An induction machine model for predicting inverter ... - IEEE Xplore

IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 17, NO. 2, JUNE 2002

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An Induction Machine Model for Predicting Inverter–Machine Interaction Scott D. Sudhoff, Member, IEEE, Dionysios C. Aliprantis, Student Member, IEEE, Brian T. Kuhn, Member, IEEE, and Patrick L. Chapman, Member, IEEE

Abstract—The conventional induction motor model typically used in drive simulations is very inaccurate in predicting machine performance, except perhaps for the fundamental component of the current and the average torque near rated operating conditions. Predictions of current and torque ripple are often in error by a factor of two to five. This work sets forth an induction machine model specifically designed for use with inverter models to study machine–inverter interaction. Key features include stator and rotor leakage saturation as a function of current and magnetizing flux, distributed effects in the rotor circuits, and a highly computationally efficient implementation. The model is considerably more accurate than the traditional model, particularly in its ability to predict switching frequency phenomena. The predictions of the proposed model are compared with those of the standard model and to experimental measurements on a 37-kW induction motor drive. Index Terms—Induction motor drives, induction motors, modeling, simulation, squirrel cage motors.

I. INTRODUCTION

T

HE INACCURACIES of the standard induction motor model [1] have been recognized for some time in the literature. Although it is capable of estimating the fundamental component of the current and the average value of torque near rated operating conditions, the accuracy deteriorates at other operating points. Of particular concern in drive analysis is the fact that it underestimates the amount of current and torque ripple by a factor of two to five. A commonly employed alternative to a lumped-parameter model is finite element analysis (FEA) [2], which is widely used in the design of induction machines but to a lesser extent for drive analysis due to its increased computational burden. model is well suited to drive analysis since it is The readily linked to power converter, switching level (modulation) control, supervisory control, and mechanical system models. Simulations using this class of model execute very rapidly in comparison with time domain FEA models. However, the model has many shortcomings in terms of accuracy—particularly in the case of switching frequency effects such as Manuscript received June 7, 2001. This work was supported by the Office of Naval Research “High Frequency Vibration Analysis of Electric Machinery” under Grant N00014-98-1-01716. S. D. Sudhoff, D. C. Aliprantis, and B. T. Kuhn are with the Department of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907 USA (e-mail: [email protected]). P. L. Chapman is with the Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801 USA (e-mail: [email protected]). Publisher Item Identifier S 0885-8969(02)05413-X.

current and torque ripple. Numerous efforts to address these deficiencies have appeared in the literature. For example, one approach has focused on magnetically coupled stator and rotor circuit models, whereby the winding configuration is explicitly taken into account [3], [4]. It was recently shown in [5] that, by appropriate modification, a bar-by-bar rotor model, assuming a model can be reduced to the standard sinusoidally distributed stator winding. A major research area has been in incorporating main-axis magnetic saturation into the model [6]–[15]. Another area of improvement has been the incorporation of the deep-bar effect specific to squirrel cage rotors [16]–[18], which in effect recognizes the large rotor bars as distributed-parameter systems. In [19]–[21], both deep-bar and main-axis saturation are accounted for. Saturation of leakage inductances is frequently ignored but can have significant impact on performance [22], [23]. An interesting modeling approach is given in [24], whereby an arbitrary nonlinear framework is chosen, and parameters are found from a free acceleration test. However, the model was only experimentally verified against the free acceleration test from which the parameters were extracted. In Table I, the features of the different induction machine models are juxtaposed. Apparently, the integration in a single model of main flux path and leakage path saturation (as a function of both current and magnetizing flux linkage) with distributed circuit effects in the rotor has not been achieved previoulsly. Furthermore, validation of the established models in the switching frequency range has never been addressed. This paper sets forth a new induction machine model that is specifically designed for use in combined inverter–machine analysis and that simultaneously includes both stator and rotor leakage saturation as functions of stator current, rotor current, and - and -axis magnetizing flux linkage; magnetizing saturation as a function of magnetizing flux linkage; and distributed circuit effects in the rotor. Furthermore, this is done in such a way that the model is completely noniterative at each time step, yielding a high degree of computational efficiency. Simulation results compare very favorably with experimental results; the proposed model is not only more accurate than the traditional model in predicting the fundamental component of current and average torque but, more significantly for drive analysis, it accurately predicts inverter induced current and torque ripple. The model has been validated over a large range of operating points and is readily parameterized with laboratory tests or potentially through FEA. The parameterization procedure is discussed in [25]. The model presented is ideal for use as a tool for the design of quiet electric drives because it allows for the interaction

0885-8969/02$17.00 © 2002 IEEE

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TABLE I FEATURES OF INDUCTION MACHINE MODELS

of the supervisory controls, modulation controls, inverter, and machine that dictates the high-frequency drive performance to be accurately and rapidly studied in an integrated analysis.

where

is a speed matrix defined by (8)

II. NOTATION Throughout this work, matrix quantities will appear in bold font or have a comma in the subscript. Vector quantities will be of the following forms: (1) (2) (3) variables without superscripts, such as in (1)–(3), are in an arbitrary frame of reference with position and angular speed . An “ ” superscript of any variable denotes that it is expressed in the rotor reference frame. The electrical rotor position is designated as and electrical rotor speed as . These quantities are multiplied by the mechanical rotor position equal to and mechanical rotor speed , respectively, where is the and variables are related by the transnumber of poles. formation [1]

can be neglected. It is assumed herein that Because of the use of a squirrel cage (which has large conductors compared to the stator), the rotor circuits act in a distributed, rather than lumped parameter fashion. As a result, it is convenient to represent the voltage equations of the rotor circuits as (9) and represent the voltage drop across a linear where and impedance associated with the rotor circuits, and denote the flux linking the - and -axis rotor circuits. In order to avoid the analytically awkward step of transforming a highorder admittance operator to an arbitrary reference frame, the rotor reference frame is used for this part of the model. In the frequency domain, the rotor currents are related to the voltage drop across the linear portion of the impedance as (10) (11)

(4) where

(5) III. VOLTAGE EQUATIONS

where is the Laplace operator. In the proposed model, the linear portion of the rotor impedance is represented as a transfer function rather than the more traditional equivalent circuit. This approach offers a better fit to the admittance for a given order (number of states), and it is just as readily (in fact, more so) incorporated into a time domain simulation. The rotor admittance transfer function (which only corresponds to the linear portion of the rotor impedance) is of order and may be expressed as

In variables, the stator voltage equations may be expressed as (6) , , and denote the stator voltages, stator where currents into the machine, and flux linkages, respectively, and is the stator resistance. In (6) and throughout this work, denotes differentiation with respect to time. Transforming (6) to the arbitrary reference frame yields (7)

(12) In the time domain, (10)–(12) may be expressed as (13) (14) and (15) (16)

SUDHOFF et al.: INDUCTION MACHINE MODEL FOR PREDICTING INVERTER–MACHINE INTERACTION

For implementation purposes, it is convenient to express , and in quasicontroller canonical form as

.. .

.. .

,

In practical terms, the effect of the current arguments is relatively minor, except during conditions in which the currents are significantly greater than rated. However, the leakage inductance is significantly affected by the magnetizing flux and decreases quite rapidly as the flux level is increased [25], [26]. An implication of this on drive design is that reduction of flux level will reduce the switching frequency current and torque ripple. The rotor flux linkage equations are taken to be of the form

(17)

(24)

.. . ..

.

..

205

.

(18)

The leakage flux linkages pressed as

and

may in turn be ex-

(19)

(25)

(20)

where “ ” may be “ ” or “ .” The argument for using this form is directly analogous to the argument used in developing (23).

(21)

V. MAGNETIZING PATH MAGNETICS

It will be useful to note that

The incorporation of distributed effects in the rotor is one of the most important features of the model. In the low-frequency region, this transfer function (12) is important because it governs the effective rotor resistance and how it changes with slip frequency. In the switching frequency region, the transfer function partially takes the place of the impedance of the traditional rotor leakage inductance parameter [another portion will be taken into account by (25), which is set forth in the next section]. As such, it also mathematically captures the reduction in effective rotor leakage inductance with frequency. This dropoff is one of the most important factors in explaining why switching frequency current ripple is much higher than predicted with previous modeling techniques.

It is assumed herein that the magnetizing flux vector is in the same direction as the MMF produced by the magnetizing current (26) is so that the magnitude of the magnetizing flux linkage by the related to the magnitude of the magnetizing current relation (27) is a scalar function that denotes the absolute inverse where magnetizing inductance, and where the magnetizing current and flux linkage magnitudes are defined as

IV. LEAKAGE PATH MAGNETICS

(28)

It is convenient to break the stator flux linkage vectors into leakage and magnetizing terms as

and (29)

(22) is the stator leakage flux, and is the magwhere netizing flux. If axial flux is neglected, then Gauss’ law guaranterm is zero. tees that the The stator leakage flux linkage is a complicated function. Clearly, it depends on the stator current since the leakage flux path utilizes iron in the region of the winding where saturation can occur. It also depends on the rotor current, which serves as an MMF source able to saturate the zigzag leakage flux path. It is reasonable to assume that due to these localized effects, the leakage flux in a given axis is influenced by the stator and rotor currents of the same axis. Moreover, since the permeance of the stator teeth (which are a part of several of the stator leakage flux component paths) is determined by the flux level, it is expected that the stator leakage flux linkage is affected by the magnetizing flux linkage in both axes. This functional dependence may as be incorporated in the leakage inductance (23) where “ ” can be “ ” or “ .”

Incorporating (29) into (26) and differentiating with respect to time yields (30) where

(31) The “ ” subscript denotes incremental value. The incorporation of magnetic saturation is important in order to correctly predict the magnetizing component of the current. is In addition, since the effective incremental inductance much lower than the absolute magnetizing inductance, magnetic saturation also contributes to the underestimation of the current

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ripple in the standard induction machine model, although this is not as important an effect as the leakage saturation and distributed effects of the rotor.

A similar set of expressions can be derived from the rotor equations (9), (24), and (25):

VI. MODEL INTEGRATION It is now appropriate to integrate the various submodels into a unified induction machine model. Their equations are combined in the rotor reference frame. A convenient choice of state variables is the set . In this section, an algorithm is set forth to determine the functional dependence of the state variables and inputs as a differential equation of the form

(38) and

(32) (39) which is the fundamental problem in defining any time domain model. Although the model includes both magnetizing and leakage saturation, this can be done in a completely noniterative fashion. The first step in this procedure is to calculate the rotor, magnetizing, and stator currents as a function of the states. The rotor and are readily calculated in terms of the rotor currents and from (14) and (16), respectively. admittance states and depend on the magneThe magnetizing currents tizing flux states, as can be seen from (29) and (26). Finally, the stator currents are given by

It is necessary to eliminate the dependence of (36)–(39) on , . Using (30) the time derivatives of the stator currents and (33), we get (40) which may be used in conjunction with (36)–(39) to yield

(33) and The next step is to calculate the stator flux linkages using (22) and (23). From (7), their time derivatives are (41) (34) (35) Note that these derivatives are found as an intermediate calcuand are not states. lation—they are not integrated since The goal of the ensuing analysis is the formulation of a linear system of equations with the rotor voltages across the linear and , as well as the time derivatives of the impedances and , taken as unknown. magnetizing flux linkages The differentiation with respect to time of the stator flux linkages yields

(42)

(43) (36)

(37)

(44)


where

(45)

The solution of (49)–(52) yields the time derivatives of two of the state variables (the magnetizing flux linkages and ) and the rotor admittance voltages and . These latter quantities may then be used in conjunction with (13) and (15) to calculate the derivatives of the remaining state variables and . The electromagnetic torque of the motor is given by [27]

(46)

(47)

(48) The dependence of (41)–(44) on the time rate of change of and is readily eliminated using (20) the rotor currents and (21):

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(53) Although this formulation is more complicated than the staninduction motor model, it largely retains the compudard tational efficiency (although it can have more states depending upon the order of the rotor transfer function). In particular, despite the nonlinearities present, the model is completely noniterative at each time step, unlike several other induction motor models that incorporate magnetic saturation. This was accomplished by virtue of an appropriate choice of state variables as well as through the selection of the functional forms for the magnetizing characteristics. VII. EXPERIMENTAL VALIDATION

(49)

(50)

(51)

(52)

The predictions of a standard qd induction motor model [1] and the model set forth herein were compared with measured waveforms for a four-pole, 460-V, 37-kW, 60-Hz induction motor drive. The inverter for this drive was fed from an 800-V dc bus. The modulation strategy used was third harmonic sine-triangle, wherein the amplitude of the third harmonic injection was one sixth of that of the fundamental. The IGBT and diode voltage drops were taken as constants of approximately 1.9 V and 1.3 V, respectively. The delay times from a commanded turn-on of an IGBT to the time it started turning on, and from the commanded turn-off of an IGBT to the time it begins to actually turn off, were 4.35 s and 2.52 s, respectively. The turn-on and turn-off times were 145 ns and 320 ns, respectively. The inverter was modeled using the strategy set forth in [28]. The commanded voltage and frequency were 464 V, l-l (fund. rms), and 60 Hz, respectively. For the study shown, the speed was 1790 r/min. Fig. 1 depicts the performance as predicted by the stanmodel. The parameters were obtained from no-load dard , and blocked-rotor tests and were found to be mH, mH, and . These values reflect the machine in a relatively cold state, and all studies were conducted at a relatively cold state as well. Variables depicted include the -phase inverter current, the -phase current ripple (the -phase current less its fundamental component), the per unit -phase ripple current spectrum (the base current is 67 A), the electromagnetic torque, and the per unit electromagnetic torque ripple spectrum (the base torque is 200 N m). Fig. 2 illustrates the measured performance. Therein, the air-gap torque was obtained from (53), where the variables were calculated by post-processing recorded terminal voltage and current waveforms. Comparing Fig. 2 to Fig. 1, it is evident model drastically underestimates both the that the classical peak current and torque ripple by a factor of approximately

208

Fig. 1. Drive performance predicted by standard model.

1.5–5 over the entire frequency range. In addition, the classical model is in error in its prediction of the average torque by approximately 20 N m. Clearly, the validity of using the standard model to evaluate the power converter/modulator strategy performance or to choose a switching frequency based on the current or torque ripple level requirements is highly questionable. The performance of the machine as predicted by the proposed model is shown in Fig. 3. The induction motor parameters were obtained using the procedure presented in [25] and [26] and are shown in (54)–(57) at the bottom of the next page. Note that a subset of the general form of the leakage inductances was used herein; this is a result of the specific test procedure used and not the model itself. In accordance with earlier comments, this simplification is appropriate, provided that rated


Fig. 2.

Drive performance as measured.

current is not greatly exceeded, which is generally the case in drive applications. From Fig. 3, it can be seen that the proposed model much more accurately predicts the performance of the machine as seen in the laboratory over the entire frequency range depicted. It also accurately predicts the average torque. The most evident differences between the simulated and predicted waveforms are in low-frequency (less than 2 kHz) spatial effects such as the 120-Hz component of the torque, which are not represented in this model. The accuracy of the proposed model in predicting machine performance, both in terms of the fundamental component of the applied waveforms as well as in terms of switching frequency components, make it ideal to use for combined converter, supervisory control, switch level (modulation) control, and machine simulations. Examples of the application of the


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VIII. SUMMARY An induction motor model is set forth herein, which is particularly useful in predicting switching frequency effects. It includes stator and rotor leakage saturation, main-axis saturation, and distributed effects in the rotor circuit. It is more accurate model in every facet of the prediction of than the standard machine performance. Further, its simultaneous incorporation of leakage saturation (using a highly general form), magnetizing saturation, distributed circuit effects in the rotor, and high degree of computational efficiency (obtained by the fact that the model is completely noniterative at each time step) set it apart from other work in this area. The relative computational speed of the model and the fact that it is readily tied to power electronic converter and controls models make it ideal for studying power converter–machine interaction as well as other aspects of drive performance, such as the effectiveness of supervisory controls. ACKNOWLEDGMENT The authors appreciate the guidance and comments of technical monitors L. Peterson, T. Calvert, and R. McConnel. REFERENCES

Fig. 3.

Drive performance predicted by proposed model.

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Scott Sudhoff (M’92) received the B.S. (highest distinction), M.S., and Ph.D. degrees in electrical engineering from Purdue University, West Lafayette, IN, in 1988, 1989, and 1991, respectively. From 1991 to 1993, he served as half-time visiting faculty with Purdue University and as a half-time consultant with P. C. Krause and Associates. From 1993 to 1997, he served as a faculty member at the University of Missouri-Rolla, and in 1997, he joined the faculty of Purdue University. His interests include electric machines, power electronics, and finite-inertia power systems. He has published over 40 papers in these areas.

Dionysios Aliprantis (S’96) received the Diploma in electrical and computer engineering from the National Technical University of Athens, Athens, Greece, in 1999. He is currently pursuing the Ph.D. degree at Purdue University, West Lafayette, IN. His interests include electric machines and power electronics.

Brian Kuhn (M’93) received the B.S. and M.S. degrees in electrical engineering from the University of Missouri-Rolla in 1996 and 1997, respectively. He is presently a Research Engineer with the Electrical and Computer Engineering Department, Purdue University, West Lafayette, IN. His research interests include modeling of electrical machinery and power electronics-based distribution systems.

Patrick L. Chapman (S’94–M’97) received the Ph.D. degree from Purdue University, West Lafayette, IN, in August 2000. He is an Assistant Professor at the University of Illinois at Urbana-Champaign in the Department of Electrical and Computer Engineering. He conducts research in electromechanics, power electronics, integrated power circuits, and power semiconductor devices. He has published a number of papers on these subjects.