Author: Ion Boldea, S.A.Nasar………… ……… Chapter 13 INDUCTION MACHINE TRANSIENTS 13.1. INTRODUCTION Induction machines undergo transients when voltage, current, and (or) speed undergo changes. Turning on or off the power grid leads to starting transients an induction motor. Reconnecting an induction machine after a short-lived power fault (zero current) is yet another transient. Bus switching for large power induction machines feeding urgent loads also qualifies as large deviation transients. Sudden short-circuits, at the terminals of large induction motors lead to very large peak currents and torques. On the other hand more and more induction motors are used in variable speed drives with fast electromagnetic and mechanical transients. So, modeling transients is required for power-grid-fed (constant voltage and frequency) and for PWM converter-fed IM drives control. Modeling the transients of induction machines may be carried out through circuit models or coupled field/circuit models (through FEM). We will deal first with phase-coordinate abc model with inductance matrix exhibiting terms dependent on rotor position. Subsequently, the space phasor (d–q) model is derived. Both single and double rotor circuit models are dealt with. Saturation is also included in the space-phasor (d–q) model. The abc–dq model is then derived and applied, as it is adequate for nonsymmetrical voltage supplies and PWM converter-fed IMs. Reduced order d–q models are used to simplify the study of transients for low and large motors, respectively. Modeling transients with the computation of cage bar and end-ring currents is required when cage and/or end-ring faults occur. Finally the FEM coupled field circuit approach is dealt with. Autonomous generator transients are left out as they are treated in the chapter dedicated to induction generators. 13.2. THE PHASE COORDINATE MODEL The induction machine may be viewed as a system of electric and magnetic circuits which are coupled magnetically and/or electrically. An assembly of resistances, self inductances, and mutual inductances is thus obtained. Let us first deal with the inductance matrix. A symmetrical (healthy) cage may be replaced by a wound three-phase rotor. [2] Consequently, the IM is represented by six circuits, (phases) (Figure 14.1). Each of them is characterized by a self inductance and 5 mutual inductances. © 2002 by CRC Press LLC Author: Ion Boldea, S.A.Nasar………… ……… The stator and rotor phase self inductances do not depend on rotor position if slot openings are neglected. Also, mutual inductances between stator phases and rotor phases, respectively, do not depend on rotor position. A sinusoidal distribution of windings is assumed. Finally, stator/rotor phase mutual inductances depend on rotor position ( θ er = p 1 θ r ). The induction matrix, ( ) ercbabca rrr L θ is () [] =θ rrrrrrrrr rrrrrrrrr rrrrrrrrr rrr rrr rrr rrr cccbcaccbcac cbbbbacbbbab cabaaacabaaa cccbcaccbcac bcbbbabcbbab acabaaacabaa ercbabca LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL L (13.1) ω r a c b a c b r er r θ Figure 13.1 Three-phase IM with equivalent wound rotor with π +θ=== −=== π −θ=== +===θ=== −===+=== 3 2 cosLLLL ;2/LLLL ; 3 2 cosLLLL ;LLLLL ;cosLLLL ;2/LLLL ;LLLLL ersrmcaabbc r mrcbcabaersrmcbbaac r mr r lrccbbaaersrmccbbaa msbcacabmslsccbbaa rrr rrrrrrrrr rrrrrrrrr (13.2) Assuming a sinusoidal distribution of windings, it may be easily shown that r mrmssrm LLL ⋅= (13.3) © 2002 by CRC Press LLC Author: Ion Boldea, S.A.Nasar………… ……… Reducing the rotor to stator is useful especially for cage rotor IMs, no access to rotor variables is available. In this case, the mutual inductance becomes equal to self inductance L srm Æ L sm and the rotor self inductance equal to the stator self inductance L mr r Æ L sm . To conserve the fluxes and losses with stator reduced variables, rs sm srm r cr cr r br br r ar ar K L L i i i i i i ==== (13.4) rscr r cr br r br ar r ar r cr cr r br br r ar ar K 1 i i i i i i V V V V V V ====== (13.5) 2 rs r lr lr r r r K 1 L L R R == (13.6) The expressions of rotor resistance R r , leakage inductance L lr , both reduced to the stator for both cage and wound rotors are given in Chapter 6. The same is true for R s , L ls of the stator. The magnetization self inductance L sm has been calculated in Chapter 5. Now the matrix form of phase coordinate (variable) model is [] [][] [] [] [] [] [] [] [ ] rrrsss T cbacba T cbacba R,R,R,R,R,RDiagR i,i,i,i,i,ii V,V,V,V,V,VV dt d iRV rrr rrr = = = Ψ+= (13.7) [] () [ ] [] iL ercbabca rrr θ=Ψ (13.8) () [ ] +−−θ π +θ π −θ −+− π −θθ π +θ −−+ π +θ π −θθ θ π −θ π +θ+−− π +θθ π −θ−+− π −θ π +θθ−−+ = =θ smlssmsmersmersrmersrm smsmlssmersrmersmersrm smsmsmlsersrmersrmersm ersmersrmersrmsmlssmsm ersrmersmersrmsmsmlssm ersrmersrmersmsmsmsmls ercbabca LL2/L2/LcosL 3 2 cosL 3 2 cosL 2/LLL2/L 3 2 cosLcosL 3 2 cosL 2/L2/LLL 3 2 cosL 3 2 cosLcosL cosL 3 2 cosL 3 2 cosLLL2/L2/L 3 2 cosLcosL 3 2 cosL2/LLL2/L 3 2 cosL 3 2 cosLcosL2/L2/LLL L rrr (13.9) With (13.8), (13.7) becomes © 2002 by CRC Press LLC Author: Ion Boldea, S.A.Nasar………… ……… [] [][] [] [] [] [] [] dt d i d Ld dt id i i L LiRV er er θ θ + ∂ ∂ ++= (13.10) Multiplying (13.10) by [i] T we get [] [ ] [] [] [][][] [] [][] r er TTTT iL d d i 2 1 iiL 2 1 dt d iRiVi ω θ + += (13.11) The first term represents the winding losses, the second, the stored magnetic energy variation, and the third, the electromagnetic power P e . [] [] [] r er T 1 r ee i d Ld i 2 1 p TP ω θ = ω = (13.12) The electromagnetic torque T e is [] [] [] i d Ld ip 2 1 T er T 1e θ = (13.13) The motion equation is r er loade r 1 dt d ;TT dt d p J ω= θ −= ω (13.14) An 8 th order nonlinear model with time-variable coefficients (inductances) has been obtained, even with core loss neglected. Numerical methods are required to solve it, but the computation time is prohibitive. Consequently, the phase coordinate model is to be used only for special cases as the inductance and resistance matrix may be assigned any values and rotor position dependencies. The complex or space variable model is now introduced to get rid of rotor position dependence of parameters. 13.3. THE COMPLEX VARIABLE MODEL Let us use the following notations: [] [] [] [] ;eaRe 3 4 cos ;aeRe 3 2 cos aRe 3 4 cos ;aRe 3 2 cos ;ea erer j 2 er j er 2 3 2 j θθ π = π +θ= π +θ = π = π = (13.15) Based on the inductance matrix, expression (13.9), the stator phase a and rotor phase a r flux linkages Ψ a and Ψ ar are [] () [ ] er rrr j c 2 bamsc 2 bamsalsa eiaaiiReLiaaiiReLiL θ ++++++=Ψ (13.16) © 2002 by CRC Press LLC Author: Ion Boldea, S.A.Nasar………… ……… [] () [ ] er rrrrr j c 2 bamsc 2 bamsalra eiaaiiReLiaaiiReLiL θ− ++++++=Ψ (13.17) We may now introduce the following complex variables as space phasors: [1] () c 2 ba s s iaaii 3 2 i ++= (13.18) () rrr c 2 ba r r iaaii 3 2 i ++= (13.19) Also, () cbaa s s iii 3 1 iiRe ++−= (13.20) () rrrr cbaa r r iii 3 1 iiRe ++−= (13.21) In symmetric steady-state and transient regimes, 0iiiiii rrr cbacba =++=++ (13.22) With the above definitions, Ψ a and Ψ ar become msm j r r s s m s s lsa L 2 3 L ;eiiReLiReL er = ++ =Ψ θ (13.23) ++ =Ψ θ− er r j s s r r m r r lra eiiReLiReL (13.24) Similar expressions may be derived for phases b r and c r . After adding them together, using the complex variable definitions (13.18) and (13.19) for flux linkages and voltages, also, we obtain er er j s s m r r r r r r r r r r r r j r r m s s s s s s s s s s s s eiLiL ; dt d iRV ;eiLiL ; dt d iRV θ− θ +=Ψ Ψ += +=Ψ Ψ += (13.25) where mrlrmsls LLL ;LLL +=+= (13.26) ()() rrr c 2 ba r r c 2 ba s s VaaVV 3 2 V ; VaaVV 3 2 V ++=++= (13.27) © 2002 by CRC Press LLC Author: Ion Boldea, S.A.Nasar………… ……… In the above equations, stator variables are still given in stator coordinates and rotor variables in rotor coordinates. Making use of a rotation of complex variables by the general angle θ b in the stator and θ b – θ er in the rotor, we obtain all variables in a unique reference rotating at electrical speed ω b , dt d b b θ =ω (13.28) () () () erberberb bbb j b r r r j b r r r j b r r r j b s s s j b s s s j b s s s eVV ;eii ;e ;eVV ;eii ;e θ−θθ−θθ−θ θθθ ==Ψ=Ψ ==Ψ=Ψ (13.29) With these new variables Equations (13.25) become () s m r r rr rb r r r r r m s s ss b s s s s iLiL ;j dt d iRV iLiL ;j dt d iRV +=ΨΨω−ω+ Ψ += +=ΨΨω+ Ψ += (13.30) For convenience, the superscript b was dropped in (13.30). The electromagnetic torque is related to motion-induced voltage in (13.30). () () * r r 1 * s s 1e ijRep 2 3 ijRep 2 3 T ⋅ψ⋅⋅⋅−=⋅ψ⋅⋅⋅= (13.31) Adding the equations of motion, the complete complex variable (space- phasor) model of IM is obtained. r er loade r 1 dt d ;TT dt d p J ω= θ −= ω (13.32) The complex variables may be decomposed in plane along two orthogonal d and q axes rotating at speed ω b to obtain the d–q (Park) model. [2] qrdr r qrdr r qrdr r qd s qd s qd s j ;ijii ;VjVV j ;ijii ;VjVV Ψ⋅+Ψ=Ψ⋅+=⋅+= Ψ⋅+Ψ=Ψ⋅+=⋅+= (13.33) With (13.33), the voltage Equations (13.30) become © 2002 by CRC Press LLC Author: Ion Boldea, S.A.Nasar………… ……… () () ()() qrddrqm1dqqd1e drrbqrrqr qr qrrbdrrdr dr dbqsq q qbdsd d iiiiLp 2 3 iip 2 3 T iRV dt d iRV dt d iRV dt d iRV dt d −=Ψ−Ψ= Ψ⋅ω−ω−⋅−= Ψ Ψ⋅ω−ω+⋅−= Ψ Ψ⋅ω−⋅−= Ψ Ψ⋅ω+⋅−= Ψ (13.34) Also from (13.27) with (13.19), the Park transformation for stator P(θ b ) is derived. () [] ⋅θ= c b a b 0 q d V V V P V V V (13.35) () [] () () π −θ− π +θ−θ− π −θ− π +θ−θ− ⋅=θ 2 1 2 1 2 1 3 2 sin 3 2 sinsin 3 2 cos 3 2 coscos 3 2 P bbb bbb b (13.36) The inverse Park transformation is () [] () [] T b 1 b P 2 3 P θ⋅=θ − (13.37) A similar transformation is valid for the rotor but with θ b – θ er instead of θ b . It may be easily proved that the homopolar (real) variables V 0 , i 0 , V 0r , i 0r , Ψ 0 , Ψ 0r do not interface in energy conversion 0r0r0r0rr0 r0 0s000s0 0 iL ;iRV dt d iL ;iRV dt d ⋅≈Ψ⋅−= Ψ ⋅≈Ψ⋅−= Ψ (13.38) L 0s and L 0r are the homopolar inductances of stator and rotor. Their values are equal or lower (for chorded coil windings) to the respective leakage inductances L ls and L lr . A few remarks on the complex variable (space phasor) and d–q models are in order. © 2002 by CRC Press LLC Author: Ion Boldea, S.A.Nasar………… ……… • Both models include, in the form presented here, only the space fundamental of mmfs and airgap flux distributions. • Both models exhibit inductances independent of rotor position. • The complex variable (space phasor) model is credited with a reduction in the number of equations with respect to the d–q model but it operates with complex variables. • When solving the state space equations, only the d–q model, with real variables, benefits the existing commercial software (Mathematica, Matlab– Simulink, Spice, etc.). • Both models are very practical in treating the transients and control of symmetrical IMs fed from symmetrical voltage power grids or from PWM converters. • Easy incorporation of magnetic saturation and rotor skin effect are two additional assets of complex variable and d–q models. The airgap flux density retains a sinusoidal distribution along the circumferential direction. • Besides the complex variable which enjoys widespread usage, other models (variable transformations), that deal especially with asymmetric supply or asymmetric machine cases have been introduced (for a summary see Reference. [3, 4]) 13.4. STEADY-STATE BY THE COMPLEX VARIABLE MODEL By IM steady-state we mean constant speed and load. For a machine fed from a sinusoidal voltage symmetrical power grid, the phase voltages at IM terminals are () 1,2,3=i ; 3 2 1itcos2VV 1c,b,a π ⋅−−ω⋅= (13.39) The voltage space phasor b s V in random coordinates (from (13.27)) is () () () () er j c 2 ba b s etVataVtV 3 2 V θ− ++= (13.40) From (13.39) and (13.40), ()() [] b1b1 b s tsinjtcos2VV θ−ω+θ−ω= (13.41) Only for steady-state, 0bb t θ+ω=θ (13.42) Consequently, () [] 0b1 tj b s e2VV θ+ω−ω = (13.43) © 2002 by CRC Press LLC Author: Ion Boldea, S.A.Nasar………… ……… For steady-state, the current in the space phasor model follows the voltage frequency: (ω 1 - ω b ). Steady-state in the state space equations means replacing d/dt with j(ω 1 - ω b ). Using this observation makes Equations (13.30) become () 0m m 0m r r1 0r 1 0r r 0r 0r0s0m0m0r rl 0r 0m0s sl 0s0s 1 0s s 0s iL ;S ;jSiRV iii ;iL ;iL ;jiRV =Ψ ω ω−ω =Ψω+= +=Ψ+=Ψ Ψ+=ΨΨω+= (13.44) So the form of space phasor model voltage equations under the steady-state is the same irrespective of the speed of the reference system ω b . When ω b , only the frequency changes of voltages, currents, flux linkages in the space phasor model varies as it is ω 1 – ω b . No wonder this is so, as only Equations (13.44) exhibit the total emf, which should be independent of reference system speed ω b . S is the slip, a variable well known by now. Notice that for ω b = ω 1 (synchronous coordinates), d/dt = (ω 1 - ω b ) = 0. Consequently, for synchronous coordinates the steady-state means d.c. variables. The space phasor diagram of (13.44) is shown in Figure 13.2 for a cage rotor IM. motoring ca g e rotor I m0 I r0 I s0 I r0 - V s0 I s0 Ψ s0 R s j ω 1 Ψ s0 Ψ m0 j R s ω 1 I r0 = Ψ r0 -L lr I r0 L ls I s0 I s0 I m0 Ψ s0 S>0 Ψ r0 Ψ m0 L lr I r0 I r0 Ψ s0 j ω 1 I s0 R s V s0 generating S<0 L ls I s0 Figure 13.2 Space phasor diagram for steady-state From the stator space Equations (13.44) the torque (13.31) becomes 0r0r1 * 0r0r 1e iP 2 3 ijp 2 3 T Ψ= Ψ= (13.45) © 2002 by CRC Press LLC Author: Ion Boldea, S.A.Nasar………… ……… Also, from (13.44), r 0r 1 0r R jSi Ψ ω−= (13.46) With (13.46), alternatively, the torque is 1 r 2 0r 1e S R p 2 3 T ω Ψ = (13.47) Solving for Ψ r0 in Equations (13.44) lead to the standard torque formula. () m ls 1 2 lr1ls 2 1 2 r 1s r 2 s 1 1 e L L 1C ; LCL S R CR S R V p3 T += +ω+ + ω ≈ (13.48) Expression (13.47) shows that, for constant rotor flux space-phasor amplitude, the torque varies linearly with speed as it does in a separately excited d.c. motor. So all steady-state performance may be calculated using the space- phasor model as well. 13.5. EQUIVALENT CIRCUITS FOR DRIVES Equations (13.30) lead to a general equivalent circuit good for transients, especially in variable speed drives (Figure 13.3). ( ) ( ) ()()()() m rb r rlrb r r r m b s slb s s s jpiLjpiRV jpiLjpiRV Ψω−ω++ω−ω++= Ψω++ω++= (13.49) The reference system speed ω b may be random, but three particular values have met with rather wide acceptance. • Stator coordinates: ω b = 0; for steady-state: p Æ jω 1 • Rotor coordinates: ω b = ω r ; for steady-state: p Æ jSω 1 • Synchronous coordinates: ω b = ω 1 ; for steady-state: p Æ 0 I s I r I m V s (p+j )L ω sl b (p+j )L ω m b (p+j( )L ω −ω ) rl br R r -j L I ω rmm R s a.) Figure 13.3 The general equivalent circuit a.) and for steady-state b.) (continued) © 2002 by CRC Press LLC [...]... smaller than the normal (d.c.) inductances, the machine behavior at high currents is expected to show further increased currents Furthermore, as shown in Chapter 6, the leakage flux circumferential flux lines at high currents influence the main (radial) flux and contribute to the resultant flux in the machine core The saturation in the stator is given by the stator flux Ψs and in the rotor by the rotor... saturation while the conventional circuit (Figure 13.8) indicates moderate saturation of the main path flux and some saturation of the stator leakage path So when high current (torque) transients occur, the real machine, due to the Lms reduction, produces torque performance quite different from the predictions by the conventional equivalent circuit (Figure 13.8), up to 2 to 2.5 p.u current, however, the differences... Depending on the machine design and load, the relative importance of the two variable inductances Lsi and Lri may be notable or negligible Consequently, the equivalent circuit may be simplified For example, for heavy loads the rotor saturation may be ignored and thus only Lsi remains Then Lsi and Lg in parallel may be lumped into an equivalent variable inductance Lms and Llse and Llre into the total constant... ……… To include the core loss in the space phasor model of IMs, we assume that the core losses occur, both in the stator and rotor, into equivalent orthogonal windings: cd – cq, cdr – cqr (Figure 13.12) Alternatively, when rotor core loss is neglected (cage rotor IMs), the cdr – cqr windings account for the skin effect via the double-cage equivalence principle Let us consider also that the core loss... effect also influences the transients of IMs and the model on Figure 13.13 can handle it for cage rotor IMs directly as shown earlier in this paragraph 13.9 REDUCED ORDER MODELS The rather involved d–q (space phasor) model with skin effect and saturation may be used directly to investigate the induction machine transients More practical (simpler) solutions have also been proposed They are called reduced... known The inputs are the two voltage space phasors V s and V r and the outputs are the two flux linkage space phasors, Ψ s and Ψ r The structural diagram of Equations (13.55) is shown in Figure 13.7 The transient behavior of stator and rotor flux linkages as complex variables, at constant speed ωb and ωr, for standard step or sinusoidal voltages V s , V r signals has analytical solutions Finally, the. .. sp eed - Figure 13.7 IM space-phasor diagram for constant speed The two complex eigenvalues of (13.55) are obtained from τ s ' p + 1 + jωb τ s ' − Kr =0 − Ks τr ' p + 1 + j(ωb − ωr )τ r ' (13.58) As expected, the eigenvalues p1,2 depend on the speed of the motor and on the speed of the reference system ωb Equation (13.58) may be put in the form p 2 τs ' τr '+ p[(τs '+ τ r ') + jτs ' τ r ' (2ωb − ωr... based on (13.51), Figure 13.4 The generalized equivalent circuit in Figure 13.4 warrants the following comments: • For a = 1, the general equivalent circuit of Figure 13.4 is reobtained and a a Ψ m = Ψ m : the main flux © 2002 by CRC Press LLC Author: Ion Boldea, S.A.Nasar………… ……… • For a = Lm/Lr the inductance term in the “rotor section” “disappears”, being moved to the primary section and a aΨm =... expected, the functions Ψs(isi) = Lsiisi and Ψr(iri) = Lriiri have to be known together with Llse, Llre, Rs, Rr, Lg which are hereby considered constant In the presence of skin effect, Llre and Rr are functions of slip frequency ωsr = ω1 – ωr Furthermore we should notice that it is not easy to measure all parameters in the equivalent circuit on Figure 13.10 It is, however, possible to calculate them either... may be obtained by considering only the amplitude transients of Ψ r in (13.73) [11], but the results are not good enough in the sense that the torque fast (grid frequency) transients are present during start up at higher speed than for the full model Modified second-order models have been proposed to improve the precision of torque results [11] (Figure 13.19c), but the results are highly dependent on . to the resultant flux in the machine core. The saturation in the stator is given by the stator flux Ψ s and in the rotor by the rotor flux for high levels. expected, the eigenvalues p 1,2 depend on the speed of the motor and on the speed of the reference system ω b . Equation (13.58) may be put in the form