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926 M. F. Rahman et al.

from electromagnetic noise and thus further adds to

the expense of the drive system.

(e) The allocation of space for the mounting of the posi-

tion sensor may be a problem for small applications

(such as for motors used in consumer products).

Hence, to overcome the problems induced by rotor-position

transducers, researchers have developed a number of methods

to eliminate the electromechanical sensor for deriving posi-

tion information. This is achieved by indirectly determining

the rotor position. Such methods are commonly termed sen-

sorless rotor-position estimation methods. The term sensorless

seems to imply that there are no sensors at all. However,

there must be some form of sensor used to measure the rotor

position. In fact, the term sensorless position estimation in

reality implies that there are no additional sensors required to

determine position apart from those that measure the motor

electrical parameters to control the motor. These are normally

current- or voltage-measuring circuits.

Hence, all sensorless position estimation methods for the

SR motor use some form of processing on electrical wave-

forms of the motor windings. In essence, the major difference

between sensorless position detection, and the electromechan-

ical sensors mentioned above, is that there is no mechanical

connection of the sensor to the motor shaft. Therefore, sen-

sorless position detection involves electrical measurements

only.

33.10 Synchronous Reluctance

Motor Drives

33.10.1 Introduction

In recent years, there has been a revival of interest in reluc-

tance machines. Two main machines have been the focus

of this interest: the switched-reluctance machine (SRM) and

the synchronous reluctance machine (Syncrel). The SRM is a

machine that does not have sine-wave spatial distributed wind-

ings, but instead has concentrated coils and a doubly salient

rotor and stator structure. The operation of this machine is

highly nonlinear in character, and normal ac machine model-

ing techniques cannot be applied in a straightforward manner

to describe its operation. The SRM drive has been considered

in detail in an earlier section.

The Syncrel, on the other hand, has conventional three-

phase sinusoidally distributed windings on the stator. The

word “synchronous” in the machine’s name emphasizes the

fact that the stator windings generate a spatial sinusoidally

distributed magnetomotive force (mmf) in the airgap between

the stator and the rotor, and under steady-state conditions the

rotor rotates in synchronism with this ﬁeld. Therefore, the

stator winding conﬁguration of this machine is virtually

exactly the same as that of the induction machine or the

conventional synchronous machine. The major difference

between the Syncrel and conventional synchronous and induc-

tion machines is in the rotor structure. In both the induction

machine and the synchronous machine, there is a source of

ﬂux in the rotor itself. In the case of the induction machine,

this ﬂux is produced by currents resulting from an induction

mechanism, and for the synchronous machine, there is a ﬁeld

winding wound on the rotor that is fed with the dc current to

produce ﬂux. The permanent-magnet synchronous machine

replaces the wound ﬁeld on the rotor with a permanent mag-

net. The Syncrel, on the other hand, does not have any source

of ﬂux on the rotor, but instead the rotor is designed to dis-

tort the ﬂux density distribution produced by the sinusoidally

distributed mmf.

Sinusoidally wound reluctance machines were traditionally

used in the ﬁber-spinning industry because of their syn-

chronous nature. This made it simple to keep a large number

of machines running at the same speed using just the frequency

of the supply to the machines. These machines were direct-on-

line-start machines. This was facilitated by the presence of an

induction machine starting cage on the rotor. This cage was

also essential to damp out oscillations in the rotor speed when

running at synchronous speed. It should be pointed out that

these machines are not considered to be Syncrels – a Syncrel

does not have an induction machine cage on the rotor. A Syncrel

is absolutely dependent on an intelligent inverter drive in order

to start the machine and to stabilize it when running. The lack

of a requirement for a starting cage means that the rotor design

can be optimized for best torque and power performance.

The revival of interest in the Syncrel in the early 1980s was

motivated by the development of low-cost microprocessors

and reliable power electronics, coupled with the perception

that the Syncrel may be more efﬁcient and simpler to con-

trol in variable-speed applications compared to the induction

machine. The control simplicity is achieved in practice, mainly

because one does not have to locate the ﬂux vector in order

to implement vector control. The potential for improved efﬁ-

ciency and torque density compared to the induction machine

is very much dependent on the rotor design. The Syncrel has

the advantage over the switched-reluctance machine; in that it

produces relatively smooth torque naturally, and it uses a con-

ventional three-phase inverter. Therefore, inverter technology

developed for the induction machine can be applied directly.

33.10.2 Basic Principles

Reluctance machines are one of the oldest electric machine

structures, since they are based on the basic physical fact that

a magnet attracts a piece of iron. In fact, Syncrel structures

were being published in the early 1920s [1]. The essential idea

behind the operation of all reluctance machines is that the

windings of the machine produce magnetic poles that are used

to attract the reluctance rotor. If the magnetic poles are moved

around the periphery of the machine at the rate at which the

33 Motor Drives 927

A phase axis

B phase axis

C phase axis

Reluctance

rotor

FIGURE 33.109 Conceptual diagram of a synchronous reluctance machine.

rotor is moving, then sustained torque and rotation can be

achieved.

A conceptual diagram of a Syncrel is shown in Fig. 33.109.

In this ﬁgure, the rotor is represented as a simple “dumbbell”-

type rotor. The axes of the three-phase sinusoidally distributed

windings are indicated by the dashed lines. If these wind-

ings are being fed with currents, then a spatial sinusoidally

distributed, mmf results. Because this mmf is sinusoidally dis-

tributed, it can be represented by a “space vector” (similar

to sinusoidal time-varying quantities being represented by a

time phasor). In Fig. 33.109, this resultant space vector is indi-

cated by F, F

, and F

are the components of this vector that

lie along the high-permeance and low-permeance axes of the

machine, respectively.

If one considers the situation shown in Fig. 33.109, then

the rotor would tend to rotate in the direction indicated. This

rotation would continue until the high permeance d

axis (i.e.

the least-reluctance axis) of the machine aligns with the mmf

vector. When this alignment occurs, the ﬂux produced by the

stator mmf vector would be maximized. If the vector F also

rotates as the rotor rotates, then as mentioned previously, the

angle between F and d

will remain constant and the rotor will

continue to chase the F vector, continuous rotation being the

result.

One can ask even more fundamental questions such as

“Why does the rotor rotate to the position that maximizes

the ﬂux density?” This is essentially asking, why does a magnet

attract a piece of iron. To completely answer this question one

has to delve into the ﬁeld of quantum physics, which is beyond

the scope of this presentation. A less complicated explanation

is based on the fact that the stator ﬂux density tends to align

the domains in the ferromagnetic rotor material, which pro-

duces an effect similar to having a current-carrying winding

wrapped around the rotor. This effective current then inter-

acts with the stator ﬂux density to produce a force that has a

component that is oriented radially around the periphery of

the machine. It is this component that produces the torque on

the rotor that causes the alignment with the stator mmf vector.

It was mentioned previously that in order to have a continu-

ous motion, the mmf vector F must rotate at the same angular

velocity as the rotor, so that the angle between the mmf vector

and the d

axis of the rotor is kept at a constant value. The

rotation of the mmf vector is achieved by feeding the three-

phase windings of the machine with time-varying currents.

928 M. F. Rahman et al.

It can be shown that if these form a balanced 120

◦

temporally

phase-shifted set of sinusoidal currents, then the resultant mmf

vector will rotate at a constant velocity related to the frequency

of the input current waveforms and the number of pole pairs

in the machine.

In order to get a more precise ﬁgure for the torque pro-

duced by a Syncrel, one has to develop techniques of modeling

the machine. Because the Syncrel is a reluctance machine, the

coenergy technique for developing the torque expressions can

be used [2, 3]. The coenergy technique is a very accurate way of

determining the torque as it explicitly takes into account, the

saturation nonlinearities in the iron of the machine. However,

the technique does not lend itself to mathematical analysis

and is not a good way of understanding the basic dynamic

properties of the machine. The coenergy approach will not be

pursued any further in this presentation.

33.10.3 Machine Structure

The essential difference between the Syncrel and, say, the

induction machine is the design of the rotor. Most experi-

mental Syncrel systems that have been built use the stator of

an induction machine, including the same windings. The rotor

designs can take on a number of different forms, from the very

simple and basic dumbbell-shaped rotor (such as that sketched

in Fig. 33.109) to more complex designs. Unfortunately, the

designs that are simple to manufacture (such as the dumb-

bell design) do not give good performance; therefore, one is

forced into more complex designs. The design of the rotor in

a Syncrel is the key to whether it is economic to manufacture

and has competitive performance with similar machines.

The design of Syncrel rotors fall into four main categories

of increasing manufacturing complexity and performance:

dumbbell or higher pole-number equivalent designs, ﬂux-

barrier designs, radially laminated ﬂux-barrier designs, and

axially laminated designs. The ﬁrst two of these design

methodologies are old and lead to designs with poor to mod-

est performance. Therefore, they will not be considered any

further. The latter two, however, lead to machine designs with

performance comparable to that of the induction machine.

Figures 33.110 and 33.111 show the cross section of a four-

pole machine with a radial lamination ﬂux barrier designed

rotor and an axially laminated rotor. The radial lamination

design allows the rotor to be built using similar techniques

to standard radial laminations for other machines. The ﬂux

barriers can be punched for mass production, or wire-eroded

for low production numbers. These laminations are simply

stacked onto the shaft to form the rotor. The punched areas

can be ﬁlled with plastic or epoxy materials for extra strength,

if required. The iron bridges at the outside of the rotor are

designed to saturate under normal ﬂux levels and therefore do

not adversely affect the performance of the machine. They are

there to provide mechanical strength.

D-axis

Q-axis

Air flux barriers

Narrow iron

bridge

Shaft

FIGURE 33.110 Cross section of a radially laminated Syncrel.

D-axis

Inter-lamination

space

Lamination

Pole-piece

Shaft

Q-axis

FIGURE 33.111 Cross section of an axially laminated Syncrel.

The axially laminated rotor is constructed with lamina-

tions running the length of the rotor (i.e. into the page

on Fig. 33.111). In between the laminations, a nonmagnetic

packing material is used. This can be aluminum or bronze,

33 Motor Drives 929

for example, but a nonconductive material such as slot insula-

tion is better since eddy currents can be induced in conductive

materials. The ratio of the steel laminations to nonmagnetic

material is usually about 1:1. The axial laminations are all

stacked on top of each other, and a nonmagnetic pole piece is

bolted on top of the stack to hold the laminations to the shaft.

The strength of these bolts is usually the main limitation on

the mechanical strength and hence the speed of rotation of this

rotor. If more or thicker bolts are used to increase strength,

the magnetic properties of the rotor are compromised because

of the amount of lamination that has to be cut out to make

room for them.

Radial and axial laminated rotors are usually limited to

four-pole or higher machines because of the difﬁculty of

accommodating the shaft in two-pole designs. An axially lam-

inated two-pole rotor has been built with the shafts effectively

bonded onto the end of the rotor. Another design was con-

structed of a block of alternating steel and bronze laminations,

the whole structure being brazed together and the resultant

stack then being machined into a round rotor and shafts (this

rotor was used for high-speed generator applications).

Of the two rotor designs, the radially laminated one has

the best potential for economic production. The axially lami-

nated rotor in general gives the best performance, but the mass

production difﬁculties with folding and assembling the lami-

nations make its adoption by industry unlikely. On the other

hand, improved designs for radially laminated rotors mean

that they can now produce performance very close to that

of the axial-laminated designs, and the ease of manufacture

would indicate that these rotors are the future of Syncrel

rotors.

33.10.4 Basic Mathematical Modeling

In order to give a more quantitative understanding of the

machine, a basic mathematical dynamic model of the machine

will be introduced.

This model will assume that the iron

material in the machine does not saturate. This means that the

ﬂux density and ﬂux linkage of the windings in the machine

are linear functions of the currents in the machine.

To derive the electrical dynamic model of a machine, one

usually uses Faraday’s ﬂux linkage expressions. In the case of

the Syncrel, the self and mutual ﬂux linkage between the phases

is obviously a function of the angular position of the rotor;

therefore, one needs to have expressions for these inductances

in terms of rotor position. The fundamental assumption used

to make this mathematically tractable is that the inductances

vary as a sinusoidal function of the rotor position.

The other

Note that the model is not derived but instead just stated. The structure

of the model will be heuristically explained.

The sinusoidal variation of inductance with rotor position turns out to

be very accurate because of the fact that the stator windings are sinusoidally

wound. This forces the ﬂux linkage to behave in a sinusoidal fashion.

major part of the modeling process is the conversion from a

three-phase model to a two-phase model. This is a process that

is carried out for most sinusoidally wound machines, since it

allows a variety of machines to be represented by very similar

models.

A further complication in this process is that the two-phase

model is derived in a “rotating reference frame,” as opposed

to a stationary reference frame. Developing the equations in

a rotating reference frame has the advantage that the normal

sinusoidal currents feeding the machine are transformed into

dc currents in steady state, and the angular dependence of the

machine’s inductances disappear.

One way of heuristically understanding the effect of the

rotating frame transformation is to imagine that we are observ-

ing the machine’s behavior from the vantage point of the

rotor. Because the sinusoidal ﬂux density waveform is rotating

around the machine in synchronism with the rotor, it appears

from the rotor that the ﬂux density is not changing with time –

i.e. it is a ﬂux density created by dc currents ﬂowing in a single

sinusoidally distributed winding. This single sinusoidal wind-

ing is effectively rotating with the rotor. It should be noted

that the transformation process of the ﬂuxes, currents, volt-

ages, and machine parameters to the two-phase rotating frame

is an invertible process; therefore one can apply the inverse

transformation to ascertain what is happening in the original

three-phase machine.

The models derived using the three-phase to two-phase

transformations are known as dq models, the d and q referring

to the two axes of the machine in the two-phase model (both

stationary and rotating frame dq axes are shown in Fig. 33.109).

The linear

dq equations for the Syncrel can be derived as [4]

= Ri

−ωL

(33.99)

= Ri

−ωL

(33.100)

where

= the d-axis voltage and current,

= the q-axis voltage and current,

, L

= the d-and q-axes inductances, respectively,

ω = the electrical angular velocity of the rotor.

Thus far we have concentrated on the electrical dynamics

of the machine. The other very important aspect is the torque

produced by the machine. It is possible to derive the torque

of the Syncrel using the principle of virtual work based on

coenergy as:

−L

(33.101)

Linear refers to the fact that the equations are derived assuming that the

iron circuit behaves linearly in relation to applied mmf and the ﬂux produced.

930 M. F. Rahman et al.

−

FIGURE 33.112 Two-phase equivalent circuit of the Syncrel in a rotating frame.

The 3/2 factor is to account for the fact that the two-phase

machine produces two-thirds the torque of the three-phase

machine.

The only other remaining equation is the mechanical

equation for the system:

J ˙ω

+Dω

= T

(33.102)

where

J ≡ the rotational inertia of the rotor/load,

D ≡ the friction coefﬁcient for the load,

≡ the ﬁxed load torque of the load,

≡ the rotor mechanical angular velocity (= ω/p

Remark

Equation (33.101) shows that the machine must be designed so

that L

−L

is as large as possible. This will maximize the torque

that is produced by the machine for given d- and q-axis currents.

To lower L

, one must design the q-axis so that it has as much

air obstructing the ﬂow of ﬂux as possible, and the d-axis must

be designed so that it has as much iron as possible. In practice

these quantities cannot be varied independently.

Figure 33.112 shows the equivalent circuit for the Syncrel

corresponding to Eqs. (33.99) and (33.100). One can see that

the dynamic equations for the Syncrel are intrinsically simple.

In contrast, the induction machine electrical equations consist

of a set of four complex coupled differential equations.

33.10.5 Control Strategies and Important

Parameters

The dq model captured in Eqs. (33.99)–(33.101) can be used

to explain a number of control strategies for the Syncrel. It is

beyond the scope of this section to present the derivation of

The 3/2 conversion factor is required if the transformations are power-

variant transformations, as opposed to the power-invariant transformations.

The power-variant transformations are the most common ones used because

the single-phase machine parameters can be used directly in the resultant

models, and the two-phase voltages and currents are identical in magnitude

to their three-phase counterparts.

Rotor

FIGURE 33.113 Space phasor diagram of a Syncrel.

these. The interested reader should consult references [4–6]

cited at the end of this chapter.

One of the most common control strategies for any elec-

trical machine is to maximize the torque per ampere of input

current. The following discussion should be considered in con-

junction with Fig. 33.113, which shows the relationship of the

various vectors in the machine to the d

- and d

-axes.

It turns out that one of the critical parameters for the control

of the Syncrel is the angle of the resultant current vector in the

machine in relation to the d-axis of the machine. It is possible

to write the torque expression for the Syncrel in terms of this as



−L



sin 2δ (33.103)

It is obvious that this expression is maximized for a given

value of i if δ =

. Therefore, one should control the currents

so that δ stays at this angle, if maximum torque per ampere is

desired.

Another control objective for the Syncrel is to maximize the

power factor for the machine. This is important to minimize

the kVA for the inverter. It can be shown that the current angle

33 Motor Drives 931

to maximize the power factor is [4]

δ = tan

−1



ξ (33.104)

where

ξ = L

(the inductance ratio).

Remark

Equation (33.104) indicates that ξ is the important parameter in

relation to power factor. In order to obtain a power factor of 0.8,

one requires an inductance ratio of approximately 10.

Finally, we shall consider another control objective – max-

imize the rate of change of torque with a ﬁxed-current-angle

control strategy. In effect, this means that one is maximizing

the rate of change of the currents in the machine for a given

voltage applied to it. The analysis of this requirement results

in [4]

δ = tan

−1

ξ (33.105)

Remark

As with the maximum-power-factor case, ξ is the most important

parameter in relation to the rate of change of torque. Because

this control effectively optimizes the current into the machine for

a given voltage and angular velocity, this angle also corresponds

to that required to maximize the ﬁeld-weakening range of the

machine.

Other control strategies for the machine can be devised, as

well as the current angles required to obtain the maximum

power from the machine during ﬁeld-weakening operation.

Remark

If one carries out a thorough analysis of all the control properties

of the Syncrel, then it emerges that all performance measures for

the machine are enhanced by a large value of the ξ ratio.

33.10.6 Practical Considerations

The control strategies discussed in the previous section were all

derived assuming that the machine does not exhibit saturation

and there are no iron losses. The q-axis of the machine does not

have any saturation, as the ﬂux path on this axis is dominated

by air. However, the d-axis of the machine does exhibit sub-

stantial saturation under operational ﬂux levels, and this effect

must be accounted for to optimize the drive performance.

The effect that saturation has on the ideal current angles

is to increase them. This increase is most pronounced for the

maximum torque per ampere control strategy, since this strat-

egy results in a larger component of current in the d-axis,

and consequently more saturation. Maximum power factor

and maximum rate of change of torque are not affected as

much. In order to get the correct current angle for maximum

torque per ampere, a lookup table of the saturation character-

istic of the machine must be stored in the controller, which is

consulted in order to calculate the desired current angle [7].

It has been found that iron losses in the stator and the

rotor also affect the optimal current angles. However, usually

saturation effects dominate, and the effects of iron losses can

be ignored.

33.10.7 A Syncrel Drive System

The basic structure of a variable-speed drive system based on

using the Syncrel is shown in Fig. 33.114. Many components

of this drive are very similar to those found in an induction

machine drive system. One notable exception is the L

lookup

table block and the current reference generator. The L

lookup

stores the current vs d-axis inductance table for the machine,

thereby allowing the inductance to be determined for various

current levels. This table is also used to generate the incremen-

tal d-axis inductance. The inductance values generated from

this table are used in the state feedback block and the torque

estimator.

The state feedback block effectively generates an offset

voltage to the PWM generator so that the voltage it pro-

duces is at least enough to counter the back-emf. This

technique effectively eliminates the back-emf disturbance from

the current-control loops.

The current reference generator takes the desired torque as

an input and generates the required d- and q-axis currents at

the output. This block uses a lookup-table technique together

with an inverse of the torque equation to generate these cur-

rents and takes into account the saturation characteristics of

the machine.

The three-to-two-phase block converts the currents from a

three-phase stationary frame to a two-phase rotating frame.

This is a standard block in induction machine drives, and as

with induction machine drives, this means that the Syncrel

control algorithm is implemented in a rotating reference

frame. The conversion from this frame back to the stationary

frame occurs implicitly in the space vector PWM generator.

The Syncrel control algorithm is essentially a simpliﬁed vec-

tor controller, and consequently the computational require-

ments are not high. This means that a Syncrel controller can

be implemented on a modest microprocessor. As far as input

and output hardware is concerned, the requirements are basi-

cally the same as those for an induction machine system – i.e.

sample two of the phase currents, the link voltage, and the

rotor position.

33.10.8 Conclusion

The Syncrel-based drive system offers simplicity in control,

excellent performance for variable speed and position-control

932 M. F. Rahman et al.

Three phase mains

Space

Vector

PWM

Inverter

Current

Reference

Generator

+−

−

State

Feedback

lookup

Torque

Estimator

Velocity

Observer

Three to

two

phase

Syncrel

wλ

FIGURE 33.114 Block diagram of a Syncrel drive system.

applications, good torque and power density, and efﬁciency

that is more than competitive with that of the induction

machine drive systems. To date very few commercial drive sys-

tems are available using Syncrels, this being mainly due to the

slow emergence of easy-to-manufacture rotors that give good

performance, and the conservatism of the motor-drive indus-

try. One commercial application that has emerged is ac-servo

applications, where the Syncrel offers low torque ripple (with

appropriate rotor design) together with a small moment of

inertia. Other applications under consideration are in the area

of drives for electric vehicles and generators for ﬂywheel energy

storage systems. It remains to be seen whether the Syncrel

can ever challenge the supremacy of the vector-controlled

induction machine in mainstream industrial applications.

The interested reader who wishes to pursue Syncrel drives

in more detail can ﬁnd a good coverage of the control and

motor design issues in [8].

References

1. J. Kostko, “Polyphase reaction synchronous motors,” J. Amer. Inst.

Elec. Eng., 42, 1162–1168, 1923.

2. D. O’Kelly and S. Simmons, Introduction to Generalised Electrical

Machine Theory, McGraw Hill, U.K., 1968.

3. D. Staton, W. Soong, and T.J. Miller, “Uniﬁed theory of torque

production in switched reluctance motors,” IEEE Trans. on Industry

Applications, IA-31, 329–337, 1995.

4. R.E. Betz, “Theoretical aspects of control of synchronous reluctance

machines,” IEE Proc. B, 139, 355–364, 1992.

5. A. Chiba and T. Fukao, “A closed-loop operation of super high speed

reluctance motor for quick torque response,” IEEE Trans. on Industry

Applications, IA-28, 600–606, 1992.

6. M.G. Jovanovic and R.E. Betz, “Theoretical aspects of the control of

synchronous reluctance machine including saturation ad iron losses,”

Proceedings of IEEE-IAS Annual Meeting, Houston, October, 1992.

7. M.G. Jovanovic, “Sensorless Control of Synchronous Reluctance

Machines,” Ph.D. thesis, University of Newcastle, Australia, 1997.

8. I. Boldea, Reluctance Synchronous Machines and Drives. Oxford

University Press, 1996.

Further Reading

DC Motor Drives

9. G. K. Dubey, Power Semiconductor Controlled Drives. Prentice Hall,

1989.

10. W. Leonard, Control of Electric Drives. Springer-Verlag, 1985.

11. V. Subrahmanyam, Electric Drives; Concepts and Applications.

McGraw Hill, 1994.

33 Motor Drives 933

12. M. A. El-Sharkawi, Fundamentals of Electric Drives. Thompson

Learning, 2000.

Induction Motor Drives

13. B. K. Bose, Power Electronics and AC Drives. Prentice Hall, 1987.

14. J. M. D. Murphy and G. G. Turnbull, Power Electronic Control of AC

Motors. Pergamon Press, 1988.

15. D. W. Novotny and T. A. Lipo, Vector Control and Dynamics of Drives.

Oxford Science Publications, 1996.

16. P. Vas, Electrical Machines and Drives. a Space Vector Theory

Approach, Clarendon Press, 1992.

17. G. K. Dubey, Power Semiconductor Controlled Drives. Prentice Hall,

1989.

Synchronous Motor Drives

18. W. Leonard, Control of Electrical Drives. Springer-Verlag, 1985.

19. J. M. D. Murphy and G. G. Turnbull, Power Electronic Control of AC

Drives. Pergamon Press, 1988.

20. G. R. Slemon, Electric Machines and Drives. Addison-Wesley, 1992.

21. W. W. Novotny and T. A. Lipo, Vector Control and Dynamics of AC

Drives. Oxford Science Publications, 1996.

Permanent-magnet AC Synchronous

Motor Drives

22. T. J. E. Miller, Brushless Permanent Magnet and Reluctance Motor

Drives. Oxford Science Publications, 1989.

23. “Performance and Design of Permanent Magnet AC Motor Drives”,

Tutorial course, IEEE Industry Application Society, 1989.

24. R. M. Crowder, Electric Drives and Their Controls. Oxford Science

Publications, 1995.

Permanent-magnet Brushless DC (BLDC) Motor

Drives

25. T. J. E. Miller, Brushless Permanent-magnet and Reluctance Motor

Drives. Oxford University press, 1989. For “square-wave brushless dc

motors.”

26. T. Kenjo and S. Nagamori, Permanent-magnet and Brushless DC

Motors. Oxford University Press, 1985. For “small machines used

particularly in consumer electronics.”

27. D. C. Hanselman, Brushless Permanent-magnet Motor Design.

Mc Graw Hill, 1994. For “the possible range of controllers in the

last chapter.”

28. Application notes from the various integrated circuit companies,

particularly those that specialize in motor control.

Servo Drives

29. G. W. Younkin, Industrial Servo Control Systems: Fundamentals and

Applications. Marcel Dekker, 1996.

30. B. C. Kuo and J. Tal, Incremental Motion Control 1: DC motors and

systems. SRL Publishing, 1978.

31. D. Shetty and R. Kolk, Mechatronics Systems Design. PWS Publishing,

1997.

32. A. Fransua and R. Magureanu, Electric machines and Drive Systems.

Techical Press, Oxford, 1984.

Stepper Motor Drives

33. H. B. Ertan, A. Hughes, and P. J. Lawrenson, “Efﬁcient numerical

method for predicting the torque-displacement curve of saturated

VR stepping motors,” Proc. IEE, 133, Part B, 1980.

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Control Methods for Switching

Power Converters

J. Fernando Silva, Ph.D. and

Sónia Ferreira Pinto, Ph.D.

Instituto Superior Técnico, DEEC,

A.C. Energia, Laboratório de

Máquinas Eléctricas e

Electrónica de Potência, Centro

de Automática da Universidade

Técnica de Lisboa, AV. Rorisco

Pais 1, 1049-001 Lisboa, Portugal

34.1 Introduction .......................................................................................... 935

34.2 Switching Power Converter Control Using State-space Averaged Models .......... 936

34.2.1 Introduction • 34.2.2 State-space Modeling • 34.2.3 Converter Transfer Functions

• 34.2.4 Pulse Width Modulator Transfer Functions • 34.2.5 Linear Feedback Design

Ensuring Stability • 34.2.6 Examples: Buck–Boost DC/DC Converter, Forward DC/DC

Converter, 12 Pulse Rectiﬁers, Buck–Boost DC/DC Converter in the Discontinuous Mode

(Voltage and Current Mode), Three-phase PWM Inverters

34.3 Sliding-mode Control of Switching Converters............................................. 955

34.3.1 Introduction • 34.3.2 Principles of Sliding-mode Control • 34.3.3 Constant-frequency

Operation • 34.3.4 Steady-state Error Elimination in Converters with Continuous Control Inputs

• 34.3.5 Examples: Buck–Boost DC/DC Converter, Half-bridge Inverter, 12-pulse Parallel

Rectiﬁers, Audio Power Ampliﬁers, Near Unity Power Factor Rectiﬁers, Multilevel Inverters,

Matrix Converters

34.4 Fuzzy Logic Control of Switching Converters............................................... 993

34.4.1 Introduction • 34.4.2 Fuzzy Logic Controller Synthesis • 34.4.3 Example: Near Unity

Power Factor Buck–Boost Rectiﬁer

34.5 Conclusions ........................................................................................... 996

References ............................................................................................. 997

34.1 Introduction

Switching power converters must be suitably designed and

controlled in order to supply the voltages, currents, or fre-

quency ranges needed for the load and to guarantee the

requested dynamics [1–4]. Furthermore, they can be designed

to serve as “clean” interfaces between most loads and the elec-

trical utility system. Thereafter, the set switching converter

plus load behaves as an almost pure electrical utility resistive

load.

This chapter provides basic and some advanced skills to con-

trol electronic power converters, taking into account that the

control of switching power converters is a vast and interdis-

ciplinary subject. Control designers for switching converters

should know the static and dynamic behavior of the elec-

tronic power converter and how to design its elements for the

intended operating modes. Designers must be experts on con-

trol techniques, especially the nonlinear ones, since switching

converters are nonlinear, time-variant, discrete systems, and

designers must be capable of analog or digital implementa-

tion of the derived modulators, regulators, or compensators.

Powerful modeling methodologies and sophisticated control

processes must be used to obtain stable-controlled switching

converters, not only with satisfactory static and dynamic per-

formance, but also with low sensitivity against load or line

disturbances or, preferably, robustness.

In Section 34.2, the techniques to obtain suitable nonlinear

and linear state-space models, for most switching converters,

are presented and illustrated through examples. The derived

linear models are used to create equivalent circuits, and to

design linear feedback controllers for converters operating in

the continuous or discontinuous mode. The classical linear

time-invariant systems control theory, based on Laplace trans-

form, transfer function concepts, Bode plots or root locus, is

best used with state-space averaged models, or derived circuits,

and well-known triangular wave modulators for generating

the switching variables or the trigger signals for the power

semiconductors.

935

936 J. F. Silva and S. F. Pinto

Nonlinear state-space models and sliding-mode controllers,

presented in Section 34.3, provide a more consistent way of

handling the control problem of switching converters, since

sliding mode is aimed at variable structure systems, as are

switching power converters. Chattering, a characteristic of slid-

ing mode, is inherent to switching power converters, even if

they are controlled with linear methods. Chattering is very

hard to remove and is acceptable in certain converter vari-

ables. The described sliding-mode methodology deﬁnes exactly

the variables that need to be measured, while providing the

necessary equations (control law and switching law) whose

implementation gives the robust modulator and compensator

low-level hardware (or software). Therefore, the sliding-mode

control integrates the design of the switching converter modu-

lator and controller electronics, reducing the needed designer

expertise. This approach requires measurement of the state

variables, but eliminates conventional modulators and lin-

ear feedback compensators, enabling better performance and

robustness. It also reduces the converter cost, control com-

plexity, volume, and weight (increasing power density). The

so-called main drawback of sliding mode, variable switch-

ing frequency, is also addressed, providing ﬁxed-frequency

auxiliary functions and suitable augmented control laws to

null steady-state errors due to the use of constant switching

frequency.

Fuzzy control of switching converters (Section 34.4) is a

control technique needing no converter models, parameters,

or operating conditions, but only an expert knowledge of the

converter dynamics. Fuzzy controllers can be used in a diverse

array of switching converters with only small adaptations, since

the controllers, based on fuzzy sets, are obtained simply from

the knowledge of the system dynamics, using a model reference

adaptive control philosophy. Obtained fuzzy control rules can

be built into a decision-lookup table, in which the control

processor simply picks up the control input corresponding

to the sampled measurements. Fuzzy controllers are almost

immune to system parameter ﬂuctuations, since they do not

take into account their values. The steps to obtain a fuzzy

controller are described, and the example provided compares

the fuzzy controller performance to the current-mode control.

34.2 Switching Power Converter Control

Using State-space Averaged Models

34.2.1 Introduction

State-space models provide a general and strong basis for

dynamic modeling of various systems including switching con-

verters. State-space models are useful to design the needed

linear control loops, and can also be used to computer simulate

the steady state, as well as the dynamic behavior, of the switch-

ing converter, ﬁtted with the designed feedback control loops

and subjected to external perturbations. Furthermore, state-

space models are the basis for applying powerful nonlinear

control methods such as sliding mode. State-space averaging

and linearization provides an elegant solution for the appli-

cation of widely known linear control techniques to most

switching converters.

34.2.2 State-space Modeling

Consider a switching converter with sets of power

semiconductor structures, each one with two different cir-

cuit conﬁgurations, according to the state of the respective

semiconductors, and operating in the continuous mode of

conduction. Supposing the power semiconductors as con-

trolled ideal switches (zero on-state voltage drops, zero off-

state currents, and instantaneous commutation between the

on- and off-states), the time (t ) behavior of the circuit, over

period T, can be represented by the general form of the

state-space model (34.1):

˙x = Ax +Bu

y = Cx +Du

(34.1)

where x is the state vector, ˙x = dx/dt , u is the input or

control vector, and A, B, C, D are respectively the dynamics

(or state), the input, the output, and the direct transmission

(or feedforward) matrices.

Since the power semiconductors will either be conducting

or blocking, a time-dependent switching variable δ(t ) can be

used to describe the allowed switch states of each structure

(i.e. δ(t) = 1 for the on-state circuit and δ(t) = 0 for the

off-state circuit). Then, two subintervals must be considered:

subinterval 1 for 0 ≤ t ≤ δ

T, where δ(t ) = 1 and subinterval

2 for δ

T ≤ t ≤ T where δ(t) = 0. The state equations of the

circuit, in each of the circuit conﬁgurations, can be written as:

˙x = A

x +B

y = C

x +D

for 0 ≤ t ≤ δ

T where δ

(

)

= 1

(34.2)

˙x = A

x +B

y = C

x +D

for δ

T ≤ t ≤ T where δ

(

)

= 0

(34.3)

34.2.2.1 Switched State-space Model

Given the two binary values of the switching variable δ(t ),

Eqs. (34.2) and (34.3) can be combined to obtain the nonlinear

and time-variant switched state-space model of the switching