The University of Texas at Austin,. Austin, TX ... From this bond graph. state equations were extracted ... rents flow in the sequence: barâend ringâopposite side.
Jongbaeg Kim Michael D. Bryant Mechanical Engineering, The University of Texas at Austin, Austin, TX 78712-1063
Bond Graph Model of a Squirrel Cage Induction Motor With Direct Physical Correspondence An existing bond graph of a squirrel cage induction motor was modified to make the bond graph physically more representative. The intent was to form a one-to-one correspondence between motor components and bond graph elements. Components explicitly represented include the stator coils, the squirrel cage rotor bars, and the magnetic flux routing section. The final bond graph spans electrical, magnetic, and mechanical energy domains, and contains common motor faults. From this bond graph. state equations were extracted and simulations performed. Simulated were the response of healthy motors, and motors with shorted stator coils and broken rotor bars. 关S0022-0434共00兲01203-X兴 Keywords: Induction Motor, Bond Graph, Squirrel Cage, Faults
Introduction A motor has two major sub-systems: a rotating rotor and a nonrotating stator. Widely used squirrel cage induction machines 关1,2兴 exhibit great utility for variable speed systems and are simple, rugged, and inexpensive. Mounted 共in slots兲 on the shaft of a squirrel cage rotor are solid conductive bars that extend axially and connect to end rings. Steel laminations beneath the rotor bars route magnetic flux through the rotor. In large machines, copper alloy rotor bars are driven into the rotor slots and brazed to the end rings. Rotors of up to 50 cm diameter usually have diecast aluminum bars. The core laminations are stacked in a mold, which is then filled with molten aluminum. In this single economical process, the rotor bars, end rings, and cooling fan blades are cast simultaneously 关3兴. Figure 1 depicts a squirrel cage induction motor. A substantial literature modeling induction motors employs Park’s 关4兴 tworeaction theory, which accounts for magneto-mechanical energy transduction via multi-port inductances. From Park’s model, Ghosh and Bhadra 关5兴 formulated the bond graph in Fig. 2. Although this bond graph accurately represents Park’s 关4兴 equations, the bond graph elements in Fig. 2 are only implicitly linked to the real physical components in a motor. In addition, only the electrical and mechanical energy domains are present. Magnetic effects are only implicit in the pair of 2-port inductances, which lack magnetic losses, and flux return effects such as flux leakage. We altered Ghosh and Bhadra’s bond graph to partition and make explicit the electrical, magnetic, and mechanical energy domains; to form a one-to-one correspondence between physical components in the machine, and elements in the bond graph, and to append additional elements to the bond graph to make it more consistent with real induction motors. When energized by an AC supply voltage, the stator coils form a radial magnetic field vector that rotates within the stator, about its central axis. The stator magnetic field passes through the squirrel cage rotor, including the conductive rotor bars. This time varying field induces a voltage over the rotor bars. Resulting bar currents flow in the sequence: bar→end ring→opposite side bar→opposite end ring→original bar. Induced by this time varying current loop is a secondary magnetic field which attempts to align with the stator field. However, because this rotor field was Contributed by the Dynamic Systems and Control Division for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received by the Dynamic Systems and Control Division January 4, 2000. Associate Technical Editor: Jay F. Tu.
induced by the rotating stator field, the rotor field, and hence the rotor, follows the stator field. This is motor action 关6兴. The induction motor speed depends on the speed of the rotating stator field.
The Equivalence of Three-Phase and Two-Phase Systems The real system we will consider is a two pole, ‘‘Y’’ connected three phase squirrel cage induction motor. In references 关5,7,8兴, a multi-phase induction motor was modeled with an equivalent twoaxis representation. Each phase winding generates its own magnetic field, which can be represented as a vector aligned along the axis of the winding. The sum of these phase vectors produces a phasor vector. If the phase vectors vary properly with time, the phasor rotates. A transformation from three phases (a,b,c) to two phases 共␣,兲 was represented in reference 关8兴 in matrix form. If the ‘‘a’’ and ‘‘␣’’ phase windings are coaxial, the induced Magneto Motive Forces 共MMF兲 of the a and ␣ phases of the three and twophase systems are codirectional. By appropriate changes to the two-phase currents, the magnitude of the phasors of the three and two-phase systems can be made equal. Ghosh and Bhadra 关5兴 represented this via transformer elements in the stator section. The two-phase currents were represented in terms of three phases as
Fig. 1 Cross-sectional view of squirrel cage induction motor
Journal of Dynamic Systems, Measurement, and Control Copyright © 2000 by ASME
SEPTEMBER 2000, Vol. 122 Õ 461
Direct Bond Equations
Fig. 2 Ghosh and Bhadra’s †5‡ bond graph of a squirrel cage induction motor
冋 册 冑冑 冋
2 cos 0 i␣ ⫽ i 3 sin 0 ⫽
冋
冑2/ 冑3
cos 2 /3
cos 4 /3
sin 2 /3
sin 4 /3
⫺/ 冑6 / 冑2
0
⫺/ 冑6 ⫺/ 冑2
册冋 册
册冋
ia ib ic
册
s
(1)
冋册 ⫽
冋
R s ⫹L s d/dt
0
L m d/dt
0
0
R s ⫹L s d/dt
0
L m d/dt
L m d/dt
L m r
R r ⫹L r d/dt
L r r
⫺L m r
L m d/dt
⫺L r r
R r ⫹L r d/dt
冋册
冋 册冋 冋 册冋 ␣r
ia ib ic
i ␣s i s ⫻ i ␣r i r
册
(2)
Representation
of
Dynamic
Ghosh and Bhadra 关5兴 represented Eqs. 共1兲–共4兲 in their bond graph, reproduced in Fig. 2. They used modulated gyrators MGY:r 1 ⫽L m i  s , MGY:r 2 ⫽L r i  r , MGY:r 3 ⫽L m i ␣ s , MGY:r 4 ⫽L r i ␣ r , to represent the electro-magnetic torque of Eq. 共3兲; employed transformers TF:m 1 , TF:m 2 , TF:m 3 , TF:m 4 , TF:m 5 with moduli m 1 ⫽ 冑3/2,m 2 ⫽m 3 ⫽⫺ 冑6,m 4 ⫽ 冑2,m 5 ⫽⫺ 冑2 to implement the mathematical transform of Eq. 共1兲; and excited the system with effort sources MSe :V a , MSe :V b , and MSe :V c having sinusoidal voltages with equal amplitudes but 0, /3, and 2/3 phase lags, respectively. Although this correctly programs the governing equations for a three-phase induction motor, it lacks a correspondence between bond graph elements and real system components. Moreover, elements and their constitutive laws involve only electrical and mechanical energy domains. Faults or design parameters relevant to the magnetic domain are only implicit in the mutual inductances, posed as 2-port inertances I: ␣ and I:  with constitutive laws ␣s
Under assumptions of a spatially sinusoidal distribution of MMFs, and ignoring magnetic losses and saturation, Ghosh and Bhadra expressed a symmetric induction motor in an orthogonal stationary reference frame with ␣ and  phases fixed on the stator as V ␣s V s 0 0
Graph
r
⫽
⫽
册冋 册 册冋 册
Ls
L m i ␣s
Lm
Lr
Ls
L m i s
Lm
Lr
i ␣r
i r
(5)
.
In ␣ s ,  s , ␣ r , and  r are flux linkage of the respective windings. In Fig. 2, five integral 共independent兲 causalities exist on inertance energy storage elements, with system state variables ␣ s ,  s , ␣ r ,  r , and h, where h is the rotor angular momentum.
Moving and Adding Bond Graph Elements for Explicit Representation To represent real system elements or components explicitly, certain bond graph elements should be moved, altered, or added. In Fig. 2, ␣ and  phase stator resistance elements, R s ␣ and R s  should be split into three stator coil resistances R sa , R sb , and R sc , to make explicit the resistance of each of the stator coils. This was done without altering the governing equations. The revised bond graph shown in Fig. 3 moved R s ␣ and R s  back through the transformers in front of the phases. To maintain an equivalence between Fig. 2 and Fig. 3, related R sa , R sb , and R sc to R s ␣ and R s  . Since most motors possess symmetry between phases, let R sa ⫽R sb ⫽R sc ⫽R, and R s ␣ ⫽R s  ⫽R s . For the bond
Equation 共2兲 relates stator voltages to stator and rotor currents. In addition, needed is the electro-magnetic motor torque for a P-pole machine, expressed as P T e ⫽ 关 i ␣ r 共 L m i  s ⫹L r i  r 兲 ⫺i  r 共 L m i ␣ s ⫹L r i ␣ r 兲兴 2
(3)
This motor torque is balanced against other torques via T e ⫽J
dm ⫹c m ⫹T L dt
(4)
Terms on the right side of Eq. 共4兲 represent rotor inertial torque, shaft/bearing damping torque, and load torque, respectively. In Eqs. 共2兲–共4兲, V ␣ s and V  s are ␣ and  axis stator voltages; i ␣ s and i  s are ␣ and  axis stator currents; i ␣ r and i  r are ␣ and  axis rotor currents; R s and R r are stator and rotor resistances; L s , L m , and L r are stator self-inductance, mutual inductance, and rotor self-inductance; T e and T L are electro-magnetic torque and mechanical load torque; J is the moment of inertia of the rotor, c is the viscous resistance coefficient; r and m are electrical and mechanical angular velocities of the rotor; and P is number of pole pairs. 462 Õ Vol. 122, SEPTEMBER 2000
Fig. 3 The stator resistances in Fig. 2 were redistributed to each of the stator coils, and modulated GY elements were simplified
Transactions of the ASME
graphs of Figs. 2 and 3 to be equivalent, the voltages 共efforts兲 on the 2-port inertances on the stator sides must be equal. The causality in both Figs. 2 and 3 asserts that the voltages to the 2-port inertances arise from the neighboring 1-junctions. Summing voltages from other bonds to these 1-junctions, and equating these respective voltages between Fig. 2 and Fig. 3 gives
再 冉 冊冎 再 冉 冊冎
Vb Vc 1 i ␣s i s ⫹ ⫺R s i  s ⫽ V b ⫺R ⫹ m4 m5 m4 m2 m4 ⫹
i ␣s i s 1 V c ⫺R ⫹ m5 m3 m5
(6)
再 冉 冊冎 再 冉 冊冎 再 冉 冊冎
1 i ␣s Va Vb Vc ⫹ ⫹ ⫺R s i ␣ s ⫽ V ⫺R m1 m2 m3 m1 a m1 ⫹
i ␣s i s 1 V b ⫺R ⫹ m2 m2 m4
⫹
i ␣s i s 1 V c ⫺R ⫹ m3 m3 m5
(7)
By solving for i  s /i ␣ s , we obtain equations in terms of resistances and transformer moduli
m 21 m 22 m 23 m 4 m 5 R s ⫺ 共 m 22 m 23 m 4 m 5 ⫹m 21 m 23 m 4 m 5 ⫹m 21 m 22 m 4 m 5 兲 R 共 m 3 m 4 m 25 ⫹m 2 m 4 m 25 兲 R i s ⫽ ⫽ 2 2 2 2 i ␣ s m 2 m 3 m 4 m 5 R s ⫺ 共 m 2 m 3 m 5 ⫹m 2 m 3 m 4 兲 R 共 m 21 m 2 m 23 m 5 ⫹m 21 m 22 m 3 m 4 兲 R
By replacing the transformer moduli, m 1 – m 5 of the three-phase to two-phase transformation with real numbers, m 1 ⫽ 冑3/2,m 2 ⫽m 3 ⫽⫺ 冑6,m 4 ⫽ 冑2,m 5 ⫽⫺ 冑2, which is given in Eq. 共1兲, we find that R s ⫽R, i.e., R s ␣ ⫽R s  ⫽R sa ⫽R sb ⫽R sc .
Simplified Representation of the Signal and Modulated GY Elements In terms of the 2-port I field of Eq. 共5兲, Eq. 共3兲 can be rewritten as P T e ⫽ 共  r i ␣ r ⫺ ␣ r i  r 兲 2
(9)
From this relation, Fig. 2 can be rearranged into the form of Fig. 3, where the modulated gyrators MGY:r l1 ⫽  r and MGY:r l2 ⫽ ␣ r are modulated by the flux linkages ␣ r and  r of the 2-port inertances.
Squirrel Cage Rotor The number of squirrel cage rotor bars depends on the rotor’s size, and usually, tens of bars 共10–30兲 are in a rotor. In this study, we consider the squirrel cage rotor with five bars 共numbered 1–5兲 depicted in Fig. 4. Shown also is the rotor magnetic field 共dashed line兲, with north poles 共N兲 on top, and south poles 共S兲 beneath, and bar currents. Currents directed out of plane are denoted by a ‘‘•’’, and currents flowing into the plane are denoted by a ‘‘⫻’’. Each end of each rotor bar is attached to a solid end ring. Induced currents flow through each bar, and then into the end rings. With five bars, there exist five different currents in this rotor. At the instant of the rotor position shown in Fig. 4共a兲, the sum of the currents in bars 1, 2, and 5—induced by the rotating magnetic field of the stator—must equal the sum of the currents in bars 3 and 4. Likewise, the sum of bar currents 1 and 5 at the position of
Fig. 4 Squirrel cage rotor with five bars
Journal of Dynamic Systems, Measurement, and Control
(8)
Fig. 4共b兲 must equal the sum of bar currents 2, 3, and 4. In Fig. 4, the thickness of each ⫻ and • shows the relative current magnitude in each bar. To incorporate individual rotor bars into the bond graph, the rotor’s ␣ and  phase currents and voltages should be split into separate bar currents and voltages. The a, b, c and ␣,  axes are stationary with respect to the stator, but because the rotor rotates, bar currents must depend on the angular position of the rotor. Using results in Hancock 关8兴, rotor bar currents can be related to the ␣,  phase currents as
冋 再
i rk ⫽m i ␣ r cos ⫹
冎
再
2 共 k⫺1 兲 2 共 k⫺1 兲 ⫹i  r sin ⫹ n n
冎册
(10)
In Eq. 共10兲, i rk represents the current in the kth rotor bar (k ⫽1,2, . . . n), ␣ r and  r are rotor currents from Fig. 2, and magnitude modulus m depends on the total number of bars, n. For n⫽5 bars, we will have currents i r1 to i r5 . Accordingly, rotor bars can be incorporated into the bond graph of Fig. 3 via -modulated transformers. Figure 5 shows the transformation of ␣ and  phase currents into individual rotor bar currents, where the transformer moduli are
再
mr k ⫽m cos ⫹
2 共 k⫺1 兲 n
冎
k⫽1,2, . . . ,n
(11)
Fig. 5 Transformation of ␣ and  phase currents into rotor bar currents, and inclusion of rotor bar action
SEPTEMBER 2000, Vol. 122 Õ 463
再
mr k⫹n ⫽m sin ⫹
2 共 k⫺1 兲 n
冎
with n⫽5.
(12)
(13)  r ⫹ 共 mr 6 兲 6 ⫹ 共 mr 7 兲 7 ⫹¯⫹ 共 mr 10兲 10⫽0
冉 冉 冉 冉
2 cos ⫹ 5
i r1 4 i r2 cos ⫹ i r3 ⫽m 5 i r4 6 cos ⫹ i r5 5
In Fig. 5, the battery of 0-junctions on the right-side completes the summation of ␣ and  phase currents demanded by the right side of Eq. 共10兲. The voltages that sum over the two 1 junctions located between the I fields and the MTF’s give rise to
␣ r ⫹ 共 mr 1 兲 1 ⫹ 共 mr 2 兲 2 ⫹¯⫹ 共 mr 5 兲 5 ⫽0
冤
冋册
cos
8 cos ⫹ 5
冋 册
冑2 i ␣r ⫽ i r 冑5
冋
冉 冉
2 5
cos
cos ⫹
sin
2 sin ⫹ 5
冊 冉 冊 冉
cos ⫹
5
T e⫽
兺
k⫽1
5
T k⫽
P 2 k⫽1
兺
冉
⫺ ␣ r sin ⫹
冑2 冑5
冋
冉
2 共 k⫺1 兲 n
 r cos ⫹
2 共 k⫺1 兲 n
冊册
冊
i rk
(17)
2 5
sin ⫹
4 5
sin ⫹
6 5
8 sin ⫹ 5
冥
冊 冊冋 册 冊 冊 i ␣r i r
冋 册
5 1 2 0
0 1
(15)
and the left-inverse of A is AT if m 2 ⫽2/5, i.e., the transformer modulus m has a value which normalizes AT A. For a rotor of n bars, m⫽ 冑2/n. The proof is shown in the Appendix. From Eqs. 共14兲 and 共15兲, the inverse transformation is
4 5
4 sin ⫹ 5
If substituted into the rotor output torque equation 共9兲, the electromagnetic torque becomes
冉 冉 冉 冉
sin ⫹
i.e., irotor⫽Aitwo phase . (14) The two column vectors of the 5⫻2 transformation matrix A form an orthogonal set for any value of rotor angle : the rank of A is 2. For the m⫻n (m⬎n) matrix A having rank n, there exists 关9兴 an n⫻m left-inverse B such that BA⫽In , where In is the identity matrix of order n. In our model AT A⫽m 2
Here the flux linkage 1 , 2 , . . . , 10 associated with rotor bars are located to the right of the MTF’s. To obtain the torque contributed by each bar, Eq. 共10兲 for k⫽1,2, . . . ,5 is rewritten in matrix form as
冊 冊 冊 冊
sin
冊 冉 冊 冉
cos ⫹
6 5
6 sin ⫹ 5
冊 冉 冊 冉
cos ⫹
8 5
8 sin ⫹ 5
冊 冊
册冋 册 i r1 i r2 i r3 i r4 i r5
(16)
tions TF:m—GY:n and GY:n—TF:1/m. Here the gyrator modulus n is the effective number of coil turns; m is the transformer modulus; and P is the permeance of the magnetic circuit element 关H兴. Using the constitutive law of the C element, M ⫽ /P, the two port I elements pertaining to the ␣ and  phases were converted into 2-port C elements that now represent interactions between magnetic flux and magnetomotive force of the stator and rotor. Figure 6 shows the new bond graph with five rotor bars and
These revisions are shown on the far right side of Fig. 5, which includes stator and rotor bar interactions based on Eq. 共17兲. Here the moduli of the kth modulated gyrator is r k⫽
冑冋
冉
冊
冉
2 共 k⫺1 兲 2 共 k⫺1 兲 2  r cos ⫹ ⫺ ␣ r sin ⫹ n n n
冊册
(18) where n⫽5 for Fig. 5. Finally, the electric resistances of the rotor were grouped with each rotor bar in a manner similar to that of the stator resistances.
Modifications for the Magnetic Circuit The bond graph in Fig. 5 models the interaction between stator coils and rotor bars with 2-port I elements—inductances—in the electrical energy domain. An inductance only describes storage of magnetic energy. Neglected are power losses and leakage effects in the magnetic domain, which may be caused by component deterioration. To describe these interactions, we replace all I inductance elements with equivalent combinations of gyrators and C elements, without violating causality. The bond graph representations I:L and GY:n—C: P are equivalent, as are the combina464 Õ Vol. 122, SEPTEMBER 2000
Fig. 6 Bond graph representing stator and rotor bar action in the magnetic circuit
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Table 1 System parameters of a two pole, three-phase squirrel cage induction motor R sa ,R sb ,R sc 关 ⍀ 兴 R r1 ,R r2 , . . . ,R r10关 ⍀ 兴 V a ,V b ,V c 关 V兴 关Hz兴 L s 关 H兴 L r 关 H兴 L m 关 H兴 n ns nr c 关N•s/m兴 J 关kg•m2兴
Stator coil resistance Rotor bar resistance Input voltage amplitude Input voltage frequency Stator inductance Rotor inductance Mutual inductance Number of rotor bars Number of effective stator coil turns Number of effective rotor coil turns 共bar兲 Mechanical resistance Mechanical inertia
the GY-C-GY combination that replaced the 2-port I. The gyrators were then moved through the bond graph to new locations more consistent with motor components. The GY to the left of the 2-port C was moved into the electrical section, where it now represents the action and number of turns of the stator coils. The GY leap-frogged the transformers that were based on equation 共1兲, changing the transformer moduli from m to 1/m. The GY to the right of the 2-port C skipped over a 1-junction, converting that 1-junction into the 0-junction shown in Fig. 6. Similarly, a 0- and 1-junction to the left of the 2-port C in Fig. 5 were converted to a 1- and 0-junction in Fig. 6. In the bond graph of Fig. 6, electrical energy inputs, transformation of energy from electrical domain to magnetic domain, mathematical phase transformations, power interactions between stator and rotor bars in terms of magnetic flux and magneto motive force, and mechanical rotor output are all represented and labeled. In Fig. 6, the two sets of gyrator moduli n s and n r stand for the effective coil turns which relate electrical and magnetic variables of stator and rotor, respectively.
Deriving State Equations State equations were derived from the bond graph of Fig. 6 with n s1 ⫽n s2 ⫽n s3 ⫽n s , R sa ⫽R sb ⫽R sc ⫽R s . In terms of magnetic variables, the state equations are
˙ ␣ s ⫽
number of effective rotor coil turns, the magnetic flux 关Wb兴, M the magnetomotive force 关A兴, P is the Permeance 关H兴, and h the angular momentum 关N•m•s⫽kg•m2/s兴.
Simulation Results Simulations of a squirrel cage induction motor used the bond graph simulation tool, 20-SIM 关10兴. For integration of state equations, a Runge-Kutta fourth-order method was adopted. Values of the system parameters for the simulations are presented in Table 1, some were identical to those used by Ghosh and Bhadra 关5兴. Shown in Figs. 7 and 8 are plots of rotor angular velocity and stator currents versus time. Initially, the rotor velocity rises slowly to a steady state value of about 377 rad/s, while the stator currents oscillate at the input frequency with initial large amplitude. After about 1.5 seconds, the motor reaches steady state. Here the stator currents decrease to a steady value, and rotor velocity oscillations vanish. Figure 7 plots the rotor axis angular velocity versus time, for 230 V, 60 Hz three phase AC voltages applied to the stator coils. These inputs applied to a two pole AC motor yields a theoretical steady state angular velocity of 3600 rpm 共377 rad/s兲. The numerical simulation produced a steady state value very close: the
Va Vb Vc ⫹ ⫹ ⫺R s n r2 l 共 Pr ␣ s ⫺Pm ␣ r 兲 n sm 1 n sm 2 n sm 3
˙  s ⫽
Vb Vc ⫹ ⫺R s n r2 l 共 Pr  s ⫺Pm  r 兲 n sm 4 n sm 5
˙ ␣ r ⫽⫺R r n s2 l 共 ⫺Pm ␣ s ⫹Ps ␣ r 兲 ⫹  r
h m 6J
˙  r ⫽⫺R r n s2 l 共 ⫺Pm  s ⫹Ps  r 兲 ⫹ ␣ r
h m 6J
h˙ ⫽
0.0788 0.0408 230 60 0.0153 0.0159 0.0147 5 100 1 0.15 0.4
Pm 2 m 6 共 Ps Pr ⫺Pm 兲
关 ⫺ ␣ s  r ⫹ ␣ r  s 兴 ⫺c
h m 6J
(19)
where the magnetic state variables are stator and rotor phase fluxes ␣ s ,  s , ␣ r ,  r , and rotor angular momentum h. The constitutive law of the 2-port C element is
冋 册冋 冋 册冋 ␣s
␣r
s
r
⫽ ⫽
册冋 册 册冋 册
Ps
Pm M ␣ s
Pm
Pr
Ps
Pm M  s
Pm
Pr
M ␣r
Fig. 7 Angular velocity of rotor axis and stator currents in stator winding
(20)
M r
In the state equations, l⫽
1 2 L s L r ⫺L m
⫽
1
1
n s2 n r2
2 Ps Pr ⫺Pm
(21)
where the permeances Ps ⫽L s /n s2 , Pm ⫽L m /n s n r , Pr ⫽L r /n r2 are expressed in terms of coil turns and inductances of stator and rotor. Here n s is the number of effective stator coil turns, n r the Journal of Dynamic Systems, Measurement, and Control
Fig. 8 Angular velocity of rotor axis and stator currents in stator windings, at startup
SEPTEMBER 2000, Vol. 122 Õ 465
Fig. 9 Angular velocity of rotor axis, slip, and currents in each rotor bar, at startup
Fig. 10 Angular velocity of rotor axis and 5-currents in each rotor bar, at startup
Fig. 11 Angular velocity of rotor axis and 5-currents in each rotor bar, from startup to steady state
difference was due to the mechanical resistance load. Figure 8 expands the Fig. 7 time scale to show the three stator currents with 120 deg phase difference, during motor start-up. Figures 9–11 show the currents in the five rotor bars and the rotor velocity. Recall there exists a 2/5 phase difference between currents in neighboring bars. This is clearly shown in Fig. 10, which represents the motor at startup. While the 60 Hz frequency of the stator currents generated a magnetic field rotating at constant speed, the frequency of the currents in the rotor bars decreases continuously as the rotor velocity increases. This is related to ‘‘slip’’ in induction motors, the normalized difference between the electrical angular velocity of the air gap MMF established by the stator currents, and the electrical angular velocity of the rotor 关11兴. Slip is defined as s⫽
s⫺ r s
466 Õ Vol. 122, SEPTEMBER 2000
(22)
Fig. 12 Stator currents and rotor velocity of a machine with a broken rotor bar
Fig. 13 Stator current of second phase and rotor velocity of a healthy machine at steady state
where s is the synchronous speed, or the speed of the stator currents, and r is the speed of the rotor. The magnitude and frequency of the currents and voltages of the rotor depends on the relative velocity between the rotating magnetic field and the rotor. In these simulations, this relative velocity maximizes at t⫽0, where the slip is unity. As the rotor velocity increases, the relative velocity and the slip decrease, suggesting that the decrease of amplitude and frequency of rotor bar currents in Fig. 9 are probably due to the decrease of slip, also shown in Fig. 9. If s ⫽ r , slip s⫽0 and no current is induced in the rotor bars, hence no torque. However, the steady state currents of the rotor bars in Fig. 9 are not zero, even though there is no external load, because of the frictional load of the bearing, modeled as the resistance R:c in Fig. 6. If an external load is applied to the motor axis, the slip should increase and therefore the current and voltage in the rotor bars should also increase. Figure 11 shows the currents in the rotor bars during steady state. All simulation results shown above are for a healthy motor. When rotor bars break, currents, velocity, and torque will deviate. Because we have a one-to-one correspondence between bond graph elements and machine components, it is possible to represent broken rotor bars by increasing the rotor bar resistance R r . In modern squirrel cage induction motors, bars and end-rings contact the rotor core. Due to this available current shunt, currents in a broken bar are not zero 关12兴, and the resistance is not infinity. Figure 12 shows the stator currents and rotor velocity for a rotor with rotor bar 3 broken. During startup, the rotor velocity increases and oscillates. These oscillations persist at steady state. With these deviations, the currents in the stator coils also change. For comparison, a corresponding healthy machine simulation is shown in Fig. 13. The changes are more clearly presented in Fig. 14. Figure 15 plots the currents in each rotor bar, with bar 3 assumed broken. From Fig. 15, the induced currents are largest in Transactions of the ASME
Fig. 14 Stator current of second phase and rotor velocity of a machine with a broken rotor bar at steady state
Fig. 15 Angular velocity of rotor axis and 5 currents in each rotor bar when the third bar is broken
Fig. 17 Rotor velocities of healthy and shorted machines
Fig. 18 Rotor torques of healthy and shorted machines
Fig. 19 Rotor bar currents of shorted machine Fig. 16 Torque-time plot of healthy machine and one rotor bar-broken machine
the two rotor bars nearest the broken bar. Figure 16 compares the torque characteristics of the healthy machine and broken bar machine. The rotor torque oscillates in the broken bar machine, even at steady state. During startup, the oscillation of torque is larger in the broken bar machine than the healthy machine. Simulations of an induction motor with a short circuited stator coil are shown in Figs. 17–19. In these simulations, the resistance of the shorted coil decreases, and the coil current, the magnetic fields, and the induced currents in the rotor bars also change. Figure 17 shows a difference in rise time of rotor velocity between the healthy machine and the short-circuited stator coil machine. Figure 18 shows the rotor torque for both healthy and Journal of Dynamic Systems, Measurement, and Control
shorted machines. The overall trend of the torques are similar, but there exists small amplitude and relatively high frequency oscillations in the short-circuited case. These oscillations are also seen in the rotor bar currents of the short circuited machine. Fig. 19, compared with the rotor bar currents of the healthy machine, Fig. 11.
Summary, Discussion, and Conclusions A bond graph model of a squirrel cage induction motor was constructed that exhibited a one-to-one correspondence between the bond graph elements and real system components. This bond graph was based on a prior bond graph by Ghosh and Bhadra 关5兴. Included were stator coil windings for three phases, mathematical transformations to incorporate two reaction theory, magnetic state variables to represent magnetic interactions between stator and SEPTEMBER 2000, Vol. 122 Õ 467
rotor, individual rotor bars and contributions to the total rotor torque and velocity, and mechanical inertias and resistances. The simulations in this article had five rotor bars. Using this model, simulations of a healthy machine were compared to simulations of machines with a broken rotor bar and a shorted stator coil. The degraded machine simulations predicted oscillations in currents and angular velocities, seen in real motors. Most induction motor designs employ three phase excitation of the stator. For a rotor with more bars, the bond graph of Fig. 6 can be easily altered. More rotor bars can be included in Fig. 6 by adding additional pairs of power pathways to the right of the 2-port C’s, such that n power pathways fan out from both ␣ and  rotor phases. For the new value of n, these power pathways must update Eqs. 共11兲 and 共12兲 for moduli mr k for the modulated transformers MTF:mr k , and Eq. 共18兲 for the modulus r k of the modulated gyrators MGY:r k . To update the electromechanical torque in Eq. 共17兲, we must replace the 5 in the upper index of the sum and the square root argument in the denominator with the new value of n.
Acknowledgments The authors would like to thank the National Science Foundation, grant DMI-9713605. Manufacturing Machines and Equipment Program, Ming C. Leu program officer, Division of Design, Manufacture, and Industrial Innovation for support for this work.
Nomenclature
冋
Appendix In this section, for a rotor with n bars, we prove
c ⫽ viscous resistance coefficient h ⫽ angular momentum 关N•m•s⫽kg•m2•s兴 i ␣ s ,i  s ⫽ ␣ and  axis stator currents i ␣ r ,i  r ⫽ ␣ and  axis rotor currents i rk ⫽ current in the kth rotor bar
AT A⫽m
J ⫽ moment of inertia L s ,L m ,L r ⫽ stator self inductance, mutual inductance, and rotor self-inductance m ⫽ magnitude modulus that depends on the total number of bars M ⫽ magneto motive force 关ampere 共A兲兴 m 1 – m 5 ⫽ moduli of transformers for three-phase to two-phase transformation n ⫽ modulus of gyrator 共number of coil turns兲 n s ⫽ number of effective stator coil turn n r ⫽ number of effective rotor coil turn P ⫽ number of pole pairs R s ,R r ⫽ stator and rotor resistances R s ␣ ,R s  ,R sa , R sb ,R sc ,R ⫽ electrical resistances T e ,T L ⫽ electro-magnetic torque and mechanical load torque V a ,V b ,V b ⫽ sinusoidal input voltages V ␣ s ,V  s ⫽ ␣ and  axis stator voltages ⫽ magnetic flux 关Weber 共Wb兲兴 ⫽ flux linkage r , m ⫽ electrical and mechanical angular velocities of the rotor P ⫽ permeance of circuit element 关Henry 共H兲兴
冉 冉
cos
2 cos ⫹ n
sin
2 sin ⫹ n
冊 冉 冊 冉
4 cos ⫹ n 4 sin ⫹ n
冊 冊
m⫽
冋
s 11
s 12
s 21
s 22
冉 冉
¯
2 共 n⫺1 兲 cos ⫹ n
¯
2 共 n⫺1 兲 sin ⫹ n
册
冊 冊
册
m
冤
cos
冉 冉
2 n
cos ⫹
4 n
n
兺
k⫽1
兺
再
k⫽1
cos2 ⫹
再
s 22⫽
兺
冎
k⫽1
再
2 n
sin ⫹
4 n
冊 冊
⯗
冊 冉
2 共 n⫺1 兲 sin ⫹ n
冊
冥
(A.2)
s 11⫽
冎
兺
n
再
再
1⫹cos 2 ⫹
4 共 k⫺1 兲 n
n⫺1
兺
j⫽0
再
cos 2 ⫹ j
冎
1 4 共 k⫺1 兲 1 s 12⫽ sin 2 ⫹ ⫽ 2 k⫽1 n 2
兺
冎
2
k⫽1
n 1 ⫽ ⫹ 2 2
冎
2 共 k⫺1 兲 sin ⫹ . n
468 Õ Vol. 122, SEPTEMBER 2000
sin ⫹
Equations 共A.4兲 can be rewritten using double angle trigonometric formulas:
2 共 k⫺1 兲 , n
冎 再
2
冉 冉
sin
(A.3)
2 共 k⫺1 兲 2 共 k⫺1 兲 cos ⫹ sin ⫹ , n n (A.4) n
(A.1)
冊 冊
⯗
冉
n
s 11⫽
s 12⫽s 21⫽
cos ⫹
2 共 n⫺1 兲 cos ⫹ n
where n
2 . n
From Eqs. 共15兲 and 共16兲, the transformation matrix times its transpose is
The result of the multiplication is a square matrix of dimension 2, AT A⫽m 2
冑
4 n
n⫺1
兺
j⫽0
冎 再
sin 2 ⫹ j
4 n
冎 (A.5)
Transactions of the ASME
n
s 22⫽
兺
再
1⫺cos 2 ⫹
4 共 k⫺1 兲 n
冎
Therefore, we can conclude
2
k⫽1
n 1 ⫽ ⫺ 2 2
n⫺1
兺
j⫽0
再
4 cos 2 ⫹ j n
T
A A⫽m
冎
Via formulas 1.341-1 and 1.341-3 in reference 关13兴, the sums of sine and cosine terms on the right sides of Eq. 共A.5兲 are zero, for n⭓3. Thus
s 11⫽
s 22⫽
冦 冦
1 1 ⫹ cos 2 ; 2 2
n⫽1
1⫹cos 2 ;
n⫽2
n ; 2
n⭓3
1 1 ⫺ cos 2 ; 2 2
n⫽1
1⫺cos 2 ;
n⫽2
n ; 2
n⭓3
s 12⫽s 21⫽
冦
1 sin 2 ; 2
n⫽1
sin 2 ;
n⫽2
0;
n⭓3
(A.6)
(A.7)
(A.8)
Journal of Dynamic Systems, Measurement, and Control
2
冋
s 11
s 12
s 21
s 22
册
⫽m
2
冋 册 n 2
0
0
n 2
⫽m 2
冋 册
n 1 2 0
0 1
(A.9)
for the rotor with more than 2 rotor bars, i.e., n⭓3.
References 关1兴 Traister, J. E., 1988, Handbook of Polyphase Motors, Englewood Cliffs, N.J. 关2兴 Pansini, A. J., 1989, Basics of Electric Motors: Including Polyphase Induction and Synchronous Motors, Prentice Hall, N.J. 关3兴 McPherson, G., 1981, An Introduction to Electrical Machines and Transformers, Wiley, New York. 关4兴 Park, R. H., 1929, ‘‘Two-Reaction Theory of Synchronous MachinesGeneralized Method of Analysis, Part I,’’ AIEE Trans., 48, pp. 719–727. 关5兴 Ghosh, B. C., and Bhadra, S. N., 1993, ‘‘Bond Graph Simulation of a Current Source Inverter Driven Induction Motor 共Csi-Im兲 System,’’ Elect. Mac. Power Syst., 21, pp. 51–67. 关6兴 Lawrie, R. J., 1987, Electric Motor Manual: Application, Installation, Maintenance, Troubleshooting, McGraw-Hill, New York. 关7兴 Sahm, D. A., 1979, ‘‘Two-Axis Bond Graph Model of the Dynamics of Synchronous Electrical Machines,’’ J. Franklin Inst., 308, No. 3, pp. 205–218. 关8兴 Hancock, N. N., 1974, Matrix Analysis of Electrical Machinery, 2nd ed., Pergamon Press, New York. 关9兴 Strang, G., 1988, Linear Algebra and Its Applications, 3rd ed., Harcourt Brace. 关10兴 Control Lab Products B. V., 20-SIM Reference Manual, University of Twente, Enschede, Netherlands. 关11兴 Krause, P. C., and Wasynczuk, O., 1989, Electromechanical Motion Devices, McGraw-Hill, New York, pp. 183–184. 关12兴 Manolas, S. T., and Tegopolous, J. A., 1997, ‘‘Analysis of Squirrel Cage Induction Motors with Broken Bars and Rings,’’ IEEE International Electric Machines and Drives Conference, May 18–21, 1997, Milwaukee, WI. 关13兴 Gradshteyn, I. S., and Ryzhik, I. M., 1980, Table of Integrals, Series, and Products, Academic Press, New York, p. 29.
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