IEICE TRANS. FUNDAMENTALS, VOL.E88–A, NO.10 OCTOBER 2005

2521

PAPER

Special Section on Nonlinear Theory and its Applications

Limit Cycle of Induction Motor Drive and Its Control Hongmei LI†a) , Nonmember and Takashi HIKIHARA†† , Member

SUMMARY Limit cycle oscillations of rotor speed are substantially caused by inverter’s dead time, when an induction motor (IM) drive operates in low frequency condition. In this paper, without any hardware modification, discontinuous PWM (DPWM3) modulate strategy possibly controls the unfavorable rotor speed limit cycle under no load operation condition. Simulated results are presented to demonstrate the eﬀectiveness. key words: IM drive, limit cycle, DPWM3, control of limit cycle

1.

Introduction

With the introduction of solid-state inverters, three phase IM drive systems fed by V/f PWM voltage source inverter (VSI) become popular. The great majority of operating variable speed drives is of this type today. They have often experienced unexpected sustained oscillations in low speed range and in light load conditions [1]. Although these oscillations have been observed and analyzed [2]–[5], few papers deal with the essence of these oscillations and improving stability of the V/f controlled IM drive system. In this paper, a nonlinear mathematical model of IM is set up, allowing for the eﬀect of main magnetic circuit saturation. The study shows the oscillatory states depend on inverter’s dead time. In order to clarify the inherent relationship between sustained oscillations and dead time, the bifurcation theory is introduced to explain the mechanism of oscillations. The study indicates dead time can cause IM drive to appear Hopf bifurcation, and the rotor speed motion has limit cycles. Finally, preserving the merit of simplicity in IM drive, DPWM3 modulate strategy is proposed to control the limit cycles of rotor speed without any additional hardware modification. Simulated results demonstrate the eﬀectiveness. 2.

Mathematical Model of IM Drive

2.1 Model of IM VSI is widely utilized in IM drive, as shown in Fig. 1. VSI generates output voltage with controllable magnitude and frequency and drives an IM. The magnetization curve of IM Manuscript received January 19, 2005. Final manuscript received June 8, 2005. † The author is with Hefei University of Technology, Hefei 230009, China. †† The author is with Kyoto University, Kyoto-shi, 615-8510 Japan. a) E-mail: [email protected] DOI: 10.1093/ietfec/e88–a.10.2521

Fig. 1

Fig. 2

Circuit diagram of PWM-VSI.

Magnetizing curve of IM.

is assumed to have the characteristic described by the following formula: Im = K1 λm + K3 λ3m + K5 λ5m ,

(1)

where Im denotes magnetizing current and λm magnetizing linkage. We can fit experimental datum shown in Fig. 2 and obtain coeﬃcients K1 , K3 , and K5 . The voltage matrix equation of IM, fixed in stator d-q reference frame, is given by [5]: I1 I1 −1 = − [Z2 ] [Z1 ] + [Z2 ]−1 [U] , p λ λ (2) pωr = n(T e − T l )/J, 3 T e = n(Iqs λmd − Ids λmq ), 2

c 2005 The Institute of Electronics, Information and Communication Engineers Copyright

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[Z1 ] =

0 r1 L2σ ωr −r2

0 0 K1 r 2 + r 2 Y K1 L2σ ωr + ωr + L2σ ωr Y 0 0 −K1 L2σ ωr − ωr − L2σ ωr Y K1 r 2 + r 2 Y r1 0 −r2 −L2σ ωr

L1σ 0 [Z2 ] = −L2σ 0

0 L1σ 0 −L2σ

1 0 K1 L2σ + 1 + L2σ A L2σ B 0 1 L2σ B K1 L2σ + 1 + L2σ C

where n denotes number of pole pairs, T e electromagnetic torque, J moment of inertia of motor, T l load torque, r resistor, Lσ leakage impedance, ω rotor electrical angular velocity, λdm d axial magnetizing linkage, and λqm q axial magnetizing linkage, respectively. Subscripts 1 and 2 correspond to stator and rotor. p depicts diﬀerential operator d/dt. Moreover, A = 2K3 λ2mq + 4K5 λ4mq + 4K5 λ2mq λ2md + K3 λ2m + K5 λ4m ,

Fig. 3

Switch functions of VSI inverter.

B = 2K3 λmq + 4K5 λmd λ3mq + 4K5 λ3md λmq , C = 2K3 λ2md + 4K5 λ4mq + 4K5 λ2mq λ2md + K3 λ2m + K5 λ4m , Y = K3 λ2m λ4m , λ2m = λ2md + λ2mq , λ4m = λ4md + 2λ2md λ2mq + λ4mq , are satisfied. (a) I1 ≥ 0

2.2 Model of Inverter First, switch function concept is introduced to set up the model of VSI inverter. Sinusoidal modulate wave is compared with triangular carrier wave and the intersections define switch instants. Then switch functions S 1 (ωt), S 3 (ωt), and S 5 (ωt) are obtained and shown in Fig. 3. The phase voltages of inverter (output phase of inverter with respect to the fictional center point o of the dc supply) are figured by Uao , Ubo and Uco , and U D denotes the DC-link voltage. Then, the phase voltages are derived as follows [6]: Uao (ωt) = U D S 1 (ωt), Ubo (ωt) = U D S 3 (ωt), Uco (ωt) = U D S 5 (ωt), Taking inverter’s dead time into consideration, the deviation of Ua from the exact potential Uao becomes pulse-wise voltage shown in Fig. 4. The deviation voltage Uaε = Uao − Ua has the following features: (1) Constant height is equal to dc source voltage U D ,

(b) I1 < 0 Fig. 4

Ideal pulse and practical pulse of inverter.

(2) Pulse width is always kept at dead time td , (3) Polarity of pulse depends on the motor current’s polarity. According to abovementioned three features, deviation voltage of inverter can be acquired and shown in Fig. 5. Then voltage matrix U of Eq. (2) is acquired as follows:

LI and HIKIHARA: LIMIT CYCLE OF INDUCTION MOTOR DRIVE AND ITS CONTROL

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Fig. 5

Voltages of VSI inverter.

Table 1

Uq1 U [U] = d1 Uq2 Ud2

IM parameters.

1 (Uab − Uca ) 3 √ = − 3 U bc 3 0 0

,

(3)

Fig. 6

Acceleration response without dead time.

Taking main magnetic circuit saturation of IM and dead time of inverter into consideration, formulas (2) and (3) address nonlinear mathematical model of IM drive controlled by SPWM modulate strategy. 3.

Simulation Results

In the simulation of IM drive based on the obtained model, inverter’s carrier wave frequency is fc =900 Hz, output frequency f = 10 Hz, td = 20 µs and U D =513 V. The IM parameters are listed in Table 1. Figure 6 shows results of acceleration response in IM drive without dead time. The system is stable except a little pulsation of rotor speed, inexistence of distorted stator current and a little torque pulsation. Figure 7 shows results of acceleration response in IM drive with dead time, the system is unstable. The sustained rotor speed oscillation, distorted stator current and large-scale torque pulsation all shows the unstable states. A new sustained periodicity is generated. 4.

Hopf Bifurcation in IM Drive

Simulation results indicate that dead time can cause IM drive to appear sustained oscillation of rotor speed. In order to explain the phenomenon physically, Hopf bifurcation theory is introduced to indicate the essence of these oscillations. Using the reference frame fixed on stator, transient rotating variable formed by transient qualities of three phases

Fig. 7

Acceleration response with dead time.

is given by: F = 2(Fa + αFb + α2 Fc )/3 The IM equation is expressed by:

α = exp( j2π/3) ,

(4)

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i1 1 r1 + L11 p L12 p U = , L12 (p − jωr ) r2 + L22 (p − jωr ) i2 0

(5)

1 , i1 , and i denote spatial rotating variable of stator where U 2 voltage, current and rotor current, respectively. i0 = i1 + (L22 /Lm )i2 ,

(6)

From Eq. (5) and Eq. (6), we can obtain: → − 1 L0 p r1 + L1 p i1 U → = − , (7) −1/T 0 1/T 0 + (p − jωr ) 0 i0

α3 = (Au + u1 )k2 + 2Au1 + 2uθ˙2 , α4 = [θ˙2 u + (us + 2u)u1 ]k2 + u21 + u2 θ˙2 , α5 = (θ˙2 u2 + u21 )k2 , A = us + u, u = L11 /(L1 T 0 ), us = 1/T s , T s = L1 /r1 , u1 = r1 /(L1 T 0 ). Allowing for inverter’s dead time, if the fundamental wave of deviation voltage caused by dead time is considered, then the following expression is obtained: ε1 = ξU D exp j(θ˙ + φ0 ) , U (12) where ξ = 4 fc td /π, according to equivalent circuit of IM, the following representation is given:

where L0 = L212 /L22 , L1 = L11 − L0 , T 0 = L22 /r2 .

˙ 11 )∠φ0 , jU − ξU D ∠φ0 = i0 (r1 + jθL

From Eq. (13), we know that dead time corresponds to the increase of equivalent stator resistance, which can be calculated by:

Torque equation of IM is expressed as: − → − → T e = qL0 Im ( i1 × i∗0 ),

(8)

ξU D ∠φ0 = req i0 ∠φ0 req =

Dynamic equation of IM is expressed as: (J/n)pωr = T e − T l , q = 3/2n

(9)

On the supposition that harmonics of inverter output 1 = voltage are neglected, the stator voltage given by V ˙ drives an IM, then the corresponding magnejU exp ( jθ) tizing current is i0 = i0 exp j(θ˙ + φ0 ) , where θ˙ = 2π f and φ0 is the initial phase angle of magnetizing current. Small deviations in the variables (i0 , φ0 , ωr ) from an equilibrium point (i00 , φ00 , ωr0 ) are assumed to be as i0 = i00 +∆i0 , ˙i0 = ∆˙i0 , φ0 = φ00 +∆φ0 , φ˙ 0 = ∆φ˙ 0 , ωr = ωr0 +∆ωr , ˙ r , then the linear mathematical model of IM and ω ˙ r = ∆ω drive is given by [7]: −α12 p − β12 L1 T 0 θ˙ α11 p2 + β11 p + γ11 α11 p2 + β11 p + γ11 −L1 T 0 p − r1 T 0 α12 p + β12 −k1 −k2 p p + k2 ∆i0 ∆φ0 = 0 (10) ∆ωr

−ξU D r1 +

(14) ξU D (r12

+

θ˙2 L211 )

(ξU D r1 )2 +(r12 + θ˙2 L211 )(U 2 −ξ 2 U D2 )

Eq. (14) clearly shows that the existence of inverter’s dead time results in an increase of equivalent stator resistance. If characteristic equation (11) has a single pair of purely imaginary eigenvalues λ(u0 ) = ± jω, and no other eigenvalues with zero real part, and furthermore, d Re(λ(u))/du|u=u0 0. The conditions for the existence of a pair of pure imaginary eigenvalues can be formulated for any dimension n in the following way [8]: Hn−1 (u) = 0, Hn−2 > 0, Hn−3 > 0, . . . ,

H1 (u) > 0,

∆φ0 = i00 ∆φ0 , α11 = L1 T 0 , β11 = r1 T 0 + L11 , γ11 = [r1 − L1 T 0 θ˙(θ˙ − ωr0 ), α12 = L1 T 0 (2θ˙ − ωr0 ), ˙ 11 ], ∆ωr = i00 ∆ωr , β12 = [r1 T 0 (θ˙ − ωr0 ) + θL 2 ˙ k1 = 2ki00 (θ − ωr0 )L0 T 0 /J, k2 = kL0 T 0 i200 /J, k = 3n/J. The characteristic equation is derived as: (11)

where α0 = 1,

α1 = 2A,

α2 = uk2 + A2 + 2u1 + θ˙2 ,

p0 (u) > 0,

where Hi (u) stands for the i principal minor of the Hurwitz matrix of the characteristic polynomial and p0 (u) is the zeroorder term of this polynomial. Using the abovementioned model and analytical results, Hopf bifurcation set for dead time is predetermined and illustrated in Fig. 8. Taking dead time into consideration, the IM system results in an increase of equivalent stator

where

α0 p5 + α1 p4 + α2 p3 + α3 p3 + α4 p + α5 = 0

(13)

Fig. 8

Hopf bifurcation set with dead time.

LI and HIKIHARA: LIMIT CYCLE OF INDUCTION MOTOR DRIVE AND ITS CONTROL

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Fig. 9 Triangle-intersection-technique-based PWM employing the zerosequence principle.

Fig. 10 DPWM3 modulate strategy. —— Sinusoidal modulate wave, — - — Zero sequence signal, — — DPWM3 modulate wave.

resistance and appears Hopf bifurcation, and then limit cycle oscillation of rotor speed is inevitable. The range of Hopf bifurcation set will expand with increase of dead time. When td = 0 µs and f = 10 Hz are satisfied, the system becomes stable, in same conditions, taking dead time into consideration, the system appears Hopf bifurcation and limit cycle oscillations of rotor speed will appear. 5.

Control Limit Cycle of Rotor Speed

The phenomenon that Hopf bifurcation set will be expanded with increase of dead time is a clue to control limit cycle through inverter’s operation. As we know, when an inverter operates in the condition of discontinuous PWM (DPWM) modulate strategy, sectors of two 60◦ have not switch in per phase and per cycle, therefore haven’t dead time eﬀect. Therefore DPWM modulate strategy can be utilized eﬀectively to improve stability of IM drive. The potential diﬀerence between the three-wire load neural point and the center point of DC-link capacitor, Uo , is the zero-sequence voltage, and can be arbitrarily selected. The zero-sequence signal injecting technique block diagram is illustrated in Fig. 9. Assume that three phase sinusoidal modulate waves are expressed as: ∗ U = M cos(2π f t), a∗ = M cos(2π f t − 120◦ ), U (15) U b∗ = M cos(2π f t + 120◦ ),

Fig. 11

Umax = max(Ua , Ub , Uc ), Umin = min(Ua , Ub , Uc ),

c

where M denotes modulate factor. Then rotary transform is implemented to three phase sinusoidal modulate waves, according to power factor angle φ of load, the following is derived. U = U ∗ e jφ

(16)

The zero-sequence signal of DPWM3 modulate strategy is given as follows: ∗ , Umin + Umax ≥ 0 ⇒ Uo = Ut /2 − Umax ∗ Umin + Umax < 0 ⇒ Uo = −Ut /2 − Umin , ∗ ∗ ∗ ∗ Umax = max(Ua , Ub , Uc ), ∗ = min(Ua∗ , Ub∗ , Uc∗ ), Umin

(17)

Acceleration response using DPWM3 modulate strategy.

Ut corresponds to amplitude of triangular carrier wave. When an IM drive has no load, power factor φ = 90◦ , we select φ = 60◦ , then DPWM3 modulate strategy is brought and shown in Fig. 10. Figure 11 shows results for acceleration response of IM drive using DPWM3 modulate strategy. Limit cycle oscillation of rotor speed is eﬀectively suppressed by DPWM3 modulate strategy. 6.

Conclusion

In this paper, we have discussed that the sustained oscillation of rotor speed in IM drive is caused by inverter’s dead

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time. Hopf bifurcation theory is introduced to explain the phenomenon physically and clarify the inherent relationship between dead time and sustained oscillation. Without any additional hardware modification, limit cycle oscillation of rotor speed caused by dead time can be eﬀectively suppressed by DPWM3 modulate strategy. References [1] A. Munoz-Garcia, T.A. Lipo, and D.W. Novotny, “A new induction motor v/f control method capable of high-performance regulation at low speeds,” IEEE Trans. Ind. Appl., vol.34, no.4, pp.813–821, 1998. [2] R. Ueda, T. Sonoda, Y. Inoue, and T. Umezu, “Unstable oscillation mode in PWM variable speed drive of induction motor and its stabilization,” Conf. Rec. Ann. Meeting IEEE-IAS, pp.686–691, 1982. [3] N. Mutoh, A. Ueda, K. Sakai, M. Hattori, and K. Nandoh, “Stabilizing control method for suppressing oscillation of induction motors driven by PWM inverters,” IEEE Trans. Ind. Electron., vol.37, no.1, pp.48– 56, 1990. [4] M. Lockwood, “Simulation of unstable oscillation in PWM variablespeed drives,” IEEE Trans. Ind. Appl., vol.24, no.1, pp.137–141, 1988. [5] H.M. Li, Zh.J. Li, and L.Ch. Liu, “Analysis the low frequency oscillation of inverter fed asynchronous motor,” Trans. China Electro Technical Society, vol.15, no.3, pp.16–19, 2000. [6] H.M. Li, Zh.J. Li, and Sh.J. Du, “Dynamic performance simulation of SVPWM inverter fed induction motor,” Electr. Mach. Contr., vol.5, no.3, pp.145–148, 2001. [7] R. Ueda, T. Sonoda, K. Koga, and M. Ichikawa, “Stability analysis in induction motor driven by V/f controlled general purpose inverter,” IEEE Trans. Ind. Appl., vol.28, no.2, pp.472–481, 1988. [8] W.M. Liu, “Criterion of Hopf bifurcations without using eigenvalues,” J. Math. Anal. Appl., vol.182, pp.250–256, 1994.

Hongmei Li received the B.S. and M.S. degrees in Electrical Engineering from Hefei University of Technology in 1991 and 1996, respectively, and Ph.D. degree in Electrical Engineering from Shenyang University of Technology in 2003. Since 1996, she has been with the Department of Electrical Engineering in Hefei University of Technology, where she is currently Associate Professor. Her areas of research interest are including nonlinear dynamics, power electronics and motor control. She now is in Kyoto University as a visiting scholar.

Takashi Hikihara was born in Kyoto, Japan, on August 9, 1958. He received the B.E. from Kyoto Institute of Technology and M.E. and Ph.D. degree in Electrical Engineering from Kyoto University, Kyoto, Japan in 1982, 1984, and 1990, respectively. He is currently a professor in Department of Electrical Engineering in Kyoto University. His areas of research interest are including structural analysis of nonlinear system, applications and control of nonlinear dynamics, power electronics, and power engineering. He is also a member of the IEEE, IEE, APS, SICE, ISICE and IEE, Japan.

2521

PAPER

Special Section on Nonlinear Theory and its Applications

Limit Cycle of Induction Motor Drive and Its Control Hongmei LI†a) , Nonmember and Takashi HIKIHARA†† , Member

SUMMARY Limit cycle oscillations of rotor speed are substantially caused by inverter’s dead time, when an induction motor (IM) drive operates in low frequency condition. In this paper, without any hardware modification, discontinuous PWM (DPWM3) modulate strategy possibly controls the unfavorable rotor speed limit cycle under no load operation condition. Simulated results are presented to demonstrate the eﬀectiveness. key words: IM drive, limit cycle, DPWM3, control of limit cycle

1.

Introduction

With the introduction of solid-state inverters, three phase IM drive systems fed by V/f PWM voltage source inverter (VSI) become popular. The great majority of operating variable speed drives is of this type today. They have often experienced unexpected sustained oscillations in low speed range and in light load conditions [1]. Although these oscillations have been observed and analyzed [2]–[5], few papers deal with the essence of these oscillations and improving stability of the V/f controlled IM drive system. In this paper, a nonlinear mathematical model of IM is set up, allowing for the eﬀect of main magnetic circuit saturation. The study shows the oscillatory states depend on inverter’s dead time. In order to clarify the inherent relationship between sustained oscillations and dead time, the bifurcation theory is introduced to explain the mechanism of oscillations. The study indicates dead time can cause IM drive to appear Hopf bifurcation, and the rotor speed motion has limit cycles. Finally, preserving the merit of simplicity in IM drive, DPWM3 modulate strategy is proposed to control the limit cycles of rotor speed without any additional hardware modification. Simulated results demonstrate the eﬀectiveness. 2.

Mathematical Model of IM Drive

2.1 Model of IM VSI is widely utilized in IM drive, as shown in Fig. 1. VSI generates output voltage with controllable magnitude and frequency and drives an IM. The magnetization curve of IM Manuscript received January 19, 2005. Final manuscript received June 8, 2005. † The author is with Hefei University of Technology, Hefei 230009, China. †† The author is with Kyoto University, Kyoto-shi, 615-8510 Japan. a) E-mail: [email protected] DOI: 10.1093/ietfec/e88–a.10.2521

Fig. 1

Fig. 2

Circuit diagram of PWM-VSI.

Magnetizing curve of IM.

is assumed to have the characteristic described by the following formula: Im = K1 λm + K3 λ3m + K5 λ5m ,

(1)

where Im denotes magnetizing current and λm magnetizing linkage. We can fit experimental datum shown in Fig. 2 and obtain coeﬃcients K1 , K3 , and K5 . The voltage matrix equation of IM, fixed in stator d-q reference frame, is given by [5]: I1 I1 −1 = − [Z2 ] [Z1 ] + [Z2 ]−1 [U] , p λ λ (2) pωr = n(T e − T l )/J, 3 T e = n(Iqs λmd − Ids λmq ), 2

c 2005 The Institute of Electronics, Information and Communication Engineers Copyright

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[Z1 ] =

0 r1 L2σ ωr −r2

0 0 K1 r 2 + r 2 Y K1 L2σ ωr + ωr + L2σ ωr Y 0 0 −K1 L2σ ωr − ωr − L2σ ωr Y K1 r 2 + r 2 Y r1 0 −r2 −L2σ ωr

L1σ 0 [Z2 ] = −L2σ 0

0 L1σ 0 −L2σ

1 0 K1 L2σ + 1 + L2σ A L2σ B 0 1 L2σ B K1 L2σ + 1 + L2σ C

where n denotes number of pole pairs, T e electromagnetic torque, J moment of inertia of motor, T l load torque, r resistor, Lσ leakage impedance, ω rotor electrical angular velocity, λdm d axial magnetizing linkage, and λqm q axial magnetizing linkage, respectively. Subscripts 1 and 2 correspond to stator and rotor. p depicts diﬀerential operator d/dt. Moreover, A = 2K3 λ2mq + 4K5 λ4mq + 4K5 λ2mq λ2md + K3 λ2m + K5 λ4m ,

Fig. 3

Switch functions of VSI inverter.

B = 2K3 λmq + 4K5 λmd λ3mq + 4K5 λ3md λmq , C = 2K3 λ2md + 4K5 λ4mq + 4K5 λ2mq λ2md + K3 λ2m + K5 λ4m , Y = K3 λ2m λ4m , λ2m = λ2md + λ2mq , λ4m = λ4md + 2λ2md λ2mq + λ4mq , are satisfied. (a) I1 ≥ 0

2.2 Model of Inverter First, switch function concept is introduced to set up the model of VSI inverter. Sinusoidal modulate wave is compared with triangular carrier wave and the intersections define switch instants. Then switch functions S 1 (ωt), S 3 (ωt), and S 5 (ωt) are obtained and shown in Fig. 3. The phase voltages of inverter (output phase of inverter with respect to the fictional center point o of the dc supply) are figured by Uao , Ubo and Uco , and U D denotes the DC-link voltage. Then, the phase voltages are derived as follows [6]: Uao (ωt) = U D S 1 (ωt), Ubo (ωt) = U D S 3 (ωt), Uco (ωt) = U D S 5 (ωt), Taking inverter’s dead time into consideration, the deviation of Ua from the exact potential Uao becomes pulse-wise voltage shown in Fig. 4. The deviation voltage Uaε = Uao − Ua has the following features: (1) Constant height is equal to dc source voltage U D ,

(b) I1 < 0 Fig. 4

Ideal pulse and practical pulse of inverter.

(2) Pulse width is always kept at dead time td , (3) Polarity of pulse depends on the motor current’s polarity. According to abovementioned three features, deviation voltage of inverter can be acquired and shown in Fig. 5. Then voltage matrix U of Eq. (2) is acquired as follows:

LI and HIKIHARA: LIMIT CYCLE OF INDUCTION MOTOR DRIVE AND ITS CONTROL

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Fig. 5

Voltages of VSI inverter.

Table 1

Uq1 U [U] = d1 Uq2 Ud2

IM parameters.

1 (Uab − Uca ) 3 √ = − 3 U bc 3 0 0

,

(3)

Fig. 6

Acceleration response without dead time.

Taking main magnetic circuit saturation of IM and dead time of inverter into consideration, formulas (2) and (3) address nonlinear mathematical model of IM drive controlled by SPWM modulate strategy. 3.

Simulation Results

In the simulation of IM drive based on the obtained model, inverter’s carrier wave frequency is fc =900 Hz, output frequency f = 10 Hz, td = 20 µs and U D =513 V. The IM parameters are listed in Table 1. Figure 6 shows results of acceleration response in IM drive without dead time. The system is stable except a little pulsation of rotor speed, inexistence of distorted stator current and a little torque pulsation. Figure 7 shows results of acceleration response in IM drive with dead time, the system is unstable. The sustained rotor speed oscillation, distorted stator current and large-scale torque pulsation all shows the unstable states. A new sustained periodicity is generated. 4.

Hopf Bifurcation in IM Drive

Simulation results indicate that dead time can cause IM drive to appear sustained oscillation of rotor speed. In order to explain the phenomenon physically, Hopf bifurcation theory is introduced to indicate the essence of these oscillations. Using the reference frame fixed on stator, transient rotating variable formed by transient qualities of three phases

Fig. 7

Acceleration response with dead time.

is given by: F = 2(Fa + αFb + α2 Fc )/3 The IM equation is expressed by:

α = exp( j2π/3) ,

(4)

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i1 1 r1 + L11 p L12 p U = , L12 (p − jωr ) r2 + L22 (p − jωr ) i2 0

(5)

1 , i1 , and i denote spatial rotating variable of stator where U 2 voltage, current and rotor current, respectively. i0 = i1 + (L22 /Lm )i2 ,

(6)

From Eq. (5) and Eq. (6), we can obtain: → − 1 L0 p r1 + L1 p i1 U → = − , (7) −1/T 0 1/T 0 + (p − jωr ) 0 i0

α3 = (Au + u1 )k2 + 2Au1 + 2uθ˙2 , α4 = [θ˙2 u + (us + 2u)u1 ]k2 + u21 + u2 θ˙2 , α5 = (θ˙2 u2 + u21 )k2 , A = us + u, u = L11 /(L1 T 0 ), us = 1/T s , T s = L1 /r1 , u1 = r1 /(L1 T 0 ). Allowing for inverter’s dead time, if the fundamental wave of deviation voltage caused by dead time is considered, then the following expression is obtained: ε1 = ξU D exp j(θ˙ + φ0 ) , U (12) where ξ = 4 fc td /π, according to equivalent circuit of IM, the following representation is given:

where L0 = L212 /L22 , L1 = L11 − L0 , T 0 = L22 /r2 .

˙ 11 )∠φ0 , jU − ξU D ∠φ0 = i0 (r1 + jθL

From Eq. (13), we know that dead time corresponds to the increase of equivalent stator resistance, which can be calculated by:

Torque equation of IM is expressed as: − → − → T e = qL0 Im ( i1 × i∗0 ),

(8)

ξU D ∠φ0 = req i0 ∠φ0 req =

Dynamic equation of IM is expressed as: (J/n)pωr = T e − T l , q = 3/2n

(9)

On the supposition that harmonics of inverter output 1 = voltage are neglected, the stator voltage given by V ˙ drives an IM, then the corresponding magnejU exp ( jθ) tizing current is i0 = i0 exp j(θ˙ + φ0 ) , where θ˙ = 2π f and φ0 is the initial phase angle of magnetizing current. Small deviations in the variables (i0 , φ0 , ωr ) from an equilibrium point (i00 , φ00 , ωr0 ) are assumed to be as i0 = i00 +∆i0 , ˙i0 = ∆˙i0 , φ0 = φ00 +∆φ0 , φ˙ 0 = ∆φ˙ 0 , ωr = ωr0 +∆ωr , ˙ r , then the linear mathematical model of IM and ω ˙ r = ∆ω drive is given by [7]: −α12 p − β12 L1 T 0 θ˙ α11 p2 + β11 p + γ11 α11 p2 + β11 p + γ11 −L1 T 0 p − r1 T 0 α12 p + β12 −k1 −k2 p p + k2 ∆i0 ∆φ0 = 0 (10) ∆ωr

−ξU D r1 +

(14) ξU D (r12

+

θ˙2 L211 )

(ξU D r1 )2 +(r12 + θ˙2 L211 )(U 2 −ξ 2 U D2 )

Eq. (14) clearly shows that the existence of inverter’s dead time results in an increase of equivalent stator resistance. If characteristic equation (11) has a single pair of purely imaginary eigenvalues λ(u0 ) = ± jω, and no other eigenvalues with zero real part, and furthermore, d Re(λ(u))/du|u=u0 0. The conditions for the existence of a pair of pure imaginary eigenvalues can be formulated for any dimension n in the following way [8]: Hn−1 (u) = 0, Hn−2 > 0, Hn−3 > 0, . . . ,

H1 (u) > 0,

∆φ0 = i00 ∆φ0 , α11 = L1 T 0 , β11 = r1 T 0 + L11 , γ11 = [r1 − L1 T 0 θ˙(θ˙ − ωr0 ), α12 = L1 T 0 (2θ˙ − ωr0 ), ˙ 11 ], ∆ωr = i00 ∆ωr , β12 = [r1 T 0 (θ˙ − ωr0 ) + θL 2 ˙ k1 = 2ki00 (θ − ωr0 )L0 T 0 /J, k2 = kL0 T 0 i200 /J, k = 3n/J. The characteristic equation is derived as: (11)

where α0 = 1,

α1 = 2A,

α2 = uk2 + A2 + 2u1 + θ˙2 ,

p0 (u) > 0,

where Hi (u) stands for the i principal minor of the Hurwitz matrix of the characteristic polynomial and p0 (u) is the zeroorder term of this polynomial. Using the abovementioned model and analytical results, Hopf bifurcation set for dead time is predetermined and illustrated in Fig. 8. Taking dead time into consideration, the IM system results in an increase of equivalent stator

where

α0 p5 + α1 p4 + α2 p3 + α3 p3 + α4 p + α5 = 0

(13)

Fig. 8

Hopf bifurcation set with dead time.

LI and HIKIHARA: LIMIT CYCLE OF INDUCTION MOTOR DRIVE AND ITS CONTROL

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Fig. 9 Triangle-intersection-technique-based PWM employing the zerosequence principle.

Fig. 10 DPWM3 modulate strategy. —— Sinusoidal modulate wave, — - — Zero sequence signal, — — DPWM3 modulate wave.

resistance and appears Hopf bifurcation, and then limit cycle oscillation of rotor speed is inevitable. The range of Hopf bifurcation set will expand with increase of dead time. When td = 0 µs and f = 10 Hz are satisfied, the system becomes stable, in same conditions, taking dead time into consideration, the system appears Hopf bifurcation and limit cycle oscillations of rotor speed will appear. 5.

Control Limit Cycle of Rotor Speed

The phenomenon that Hopf bifurcation set will be expanded with increase of dead time is a clue to control limit cycle through inverter’s operation. As we know, when an inverter operates in the condition of discontinuous PWM (DPWM) modulate strategy, sectors of two 60◦ have not switch in per phase and per cycle, therefore haven’t dead time eﬀect. Therefore DPWM modulate strategy can be utilized eﬀectively to improve stability of IM drive. The potential diﬀerence between the three-wire load neural point and the center point of DC-link capacitor, Uo , is the zero-sequence voltage, and can be arbitrarily selected. The zero-sequence signal injecting technique block diagram is illustrated in Fig. 9. Assume that three phase sinusoidal modulate waves are expressed as: ∗ U = M cos(2π f t), a∗ = M cos(2π f t − 120◦ ), U (15) U b∗ = M cos(2π f t + 120◦ ),

Fig. 11

Umax = max(Ua , Ub , Uc ), Umin = min(Ua , Ub , Uc ),

c

where M denotes modulate factor. Then rotary transform is implemented to three phase sinusoidal modulate waves, according to power factor angle φ of load, the following is derived. U = U ∗ e jφ

(16)

The zero-sequence signal of DPWM3 modulate strategy is given as follows: ∗ , Umin + Umax ≥ 0 ⇒ Uo = Ut /2 − Umax ∗ Umin + Umax < 0 ⇒ Uo = −Ut /2 − Umin , ∗ ∗ ∗ ∗ Umax = max(Ua , Ub , Uc ), ∗ = min(Ua∗ , Ub∗ , Uc∗ ), Umin

(17)

Acceleration response using DPWM3 modulate strategy.

Ut corresponds to amplitude of triangular carrier wave. When an IM drive has no load, power factor φ = 90◦ , we select φ = 60◦ , then DPWM3 modulate strategy is brought and shown in Fig. 10. Figure 11 shows results for acceleration response of IM drive using DPWM3 modulate strategy. Limit cycle oscillation of rotor speed is eﬀectively suppressed by DPWM3 modulate strategy. 6.

Conclusion

In this paper, we have discussed that the sustained oscillation of rotor speed in IM drive is caused by inverter’s dead

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time. Hopf bifurcation theory is introduced to explain the phenomenon physically and clarify the inherent relationship between dead time and sustained oscillation. Without any additional hardware modification, limit cycle oscillation of rotor speed caused by dead time can be eﬀectively suppressed by DPWM3 modulate strategy. References [1] A. Munoz-Garcia, T.A. Lipo, and D.W. Novotny, “A new induction motor v/f control method capable of high-performance regulation at low speeds,” IEEE Trans. Ind. Appl., vol.34, no.4, pp.813–821, 1998. [2] R. Ueda, T. Sonoda, Y. Inoue, and T. Umezu, “Unstable oscillation mode in PWM variable speed drive of induction motor and its stabilization,” Conf. Rec. Ann. Meeting IEEE-IAS, pp.686–691, 1982. [3] N. Mutoh, A. Ueda, K. Sakai, M. Hattori, and K. Nandoh, “Stabilizing control method for suppressing oscillation of induction motors driven by PWM inverters,” IEEE Trans. Ind. Electron., vol.37, no.1, pp.48– 56, 1990. [4] M. Lockwood, “Simulation of unstable oscillation in PWM variablespeed drives,” IEEE Trans. Ind. Appl., vol.24, no.1, pp.137–141, 1988. [5] H.M. Li, Zh.J. Li, and L.Ch. Liu, “Analysis the low frequency oscillation of inverter fed asynchronous motor,” Trans. China Electro Technical Society, vol.15, no.3, pp.16–19, 2000. [6] H.M. Li, Zh.J. Li, and Sh.J. Du, “Dynamic performance simulation of SVPWM inverter fed induction motor,” Electr. Mach. Contr., vol.5, no.3, pp.145–148, 2001. [7] R. Ueda, T. Sonoda, K. Koga, and M. Ichikawa, “Stability analysis in induction motor driven by V/f controlled general purpose inverter,” IEEE Trans. Ind. Appl., vol.28, no.2, pp.472–481, 1988. [8] W.M. Liu, “Criterion of Hopf bifurcations without using eigenvalues,” J. Math. Anal. Appl., vol.182, pp.250–256, 1994.

Hongmei Li received the B.S. and M.S. degrees in Electrical Engineering from Hefei University of Technology in 1991 and 1996, respectively, and Ph.D. degree in Electrical Engineering from Shenyang University of Technology in 2003. Since 1996, she has been with the Department of Electrical Engineering in Hefei University of Technology, where she is currently Associate Professor. Her areas of research interest are including nonlinear dynamics, power electronics and motor control. She now is in Kyoto University as a visiting scholar.

Takashi Hikihara was born in Kyoto, Japan, on August 9, 1958. He received the B.E. from Kyoto Institute of Technology and M.E. and Ph.D. degree in Electrical Engineering from Kyoto University, Kyoto, Japan in 1982, 1984, and 1990, respectively. He is currently a professor in Department of Electrical Engineering in Kyoto University. His areas of research interest are including structural analysis of nonlinear system, applications and control of nonlinear dynamics, power electronics, and power engineering. He is also a member of the IEEE, IEE, APS, SICE, ISICE and IEE, Japan.