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1388 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 40, NO. 5, SEPTEMBER/OCTOBER 2004

Design of Fast and Robust Current Regulatorsfor High-Power Drives Based on

Complex State VariablesJoachim Holtz, Fellow, IEEE, Juntao Quan, Member, IEEE, Jorge Pontt, Member, IEEE,

José Rodríguez, Senior Member, IEEE, Patricio Newman, and Hernán Miranda

Abstract—High-power pulsewidth-modulated inverters formedium-voltage applications operate at switching frequenciesbelow 1 kHz to keep the dynamic losses of the power devices ata permitted level. Also, the sampling rate of the digital signalprocessing system is then low, which introduces considerablesignal delays. These have adverse effects on the dynamics of thecurrent control system and introduce undesired cross couplingbetween the current components and .

To overcome this problem, complex state variables are used toderive more accurate models of the machine and the inverter. Fromthese, a novel current controller structure employing single-com-plex zeros is synthesized.

Experimental results demonstrate that high dynamic per-formance and zero cross coupling is achieved even at very lowswitching frequency.

Index Terms—Complex state variables, current control, induc-tion motor, low switching frequency, medium-voltage inverters,root locus techniques.

I. INTRODUCTION

THE dynamic analysis of induction motors is traditionallybased on state equations, with the - and -axis compo-

nents of the stator currents and the rotor flux linkage as statevariables [1]. The electromagnetic subsystem of the machineis usually modeled as a fourth-order dynamic system in termsof scalar state variables. Although the root locus plot is easilyobtained in this approach, it appeared difficult so far to utilizethis powerful tool of dynamic system analysis for the design ofcurrent controllers in ac drives. Root locus techniques are in-deed being considered being hardly useful for this purpose. Itis frequently commented in the literature that the obstacle is thedynamic complexity of the induction motor [2]. The design ofthe current control system of an induction motor drive is there-fore commonly based on an approximation: the intercoupled

Paper IPCSD-04-040, presented at the 2003 Industry Applications SocietyAnnual Meeting, Salt Lake City, UT, October 12–16, and approved for publica-tion in the IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS by the IndustrialDrives Committee of the IEEE Industry Applications Society. Manuscript sub-mitted for review February 13, 2004 and released for publication June 14, 2004.

J. Holtz is with the Electrical Machines and Drives Group, Wuppertal Uni-versity, 42097 Wuppertal, Germany (e-mail: [email protected]).

J. Quan is with Rockwell Automation AG, CH 6036 Dierikon, Switzerland(e-mail: [email protected]).

J. Pontt, J. Rodríguez, P. Newman, and H. Miranda are with the De-partamento de Electronica, Universidad Federico Santa María, Valparaíso,Chile (e-mail: [email protected]; [email protected]; [email protected];[email protected]).

Digital Object Identifier 10.1109/TIA.2004.834049

fourth-order system is replaced by two independent first-ordersystems, one for each current component.

This paper uses space vector state variables to model the in-duction motor. A rigorous approach enables a root locus baseddesign of the current control system and facilitates the defini-tion of new control structures. A comparison is made betweenconventional and the new complex current controllers. The im-provements are documented by experiments.

II. MACHINE MODELS

A. Scalar Versus Complex Machine Model

The complex state variable approach for the dynamic anal-ysis of ac machines was presented in two earlier publications,[3] and [4]. It was demonstrated that the complex eigenvalues ofthe system represent dynamically independent components oftransient energy. The definition of transient energy of an ac ma-chine is given in [4]. Not considering the mechanical inertia, thetransient energy of an induction motor is stored in its magneticfields. There are only two dynamically independent entities ofmagnetic energy. Physically, these relate to the main flux andthe leakage fluxes, respectively. Given the predominantly sinu-soidal distribution in space of the magnetic fields, each entity ischaracterized by an amount of energy and its spatial orientation,being represented by the amplitude of a sinusoidal flux densitydistribution and its phase angle. It appears stringent, therefore,to use space vectors as the state variables to represent the totaltransient energy. The electromagnetic subsystem of an induc-tion machine is thus modeled as a second-order system in termsof two complex state variables [3].

Subsequent linear transformations permit representing themain flux and the leakage fluxes by the respective space vectorsof the stator flux linkage and the rotor flux linkage, or the statorcurrent and the rotor flux linkage, or other appropriate spacevectors.

Modeling of ac machines in terms of complex space vec-tors has been previously reported in the literature. Many authorsconsider the complex vector representation by a second-ordermodel equivalent to the scalar representation of a fourth-ordermodel, arguing that both models give the same results [5]. It isindeed a very straightforward mathematical operation that con-verts a second-order complex vector model of the machine to thescalar fourth-order model, and vice versa. Although apparentlymathematically correct, the operation changes the physical sig-nificance of the system eigenvalues. This is a strong argumentagainst the equivalence of the two models. It is in fact only the

0093-9994/04$20.00 © 2004 IEEE

HOLTZ et al.: DESIGN OF FAST AND ROBUST CURRENT REGULATORS FOR HIGH-POWER DRIVES 1389

complex state variable model that permits a physical interpreta-tion of transient machine behavior, as its two eigenvalues referdirectly to the two entities of transient energy inside the machine[4]. Against this, the fourth-order scalar model must be con-sidered an approximation. It replaces the spatially distributedmagnetic energy by the energy of concentrated inductors. Theseform the elements of coupled, lumped-parameter circuits [1].The response to a transient excitation is time-dependent oscilla-tions, while the original dynamic process is commanded by theinteraction between distributed magnetic energies which resultsin their spatial displacement.

To understand the difference between the scalar model andthe complex state-variable model, a more rigorous definition isneeded of a space vector than that originally given by Kovácsand Rácz [6]. Neglecting space harmonics, the flux density dis-tribution in the air gap of an ac machine is sinusoidal. Accordingto the original concept, such sinusoidal distribution is uniquelydescribed by a space vector. It was found convenient to usethe flux linkages of the distributed machine windings for ma-chine analysis instead of the distributed flux densities, whichis equivalent. Current and voltage space vectors were subse-quently defined with reference to the respective phase currentsand phase voltages, thus introducing scalar circuit parameters.The two-axis-transformation [1] eliminates the zero-sequencecomponent of a three phase system, representing the three phasecurrents by their two orthogonal - and -components. How-ever, two scalar components cannot be considered an equiva-lent to a complex space vector. The use of scalar components asstate variables entails a division in two portions of the contin-uous spatial distributions that represent the transient state of thesystem only as a whole.

Following from this, a space vector is not just a complex vari-able which could be formally separated in two independent or-thogonal components; instead, it is an entity of its own class,which describes a continuous, sinusoidal distribution in space.It is the total energy assigned to such distribution that representsthe transient system state; this energy is inherently indivisible.

To elaborate on this, a current shall be considered that flows inonedistributedphasewinding,while theotherphasewindingsaredeenergized. Since space harmonics are neglected, the windingconductors are necessarily arranged in a spatially sinusoidaldensity pattern. Hence the phase current under considerationgenerates a sinusoidal current density distribution, thus defininga current space vector of determined magnitude and phase angle.The phase angle is fixed in space as the winding axis is fixed.The resulting magnetic field represents the associated energy.

Expanding on this, there is no basic difference when all phasecurrents in a polyphase winding are nonzero. The current den-sity distributions of the individual phase currents superimposeto form a resulting distribution, which is again spatially sinu-soidal; it is described by the space vector of the polyphase cur-rent system. The resulting magnetic field establishes a spatialdistribution of magnetic energy. The magnitude and spatial ori-entation of this distribution can assume any values, dependingon the respective magnitudes of the phase currents.

Also, voltage vectors are representatives of sinusoidal distri-butions. The resistive voltage drop in a distributed winding ar-rangement of sinusoidal winding density exhibits necessarily a

Fig. 1. Signal flow graph of an induction motor.

sinusoidal variation in space. The same is true for the inducedvoltage, which is also distributed in space; it is proportional tothe local winding density.

B. Complex State Variable Model

The spatial information is accurately conserved when thecomplex state variable approach is followed. The state equa-tions of the electromagnetic subsystem of an induction motorare written as

(1a)

(1b)

where the normalized space vectors of the stator current andof the rotor flux linkage are the two state variables of a

second-order system. is the mechanical angular velocity ofthe motor shaft. The parameters in (1) are

, and , where is thetotal leakage coefficient. The angular velocity of the referenceframe is denoted by a general value . It is assumed that themechanical time constant of the motor is much larger than thetransient time constants of the electromagnetic subsystem, andhence const. is a valid approximation. Note that time isalso normalized: , where is the nominal statorfrequency.

A visual representation of (1) is the signal flow graph Fig. 1,where is the rotor induced voltage in the stator winding. Thesignificance of complex signal flow graphs is detailed in [3].

When designing the current control system, is consid-ered as the input, and is the output variable of the motor.The transfer function of the induction motor is derived from(1) choosing synchronous coordinates, as (2), shownat the bottom of the next page, where and

are the respective Laplace transforms,is the electrical rotor frequency, and

is a constant machine parameter. in boldface notation un-derlines the complex nature of this transfer function. This prop-erty is owed to the existence of complex coefficients.


Fig. 2. Root loci of the induction motor, ! = 0 . . . 1 par, stationarycoordinates; lower curve; root loci transformed to synchronuos coordinates.

As the dynamics of the load are not modeled, is as-sumed throughout. The solution of the characteristic equationof (2) defines the locations in the complex plane of the twosingle-complex poles and , and the numerator con-tributes a zero . These are plotted in Fig. 2 in stationary co-ordinates, , as functions of the angular velocity ofthe rotor. The zeros of the transfer function (2) are marked bycircles.

A physical interpretation given in [4] assigns a particular dis-tribution in space of transient magnetic energy to each pole.The rate of energy decay is determined by the respective realpart. The imaginary part defines the direction and angular ve-locity at which the magnetic energy displaces during a transientprocess with respect to the chosen reference frame. This defini-tion of the angular velocity is unique as eigenvalues of complexstate variables have only one imaginary part. Contrasting to this,the eigenvalues of the scalar machine model result as conjugatecomplex pairs that do not define a particular direction of rota-tion. This is one of the deficiencies of the scalar model.

A transformation of the root loci to synchronous coordinatescan be done by simply substracting the respective -valuesfrom the imaginary component of all roots as indicated in thelower portion of Fig. 2, and exemplified by arrows. The resultingcurves are the root loci in synchronous coordinates, .The same result is obtained by letting in (1) and sub-sequently computing the system eigenvalues.

The graphs demonstrate that the transient magnetic fields,characterized by the poles and , rotate in a posi-tive direction in the stationary reference frame, and in a nega-tive direction in synchronous coordinates [4]. The constant zero

in stator coordinates gets expanded to the trajec-tory when the system is viewed in syn-chronous coordinates.

Fig. 3. Cross coupling between the current components in aproportional-plus-integral (PI)-controlled induction motor drive; operation atrated speed and 500-Hz switching frequency (simulation).

C. Complex Transfer Functions

Having introduced complex state variables to analyze the dy-namics of the induction motor and its control, the pertainingtransfer functions have complex coefficients and, hence, are alsocomplex in general. This can be easily seen from the transferfunction (2) of the induction motor.

The complex transfer function of the closed-loop current con-trol system can be written as

(3)

where and are the respective Laplace transforms ofand . The transfer function (3) shall be used to assess

the dynamic performance of the current control system.In a transient process, the adjustment of the current space

vector to a change of its reference value is retarded by variousdelays. These are caused by the eigenbehavior of the machine,and by the delay of the inverter and the time-discrete samplingof the digital control. It is important that the trajectory of the cur-rent space vector gets adjusted such that both the commandedtorque and magnetic excitation of the machine are established atminimum time delay. Contra-productive to this aim is the com-plex nature of the transfer function (3). It causes the adverseeffect that the output vector of the system moves in a differentspatial direction than the exciting input vector. As an example,Fig. 3 shows how the current control can react on a commandedchange of only the imaginary current component . While isexpected to maintain its commanded value, it gets heavily dis-turbed; it temporarily even reverses its polarity. The effect isamplified by a low value of switching frequency which is 500Hz in this example.

A figure of merit to evaluate such undesired cross couplingbetween the current components is developed next.

(2)


Fig. 4. Cross-frequency response of the induction motor; ! = par.

D. Cross-Frequency Response

It is easily seen from (3) that the cross coupling is formallyintroduced by the complex factor . In general, it is the imagi-nary component of a complex transfer function thatdetermines the amount of cross coupling of a dynamic system

. To obtain the percentage of cross coupling,shall be referred to the term that defines the directthroughput of the input signal. Of particular interest is the fre-quency response, , which leads to the definition of thenormalized frequency response of cross coupling

(4)

as a figure of merit to express the amount of cross coupling.Applying (4) to the transfer function (2) yields the cross-fre-

quency response of an induction motor. It is displayed in Fig. 4for different mechanical angular rotor velocities .

These characteristics can be explained by examining the rootloci of the motor, Fig. 2. A particular situation is , whereboth roots are real valued since . Since also thezero of (2) is a real-valued constant, the transfer function hasno imaginary component at . This indicates that nocross coupling exists. Such situation does not reflect in Fig. 4since zero values cannot be displayed in a logarithmic scale.As the velocity of rotation increases, both roots move intothe third quadrant of the complex plane; their imaginary partsincrease in magnitude. This lets the imaginary component of thecross-frequency response (2) increase, and the magnitude of thecross-frequency response (4), at given frequency of excitation,rises in proportion to .

The variation with frequency of these curves is explainedlooking at the poles and zeros of (2), the locations of whichare shown in Fig. 2. The magnitude of is constantat low frequencies and changes to a 20-dB/decade decline at

. The slope increases slightly around ,while readjusting to dB/decade at higher frequency. Thisis because which almost cancels their effect. The crossfrequency response is then dominated by .

It is an adverse property of this system that the cross-coupling gain is high over most of the frequency range, peakingup to about 34 dB at low excitation frequency and rated speed,

.

Fig. 5. Pl current control system; simplified machine model.

The problem persists, in principle, when the machine is oper-ated at closed-loop current control. Existing current controllersexhibit real-valued roots and zeros and, thus, do not satisfacto-rily manipulate the single-complex eigenvalues of the machineand the inverter. It is only by virtue of the zero steady-state errorproperty of PI controllers that the existing cross coupling getscompensated when the transients have died out. It depends onthe bandwidth of the current control system how fast the tran-sient error is eliminated. The dynamic error is tolerable whenthe control bandwidth is much higher than the eigenfrequenciesof the machine. The critical margin, however, narrows:

• when the control bandwidth reduces, as in high-power in-verters that operate at low switching frequency; or

• when the machine eigenfrequencies increase, which hap-pens at higher speed in the field-weakening range.

Both conditions tend to increase the undesired cross coupling.State-of-the-art current control techniques [7] cannot handle

this problem in a satisfying manner. Before developing an im-proved strategy, the existing current control methods will beevaluated against the performance requirements.

III. METHODS OF CURRENT CONTROL

A. Hysteresis Current Control

A hysteresis controller instantaneously applies high ampli-tude voltages to the machine terminals whenever a preset errormargin in space in exceeded, thus forcing the stator currentvector to return close to the neighborhood of its reference vector.The high stator voltages completely override the machine dy-namics which makes hysteresis current control an extremely ro-bust approach. Cross coupling is also eliminated. The drawbackof this method is the high harmonic content of the machine cur-rents which renders hysteresis control inapplicable at mediumand low switching frequency [8]. Hence, it is mostly preferredto use linear current controllers and to generate the switchingsequence by a pulsewidth modulator.

B. PI Current Controller With Real-Valued Time Constant

The most common implementation uses Pl current controllersin synchronous coordinates for the two current components infield coordinates, and . The complex notation merges the twocontrollers into a single controller having the transfer function

(5)

where the controller time constants are real valued,is a coefficient, and is the open-loop gain. The complex signalflow graph of the PI controller is shown in the left of Fig. 5.


Fig. 6. Signal flow graph of the current control system.

A first-order delay models the PWM inverter and the timedelays that result from processing the signals as sampled data.The sampling delay is commonly represented by

(6)

where is the delay time, is the sampling fre-quency of the microprocessor, and is the switching frequencyof the inverter.

Neglecting the inverter and sampling delay [5], [9] is onlypermitted if the switching frequency is much higher than thecharacteristic eigenfrequencies of the motor.

The design of the controller is conventionally based on a sim-plified open-loop transfer function. It is obtained from Fig. 5with reference to (5) and (6)

(7)

The controller time constant in (7) is determined with refer-ence to the per-phase equivalent circuit of the machine, in whichthe stator current dynamics are dominated by the transient timeconstant . Such approximation neglects the imaginary parts ofthe coefficients in (2). Accordingly, all cross-coupling links inFig. 1 disappear in Fig. 5. The PI controllers for the respectivecurrent components and then operate independently fromeach other. Their time constants are selected as , whichmakes the sampling frequency determine the controllerbandwidth. The open-loop gain is set for a given damping

of the closed-loop system. Choosing yields.

The performance of the simplified system is analyzed usingthe accurate machine model Fig. 1. Synchronous coordinatesare selected, . The resulting signal flow graph of thecurrent control system is shown in Fig. 6.

The delays introduced by the PWM inverter and the time-discrete sampling are approximated by a first-order system

(8)

where is the reference voltage vector generated by the currentcontroller. The variables are marked by the superscript asbeing referred to in stationary coordinates.

Equation (8) is multiplied by the unity vector rotatorto effect a transformation to synchronous coordi-

nates

(9)

Space vectors referred to in synchronous coordinates are notspecifically marked in the following.

A transformation of (9) to the frequency domain yields thetransfer function of the inverter and sampling delay

(10)

where and are the re-spective Laplace transforms of the input and the output signal.Note that is a transfer function having complex coeffi-cients.

The open-loop transfer function of the current control systemis obtained from (2) and (10) as (11), shown at the bottom ofthe page, where is the transfer function (5) of the cur-rent controller. is of fourth order in terms of complexstate variables. With a view toward high power applications, theswitching frequency is chosen as Hz which corre-sponds to a normalized time constant . The time con-stant of the current controller is again chosen as ,which positions the zero in Fig. 7. This plot shows the lo-cations of the four poles and two zeros in the base speed range.Only at zero speed does the machine pole get canceled by thenumerator term of the controller (zero ), although not exactly.

Fig. 8 shows the root loci of the closed-loop system. As perthe design, the roots and are critically damped at zerospeed, being located symmetrically at 45 with respect to thenegative real axis. The situation is much different at nonzerospeed. The complex nature of the state variables is then re-flected in asymmetric pole locations with respect to the realaxis. Pole positions having unbalanced imaginary parts indicate

(11)


Fig. 7. Root loci of the open-loop system, Pl control, synchronous coordinates;par: ! = 0 . . . 1.

Fig. 8. Root loci of the closed-loop system, Pl control, synchronouscoordinates; f = 500 Hz, ! = 0 . . . 1 parameter.

that cross coupling between the current components exists. Thecross-frequency response function (4) then becomes nonzero,which makes the current space vector deviate from its com-manded trajectory as in Fig. 3. A balance between the imaginaryparts of the single-complex poles in Fig. 8 exists only atwhere form what looks like a conjugate complex pair,and and are real valued.

The particular situation is different from what isnormally known as a conjugate complex pair

, being roots of a characteristic equation with realcoefficients. Such roots form a conjugate complex pair.Against this, each of the roots is single-complex. Only at

do these roots happen to have equal imaginary parts ofopposed sign for the system under consideration. They do notform a pair in the sense that do.

The closed-loop cross-coupling frequency response(Fig. 9) shows reduced gain at frequencies below 5

Hz as compared with the behavior of the uncontrolled machine,Fig. 4. On the abscissa, corresponds tosince is normalized by .

Experimental results were obtained from a 30-kW inductionmotor drive operating at the relatively high switching frequencyof 1100 Hz. The machine was run in a current-controlled mode

Fig. 9. Cross-frequency response of the closed-loop system, Pl controller.

Fig. 10. Recorded step response of the current control system, Pl controller,f = 1100 Hz.

at nominal speed and nominal excitation. The load torque wasfirst set to its nominal positive value, and the command wasthen reversed to its nominal negative value. Fig. 10 shows howthe stator current trajectory deviates from its commanded trackas temporarily also reverses. It takes about 10 ms until theundesired cross coupling is compensated.

The controlling microprocessor was programmed to acquirethe respective current values in field coordinates. The data werepassed through D/A converters to be visualized on an oscillo-scope. It is for this reason that time-discrete values are seen inFig. 10 at a sampling rate of kHz.

C. PI Current Controller With Feedforward Compensation

An improved model of the induction motor reflects thefact that cross coupling exists in the stator winding whensynchronous coordinates are used, Fig. 11. A feedforwardcompensation signal is added to the PI controller. Thevoltage is intended to cancel the internal, motioninduced voltage in the stator winding. Notethat . The method works well at high switchingfrequency where the inverter and sampling delay is negligible.The compensation signal is retarded at low switching frequencyby a large inverter and sampling delay ; the compensation isthen erroneous and cross coupling persists. Another source ofuncompensated cross coupling is the voltage that the rotorinduces in the stator winding, Fig. 11.

The performance of feedforward compensation is studiedby inserting the compensated PI controller of Fig. 11 into thecurrent control system Fig. 6. The resulting closed-loop


Fig. 11. Pl current control with feedforward compensation.

Fig. 12. Root loci using Pl control with feedforward compensation,synchronous coordinates; f = 500 Hz, ! = 0 . . . 2 parameter.

transfer function is shown in (12) at the bottom of thepage. The root locus plot Fig. 12 was computed for 500-Hzswitching frequency. There is still a clear tendency of the threesingle-complex roots to show unbalanced imaginary parts asthe angular mechanical velocity increases. Moreover, thesystem gets unstable at higher speed. The reason is that thefeedforward decoupling is counteracted by the phase shiftof the inverter at higher frequencies. The performance is notimproved as compared with that of the PI-controlled system,nor is the cross-coupling response at nominal speed Fig. 13much better than that of Fig. 9.

The oscillograms taken experimentally at 1100-Hz switchingfrequency confirm these results. The current trajectory in Fig. 14still deviates from the desired path. An added problem is theinherent parameter sensitivity of feedforward control. A 20%error of the leakage inductance leads to the much higherdeviation shown in Fig. 15.

Fig. 13. Cross-frequency response of the closed-loop system, Pl controllerwith feedforward compensation.

Fig. 14. Recorded step response of the current control system, Pl control withfeedforward compensation, synchronous coordinates; f = 1100 Hz.

Fig. 15. Recorded step response as in Fig. 14, 20% parameter error.

(12)


Fig. 16. Current control system using a controller with single-complex poles. The induction motor is represented in the right.

D. Current Controller With Single-Complex Poles

The foregoing analysis of established current control schemeshas served to provide a better understanding of the dynamicproperties of ac machines. This is owed to the use complex statevariable models. The root loci of the resulting single-complexsystem eigenvalues have been introduced as a valuable tool inthis respect.

This technique is now expanded to create a novel class ofcontrollers. Such controller is constructed using single-complexpoles so as to better correspond to the unique dynamic propertiesof ac machines.

The components of the current control system are modeledwithout approximation. Referring to the signal flow graphFig. 6, an open-loop transfer function is derived from (2) and(10), considering no-load, , and

(13)

where represents the controller to be designed. Its transferfunction is constructed such that the single-complex eigenvaluesof the plant are displaced onto the real axis of the complex plane.The current controller

(14)

meets this requirement. The choice of the time constants

(15)

is appropriate. is a coefficient and is the open-loopgain.

The visualization of (14) in the left of the signal flow graphFig. 16 illustrates how the complex current controller mirrorsthe system roots into its poles. The original signal flow graph ofthe induction motor Fig. 1 appears in the right of Fig. 16. A rear-ranged layout serves for a better understanding of the elementsof the complex controller in the left. The structure in the lowerright of the current controller box in Fig. 16 compensates the

Fig. 17. Roots and poles of the plant and the complex current controller,synchronous coordinates; f = 500 Hz, ! = 0 . . . 1 parameter.

complex inverter and sampling delay . It generates the chainof zeros shown in the left of Fig. 17 at as a func-tion of , thus replacing the chain of poles by a single pole at

on the real axis.The portion in the center aims at compensating the eigen-

behavior of the stator. Its contributions are a chain of zeros atas function of , and an added pole in the

origin which serves as an error integrator. The isolated transferfunction of this portion is

(16)

An additional signal from the upper structure of the complexcontroller models the rotor, and in combination with (16), the dy-namic interaction between the stator and the rotor of the machine.It adds a chain of zeros as function of , and apole at . In its interaction with (16), it displaces the zerosof (16) and its own zeros to be exactly located on the two typ-ical single-complex eigenvalues (Fig. 2) of the induction motor.These are then seen eliminated in the right of the root locus plot,Fig. 17. The added pole eliminates the zero of (13).

With all single-complex zeros in the third quadrant of the rootlocus plane being cancelled, the open-loop transfer function re-sulting from (13)–(15)

(17)


Fig. 18. Roots of the open-loop transfer function, complex current controller.

Fig. 19. Recorded step response of the current control system, complex currentcontroller, f = 1100 Hz.

has no more complex coefficients; cross coupling is, thus, elim-inated. The location of the two real roots of (17) is shown inFig. 18.

The closed-loop system is designed for critical dampingwhich determines the open-loop gain .

The cross-frequency response function assumes minus infinitydB for all values of .

The step response (Fig. 19) obtained from the experimentalsetup shows the current trajectory moving on the shortest trajec-tory between the two reference points.

IV. SUMMARY

State-of-the-art current controllers perform unsatisfactorily ifapplied to PWM-controlled voltage-source inverters operated atvery low switching frequency. The reason is that the dynamicsof the drive motor and the inverter are inaccurately modeled.The penalty is a high degree of cross coupling between thetorque-building and the field-generating current component ofa vector-controlled drive. The common remedy of feedforwardcompensation of the stator cross coupling gives little improve-ment at very low switching frequency.

Modeling the induction motor and the inverter in terms ofcomplex state variables provides a better insight in the dynamicsof controlled ac drives. The roots and poles of the respectivetransfer functions then have the property of being single-com-plex; the root loci become physically interpretable. It is shownthat the sources of cross coupling are reflected in the unbalancedimaginary parts of three single-complex system roots. The com-plete elimination of cross coupling is enabled by a novel typeof current controller having three single-complex zeros. Thetheory is confirmed by experimental results.

REFERENCES

[1] P. C. Krause, O. Wasynczuk, and S. D. Sudhoff, Analysis of ElectricMachinery. New York: IEEE Press, 1995.

[2] W. Leonhard, Control of Electrical Drives. Stuttgart, Germany:Teubner, 1973.

[3] J. Holtz, “The representation of AC machine dynamics by complexsignal flow graphs,” IEEE Trans. Ind. Electron., vol. 42, pp. 263–271,June 1995.

[4] , “On the spatial propagation of transient magnetic fields in ACmachines,” IEEE Trans. Ind. Applicat., vol. 32, pp. 927–937, July/Aug.1996.

[5] F. Briz, M. W. Degener, and R. D. Lorenz, “Analysis and design of cur-rent regulators using complex vectors,” IIEEE Trans. Ind. Applicat., vol.32, pp. 817–825, May/June 2000.

[6] P. K. Kovács and E. Rácz, Transient Phenomena in Electrical Ma-chines. Amsterdam, The Netherlands: Elsevier, 1984.

[7] M. P. Kazmierkowski and L. Malesani, “Current control techniques forthree-phase voltage-source PWM converters: A survey,” IEEE Trans.Ind. Electron., vol. 45, pp. 691–703, Oct. 1998.

[8] J. Holtz, “Pulsewidth modulation for electronic power converters,” Proc.IEEE, vol. 82, pp. 1194–1214, Aug. 1994.

[9] F. Britz, M. W. Degener, and R. D. Lorenz, “Dynamic analysis of cur-rent regulators for AC motors using complex vectors,” IEEE Trans. Ind.Applicat., vol. 35, pp. 1424–1432, Nov./Dec. 1999.

Joachim Holtz (M’87–SM’88–F’93) graduated in1967 and received the Ph.D. degree in 1969 from theTechnical University Braunschweig, Braunschweig,Germany.

In 1969, he became an Associate Professor and, in1971, he became a Full Professor and Head of theControl Engineering Laboratory, Indian Institute ofTechnology, Madras, India. In 1972, he joined theSiemens Research Laboratories, Erlangen, Germany.From 1976 to 1998, he was a Professor and Head ofthe Electrical Machines and Drives Laboratory, Wup-

pertal University, Wuppertal, Germany. He is currently Professor Emeritus anda Consultant. He has extensively published, among others, ten invited papersin journals and 17 invited conference papers. He has received 11 Prize PaperAwards. He is the coauthor of four books, and holds 30 patents.

Dr. Holtz was the recipient of the IEEE Industrial Electronics Society Dr.Eugene Mittelmann Achievement Award, the IEEE Industry Applications So-ciety Outstanding Achievement Award, the IEEE Power Electronics SocietyWilliam E. Newell Field Award, the IEEE Third Millenium Medal, and the IEEELamme Gold Medal. He has earned ten IEEE Prize Paper Awards. He is PastEditor-in-Chief of the IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, aDistinguished Lecturer of the IEEE IndustryApplications and IEEE IndustrialElectronics Societies, a Senior AdCom Member of the IEEE Industrial Elec-tronics Society, Chair, IEEE Industrial Electronics Society Fellow Committee,and member of the Static Power Converter Committee of the IEEE Industry Ap-plications Society.

Juntao Quan (M’04) was born in Jiangxi, China,in 1964. He received the B.Eng. degree fromJiangxi Polytechnic College, Nanchang University,Nanchang, China, in 1983, the M.Eng. degreefrom Northeast-Heavy Mechanic Institute, YanshanUniversity, Qinhuangdao, China, in 1989, and thePh.D. degree from Wuppertal University, Wuppertal,Germany, in 2002, all in electrical engineering.

He was an Assistant Electrical Engineer for threeyears at the Nanchang Bus Factory, Nanchang, China.From 1989 to 1994, he was a Lecturer at Yanshan

University. During this time, he also worked on various projects for applicationsof power electronics. In 1995, he joined the Electrical Machines and Drives Lab-oratory, Wuppertal University, where he worked and studied toward the Ph.D.degree. From 2000 to 2003, he was an R&D Engineer in the Danaher MotionGroup, Kollmorgen-Seidel, Düsseldorf, Germany. He is presently with Rock-well Automation AG, Dierikon, Switzerland. His main interests are in the areasof adjustable-speed drives, microprocessor-embedded real-time control, powerelectronics applications, and advanced motion control.


Jorge Pontt (M’00) received the Engineer andMaster degrees in electrical engineering fromthe Universidad Técnica Federico Santa María(UTFSM), Valparaíso, Chile, in 1977.

Since 1977, he has been a Professor in the Depart-ment of Electrical Engineering and Department ofElectronic Engineering, UTFSM. He is the coauthorof the software Harmonix used in harmonic studiesin electrical systems. He has authored more than 60international refereed journal and conference papers.He is a Consultant to the mining industry, in partic-

ular, in the design and application of power electronics, drives, instrumentationsystems, and power quality issues, with management of more than 80 consultingand R&D projects. He has had scientific stays at the Technische HochschuleDarmstadt (1979–1980), University of Wuppertal (1990), and University ofKarlsruhe (2000–2001), all in Germany. He is currently Director of the Centerfor Semiautogenous Grinding and Electrical Drives at the UTFSM.

José Rodríguez (M’81–SM’94) received the Engi-neer degree from the Universidad Técnica FedericoSanta María, Valparaiso, Chile, in 1977, and theDr.-Ing. degree from the University of Erlangen,Erlangen, Germany, in 1985, both in electricalengineering.

Since 1977, he has been with the UniversityTécnica Federico Santa María, where he is currentlyVice Rector of Academic Affairs and a Professorin the Electronics Engineering Department. Duringhis sabbatical leave in 1996, he was responsible

for the Mining Division of Siemens Corporation in Chile. He has extensiveconsulting experience in the mining industry, especially in the applicationof large drives like cycloconverter-fed synchronous motors for SAG mills,high-power conveyors, controlled drives for shovels, and power quality issues.His research interests are mainly in the areas of power electronics and electricaldrives. Recently, his main research interests have been multilevel invertersand new converter topologies. He has authored or coauthored more than 130refereed journal and conference papers and contributed to one chapter in thePower Electronics Handbook (New York: Academic, 2001).

Patricio Newman was born in Concepción, Chile,in 1979. He is currently working toward the Masterdegree in electronics engineering at the UniversidadTecnica Federico Santa Maria, Valparaiso, Chile.

Since 2002, he has been with the Power ElectronicsResearch Group of the Departamento de Electronica,Universidad Tecnica Federico Santa Maria. His maininterests in power electronics are AFE rectifiers andmultipulse drives.

Hernán Miranda was bom in Valparaiso, Chile, in1979. He is currently working toward the Master de-gree in automatic control at the Universidad TecnicaFederico Santa Maria, Valparaiso, Chile.

Since 2003, he has been with the Power ElectronicsResearch Group of the Departamento de Electronica,Universidad Tecnica Federico Santa Maria.

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