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Analysis of Electric Machinery and Drive Systems Editor(s): Paul Krause, Oleg Wasynczuk, Scott Sudhoff, Steven Pekarek
215 6.1. INTRODUCTION The induction machine is used in a wide variety of applications as a means of convert- ing electric power to mechanical work. It is without doubt the workhorse of the electric power industry. Pump, steel mill, and hoist drives are but a few applications of large multiphase induction motors. On a smaller scale, induction machines are used as the controlled drive motor in vehicles, air conditioning systems, and in wind turbines, for example. Single-phase induction motors are widely used in household appliances, as well as in hand and bench tools. In the beginning of this chapter, classical techniques are used to establish the voltage and torque equations for a symmetrical induction machine expressed in terms of machine variables. Next, the transformation to the arbitrary reference frame pre- sented in Chapter 3 is modifi ed to accommodate rotating circuits. Once this groundwork has been laid, the machine voltage equations are written in the arbitrary reference frame directly without a laborious exercise in trigonometry that one faces when substituting the equations of transformations into the voltage equations expressed in machine variables. The equations may then be expressed in any reference frame by appropriate assignment of the reference-frame speed in the arbitrary reference-frame Analysis of Electric Machinery and Drive Systems, Third Edition. Paul Krause, Oleg Wasynczuk, Scott Sudhoff, and Steven Pekarek. © 2013 Institute of Electrical and Electronics Engineers, Inc. Published 2013 by John Wiley & Sons, Inc. SYMMETRICAL INDUCTION MACHINES 6 216 SYMMETRICAL INDUCTION MACHINES voltage equations. Although the stationary reference frame, the reference frame fi xed in the rotor, and the synchronously rotating reference frame are the most frequently used, the arbitrary reference frame offers a direct means of obtaining the voltage equa- tions in these and all other reference frames. The steady-state voltage equations for an induction machine are obtained from the voltage equations in the arbitrary reference frame by direct application of the material presented in Chapter 3 . Computer solutions are used to illustrate the dynamic perfor- mance of typical induction machines and to depict the variables in various reference frames during free acceleration. Finally, the equations for an induction machine are arranged appropriate for computer simulation. The material presented in this chapter forms the basis for solution of more advanced problems. In particular, these basic concepts are fundamental to the analysis of induction machines in most power system and controlled electric drive applications. 6.2. VOLTAGE EQUATIONS IN MACHINE VARIABLES The winding arrangement for a two-pole, three-phase, wye-connected, symmetrical induction machine is shown in Figure 6.2-1 (which is Fig. 1.4-3 repeated here for conve- nience). The stator windings are identical, sinusoidally distributed windings, displaced 120°, with N s equivalent turns and resistance r s . For the purpose at hand, the rotor wind- ings will also be considered as three identical sinusoidally distributed windings, displaced 120°, with N r equivalent turns and resistance r r . The positive direction of the magnetic axis of each winding is shown in Figure 6.2-1 . It is important to note that the positive direction of the magnetic axes of the stator windings coincides with the direction of f as , f bs , and f cs as specifi ed by the equations of transformation and shown in Figure 3.3-1 . The voltage equations in machine variables may be expressed vri abcs s abcs abcs p=+l (6.2-1) vri abcr r abcr abcr p=+l (6.2-2) where f abcs T as bs cs fff ( ) = [] (6.2-3) f abcr T ar br cr fff ( ) = [] (6.2-4) In the above equations, the s subscript denotes variables and parameters associated with the stator circuits, and the r subscript denotes variables and parameters associated with the rotor circuits. Both r s and r r , are diagonal matrices each with equal nonzero ele- ments. For a magnetically linear system, the fl ux linkages may be expressed as l l abcs abcr ssr sr T r abcs abcr ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ LL LL i i() (6.2-5) VOLTAGE EQUATIONS IN MACHINE VARIABLES 217 The winding inductances are derived Chapter 2 . Neglecting mutual leakage between the stator windings and also between the rotor windings, they can be expressed as L s ls ms ms ms ms ls ms ms ms ms ls LL L L LLL L LLLL = +− − −+− −− + 1 2 1 2 1 2 1 2 1 2 1 2 mms ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ (6.2-6) Figure 6.2-1. Two-pole, three-phase, wye-connected symmetrical induction machine. q r f s f r ar-axis as-axis cs-axis cr-axis bs-axis br-axis bs bs¢ br¢ cr¢ cs¢ ar¢ as¢ ar as br cr cs w r v as v br v cr v ar v bs v cs i as i bs i ar i cs i cr i br r s r s r s r r r r r r N s N s N s N r N r N r + + + + + + 218 SYMMETRICAL INDUCTION MACHINES L r lr mr mr mr mr lr mr mr mr mr lr LL L L LLL L LLLL = +− − −+− −− + 1 2 1 2 1 2 1 2 1 2 1 2 mmr ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ (6.2-7) L sr sr rr r r L= + ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ − ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ − ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ cos cos cos cos c θθ π θ π θ π 2 3 2 3 2 3 oos cos cos cos cos θθ π θ π θ π θ rr rr r + ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ + ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ − ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎡ ⎣ ⎢ 2 3 2 3 2 3 ⎢⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ (6.2-8) In the above inductance equations, L ls and L ms are, respectively, the leakage and mag- netizing inductances of the stator windings; L lr and L mr are for the rotor windings. The inductance L sr is the amplitude of the mutual inductances between stator and rotor windings. A majority of induction machines are not equipped with coil-wound rotor wind- ings; instead, the current fl ows in copper or aluminum bars that are uniformly distrib- uted and are embedded in a ferromagnetic material with all bars terminated in a common ring at each end of the rotor. This type of rotor confi guration is referred to as a squirrel-cage rotor. It may at fi rst appear that the mutual inductance between a uni- formly distributed rotor winding and a sinusoidally distributed stator winding would not be of the form given by (6.2-8) . However, in most cases, a uniformly distributed winding is adequately described by its fundamental sinusoidal component and is rep- resented by an equivalent three-phase winding. Generally, this representation consists of one equivalent winding per phase; however, the rotor construction of some machines is such that its performance is more accurately described by representing each phase with two equivalent windings connected in parallel. This type of machine is commonly referred to as a double-cage rotor machine. Another consideration is that in a practical machine, the rotor conductors are often skewed. That is, the conductors are not placed in the plane of the axis of rotation of the rotor. Instead, the conductors are skewed slightly (typically one slot width) with the axis of rotation. This type of conductor arrangement helps to reduce the magnitude of harmonic torques that result from harmonics in the MMF waves. Such design fea- tures are not considered here. Instead, it is assumed that all effects upon the amplitude of the fundamental component of the MMF waveform due to skewing and uniformly distributed rotor windings are accounted for in the value of N r . The assumption that the induction machine is a linear (no saturation) and MMF harmonic-free device is an oversimplifi cation that cannot describe the behavior of induction machines in all modes of operation. However, in the majority of applications, its behavior can be adequately predicted with this simplifi ed representation. VOLTAGE EQUATIONS IN MACHINE VARIABLES 219 When expressing the voltage equations in machine variable form, it is convenient to refer all rotor variables to the stator windings by appropriate turns ratios. ′ =ii abcr r s abcr N N (6.2-9) ′ =vv abcr s r abcr N N (6.2-10) ′ =ll abcr s r abcr N N (6.2-11) The magnetizing and mutual inductances are associated with the same magnetic fl ux path; therefore L ms , L mr , and L sr are related as set forth by (1.2-21) with 1 and 2 replaced by s and r , respectively, or by (2.8-57) – (2.8-59) . In particular L N N L ms s r sr = (6.2-12) Thus, we will defi ne ′ = = + ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ − ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ − LL sr s r sr ms rr r r N N L cos cos cos cos θθ π θ π θ 2 3 2 3 2 ππ θθ π θ π θ π 3 2 3 2 3 2 3 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ + ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ + ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ − ⎛ ⎝ ⎜ ⎞ cos cos cos cos rr rr ⎠⎠ ⎟ ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ cos θ r (6.2-13) Also, from (1.2-18) or (2.8-57) and (2.8-58) , L mr may be expressed as L N N L mr r s ms = ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 2 (6.2-14) and if we let ′ = ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ LL r s r r N N 2 (6.2-15) then, from (6.2-7) 220 SYMMETRICAL INDUCTION MACHINES ′ = ′ +− − − ′ +− −− ′ L r lr ms ms ms ms lr ms ms ms ms LL L L LLL L LLL 1 2 1 2 1 2 1 2 1 2 1 2 llr ms L+ ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ (6.2-16) where ′ = ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ L N N L lr s r lr 2 (6.2-17) The fl ux linkages may now be expressed as l l abcs abcr sr sr T r abcs abcr ′ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = ′ ′′ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ′ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ LL LL i i s () (6.2-18) The voltage equations expressed in terms of machine variables referred to the stator windings may now be written as v v rL L LrL i i abcs abcr ss sr sr T rr abcs pp pp ′ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = + ′ ′′ + ′ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ′ () aabcr ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ (6.2-19) where ′ = ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ rr r s r r N N 2 (6.2-20) 6.3. TORQUE EQUATION IN MACHINE VARIABLES Evaluation of the energy stored in the coupling fi eld by (1.3-51) yields the familiar expression for energy stored in a magnetically linear system. In particular, the stored energy is the sum of the self-inductance of each winding times one-half the square of its current and all mutual inductances, each times the currents in the two windings coupled by the mutual inductance. Thus, the energy stored in the coupling fi eld may be written W f abcs T s abcs abcs T sr abcr abcr T rab =+ ′′ + ′′′ 1 2 1 2 () () () iLi iLi iLi ccr (6.3-1) TORQUE EQUATION IN MACHINE VARIABLES 221 Since the machine is assumed to be magnetically linear, the fi eld energy W f is equal to the coenergy W c . Before using the second entry of Table 1.3-1 to express the electromagnetic torque, it is necessary to modify the expressions given in Table 1.3-1 to account for a P -pole machine. The change of mechanical energy in a rotational system with one mechanical input may be written from (1.3-71) as dW T d merm =− θ (6.3-2) where T e is the electromagnetic torque positive for motor action (torque output) and θ rm is the actual angular displacement of the rotor. The fl ux linkages, currents, W f , and W c , are all expressed as functions of the electrical angular displacement θ r . Since θθ rrm P = ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 2 (6.3-3) where P is the number of poles in the machine, then dW T P d me r =− ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 2 θ (6.3-4) Therefore, to account for a P -pole machine, all terms on the right-hand side of Table 1.3-1 should be multiplied by P /2. Therefore, the electromagnetic torque may be evalu- ated from T PW er cr r (, ) (, ) i i θ θ θ = ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ∂ ∂2 (6.3-5) The abbreviated functional notation, as used in Table 1.3-1 , is also used here for the currents. Since L s and ′ L r are not functions of θ r , substituting W f from (6.3-1) into (6.3- 5) yields the electromagnetic torque in Newton meters (N·m) T P e abcs T r sr abcr = ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ∂ ∂ ′′ 2 () []iLi θ (6.3-6) In expanded form, (6.3-6) becomes T P Lii i i ii i e msasarbrcrbsbrar =− ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ′ − ′ − ′ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ + ′ − ′ − 2 1 2 1 2 1 2 11 2 1 2 1 2 3 2 ′ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎡ ⎣ ⎢ { + ′ − ′ − ′ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎤ ⎦ ⎥ + i ii i i i cr cs cr br ar r sin [ θ aas br cr bs cr ar cs ar br r ii iii iii()()()]cos ′ − ′ + ′ − ′ + ′ − ′ ⎫ ⎬ ⎭ θ (6.3-7) 222 SYMMETRICAL INDUCTION MACHINES The torque and rotor speed are related by TJ P pT erL = ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ + 2 ω (6.3-8) where J is the inertia of the rotor and in some cases the connected load. The fi rst term on the right-hand side is the inertial torque. In (6.3-8) , the units of J are kilogram·meter 2 (kg·m 2 ) or Joules·second 2 (J·s 2 ). Often the inertia is given as a quantity called WR 2 , expressed in units of pound mass·feet 2 (lbm·ft 2 ). The load torque T L is positive for a torque load on the shaft of the induction machine. 6.4. EQUATIONS OF TRANSFORMATION FOR ROTOR CIRCUITS In Chapter 3 , the concept of the arbitrary reference frame was introduced and applied to stationary circuits. However, in the analysis of induction machines, it is necessary to transform the variables associated with the symmetrical rotor windings to the arbi- trary reference frame. A change of variables that formulates a transformation of the three-phase variables of the rotor circuits to the arbitrary reference frame is ′ = ′ fKf qd r r abcr0 (6.4-1) where () ′ = ′′′ [] f qd r T qr dr r fff 00 (6.4-2) () ′ = ′′′ [] f abcr T ar br cr fff (6.4-3) K r = − ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ + ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ − ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 2 3 2 3 2 3 2 3 cos cos cos sin sin sin ββ π β π ββ π ββ π + ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ 2 3 1 2 1 2 1 2 (6.4-4) βθθ =− r (6.4-5) The angular displacement θ is defi ned by (3.3-5) – (3.3-8) , and θ r is defi ned by d dt r r θ ω = (6.4-6) The inverse of (6.4-4) is EQUATIONS OF TRANSFORMATION FOR ROTOR CIRCUITS 223 () cos sin cos sin cos K r − =− ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ − ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ + ⎛ ⎝ ⎜ ⎞ ⎠ 1 1 2 3 2 3 1 2 3 ββ β π β π β π ⎟⎟ + ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ sin β π 2 3 1 (6.4-7) The r subscript indicates the variables, parameters, and transformation associated with rotating circuits. Although this change of variables needs no physical interpreta- tion, it is convenient, as in the case of stationary circuits, to visualize these transforma- tion equations as trigonometric relationships between vector quantities, as shown in Figure 6.4-1 . It is clear that the above transformation equations for rotor circuits are the trans- formation equations for stationary circuits, with β used as the angular displacement of the arbitrary reference frame rather than θ . In fact, the equations of transformation for stationary and rotor circuits are special cases of a transformation for all circuits, station- ary or rotating. In particular, if in β , θ r is replaced by θ c , where d dt c c θ ω = (6.4-8) then ω c , the angular velocity of the circuits, may be selected to correspond to the circuits being transformed, that is, ω c = 0 for stationary circuits and ω c = ω r for rotor circuits. Although this more general approach could have been used in Chapter 3 , it does add to the complexity of the transformation, making it somewhat more diffi cult to follow without deriving any advantage from the generality of the approach, since only two types of circuits, stationary or fi xed in the rotor, are considered in this chapter. Figure 6.4-1. Transformation for rotating circuits portrayed by trigonometric relationships. f br ¢ f cr ¢ f dr ¢ f ar ¢ f qr ¢ w r q r w q b 224 SYMMETRICAL INDUCTION MACHINES It follows that all equations for stationary circuits in Section 3.3 and Section 3.4 are valid for rotor circuits if θ is replaced by β and ω by ω − ω r . The phasor and steady- state relations for stationary circuits, given in Section 3.7 , Section 3.8 , and Section 3.9 , also apply to rotor circuits of an induction machine if we realize that the rotor variables, during balanced, steady-state operation are of the form ′ = ′ −+ [] FF t ar r e r erf 20cos ( ) ( ) ωω θ (6.4-9) ′ = ′ −+ − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ FF t br r e r erf 20 2 3 cos ( ) ( ) ωω θ π (6.4-10) ′ = ′ −+ + ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ FF t cr r e r erf 20 2 3 cos ( ) ( ) ωω θ π (6.4-11) where θ erf (0) is the phase angle of ′ F ar at time zero. 6.5. VOLTAGE EQUATIONS IN ARBITRARY REFERENCE-FRAME VARIABLES Using the information set forth in Chapter 3 and in the previous section, we know the form of the voltage equations in the arbitrary reference frame without any further analysis [1] . In particular vri qd s s qd s dqs qd s p 00 0 =++ ω ll (6.5-1) ′ = ′′ +− ′ + ′ vri qd r r qd r r dqr qd r p 00 0 () ωω ll (6.5-2) where ()l dqs T ds qs =− [] λλ 0 (6.5-3) () ′ = ′ − ′ [] l dqr T dr qr λλ 0 (6.5-4) The set of equations is complete once the expressions for the fl ux linkages are deter- mined. Substituting the equations of transformation, (3.3-1) and (6.4-1) , into the fl ux linkage equations expressed in abc variables (6.2-18) , yields the fl ux linkage equations for a magnetically linear system l l qd s qd r ss s ssr r rsr T sr 0 0 11 1 ′ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = ′ ′′ −− − KL K KL K KL K K () () ()() LLK i i rr qd s qd r () − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ′ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 1 0 0 (6.5-5) We know from Chapter 3 that for L s of the form given by (6.2-6) [...]... ohms kg·m2 3 50 500 2250 220 460 2300 2300 1710 1705 1773 17 86 11.9 198 1.98 × 103 8.9 × 103 5.8 46. 8 93 .6 421.2 0.435 0.087 0. 262 0.029 0.754 0.302 1.2 06 0.2 26 26. 13 13.08 56. 02 13.04 0.754 0.302 1.2 06 0.2 26 0.8 16 0.228 0.187 0.022 0.089 1 .66 2 11. 06 63.87 245 FREE ACCELERATION CHARACTERISTICS 118.7 94. 96 Te, N·m 71.22 47.48 23.74 0 900 1800 Speed, r/min –23.74 Figure 6. 10-1 Torque–speed characteristics... LM (iqsidr − idsiqr ) ′ ′ ⎝ 2⎠ ⎝ 2 ⎠ (6. 6-2) 230 SYMMETRICAL INDUCTION MACHINES where Te is positive for motor action Other equivalent expressions for the electromagnetic torque of an induction machine are ⎛ 3⎞ ⎛ P ⎞ Te = ⎜ ⎟ ⎜ ⎟ (λqr idr − λ dr iqr ) ′ ′ ′ ′ ⎝ 2⎠ ⎝ 2 ⎠ (6. 6-3) ⎛ 3⎞ ⎛ P ⎞ Te = ⎜ ⎟ ⎜ ⎟ (λ dsiqs − λqsids ) ⎝ 2⎠ ⎝ 2 ⎠ (6. 6-4) Equations (6. 6-3) and (6. 6-4) may be somewhat misleading since... velocity of the rotor as ⎛ 2⎞ pWm = −Te ⎜ ⎟ pθ r ⎝ P⎠ (6. 6-11) Therefore, (6. 6 -6) can be written as ⎛ 2⎞ ⎛ 2⎞ ⎜ ⎟ pWe = pW f + Te ⎜ ⎟ pθ r ⎝ 3⎠ ⎝ P⎠ (6. 6-12) If we compare (6. 6-12) with (6. 6-8), and if we equate coefficients of pθr, we can express torque, positive for motor action, as ⎛ 3⎞ ⎛ P ⎞ Te = ⎜ ⎟ ⎜ ⎟ (λqr idr − λ dr iqr ) ′ ′ ′ ′ ⎝ 2⎠ ⎝ 2 ⎠ (6. 6-13) It is important to note that this expression for... )λqr + pλ dr ′ ′ ′ ′ (6. 5-14) v0 r = rr′i0 r + pλ0 r ′ ′ ′ (6. 5-15) Substituting (6. 5 -6) , (6. 5-8), and (6. 5-9) into (6. 5-5) yields the expressions for the flux linkages In expanded form λqs = Llsiqs + LM (iqs + iqr ) ′ (6. 5- 16) λ ds = Llsids + LM (ids + idr ) ′ (6. 5-17) λ0 s = Llsi0 s (6. 5-18) 2 26 SYMMETRICAL INDUCTION MACHINES rs wlds + (w – wr) l¢ dr – + Llr ¢ Lls – r¢ r + + i¢ qr iqs vqs vqr ¢ LM... Figure 6. 10 -6 Machine variables during free acceleration of a 2250-hp induction motor EXAMPLE 6B Let us calculate the steady-state torque and current at stall for the 3-hp machine given in Table 6. 10-1 and compare these values to those shown in Figure 6. 10-1 and Figure 6. 10-5 From (6. 9-19) and Table 6. 10-1 with s = 1 Te = (3)(4 / 2)(1)[( 26. 13)2 / 377](0.8 16) (1)(220 / 3 )2 [(0.435)(0.8 16) + (1)(1)2 ( 26. 132... 12.9 Z br = (6A-8) Now (rs + rr′ ) + j 15 ( Xls + Xlr ) = 1.052 Ω ′ 60 (6A-9) From that 2 ⎡ 15 ⎤ 2 2 ′ ⎢ 60 ( Xls + Xlr ) ⎥ = (1.052) − (rs + rr′ ) ⎣ ⎦ = (1.052)2 − (0.531 + 0.408)2 = 0.225 Ω (6A-10) 244 SYMMETRICAL INDUCTION MACHINES Thus Xls + Xlr = 1.9 Ω ′ (6A-11) Generally Xls and Xlr are assumed equal; however, in some types of induction machines, ′ a different ratio is suggested [6] We will assume... 3⎠ (6. 6-7) Removing the ir voltage drop is not necessary; we have done this to be consistent with the work in Chapter 1 Appropriate substitution of (6. 5-10)– (6. 5-15) yields ⎛ 2⎞ ′ ′ ′ ′ ′ ′ ⎜ ⎟ pWe = iqs pλqs + ids pλ ds + i0 s pλ0 s + iqr pλqr + idr pλ dr + i0 r pλ0 r ⎝ 3⎠ + ω (λ dsiqs − λqsids + λ dr iqr − λqr idr ) − (λ dr iqr − λqr idr ) pθ r ′ ′ ′ ′ ′ ′ ′ ′ (6. 6-8) Comparing (6. 6-8) with (6. 6 -6) ,... in the case of a linear magnetic system, the coefficient of pθ in (6. 6-8) becomes zero Whereupon, (6. 6-13) becomes valid in all reference frames This would seem to fly in the face of (6. 6-14) However, for a linear magnetic system, (6. 6-13) and (6. 6-14) are equal Moreover, ⎛ 3⎞ ⎛ P ⎞ Te = ⎜ ⎟ ⎜ ⎟ LM (iqsidr − idsiqr ) ′ ′ ⎝ 2⎠ ⎝ 2 ⎠ (6. 6-15) is also a valid expression for torque if the magnetic system... 26. 884 × 26. 884)]2 + (1)2 (0.8 16 × 26. 884 + 1 × 0.435 × 26. 884)2 = 51.9 N ⋅ m (6B-1) This is approximately the average of the pulsating torque at ωr = 0 depicted in Figure 6. 10-1 and Figure 6. 10-5 The stall, steady-state current may be calculated from I as = Vas (rs + rr′ ) + j ( Xls + Xlr ) ′ (220 / 3 )/0° (0.435 + 0.8 16) + j (0.754 + 0.754) = 64 .8 / − 50.3°A = (6B-2) 5.94 Steady-state torque 3. 96. .. variables that are identical; (6. 6-13), (6. 6-14) with the raised r removed, and (6. 6-15) The reader may wish to verify these statements Also, the above expressions for torque are often written in terms of flux linkages per second and currents For example, (6. 6-13) can be written as ⎛ 3⎞ ⎛ P ⎞ ⎛ 1 ⎞ Te = ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ (ψ qr idr − ψ dr iqr ) ′ ′ ′ ′ ⎝ 2 ⎠ ⎝ 2 ⎠ ⎝ ωb ⎠ (6. 6- 16) It is left to the reader to . i ′ +−+ ′′ − ′′ − ′′ − ′ λ ωλ λ λ λ λ λ ()( ′′ ip dr r ) θ (6. 6-8) Comparing (6. 6-8) with (6. 6 -6) , it is clear that the right-hand side of (6. 6-4) is pW f − pW m . Now TORQUE. qr Li L i i=+ + ′ () (6. 5- 16) λ ds ls ds M ds dr Li L i i=+ + ′ () (6. 5-17) λ 00slss Li= (6. 5-18) 2 26 SYMMETRICAL INDUCTION MACHINES ′ = ′′ ++ ′ λ qr