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Speed regulation of induction motors with wound rotor
5th IFAC International Workshop on Periodic Control Systems The International Federation of Automatic Control July 3-5, 2013. Caen, France
Speed regulation of induction motors with wound rotor G.A. Leonov, E.P. Solovyeva, A.M. Zaretskiy Department of Mathematical Information Technology, University of Jyv¨ askyl¨ a P.O. Box 35 (Agora), Finland, FI-40014 Department of Applied Cybernetics, Saint-Petersburg State University Universitetsky pr. 28, Saint-Petersburg, Russia, 198504 Abstract: The regulation of speed is required in many applications of induction motors, including in electric locomotives and electric driving vehicles. In this paper the speed control problem for a wound rotor induction motor is considered. The speed is controlled by changing the external resistance in the rotor circuit. It is assumed that induction motors operate under constant load conditions during changes of the external resistance. A mathematical model of the induction motor with wound rotor proposed in this paper has periodical solutions. In such model the complex effects, which can’t be found numerically, may appear such as hidden oscillations (see, e.g., [J. de Bruin et al., 2009; M.A. Kiseleva et al., 2012]). Using the non-local reduction method the system of induction motors is studied and analytical estimations of speed control range are obtained. Analytical results are compared with computer experiments. Keywords: induction motor, wound rotor, speed control, non-local reduction method 1. INTRODUCTION
2. MODELLING OF WOUND ROTOR INDUCTION MOTOR
In engineering practice it is frequently needed to regulate the speed of induction motors. For example, an induction motor driving the metal-working machine should operate at various speeds because the blanks are processed by rough cutting at low speed and by finishing operations at high speed. The speed control is necessary in industrial applications (crane, hoist) and in electric vehicles (electric locomotives). The speed of induction motors can be controlled by variation of frequency of power supply, by variation of supply voltage, or by varying the slip frequency (Leonhard, 2001; Herman, 2009). For a wound rotor induction motor, the speed control is frequently achieved by varying the additional (external) resistance in the rotor circuit (Agrawal, 2001; Bakshi and Bakshi, 2009; Subrahmanyam, 2011). This way to control the speed of the wound rotor induction motor is used in applications requiring the motor to operate over a wide range of speeds.
In this paper several assumptions are made to develop a mathematical model of the wound rotor induction motor: magnetic permeability of materials of the stator and rotor is infinite; saturation, steel losses and mutual inductance are ignored; electromagnetic processes in the rotor windings do not affect the currents in the stator windings, i.e. magnetic field vector is constant in magnitude and rotates with constant angular speed. The last assumption dates back to classical ideas of N. Tesla and G. Ferraris (Tesla, 1888; Ferraris, 1888) and allows to describe the dynamics of the wound rotor induction motor by the dynamics of its rotor.
In this paper a mathematical model of a wound rotor induction motor, which fully takes into account the geometry of its rotor, is proposed. The speed control of the wound rotor induction motor by varying the rotor resistance for constant torque is considered. Using the non-local reduction method (Leonov et al., 1992; Gelig et al., 2004; Leonov, 2006; Leonov and Solovyeva, 2012) analytical estimations of speed control range are obtained. The results of computer experiments are given.
Fig. 1. Scheme of the wound rotor with the additional resistance r The wound rotor winding is considered as three identical coils displaced 120 electrical degrees apart and connected to the star (in Fig. 1). Free ends of the rotor winding are connected through slip rings and brushes to external resistances. These resistances are varied in order to control the motor speed. The motion of the wound rotor is considered in a rotating system of coordinates rigidly → − connected to the magnetic induction vector B . Using
? This work was supported by the Government of the Russian Federation (Ministry of Education and Science of the Russian Federation), Russian Foundation for Basic Research and Saint-Petersburg State University (Russia), the Faculty of Information Technology (University of Jyv¨ askyl¨ a, Finland).
978-3-902823-38-0/2013 © IFAC
90
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2013 IFAC PSYCO July 3-5, 2013. Caen, France
of the parameter c∗ is changed to c. Hence, a new stable equilibrium point is as follows: p c a − a2 − 4γ 2 γs0 γ s0 = , x0 = − , y0 = − . 2γ ac a
the approach suggested in (Leonov, 2006; Leonov and Kondrateva, 2009; Leonov and Solovyeva, 2012), we obtain the system of differential equations of induction motor with the wound rotor: ˙ Li˙ 1 + (R + r) i1 = −2nSB cos(θ) θ, Li˙ 2 + (R + r) i2 = −2nSB cos(θ +
π ˙ ) θ, 3
Li˙ 3 + (R + r) i3 = −2nSB cos(θ +
2π ˙ ) θ, 3
The mathematical formulation of the speed regulation problem for an induction motor with the wound rotor under fixed load conditions is as follows: to find conditions under which the solution of the system (2) x(t), y(t), s(t) with initial data s = s∗ , x∗ , y∗ belongs to the attraction domain of the stationary solution s0 , x0 , y0 . It means that relations
(1)
π ) i2 3 2π + cos(θ + ) i3 − M, 3
J θ¨ = 2nSB cos θ i1 + cos(θ +
lim s(t) = s0 ,
lim y(t) = y0 (3)
t→+∞
Let us introduce the notation
c , c∗ p c a + a2 − 4γ 2 ρ=
s1 =
Using nonsingular change of coordinates ˙ s = θ, π 2π L (cos θ i1 + cos(θ + )i2 + cos(θ + )i3 ), x= 3nSB 3 3
,
2γ
γ ψ(s) = − s2 + as − cγ, c 12 γ2 Γ = 2 max λ c − λ − 2 . 4c (c − λ) λ∈(0,c) Theorem 1. Suppose that γ < 2c2 and the following inequalities
L π 2π (sin θ i1 + sin(θ + )i2 + sin(θ + )i3 ), 3nSB 3 3 z = i1 + i3 − i2 ,
cΓ > γ, p a2 − 4γ 2 ρ>1− a
system (1) reduces to the following form θ˙ = s, s˙ = ay + γ, x˙ = −cx + ys, y˙ = −cy − xs − s, z˙ = −cz,
(4) (5)
are fulfilled. Then the solution of system (2) with initial γ ∗ data s = s∗ , x = − γs ac∗ , y = − a satisfies relations (3). Proof. The change of variables
2
where a = 6 (nSB) γ = M c = R+r JL , J , L . Note that the last equation can be integrated easily, and the remaining equations, except the first one, do not depend on θ, therefore, it suffices to consider the system s˙ = ay + γ, x˙ = −cx + ys, y˙ = −cy − xs − s.
t→+∞
must be satisfied.
where n – number of turns of one coil; S – area of one turn; R – resistance; r – additional resistance; L – inductance; i1 , i2 , i3 – currents; θ – angle between plane of turns of coil with current i1 and plane perpendicular to magnetic → − induction vector B ; J – inertia moment of the rotor; M – external load torque.
y=
lim x(t) = x0 ,
t→+∞
η = ay + γ,
z = −x −
γ s ac
reduces system (2) to the form s˙ = η, η˙ = −cη + azs − ψ(s), 1 γ z˙ = −cz − sη − η a ac
(2)
(6)
and transforms initial data s∗ , x∗ , y∗ of system (2) into ∗ initial data s = s∗ , η = 0, z = (ρ − 1) γs ac of system (6).
3. THE SPEED REGULATION PROBLEM FOR A WOUND ROTOR INDUCTION MOTOR
Under the condition 0 < γ < a/2, the stationary set of system (6) consists of two points: the asymptotically stable point (s0 , 0, 0) and the unstable point (s1 , 0, 0). Then for system (6), relations (3) take the form
Let us consider the speed regulation problem for system (2) of the induction motor with the wound rotor. The speed is regulated by varying the additional resistance in the rotor circuit. It is assumed that the induction motor operates in steady state conditions under fixed load γ and c = c∗ . Suppose that this steady state operation corresponds to the stable equilibrium of system (2): p c∗ a − a2 − 4γ 2 γs∗ γ s∗ = , x∗ = − , y∗ = − . 2γ ac∗ a Then, at time t = τ the external resistance in the rotor circuit is varied without changing the load γ, i.e. the value
lim s(t) = s0 ,
t→+∞
lim η(t) = 0,
t→+∞
lim z(t) = 0. (7)
t→+∞
Let us consider the equation F (s)F 0 (s) = −ΓF (s) − ψ(s)
(8)
with initial data F (s1 ) = 0. It is easy to show that inequalities (4) and (5) are equivalent to the following inequalities
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2013 IFAC PSYCO July 3-5, 2013. Caen, France
p c2 1 (1 − )2 (a − a2 − 4γ 2 )2 > 2 γ ρ p 2 1 2 a − a2 − 4γ 2 1 − , ρ p p 1 (a2 − 4γ 2 ) > ( − 1)(a − a2 − 4γ 2 ) a2 − 4γ 2 . ρ
Relations (12) and (13) imply the positive invariance of sets Ω = {W (s, η, z) < 0, s ∈ [s2 , s1 ]}, Ω1 = {W (s, η, z) < 0, s ∈ (−∞, s1 ]}, Ω2 = {V (s, η, z) < C}, Ω∗ = Ω1 ∩ Ω2 , where p 1 1 C > max{ (1 − )2 (a − a2 − 4γ 2 )2 + 8 ρ Zs∗ (14) + ψ(s)ds, 0}.
Γ2
(9)
(10)
Hense, 4Γ2 (s1 − s∗ )2 = p p 1 h a2 − 4γ 2 ) i2 a2 − 4γ 2 ) ρ c(a − 2 c(a + = 4Γ − = 2γ 2γ i 2h p p 2 c 1 = Γ2 2 (1 − )(a − a2 − 4γ 2 ) + 2 a2 − 4γ 2 = γ ρ
s1
Using (11), we obtain γs∗ ) < 0. ac Taking into account condition (14), we have γs∗ V (s∗ , 0, (ρ − 1) ) < C. ac Consequently, γs∗ γs∗ (s∗ , 0, (ρ − 1) ) ∈ Ω, (s∗ , 0, (ρ − 1) ) ∈ Ω∗ . ac ac
p c2 h 1 = Γ2 2 (1 − )2 (a − a2 − 4γ 2 )2 + γ ρ p p 1 +4(1 − )(a − a2 − 4γ 2 ) a2 − 4γ 2 + ρ i p 2 1 2 2 +4(a − 4γ 2 ) > a − a2 − 4γ 2 1 − . ρ Using the last inequality and the estimate obtained for the solution F (s) of equation (8) at s ∈ (0, s1 ) in (Barbashin and Tabueva, 1969) F (s) > Γ(s1 − s), we obtain p 1 1 F (s∗ ) > a − a2 − 4γ 2 1 − . (11) 2 ρ
W (s∗ , 0, (ρ − 1)
Let x(t) = (s(t), η(t), z(t)) be a solution of system (6) bounded at t ≥ 0. Then the function V (x(t)) is bounded at t ≥ 0 as well. Under the condition γ < 2c2 , by virtue of Sylvester’s criterion, the quadratic form from η and z of (13) is negative definite. Therefore, it follows from (13) that the function V (s(t), η(t), z(t)) does not increase in t for t ≥ 0. This fact and the boundedness of the function V (s(t), η(t), z(t)) for t ≥ 0 imply the existence of a finite limit lim V (s(t), η(t), z(t)) = const.
Thus, equation (8) has either a solution F (s) (curve 1 in Fig. 2) such that F (s2 ) = 0, s2 < s∗ , F (s) > 0, ∀s ∈ (s2 , s1 ); or a solution F (s) (curve 2 in Fig. 2) such that F (s) > 0, ∀s ∈ (−∞, s1 ).
t→+∞
It follows from the boundedness of the trajectory x(t) that set Ω0 of its omega-limit points is nonempty. Let y ∈ Ω0 . Then, by virtue of the invariance of Ω0 , the trajectory going from y is contained in Ω0 for all t ∈ R. Therefore, for all t ∈ R, we have V (s(t, y), η(t, y), z(t, y)) ≡ L. Using estimate (13), we obtain identities η(t, y) ≡ 0 and z(t, y) ≡ 0. Relations (6) and (13) imply s(t, ˙ y) ≡ 0. Therefore, s(t, y) ≡ const, and Ω0 is a subset of the stationary set of system (6).
Fig. 2. Two possible cases of behavior of solution of equation (8)
Thus, any bounded solution of system (6) tends to an equilibrium point. This fact, the boundedness and the positive invariance of sets Ω, Ω∗ and the inclusions γs∗ γs∗ (s∗ , 0, (ρ − 1) ) ∈ Ω, (s∗ , 0, (ρ − 1) ) ∈ Ω∗ , ac ac (s0 , 0, 0) ∈ Ω, (s0 , 0, 0) ∈ Ω∗ imply relations (7), which are equivalent relations (3). This completes the proof of the theorem.
Let us introduce functions a2 1 1 W (s, η, z) = z 2 + η 2 − F 2 (s), 2 2 2 Zs 2 a 1 V (s, η, z) = z 2 + η 2 + ψ(s)ds. 2 2 s1
Under conditions γ < 2c2 and (6), functions W (s, η, z), V (s, η, z) satisfy relations ˙ (s, η, z) + 2λW (s, η, z) ≤ 0, W (12)
Conditions (4) and (5) of Theorem 1 allow to obtain the lower estimate of the range of additional resistance regulation for induction motor with wound rotor under constant load. In Fig. 3 area 1 corresponds to the range of regulation of parameter ρ (the initial additional resistance r∗ = rf ix ) for varied parameter a of system (2) and for
√ γ γ2 V˙ (s, η, z) = −( √ η + caz)2 − (c − 3 )η 2 ≤ 0. (13) 4c 2c c
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2013 IFAC PSYCO July 3-5, 2013. Caen, France
given load γ = γ0 ; area 2 represents the range of regulation of parameter ρ as the initial additional resistance increases (r∗ > rf ix ) for given load γ = γ0 ; area 3 corresponds to the range of regulation of parameter ρ as the load decreases (permissible load γ < γ0 ). Thus, from the range of regulation of parameter ρ we can obtain the range ∗) of regulation of the additional resistance r = ρ(R+r R (r∗ > rf ix ) and the range of speed regulation s.
0.06
0.04
i2
0.02
0
−0.02
−0.04
−0.06 0.1 0 −0.1 i
2
1.5
1
0.5
0
1
2.5
3
3.5
s
Fig. 5. Trajectory of system (1) as the additional resistance decreases
0.2
0.15
i2
0.1
Fig. 3. The range of regulation of parameter ρ for different load and for different additional resistance: area 1 c∗ = 2, γ = 1; area 2 - c∗ = 3, γ = 1; area 3 - c∗ = 3, γ = 1.8
0.05
0
−0.05 0.2 0.1 0
A typical situation during the operation of an induction motor is as follows: the motor runs in no-load mode, during transient process it is pulled in synchronism, next load-on occurs and finally the speed control of motor is required without changing the load. An operating mode of the wound rotor induction motor under no-load conditions (γ = 0) corresponds to a stationary solution θ = const, θ˙ = s = 0, i1 = 0, i2 = 0, i3 = 0 of system (1). Suppose that the initial condition is θ = 0, θ˙ = s = 0, i1 = 0, i2 = 0, i3 = 0. After the permissible sudden change of load (Leonov and Solovyeva, 2012) the trajectory, after some transient process, tends to the periodical solution of system (1) (see blue part of trajectories in Figs. 4-6). This solution corresponds to an operating mode of wound rotor induction motor under load.
−0.1 i1
i2
−0.04 0.05
0
3
2
4
5
6
4
5
6
s
The speed control of induction motors is one of the most important engineering problems. In this paper the speed control problem is considered for an induction motor with wound rotor. It is shown that one of simple ways to regulate the speed of wound rotor induction motor is variation of the additional resistance in the rotor circuit. The found estimations of range of the additional resistance regulation prove that the motor can operate over a wide range of speed. This property allows to use the wound rotor induction motor in many applications. Results of computer experiments show that varying the additional resistance not only decreases or increases motor speed (in Figs. 4 5) but also may lead to an unstable operating mode (Fig. 6). However, in more complex system such as drilling machine, grinding machine, the complex dynamical effects may appear. In the works (Bruin et al., 2009; Leonov and Kuznetsov, 2013a,b; Kiseleva et al., 2012) the drilling system actuated by the induction motor is considered and hidden oscillations (Leonov et al., 2012, 2011; Bragin et al., 2011; Kuznetsov et al., 2011; Leonov and Kuznetsov, 2011) were found. Thus, more careful analysis of induction machines and their control methods are required.
−0.02
1
3
2
4. CONCLUSION
0
0
1
the additional resistance is illustrated. This means that the induction motor does not pull in synchronism and the motor stops.
0.02
−0.05
0
Fig. 6. Trajectory of system (1) at impermissible change of the additional resistance
0.04
i1
−0.2
7
s
Fig. 4. Trajectory of system (1) as the additional resistance increases For different additional resistance values (parameter r of system (1)) computer modeling is done. In Figs. 4-6 the green parts of trajectories correspond to transient process, arising from variations of the additional resistance. For permissible given load, the higher the additional resistance is, the lower the rotation speed of induction motor becomes (see in Figs. 4 - 5). In Fig. 6 an impermissible change of
REFERENCES Agrawal, J. (2001). Power electronic systems: theory and design. Prentice Hall.
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Leonov, G., Reitmann, V., and Smirnova, V. (1992). Non-Local methods for pendulum-like feedback systems. Stuttgart-Leipzig, Teubner VerlagsgesellSchaft. Leonov, G. and Solovyeva, E. (2012). The nonlocal reduction method in analyzing the stability of differential equations of induction machines. Doklady Mathematics, 85(3), 375–379. Subrahmanyam, V. (2011). Electric drives: concepts and applications. Tata McGraw Hill. Tesla, N. (1888). Patent application: Electro-magnetic motor.
Bakshi, U. and Bakshi, M. (2009). Electrical drives and control. Technical Publications. Barbashin, E. and Tabueva, V. (1969). Dynamical system with cylindrical phase space. Nauka. Bragin, V.O., Vagaitsev, V.I., Kuznetsov, N.V., and Leonov, G.A. (2011). Algorithms for finding hidden oscillations in nonlinear systems. The Aizerman and Kalman conjectures and Chua’s circuits. Journal of Computer and Systems Sciences International, 50(4), 511–543. doi:10.1134/S106423071104006X. Bruin, J.D., Doris, A., van de Wouw, N., Heemels, W., and Nijmeijer, H. (2009). Control of mechanical motion systems with non-collocation of actuation and friction: A popov criterion approach for input-to-state stability and set-valued nonlinearities. Automatica, 45(2), 405– 415. Ferraris, G. (1888). Rotazioni elettrodinamiche prodotte per mezzo di correnti alternate. Il Nuovo Cimento, 23, 246–263. Gelig, A., Leonov, G., and Yakubovich, V. (2004). Stability of stationary sets in control systems with discontinuous nonlinearities. World Scientific. Herman, S. (2009). Electric motor control. Cengage Learning. Kiseleva, M.A., Kuznetsov, N.V., Leonov, G.A., and Neittaanm¨ aki, P. (2012). Drilling systems failures and hidden oscillations. In IEEE 4th International Conference on Nonlinear Science and Complexity, NSC 2012 - Proceedings, 109–112. doi:10.1109/NSC.2012.6304736. Kuznetsov, N.V., Leonov, G.A., and Seledzhi, S.M. (2011). Hidden oscillations in nonlinear control systems. IFAC Proceedings Volumes (IFAC-PapersOnline), 18(1), 2506–2510. doi:10.3182/20110828-6-IT-1002.03316. Leonhard, W. (2001). Control of electric drives. Springer. Leonov, G.A. and Kuznetsov, G.V. (2013a). Hidden oscillations in drilling systems: torsional vibrations. Journal of Applied Nonlinear Dynamics, 2(1), 83–94. doi: 10.5890/JAND.2012.09.006. Leonov, G.A. and Kuznetsov, N.V. (2011). Analyticalnumerical methods for investigation of hidden oscillations in nonlinear control systems. IFAC Proceedings Volumes (IFAC-PapersOnline), 18(1), 2494–2505. doi: 10.3182/20110828-6-IT-1002.03315. Leonov, G.A. and Kuznetsov, N.V. (2013b). Hidden attractors in dynamical systems. From hidden oscillations in Hilbert-Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractors in Chua circuits. International Journal of Bifurcation and Chaos, 23(1). doi:10.1142/S0218127413300024. art. no. 1330002. Leonov, G.A., Kuznetsov, N.V., and Vagaitsev, V.I. (2011). Localization of hidden Chua’s attractors. Physics Letters A, 375(23), 2230–2233. doi: 10.1016/j.physleta.2011.04.037. Leonov, G.A., Kuznetsov, N.V., and Vagaitsev, V.I. (2012). Hidden attractor in smooth Chua systems. Physica D: Nonlinear Phenomena, 241(18), 1482–1486. doi:10.1016/j.physd.2012.05.016. Leonov, G. (2006). Phase synchronization: Theory and applications. Avtomat. i Telemekh., (10), 47–85. Leonov, G. and Kondrateva, N. (2009). Analysis of stability of alternating current electric machines. Izd. St.Peterburg. Univ.
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Speed regulation of induction motors with wound rotor G.A. Leonov, E.P. Solovyeva, A.M. Zaretskiy Department of Mathematical Information Technology, University of Jyv¨ askyl¨ a P.O. Box 35 (Agora), Finland, FI-40014 Department of Applied Cybernetics, Saint-Petersburg State University Universitetsky pr. 28, Saint-Petersburg, Russia, 198504 Abstract: The regulation of speed is required in many applications of induction motors, including in electric locomotives and electric driving vehicles. In this paper the speed control problem for a wound rotor induction motor is considered. The speed is controlled by changing the external resistance in the rotor circuit. It is assumed that induction motors operate under constant load conditions during changes of the external resistance. A mathematical model of the induction motor with wound rotor proposed in this paper has periodical solutions. In such model the complex effects, which can’t be found numerically, may appear such as hidden oscillations (see, e.g., [J. de Bruin et al., 2009; M.A. Kiseleva et al., 2012]). Using the non-local reduction method the system of induction motors is studied and analytical estimations of speed control range are obtained. Analytical results are compared with computer experiments. Keywords: induction motor, wound rotor, speed control, non-local reduction method 1. INTRODUCTION
2. MODELLING OF WOUND ROTOR INDUCTION MOTOR
In engineering practice it is frequently needed to regulate the speed of induction motors. For example, an induction motor driving the metal-working machine should operate at various speeds because the blanks are processed by rough cutting at low speed and by finishing operations at high speed. The speed control is necessary in industrial applications (crane, hoist) and in electric vehicles (electric locomotives). The speed of induction motors can be controlled by variation of frequency of power supply, by variation of supply voltage, or by varying the slip frequency (Leonhard, 2001; Herman, 2009). For a wound rotor induction motor, the speed control is frequently achieved by varying the additional (external) resistance in the rotor circuit (Agrawal, 2001; Bakshi and Bakshi, 2009; Subrahmanyam, 2011). This way to control the speed of the wound rotor induction motor is used in applications requiring the motor to operate over a wide range of speeds.
In this paper several assumptions are made to develop a mathematical model of the wound rotor induction motor: magnetic permeability of materials of the stator and rotor is infinite; saturation, steel losses and mutual inductance are ignored; electromagnetic processes in the rotor windings do not affect the currents in the stator windings, i.e. magnetic field vector is constant in magnitude and rotates with constant angular speed. The last assumption dates back to classical ideas of N. Tesla and G. Ferraris (Tesla, 1888; Ferraris, 1888) and allows to describe the dynamics of the wound rotor induction motor by the dynamics of its rotor.
In this paper a mathematical model of a wound rotor induction motor, which fully takes into account the geometry of its rotor, is proposed. The speed control of the wound rotor induction motor by varying the rotor resistance for constant torque is considered. Using the non-local reduction method (Leonov et al., 1992; Gelig et al., 2004; Leonov, 2006; Leonov and Solovyeva, 2012) analytical estimations of speed control range are obtained. The results of computer experiments are given.
Fig. 1. Scheme of the wound rotor with the additional resistance r The wound rotor winding is considered as three identical coils displaced 120 electrical degrees apart and connected to the star (in Fig. 1). Free ends of the rotor winding are connected through slip rings and brushes to external resistances. These resistances are varied in order to control the motor speed. The motion of the wound rotor is considered in a rotating system of coordinates rigidly → − connected to the magnetic induction vector B . Using
? This work was supported by the Government of the Russian Federation (Ministry of Education and Science of the Russian Federation), Russian Foundation for Basic Research and Saint-Petersburg State University (Russia), the Faculty of Information Technology (University of Jyv¨ askyl¨ a, Finland).
978-3-902823-38-0/2013 © IFAC
90
10.3182/20130703-3-FR-4039.00031
2013 IFAC PSYCO July 3-5, 2013. Caen, France
of the parameter c∗ is changed to c. Hence, a new stable equilibrium point is as follows: p c a − a2 − 4γ 2 γs0 γ s0 = , x0 = − , y0 = − . 2γ ac a
the approach suggested in (Leonov, 2006; Leonov and Kondrateva, 2009; Leonov and Solovyeva, 2012), we obtain the system of differential equations of induction motor with the wound rotor: ˙ Li˙ 1 + (R + r) i1 = −2nSB cos(θ) θ, Li˙ 2 + (R + r) i2 = −2nSB cos(θ +
π ˙ ) θ, 3
Li˙ 3 + (R + r) i3 = −2nSB cos(θ +
2π ˙ ) θ, 3
The mathematical formulation of the speed regulation problem for an induction motor with the wound rotor under fixed load conditions is as follows: to find conditions under which the solution of the system (2) x(t), y(t), s(t) with initial data s = s∗ , x∗ , y∗ belongs to the attraction domain of the stationary solution s0 , x0 , y0 . It means that relations
(1)
π ) i2 3 2π + cos(θ + ) i3 − M, 3
J θ¨ = 2nSB cos θ i1 + cos(θ +
lim s(t) = s0 ,
lim y(t) = y0 (3)
t→+∞
Let us introduce the notation
c , c∗ p c a + a2 − 4γ 2 ρ=
s1 =
Using nonsingular change of coordinates ˙ s = θ, π 2π L (cos θ i1 + cos(θ + )i2 + cos(θ + )i3 ), x= 3nSB 3 3
,
2γ
γ ψ(s) = − s2 + as − cγ, c 12 γ2 Γ = 2 max λ c − λ − 2 . 4c (c − λ) λ∈(0,c) Theorem 1. Suppose that γ < 2c2 and the following inequalities
L π 2π (sin θ i1 + sin(θ + )i2 + sin(θ + )i3 ), 3nSB 3 3 z = i1 + i3 − i2 ,
cΓ > γ, p a2 − 4γ 2 ρ>1− a
system (1) reduces to the following form θ˙ = s, s˙ = ay + γ, x˙ = −cx + ys, y˙ = −cy − xs − s, z˙ = −cz,
(4) (5)
are fulfilled. Then the solution of system (2) with initial γ ∗ data s = s∗ , x = − γs ac∗ , y = − a satisfies relations (3). Proof. The change of variables
2
where a = 6 (nSB) γ = M c = R+r JL , J , L . Note that the last equation can be integrated easily, and the remaining equations, except the first one, do not depend on θ, therefore, it suffices to consider the system s˙ = ay + γ, x˙ = −cx + ys, y˙ = −cy − xs − s.
t→+∞
must be satisfied.
where n – number of turns of one coil; S – area of one turn; R – resistance; r – additional resistance; L – inductance; i1 , i2 , i3 – currents; θ – angle between plane of turns of coil with current i1 and plane perpendicular to magnetic → − induction vector B ; J – inertia moment of the rotor; M – external load torque.
y=
lim x(t) = x0 ,
t→+∞
η = ay + γ,
z = −x −
γ s ac
reduces system (2) to the form s˙ = η, η˙ = −cη + azs − ψ(s), 1 γ z˙ = −cz − sη − η a ac
(2)
(6)
and transforms initial data s∗ , x∗ , y∗ of system (2) into ∗ initial data s = s∗ , η = 0, z = (ρ − 1) γs ac of system (6).
3. THE SPEED REGULATION PROBLEM FOR A WOUND ROTOR INDUCTION MOTOR
Under the condition 0 < γ < a/2, the stationary set of system (6) consists of two points: the asymptotically stable point (s0 , 0, 0) and the unstable point (s1 , 0, 0). Then for system (6), relations (3) take the form
Let us consider the speed regulation problem for system (2) of the induction motor with the wound rotor. The speed is regulated by varying the additional resistance in the rotor circuit. It is assumed that the induction motor operates in steady state conditions under fixed load γ and c = c∗ . Suppose that this steady state operation corresponds to the stable equilibrium of system (2): p c∗ a − a2 − 4γ 2 γs∗ γ s∗ = , x∗ = − , y∗ = − . 2γ ac∗ a Then, at time t = τ the external resistance in the rotor circuit is varied without changing the load γ, i.e. the value
lim s(t) = s0 ,
t→+∞
lim η(t) = 0,
t→+∞
lim z(t) = 0. (7)
t→+∞
Let us consider the equation F (s)F 0 (s) = −ΓF (s) − ψ(s)
(8)
with initial data F (s1 ) = 0. It is easy to show that inequalities (4) and (5) are equivalent to the following inequalities
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2013 IFAC PSYCO July 3-5, 2013. Caen, France
p c2 1 (1 − )2 (a − a2 − 4γ 2 )2 > 2 γ ρ p 2 1 2 a − a2 − 4γ 2 1 − , ρ p p 1 (a2 − 4γ 2 ) > ( − 1)(a − a2 − 4γ 2 ) a2 − 4γ 2 . ρ
Relations (12) and (13) imply the positive invariance of sets Ω = {W (s, η, z) < 0, s ∈ [s2 , s1 ]}, Ω1 = {W (s, η, z) < 0, s ∈ (−∞, s1 ]}, Ω2 = {V (s, η, z) < C}, Ω∗ = Ω1 ∩ Ω2 , where p 1 1 C > max{ (1 − )2 (a − a2 − 4γ 2 )2 + 8 ρ Zs∗ (14) + ψ(s)ds, 0}.
Γ2
(9)
(10)
Hense, 4Γ2 (s1 − s∗ )2 = p p 1 h a2 − 4γ 2 ) i2 a2 − 4γ 2 ) ρ c(a − 2 c(a + = 4Γ − = 2γ 2γ i 2h p p 2 c 1 = Γ2 2 (1 − )(a − a2 − 4γ 2 ) + 2 a2 − 4γ 2 = γ ρ
s1
Using (11), we obtain γs∗ ) < 0. ac Taking into account condition (14), we have γs∗ V (s∗ , 0, (ρ − 1) ) < C. ac Consequently, γs∗ γs∗ (s∗ , 0, (ρ − 1) ) ∈ Ω, (s∗ , 0, (ρ − 1) ) ∈ Ω∗ . ac ac
p c2 h 1 = Γ2 2 (1 − )2 (a − a2 − 4γ 2 )2 + γ ρ p p 1 +4(1 − )(a − a2 − 4γ 2 ) a2 − 4γ 2 + ρ i p 2 1 2 2 +4(a − 4γ 2 ) > a − a2 − 4γ 2 1 − . ρ Using the last inequality and the estimate obtained for the solution F (s) of equation (8) at s ∈ (0, s1 ) in (Barbashin and Tabueva, 1969) F (s) > Γ(s1 − s), we obtain p 1 1 F (s∗ ) > a − a2 − 4γ 2 1 − . (11) 2 ρ
W (s∗ , 0, (ρ − 1)
Let x(t) = (s(t), η(t), z(t)) be a solution of system (6) bounded at t ≥ 0. Then the function V (x(t)) is bounded at t ≥ 0 as well. Under the condition γ < 2c2 , by virtue of Sylvester’s criterion, the quadratic form from η and z of (13) is negative definite. Therefore, it follows from (13) that the function V (s(t), η(t), z(t)) does not increase in t for t ≥ 0. This fact and the boundedness of the function V (s(t), η(t), z(t)) for t ≥ 0 imply the existence of a finite limit lim V (s(t), η(t), z(t)) = const.
Thus, equation (8) has either a solution F (s) (curve 1 in Fig. 2) such that F (s2 ) = 0, s2 < s∗ , F (s) > 0, ∀s ∈ (s2 , s1 ); or a solution F (s) (curve 2 in Fig. 2) such that F (s) > 0, ∀s ∈ (−∞, s1 ).
t→+∞
It follows from the boundedness of the trajectory x(t) that set Ω0 of its omega-limit points is nonempty. Let y ∈ Ω0 . Then, by virtue of the invariance of Ω0 , the trajectory going from y is contained in Ω0 for all t ∈ R. Therefore, for all t ∈ R, we have V (s(t, y), η(t, y), z(t, y)) ≡ L. Using estimate (13), we obtain identities η(t, y) ≡ 0 and z(t, y) ≡ 0. Relations (6) and (13) imply s(t, ˙ y) ≡ 0. Therefore, s(t, y) ≡ const, and Ω0 is a subset of the stationary set of system (6).
Fig. 2. Two possible cases of behavior of solution of equation (8)
Thus, any bounded solution of system (6) tends to an equilibrium point. This fact, the boundedness and the positive invariance of sets Ω, Ω∗ and the inclusions γs∗ γs∗ (s∗ , 0, (ρ − 1) ) ∈ Ω, (s∗ , 0, (ρ − 1) ) ∈ Ω∗ , ac ac (s0 , 0, 0) ∈ Ω, (s0 , 0, 0) ∈ Ω∗ imply relations (7), which are equivalent relations (3). This completes the proof of the theorem.
Let us introduce functions a2 1 1 W (s, η, z) = z 2 + η 2 − F 2 (s), 2 2 2 Zs 2 a 1 V (s, η, z) = z 2 + η 2 + ψ(s)ds. 2 2 s1
Under conditions γ < 2c2 and (6), functions W (s, η, z), V (s, η, z) satisfy relations ˙ (s, η, z) + 2λW (s, η, z) ≤ 0, W (12)
Conditions (4) and (5) of Theorem 1 allow to obtain the lower estimate of the range of additional resistance regulation for induction motor with wound rotor under constant load. In Fig. 3 area 1 corresponds to the range of regulation of parameter ρ (the initial additional resistance r∗ = rf ix ) for varied parameter a of system (2) and for
√ γ γ2 V˙ (s, η, z) = −( √ η + caz)2 − (c − 3 )η 2 ≤ 0. (13) 4c 2c c
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given load γ = γ0 ; area 2 represents the range of regulation of parameter ρ as the initial additional resistance increases (r∗ > rf ix ) for given load γ = γ0 ; area 3 corresponds to the range of regulation of parameter ρ as the load decreases (permissible load γ < γ0 ). Thus, from the range of regulation of parameter ρ we can obtain the range ∗) of regulation of the additional resistance r = ρ(R+r R (r∗ > rf ix ) and the range of speed regulation s.
0.06
0.04
i2
0.02
0
−0.02
−0.04
−0.06 0.1 0 −0.1 i
2
1.5
1
0.5
0
1
2.5
3
3.5
s
Fig. 5. Trajectory of system (1) as the additional resistance decreases
0.2
0.15
i2
0.1
Fig. 3. The range of regulation of parameter ρ for different load and for different additional resistance: area 1 c∗ = 2, γ = 1; area 2 - c∗ = 3, γ = 1; area 3 - c∗ = 3, γ = 1.8
0.05
0
−0.05 0.2 0.1 0
A typical situation during the operation of an induction motor is as follows: the motor runs in no-load mode, during transient process it is pulled in synchronism, next load-on occurs and finally the speed control of motor is required without changing the load. An operating mode of the wound rotor induction motor under no-load conditions (γ = 0) corresponds to a stationary solution θ = const, θ˙ = s = 0, i1 = 0, i2 = 0, i3 = 0 of system (1). Suppose that the initial condition is θ = 0, θ˙ = s = 0, i1 = 0, i2 = 0, i3 = 0. After the permissible sudden change of load (Leonov and Solovyeva, 2012) the trajectory, after some transient process, tends to the periodical solution of system (1) (see blue part of trajectories in Figs. 4-6). This solution corresponds to an operating mode of wound rotor induction motor under load.
−0.1 i1
i2
−0.04 0.05
0
3
2
4
5
6
4
5
6
s
The speed control of induction motors is one of the most important engineering problems. In this paper the speed control problem is considered for an induction motor with wound rotor. It is shown that one of simple ways to regulate the speed of wound rotor induction motor is variation of the additional resistance in the rotor circuit. The found estimations of range of the additional resistance regulation prove that the motor can operate over a wide range of speed. This property allows to use the wound rotor induction motor in many applications. Results of computer experiments show that varying the additional resistance not only decreases or increases motor speed (in Figs. 4 5) but also may lead to an unstable operating mode (Fig. 6). However, in more complex system such as drilling machine, grinding machine, the complex dynamical effects may appear. In the works (Bruin et al., 2009; Leonov and Kuznetsov, 2013a,b; Kiseleva et al., 2012) the drilling system actuated by the induction motor is considered and hidden oscillations (Leonov et al., 2012, 2011; Bragin et al., 2011; Kuznetsov et al., 2011; Leonov and Kuznetsov, 2011) were found. Thus, more careful analysis of induction machines and their control methods are required.
−0.02
1
3
2
4. CONCLUSION
0
0
1
the additional resistance is illustrated. This means that the induction motor does not pull in synchronism and the motor stops.
0.02
−0.05
0
Fig. 6. Trajectory of system (1) at impermissible change of the additional resistance
0.04
i1
−0.2
7
s
Fig. 4. Trajectory of system (1) as the additional resistance increases For different additional resistance values (parameter r of system (1)) computer modeling is done. In Figs. 4-6 the green parts of trajectories correspond to transient process, arising from variations of the additional resistance. For permissible given load, the higher the additional resistance is, the lower the rotation speed of induction motor becomes (see in Figs. 4 - 5). In Fig. 6 an impermissible change of
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