Generalized Single Stage Class C Ampliﬁer: Analysis from the Viewpoint of Chaotic Behavior

Featured Application: Results arising from research on an isolated cell of a bipolar-transistor-based class C amplifier show that chaos belongs to natural behavior of such simple dynamical systems. Since this network topology can be found in a huge number of complex electronic systems, the pronounced conclusion can be extrapolated to a variety of common radio frequency circuits. Abstract: This paper brieﬂy describes a recent discovery that occurred during the study of the simplest mathematical model of a class C ampliﬁer with a bipolar transistor. It is proved both numerically and experimentally that chaos can be observed in this simple network structure under three conditions: (1) the transistor is considered non-unilateral, (2) bias point provides cubic polynomial feedforward and feedback transconductance, and (3) the LC tank has very high resonant frequency. Moreover, chaos is generated by an autonomous class C ampliﬁer; i.e., an isolated system without a driving force is analyzed. By the connection of a harmonic input signal, much more complex behavior can be observed. Additionally, due to the high degree of generalization of the ampliﬁer cell, similar fundamental circuits can be ordinarily found as subparts of typical building blocks of a radio frequency signal path.


Introduction
So far, a significant number of research papers have been focused on numerical analysis of conventional electrical circuits from the viewpoint of chaos evolution. The general definition of chaos speaks about long-time unpredictable behavior, dense state attractors with a fractal dimension, generated waveforms with increased entropy, and extreme sensitivity to small changes in initial conditions. The mentioned properties can be detected and quantified using several well-established numerical algorithms. For example, a routine known as approximate entropy [1][2][3] can be used to measure self-similarity of the data sequence produced by any dynamical system. The time sequence can be used for calculation of the largest Lyapunov exponent as well; see [4,5] for more details. More precise results of the quantification of dynamical motion can be reached if describing a mathematical model (set of ordinary differential equations) is known. Spectra of Lyapunov exponents can be also calculated using linearization of the flow along the evolved state trajectory [6][7][8][9]. The same method can be utilized for optimization, i.e., to increase robustness of the chaotic operational regime [10], to find the parameter subspace leading to chaos [11] in the analyzed dynamical system, or to replace parts of chaotic oscillators without changing of global dynamics [12]. Some research papers showed the possibility to study chaotic systems via patterns of return maps [13], bifurcation diagrams associated with real circuits [14] and processes [15], etc. Some recent papers were focused on analysis and circuit where a state vector is x = (v 1 , v 2 , i L ) T , y 11 and y 22 are the input and output admittance of the bipolar transistor, and y 12 and y 21 are transconductance-type nonlinear smooth functions of the form y 21 (v 1 ) = a 1 ·v 1 + a 2 ·v 1 3 , y 12 (v 2 ) = b 1 ·v 2 + b 2 ·v 2 3 .
Appl. Sci. 2020, 10, x FOR PEER REVIEW 3 of 13 Note that nonlinearities are considered lower-order polynomials; i.e., approximations are valid in the close neighborhood of the calculated bias point. Fixed points are all real solutions of nonlinear algebraic equations dx/dt = 0, i.e., Therefore, Dynamical System (1) with (2) possesses only a single equilibrium located at zero. Associated eigenvalues can be established as roots of characteristic polynomial Small letters denote normalized values of circuit components. In general, Math Model (1) is analyzed using normalized values of accumulation elements and admittances. For a particular application, time and impedance rescaling need to be considered. Note that capacitor C1 de facto represents parasitic capacitance between the base and emitter Cbe, while C2 is a sum of parasitic Cce and tank capacitance. Detailed numerical investigations reveal the advantageous possibility to keep C1 = C2 = 1 F and L = 1 H. For further analysis, let us assume that linear term a1 in Function (2) is very small. Symbolic formulas for eigenvalues can be expressed as i.e., we experience a pair of complex conjugated eigenvalues if y22 2 ⋅l < 4⋅c2. In this case, local vector field geometry near the origin is spanned by the eigenvector and eigenplane defined by two lines, namely Nonlinear scalar functions y12(v2) and y21(v1) have been considered as cubic polynomials without offset, while all remaining parameters were freely enabled for optimization. During the search procedure, the bipolar transistor was fixed in an active regime of operation where y22 = 0 S. Therefore, fifth-dimensional hyperspace of the internal system parameters undergoes the optimization algorithm where the largest Lyapunov exponent (LLE) together with a simple condition for a bounded attractor has been combined and adopted as a fitness function. Finally, the following set of internal parameters was discovered: Note that nonlinearities are considered lower-order polynomials; i.e., approximations are valid in the close neighborhood of the calculated bias point. Fixed points are all real solutions of nonlinear algebraic equations dx/dt = 0, i.e., Therefore, Dynamical System (1) with (2) possesses only a single equilibrium located at zero. Associated eigenvalues can be established as roots of characteristic polynomial Small letters denote normalized values of circuit components. In general, Math Model (1) is analyzed using normalized values of accumulation elements and admittances. For a particular application, time and impedance rescaling need to be considered. Note that capacitor C 1 de facto represents parasitic capacitance between the base and emitter C be , while C 2 is a sum of parasitic C ce and tank capacitance. Detailed numerical investigations reveal the advantageous possibility to keep C 1 = C 2 = 1 F and L = 1 H. For further analysis, let us assume that linear term a 1 in Function (2) is very small. Symbolic formulas for eigenvalues can be expressed as i.e., we experience a pair of complex conjugated eigenvalues if y 22 2 ·l < 4·c 2 . In this case, local vector field geometry near the origin is spanned by the eigenvector and eigenplane defined by two lines, namely Nonlinear scalar functions y 12 (v 2 ) and y 21 (v 1 ) have been considered as cubic polynomials without offset, while all remaining parameters were freely enabled for optimization. During the search procedure, the bipolar transistor was fixed in an active regime of operation where y 22 = 0 S. Therefore, fifth-dimensional hyperspace of the internal system parameters undergoes the optimization algorithm where the largest Lyapunov exponent (LLE) together with a simple condition for a bounded attractor Appl. Sci. 2020, 10, 5025 4 of 13 has been combined and adopted as a fitness function. Finally, the following set of internal parameters was discovered: Using these values, eigenvalues near the equilibrium point form unstable geometry 3 ∈ 2 unstable ⊕ 1 stable ; i.e., the dynamical flow near the origin acts according to saddle-spiral vector field geometry. Eigenvalues are δ 1 = −0.486 and δ 2,3 = 0.033 ± j0.929.
Note that the final set of ordinary differential equations has eight terms, including two polynomials. Therefore, it does not belong to the family of the simplest chaotic flows, as discussed in papers [35][36][37].

Numerical Results
Discovered Novel Chaotic System (1) with Parameters (7) has been numerically analyzed using Mathcad and MATLAB. For numerical integration, the fourth-order Runge-Kutta method has been adopted. Figure 2 demonstrates the typical strange attractor generated by a bipolar transistor cell under bias point conditions mentioned above. The final time is 10 4 , time step is 10 −2 , and sensitivity to changes in initial conditions is also visualized. The group of 10 4 initial conditions were randomly generated around point x 0 = (1, 0, 0) T using normal distribution with deviation of 10 −2 (red dots); short time evolutions after 1 s are plotted (blue), and then states after 100 s of the integration process are depicted (orange dots), and the final state after 1000 s is provided (green dots). Note that neighborhood state trajectories diverge slowly, and this should be respected when setting input parameters of the upcoming analysis. Figure 3 shows dynamic energy calculated in the state space volume where the strange attractor evolves. In this plot, a rainbow scale is used: red denotes high energy, green stands for average value, and blue marks regions with very low dynamic energy. Figure 4 illustrates basins of attraction for the typical chaotic attractor generated by a transistor-based system. Here, each plot contains a linear grid having 101 × 101 points, and the color scale is as follows: the limit cycle is marked by green, chaotic attractor by red, and fixed-point solution is blue. Ranges of horizontal and vertical axes are the same for each plot: v 1 ∈(−5, 5) and v 1 ∈(−5, 5), respectively.
Note that the final set of ordinary differential equations has eight terms, including two polynomials. Therefore, it does not belong to the family of the simplest chaotic flows, as discussed in papers [35][36][37].

Numerical Results
Discovered Novel Chaotic System (1) with Parameters (7) has been numerically analyzed using Mathcad and MATLAB. For numerical integration, the fourth-order Runge-Kutta method has been adopted. Figure 2 demonstrates the typical strange attractor generated by a bipolar transistor cell under bias point conditions mentioned above. The final time is 10 4 , time step is 10 −2 , and sensitivity to changes in initial conditions is also visualized. The group of 10 4 initial conditions were randomly generated around point x0 = (1, 0, 0) T using normal distribution with deviation of 10 −2 (red dots); short time evolutions after 1 s are plotted (blue), and then states after 100 s of the integration process are depicted (orange dots), and the final state after 1000 s is provided (green dots). Note that neighborhood state trajectories diverge slowly, and this should be respected when setting input parameters of the upcoming analysis. Figure 3 shows dynamic energy calculated in the state space volume where the strange attractor evolves. In this plot, a rainbow scale is used: red denotes high energy, green stands for average value, and blue marks regions with very low dynamic energy. Figure 4 illustrates basins of attraction for the typical chaotic attractor generated by a transistor-based system. Here, each plot contains a linear grid having 101 × 101 points, and the color scale is as follows: the limit cycle is marked by green, chaotic attractor by red, and fixed-point solution is blue. Ranges of horizontal and vertical axes are the same for each plot: v1∈(−5, 5) and v1∈(−5, 5), respectively.

Circuit Design
Synthesis of lumped chaotic oscillators based on knowledge of describing ordinary differential equations is a simple task with multiple solutions. The most straightforward approach is known as the concept of analog computers. Following this approach, only three basic building blocks are necessary: inverting integrators, differential amplifiers, and two ports with the prescribed nonlinear transfer curve. These circuits can be designed either in voltage mode [38,39] or current mode [40]. The first case is much more common because of the commercial availability of active elements and easily measurable state variables. The corresponding circuit implementation of the analyzed dynamical system is provided via Figure 6a, and its behavior is uniquely determined by the following set of ordinary differential equations: where C 1 , C 2 , and C 3 are the left, middle, and right capacitor, respectively. Note that seven active devices are needed for network design. The time constant is a compromise between the speed of integration (should be high due to long transients) and parasitic properties of active elements (should be low such that parasitic properties can be neglected). In our case, it is chosen as τ = 10 4 × 10 −8 = 100 µs. In the first experiments, red and blue resistors were supposed to be of variable values, i.e., potentiometers.
Much better results were achieved if system dissipation also became an adjustable quantity, i.e., the green resistor was substituted by a potentiometer. It is worth noting that OrCAD PSpice circuit simulator has been used to validate the upcoming experimental results before construction of the oscillator.
Appl. Sci. 2020, 10, x FOR PEER REVIEW 8 of 13 the concept of analog computers. Following this approach, only three basic building blocks are necessary: inverting integrators, differential amplifiers, and two ports with the prescribed nonlinear transfer curve. These circuits can be designed either in voltage mode [38,39] or current mode [40]. The first case is much more common because of the commercial availability of active elements and easily measurable state variables. The corresponding circuit implementation of the analyzed dynamical system is provided via Figure 6a, and its behavior is uniquely determined by the following set of ordinary differential equations: where C1, C2, and C3 are the left, middle, and right capacitor, respectively. Note that seven active devices are needed for network design. The time constant is a compromise between the speed of integration (should be high due to long transients) and parasitic properties of active elements (should be low such that parasitic properties can be neglected). In our case, it is chosen as τ = 10 4 ⋅10 −8 = 100 μs.
In the first experiments, red and blue resistors were supposed to be of variable values, i.e., potentiometers. Much better results were achieved if system dissipation also became an adjustable quantity, i.e., the green resistor was substituted by a potentiometer. It is worth noting that OrCAD PSpice circuit simulator has been used to validate the upcoming experimental results before construction of the oscillator. The oscillator can be simplified by the following different idea. Assume that each differential equation represents the sum of the currents measured at independent nodes. One can easily obtain the electronic system visualized in Figure 6b and described by mathematical expression where K = 0.1 is the internally trimmed constant of a four-quadrant analog multiplier. Obviously, the number of active elements decreases to five. On the other hand, the third state variable is not easily measurable. Note that, without the introduction of state variable rescaling, transadmittance amplifier g1 should be able to handle high input voltages without output current distortion. The oscillator can be simplified by the following different idea. Assume that each differential equation represents the sum of the currents measured at independent nodes. One can easily obtain the electronic system visualized in Figure 6b and described by mathematical expression

Experimental Verification
where K = 0.1 is the internally trimmed constant of a four-quadrant analog multiplier. Obviously, the number of active elements decreases to five. On the other hand, the third state variable is not easily measurable. Note that, without the introduction of state variable rescaling, transadmittance amplifier g 1 should be able to handle high input voltages without output current distortion.

Experimental Verification
The new structure of the chaotic oscillator was designed using universal breadboard; see Figure 7, where the measurement setup is demonstrated. Interestingly, variously shaped strange attractors captured by oscilloscope during experimental verification of the proposed chaotic oscillator are shown in Figure 8. Some of them were not discovered during numerical analysis but were repeatedly observable within the dynamics of the lumped chaotic oscillator.
The new structure of the chaotic oscillator was designed using universal breadboard; see Figure  7, where the measurement setup is demonstrated. Interestingly, variously shaped strange attractors captured by oscilloscope during experimental verification of the proposed chaotic oscillator are shown in Figure 8. Some of them were not discovered during numerical analysis but were repeatedly observable within the dynamics of the lumped chaotic oscillator.  For given numerical values, the designed chaotic circuit generated waveforms with a continuous scale of the nonnegligible frequency components up to 100 kHz. Therefore, this oscillator can be used in many applications instead of truly random (noise) signal generators [41], to secure communication channels and information sources [42], as the core engine for chaos-based modulation and cryptography techniques [43], etc.
Appl. Sci. 2020, 10, x FOR PEER REVIEW 9 of 13 The new structure of the chaotic oscillator was designed using universal breadboard; see Figure  7, where the measurement setup is demonstrated. Interestingly, variously shaped strange attractors captured by oscilloscope during experimental verification of the proposed chaotic oscillator are shown in Figure 8. Some of them were not discovered during numerical analysis but were repeatedly observable within the dynamics of the lumped chaotic oscillator.  For given numerical values, the designed chaotic circuit generated waveforms with a continuous scale of the nonnegligible frequency components up to 100 kHz. Therefore, this oscillator can be used in many applications instead of truly random (noise) signal generators [41], to secure communication channels and information sources [42], as the core engine for chaos-based modulation and cryptography techniques [43], etc.

Discussion
The work described in this paper, that is, the discovery of chaotic dynamics within a generalized class C amplifier, leaves a significant place for future research activities. Let us mention a few unanswered questions and challenges: a) The presence of chaos in the case of nonlinear transconductances approximated by piecewiselinear functions. From a circuit design point of view, the analog multipliers can be substituted by series and/or parallel connections of diodes. b) The existence of chaos in a class D single-transistor-based amplifier. Three degrees of freedom required for chaos evolution are presented in fundamental topology (thanks to an output LC low-pass passive filter) without need of considering intrinsic parasitic capacitances of the transistor itself.

Discussion
The work described in this paper, that is, the discovery of chaotic dynamics within a generalized class C amplifier, leaves a significant place for future research activities. Let us mention a few unanswered questions and challenges: This list is by no means complete. Further interesting reading about chaotic circuits having one or two bipolar junction transistors can be found in [44].

Conclusions
This popularizing paper represents a simultaneous contribution to two areas of chaos theory: (a) the discovery of a new chaotic dynamical system with a specific type of equilibria, namely a single unstable fixed point, and (b) the numerical description of a very simple single-transistor-based network structure from the viewpoint of chaos evolution.
The analyzed amplification topology can be commonly found in many complex electronic systems. The formation of strange attractors is conditioned by the non-unilateral nature of the bipolar transistor; backward trans-conductance especially needs to be active. Typical values of admittance parameters depend not only on the bias point but also on the transistor used. Unipolar transistors are usually modeled differently: by input admittance y 11 and trans-conductance y 12 equaling zero. However, matching circuits can cause feedback required for chaos evolution. In such a situation, chaos can be observed also in the case of a class C amplifier with a field-effect transistor. Matching networks with accumulation elements will increase the order of the final circuit; i.e., the probability of chaos can be boosted as well.
Based on the findings provided in this paper, chaotic motion can be mitigated by unilaterization of a bipolar transistor, that is, by the reduction of its backward trans-conductance.