Nano and Microelectromechanical Systems P5

3.4. INDUCTION MOTORS In this section, the following variables and symbols are used: u u as bs , and u cs are the phase voltages in the stator windings as, bs and cs; u u qs ds , and u os are the quadrature-, direct-, and zero-axis components of stator voltages; i i as bs , and i cs are the phase currents in the stator windings as, bs and cs; i i qs ds , and i os are the quadrature-, direct-, and zero-axis components of stator currents; ψ ψ as bs , and ψ cs are the stator flux linkages; ψ ψ qs ds , and ψ os are the quadrature-, direct-, and zero-axis components of stator flux linkages; u u ar br , and u cr are the voltages in the rotor windings ar, br and cr; u u qr dr , and u or are the quadrature-, direct-, and zero-axis components of rotor voltages; i i ar br , and i cr are the currents in the rotor windings ar, br and cr; i i qr dr , and i or are the quadrature-, direct-, and zero-axis components of rotor currents; ψ ψ ar br , and ψ cr are the rotor flux linkages; ψ ψ qr dr , and ψ or are the quadrature-, direct-, and zero-axis components of rotor flux linkages; r ω and rm ω are the electrical and mechanical angular velocities; r θ and rm θ are the electrical and mechanical angular displacements; T e is the electromagnetic torque developed by the motor; T L is the load torque applied; s r and r r are the resistances of the stator and rotor windings; L ss and rr L are the self-inductances of the stator and rotor windings; ms L is the stator magnetizing inductance; ls L and lr L are the stator and rotor leakage inductances; N s and N r are the number of turns of the stator and rotor windings; P is the number of poles; m B is the viscous friction coefficient; J is the equivalent moment of inertia; ω and θ are the angular velocity and displacement of the reference frame. © 2001 by CRC Press LLC 3.4.1. Two-Phase Induction Motors Two-phase induction motors, shown in Figure 3.4.1, have two stator and rotor windings. r s + − +− L ss r s L ss i bs u bs u as i as N s ω r e T, T L θ ω θ r r r t= + 0 B m as' bs as bs' br br' ar ar' as Magnetic Axis ar Magnetic Axis br Magnetic Axis bs Magnetic Axis r r + − + − L rr r r L rr i br u br u ar i ar N r Stator Rotor Load Magnetic Coupling Figure 3.4.1. Two-phase symmetrical induction motor To develop a mathematical model of two-phase induction motors, we model the stator and rotor circuitry dynamics. As the control and state variables we use the voltages applied to the stator (as and bs) and rotor (ar and br) windings, as well as the stator and rotor currents and flux linkages. Using Kirchhoff’s voltage law, four differential equations are u r i d dt as s as as = + ψ , u r i d dt bs s bs bs = + ψ , u r i d dt ar r ar ar = + ψ , u r i d dt br r br br = + ψ . Hence, in matrix form we have u r i abs s abs abs d dt = + ψψ , u r i abr r abr abr d dt = + ψψ , (3.4.1) © 2001 by CRC Press LLC where u abs as bs u u =       , u abr ar br u u =       , i abs as bs i i =       , i abr ar br i i =       , ψψ abs as bs =       ψ ψ , and ψψ abr ar br =       ψ ψ are the phase voltages, currents, and flux linkages; r s s s r r =       0 0 and r r r r r r =       0 0 are the matrices of the stator and rotor resistances. Studying the magnetically coupled motor circuits, the following matrix equation for the flux linkages is found ψψ ψψ abs abr s sr sr T r abs abr       =             L L L L i i , where L s is the matrix of the stator inductances, L s ss ss L L =       0 0 , L L L ss ls ms = + , L N ms s m = ℜ 2 ; L r is the matrix of the rotor inductances, L r rr rr L L =       0 0 , L L L rr lr mr = + , L N mr r m = ℜ 2 ; L sr is the matrix of the stator-rotor mutual inductances, L sr sr r sr r sr r sr r L L L L = −       cos sin sin cos θ θ θ θ , L N N sr s r m = ℜ . Using the number of turns in the stator and rotor windings, we have i i abr r s abr N N ' = , u u abr s r abr N N ' = , and ψψ ψψ abr s r abr N N ' = . Then, taking note of the turn ratio, the flux linkages are written in matrix form as ψψ ψψ abs abr s sr sr T r abs abr ' ' ' ' '       =             L L L L i i , (3.4.2) where L L r s r r rr rr N N L L ' ' ' =       =       2 0 0 , L L L rr lr mr ' ' ' = + ; © 2001 by CRC Press LLC L L sr s r sr ms r r r r N N L ' cos sin sin cos =       = −       θ θ θ θ , L N N L ms s r sr = , L N N L mr s r mr ' =       2 , L L N N L mr ms s r sr ' = = , L L L rr lr ms ' ' = + . Substituting the matrices for self- and mutual inductances L s , L r ' and L sr ' in (3.4.2), one obtains ψ ψ ψ ψ θ θ θ θ θ θ θ θ as bs ar br ss ms r ms r ss ms r ms r ms r ms r rr ms r ms r rr as bs ar br L L L L L L L L L L L L i i i i ' ' ' ' ' ' cos sin sin cos cos sin sin cos             = − −                         0 0 0 0 . Therefore, the circuitry differential equations (3.4.1) are rewritten as u r i abs s abs abs d dt = + ψψ , u r i abr r abr abr d dt ' ' ' ' = + ψψ . where r r r s r r s r r r N N N N r r ' ' ' = =       2 2 2 2 0 0 . Assuming that the self- and mutual inductances L L L ss rr ms , , ' are time- invariant and using the expressions for the flux linkages, one obtains a set of nonlinear differential equations to model the circuitry dynamics ( ) ( ) L di dt L d i dt L d i dt r i u ss as ms ar r ms br r s as as + − = − + ' ' cos sinθ θ , ( ) ( ) L di dt L d i dt L d i dt r i u ss bs ms ar r ms br r s bs bs + + = − + ' ' sin cosθ θ , ( ) ( ) L d i dt L d i dt L di dt r i u ms as r ms bs r rr ar r ar ar cos sin ' ' ' ' ' θ θ + + = − + , ( ) ( ) − + + = − +L d i dt L d i dt L di dt r i u ms as r ms bs r rr br r br br sin cos ' ' ' ' ' θ θ . Cauchy’s form of these differential equations is found. In particular, we have the following nonlinear differential equations to model the stator-rotor circuitry dynamics for two-phase induction motors © 2001 by CRC Press LLC di dt L r L L L i L L L L i L L L L L i r L L L L L L i r L L L L L u L L L L u as rr s ss rr ms as ms ss rr ms bs r ms rr ss rr ms ar r r r rr r ms rr ss rr ms br r r r rr r rr ss rr ms as ms ss rr ms r ar = − − + − + − +       + − −       + − − − + ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' sin cos cos sin cos 2 2 2 2 2 2 2 ω ω θ θ ω θ θ θ L L L L u ms ss rr ms r br ' ' sin , − 2 θ di dt L r L L L i L L L L i L L L L L i r L L L L L L i r L L L L L u L L L L u bs rr s ss rr ms bs ms ss rr ms as r ms rr ss rr ms ar r r r rr r ms rr ss rr ms br r r r rr r rr ss rr ms bs ms ss rr ms r ar = − − − − − − −       + − +       + − − − − ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' cos sin sin cos sin 2 2 2 2 2 2 2 ω ω θ θ ω θ θ θ L L L L u ms ss rr ms r br ' ' cos , − 2 θ di dt L r L L L i L L L L L i r L L L L L L i r L L L L L i L L L L u L L L L u L L ar ss r ss rr ms ar ms ss ss rr ms as r r s ss r ms ss ss rr ms bs r r s ss r ms ss rr ms br r ms ss rr ms r as ms ss rr ms r bs ss ' ' ' ' ' ' ' ' ' ' sin cos cos sin cos sin = − − + − +       − − −       − − − − − − + 2 2 2 2 2 2 2 ω θ θ ω θ θ ω θ θ ss rr ms ar L L u ' ' , − 2 di dt L r L L L i L L L L L i r L L L L L L i r L L L L L i L L L L u L L L L u L L br ss r ss rr ms br ms ss ss rr ms as r r s ss r ms ss ss rr ms bs r r s ss r ms ss rr ms ar r ms ss rr ms r as ms ss rr ms r bs ss ' ' ' ' ' ' ' ' ' ' cos sin sin cos sin cos = − − + − −       + − +       + − + − − − + 2 2 2 2 2 2 2 ω θ θ ω θ θ ω θ θ ss rr ms br L L u ' ' . − 2 (3.4.3) The electrical angular velocity ω r and displacement θ r are used in (3.4.3) as the state variables. Therefore, the torsional-mechanical equation of motion must be incorporated to describe the evolution of ω r and θ r . From Newton’s second law, we have T T B T J d dt e m rm L rm ∑ = − − =ω ω , d dt rm rm θ ω= . The mechanical angular velocity ω rm is expressed by using the electrical angular velocity ω r and the number of poles P. In particular, ω ω rm r P = 2 . The mechanical and electrical angular displacements θ rm and θ r are related as θ θ rm r P = 2 . Taking note of Newton’s second law of motion, one obtains © 2001 by CRC Press LLC d dt P J T B J P J T r e m r L ω ω= − − 2 2 , d dt r r θ ω= . To find the expression for the electromagnetic torque developed by two- phase induction motors, the coenergy ( ) rabrabsc W θ,, ' ii is used, and ( ) r rabrabsc e WP T ∂θ θ∂ ,, 2 ' ii = . Assuming that the magnetic system is linear, one has ( ) ( ) W W L L c f abs T s ls abs abs T sr abr abr T r lr abr = = − + + − 1 2 1 2 i L I i i L i i L I i ' ' ' ' ' ' . The self-inductances L ss and L rr ' , as well as the leakage inductances L ls and L lr ' , are not functions of the angular displacement θ r , while the following expression for the matrix of stator-rotor mutual inductances L sr ' was derived L sr ms r r r r L ' cos sin sin cos = −       θ θ θ θ . Then, for P-pole two-phase induction motors, the electromagnetic torque is given by ( ) [ ] ( ) ( ) [ ] T P W P P L i i i i P L i i i i i i i i e c abs abr r r abs T sr r r abr ms as bs r r r r ar br ms as ar bs br r as br bs ar r = = = − − −             = − + + − 2 2 2 2 ∂ θ ∂θ ∂ θ ∂θ θ θ θ θ θ θ i i i L i , , ( ) sin cos cos sin sin cos . ' ' ' ' ' ' ' ' ' (3.4.4) Using (3.4.4) for the electromagnetic torque T e in the torsional- mechanical equations of motion, one obtains ( ) ( ) [ ] d dt P J L i i i i i i i i B J P J T r ms as ar bs br r as br bs ar r m r L ω θ θ ω= − + + − − − 2 4 2 ' ' ' ' sin cos d dt r r θ ω= . (3.4.5) Augmenting differential equations (3.4.3) and (3.4.5), the following set of highly nonlinear differential equations results © 2001 by CRC Press LLC ,sincossincos cossin '' ' ' ' ' ' ' ' ' '2' brr ms arr ms as rr r rr r rrbr rrms r rr r rrar rrms rbs ms as srras u L L u L L u L L L r i L LL L r i L LL i L L i L rL dt di θθθθω θθωω ΣΣΣΣ ΣΣΣ +−+         −+         +++−= ,cossincossin sincos '' ' ' ' ' ' ' ' ' '2' brr ms arr ms bs rr r rr r rrbr rrms r rr r rrar rrms ras ms bs srrbs u L L u L L u L L L r i L LL L r i L LL i L L i L rL dt di θθθθω θθωω ΣΣΣΣ ΣΣΣ −−+         ++         −−−−= ,sincos sincoscossin '' 2 ' '' ar ss bsr ms asr ms rbr ms r ss s rrbs ssms r ss s rras ssms ar rssar u L L u L L u L L i L L L r i L LL L r i L LL i L rL dt di ΣΣΣΣ ΣΣΣ +−−−         −−         ++−= θθω θθωθθω ,cossin cossinsincos '' 2 ' '' br ss bsr ms asr ms rar ms r ss s rrbs ssms r ss s rras ssms br rssbr u L L u L L u L L i L L L r i L LL L r i L LL i L rL dt di ΣΣΣΣ ΣΣΣ +−++         ++         −+−= θθω θθωθθω ( ) ( ) [ ] d dt P J L i i i i i i i i B J P J T r ms as ar bs br r as br bs ar r m r L ω θ θ ω= − + + − − − 2 4 2 ' ' ' ' sin cos , d dt r r θ ω= , (3.4.6) where L L L L ss rr msΣ = − ' 2 . In matrix form, a set of six highly coupled nonlinear differential equations (3.4.6) is © 2001 by CRC Press LLC di dt di dt di dt di dt d dt d dt L r L L r L L r L L r L B J as bs ar br r r rr s rr s ss r ss r m ' ' ' ' ' ' ω θ                                 = − − − − −                               Σ Σ Σ Σ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0                           i i i i as bs ar br r r ' ' ω θ + + +       + −       − − −       + +       L L i L L L i r L L L L i r L L L i L L L i r L L L L i r L L L ms bs r ms rr ar r r r rr r ms rr br r r r rr r ms as r ms rr ar r r r rr r ms rr br r r r rr r ms 2 2 Σ Σ Σ Σ Σ Σ ω ω θ θ ω θ θ ω ω θ θ ω θ θ ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' sin cos cos sin cos sin sin cos ( ) ss as r r s ss r ms ss bs r r s ss r ms br r ms ss as r r s ss r ms ss bs r r s ss r ms ar r ms as ar bs br r L i r L L L L i r L L L i L L L i r L L L L i r L L L i P J L i i i i Σ Σ Σ Σ Σ Σ ω θ θ ω θ θ ω ω θ θ ω θ θ ω θ sin cos cos sin cos sin sin cos sin ' ' ' ' +       − −       − −       + +       + − + 2 2 2 4 ( ) [ ] + −                                       i i i i as br bs ar r ' ' cosθ 0 +                                                   + − + − − − L L L L L L L L u u u u L L u L L u L L u L L u L L rr rr ss ss as bs ar br ms r ar ms r br ms r ar ms r br ms ' ' ' ' ' ' ' ' cos sin sin cos cos Σ Σ Σ Σ Σ Σ Σ Σ Σ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 θ θ θ θ θ r as ms r bs ms r as ms r bs L u L L u L L u L L u P J T − −                                   −                               Σ Σ Σ sin sin cos θ θ θ 0 0 0 0 0 0 2 0 (3.4.7) © 2001 by CRC Press LLC Modeling Two-Phase Induction Motors Using the Lagrange Equations The mathematical model can be derived using Lagrange’s equations. The generalized independent coordinates and the generalized forces are q i s as 1 = , q i s bs 2 = , q i s ar 3 = ' , q i s br 4 = ' , q r5 = θ , and Q u as1 = , Q u bs2 = , Q u ar3 = ' , Q u br4 = ' , Q T L5 = − Five Lagrange equations are written as d dt q q D q q Q ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ Γ Γ Π & & 1 1 1 1 1       − + + = , d dt q q D q q Q ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ Γ Γ Π & & 2 2 2 2 2       − + + = , d dt q q D q q Q ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ Γ Γ Π & & 3 3 3 3 3       − + + = , d dt q q D q q Q ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ Γ Γ Π & & 4 4 4 4 4       − + + = , d dt q q D q q Q ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ Γ Γ Π & & 5 5 5 5 5       − + + = . The total kinetic, potential, and dissipated energies are ,cossin sincos 2 5 2 1 2 4 ' 2 1 2 3 ' 2 1 542532 2 2 2 1 541531 2 1 2 1 qJqLqLqqqLqqqL qLqqqLqqqLqL rrrrmsms ssmsmsss &&&&&&& &&&&&& +++++ +−+=Γ Π = 0 , ( ) D r q r q r q r q B q s s r r m = + + + + 1 2 1 2 2 2 3 2 4 2 5 2 & & & & & ' ' . Thus, ∂ ∂ Γ q 1 0 = , ∂ ∂ Γ & & & cos & sin q L q L q q L q q ss ms ms 1 1 3 5 4 5 = + − , ∂ ∂ Γ q 2 0 = , ∂ ∂ Γ & & & sin & cos q L q L q q L q q ss ms ms 2 2 3 5 4 5 = + + , ∂ ∂ Γ q 3 0 = , ∂ ∂ Γ & & & cos & sin ' q L q L q q L q q rr ms ms 3 3 1 5 2 5 = + + , ∂ ∂ Γ q 4 0 = , ∂ ∂ Γ & & & sin & cos ' q L q L q q L q q rr ms ms 4 4 1 5 2 5 = − + , © 2001 by CRC Press LLC ( ) ( ) [ ] ,cossin sincos cossin 5324154231 542532 541531 5 qqqqqqqqqqL qqqLqqqL qqqLqqqL q ms msms msms &&&&&&&& &&&& &&&& −++−= −+ −−= Γ ∂ ∂ ∂ ∂ Γ & & q Jq 5 5 = , ∂ ∂ Π q 1 0= , ∂ ∂ Π q 2 0= , ∂ ∂ Π q 3 0= , ∂ ∂ Π q 4 0= , ∂ ∂ Π q 5 0= , ∂ ∂ D q r q s & & 1 1 = , ∂ ∂ D q r q s & & 2 2 = , ∂ ∂ D q r q r & & ' 3 3 = , ∂ ∂ D q r q r & & ' 4 4 = , ∂ ∂ D q B q m & & 5 5 = . Taking note of & q i as1 = , & q i bs2 = , & ' q i ar3 = , & ' q i br4 = and & q r5 = ω , one obtains ( ) ( ) L di dt L d i dt L d i dt r i u ss as ms ar r ms br r s as as + − + = ' ' cos sinθ θ , ( ) ( ) L di dt L d i dt L d i dt r i u ss bs ms ar r ms br r s bs bs + + + = ' ' sin cosθ θ , ( ) ( ) L d i dt L d i dt L di dt r i u ms as r ms bs r rr ar r ar ar cos sin ' ' ' ' ' θ θ + + + = , ( ) ( ) − + + + =L d i dt L d i dt L di dt r i u ms as r ms bs r rr br r br br sin cos ' ' ' ' ' θ θ , ( ) ( ) [ ] J d dt L i i i i i i i i B d dt T r ms as ar bs br r as br bs ar r m r L 2 2 θ θ θ θ + + + − + = − ' ' ' ' sin cos For P-pole induction motors, by making use of r r dt d ω θ = , six differential equations, as found in (3.4.6), result. Control of Induction Motors The angular velocity of induction motors must be controlled, and the torque-speed characteristic curves should be thoroughly examined. The electromagnetic torque developed by two-phase induction motors is given by equation (3.4.4). To guarantee the balanced operating condition for two- phase induction motors, one supplies the following phase voltages to the stator windings ( ) u t u t as M f ( ) cos= 2 ω , ( ) u t u t bs M f ( ) sin= 2 ω , © 2001 by CRC Press LLC [...]... following variables and symbols are used: uas , ubs and ucs are the phase voltages in the stator windings as, bs and cs; uqs , uds and uos are the quadrature-, direct-, and zero-axis stator voltage components; ias , ibs and ics are the phase currents in the stator windings as, bs and cs; iqs , ids and ios are the quadrature-, direct-, and zero-axis stator current components; ψ as , ψ bs and ψ cs are the... windings; Lms and Lls are the stator magnetizing and leakage inductances; Lmq and Lmd are the magnetizing inductances in the quadrature and direct axes; ℜ md and ℜ mq are the magnetizing reluctances in the direct and quadrature axes; N s is the number of turns of the stator windings; P is the number of poles; ω and θ are the angular velocity and displacement of the reference frame Micro- and miniscale... applied to adjust the magnitude u M fi following relation and frequency f of the supplied voltages To attain the acceleration and settling time specified, overshoot and rise time needed, the general purpose (standard), soft- and high-starting torque patterns are implemented based upon the requirements and criteria imposed (see the standard, soft- and high-torque patterns as illustrated in Figure 3.4.2.d)... components; ψ as , ψ bs and ψ cs are the stator flux linkages; ψ qs , ψ ds and ψ 0s are the quadrature-, direct-, and zero-axis stator flux linkages components; ψ m is the magnitude of the flux linkages established by the permanentmagnets; ω r and ω rm are the electrical and rotor angular velocities; θ r and θ rm are the electrical and rotor angular displacements; Te is the electromagnetic torque developed;... Press LLC Multiplying left and right sides of equations (3.4.20) by K s and K r , one has dK −1 − dψ qdos s ψ qdos + K s K s 1 , dt dt ' − dK r 1 ' − ' − dψ qdor u 'qdor = K r rr' K r 1i qdor + K r ψ qdor + K r K r 1 (3.4.21) dt dt The matrices of the stator and rotor resistances rs and rr' are diagonal, − u qdos = K s rs K s 1i qdos + K s and hence, − K s rs K −1 = rs and K r rr' K r 1 = rr' s Performing... synchronous reference frames, the reference frame angular velocities are ω = 0 , ω = ω r and ω = ω e , and the corresponding angular displacement θ results In particular, for zero initial conditions for stationary, rotor, and synchronous reference frames one finds θ = 0 , θ = θ r and θ = θ e Hence, the quadrature-, direct-, and zero-axis components of voltages can be obtained to guarantee the balance operation... ψ ar    and ψ abcr = ψ br   ψ cr    In (3.4.9), the diagonal matrices of the stator and rotor resistances are rs rs =  0  0  0 rs 0 0 rr  and r =  0 0 r   0 rs    0 rr 0 0 0  rr   The flux linkages equations must be thoroughly examined, and one has  ψ abcs   L s ψ  = L T  abcr   sr L sr  i abcs  , L r  i abcr    where the matrices of self- and mutual... (3.4.13) and (3.4.15), the resulting model for three-phase induction motors in the machine variables, is found Mathematical Model of Three-Phase Induction Motors in the Arbitrary Reference Frame The abc stator and rotor variables must be transformed to the quadrature, direct, and zero quantities To transform the machine (abc) stator voltages, currents, and flux linkages to the quadrature-, direct-, and. .. (3.4.8) uar = rr iar + dt dt dt It is clear that the abc stator and rotor voltages, currents, and flux linkages are used as the variables, and in matrix form equations (3.4.8) are rewritten as © 2001 by CRC Press LLC dψ abcs , dt dψ abcr + , dt u abcs = rs i abcs + u abcr = rr i abcr (3.4.9) where the abc stator and rotor voltages, currents, and flux linkages are i ar  uas  uar  ias  ψ as  i... synchronous machines can be used as motors and generators Generators convert mechanical energy into electrical energy, while motors convert electrical energy into mechanical energy A broad spectrum of synchronous electric machines can be used in electric drives, servos, and power systems applications We will develop nonlinear mathematical models, and perform nonlinear modeling and analysis of synchronous machines . the stator and rotor circuitry dynamics. As the control and state variables we use the voltages applied to the stator (as and bs) and rotor (ar and br) windings,. purpose (standard), soft- and high-starting torque patterns are implemented based upon the requirements and criteria imposed (see the standard, soft- and high-torque