Chapter 9

Application of Chaotic Modulation

With the advent of pulse width modulation (PWM), switching power converters have received a great attraction for application in electric drive systems, because of the advantages of flexible power control, compact size, and high efficiency. In general, the PWM DC–DC converter (usually called the DC chopper) functions to control the applied voltage or current for DC drive systems, whereas the PWM DC–AC converter (usually called the AC inverter) is to simultaneously control the applied voltage or current and frequency for AC drive systems. However, because of the nature of switching, those advantages are counterbalanced by the generation of harmonics, electromagnetic interference (EMI), and acoustic noise. Increasingly, there is a trend for pushing up the switching frequency of modern electric drive systems, hence reducing their volume and weight. This trend inevitably contributes to an increasing level of EMI. It also degrades the electromagnetic compatibility (EMC) of electronic devices.

Conventionally, the aforementioned EMI problems are alleviated by filtering the output or shielding the setup. In recent years, attention has been focused on using signal processing rather than filtering or shielding – namely, developing various PWM schemes for inverter-fed AC drive systems.

In this chapter, chaos is applied to PWM schemes for inverter-fed AC drive systems, hence reducing the corresponding audible noise and mechanical vibration. The open-loop and closed-loop control of chaotic PWM inverter drive systems are discussed.

9.1 Overview of PWM Schemes

PWM schemes have been the subject of intensive research during the last few decades. A large variety of methods, different in concept and performance, have been developed and described. As there are many power electronics books and survey papers comprehensively discussing various PWM schemes (Bowes and Clements, 1982; Holtz, 1992), this section aims to give a brief overview of various PWM schemes.

Basically, PWM schemes can be categorized as voltage control and current control. The voltage-controlled schemes generally operate in an open-loop feed-forward fashion, while the current-controlled schemes generally operate in a closed-loop feedback fashion.

9.1.1 Voltage-Controlled PWM Schemes

Over the years, numerous voltage-controlled PWM schemes have been developed. In this section, five representative schemes that are widely accepted for inverter-fed AC drive systems are selected for discussion: namely, the sinusoidal PWM, regular PWM, optimal PWM, delta PWM, and random PWM.

9.1.1.1 Sinusoidal PWM

The sinusoidal PWM is the earliest PWM scheme and was developed for analog implementation (Mokrytzki, 1967; Bowes, 1975). Basically, a triangular carrier wave (known as a sampling signal) is compared directly with a sinusoidal modulating wave (known as a reference signal) to determine the switching instants and, hence, the resultant pulse widths. Since the switching edges of the pulse are determined by the instantaneous intersection of the two waves, the resultant pulse widths are proportional to the amplitudes of the modulating wave at the switching instants. This causes the centers of the pulses in the PWM waveform to be unequally spaced, and presents difficulty in analytically expressing the pulse widths of the PWM waveform. Actually, such pulse widths can only be expressed by using a transcendental equation, and in terms of a series of Bessel functions. So, it is not feasible to calculate the pulse widths directly in real time.

This sinusoidal PWM takes the advantages of real-time generation using low-cost analog hardware, linear amplification, and constant average switching frequency. However, it is ill-suited for digital implementation, which is actually the trend of development for modern electric drives.

9.1.1.2 Regular PWM

The regular PWM is recognized to have a definite advantage over the sinusoidal PWM when implemented using digital or microprocessor techniques (Bowes, 1975; Bowes and Mount, 1981). Basically, the amplitude of the modulating wave at the sample instant is stored by a sampling circuit (which is also operated at the carrier frequency), and is held constant until the next sample is taken. This produces a sample-and-hold version of the modulating wave. Then, the intersection of this “staircase” modulating wave and the triangular carrier wave determine the switching instants, and hence the pulse widths. Since the staircase modulating wave has constant amplitude when each sample is taken, the pulse widths are proportional to the amplitudes of the modulating wave at regularly spaced sampling intervals.

It is an important feature of regular PWM that the sampling positions and sampled values can be defined unambiguously – that is, the pulses produced are well defined both in width and position, which is not the case for the sinusoidal PWM. Because of this feature, it is possible to derive a simple trigonometric function to calculate the pulse widths of regular PWM. Hence, the real-time generation of a regular PWM can readily be performed by using a low-cost microprocessor or microcontroller.

Rather than modulating the pulse edge symmetrically, the regular PWM can be further extended to modulate the pulse edges by different amounts. Namely, the leading and trailing edges of each pulse are determined by using two different samples of modulating wave. Although the pulse widths of this asymmetric regular PWM waveform can still be expressed as a simple trigonometric function, the required number of calculations is almost double that for the original regular PWM waveform. This will significantly increase the computational time for PWM generation, and thus reduce the maximum allowable inverter output frequency. Nevertheless, since more information about a modulating wave is associated with an asymmetric regular PWM waveform, its harmonic spectrum is superior to that produced using the symmetric one. Because of the sample-and-hold process, the performance of a regular PWM is inevitably inferior to that of a sinusoidal PWM, especially at low pulse numbers.

9.1.1.3 Optimal PWM

The optimal PWM offers the definite advantage of the capability of optimization (Buja and Indri, 1977). With the advent of microprocessor technology, the implementation of this scheme is becoming feasible. Contrary to both sinusoidal PWM and regular PWM, which are generated on the basis of well-defined modulation processes, the optimal PWM is first defined by a general PWM waveform in terms of a set of switching angles which are then determined using numerical methods.

A typical optimal PWM waveform has an odd pulse number N and is of quarter-wave symmetry. Thus, only odd harmonics exist. Optimization is generally classified as harmonic elimination and an objective function. For harmonic elimination, it aims to eliminate a well-defined number of lower order harmonics from the spectrum. Hence, it can eliminate all torque harmonics having six times the fundamental frequency at N = 5 and so on (Patel and Hoft, 1973). A well-accepted objective function is the minimization of the harmonic current distortion or the peak current at very low pulse numbers. The objective function that defines the optimization problem generally exhibits a large number of local minimums, which makes the numerical calculation of the optimal switching angles extremely time consuming, even with modern computers. These angles can subsequently be preprogrammed into the microprocessor memory and used to generate the optimal PWM waveform in real-time.

9.1.1.4 Delta PWM

The delta PWM is particularly attractive for low-cost inverter-fed AC drive systems in which variable-voltage variable-frequency (VVVF) control is adopted (Ziogas, 1981). Basically, it utilizes a sinusoidal modulating wave and a delta-shaped carrier wave which is forced to oscillate within a predefined window extending equally above and below the sinusoidal modulating wave. The minimum window width and the maximum carrier slope determine the maximum switching frequency of the resultant PWM waveform. This forced oscillation ensures that the fundamental components of the carrier wave and the modulating wave have the same amplitude, and the dominant harmonics of the carrier wave and the resultant PWM waveform cluster close to the carrier frequency. Since the ratio of the fundamental component to the frequency of the resultant PWM waveform is directly proportional to the amplitude of the modulating wave, this ratio is constant until the output frequency reaches its critical value – that is, the base frequency at which the PWM waveform becomes a square wave. After the base frequency, the voltage amplitude remains constant.

The delta PWM offers three advantageous intrinsic features. Firstly, it provides inherent constant volts per hertz control for output frequencies below the base frequency, and constant voltage above the base frequency. Also, the transition between these two modes of operation is inherently smooth. It should be noted that all other PWM schemes have to rely on additional control circuitry or algorithms to perform VVVF control. Secondly, as the corresponding dominant harmonic frequencies are close to the carrier frequency, low-order harmonics are attenuated. Thirdly, it can be implemented by using a very simple and low-cost circuit, which involves only three operational amplifiers. Nevertheless, the delta PWM suffers from some drawbacks, such as a reduction of the fundamental output voltage and the possible generation of subharmonics (Rahman, Quacioe, and Chowdhury, 1987).

9.1.1.5 Random PWM

The random PWM is particularly attractive to suppress the annoying acoustic noise in AC drive systems, which is caused by the interaction of the fundamental and harmonic flux densities in the motor (Habetler and Divan, 1991; Blaabjerg et al., 1996). Basically, by randomly modulating the triangular carrier wave in sinusoidal PWM, the spectral tones around the switching frequency are spread out with a subsequent reduction in peak values, hence eliminating the annoying whine. Although the total level of the acoustic noise emitted by the motor remains constant, the acoustic noise is more pleasing to the ear since the noise is now random. This random PWM can maintain the advantages of sinusoidal PWM, including the capability of real-time generation, linear amplification, and constant average switching frequency.

The adoption of a very high bandwidth white noise to modulate the triangular carrier wave can result in a very wide spectrum for the PWM waveform, including substantial content at low frequencies. These low-order harmonics are difficult to filter, and will cause low-frequency currents flowing in the motor. In order to eliminate these low-order harmonics, the noise source needs to be a pink noise in which the power spectral density is inversely proportional to the frequency, and each octave carries an equal amount of noise power. Apart from controlling the bandwidth and amplitude of the noise, it may also be desired to vary other characteristics of the noise, such as adding pre-emphasis. However, the use of such techniques to reduce the low-order harmonics definitely complicates the implementation of random PWM.

The major shortcoming of random PWM is that it is difficult to generate a true random signal. Also, the corresponding switching frequency may clash with the system's natural frequency, which inevitably increases the possibility of creating mechanical resonance (Lo et al., 2000).

9.1.2 Current-Controlled PWM Schemes

Although voltage-controlled PWM schemes have been widely accepted for industrial AC drive systems, current control is becoming more and more attractive because of the following advantages (Kazmierkowski and Malesani, 1998):

• direct control of instantaneous current and hence the developed torque;
• inherent peak current protection;
• inherent overload rejection;
• good dynamic response;
• compensation for load parameter variations;
• compensation for device voltage drops and converter dead times;
• compensation for voltage changes.

In recent years, many current-controlled PWM schemes have been developed. In this section, two of the most representative schemes are selected for discussion – namely, hysteresis-band PWM and space vector PWM, which are well accepted for high-performance AC drive systems.

9.1.2.1 Hysteresis-Band PWM

The hysteresis-band PWM is the most popular current-controlled PWM scheme for inverter-fed AC drive systems because of its simple hardware implementation, fast response, and inherent peak current limiting capability (Plunkett, 1979). Basically, the motor phase currents are measured and compared with the respective sinusoidal command currents so that the resulting errors are fed into the hysteresis-band current controllers which, in turn, drive the PWM inverter. Consequently, the motor phase currents are forced to swing between the upper and lower hysteresis bands and hence track with the sinusoidal command currents.

The hysteresis bands are normally fixed and are the same for all phases. However, the use of fixed hysteresis bands has the drawback that the modulation frequency varies within a band. As a result, the phase current contains rich harmonics, which cause additional machine heating. The difficulty of vector conversion of those harmonic-rich feedback currents also causes control problems of the drive system. Moreover, since the fixed hysteresis band has to be designed on the worst-case basis, the drive system generally operates with nonoptimal phase current ripples (Bose, 1990). Therefore, adaptive control may be incorporated into this hysteresis-band PWM in which the band is modulated as a function of system parameters to maintain an almost constant modulation frequency.

9.1.2.2 Space Vector PWM

The space vector PWM is well accepted for high-performance AC drive systems because it can offer low harmonic current content in steady state and fast current response in transient state (Nabae, Ogasawara, and Akagi, 1986). Traditionally, these two requirements contradict one another. This space vector PWM utilizes the current deviation vector to satisfy both requirements. For steady-state operation, the switching mode with the smallest current-deviation derivative is chosen to suppress the current harmonic content and hence the torque ripple and acoustic noise. For transient operation, the switching mode with the largest current deviation derivative is chosen to produce a high-speed current response.

Basically, the inverter output voltage is expressed as eight possible voltage vectors, based on the on–off state of the six switching devices. After detecting the back EMF vector, the current-deviation derivative can be deduced, which is the most important control variable to command the harmonic current content or current response. However, this space vector PWM may suffer from inaccurate estimation of the back EMF vector due to the derivative of high-frequency current components.

In recent years, many improved versions of space vector PWM have been developed, such as the use of three-level hysteresis comparators to select proper voltage vectors via switching electrically programmable read-only memory (EPROM) table (Kazmierkowski, Dzieniakowski, and Sulkowski, 1991), or the incorporation of an asymmetrical modulating function and randomly varied pulse rate to improve the power spectrum and reduce switching losses (Trzynadlowski, Kirlin, and Legowski, 1997).

9.2 Noise and Vibration

In general, the total conducted EMI is caused by two mechanisms: the common-mode (CM) noise is related to the capacitive coupling of voltages with the line impedance stabilizing network (LISN), and the differential-mode (DM) noise is related to the voltage difference among phases. It has been verified that the high slew rate () of the PWM inverter voltages are mainly responsible for the conducted EMI in electric drives. Figure 9.1 illustrates the conducted EMI of an induction drive system fed from a voltage source PWM inverter. The CM current flows between the phases and the ground, and the corresponding excitation source is given by (Ran et al., 1998b b). On the other hand, the DM current flows between different phases, and the corresponding excitation source is given by (Ran et al., 1998a a).

The mechanical vibration of the induction drive system is mainly due to the electromagnetic force that occurred at the stator inner surface of the induction motor (Stemmler and Eilinger, 1994). The corresponding electromagnetic force density , which is both time and position dependent, can be expressed as:

(9.1)

(9.2)

where

(9.3)

(9.4)

(9.5)

where and are respectively the fundamental and nth harmonic components of the inverter output voltage vector; and are respectively the leakage factor of the stator and rotor; and are respectively the fundamental and nth harmonic components of the angular speed of the inverter output voltage vector; , and are respectively the stator number of turns of phase winding, inner diameter and active length; , and are respectively the magnetic flux linkage vector, magnetic flux density vector and magnetic flux density at and ; and , , and are respectively the magnitude, angular frequency, and initial phase angle of the electromagnetic force of each vibration mode . From (9.1) –(9.5), it can be observed that is governed by the inverter output voltages , , and . If the spectrum of overlaps with the natural frequencies of the induction motor, mechanical resonance will occur, causing annoying audible noise and even disastrous mechanical damage. Thus, a proper PWM scheme is highly desirable to avoid the mechanical resonance in the induction drive system.

To design a PWM strategy to avoid the mechanical resonance, the natural frequencies of the induction motor should be predicted in advance. In general, the natural frequencies of the stator of the induction motor can be determined by the motor parameters (Tímár, 1989):

(9.6)

(9.7)

(9.8)

where

(9.9)

(9.10)

(9.11)

(9.12)

(9.13)

where is the stator yoke mean radius, is the stator yoke height, is the modulus of elasticity for iron, is the outside radius of rotor core, is the length of rotor core, is the bearing distance, is the shaft diameter, is the stator tooth weight, is the stator winding weight, is the stator yoke weight, is the stator tooth length, is the mean width of stator tooth, and is the stator slot number.

9.3 Chaotic PWM

Among the many available PWM schemes, random PWM is becoming more and more attractive. Random PWM can effectively spread the discrete spectral power over a continuous spectrum and significantly suppress the maximum acoustic noise that occurs at the switching frequency. However, it generally ignores the consideration of low-order harmonic frequencies and the system natural frequency, thus introducing low-order noises and increasing the possibility of creating mechanical resonance.

Since chaos possesses a random-like but bounded feature, chaotic PWM is becoming an alternative to (or even supersedes) random PWM for application to electric drive systems. Chaotic PWM is useful for both DC and AC motor drives. Among them, the application to the induction drive system is more desirable and challenging, since it has long been affected by the EMI problem and acoustic noise. Recently, it has been identified that the use of chaotic PWM to replace sinusoidal PWM can reduce the EMI in drive systems (Bellini et al., 2001; Balestra et al., 2004). This chaotic PWM scheme employs a Bernoulli shift map to chaoize an amplitude-modulated signal which then modulates the carrier frequency of sinusoidal PWM. The corresponding harmonic reduction can not only reduce the size of power filters, but also suppress the acoustic noise in PWM drive systems.

Compared with random PWM, the available chaotic PWM offers the following advantages:

• Some chaotic signals can provide better spectral performance than some random signals, namely better range resolution than the Gaussian distributed signal and equal to the range resolution of the uniformly distributed signal (Ashtari et al., 2003).
• Chaotic PWM exhibits a 9–14 dB improvement over random PWM which randomizes the pulse positions and/or widths (Balestra et al., 2004).
• Chaotic PWM performs no worse than random PWM in terms of harmonic spectra, whereas the implementation of chaotic generators is much easier than that of random generators. Notice that chaos can be governed by a simple equation, while a chaotic generator can be easily implemented by a few tens of transistors (Delgado-Restituto and Rodíguez-Vázquez, 2002).
• In the case of fast modulations, chaotic PWM can outperform random PWM in terms of harmonic spectra (Callegari, Rovatti, and Setti, 2002).
• The truly random sources are rigorously unrealizable. All the viable random techniques only generate pseudorandom sequences. Sophisticated pseudorandom sources enable a roughly continuous spectrum, but it will be more costly (Callegari, Rovatti, and Setti, 2003). Chaos has an inherent continuous spectrum which is very similar to that of true randomness. The correlation between the chaotic series will be decreased with the elapse of time. Consequently, there is no performance loss for the chaotic series to take the place of the truly random sources.

Although the available chaotic PWM offers a better advantage of easier implementation than random PWM, which needs a truly random source, it is inflexible to tune the spectral power distribution and is limited to those chaotic maps satisfying some specific characteristics, namely the mixing rate and probability density function. Also, the corresponding switching frequency may clash with the natural frequency of mechanical resonance.

In this section, two newly developed chaotic modulation schemes (Wang and Chau, 2007), namely chaotically amplitude-modulated frequency modulation (CAFM) and chaotically frequency-modulated frequency modulation (CFFM) are introduced. They are then used to modulate the switching frequency of both sinusoidal PWM and space vector PWM (Wang, Chau, and Liu, 2007). Consequently, they are applied to both the open-loop control and the closed-loop control of induction drive systems (Wang, Chau, and Cheng, 2008).

The two chaotic modulation schemes, namely CAFM and CFFM, are derived from the standard sinusoidal frequency modulation. Figure 9.2 shows the corresponding schematic diagram in which is the PWM signal for switching the power inverter, is the reference signal for power flow control, and is the carrier signal operated at a frequency much higher than the bandwidth of . They are related by:

(9.14)

For induction drive systems, is generally expressed as:

(9.15)

where is the modulation index as defined by , V is the desired output voltage, is the input DC voltage, and f is the desired output frequency. These two parameters are generally controlled in such a way that the ratio V/f is kept constant at speeds below the rated speed for so-called constant-torque operation, or V is kept at the rated voltage while f is increased beyond the rated frequency at speeds above the rated speed for so-called constant-power operation. On the other hand, is a jittered triangular wave which is expressed as:

(9.16)

where is the carrier frequency which is expressed as:

(9.17)

where is the nominal carrier frequency, is the carrier deviation frequency, and is the carrier modulation frequency. So, the spectral power of is given by:

(9.18)

(9.19)

(9.20)

where is the nth harmonic coefficient of the unmodulated carrier signal, is the frequency modulation index, is the kth-order Bessel function, and is the impulse function. So, the original spectral power around is distributed around the discrete terms at (). According to Carson's rule, 98% of the spectral power lies within the frequency range []. If 1, this frequency range will be [].

9.3.1 Chaotic Sinusoidal PWM

Based on the use of the standard sinusoidal frequency modulation scheme to modulate the switching frequency of sinusoidal PWM, the inverter output voltages can be expressed as:

(9.21)

(9.22)

(9.23)

where is the amplitude modulation index, is the amplitude of the sinusoidal reference signal, is the amplitude of the carrier signal, is the frequency of the sinusoidal reference signal, and , , are, respectively, the initial phase angles of the phase A, B, C sinusoidal reference signals. From (9.21) –(9.23), it can be found that the spectral power of the output voltage clusters around the frequencies at (). Also, based on Carson's rule, the majority of spectral power lies within the frequency range []. If and , this frequency range will become [].

In order to even out the spectral power around the frequencies at () and hence to further reduce the EMI of the drive system, a chaotic sequence is utilized to chaoize the sinusoidal frequency modulator. The first method is to substitute in (9.17) by . Hence, the amplitude of the frequency modulator can be chaoized: the so-called CAFM. The second method is to substitute in (9.17) by . Hence, the frequency of the frequency modulator is chaoized: the so-called CFFM. While the chaotic sequence exhibits the nature of continuous spectral power, the power spectrum of the inverter output voltage can be smoothed out. Since , the power spectrum of the inverter output voltage can be maintained in the frequency range [] when and . This property can not only effectively reduce the EMI, but also avoid overlapping with the natural frequency of mechanical vibration.

There are many ways to generate the desired chaotic signal. In general, these chaos generators can be classified into two categories (Delgado-Restituto and Rodíguez-Vázquez, 2002):

• Discrete-time chaos generators (chaotic discrete maps), such as the Logistic map, tent map, Bernoulli map, and Henon map.
• Continuous-time chaos generators (chaotic oscillators), such as the double-scroll-like oscillator, Colpitts oscillator, Chua's oscillator, and Lorenz system.

The use of a Logistic map is the most commonly adopted method to generate the chaotic sequence for chaotic PWM inverter-fed induction drive systems. It is governed by the map , where . Figure 9.3(a) depicts the bifurcation diagram of versus of the Logistic map. The corresponding Lyapunov exponent versus is shown in Figure 9.3(b). It can be seen that when , exhibits a zero value with . This denotes that the chaotic frequency modulator does not modulate the frequency of the carrier signal, which is equivalent to the traditional sinusoidal PWM. When , takes a fixed value with . This denotes that the chaotic frequency modulator works as a standard sinusoidal frequency modulator. When is further increased, begins to bifurcate with multiple values. When , exhibits infinite values. Also, the boundary of such infinite values changes with the value of . As the corresponding Lyapunov exponents are positive, this confirms that it is a chaotic series. Thus, by properly tuning the value of , this chaotic frequency modulator can offer various power spectra.

The use of a Bernoulli map is another commonly adopted method to generate the chaotic sequence for chaotic PWM inverter-fed induction drive systems. It is attractive because of its simplicity for the evaluation of its rate of mixing (Setti et al., 2002) and the implementation of its integrated circuit (Delgado-Restituto and Rodíguez-Vázquez, 2002). Figure 9.4 shows a typical four-way Bernoulli shift map, which is expressed as : .

The use of a Chua circuit is another method to generate the chaotic sequence for chaotic PWM inverter-fed induction drive systems (Cui et al., 2006). Since the sequence is directly obtained from sampling the Chua circuit, it inherits the stochastic nature of the Chua circuit, hence exhibiting the ideal nature of autocorrelation and cross-correlation. Figure 9.5 shows a typical Chua circuit in which the nonlinear resistor is chosen to have a piecewise linear V-I characteristic. Its major drawback is that the digital implementation of a Chua circuit is much more difficult than that of a Logistic map or Bernoulli map.

By applying CAFM and CFFM to modulate the switching frequency of the sinusoidal PWM (SPWM), CAFM-SPWM and CFFM-SPWM can be obtained. Figures 9.6 and 9.7 show the power spectra of the PWM output voltage when using CAFM-SPWM and CFFM-SPWM with various values of , respectively, in which , , and are adopted. It can be observed that when , the power spectra have significant spectral spikes, which cause undesirable peaky EMI. When , the magnitude of such spectral spikes are alleviated. When , such spectral spikes can be further reduced. When , the power spectra are significantly smoothed out so that the EMI is effectively suppressed.

9.3.2 Chaotic Space Vector PWM

Compared with sinusoidal PWM, space vector PWM offers the definite advantages of lower harmonic distortion, lower switching loss, and better utilization of DC link voltage. Figure 9.8 shows the diagram of the inverter output voltage space vectors. There are eight switching states of the three upper-leg power switches of the inverter: , , , , , , , . The three lower-leg power switches have inverted switching states. correspond to the basic voltage vector . In each switching period, the sequence of the switching states are:

where and are the active switching states. The active switching states correspond to the two adjacent basic voltage vectors and between which locates. For the sectors I to VI, they are represented by (), (), (), (), (), and (), respectively. Then, the switching times , and can be computed by:

(9.24)

(9.25)

(9.26)

where is the phase angle of , and is the switching period.

By applying CAFM and CFFM to space vector PWM (SVPWM), CAFM-SVPWM and CFFM-SVPWM are obtained. Figure 9.9 shows the flowcharts of generation of the switching period. For CAFM-SVPWM, the modulation frequency is updated at the end of each interval (). On the other hand, the modulation frequency of CFFM-SVPWM is updated at the end of each interval . In order to avoid waiting too long for updating, the longest limit of each update is with . Namely, if , the modulation frequency will be updated at ; otherwise, if , it will be updated at . For both CAFM-SVPWM and CFFM-SVPWM, is updated at the end of the jth switching period . As a symmetric regular sampling is used, the sampling period is kept synchronous with . This synchronous sampling can enable good dynamic performance for induction drive systems.

9.4 Chaotic PWM Inverter Drive Systems

To investigate the performances of the aforementioned chaotic SVPWM inverters for induction drive systems, a practical 3-phase induction drive system is used for exemplification. The parameters of the induction motor are given in Table 9.1. Throughout the experiment, a power analyzer is used to measure the power spectrum of the PWM output voltage, a current transducer is used to measure the instantaneous stator current, and an encoder is used to measure the instantaneous rotor speed.

Table 9.1 Parameters of induction motor.

Rated power	1.5 kW
Rated voltage	220 V
Rated speed	1430 rpm
Poles	4
Stator resistance	3.3 Ω
Stator leakage inductance	43.9 mH
Rotor resistance	3 Ω
Rotor leakage inductance	43.9 mH
Mutual inductance	278 mH
Rotor inertia	6.65 × 10−3 kgm2
Viscous friction coefficient	5.5 × 10−6 Nm/rad/s

9.4.1 Open-Loop Control Operation

In order to evaluate the performances of CAFM-SVPWM and CFFM-SVPWM, they are compared with the traditional fixed frequency SVPWM (FF-SVPWM) and the recently developed random frequency SVPWM (RF-SVPWM) for the same IGBT based voltage-source PWM inverter and induction motor. For CAFM-SVPWM and CFFM-SVPWM, when selecting , , and , it deduces and in order to avoid overlapping with the mechanical natural frequency of around 13.5 kHz. On the other hand, the switching frequency of FF-SVPWM is fixed at 10 kHz, while that of RF-SVPWM is randomly distributed within 10 ± 3 kHz.

The power spectra of the PWM output phase voltage and line voltage using FF-SVPWM, RF-SVPWM, CAFM-SVPWM, and CFFM-SVPWM are compared under the same conditions as shown in Figures 9.10 and 9.11. It can be seen that there exist significant spectral peaks in the spectra of FF-SVPWM. Meanwhile, RF-SVPWM can effectively smooth out the peaky harmonics and provide a flat spectrum. Because of this nature, the possibility of overlapping with the natural frequency of mechanical resonance is significantly increased. On the contrary, both CAFM-SVPWM and CFFM-SVPWM can exhibit an essentially flat spectrum, but associated with a spectral notch around the natural frequency. Thus, these two chaotic SVPWM schemes can reduce the peaky EMI while avoiding the mechanical resonance.

In order to quantitatively assess the effectiveness of the chaotic SVPWM schemes, two important indicators are adopted for comparison. Firstly, since the conducted EMI with frequency exceeding 9 kHz is stringently limited in many countries, the maximum power spectral density (PSD) of and in that frequency range is used as one indicator to compare FF-SVPWM, RF-SVPWM, CAFM-SVPWM, and CFFM-SVPWM. Secondly, since the occurrence of mechanical resonance should be avoided, the spectral power of and within the sideband of 13.4–13.6 kHz around the natural frequency is used as another indicator for comparison. Table 9.2 gives a quantitative comparison of the above SVPWM schemes, confirming that the chaotic SVPWM schemes can effectively reduce the conducted EMI and avoid the mechanical resonance.

Table 9.2 Quantitative comparison of various SVPWM schemes.

To assess whether the use of chaotic SVPWM schemes causes an adverse effect on the open-loop performances of the induction drive system, the steady-state current waveforms, as well as the start-up transient current and speed responses, are recorded as shown in Figures 9.12 and 9.13, respectively. It can be observed that the steady-state current waveforms are very sinusoidal, while the transient responses are very fast, hence confirming that the chaotic SVPWM schemes do not cause any adverse effect on the open-loop performances.

9.4.2 Closed-Loop Vector Control Operation

The two chaotic SVPWM schemes are then applied to the vector-controlled induction drive system in order to assess their closed-loop performances. The dynamical equations of the induction drive resulting from rotor field orientation are expressed as:

(9.27)

(9.28)

(9.29)

(9.30)

(9.31)

(9.32)

(9.33)

where and are respectively the d-axis and q-axis components of stator current; , , and are respectively the rotor magnetizing current, rotor flux speed, and rotor speed; and are respectively the d-axis and q-axis components of stator voltage; and are respectively the stator resistance and rotor resistance; , , , , and are respectively the stator inductance, rotor inductance, stator leakage inductance, rotor leakage inductance, and mutual inductance; and σ is the leakage coefficient. Figure 9.14 shows the corresponding block diagram in which the d-axis and q-axis constitute the reference frame rotating synchronously with the rotor flux, whereas the α-axis and β-axis constitute the stationary reference frame.

There are two closed-loop controllers, namely the inner current loop and the outer speed loop. The proportional-integral (PI) control method is adopted for both the current controller and speed controller. The sampling rate of the current controller varies with the switching period of the SVPWM inverter, while the sampling rate of the speed controller is kept constant. Due to the variable sampling rate of the current controller, the corresponding PI parameters should be updated at each sampling interval. The criteria are and in which and are respectively the discrete proportional parameter and integral parameter, and and are respectively the continuous proportional parameter and integral parameter.

Firstly, the measured power spectra of the PWM output phase voltage and line voltage are the same as the open-loop case, hence confirming that the two chaotic SVPWM schemes can reduce the peaky EMI while avoiding the mechanical resonance.

Secondly, Figure 9.15 shows the measured steady-state waveforms and trajectories of the α-axis and β-axis components of stator current. It can be seen that there is no noticeable distortion in the current waveforms and trajectories.

Thirdly, Figures 9.16–9.18 show the dynamic responses of the two chaotic SVPWM schemes as compared with the traditional FF-SVPWM. Their current responses are based on the d-axis current component step command of 0.5 A → 1 A → 0.5 A → 0.75 A and the q-axis current component step command of 0.58 A → 0.25 A → 1 A → 0.33 A. The speed responses are based on the step command of 28.3 rad/s → 12.4 rad/s → 0 rad/s → 28.3 rad/s. They illustrate that both the current controller and speed controller of the two chaotic SVPWM schemes can track the commands as accurately and as quickly as that of the FF-SVPWM scheme. Hence, it verifies that the chaotic PWM motor drive can not only suppress the conducted EMI while avoiding the mechanical resonance, but also retain the outstanding steady-state and transient performances of vector control.

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