In, respectively) that pass through the junction interface. This relationship is expressed by equation 4.5.

ID = IP + IN

(4.5)

The total storage charge of the equivalent diffusion capacitor can be marked as Q and the function of Q can be written as function 4.6,

Q = ID τ

(4.6)

The parameter ”τ ” denotes the average existing time of the majority carriers before they recombined with the opposite polarity charges. The existing times of holes and electrons are defined the same. [5] According to the capacitance calculation function, the diffusion capacitance should be represented as equation 4.7. d (ID τ ) d V 1 dQ τ I0 exp = = dν dV dv VT n

Cd =

(4.7)

For a certain diode, the parameter I0 (the diffusion current at thermal equilibrium state), VT and η are constants and will not change with external environ40

ment. Therefore, the variable that defines the diffusion capacitance should be the applied forward bias voltage. According to equation 4.7, the relationship between the forward bias voltage and diffusion capacitance is also non-linear. Thus, since both the junction and diffusion capacitances are non-linear with changeable external voltage, their sumthe total capacitance of diode is non-linear with applied voltage. This is the main reason to explain why diode is a non-linear component. Moreover, the RLD circuit can then be considered as a non-linear system.

4.1.2

Chaotic behaviour of RLD circuit

For RLD circuit, the fundamental factor that caused the chaotic behaviour is the unrecombined diffusion majority carriers (includes electrons and holes) that pass through the forward-biased p-n junction interface. [6] since the applied voltage of the RLD circuit is AC sine signal, the bias status is therefore switched between forward and reserve bias state. Assume the first bias state is forward bias and then there is diffusion current cross the diode p-n interface. Then for the next half period of sine signal, the diode should under reverse bias condition and the majority carriers should also diffuse back to the original status. If the diffused back majority carrier number is different with the diffusion carrier number under forward bias condition, the initial condition for the next period of forward bias is therefore changed from the origin condition. Thus, because the RLD non-linear system is highly depend on the initial condition, the output signal of second period AC wave will be discrepant with the first period. The larger applied forward bias voltage of diode will lead to lager diffusion current. Namely, greater amount of majority carrier will pass to the opposite side. This will result in longer time to allow the charges diffuse back to the regions and rehabilitate the initial status. [6] If the input AC signals frequency is too high, the carriers will have no sufficient time to recover the initial status. Then, the output signal waveforms become different with previous one. Finally, the accumulative divergence generates the chaotic behaviour. In addition, if the applied AC signal amplitude is large, then larger amount of carriers need to diffuse back. If the applied AC signal frequency is high, the provide time for carriers backward diffusion is shorter. Therefore, the larger input AC amplifier and frequency will make the chaotic behaviour easier to detect. 41

This explained the results of the previous section that under low frequency or low amplitude situation, the oscilloscope could not detect complete chaotic state.

4.2

The resonance frequency

For a series connected RLD circuit, since the diode can be equivalent to a capacitor, the resonance frequency is defined as the frequency makes the magnitude of inductor and capacitors impedance the same. Thus, at the resonant frequency, the impedances are equal in magnitude. However, because their phase difference is 180 degrees, they actually cancel each others impedance. In other word, the total impedance of the RLD circuit becomes smallest (only resistance R involves) at resonance frequency. From the previous resonance frequency plots, the resonance frequency corresponds to the highest output voltage point. Recall the definition function of resonance frequency (equation ??), the value depends on the inductance and capacitance and the resistance is not involved. But the sensitiveness of the circuit and the highest output voltage all relies on the resistance. [7] If the resistance increased, both the maximum output voltage and the circuits sensitiveness will decrease. Therefore, for larger resistor, the resonance frequency plots showed wider peak of output value against frequency.

4.3

Identify the chaotic behaviour

The chaotic behaviour can be identified with time domain output plots, output voltage spectrums and X-Y mode plots.

4.3.1

Time domain output waveform plots

As the figure below, both the input and output voltage waveforms were displayed. The upper sinusoidal wave is the input signal while the lower waveform stands for output signal. Since the amplitude and frequency of the input signal were designed, the oscilloscope detected a stable sinusoidal input wave. On the contrary, because the plot expressed the complete chaotic status of the RLD circuit, the plot indicated number of output waveforms was infinite. The typical chaotic behaviour in time domain can be easily identified at the wave crest which is marked with 42

the red circle. The appearance of the chaos phenomenon indicated the system was under complete unstable state.

Figure 4.1: the time domain plot of complete chaotic status (input frequency was 95 kHz and the amplitude was 4.5V)

4.3.2

Output signal spectrum plots

The figure below showed a typical complete chaotic output spectrum and the discussion of it can be divided into three parts.

Figure 4.2: the spectrum plot of complete chaotic status (input frequency was 95 kHz and the amplitude was 4.5V) The highest component of the spectrum represents the fundamental (or input) frequency. As the red circle marked, the width of every lattice represents 125 kHz. Observe the figure, the place of the highest component demonstrated the driving frequency was approximately 95 kHz. 43

In the right side of the driving frequency, there exist several sub-harmonic components. Their appearance is consistent with Fourier Transform theory. That is, the theory points out a time domain sinusoidal wave can be recognized as the harmonic of fundamental and sub-harmonic components. Thus, the appearance of the sub-harmonic components is caused by the input sine wave. In the left side of the driving component, the spectrum was continuous in that region. This behaviour indicated there existed infinite under harmonic components. According to the previous conclusion, the number of the circles and output waveforms should always equal to the number of under harmonic components plus one. In addition, the Fourier Transform theory indicates that, when spectrum displays continuous wave, then the corresponding signal must be aperiodic. The spectrum can be used to identify the correct chaotic behaviour for non-linear system. When the bifurcation happened, the outcomes showed the interval between two adjacent under harmonic components were deduced to half of the previous value. This is resulted from the halved output signals period. For example, assume the input signals amplitude was big enough and the related frequency was quite high, then the diffused carriers would have no sufficient time to move back under reverse bias voltage. However, for the next period, the carriers still moving back to the origin place when the second forward bias period applied. Therefore, a time delay appeared and it caused shorter time for the second period diffusion lasting time and finally made the amplitude of second period output wave smaller. But, for the second reverse bias period, since the diffusion current was smaller than the first period, the required time for carriers move back to origin region was shorter. Thus, the shorter required back moving time just counteracted the time delay. For the third period, the output waveform should be equal to the first period. Therefore, for the first bifurcation, two different output waveforms were generated and the frequency of each kind of output was the half of the input frequency. [Reference 6] This result caused the first under harmonic component in the spectrum and this reason explained why the intervals between adjacent two components were equal. The explanation can also be used for other bifurcations (second, third ) and finally, the circuit reached chaotic status.

44

4.3.3

The X-Y mode plots

According to analysis and discussion from previous sections, the next figure displayed the correct X-Y mode plot for complete chaotic status since there were infinite circles. The shape of the output against input voltage plot is a circle. For a linear system which can generate periodic output signal, the energy conservation law should be obeyed. Thus, the related X-Y mode plot for this linear system should display a circle. For a non-linear system, the generated output signal is aperiodic, the overall plot shape will not change but there are many distinct circles appeared and indicated the unstable status of the system. [8]

Figure 4.3: the X-Y mode plot of complete chaotic status (input frequency was 95 kHz and the amplitude was 4.5V)

4.4

Discussion of bifurcation and verification of Fiegenbaum constant

Bifurcation phenomenon is considered as a general route to approach chaos for non-linear system. In this project, the bifurcation behaviour was detected and the variation tendency proved with the increasing times of bifurcation, the system finally reached complete chaotic status. Recall the table 3.2 and 3.3 in result section, the differences of changeable parameter (adjustable frequency or amplitude) between two adjacent bifurcation 45

were different in every group (keep one parameter stable and adjust the other one). The Fiegenbaum constant theory concludes the variation tendency of these differences. The theory pointed out that if the period doubling bifurcation happened to a non-linear system by changing a single related parameter, there will be a universal constant δ for this system. [9] The value of can be verified with equation 4.8. an − an−1 an+1 − an

δ = lim

n→∞

(4.8)

To explain the above equation, the parameter an denoted the parameter that causes the nth bifurcation. For the chaotic behaviour study with stationary input signal amplitude and changeable frequency, the values of δ for different amplitudes were calculated with equation 4.8 and listed in the next table. Table 4.1: the calculated Fiegenbaum constant with fixed amplitude and changeable frequency amp=0.6v amp=1.2v amp=2.5v amp=5v amp=10v Value of δ NA NA NA 6.67 10.47 Difference with δ = 4.669 NA NA NA 2.091 5.801

The symbol \ means the system did not reach complete chaotic status and therefore the collected data was not sufficient to calculate the Fiegenbaum constant. The data used to calculate Fiegenbaum constant were the frequencies that led to first and second bifurcation and complete chaotic status. The theoretical Fiegenbaum constant equals to 4.669. However, based on the experimental results, the calculated δ values are all larger than the theoretical value and when amplitude fixed at 10V, the experimental value is about twice the theoretical value. This discrepancy may be the result from the inaccuracy of the measure equipment and the non-ideal experiment environment that may cause some noise. Also, because of the limitation of measurement range and precision of the equipment, the collected complete chaotic status data that can be used to analyse Fiegenbaum constant are not enough to get a closer result of theoretical value. Then for fixed frequency and variable amplitude situation, the calculated experimental Fiegenbaum constant values were showed below. 46

Table 4.2: the calculated Fiegenbaum constant with fixed frequency and changeable amplitude f=35.05kHz f=45.09kHz f=54.98kHz f=95.12kHz NA 6.40 3.27 2.43 NA 1.731 1.399 2.239

Value of δ Difference with δ = 4.669

The meaning of symbol \ is the same as the above table. Unlike the previous situation, for fixed frequency research, the variation tendency of Fiegenbaum constant value is decreased with growing stationary frequency. Also, compared with the fixed amplitude situation, the differences between calculated and theoretical values were generally smaller. Therefore, the outcomes of fixed frequency research can be considered as more credible than fixed amplitude study.

4.5

Error discussion

The result section has mentioned the difference between the measured resonance frequency (around 95 kHz) and the theoretical value (85.399 kHz). In addition, the simulated amplitudes or frequencies that caused bifurcation are also different from the actual values. Then in this section, the reason of these errors will be discussed.

4.5.1

Equipment and environment

The equipment used in this project (such as digital oscilloscope and function generator) has been served for kinds of experiment for years. The aging of equipment will lead to some extent of inaccuracy and generate errors. Because of the precision limitation of measuring equipment, the recorded amplitude and frequency data that caused chaotic behaviour was lack of precision and some discrepancies with actual values appeared. The function generator provided a limited range of amplitude and frequency. That limitation resulted in the collected chaotic status data was not sufficient to analyse and obtain a faithful conclusion. As mentioned before, the experiment environment was unstable and non-ideal. The non-ideal environment added the unwanted noise to both input and output 47

signal and it was detected by the oscilloscope. The superposition of circuit output and noise produced some error in recording data. Since the non-linear RLD system is highly sensitive with the initial condition, the unstable environment may also change the initial status of the system and this immeasurable varying condition made the analysis of the relationship between the chaotic behaviour and the applied input more complicate.

4.5.2

The extra capacitance

Beside the equivalent capacitance of the diode, the circuit also contains other capacitances which may be caused by the inductor and the stray capacitance effect. These capacitances can be considered as the extra capacitance of the circuit and their appearance generated some errors. In order to determine the extra capacitance, the diode was replaced by a 120.4 pF (experimental value) capacitor. The new built RLC circuit was connected with the function generator which applied AC input signal to the circuit. The resonance frequency was then measured as 71.98 kHz. According to the resonance frequency function (equation ??), the function can be rearranged as: 1 L (2πfresonance )

Ctotal =

(4.9)

Then substitute the resonance frequency and inductance with their experimental value: 1 38.6 × 10−3 (2π × 70.98 × 103 )2

Ctotal =

130.3pF

(4.10)

Since the measured capacitance of the capacitor was 120.4 pF, the extra capacitance of the circuit was 9.9 pF which can be considered quite large. Now add this extra capacitance to the measured diode capacitance and recalculate the experimental resonance frequency. The resulted resonance frequency was turned to 82.04 kHz. Compare to the theoretical and previous experimental value, the recalculated result was much closer to the theoretical value and the error was deduced from 8.5 kHz to 4.6 kHz (as the theoretical value was 86.6 kHz). Therefore, the extra capacitance was contributed to the errors between the 48

actual and theoretical value. The rest part of error may be caused by the equipment and environment is error is difficult to measure.

4.6

Assessment and further research

The overall outcomes (include the resonant frequency measurement and chaotic behaviour detection) of the project are supportive to the theme of the research. Although there existed discrepancies between the expected results and the experiment outcomes, the results are still reliable. The reasons of differences were discussed in the last section. The limitation of equipments measurement range and precision caused part of the error and this error can be diminished with more advanced equipment. However, the other part of error which caused by the extra capacitance is difficult to avoid. Because of the insufficient precision of the function generator and the RLD circuits highly sensitiveness to the input signal, the third and high times bifurcations (8 circles, 16 circles) did not appear on the oscilloscopes screen. The limited scale of supplied inputs amplitude and frequency only allowed the first complete chaos region can be observed. Therefore, if the advanced equipment is applied in further research, the higher times of bifurcation and more complete chaotic region will be able to study. The previous result chapter displayed three circles and six circles were displayed in X-Y mode plots. These plots just demonstrated that after the first complete chaos region, the approaching route for the second complete chaotic status should start from three circles. And the first bifurcation of the second chaos approaching process made the number of circles doubled to six. Based on the fact that the first and second chaotic behaviour was started with one circle and three circles, the third chaotic behaviour can be conjectured as starting with five circles. Add the bifurcation phenomenon, for the third chaos approaching route, the number of circles should be five, ten, twenty and keep on doubling until infinite. Therefore, for further research, this correctness and of this guess should be proved. Moreover, if the guess was proved to be true, the exploration of the contributed causes is suggested.

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4.7

Applications of chaotic circuit

The chaos can be seen is every field and the application of chaos is also wide. One of the chaotic circuit applications is the signal encryption in telecommunication. The encryption process is significantly effective for avoiding the other peoples usurpation and distortion of transmitting signal. The working principles are explained as following descriptions. The chaotic circuit is often designed to combine with the carrier signal generator. Therefore, when the transducer produced the baseband signal, the carrier signal generator with chaotic circuit will add the unpredictable carrier signal to the baseband signal. After that, the combined signal will be transmitted through the channel and finally reach the receiver at the terminal of the channel. At the receiver, the carrier signal will be removed and the origin message signal that contains the required information will be passed to the client. [2] During the transmission process, since the chaotic circuit contributed an unpredictable carrier signal and the combined signal is therefore indecipherable for others. Thus, the chaotic technic encrypted the origin signal.

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Chapter 5 Conclusion
The object of the project is to study and identify the chaotic behaviour of the typical non-linear systemRLD series circuit. During the experiment, the resonance frequency of the RLD circuit and equivalent RLC circuit were measured. The experiment used two methods to approach the complete chaotic status. One is fix the amplitude of the input AC signal and change the frequency while the other method is keep the frequency stable and increase the amplitude. The recorded outcomes are the output voltage (the voltage across the resistor) plots of time domain, frequency domain and X-Y mode. When the output waveform changed, the circuit became unstable and the accompanied phenomenon is called bifurcation. Therefore, through bifurcation behaviour, the complete chaotic status was detected and the collected plots can be used to identify the chaotic behaviour. In order to prove the correctness of the experimental outcomes, the software Pspice was used to simulate the situation and the results were similar with the actual outcomes. The RLD circuit is considered as non-linear because it contains a non-linear componentthe diode. The non-linear equivalent capacitance of the diode is the main cause of the non-linear feature of the diode. As a feature of non-linear systems, the chaotic behaviour of the RLD circuit is the result of unrecombined diffusion carriers during forward bias period. The collected three types of output plots were discussed in discussion chapter. Especially, the performance and generation of the under harmonic components of the spectrum are explained detailed. Based on the detected bifurcation behaviour, the Fiegenbaum constant was calculated but the experimental results were different with the theoretical value. Therefore, to explain the discrepancies between actual measured outcomes and theoretical 51

values, the error analysis section showed the possible causes of these errors. Because the limitation of experiment equipment and environment, only the first, second bifurcation and the first complete chaotic status were observed. Therefore, for further research, the higher order bifurcation and complete chaotic status should be explored with advanced equipment. The chaotic technology will be applied In many fields, such as the telecommunication field, the chaotic circuit will contribute to signal encryption. With the development of chaos phenomenon, more chaos characteristics will be explored and applied to easier human lives.

52

References
[1] T. Matsumoto, “A chaotic attractor from chua’s circuit,” http://www.eecs. berkeley.edu/∼chua/papers/Matsumoto84.pdf, 1984. [2] S. L. Brad Aimone, “Chaotic circuits and encryption,” http://physics.ucsd. edu/neurophysics/courses/physics 173 273/AimoneLarsonChaos.pdf, 2006. [3] B. V. Zeghbroeck, “The p-n junction capacitance,” http://ecee.colorado.edu/ ∼bart/book/contents.htm(, University of Liverpool, 1997. [4] S. S. Li, “p-n junction diodes,” www.sli.ece.ufl.edu/eel6383/Chapter11.pdf, 2005. [5] B. M. Wilamowski, “Semiconductor diode,” http://www.eng.auburn.edu/ ∼wilambm/pap/2011/K10147 C008.pdf, 2010. [6] M. H. ed al, “Period doubling, feigenbaum constant and time series prediction in an experimental chaotic rld circuit,” http://physlab.lums.edu.pk/images/d/ d8/RLD chaos.pdf, Feb. 2007. [7] R. Nave, “Selectivity and q of a circuit,” http://hyperphysics.phy-astr.gsu. edu/hbase/electric/serres.html, University of Nottingham, 2001. [8] J. Alam and S. Anwar, “Chasing chaos with an rl-diode circuit,” http: //physlab.lums.edu.pk/images/0/01/Chasing chaos v1.pdf, LUMS School of Science and Engineering, 2010. [9] M. Richert and D. Whitmer, “Chaotic dynamics of rld oscillator,” http://physics.ucsd.edu/neurophysics/courses/physics 173 273/ Chaotic Circuit 03.pdf, 2003.

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