Chapter 2

Introduction to Chaos Theory and Electric Drive Systems

As this book is a happy marriage between two disciplines – chaos theory and electric drive systems – the coverage of knowledge is so broad that it is desirable to briefly introduce basic chaos theory and the fundamentals of electric drive systems.

In this chapter, the necessary background knowledge of this book, namely the basic theory of chaos and some fundamentals of electric drive systems, are discussed.

2.1 Basic Chaos Theory

This section describes some of the basic principles of chaos, including the concept of dynamical systems, discrete maps, limit sets, attractors, stability, and manifolds. The criteria of chaos are then illustrated using the Lyapunov exponents, fractal dimensions, and entropy. Hence, bifurcations and routes to chaos will both be discussed. Finally, various methods for chaos analysis are introduced, which include waveforms, phase portraits, the Poincaré map, bifurcation diagrams, time-series reconstruction, and the calculations of Lyapunov exponents, embedded unstable periodic orbits, power spectra, fractal dimensions, and entropy.

2.1.1 Basic Principles

2.1.1.1 Dynamical Systems

Autonomous Dynamical Systems

An nth-order autonomous dynamical system is defined by , . The corresponding solution is , and is called as a flow. For autonomous continuous systems, the vector field f does not depend on time t.

Nonautonomous Dynamical Systems

An nth-order nonautonomous dynamical system is defined by , . The corresponding underlying flow is . For nonautonomous continuous systems, the vector field f depends not only on the state variable x but also on the time t.

2.1.1.2 Discrete Maps

Orbit

A discrete system can be defined by a map with the state equation , where are the states at the kth iterative time, and P maps the state to the next state . Starting from an initial condition , repeated application of P generates a sequence of points which is known as an orbit (Parker and Chua, 1999).

Poincaré Map

A classical technique for discretization of a dynamical system is the Poincaré map. It replaces the flow of an nth-order continuous system with an (n − 1)th-order discrete map. The Poincaré map is useful to reduce the system order and bridge the gap between continuous and discrete systems (Parker and Chua, 1999).

For an nth-order nonautonomous system with a minimum period T, the resulting Poincaré map is defined by . The corresponding orbit is a sampling of a single trajectory every T seconds, which is similar to the action of a stroboscope flashing with period T.

For an nth-order autonomous system, the underlying flow is with the trajectory . When is chosen to be an (n − 1)-dimensional hyperplane so that the trajectory intersects transversely, and is a point of on the hyperplane , the trajectory starting from will intersect at point . Due to the continuity of , trajectories starting on in a sufficiently small neighborhood of will also intersect in the vicinity of . Thus, and define the Poincaré map of some neighborhood of onto a neighborhood of . Figures 2.1 (a) and 2.1(b) show the Poincaré map for the autonomous system and nonautonomous system, respectively.

2.1.1.3 Limit Sets and Attractors

If there exists a point under the condition that the trajectory repeatedly enters every neighborhood U of y for , y is known as the limit point of (Parker and Chua, 1999). The set of such points y is defined as a limit set of , which can be represented as a function of . If there exists an open neighborhood U of a limit set Y, and for all , , the limit set Y is attracting. The attracting limit set is also called an attractor (Ott, 1993). The union of all U is defined as the attracting basin of the limit set Y.

Equilibrium Point

If there exists a point which satisfies for all t, such a point is called an equilibrium point. The attractor of an equilibrium point is the equilibrium point itself (Parker and Chua, 1999). For the discrete system, there is no limit set corresponding to an equilibrium point.

Limit Cycle

If there exists a trajectory which satisfies with a minimum period for all t, such a trajectory is known as a periodic behavior. The attractor of a periodic behavior is a closed trajectory , also called a limit cycle. If, in an th-order periodic nonautonomous system with forcing period , the period T is an integer multiple k of , then the underlying flow is known as both a period-k behavior and a kth-order subharmonic (Parker and Chua, 1999). For the discrete system, the limit sets of the periodic behavior are fixed points.

Torus

If the solution trajectory of an autonomous system is the sum of a countable number of periodic trajectories whose frequencies are incommensurate, then the whole solution trajectory exhibits a quasiperiodic behavior. If the countable number is p, the quasiperiodic behavior is known as p-periodic behavior (Parker and Chua, 1999). The attractor of the p-periodic behavior is the p-torus. For the discrete system, the attractor of the p-periodic behavior is one or more embedded tori.

Strange Attractor

A limit set is said to be chaotic if the corresponding solution trajectory is exponentially sensitive to the initial conditions (Ott, 1993). The trajectory of chaos exhibits randomly mixed behaviors. Generally, the strange attractor is used to describe the geometrical attracting limit set for chaos. For a discrete system, the limit set of chaos is a strange geometry, which is different from the simple geometry of periodic and quasiperiodic behavior. It is a fine-layered structure (Sprott, 2003).

For most cases involving differential equations, chaos commonly occurs together with geometrical strangeness (Ditto et al., 1990). The strange attractors are not a finite set of points, nor a smooth curve or surface, nor a volume bounded by a piecewise smooth closed surface (Ding, Grebogi, and Ott, 1989). However, it is possible for a chaotic attractor not to be strange. For example, the logistic map under a certain parameter has a chaotic attractor with a positive Lyapunov exponent, but this attractor is not strange since it is a simple interval within [0, 1]. For Hamiltonian systems, the dynamics can be chaotic. However, these are conservative systems that have no attractors at all (Ott, 1993).

On the other hand, it has been identified that strange nonchaotic attractors are indeed typical in systems that are driven by a two-frequency quasiperiodic force (Grebogi et al., 1984). This phenomenon can be observed experimentally by Lyapunov exponents, information dimensions, Fourier amplitude spectra, and a phase portrait (Yang and Bilimgut, 1997). The power spectrum of strange chaotic attractors is comparatively broader and has a much richer harmonic content than periodic attractors or strange nonchaotic attractors (Zhou and Moss, 1992).

2.1.1.4 Stability

Eigenvalues can be used to judge the stability of the equilibrium points. If f is a vector field describing an autonomous dynamic system, the equation governing the time evolution of a perturbation in a neighborhood of the equilibrium point can be represented as:

(2.1)

where is the differential equation of f. If are the eigenvalues of , the real part of , namely , are used to judge the stability of . If for all , is asymptotically stable. If any , is unstable. The eigenvalue distributions of different equilibrium points are shown in Figure 2.2.

The stability of a limit cycle is judged by its characteristic multipliers, which are also known as Floquet multipliers. The limit cycle corresponds to a fixed point on the Poincaré map. The local behavior of the map near is determined by the differential equations of the map P at as:

(2.2)

The eigenvalues of are defined to be , which are known as the Floquet multipliers of the limit cycle. They determine the stability of the fixed point , and the corresponding limit cycle. If for all , is asymptotically stable. If any , is unstable. Actually, there are relationships between and which can be represented by , where T is the period of the limit cycle. Figure 2.3 shows the Floquet multiplier distributions of different limit cycles.

2.1.1.5 Stable and Unstable Manifolds

Outside the near neighborhood of a fixed point, the curved lines passing through the fixed point are used to evaluate the stability of the dynamical system. If the initial condition on the curves remains on the curves after several iterations of the map, the curves are said to be invariable manifolds (Banerjee and Verghese, 2001).

Stable and unstable manifolds exist for nonchaotic limit sets Y. The stable manifold is the set of all points whose trajectory approaches Y as . The unstable manifold of the limit set Y is the set of all points whose trajectory approaches as (Parker and Chua, 1999). If the initial condition is not on the invariable manifolds, the iterative states will move away from the stable manifolds and move closer to the unstable manifolds, as illustrated in Figure 2.4. As an unstable manifold attracts the points in the state space, and a stable manifold repels the points, the stable manifold of a saddle point often acts as the boundary separating the attracting basins of different attractors in the state space (Banerjee and Verghese, 2001).

2.1.2 Criteria for Chaos

2.1.2.1 Lyapunov Exponents

The Lyapunov exponent can be used to determine the stability of quasiperiodic and chaotic behavior, and also the stability of equilibrium points and periodic behaviors. It provides a way to quantify the rates of stretching and squeezing of the attractor in the state space. The Lyapunov exponent is the exponential rate of this divergence and convergence. If the maximum Lyapunov exponent of a dynamical system is positive, this system is chaotic; otherwise, it is nonchaotic. Although the Lyapunov exponents are a generation of eigenvalues, there are many differences: the eigenvalue is a local quantity while the Lyapunov exponent is a global quantity; the eigenvalue is a constant value while the Lyapunov exponent is an average value; the eigenvalue is a complex number while the Lyapunov exponent is a real number; and the eigenvalues are not usually orthogonal while the Lyapunov exponents are mutually orthogonal (Sprott, 2003).

The Lyapunov exponents are directly related to the criteria of chaos. Therefore, it is significant to calculate the Lyapunov exponents. If the analytical model of the dynamical system is known, the process to compute the Lyapunov exponents is as follows. The solution flow of the system state variables is expressed as:

(2.3)

where is the map describing the time-t evolution of X, and the solution flow of their deviation is given by:

(2.4)

where is the map describing the time-t evolution of . Then, the Lyapunov exponents of the d-dimensional system can be computed as (Shimada and Nagashima, 1979):

(2.5)

where is the evolution time, and is the ith base vector of the d-dimensional state space at the jth step. It should be noted that should be orthogonalized and normalized at each iterative step. Under the restrictions of and , can be approximated as (Benettin, Galgani, and Strelcyn, 1976):

(2.6)

Some calculation approaches have been developed for the special dynamical systems such as the dynamical system with discontinuities and the finite-dimensional time-delay system. For a dynamical system with discontinuities, including the widely used power electronic circuits and motor drives, certain transitions are supplemented to the instants of discontinuities, thus enabling the flow of to be computed (Müller, 1995). For a dynamical system with a time delay, X on the interval [, ] can be approximated by N samples taken at intervals . These N samples can equivalently be considered as the N variables of an N-dimensional discrete map. With this approximated map, the Lyapunov exponents of a system with a time delay can be computed (Farmer, 1982).

2.1.2.2 Fractal Dimensions

Identifying chaos includes searching for a strange attractor in the state-space dynamics, which can be characterized by its fractal structure. The dimension of an attractor is a measure of the number of active variables and the complexity of the equations required to model the system dynamics (Sprott, 2003). Typical characteristics of the fractal structure are: (a) they have a fine structure and no characteristic scale length; (b) they are too irregular to be described by ordinary geometry, both locally and globally; and (c) they have some degree of self-similarity, which means that small pieces of the object resemble the whole in some respects. Fractal structures can be either deterministic where they are exactly self-similar, or random where they are only statistically self-similar. The self-similarity implies that the structure is scale invariant.

Geometrical objects with fractional dimensions are called fractals and, in dynamical systems, it has been found that chaotic attractors are fractal objects. The determination of the fractal dimensions is thus one of the methods of characterizing a chaotic attractor (Banerjee and Verghese, 2001). The dimension is a kind of quantifier which describes the attractor from the geometric aspects. The dimensionality of an attractor gives us an estimate of the number of active degrees of freedom in the system. If the attractor's dimension is not an integer, the attractor is a strange attractor. The capacity dimension – which is a measure of fractal dimensions – of a dynamical system is defined as (Kolmogorov, 1958):

(2.7)

where R is the side length of the constructed “box,” is the number of boxes required to contain all the points on the attractor, and k is a proportionality constant.

2.1.2.3 Entropy

The sum of the positive Lyapunov exponents is the Kolmogorov–Sinai (K–S) entropy (Pesin, 1977). This K–S entropy has a similarity to the usual thermodynamic entropy since it measures the expansion of nearby trajectories into new regions of state space. At the same time, unlike the thermodynamic entropy, the K–S entropy has units of inverse time, or inverse iterations for maps. It is a measure of the average rate at which predictability is lost. Its inverse is a rough estimate of the time for which a reasonable prediction is expected (Sprott, 2003). A purely random system has infinite entropy, and a periodic system has zero entropy; therefore, the K–S entropy is a positive constant for a chaotic system, and the chaotic degree increases with the value of the K–S entropy.

2.1.3 Bifurcations and Routes to Chaos

2.1.3.1 Bifurcations

Pitchfork Bifurcation

The pitchfork bifurcation is also known as the cusp bifurcation. For the supercritical pitchfork bifurcation, there exists a fixed point when . When , the stable fixed point becomes unstable and gives birth to two new stable fixed points. The supercritical pitchfork points in the direction of positive . For the subcritical pitchfork bifurcation, the stable fixed point at also becomes unstable when . However, the pitchfork points in the direction of negative . Consequently, there exist two other unstable branches when . The principle of supercritical pitchfork bifurcation is shown in Figure 2.5(a).

Saddle-Node Bifurcation

The saddle-node bifurcation is also called a fold bifurcation or tangent bifurcation. When , there is no fixed point in the system. At , two fixed points are newly formed but separate when . One fixed point is stable and the other is unstable. The principle of the saddle-node bifurcation is shown in Figure 2.5(b).

Transcritical Bifurcation

For a transcritical bifurcation, there are two equilibrium points and , and their stability switches at . For , the stable equilibrium is at . For the stable equilibrium is at . Figure 2.5(c) also shows the principle of a transcritical bifurcation.

Period-Doubling Bifurcation

A period-doubling bifurcation is also known as a flip bifurcation. In the first iterative map of the system, there is a stable fixed point when . When , this fixed point becomes unstable. In the second iterative map of the system, the stable fixed point at becomes unstable when and two new stable fixed points are produced, causing a period-2 dynamics. This type of bifurcation is known as a supercritical period-doubling bifurcation. For a subcritical period-doubling bifurcation, the period-2 dynamics at is unstable. Actually, the period-doubling bifurcation is in analogy with the pitchfork bifurcation.

Hopf Bifurcation

A Hopf bifurcation is similar to a pitchfork bifurcation in a system with a dimension higher than 1. For a supercritical Hopf bifurcation, there exists a stable focus when . At , the eigenvalues, which are a complex conjugate pair, cross the imaginary axis. When , the original focus becomes unstable, and a stable limit cycle is born. For a subcritical Hopf bifurcation, there exist a stable focus and an unstable limit cycle when . At , they coalesce and annihilate each other. Figure 2.6 shows the supercritical Hopf bifurcation.

Neimark–Sacker Bifurcation

A Neimark–Sacker bifurcation occurs when the limit cycle becomes unstable, and gives birth to a torus, which produces a quasiperiodic flow. When a torus is born in the flow, the eigenvalues of the corresponding map are a complex conjugate pair with a magnitude of 1. By analogy with the Hopf bifurcation, the birth of a torus in a flow is called a secondary Hopf.

Border Collision Bifurcation

For a flow or a map that is continuous but its derivative is discontinuous at the hyperplane M or the line χ, the M or the χ is called the border, which separates the phase space into two regions and , as shown in Figure 2.7. Unlike the bifurcations that occur in one of the smooth regions, some bifurcations occur when the equilibrium or fixed point collides with the border, and there is a discontinuous jump in the eigenvalues of the Jacobian matrix. This type of bifurcation is known as a border collision bifurcation (Banerjee and Verghese, 2001). There are two kinds of border collision bifurcation. The first is known as a border collision pair bifurcation, where there is no equilibrium or fixed point for and there are two equilibrium or fixed points for – one on region and one on region . The second kind is known as a border-crossing bifurcation, where the equilibrium or fixed point crosses the border when goes through zero.

Nonstandard Bifurcations in Discontinuous Systems

In discontinuous systems, some nonstandard bifurcation phenomena occur. In continuous systems, equilibrium or fixed points appear or disappear only in pairs while, in discontinuous systems, a single equilibrium or fixed point may appear or disappear. Furthermore, the eigenvalues of an equilibrium or fixed point of discontinuous systems do not signal the occurrence of a bifurcation. For discontinuous systems, the basin boundary of two attracting equilibrium or fixed points can be formed from points of discontinuity, as depicted in Figure 2.8(a). On the other hand, the basin boundary in continuous systems can be the stable manifold of a saddle point or an unstable periodic orbit, as depicted in Figure 2.8(b) (Banerjee and Verghese, 2001).

Homoclinic and Heteroclinic Bifurcations

If and are two different equilibrium or fixed points, and are respectively the stable and unstable manifolds of , while and are respectively the stable and unstable manifolds of . If a trajectory or orbit exists in , it is called a homoclinic trajectory or orbit. Figure 2.9(a) shows the homoclinic trajectory of a saddle point . If a trajectory or orbit exists in , it is called a heteroclinic trajectory or orbit. Figure 2.9(b) shows the structure of a heteroclinic trajectory.

A homoclinic bifurcation occurs when the stable and unstable manifolds of an equilibrium or fixed point touch one another, while a heteroclinic bifurcation occurs when the stable manifold of one equilibrium or fixed point touches the unstable manifold of another (Parker and Chua, 1999). Homoclinic and heteroclinic bifurcations are both global bifurcations.

Crisis

The crisis is another class of global bifurcation that occurs when a chaotic attractor collides with an unstable periodic orbit or its attracting basin. There are three types of crises. The first is the boundary crisis where the chaotic attractor touches the basin boundary that separates it from another coexisting attractor. After this crisis, the attractor is destroyed and the trajectories in this region are first transiently chaotic and then asymptotically approach the other attractor (Sprott, 2003). The second crisis is the interior crisis when the chaotic attractor collides with a periodic trajectory or orbit within its basin. The chaotic attractor suddenly expands in size but remains bounded after the collision. After the crisis, the trajectory or orbit on the attractor spends a long time in the original smaller attractor, and intermittently jumps from this small attractor to the newly created large attractor (Banerjee and Verghese, 2001), resulting in a crisis-induced intermittency. The third crisis is the attractor-merging crisis where two or more chaotic attractors simultaneously touch a periodic trajectory or orbit on the basin boundary that separates them. The two or more attractors then merge into one multipiece chaotic attractor. Intermittency occurs when the trajectory or orbit moves among different pieces of the attractor at random intervals. Figure 2.10 depicts different types of crises (Banerjee and Verghese, 2001).

2.1.3.2 Routes to Chaos

Period-Doubling Cascade Route to Chaos

The first kind of route to chaos through local bifurcation is the period-doubling cascade route, where stable fixed points become unstable in a series of period-doubling bifurcations, and subharmonic behavior occurs. The well-known Logistic map is identified to be a typical case for illustrating the period-doubling cascade route to chaos, as shown in Figure 2.11. As the parameter A is increased, there is a period doubling cascade. After an infinite number of period doublings, chaos and periodic “windows” are finely intermixed (Ott, 1993). For the Logistic map, chaos onsets when A is beyond the accumulating point, around 3.56994.

Intermittency Transition to Chaos

Another route to chaos resulting from local bifurcations is the intermittency transition, which can be classified into three types. The first type emanates from a saddle-node bifurcation, which often occurs in maps and flows where a stable point and an unstable point come together and coalesce. The second type is the quasiperiodic route that often occurs through subcritical Hopf bifurcations. After a secondary Hopf bifurcation – namely a Neimark–Sacker bifurcation – a new torus is generated which induces chaos. The third type is the inverse period-doubling bifurcation, where an unstable periodic orbit collapses onto a stable periodic orbit of one half its period, and they are both replaced by an unstable periodic orbit of a lower period (Ott, 1993). In the intermittency transition to chaos, the attractor is a periodic orbit for . When is slightly larger than 0, there is a long duration during which the trajectory or orbit appears to be periodic and closely resembles the orbit for . However, this regular behavior is intermittently interrupted by a finite bursting duration. Figure 2.12 illustrates these three types of intermittency transition to chaos (Ott, 1993).

Chaos from Manifold Tangle

If is a point with a stable manifold and an unstable manifold , and the two manifolds intersect transversely (with nonzero angle) at another point , then is called a homoclinic point. The action of intersection of two manifolds is called a manifold tangle. Actually, when the manifolds intersect transversely once, they intersect an infinite number of times. This results in stretching and folding actions, giving an embedded horseshoe map which leads to chaos. The stable manifold cannot intersect with the stable manifold and the unstable manifold cannot intersect with the unstable manifold. Figure 2.13 shows a homoclinic tangle where the stable and unstable manifolds of a saddle point intersect with each other infinitely, where only a portion of each manifold is depicted.

Chaos from Crisis

As mentioned above, chaotic attractors can be changed discontinuously in a crisis. That is, the boundary crisis can destroy the chaotic attractors, the interior crisis can suddenly expand the size of the chaotic attractors, and the attractor-merging crisis can combine different chaotic attractors together. On the other hand, the inverse process can create chaotic attractors, shrink chaotic attractors, and split chaotic attractors with a parameter change in the opposite direction.

2.1.4 Analysis Methods

2.1.4.1 Waveforms and Phase Portraits

The waveforms of equilibrium or periodic behaviors are regular while the waveform of a chaotic behavior is irregular. The phase portraits of the equilibrium points and periodic behaviors are points and close curves respectively, while the phase portrait of chaos is randomly distributed in a bounded region. So, the waveforms and phase portraits can be observed to discern chaotic behaviors from regular behaviors. The oscilloscope can record the waveforms at any instant, and the irregular behavior of a chaotic waveform can be observed. The phase portrait can be displayed directly by using the X–Y display mode of oscilloscope. Both the waveform and the phase portrait can also be plotted when based on the measured data sampled by the oscilloscope. Figure 2.14(a) shows the measured chaotic speed waveform of a DC motor, and Figure 2.14(b) plots the phase portrait of the measured speed versus the measured armature current of the DC motor.

2.1.4.2 Poincaré map

For the autonomous system, the Poincaré map can be obtained by using a Poincaré section to cut the attractor, which is illustrated in Figure 2.15 (Banerjee and Verghese, 2001). The Poincaré section can be chosen by fixing one system state z to be constant, and the projection of the attractor is obtained on the x − y plane. Thus, by using the X–Y mode of the oscilloscope, a 2-D projection of the attractor can be displayed. For the nonautonomous system, the Poincaré map can also be obtained by using the Poincaré section to cut the attractor. The main difference is that the nonautonomous system uses the time t to be the z state, and the Poincaré section is obtained by enabling z to be many times the sampling period.

2.1.4.3 Bifurcation Diagrams

Bifurcation diagrams function to illustrate the bifurcation behaviors in the system when changing the parameters. When changing one parameter in the system for a bifurcation diagram, other parameters should remain constant. The variable parameter is along the X-axis, while the system states are along the Y-axis of the bifurcation diagram.

Based on the Poincaré map, the system states are obtained by applying the Poincaré section for autonomous systems, or by stroboscopic sampling for nonautonomous systems. The system states should be chosen arbitrarily, but should make the bifurcation behavior of the system apparent. The variable parameter is changed step by step, and the system states are observed at each value of this parameter.

In a dynamical system where multiple attractors exist, different initial conditions should be attempted for each value of the variable parameter so that different attractors can be observed. Also, the iteration times for each bifurcation diagram should be large enough to avoid some transient system behaviors such as chaotic transients.

2.1.4.4 Time Series Reconstruction

The time series reconstruction is another useful tool for chaotic analysis, especially for experimentation. It allows the attractor to be reconstructed even in an infinite-dimensional system or a system where one or more of the state variables cannot be measured directly. For a system that has unknown dynamics, and only one state variable can be observed, the time series of this variable and its time lags can be used to create a multidimensional embedding space. This embedding space has the same geometric and dynamical characteristics as the actual system (Takens, 1980). Beginning with a time series , vectors of are constructed, where m is the embedding dimension and L is the time lag. Various ways are put forward to choose m and L properly. Normally, m is chosen to be at least , where D is the actual dimension of the system and L is chosen to be equal to the autocorrelation time, which is the time required for the autocorrelation function to drop to of its initial value (Cellucci et al., 1997).

2.1.4.5 Lyapunov Exponent Calculation

In some practical systems, the equations of motion are usually unknown so that the analytical calculation of Lyapunov exponents is not applicable. Thus, one can employ a technique to estimate the Lyapunov exponents on the basis of the experimental data of a time series (Wolf et al., 1985). An attractive method to calculate the maximum Lyapunov exponent has been especially designed for small data sets (Rosenstein, Collins, and De Luca, 1993). The method is accurate because it takes advantage of all the available data.

Firstly, the reconstructed trajectory X is expressed as a matrix, with each row being a state-space vector:

(2.8)

where is the state of the system at some discrete time i. For an N-point time series , each is given by:

(2.9)

Thus, X is an matrix, and the constants m, M, L, and N have a relationship of M = N − (m − 1) L. After reconstructing the dynamics, the algorithm locates the nearest neighbor of each point on the trajectory. The nearest neighbor is obtained by searching for the point that minimizes the distance to the point , which can be expressed as:

(2.10)

where dj(0) is the initial distance from the jth point to its nearest neighbor, and denotes the Euclidean norm. An additional constraint is imposed that the nearest neighbors have a temporal separation greater than the mean period of the time series, which can be estimated as the reciprocal of the mean frequency of the power spectrum:

(2.11)

This allows each pair of neighbors to be considered as nearby initial conditions for different trajectories. It is expected that two random vectors of initial conditions will diverge exponentially at a rate given by the maximum Lyapunov exponent (Eckmann and Ruelle, 1985). Therefore, is estimated to be the mean rate of exponential divergence of the nearest neighbors:

(2.12)

where is the average divergence at time t, and C is a constant that normalizes the initial separation. For each jth pair of nearest neighbors, they diverge approximately at the rate given by :

(2.13)

where is the initial separation. Taking the logarithm of both sides of (2.13) yields:

(2.14)

which represents a set of approximately parallel lines (for j = 1, 2,, M), each with a slope roughly proportional to . Hence, is easily and accurately calculated by using a least-squares fitting to the average line, defined by:

(2.15)

where denotes the average overall values of j. This process of averaging is the key to calculating accurate values of using small, noisy data sets. It should be noted that in (2.13) performs the function of normalizing the separation of the neighbors. However, as shown in (2.14), this normalization is not necessary for estimating .

This method is easy to implement and fast because it uses a simple measure of exponential divergence that circumvents the need to approximate the tangent map. By avoiding the normalization, it also gains a computational advantage. Figure 2.16 shows the typical plot of versus time for the Lorenz attractor, where the solid curve indicates the calculated results and the slope of the dashed line is the result of (Rosenstein, Collins, and De Luca, 1993).

2.1.4.6 Embedded Unstable Periodic Orbit

In order to achieve the stabilization of chaotic behavior with small perturbations, it is necessary to use the unstable periodic orbit (UPO) embedded in the strange attractor. The period of the UPO thus needs to be estimated. Basically, the measured time series can be used to reconstruct the attractor (Takens, 1980), and hence to find the period of the UPO.

There is a well-accepted method of estimating the period of the UPO in the strange attractor (Pawelzik and Schuster, 1991). Firstly, are the reconstructed m-dimensional state series with a prerecorded time series, and is the distance for which there are returns in the reconstructed series. By sorting the distances for every t, the pth smallest distance of is . The peak values of for different t can approximately predict the periods of the UPOs. Figure 2.17 shows the distribution of against t based on the measured time series of armature current of a wiper motor, which will be described in Chaper 8. Hence, the time for the first peak value at t = 0.1066 s is evaluated to be the period of the period-one UPO.

2.1.4.7 Power Spectrum Calculation

Power spectrum is also an important character that can be used to identify chaos. For the equilibrium point, there exist spectral peaks at zero frequency. For periodic behavior, there exist spectral peaks at the fundamental frequency and their multiple frequencies. For quasiperiodic behavior, there exist spectral peaks at the incommensurable frequencies and their common multiple frequencies. For random behavior, there is no spectral peak, and the spectrum is totally continuous. For chaotic behavior, the spectrum also exhibits continuous and broadband spectrum while there exist some spectral peaks which correspond to the average periods of the orbits to travel along the strange attractors. The power spectrum can be computed by using the Fourier transform of the autocorrelation function , which is defined as:

(2.16)

where . Equivalently, it can also be defined as the modulus square of its Fourier amplitude per unit time, which is given by:

(2.17)

There is a useful approach for calculating the power spectrum on the basis of the time series (Valsakumar, Satyanarayana, and Sridhar, 1997). The time series is given by sampling with a nonzero sampling time , and with a finite length N. Thus, it yields . The discrete version of the autocorrelation is defined as:

(2.18)

where represents the average over several initial conditions and trajectories. This averaging is performed in order to ensure that the autocorrelation function of the discrete time series is identical to that of the continuous time process evaluated at in the limit . Then, is defined as:

(2.19)

which can also be represented in a form analogous to (2.17) . The discrete Fourier transform of is expressed as:

(2.20)

Hence, the corresponding power spectrum is given by:

(2.21)

The computed power spectrum is equal to the true power spectrum under the following limit:

(2.22)

2.1.4.8 Fractal Dimension Calculation

Compared with the aforementioned capacity dimension to measure the fractal dimension, the correlation dimension has a computational advantage because it uses trajectory points and does not require a separate partition of the state space. The correlation dimension is defined as (Grassberger and Procaccia, 1983a):

(2.22)

(2.23)

(2.24)

where is the Heaviside step function, which is given by:

(2.25)

For the m-dimensional time-delay embedding, this yields:

(2.26)

When analyzing a scalar time series in which the optimal embedding is unknown, should be calculated in increasing embeddings until it ceases to change. The correlation dimension may be more physically meaningful since it emphasizes regions of the attractor visited most frequently by the orbit, rather than being a purely geometric quantity like the capacity dimension (Sprott, 2003).

2.1.4.9 Entropy Calculation

The correlation entropy is a close lower bound on the aforementioned K–S entropy, which is actually similar to the case where the correlation dimension is a close lower bound on the capacity dimension (Grassberger and Procaccia, 1983b b). The correlation entropy can also be computed with trajectory points, and is given by:

(2.27)

where is the correlation sum in (2.24)–(2.26) for an embedding dimension m.

2.2 Fundamentals of Electric Drive Systems

An electric drive system is depicted in Figure 2.18, which can be divided into two parts – electrical and mechanical. The electrical part consists of the subsystems of electric motor, power converter, electronic controller, and sensor, whereas the mechanical part includes the subsystems of mechanical transmission and mechanical load. The boundary between the electrical and mechanical parts is the air-gap of the motor, where electromechanical energy conversion takes place. The electronic controller can be further divided into two functional units: interface circuitry and processor. The sensor is used to translate the measurable quantities, such as current, voltage, temperature, speed, torque, and flux, into electronic signals. Through the interface circuitry, these signals are conditioned to the appropriate level before being fed into the processor. The processor output signals are usually amplified via the interface circuitry to drive power devices of the power converter. The converter acts as a power conditioner that regulates the power flow between the energy source and the electric motor for motoring and regeneration. Finally, the motor interfaces with the mechanical load via the mechanical transmission.

2.2.1 General Considerations

The development of electric drive systems has been based on the growth of various technologies, especially electric motors, power converters, and control strategies.

Electric motors have been available for over a century. The evolution of motors, unlike that of electronics and computer science, has been long and relatively slow. Nevertheless, the development of motors is continually fueled by new materials and sophisticated topologies.

The permanent magnet (PM) is one of the most influential materials for the development of motors, which can provide electric motors with life-long excitation. Figure 2.19 shows the typical characteristics of major PM materials for motors. Notice that Gauss (G) and Oersted (Oe) are non-SI units of magnetic flux density and coercivity, respectively, which are widely adopted in the field of magnetics. They are related to the SI units by and . The ferrite PM is the lowest in cost and has a straight demagnetization characteristic. They were once widely applicable to motors, but suffered from being bulky in size because of their low remanence. The aluminum–nickel–cobalt (Al–Ni–Co) PM has very high remanence but very low coercivity, thus its application to motors is normally limited by the demagnetizing field that it can withstand. Nevertheless, this property can be positively utilized to perform online demagnetization and remagnetization for the newly invented memory motor. Although the samarium–cobalt (Sm–Co) PM has both a high remanence and a high coercivity, its high initial cost restricts its widespread application to motors. The neodymium–iron–boron (Nd–Fe–B) PM has very high remanence and coercivity, and because of its reasonable cost, it has been widely adopted in recent PM motors. Other important parameters of PM materials are the maximum energy product, which is a measure of the maximum stored energy, and the temperature coefficient, which is a measure of the variation of magnetic characteristics with respect to temperature. In general, PMs lose remanence as temperature increases, but within a limited temperature range, the changes are reversible. When exposed to a temperature known as the Curie temperature, the magnetization of a PM is reduced to zero. In general, the relative permeability of PMs is similar to that of air, and much lower than that of iron. A brief summary of typical PM properties is given in Table 2.1.

Table 2.1 Properties of PM materials.

There are various topologies of electric motors, which create various classifications of electric motors and hence various classifications of electric drive systems. Electric motors were classified into two groups: DC and AC. The DC motors are fed by DC voltage, whereas the AC motors are fed by AC voltage. The AC motors can be further split into the two subgroups of induction motors and synchronous motors. In recent years, there have been other motors that cannot be simply grouped into DC or AC. As shown in Figure 2.20, the latest classification of electric motors is to split them into commutator motors and commutatorless motors. The former simply denote that they have a commutator and carbon brushes, while the latter have neither a commutator nor carbon brushes.

All DC motors belong to the commutator motor group, which can be further split into two subgroups – self-excited and separately excited. The self-excited subgroup can be either series excited or shunt excited; the separately excited subgroup can be excited by either field winding or PMs. On the other hand, the commutatorless motors can be further split into three subgroups – induction, synchronous, and doubly salient. Sometimes, the induction motors and synchronous motors can be regrouped as AC motors. Conventionally, the induction motors are split into wound-rotor and cage-rotor types. Similarly, the synchronous motors are split into wound-rotor, PM-rotor, and reluctance-rotor types – the so-called wound-rotor synchronous motor, PM brushless motor, and synchronous reluctance motor. The doubly salient motors refer to those motors having salient poles in both the stator and the rotor. The switched reluctance motors are doubly salient motors with a solid iron rotor. By incorporating PMs into the stator structure of doubly salient motors, doubly salient PM motors have recently been developed, which can be split into two subgroups – yoke-PM and tooth-PM types. The tooth-PM motors can be further split into two types – with PMs mounted on the teeth, and with PMs buried inside the teeth. These are termed flux-reversal PM motors and flux-switching PM motors, respectively.

In order to evaluate the aforementioned motors, some viable types, namely the conventional DC motor with field winding, the PMDC motor, the cage-rotor induction motor, the synchronous reluctance (SynR) motor, the PM brushless (PMBL) motor, the switched reluctance (SR) motor, and the doubly salient PM (DSPM) motor, are compared in terms of their power density, efficiency, controllability, reliability, maturity, and cost. A point grading system (1 to 5) is used, in which 1 is the worst and 5 is the best. As listed in Table 2.2, this evaluation indicates that the commutatorless motors are preferred to the commutator types. Among those commutatorless motors, the cage-type induction motor and the PMBL motor are the most attractive. Among those commutator motors, the PMDC motor is more acceptable.

Table 2.2 Evaluation of viable motors.

The evolution of power converters aims to achieve high power density, high efficiency, high controllability, and high reliability. Power converters may be AC–DC, AC–AC, DC–DC or DC–AC. Loosely, DC–DC converters are known as DC choppers while DC–AC converters are known as inverters, and are respectively used for DC and AC motors for electric drive systems. Initially, DC choppers were introduced in the early 1960s using force-commutated thyristors that were constrained to operate at low switching frequency. Due to the advent of fast-switching power devices, they can now be operated at tens or hundreds of kilohertz. Inverters are generally classified into voltage-fed and current-fed types. Because of the need of a large series inductance to emulate a current source, current-fed inverters are seldom used for electric drive systems. In fact, voltage-fed inverters are almost exclusively used because they are very simple and can have a power flow in either direction. Their output waveforms may be rectangular, six-step or pulse width modulation (PWM), depending on the switching strategy for different applications. For example, a rectangular output waveform is produced for a PM brushless DC motor, while a six-step or PWM output waveform is produced for an induction motor. It should be noted that the six-step output is becoming obsolete because its amplitude cannot be directly controlled and its harmonics are rich. On the other hand, the PWM waveform is harmonically optimal and its fundamental magnitude and frequency can be smoothly varied for speed control.

Since the last two decades, numerous PWM switching schemes have been developed for voltage-fed inverters, focusing on the harmonic suppression, better utilization of DC voltage, tolerance of DC voltage fluctuation as well as suitability for real-time and microcontroller-based implementation. These schemes can be classified as voltage-controlled and current-controlled PWM. The state-of-the-art voltage-controlled PWM schemes are sinusoidal PWM, regular PWM, optimal PWM, delta PWM, and random PWM. On the other hand, the use of current control for voltage-fed inverters is particularly attractive for high-performance drive systems because the motor torque and flux are directly related to the controlled current. The state-of-the-art current-controlled PWM schemes are hysteresis-band PWM and space vector PWM.

Instead of using hard switching, power converters can adopt soft switching. The key of soft switching is to employ a resonant circuit to shape the current or voltage waveform such that the power device switches at zero-current or zero-voltage condition. In general, the use of soft-switching converters possesses the following advantages:

• Due to zero-current or zero-voltage switching condition, the device switching loss is almost zero, thus giving high efficiency.
• Because of low heat sinking requirement and snubberless operation, the converter size and weight are reduced, thus giving high power density.
• The device reliability is improved because of minimum switching stress during soft switching.
• The electromagnetic interference (EMI) problem is less severe and the machine insulation is less stressed because of lower dv/dt resonant voltage pulses.
• The acoustic noise is very small because of high frequency operation.

on the other hand, their key drawbacks are the additional cost of the resonant circuit and the increased complexity.

Although there have been many soft-switching DC–DC converters developed for switched-mode power supplies, these converters cannot be directly applied to DC drive systems. Apart from suffering excessive voltage and current stresses, they generally cannot handle backward power flow during regenerative braking. It should be noted that the capability of regenerative braking is very essential for electric drive systems. Nevertheless, a soft-switching DC–DC converter, having the capability of bidirectional power flow for motoring and regenerative braking as well as the minimum hardware count, was developed for DC drive systems (Chau et al., 1997).

The development of soft-switching inverters for AC motors (including the induction motor and PM brushless AC motors) has become a research direction in power electronics. The milestone of soft-switching inverters, namely the three-phase voltage-fed resonant DC link inverter, was developed in 1986 (Divan, 1986). Consequently, many improved soft-switching topologies were proposed, such as the quasiresonant DC link, series resonant DC link, parallel resonant DC link, synchronized resonant DC link, resonant transition, auxiliary resonant commutated pole, and auxiliary resonant snubber inverters.

Compared with the development of soft-switching inverters for AC motors, there has been little development with SR motors (Cho et al., 1997). A soft-switching converter, the so-called zero-voltage-transition version, was particularly developed for SR motors (Ching, Chau, and Chan, 1998). This converter possesses the advantages that all main switches and diodes can operate at zero-voltage condition, unity device voltage, and current stresses, as well as a wide operating range. Moreover, it offers simple circuit topology, minimum hardware count, and low cost, leading to achieve high switching frequency, high power density and high efficiency.

Conventional linear control such as a PID (proportional-integral-derivative) can no longer satisfy the stringent requirement placed on high-performance electric drive systems. In recent years, many modern control strategies have been proposed. The state-of-the-art control strategies that have been proposed for electric drive systems are adaptive control, variable structure control, fuzzy control, and neural network control.

Adaptive control includes self-tuning control (STC) and model-referencing adaptive control (MRAC). Using STC, the controller parameters are tuned to adapt to variations in the system parameters. The key is to employ an identification block to track changes in system parameters and to update the controller parameters through controller adaptation in such a way that a desired closed-loop performance can be obtained. Using MRAC, the output response is forced to track the response of a reference model irrespective of any variations in parameters of the system. Based on an adaptation algorithm that utilizes the difference between the reference model and system outputs, the controller parameters are adjusted to give a desired closed-loop performance.

Variable structure control (VSC) has also been applied for electric drive systems to compete with adaptive control. Using VSC, the system can be designed to provide parameter-insensitive features, prescribed error dynamics, and simplicity in implementation. Based on a set of switching control laws, the system is forced to follow a predefined trajectory in the phase plane irrespective of system parameter variations.

Emerging technologies such as fuzzy logic and neural networks have been introduced into the field of electric drive systems. Fuzzy control is essentially a linguistic process which is based on the prior experience and heuristic rules used by human operators. Making use of neural network control (NNC), the controller can possibly interpret the behavior of system dynamics, then self-learn and self-adjust accordingly. Furthermore, these state-of-the-art control strategies can incorporate one another, such as adaptive fuzzy control, fuzzy NNC, and fuzzy VSC. In near future, controllers incorporating artificial intelligence (AI) can permit diagnosis of systems and correction of faults to supplant the need of human intervention.

2.2.2 DC Drive Systems

A DC drive system is composed of the DC motor, power converter, electronic controller, sensor, mechanical transmission, and mechanical load. The classification of various DC drive systems depends on the location of field excitation in the DC motor. Namely, they can be grouped as a self-excited DC drive system or a separately excited DC drive system. Based on the source of field excitation in the DC motor, they can also be grouped as a wound-field DC drive system or a PMDC drive system. The former has a field winding so that any field excitation can be controlled by the DC current, whereas the latter has no field winding and the PM excitation is uncontrollable.

The name applied to those wound-field DC drives is usually determined by the mutual interconnection between the field winding and armature winding. As shown in Figure 2.21, common wound-field DC drives are the separately excited, series, shunt, and cumulative compound types. Without external control, their torque-speed characteristics at the rated voltage are shown in Figure 2.22. In the separately excited DC drive, the field and armature voltages can be controlled independently of each other. The torque-speed characteristic is linearly related so that speed decreases as torque increases and speed regulation depends on the armature circuit resistance. In the series DC drive, the field current is the same as the armature current. An increase in torque is accompanied by an increase in the armature current and hence an increase in flux. The speed must drop to maintain the balance between the supply and the induced voltages. The torque-speed characteristic has an inverse relationship. In the shunt DC drive, the field and armature are connected to a common voltage source. The corresponding characteristic is similar to that of the separately excited DC drive. In a cumulative compound DC drive, the MMF of the series field is in the same direction as the MMF of the shunt field. The characteristic lies between those of the series DC and shunt DC drives, depending on the relative strength of the series and shunt fields.

By replacing the field winding and pole structure with PMs, the PMDC drive can readily be generated from the separately excited DC drive. Compared with the separately excited DC drive, the PMDC drive has relatively higher power density and higher efficiency because of the space-saving benefit by PMs and the absence of field losses. Owing to the low permeability of PMs (similar to that of air), the corresponding armature reaction is usually reduced and commutation is improved. However, since the field excitation in the PMDC drive is uncontrollable, it cannot readily attain the same operating characteristics as the separately excited DC drive.

Both wound-field DC and PMDC drives suffer from the same problem due to the use of commutators and brushes in their motors. Commutators cause torque ripples and limit the motor speed, while brushes are responsible for friction and radio-frequency interference. Moreover, due to wear and tear, periodic maintenance of commutators and brushes is always required. These drawbacks make them less reliable and unsuitable for maintenance-free operation.

The major advantages of DC drive systems are their maturity and simplicity. The simplicity is mainly due to their simple control because the air-gap flux Φ and the armature current – hence the motor speed and torque T – can be independently controlled. Irrespective of whether the motors are wound-field DC or PMDC, they are governed by the following basic equations:

(2.28)

(2.29)

(2.30)

where E is the back EMF, is the armature voltage, is the armature resistance, and is named as the back EMF constant or torque constant. For those wound-field DC motors, Φ is linearly related to the field current , which may be independently controlled, dependent on , dependent on , or dependent on both and , respectively, for those separately excited, series, shunt, or cumulative compound types. In contrast, Φ is essentially uncontrollable for those PMDC motors.

The basic topology of DC motors is shown in Figure 2.23. The corresponding design consideration includes the electric loading, magnetic loading, stator outside and inside diameters, rotor outside and inside diameters, core length, air-gap length, number of poles, number of armature slots, armature tooth width and slot depth, number of turns per coil, slot fill factor, number of commutator bars, commutation arrangement, field winding or PM excitation arrangement, thermal arrangement, speed, torque, power, efficiency, torque density, and power density.

In order to perform motion control for the DC drive systems, the use of power converters is mandatory. There are two major power converters suitable for DC drive systems, namely the DC–DC converter and AC–DC converter. The former has been used extensively for various applications, whereas the latter, also termed the controlled rectifier, is rarely used for DC drive systems due to its low input power factor. Actually, even when the supply is from the grid, the AC voltage is first converted to DC voltage by using a simple diode rectifier and then controlled by the DC–DC converter, rather than directly using the AC–DC converter.

When DC–DC converters adopt the chopping mode of operation, they are usually known as DC choppers. These DC choppers are classified as first-quadrant, second-quadrant, two-quadrant, and four-quadrant versions. The first-quadrant DC chopper (shown in Figure 2.24) is suitable for motoring, and the corresponding power flow is from the source to the load, whereas the second-quadrant is for regenerative braking and the power flow is out from the load into the source. Incorporating both of the motoring and regenerative braking, the two-quadrant DC chopper is shown in Figure 2.25. Moreover, instead of using mechanical contactors to achieve reversible operation, the four-quadrant DC chopper shown in Figure 2.26 can be employed so that motoring and regenerative braking in both forward and reversible operations are controlled electronically.

As shown in Figure 2.27, there are three ways in which the chopper output voltage can be varied, namely PWM control, frequency-modulated control, and current-limit control. In the first method, the chopper frequency is kept constant and the pulse width is varied. The second method has a constant pulse width and a variable chopping frequency. In the third method, both the pulse width and frequency are varied to control the load current between certain specified maximum and minimum limits. For conventional DC drive systems, PWM control of the two-quadrant DC chopper is generally adopted. The corresponding control is based on the variation of duty cycle δ:

(2.31)

(2.32)

where is the DC supply voltage. Hence, the motoring operation () occurs when , and regenerative braking () occurs when . The no-load operation is obtained when . As the current is always flowing, a discontinuous conduction mode does not occur.

In general, speed control of DC drive systems can be accomplished by two methods – armature control and field control. When the armature voltage of the DC motor is reduced, the armature current (and hence the motor torque) decrease, causing the motor speed to decrease. In contrast, when the armature voltage is increased, the motor torque increases, causing the motor speed to increase. Since the maximum allowable armature current remains constant and the field is fixed, this armature voltage control has the advantage of retaining the maximum torque capability at all speeds. However, since the armature voltage cannot be further increased beyond the rated value, this control is used only when the DC drive system operates below its base speed. On the other hand, when the field voltage of the DC motor is weakened while the armature voltage is fixed, the motor induced EMF decreases. Because of low armature resistance, the armature current will increase by an amount much larger than the decrease in the field. Thus, the motor torque is increased, causing the motor speed to increase. Since the maximum allowable armature current is constant, the induced EMF remains constant for all speeds when the armature voltage is fixed. Hence, the maximum allowable motor power becomes constant so that the maximum allowable torque varies inversely with the motor speed. Therefore, in order to achieve wide-range speed control of DC drive systems, the armature control has to be combined with the field control. By maintaining the field constant at the rated value, armature control is employed for speeds from standstill to the base speed. Then, by keeping the armature voltage at the rated value, field control is used for speeds beyond the base speed. The corresponding maximum allowable torque and power in the combined armature and field control are shown in Figure 2.28.

It should be noted that only the separately excited DC drive system can perform the combined armature and field control. The corresponding torque-speed characteristics during motoring and regenerative braking are depicted in Figure 2.29. For the shunt, series and cumulative compound DC drive systems, both of armature voltage and field voltage are varied simultaneously, whereas the PM drive system can only perform armature control.

2.2.3 Induction Drive Systems

At present, the induction drive system is the most mature technology among various commutatorless drive systems. There are two types of induction motors, namely wound-rotor and cage-rotor. Because of high cost, need of maintenance, and lack of sturdiness, the wound-rotor induction motor is less attractive than its cage-rotor counterpart. Apart from the common advantages of commutatorless drive systems, the induction drive systems possess the definite advantages of low cost and ruggedness. These advantages can generally outweigh their major disadvantage of control complexity, and facilitate them to be widely accepted.

The most common induction drive system is composed of the cage-rotor induction motor, PWM inverter, electronic controller, sensor, mechanical transmission and mechanical load. Reasonable high-voltage low-current motor design should be employed to reduce the copper loss of motor windings as well as the cost and size of the PWM inverter. High-speed operation should also be adopted to minimize the motor size and weight, although the maximum speed of the motor is limited by the bearing friction and windage losses as well as the transaxle tolerance. Low stray reactance is also necessary to favor flux-weakening operation.

The basic topology of induction motors is shown in Figure 2.30. The corresponding design consideration includes the electric loading, magnetic loading, stator outside and inside diameters, rotor outside and inside diameters, core length, air-gap length, number of poles, number of stator slots, number of rotor slots, stator tooth width and slot depth, rotor tooth width and slot depth, number of turns per phase, slot fill factor, thermal arrangement, speed, torque, power, efficiency, torque density, and power density.

The three-phase voltage-fed PWM inverter shown in Figure 2.31 is almost exclusively used for induction drive systems. The inverter design highly depends on the technology of power devices. At present, the IGBT-based inverter is most attractive. Since the hard-switching inverter topology is almost fixed, the inverter design generally depends on the selection of power devices and PWM switching schemes. The selection of power devices is based on the criteria that (1) the voltage rating is at least twice the nominal supply voltage because of the voltage surge during switching, (2) the current rating is large enough so that there is no need to connect several power devices in parallel, and (3) the switching speed is sufficiently high to suppress motor harmonics and acoustic noise levels. The power module is normally a two-in-one or even six-in-one type, namely two or six devices are internally connected with an antiparallel diode across each device, to minimize wiring and stray impedance. On the other hand, the selection of PWM switching schemes is based on the criteria that the magnitude and frequency of the fundamental component of the output waveform can be smoothly varied, the harmonic distortion of the output waveform is minimum, the switching algorithm can be real-time implemented with minimum hardware and compact software, and the variation of input voltage can be handled. There are numerous PWM schemes that have been available, such as the sinusoidal PWM, regular PWM, optimal PWM, delta PWM, random PWM, hysteresis-band PWM, and space vector PWM. Currently, both the sinusoidal PWM and space vector PWM are widely used for induction drive systems.

Speed control in induction drive systems is considerably more complex than that of DC drive systems because the induction motors suffer from nonlinearity of the dynamic model with coupling between direct and quadrature axes. There are two representative control strategies, namely variable-voltage variable-frequency (VVVF) control and field-oriented control (FOC) which is also known as vector control or decoupling control. The basic equation of speed control is governed by:

(2.33)

where n is the motor speed in rev/s or rps, is the rotating-field synchronous speed in rev/s, s is the slip, p is the number of pole pairs, and f is the supply frequency in Hz. Thus, the motor speed can be controlled by variations in f, p, and/or s. In general, more than one control variable is adopted. On top of these control strategies, sophisticated control algorithms such as adaptive control and optimal control have also been employed to achieve faster response, higher efficiency, and wider operating ranges.

Figure 2.32 shows the functional block diagram of VVVF control of induction drive systems. This strategy is based on constant volts/hertz control for frequencies below the motor rated frequency, whereas variable frequency control with constant rated voltage for frequencies beyond the rated frequency. For very low frequencies, voltage boosting is applied to compensate the difference between the applied voltage and the induced EMF due to the stator resistance drop. As shown in Figure 2.33, the induction drive system characteristics can be divided into three operating regions. The first region is called the constant torque region in which the motor can deliver its rated torque for frequencies below the rated frequency. In the second region, called the constant power region, the slip is increased to the maximum value in a preprogrammed manner so that the stator current remains constant and the motor can maintain its rated power capability. In the high-speed region, the slip remains constant while the stator current decreases. Thus, the torque capability declines with the square of the speed. Because of the disadvantages of air-gap flux drifting and sluggish response, the VVVF control strategy is becoming less attractive for high-performance induction drive systems.

In order to improve the dynamic performance of induction drive systems, FOC is preferred to VVVF control. Figure 2.34 shows the functional block diagram of FOC of induction drive systems. By using FOC, the mathematical model of induction motors is transformed from the stationary reference frame (α–β frame) to the general synchronously rotating frame (x–y frame) as shown in Figure 2.35. Thus, at steady state, all the motor variables such as supply voltage , stator current , rotor current , and rotor flux linkage can be represented by DC quantities. When the x-axis is purposely selected to be coincident with the rotor flux linkage vector, the reference frame (d − q frame) rotates synchronously with the rotor flux, as shown in Figure 2.36, where and are the d-axis and q-axis components of stator current, respectively. Hence, the motor torque T can be obtained as:

(2.34)

where M and are, respectively, the mutual and rotor inductances per phase. Since can be written as , the torque equation can be rewritten as:

(2.35)

This torque equation is very similar to that of separately excited DC motors. Namely, resembles the field current while resembles the armature current . Thus, can be considered as the field component of , which is responsible for establishing the air-gap flux. On the other hand, can be considered as the torque component of , which produces the desired motor torque. Therefore, by means of this FOC, the motor torque can be effectively controlled by adjusting the torque component as long as the field component remains constant. Hence, induction drive systems can offer the same fast transient response as separately excited DC drive systems. In order to attain the above FOC, the rotor flux linkage vector is always aligned with the d-axis. This criterion, the so-called decoupling condition, can be attained through slip frequency control as given by:

(2.36)

where is the rotor resistance per phase.

Since the advent of FOC, a number of methods have been proposed for implementation. Basically, these methods can be classified into two groups, namely direct FOC and indirect FOC. The direct FOC requires the direct measurement of the rotor flux, which not only increases the complexity in implementation, but also suffers from unreliable measurement at low speeds. In contrast, the indirect FOC determines the rotor flux by calculation, instead of by direct measurement. This method has the definite advantage of being easier to implement than the direct FOC. Therefore, the indirect FOC is widely adopted for the high-performance motion control of induction drive systems.

Although the indirect FOC has been widely used for high-performance induction drive systems, it still suffers from some drawbacks. In particular, the rotor time constant (which has a dominant effect on the decoupling condition) changes severely, depending on operating temperature and magnetic saturation, and can lead to deterioration in the desired FOC. In general, there are two ways to solve this problem: one is to perform an online identification of the rotor time constant and, accordingly, update the parameters used in the FOC controller; the other is to adopt a sophisticated control algorithm to render the FOC controller insensitive to variations in the parameters of the motor.

The model reference adaptive control (MRAC) algorithm has been widely used for the FOC control of induction drive systems. Firstly, a reference model – which is designed to be an optimal system in general – is made to satisfy the desired dynamic performance of the drive system. An adaptive mechanism is then adopted which aims to force the drive system to follow the reference model even after some variation in the parameters of the system, such as a change of due to prolonged operation. The main criterion of the adaptive mechanism is to assure robustness with asymptotically zero error between the outputs of the reference model and the drive system. The definite advantage of this MRAC scheme is that there is no need to carry out an explicit parameter identification or estimation in the synthesis of the drive system control input. In fact, only the command input, the controlled drive system output, and the reference model are required to establish this control scheme.

2.2.4 Synchronous Drive Systems

As mentioned above, AC drive systems are composed of two major groups – induction drive systems and synchronous drive systems. Previously, the induction motor drives almost monopolized the whole market of AC drive systems, but no longer, due to the fact that the wound-rotor synchronous motor needs an external DC current to excite the field windings in the rotor via slip rings and carbon brushes, which causes many problems such as maintenance requirement, bulky accessories, and safety issues. With the advent of high-energy PM materials, the PM brushless motors (including the PM synchronous motor) are becoming attractive and can directly compete with the induction motors for the market of AC drive systems. Also, with the introduction of new magnetic material and structure, the synchronous reluctance motor can provide a decent performance without the use of field windings or PMs in the rotor.

2.2.4.1 PM Brushless Drive Systems

Among those viable electric drive systems, the PM brushless drive systems are most capable of competing with the mature induction drive systems. Their advantages are summarized below:

• Since the magnetic field is excited by high-energy PMs, the overall weight and volume can be significantly reduced for a given power output, leading to high power density.
• Because of the absence of rotor copper losses, their efficiency is inherently high.
• Since the heat mainly arises in the stator, it can be more efficiently dissipated to surroundings.
• Since PM excitation suffers from no risk of manufacturing defects, overheating, or mechanical damage, their reliability is inherently high.
• Because of the lower electromechanical time constant of the rotor, the rotor acceleration at a given input power can be increased.

Nevertheless, the PM brushless drive systems suffer from the drawbacks of relatively high PM material cost and uncontrollable PM flux.

The system configuration of PM brushless drive systems is similar to that of induction drive systems – namely, it consists of a PM brushless motor, power inverter, electronic controller, sensor, mechanical transmission, and mechanical load. Based on the waveforms feeding into the motor terminals, PM brushless motors can be divided into two types – PM brushless AC (BLAC) and PM brushless DC (BLDC). PM BLAC motors are fed by sinusoidal or near-sinusoidal AC waves. Actually, they are usually known as PM synchronous motors or, sometimes, sinusoidal-fed PM brushless motors. On the other hand, PM BLDC motors are fed by rectangular AC waves, and are sometimes known as rectangular-fed PM brushless motors. Since the interaction between a rectangular field and a rectangular current in the motor can produce a higher torque than that produced by a sinusoidal field and sinusoidal current, the PM BLDC drive system possesses a higher power density than the PM BLAC drive system. Meanwhile, the PM BLDC motor has a significant torque pulsation, whereas the PM BLAC motor produces an essentially constant instantaneous torque, or so-called smooth torque like a wound-rotor synchronous drive system.

The basic consideration of the PM brushless motor design includes the electric loading, magnetic loading, stator outside and inside diameters, rotor outside and inside diameters, core length, air-gap length, PM material characteristics, PM topology and dimensions, number of poles, number of stator slots, stator tooth width and slot depth, number of turns per phase, slot fill factor, thermal arrangement, speed, torque, power, efficiency, torque density, and power density.

On the basis of the placement of PMs, PM brushless motors can be classified as surface-mounted, surface-inset, interior-radial, and interior-circumferential types (Gan et al., 2000; Zhu and Howe, 2007). For the surface-mounted PM brushless motor topology as shown in Figure 2.37, the PMs are simply mounted on the rotor surface by using epoxy adhesives. Since the permeability of PMs is near to that of air, the effective air-gap is the sum of the actual air-gap length and the radial thickness of the PMs. Hence, the corresponding armature reaction field is small and the stator winding inductance is low. Also, since the d-axis and q-axis stator winding inductances are nearly the same, its reluctance torque is almost zero. For the surface-inset PM brushless motor topology, as shown in Figure 2.38, the PMs are inset or buried in the rotor surface. Thus, the q-axis inductance becomes higher than the d-axis inductance, hence producing the additional reluctance torque. Also, since the PMs are inside the rotor, it can withstand the centrifugal force at high-speed operation, thus offering good mechanical integrity. For the interior-radial PM brushless motor topology, as shown in Figure 2.39, the PMs are radially magnetized and buried inside the rotor. Similar to the surface-inset type, the PMs are mechanically protected, hence allowing for high-speed operation. Also, because of its d–q saliency, an additional reluctance torque is generated. Contrary to the surface-inset type, this interior-radial topology adopts linear PMs which are more easily inserted and can be easily machinable. For the interior-circumferential PM brushless motor topology shown in Figure 2.40, the PMs are circumferentially magnetized and buried inside the rotor. This gives the definite advantage that the air-gap flux density can be higher than the PM remanent flux density – which is known as flux focusing. Also, this holds the merits of good mechanical integrity and additional reluctance torque. However, because of significant flux leakage at the inner ends of the PMs, a nonmagnetic shaft or collar is generally required.

As mentioned above, PM brushless drive systems have two basic operations, namely BLAC and BLDC, as shown in Figure 2.41. Actually, each PM brushless motor can operate at both modes if the torque density, torque smoothness, and efficiency are not of great concern. As PM BLAC drive systems operate with a sinusoidal current and a sinusoidal air-gap flux, they need a high-resolution position signal for closed-loop control, hence desiring a costly position encoder or resolver. In contrast, the PM BLDC drive systems operate with a rectangular current and a trapezoidal air-gap flux, and only require a low-cost sensor for phase-current commutation. Nevertheless, the PM BLAC drive systems allow for open-loop operation, whereas the position feedback is mandatory for the PM BLDC drive systems.

Control of PM BLAC motors is similar to that of induction motors. So, the control strategies for induction drive systems, such as VVVF and FOC, are applicable to PM BLAC drive systems. Based on FOC, the generated torque can be expressed as:

(2.37)

where p is the number of pole-pairs, is the stator winding flux linkage due to the PMs, , are respectively the d-axis and q-axis stator winding inductances, and , are respectively the d-axis and q-axis currents. Moreover, by incorporating the well-known d–q axis transformation, the well-developed flux-weakening control technique can readily be applied to PM BLAC drive systems for constant power operation. The maximum flux-weakening capability is achieved when the motor is designed to have unity per-unit d-axis inductance (Soong and Ertugrul, 2002):

(2.38)

where is the PM flux linkage, is the d-axis winding inductance, and is the rated current. In general, the ratio of is less than unity, therefore the higher the ratio, the higher will be the flux-weakening capability. The flux-weakening control has been comprehensively studied in various PM BLAC drive systems (Zhu, Chen, and Howe, 2000; Uddin and Rahman, 2007). The constant-power operation for PM BLDC drive systems is more complex. Since the operating waveforms are no longer sinusoidal, d–q axis transformation and, hence, flux-weakening control are ill-suited. Nevertheless, the corresponding constant-power operation can be offered by using advanced conduction angle control (Chan et al., 1995; Kim, Kook, and Ko, 1997).

Figure 2.42 shows the torque-speed characteristics of the PM brushless drive systems without control, and with either flux-weakening control for the BLAC or advanced conduction angle control for the BLDC. It illustrates that the speed range of constant-power operation can be significantly extended. On the other hand, Figure 2.43 gives a comparison of the torque-speed characteristics of PM BLAC and PM BLDC drive systems. It can be seen that the BLAC drive systems offer higher torque and higher power capabilities than the BLDC motor drives employing two-phase 120° conduction. Nevertheless, the BLDC motor drives employing three-phase 180° conduction can offer a better high-speed power capability, but with the sacrifice of low-speed torque capability (Zhu and Howe, 2007). Moreover, for PM BLDC drive systems with multiphase polygonal windings (Wang et al., 2002), the corresponding back EMF, rather than the air-gap flux, can be directly varied to enable constant-power operation.

2.2.4.2 Synchronous Reluctance Drive Systems

PM brushless motors have been accepted to provide the highest efficiency and highest power density. However, they suffer from the drawbacks of high PM material cost, the accidental demagnetization of PMs, and the thermal instability of PMs. The synchronous reluctance (SynR) motor relies on reluctance torque, rather than the reaction torque which is dominant in the cylindrical wound-rotor synchronous motor and the surface-mounted PM brushless motor.

The system configuration of SynR drive systems is similar to that of PM BLAC drive systems. It consists of the SynR motor, PWM inverter, electronic controller, sensor, mechanical transmission, and mechanical load. There are a number of advantages associated with this system:

• The SynR motor does not require field windings or PMs in the rotor, hence offering high mechanical integrity to withstand high-speed operation.
• Since the SynR motor does not need expensive PM material, it is much less costly than PM brushless motors.
• As the SynR motor eliminates the problem of accidental demagnetization of PMs, the drive system can offer a very high-current operation.
• As the SynR motor also eliminates the problem of thermal instability of PMs, the drive system can allow for working in a high-temperature environment.

The SynR motor is actually one of the earliest types of electric motors. The first generation was to adopt a cylindrical rotor with multiple slits along the lines of the direct-axis flux (Kostko, 1923). This rotor configuration could not offer a high saliency ratio, leading to the creation of a low reluctance torque. The second generation was to adopt a segmental rotor (Lawrenson and Gupta, 1967). Its saliency ratio could be five or even larger, hence creating a higher reluctance torque. The latest third generation is to adopt an axially laminated rotor, which has steel sheets bent into a U-shape and stacked in a radial direction (Cruickshank, Anderson, and Menzies, 1971). Its saliency can achieve seven or more, which enables the SynR motor to compete favorably with an induction motor. Nevertheless, this rotor involves higher manufacturing cost which, perhaps, can be solved by mass production.

A modern axially laminated SynR motor is shown in Figure 2.44. Its rotor is constructed of thin laminations which are bent in a semicircular shape. These iron segments are separated by insulating material, such as air or plastic. By selecting the ratio between the width of each iron segment and the width of each insulating material, the saliency of this SynR motor can be optimized, enabling ten or more to be achieved (Matsuo and Lipo, 1994).

Since the stator winding of the SynR motor is sinusoidally distributed, the control of SynR motors is similar to that of PM BLAC motors. Therefore, the control strategies for PM BLAC drive systems, such as VVVF and FOC, are also applicable to SynR drive systems (Xu et al., 1991). Based on the d−q axis transformation, the SynR motor can be modeled as:

(2.39)

(2.40)

where , are the d−q components of applied voltage, , are the d−q components of stator winding current, , are the d−q components of inductance, is the stator winding resistance, and is the rotor speed. The saliency is defined as:

(2.41)

Consequently, the generated torque can be expressed as:

(2.42)

where p is the number of pole-pairs. It is obvious that the larger the value of κ, the higher will be the generated torque. Moreover, the saliency of seven can enable the SynR motor to operate at a power factor of about 0.8, which is comparable with the induction motors for electric drive systems.

2.2.5 Doubly Salient Drive Systems

A doubly salient drive system adopts a doubly salient motor, which means that boththe stator and the rotor have salient poles. The two most common doubly salient motors are the switched reluctance (SR) and doubly salient PM (DSPM) types, both of which adopt a simple solid-iron rotor with salient poles. The major difference between their structures is that the SR motor installs only armature windings in the stator, whereas the DSPM motor incorporates both armature windings and PMs in the stator. In terms of torque generation, they are very different – that is, the torque generated in the SR motor is solely a reluctance torque, whereas the torque generated in the DSPM motor is mainly a PM torque.

The DSPM motors, and hence their drive systems, are sometimes classified as a subclass of PM brushless drive systems, since they adopt PMs in the motor structure and offer the merit of brushless. As their PMs are located in the stator rather than in the rotor, they are known as stator-PM brushless drive systems, in contrast to the conventional rotor-PM brushless drive systems.

2.2.5.1 Switched Reluctance Drive Systems

Although the concept of variable reluctance was adopted for electric motors over a century ago, the SR motor did not reach its full potential until the advent of power electronics. In general, the SR drive system consists of an SR motor, power converter, electronic controller, sensor, mechanical transmission, and mechanical load.

Figure 2.45 shows a three-phase 6/4-pole SR motor. Because of the salient nature of both the stator and rotor poles, the inductance L of each phase varies with the rotor position, as shown in Figure 2.46. The operating principle of the SR motor is based on the “minimum reluctance” rule. For instance, when the phase A winding is excited, the rotor tends to rotate clockwise in order to decrease the reluctance of the flux path until the rotor poles align with the stator poles A+ and A− , where the reluctance of the flux path has a minimum value (the inductance has a maximum value). Phase A is then switched off and phase B is switched on so that the reluctance torque tends to make the relevant rotor poles align with the stator poles B+ and B− . The torque direction is always toward the nearest aligned position. Hence, by conducting the phase windings in the sequence of A–B–C according to the rotor position feedback from the position sensor, the rotor can continuously rotate clockwise.

According to the co-energy principle, the reluctance torque produced by one phase at any rotor position is given by:

(2.43)

where is the rotor position angle, i is the phase current, and is the co-energy, which is defined as the area below the magnetization curve of flux linkage versus current. This can be expressed as:

(2.44)

Since the flux linkage can be written as , the reluctance torque can be rewritten as:

(2.45)

After neglecting magnetic saturation, the inductance is independent of the phase current. The reluctance torque can thus be deduced as:

(2.46)

From the above analysis, it can be seen that the SR motor has two significant features. One is that the direction of torque is independent of the polarity of the phase current. The other is that the motoring torque can be produced only in the direction of rising inductance (), otherwise, a negative torque (or braking torque) is produced. Each phase is therefore fed with current and hence produces a positive torque only in half a rotor pole-pitch. This is also the reason why the SR motor generally suffers from a large torque ripple. Nevertheless, this torque ripple can be alleviated by increasing the number of phases.

Although SR motors possess a simplicity in construction, that does not imply simplicity in analysis and design. Because of the heavy saturation of pole tips and the fringing effect of poles and slots, the design of SR motors suffers from a great difficulty when using the magnetic circuit approach. In most cases, the electromagnetic finite element analysis is employed to determine the motor parameters and performances. Optimization is then based on the minimization of total losses while taking into account the pole arc constraint, height constraint, and maximum flux density constraint. Nevertheless, there are some basic criteria to initialize the design process of SR motors (Chan et al., 1996).

An appropriate selection of the number of phases and poles is important to satisfy the desired motor performance. In order to allow for starting and running bidirectionally, SR motors should have at least three phases with six stator poles and four rotor poles. The three-phase 6/4-pole SR motor offers the lowest cost and highest efficiency, but its corresponding large torque ripple reduces the smoothness of its operation. On the other hand, the four-phase 8/6-pole SR motor has a relatively higher cost and lower efficiency while possessing the smallest torque ripple. The three-phase 12/8-pole SR motor is a compromise design between the three-phase and four-phase types, and their selection should be based on the operating requirements and cost justification. It should be noted that higher numbers of phases and poles require more power devices and a higher switching frequency, leading to increased costs and switching losses, respectively. The numbers of both the stator and rotor poles, and , are governed by:

(2.47)

(2.48)

where m is the number of phases and k is a positive integer. When the rotor speed is rev/s or rps, the commutating frequency of a particular phase is given by:

(2.49)

In order to minimize the switching frequency and to decrease the iron losses in poles and yokes, the number of rotor poles selected should be as small as possible. Thus, the number of rotor poles is usually smaller than the stator poles. Other considerations of SR motor design include the electric loading, magnetic loading, stator outside and inside diameters, rotor outside and inside diameters, stator and rotor pole arcs, stator and pole heights, core length, air-gap length, number of stator windings per phase, thermal arrangement, speed, torque, power, efficiency, torque density, and power density.

Similar to DC drive systems, the chopping frequency of SR drive systems should be above 10 kHz to minimize acoustic noise. Many converter circuits have been developed in attempts to reduce the number of power devices and take full advantage of unipolar operation. However, when the device count is reduced, there is a penalty in the form of lower controllability, lower reliability, lower operating performance, or extra passive components. The converter circuit shown in Figure 2.47 is well suited for three-phase SR drive systems. It utilizes two power devices to independently control the current of each phase and two freewheeling diodes to return any stored magnetic energy to the DC source or link. Since this circuit topology needs two power devices per phase, the converter cost is relatively higher than that with less power devices. However, this bridge arrangement allows control of each phase winding independent of the state of other phase windings. Thus, it is possible to allow for phase overlapping to increase the torque production and extend the constant-power range.

SR drive systems have three modes of operation. When the speed is below the base speed , the current can be limited by chopping, known as current chopping control (CCC). In the CCC mode, as shown in Figure 2.48, the turn-on angle and the turn-off angle are fixed and the firing angle depends only on the speed feedback. The torque can be controlled by changing the current limits, and thus the constant-torque characteristic can be achieved by CCC. During high-speed operation, however, the peak current is limited by the EMF of the phase winding. The corresponding characteristic is essentially controlled by phasing the switching instants relative to the rotor position – known as angular position control (APC). In the APC mode as shown in Figure 2.49, the constant-power characteristic can be achieved. At the critical speed , both and reach their limit values. Thereafter, the SR drive can no longer keep constant-power operation and thus offer the characteristic similar to that of the DC series drive. The torque-speed characteristics of these three operation modes are shown in Figure 2.50. Moreover, because of the inherently high nonlinearity of SR motors, various advanced control methods have been developed for the SR drive systems, such as the adaptive control, fuzzy logic control, and sliding mode control (Zhan, Chan, and Chau, 1999).

2.2.5.2 Doubly Salient PM Drive Systems

The DSPM motor topologies have salient poles in both of the stator and rotor, and incorporate PMs located in the stator. Similar to the SR motors where their rotor has neither PMs nor windings, these DSPM motors are mechanically simple and robust, and are hence very suitable for high-speed operation. According to the shape and location of the PMs, they can be classified as the yoke-magnet and tooth-magnet types.

Yoke-magnet DSPM motors can be further split into linear-magnet and curved-magnet types. As shown in Figure 2.51, the yoke-linear-magnet type is most common and relatively more mature (Chau et al., 2005; Cheng, Chau, and Chan, 2001, 2003). Although they have salient poles in the stator and rotor, the PM torque significantly dominates the reluctance torque, hence exhibiting low cogging torque. Since the variation of flux linkage with each coil as the rotor rotates is unipolar, it is very suitable for BLDC operation. On the other hand, when the rotor is skewed, it can offer a BLAC operation. The major disadvantage of this topology is the relatively low torque density, which results from its unipolar flux linkage. As shown in Figure 2.52, the yoke-curved-magnet type is very similar to the previous one, except for the shape of the PMs. As opposed to the yoke-linear-magnet type, the periphery of this topology is essentially circular. Also, since there is more space to accommodate the PMs, this DSPM motor can achieve a higher air-gap flux density. Its major drawback is the difficulty in machining the curved PMs and inserting them into the stator core.

The variation in the air-gap flux of the DSPM motors is induced by the variation in permeance rather than by the rotation of the magnet. Thus, the flux always tends to flow through the shortest path that the stator poles align with the rotor poles from the unaligned position, causing the motor to rotate. The corresponding flux linkage per phase is composed of the PM flux linkage and the armature reaction flux linkage Li as given by:

(2.50)

where L is the self-inductance and i is the armature current per phase. The magnetic field co-energy can thus be obtained by subtracting the energy stored in the armature windings from the total input energy:

(2.51)

Then, by differentiating the co-energy, the generated torque can be obtained, as given by:

(2.52)

where is the rotor position, is the PM torque component which is due to the interaction between the armature current and the PM flux linkage, and is the reluctance torque component, which is due to the variation of self-inductance. The theoretical waveforms of and L are shown in Figure 2.53. In order to produce a continuous unidirectional torque, a bipolar armature current is used, in which a positive current is applied when the flux linkage increases, whereas a negative current is applied when the flux linkage decreases. As a result, the PM torque becomes the dominant torque component, while the reluctance torque is a parasitic pulsating torque with zero average value.

It should be noted that the bipolar armature current operation of DSPM motors is fundamentally different from that of SR motors in which a unipolar armature current is adopted to create the reluctance torque only during the period of increasing inductance. Therefore, the torque density of DSPM drive systems is inherently higher than that of SR drive systems.

Tooth-magnet DSPM motors can be further split into surface-magnet and interior-magnet types. Figure 2.54 shows the tooth-surface-magnet DSPM motor topology, which is commonly known as a flux-reversal PM motor, since the flux linkage with each coil reverses polarity as the rotor rotates (Deodhar et al., 1997; Zhu and Howe, 2007). In this topology, each stator tooth has a pair of PMs of different polarities mounted onto the surface. Hence, the flux linkage variation is bipolar so that the torque density can be higher than that of the yoke-magnet types. However, since the PMs are on the surface of the stator teeth, they are more prone to partial demagnetization. Also, a significant eddy current loss in the PMs may result. On the other hand, Figure 2.55 shows the tooth-interior-magnet motor topology which is commonly known as a flux-switching PM motor (Zhu et al., 2005; Zhu and Howe, 2007). In this topology, each stator tooth consists of two adjacent laminated segments and a PM, and each of these segments is sandwiched by two circumferentially magnetized PMs. Hence, it enables flux focusing. Additionally, this flux-switching PM motor has less armature reaction, hence offering higher electric loading. Since its back EMF waveform is essentially sinusoidal, this motor is more suitable for a BLAC operation.

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