Chapter 7
Stimulation of Chaos in Electric Drive Systems
Instead of avoiding the occurrence of chaos or stabilizing the chaotic phenomenon in electric drive systems, it is becoming attractive to stimulate the occurrence of chaos and hence to positively utilize chaos. The stimulation of chaos, termed the chaoization, can be classified into two approaches: control-oriented chaoization and design-oriented chaoization. The control-oriented approaches – which offer the definite advantage of flexibility, but need additional control – use control means to stimulate the system exhibiting chaotic behavior. On the other hand, the design-oriented approaches – which offer the definite advantage of simplicity, but lack flexibility – design the system so that it can inherently generate chaotic behavior.
In this chapter, the control-oriented and design-oriented chaoization are both discussed, hence stimulating chaotic operation in various electric drive systems.
7.1 Control-Oriented Chaoization
Many control approaches that have been proposed to stimulate chaos in various nonlinear systems. For the stimulation of chaos in electric drive systems, the available approaches are quite limited, and most of them are still under theoretical development (Ge, Cheng, and Chen, 2004). In this section, several viable control approaches that have been successfully implemented in electric drive systems are discussed, namely time-delay feedback control, proportional time-delay control, and chaotic speed reference control.
7.1.1 Time-Delay Feedback Control of PMDC Drive System
As discussed in Chapter 3, a PMDC motor can be modeled as a single-input/single-output (SISO) linear time-invariant (LTI) system:
where 
is the motor speed, T is the motor torque, B is the viscous damping coefficient, and J is the load inertia, while T and 
denote the input and output of the system, respectively. By employing the anticontrol of chaos in continuous-time systems (Wang, Chen, and Yu, 2000), the time-delay feedback control can be designed as:
where 
is the torque parameter, 
is the speed parameter, and 
is the time-delay parameter. It should be noted that all three parameters are adjustable to achieve the desired chaotic motion (Ye and Chau, 2007). Also, 
should be a bounded function to satisfy the condition that the required torque does not exceed the motor torque capability. Substituting (7.2) into (7.1), the time-delay system equation can be formulated as:
After defining the normalized speed 
, (7.3) is rewritten as:
For a SISO LTI system, this can be expressed as:
(7.5) 
where 
is the input, 
is the output, and 
and 
are constants with 
. When the system is incorporated with time-delay feedback control, 
and 
are related by:
(7.6) 
where 
is a bounded continuous function. Accordingly, a time-delay differential equation can be asymptotically approximated by a difference equation as given by:
where 
for 
, 
is sufficiently large, and 
is large.
For this chaotic motor, 
, 
and 
is represented by 
. Hence, by using (7.7), (7.4) can be asymptotically approximated by:
(7.8) 
Since a sine function is a continuous bounded function, it is naturally chosen as 
. Hence, the system equation can be expressed as:
By writing 
and 
, (7.9) can be expressed as:
(7.10) 
which has been proved to exhibit chaotic behavior with certain values of r (Strogatz, 1994). Therefore, the proposed time-delay feedback control scheme given by (7.2) is rewritten as:
which can offer chaotic motion by selecting appropriate values of 
, 
, and 
.
In order to implement the desired chaotic motor based on the above derivation, both the armature current and the rotor speed of the PMDC motor are used as feedback signals, while the torque command 
calculated by (7.11) is used to generate the current command 
for PWM control of the PMDC motor. Figure 7.1 shows the corresponding control system. First, the measured speed feedback is delayed by a preset value of 
. Then, the delayed speed is fed into the torque control block in which proper values of 
and 
are preset. Hence, it first generates 
and then 
. Subsequently, the difference between 
and the measured current feedback is fed into the current control block in which a simple PI control is adopted. Hence, it generates the desired duty ratio for the full-bridge PWM converter which provides bidirectional current control of the PMDC motor.
Figure 7.1 Time-delay feedback control of PMDC motor
Table 7.1 lists some key data of the PMDC motor. Making use of (7.11), bifurcation diagrams with respect to various adjustable parameters are readily deduced. These bifurcation diagrams can illustrate at a glance how the system behavior is affected by varying the parameters. For the sake of illustration, the case of no load torque 
is adopted for the simulation.
Table 7.1 Key parameters of PMDC motor.
| Supply voltage U | 24 V |
| Torque constant K | 0.05 Nm/A |
| Armature resistance R | 1.1 Ω |
| Armature inductance L | 0.4 mH |
| Viscous coefficient B | 1.0 × 10−4 Nm/rad/s |
| Rotor inertia J | 1.0388 × 10−5 Nm/rad/s2 |
When selecting ξ = 20 and τ = 1.5 s, both the speed and current bifurcation diagrams with respect to 
are as shown in Figure 7.2. It can be seen that the motor initially operates at a fixed point (which is equivalent to the normal or so-called period-1 operation) with a small value of 
. Figure 7.3 shows the corresponding motor speed and armature current waveforms when 
. With an increase of 
, the motor bifurcates to a period-2 operation, which is equivalent to an abnormal subharmonic operation. Figure 7.4 shows the corresponding speed and current when 
. Finally, the motor exhibits chaotic motion when 
is further increased. Figure 7.5 shows the corresponding speed and current when 
. When the motor is run in chaotic mode, both the amplitude and direction of the motor speed and armature current change with time and present ergodicity in the range illustrated in Figure 7.2. Therefore, the torque parameter 
is the key to induce chaotic motion from normal operation.
Figure 7.2 Bifurcation diagrams with respect to torque parameter. (a) Motor speed. (b) Armature current
On the other hand, when selecting 
and 
, both the speed and current bifurcation diagrams with respect to 
are shown in Figure 7.6. It can be seen that 
can be used to almost linearly adjust the speed range of the chaotic motion, which is why it is called the speed parameter. Therefore, the speed parameter 
serves to adjust the boundary of chaotic motion.
Figure 7.6 Bifurcation diagrams with respect to speed parameter. (a) Motor speed. (b) Armature current
Furthermore, when selecting 
and 
, the corresponding bifurcation diagrams with respect to 
are as shown in Figure 7.7. It can be observed that the change of 
has little effect on the speed range of the chaotic motion. Nevertheless, this parameter has a significant impact on the system realization. If the parameter is too large, the refreshing rate of the control system will be too slow; if the parameter is too small, it will require too much computational resource. Therefore, the time-delay parameter 
functions to tune the refreshing rate of the chaoization.
Figure 7.7 Bifurcation diagrams with respect to time-delay parameter. (a) Motor speed. (b) Armature current
In order to verify the above analysis, the simulation and experimentation are both performed with the same parameter settings. Firstly, Figure 7.8 shows the simulated waveforms of motor speed and armature current under normal operation (
, 
, 
), while Figure 7.9 shows the measured normal waveforms of motor speed and armature current under the same conditions. It can be seen that the agreement is very good. Secondly, Figure 7.10 shows the simulated waveforms of motor speed and armature current under chaotic operation (
, 
, 
), while Figure 7.11 shows the measured chaotic waveforms of motor speed and armature current under the same conditions. It can be seen that both the simulated and the measured waveforms exhibit chaotic behavior but with different patterns. This is expected because the chaotic behavior is not periodic, so that the period of measurement cannot be the same with that of simulation. Also, chaotic behavior is very sensitive to the initial states. A tiny discrepancy in the initial states will cause a huge difference in chaotic behavior. Nevertheless, it can be found that the measured boundaries of the chaotic waveforms are in good agreement with the simulated ones. This is the actual nature of chaos – random-like but bounded.
Figure 7.9 Measured waveforms of motor speed and armature current under normal operation (1666.7 rpm/div, 0.5 A/div, 0.2 s/div)
Figure 7.11 Measured waveforms of motor speed and armature current under chaotic operation (1666.7 rpm/div, 1 A/div, 0.2 s/div)
7.1.2 Time-delay Feedback Control of PM Synchronous Drive System
As discussed in Chapter 4, a 3-phase PM synchronous motor (PMSM) can be transformed into the d–q frame. In order to utilize the advantages of vector control, the d-axis stator current 
is set to zero so that the electromagnetic torque 
can simply be expressed as:
(7.12) 
where P is the number of poles, 
is the PM flux linkage in the rotor, and 
is the q-axis stator current. Since this torque expression resembles that of the PMDC motor, the same time-delay feedback control scheme can be used:
Hence, the torque command 
calculated by (7.13) is used to generate the torque-component current command 
for the PM synchronous motor (Gao and Chau, 2002). Figure 7.12 shows the corresponding control system.
Figure 7.12 Time-delay feedback control of PM synchronous motor
A practical 3-phase PM synchronous motor is used for exemplification. The corresponding key parameters are P = 4, 
, J = 1.44 × 10−5 kgm2, B = 5.416 × 10−4 Nm/rad/s, Ld = 11.5 mH, and Lq = 11.5 mH. Based on a time-delay feedback, the motor initially operates at a fixed point with a small value of μ. With an increase in μ, periodic motion occurs, and with a further increase in μ, the system exhibits chaotic motion. Figure 7.13 shows the simulated speed and current waveforms under period-2 operation when μ = 2.55, ξ = 10, and τ = 1 s, whereas Figure 7.14 shows the measured speed and current waveforms under the same μ and ξ. As expected, the experimental measurement closely matches the simulation waveforms. On the other hand, Figure 7.15 shows the simulated chaotic speed and current waveforms when μ = 4, ξ = 10, and τ = 1 s, while Figure 7.16 shows the measured chaotic speed and current waveforms with the same μ and ξ. It can be found that they exhibit chaotic behavior and offer similar boundaries. Notice that chaotic waveforms cannot be directly compared since they are random-like but bounded.
7.1.3 Proportional Time-Delay Control of PMDC Drive System
The principle of the proportional-time-delay (PTD) control of a PMDC drive system can be represented as:
(7.14) 
where 
is the current reference, 
is the time-delay gain, 
is the time-delay constant, 
and 
are proportional gains, and 
is the nominal rotational speed. It should be noted that a proportional term 
is added on the conventional time-delay feedback control term 
. Thus, this chaoizing approach can offer unidirectional rotation for the PMDC drive. Moreover, it possesses the ability of chaoizing the PMDC drive with a large inertia which is highly desirable for many industrial applications. Figure 7.17 shows the block diagram of this PTD control.
Figure 7.17 PTD torque controller
For the PTD control, when 
tracks 
well, the equation of motion can be written as:
Accordingly, when 
, 
is sufficiently large and 
is bounded, the approximated solution of (7.15) can be computed iteratively as (Wang, Chen, and Yu, 2000):
(7.16) 
A practical PMDC motor is used for exemplification, and its key parameters are listed in Table 7.2. Given β = 10, σ = 1, 
, and 
, the bifurcation diagram of 
versus 
under PTD control can be deduced as depicted in Figure 7.18(a). It can be seen that it is a typical period-doubling route to chaos. Figure 7.18(b) shows the corresponding Lyapunov exponent in which the positive values denote the existence of chaos mathematically. Similarly, given 
= 10, α = 400, 
, and 
, the bifurcation diagram of 
against 
and the corresponding Lyapunov exponent are shown in Figure 7.19. Also, there is a period-doubling route to chaos with respect to 
.
Table 7.2 Key parameters of PMDC motor.
| Torque constant KT | 0.2286 Nm/A |
| Back EMF constant KE | 0.2286 V/rad/s |
| Armature resistance Ra | 3.42 Ω |
| Armature inductance La | 3.4 mH |
| Viscous coefficient B | 8 × 10−5 Nm/rad/s |
| Rotor inertia J | 1.588 × 10−4 kgm2 |
Figure 7.18 Chaotic analysis of PTD with respect to 
. (a) Bifurcation diagram. (b) Lyapunov exponent
Figure 7.19 Chaotic analysis of PTD with respect to 
. (a) Bifurcation diagram. (b) Lyapunov exponent
Both computer simulation and experimentation are used to assess the performance of the chaotic drive system. The selected parameters for chaoization are Tl = 0, α = 400, β = 10, τ = 1 s, 
, and σ = 1. The simulated motor speed and armature current waveforms are shown in Figure 7.20. It can be observed that the waveforms are chaotic, while the motor speed can be maintained at positive values. Under the same operating condition, the measured waveforms are shown in Figure 7.21. It can be seen that the simulated and measured waveforms have the same boundaries although the patterns cannot fully match one another, which is actually the random-like but bounded nature of chaos.
Figure 7.21 Measured chaotic waveforms using PTD. (a) Motor speed (500 ms/div; 191 rpm/div). (b) Armature current (500 ms/div; 0.5 A/div)
7.1.4 Chaotic Signal Reference Control of PMDC Drive System
The chaotic speed reference (CSR) control of a PMDC drive essentially adopts PI control to force the motor speed 
following a chaotic speed reference 
. The PI control principle is given by:
(7.17) 
where 
is the proportional coefficient and 
is the integral coefficient. The chaotic speed reference is generated by the well-known Logistic map which is given by:
(7.19) 
where 
is the base speed, 
is the speed boundary, 
is the chaotic series, and 
is the control parameter of the Logistic map. Figure 7.22 shows the block diagram of the CSR torque controller.
Figure 7.22 CSR torque controller
For the CSR torque control, the value of 
is updated according to (7.18) at each interval of 50 ms to enable 
to track 
accurately. Thus, the motor speed of the PMDC drive can be expressed as an iterative map:
(7.20) 
Given 
and 
, Figure 7.23 shows the bifurcation diagram of 
versus 
of the PMDC motor using the CSR control, and the corresponding Lyapunov exponent. It can be observed that there is a typical period-doubling route to chaos.
Both computer simulation and experimentation are used to assess the performance. The selected parameters for chaoization are 
, 
, and A = 4. The simulated rotor speed and armature current waveforms are shown in Figure 7.24. Similar to the PTD control, this CSR control can produce the desired chaotic waveforms, namely the rotor speed can be maintained at positive values. Furthermore, by comparing their speed and current waveforms, it can be observed that the PTD control needs lower current magnitudes than the CSR control while the CSR control can offer more accurate speed boundaries than the PTD control. The reason is due to the fact that the PTD control is a model-based method without a closed-loop of speed. Therefore, it is sensitive to motor parameter variation and load torque fluctuation.
Under the same operating condition, the measured waveforms are shown in Figure 7.25. It can be seen that the simulated and measured waveforms have the same boundaries although the patterns cannot fully match one another, which is actually the random-like but bounded nature of chaos. Nevertheless, there are some discrepancies between the simulated and measured current waveforms, especially in the high-frequency region. It is mainly due to the system parasitics and noise in measurement which have been ignored in the simulation model.
Figure 7.25 Measured chaotic waveforms using CSR. (a) Motor speed (500 ms/div; 191 rpm/div). (b) Armature current (500 ms/div; 0.5 A/div)
7.2 Design-Oriented Chaoization
Instead of relying on control approaches to stimulate chaos, an electric drive system can generate chaotic behavior based on design-oriented approaches. Namely, the electric drive system can spontaneously produce chaotic motion once power on. In this section, two kinds of electric drive systems, namely the doubly salient PM (DSPM) drive and the shaded-pole induction drive, are chaoized by using design-oriented approaches.
7.2.1 Doubly Salient PM Drive System
The DSPM motor is a new kind of brushless motors (Liu et al., 2008). It combines the advantages of both the switched reluctance motor and the PM brushless motor, hence achieving robust structure, high power density, high efficiency, and high reliability, while being maintenance free (Cheng, Chau, and Chan, 2001). Thus, it is attractive for high-performance applications.
A 3-phase 12/8-pole rotor-skewed DSPM motor is adopted for exemplification. As shown in Figure 7.26, it consists of 3-phase armature windings in the stator, 4 pieces of PM material located in the stator, 12 salient poles in the stator, and 8 salient poles in the rotor. With rotor skewing, the PM flux can be expressed as:
Figure 7.26 Topology of 3-phase 12/8-pole rotor-skewed DSPM motor
where both 
and 
correspond to the A, B, and C phases, respectively, 
and 
are the average and amplitude of the PM flux variations, 
is the number of pole pairs, and 
is the spatial angle with reference to the aligned position of phase A. The mutual inductances of the windings are negligible, whereas the self-inductances can be written as:
(7.22) 
where 
and 
are the average and amplitude of self-inductance variations. Hence, the dynamic equations of the DSPM motor fed by 3-phase symmetric sinusoidal voltages are given by:
(7.23) 
(7.24) 
(7.25) 
where 
is the input phase voltage, 
is the amplitude of the phase voltage, 
is the frequency of the phase voltage, 
is the phase current, Rs is the winding resistance, ω is the motor speed, 
is the electromagnetic torque, 
is the load torque, B is the viscous coefficient, and J is the inertia. Thus, (7.21) –(7.26) form a fourth-order nonautonomous dynamical system.
The system model given by illustrates that the variation in the PM flux is an important parameter contributing to the system dynamics. So, the design of PMs – mathematically the value of 
– is used for chaoization of the DSPM drive. By adopting 
, 
, 
, 
, 
, Rs = 2.7 Ω, B = 4 × 10−4 Nm/rads−1, Tl = 0, and J = 1.35 × 10−3 kgm2, the bifurcation diagram of ω with respect to 
can be obtained, as shown in Figure 7.27. It can be seen that the motor speed changes from regular period-1 motion to subharmonic operation at 
, and then bifurcates to chaotic operation at 
(the so-called threshold of chaoization). Figure 7.28 shows the corresponding normalized maximum Lyapunov exponent value where the positive values are normalized to be unity while the negative or zero values are normalized to be zero. Thus, a value of unity indicates the existence of chaotic motion. It should be noted that the normalized maximum Lyapunov exponent has a zero value at 
, where subharmonic motion occurs according to the bifurcation diagram.
Figure 7.27 Bifurcation diagram of motor speed against PM flux of DSPM drive
Figure 7.28 Normalized maximum Lyapunov exponent against PM flux of DSPM drive
Chaotic motion in the DSPM drive can also be produced by tuning the amplitude of the input voltage. Given 
, the bifurcation diagram of the motor speed against 
is plotted as shown in Figure 7.29. Also, there is a critical value of Ux = 18.2 V where period-1 motion will bifurcate into subharmonic motion, and a critical value of Ux = 28.3 V where chaotic motion occurs. Thus, the DSPM drive can spontaneously produce chaotic motion by tuning the input voltage, which actually can be easily realized by using a transformer. Figure 7.30 shows the corresponding normalized maximum Lyapunov exponent. It should be noted that when Ux > 40.0 V, the corresponding normalized maximum Lyapunov exponent is zero, but clusters of points can be observed from the bifurcation diagram at this region. This indicates that there is a quasiperiodic motion in this region.
Figure 7.29 Bifurcation diagram of motor speed against input voltage of DSPM drive
Figure 7.30 Normalized maximum Lyapunov exponent against input voltage of DSPM drive
Firstly, the behaviors of the DSPM drive under different 
values are simulated with a fixed value of Ux = 35.4 V. Figure 7.31 shows its trajectory and Poincaré map when 
. The corresponding dynamics exhibit random-like but bounded behavior, which is a typical characteristic of chaos. Then, when 
, the system exhibits a higher order subharmonic motion, as shown in Figure 7.32. Consequently, when 
, it becomes a second-order subharmonic motion as shown in Figure 7.33. Finally, when 
, the system operates with normal period-1 motion, as shown in Figure 7.34.
. (a) Trajectory. (b) Poincaré map
. (a) Trajectory. (b) Poincaré map
. (a) Trajectory. (b) Poincaré map
. (a) Trajectory. (b) Poincaré map
Secondly, the behaviors of the DSPM drive under different 
values are simulated with a fixed value of 
. Figure 7.35 shows its trajectory and the Poincaré map when 
. It can be seen that the trajectory distributes densely, while the Poincaré map possesses a closed orbit. This indicates that the DSPM drive exhibits a quasiperiodic motion. Then, when Ux = 35.4 V, it exhibits a chaotic motion, as shown in Figure 7.36. Consequently, when Ux = 25 V, second-order subharmonic motion occurs, as shown in Figure 7.37. Finally, when Ux = 10 V, the system operates with a normal period-1 motion as shown in Figure 7.38.
. (a) Trajectory. (b) Poincaré map
. (a) Trajectory. (b) Poincaré map
. (a) Trajectory. (b) Poincaré map
. (a) Trajectory. (b) Poincaré map
7.2.2 Shaded-Pole Induction Drive System
The shaded-pole induction motor is one of the most popular single-phase induction motors for domestic electric appliances. It offers the definite advantages of simple structure and low cost, as well as being highly rugged and reliable (Veinott and Martin, 1987). Its unique feature is the use of an auxiliary single-turn winding, which differs from the distributed windings of other single-phase induction motors, to produce the desired starting torque.
As shown in Figure 7.39(a), the shaded-pole induction motor has a squirrel-cage rotor and a salient-pole stator. It does not need a starting device or switch. Wound on the stator are the main and auxiliary windings. The auxiliary winding is a short-circuited copper band located in a corner of each pole, the so-called shaded-pole winding. This shaded-pole winding has no electrical connection with the input voltage supply. When the main winding is connected to the input voltage supply, the flux through the shaded portion of the pole lags the flux through the unshaded portion owing to the fact that the induced current in the auxiliary winding tends to oppose or delay the generated flux. As a result, the stator flux has two components, one of which lags the other. This leads to the development of a kind of rotating field, thus inducing the rotor current and hence the rotor flux. The interaction of the stator and rotor fluxes produces the starting torque whose direction is from the unshaded portion to the shaded portion of the pole.
Figure 7.39(b) shows the stationary α–β axis model of a shaded-pole induction motor. When the motor is connected to the input voltage supply, its main winding voltage has the form of 
, where 
is the voltage amplitude, f is the supply frequency, and θ is the initial phase angle. The system dynamical behavior can then be expressed as (Desai and Mathew, 1971; Osheiba, Ahmed, and Rahman, 1991):
(7.28) 
(7.29) 
(7.30) 
where R is the resistance matrix and L is the inductance matrix; im and ia are the currents of the main winding and auxiliary winding, respectively; iα and iβ are the rotor currents with respect to the α–axis and β–axis, respectively; Rm, Ra, and Rr are the resistances of the main winding, auxiliary winding, and rotor winding, respectively; Lmm, Laa, Lαα, and Lββ are the self-inductances of the main winding, auxiliary winding, α–axis rotor winding, and β–axis rotor winding, respectively; Lmα is the mutual inductance between the main winding and the α–axis rotor winding; Lma is the mutual inductance between the main winding and the auxiliary winding; Lar is the mutual inductance between the auxiliary winding and the rotor winding when the rotor is aligned with the auxiliary winding axis; ωr is the motor speed; np is the number of pole pairs; δ is the angle between the main winding and the auxiliary winding; J is the rotor inertia; B is the viscous coefficient; and Te and TL are the electromagnetic torque and load torque, respectively. The model described by (7.27) –(7.31) is a periodically driven nonautonomous system, which is apt to subharmonic oscillations and chaos (Moon and Holmes, 1979).
A practical shaded-pole induction motor with its parameters listed in Table 7.3 is used for exemplification. Based on the model described by (7.27) –(7.31) and the Poincaré mapping, speed bifurcation diagrams with respect to different system parameters can be obtained. Since the motor parameters, such as the pole number, inductances, and resistances are essentially fixed after production, the frequency and amplitude of the applied voltage are selected as the bifurcation parameters. The steady-state behavior against the bifurcation parameter plane is depicted in Figure 7.40 in which the steady-state solutions of the period-1, period-k (k > 1), and the others are represented by the plain white, small dot, and large dot, respectively. The corresponding bifurcation diagrams are depicted in Figure 7.41. It can be observed that the drive behavior varies greatly with the change of these operating parameters. Periodic oscillation, quasiperiodic, and chaos can all be found.
Table 7.3 Key parameters of shaded-pole induction motor.
| Number of pole pairs np | 1 |
| Self inductance of main winding Lmm | 411.1 mH |
| Self inductance of auxiliary winding Laa | 0.01102 mH |
| Self inductance of α-axis rotor winding Lαα | 410.5 mH |
| Self inductance of β-axis rotor winding Lββ | 376.7 mH |
| Mutual inductance between main and α-axis rotor windings Lmα | 352.8 mH |
| Mutual inductance between main and auxiliary windings Lma | 1.5 mH |
| Mutual inductance between auxiliary and rotor windings when rotor aligns with auxiliary axis Lar | 1.7 mH |
| Resistance of main winding Rm | 5.630 Ω |
| Resistance of auxiliary winding Ra | 0.212 Ω |
| Resistance of rotor winding Rr | 25.0 Ω |
| Angle between main and auxiliary windings δ | 28° |
| Rotor inertia J | 2.130 × 10−5 kgm2 |
| Viscous coefficient B | 1.47 × 10−4 Nm/rad/s |
Figure 7.40 Categories of steady-state solutions in bifurcation parameter plane
Figure 7.41 Speed bifurcation diagrams. (a) Varying frequency with Vm = 220 V. (b) Varying amplitude with f = 10 Hz. (c) Varying amplitude with f = 22 Hz
As shown in Figure 7.41(a), the drive works as a normal shaded-pole induction motor in the frequency range of 40–60 Hz. Namely, it operates at constant speed with a small amplitude of period-1 oscillation. With a decrease in frequency, a period-2 oscillation occurs. When the frequency is further decreased, complex behaviors (such as quasiperiodic and chaos) arise. Despite of different patterns, similar findings can be observed from the bifurcation diagrams in Figures 7.41(b) and 7.41(c). Therefore, different chaotic boundaries and even types of chaos can result by varying either the frequency or the amplitude of the applied voltage.
Figures 7.42 and 7.43 show the measured speed and main winding current waveforms, respectively, at various periodic-speed operations, namely period-1, period-2, and period-3. It can be found that the period-1 waveform is the normal speed waveform of a conventional shaded-pole induction drive, whereas the period-2 and period-3 waveforms correspond to its abnormal subharmonic operations. Their speed ripples actually reflect the well-known phenomenon of torque pulsation caused by the auxiliary winding.
Figure 7.42 Measured periodic speed waveforms. (a) Vm = 220 V, f = 60 Hz. (b) Vm = 220 V, f = 38 Hz. (c) Vm =160 V, f = 22 Hz
Figure 7.43 Measured periodic main winding current waveforms. (a) Vm = 220 V, f = 60 Hz. (b) Vm = 220 V, f = 38 Hz. (c) Vm = 160 V, f = 22 Hz
As can be seen in the speed bifurcation diagrams, the drive exhibits chaotic behavior at certain ranges of the bifurcation parameters. Figures 7.44 and 7.45 show the measured speed and main winding current waveforms, respectively, at various chaotic speed operations. It can be found that the chaotic speed waveforms offer a well-known property of chaos, namely random-like but bounded oscillations. Also, these waveforms are aperiodic and very sensitive to the initial conditions. Although, the main winding current waveforms seem to be more regular than the speed waveforms, they are still random-like and aperiodic.
Figure 7.44 Measured chaotic speed waveforms. (a) Vm = 150 V, f = 10 Hz. (b) Vm = 220 V, f = 10 Hz. (c) Vm = 220 V, f = 18 Hz
Figure 7.45 Measured chaotic main winding current waveforms. (a) Vm = 150 V, f = 10 Hz. (b) Vm = 220 V, f =10 Hz. (c) Vm = 220 V, f = 18 Hz
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