Analysis of Electric Machinery and Drive Systems Editor(s): Paul Krause, Oleg Wasynczuk, Scott Sudhoff, Steven Pekarek
142 5.1. INTRODUCTION The electrical and electromechanical behavior of most synchronous machines can be predicted from the equations that describe the three-phase salient-pole synchronous machine. In particular, these equations can be used directly to predict the performance of synchronous motors, hydro, steam, combustion, or wind turbine driven synchronous generators, and, with only slight modifi cations, reluctance motors. The rotor of a synchronous machine is equipped with a fi eld winding and one or more damper windings and, in general, each of the rotor windings has different electri- cal characteristics. Moreover, the rotor of a salient-pole synchronous machine is mag- netically unsymmetrical. Due to these rotor asymmetries, a change of variables for the rotor variables offers no advantage. However, a change of variables is benefi cial for the stator variables. In most cases, the stator variables are transformed to a reference frame fi xed in the rotor (Park ’ s equations) [1] ; however, the stator variables may also be expressed in the arbitrary reference frame, which is convenient for some computer simulations. In this chapter, the voltage and electromagnetic torque equations are fi rst estab- lished in machine variables. Reference-frame theory set forth in Chapter 3 is then used 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. SYNCHRONOUS MACHINES 5 VOLTAGE EQUATIONS IN MACHINE VARIABLES 143 to establish the machine equations with the stator variables in the rotor reference frame. The equations that describe the steady-state behavior are then derived using the theory established in Chapter 3 . The machine equations are arranged convenient for computer simulation wherein a method for accounting for saturation is given. Computer traces are given to illustrate the dynamic behavior of a synchronous machine during motor and generator operation and a low-power reluctance motor during load changes and variable frequency operation. Nearly all of the electric power used throughout the world is generated by syn- chronous generators driven either by hydro, steam, or wind turbines or combustion engines. Just as the induction motor is the workhorse when it comes to converting energy from electrical to mechanical, the synchronous machine is the principal means of converting energy from mechanical to electrical. In the power system or electric grid environment, the analysis of the synchronous generator is often carried out assuming positive currents out of the machine. Although this is very convenient for the power systems engineer, it tends to be somewhat confusing for beginning machine analysts and inconvenient for engineers working in the electric drives area. In an effort to make this chapter helpful in both environments, positive stator currents are assumed into the machine as done in the analysis of the induction machine, and then in Section 5.10 , the sense of the stator currents is reversed, and high-power synchronous generators that would be used in a power system are considered. The changes in the machine equations necessary to accommodate positive current out of the machine are described. Computer traces are then given to illustrate the dynamic behavior of typical hydro and steam turbine-driven generators during sudden changes in input torque and during and fol- lowing a three-phase fault at the terminals. These dynamic responses, which are calcu- lated using the detailed set of nonlinear differential equations, are compared with those predicted by an approximate method of calculating the transient torque–angle charac- teristics, which was widely used before the advent of modern computers and which still offer an unequalled means of visualizing the transient behavior of synchronous genera- tors in a power system. 5.2. VOLTAGE EQUATIONS IN MACHINE VARIABLES A two-pole, three-phase, wye-connected, salient-pole synchronous machine is shown in Figure 5.2-1 . The stator windings are identical sinusoidally distributed windings, displaced 120°, with N s equivalent turns and resistance r s . The rotor is equipped with a fi eld winding and three damper windings. The fi eld winding ( fd winding) has N fd equivalent turns with resistance r fd . One damper winding has the same magnetic axis as the fi eld winding. This winding, the kd winding, has N kd equivalent turns with resis- tance r kd . The magnetic axis of the second and third damper windings, the kq 1 and kq 2 windings, is displaced 90° ahead of the magnetic axis of the fd and kd windings. The kq 1 and kq 2 windings have N kq 1 and N kq 2 equivalent turns, respectively, with resistances r kq 1 and r kq 2 . It is assumed that all rotor windings are sinusoidally distributed. In Figure 5.2-1 , the magnetic axes of the stator windings are denoted by the as, bs , and cs axes. This notation was also used for the stator windings of the induction 144 SYNCHRONOUS MACHINES machine. The quadrature axis ( q -axis) and direct axis ( d -axis) are introduced in Figure 5.2-1 . The q -axis is the magnetic axis of the kq 1 and kq 2 windings, while the d -axis is the magnetic axis of the fd and kd windings. The use of the q- and d -axes was in exis- tence prior to Park ’ s work [1] , and as mentioned in Chapter 3 , Park used the notation of f q , f d , and f 0 in his transformation. Perhaps he made this choice of notation since, in effect, this transformation referred the stator variables to the rotor where the traditional q -and d -axes are located. We have used f qs , f ds , and f 0 s , and ′ f qr , ′ f dr , and ′ f r0 to denote transformed induction machine variables without introducing the connotation of a q- or d -axis. Instead, the q- and d -axes have been reserved to denote the rotor magnetic axes of the synchronous machine where they have an established physical meaning quite independent of any transformation. For this reason, one may argue that the q and d subscripts should not be used to denote the transformation to the arbitrary reference frame. Indeed, this line of reasoning has merit; however, since the transformation to the arbitrary reference Figure 5.2-1. Two-pole, three-phase, wye-connected salient-pole synchronous machine. θr w r as-axis fd fd′ bs-axis as′ q-axis kd′ kd kq1 kq2 kq1′ kq2′ bs′ bs cs′ cs-axis d-axis as cs N kd N fd N kq2 N kq1 r s i bs i cs i kq1 i kq2 v kq2 v kq1 + + + + + + N s − − − − − − v as v bs v cs i kd i fd r kd v kd v fd i as r s + r fd N s r s N s VOLTAGE EQUATIONS IN MACHINE VARIABLES 145 frame is in essence a generalization of Park ’ s transformation, the q and d subscripts have been selected for use in the transformation to the arbitrary reference primarily out of respect for Park ’ s work, which is the basis of it all. Although the damper windings are shown with provisions to apply a voltage, they are, in fact, short-circuited windings that represent the paths for induced rotor currents. Currents may fl ow in either cage-type windings similar to the squirrel-cage windings of induction machines or in the actual iron of the rotor. In salient-pole machines at least, the rotor is laminated, and the damper winding currents are confi ned, for the most part, to the cage windings embedded in the rotor. In the high-speed, two- or four- pole machines, the rotor is cylindrical, made of solid iron with a cage-type winding embedded in the rotor. Here, currents can fl ow either in the cage winding or in the solid iron. The performance of nearly all types of synchronous machines may be adequately described by straightforward modifi cations of the equations describing the performance of the machine shown in Figure 5.2-1 . For example, the behavior of low-speed hydro turbine generators, which are always salient-pole machines, is generally predicted suf- fi ciently by one equivalent damper winding in the q -axis. Hence, the performance of this type of machine may be described from the equations derived for the machine shown in Figure 5.2-1 by eliminating all terms involving one of the kq windings. The reluctance machine, which has no fi eld winding and generally only one damper winding in the q -axis, may be described by eliminating the terms involving the fd winding and one of the kq windings. In solid iron rotor, steam turbine generators, the magnetic characteristics of the q- and d -axes are identical, or nearly so, hence the inductances associated with the two axes are essentially the same. Also, it is necessary, in most cases, to include all three damper windings in order to portray adequately the transient characteristics of the stator variables and the electromagnetic torque of solid iron rotor machines [2] . The voltage equations in machine variables may be expressed in matrix form as vri abcs s abcs abcs p=+l (5.2-1) vri qdr r qdr qdr p=+l (5.2-2) where ()[ ]f abcs T as bs cs fff= (5.2-3) ()[ ]f qdr T kq kq fd kd ffff= 12 (5.2-4) In the previous equations, the s and r subscripts denote variables associated with the stator and rotor windings, respectively. Both r s and r r are diagonal matrices, in particular r s sss rrr= diag[ ] (5.2-5) r rkqkqfdkd rrrr= diag[ ] 12 (5.2-6) 146 SYNCHRONOUS MACHINES The fl ux linkage equations for a linear magnetic system become l l abcs qdr ssr sr T r abcs qdr ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ LL LL i i () (5.2-7) From the work in Chapters 1 and 2 , neglecting mutual leakage between stator windings, we can write L s as L s lsAB r AB r AB r LLL LL LL = +− − − − ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ −− +cos cos cos2 1 2 2 3 1 2 2 3 θθ π θ π ⎛⎛ ⎝ ⎜ ⎞ ⎠ ⎟ −− − ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ +− − ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ − 1 2 2 3 2 2 3 1 2 LL LLL AB r lsAB r cos cos θ π θ π LLL LL LL L AB r AB r AB r −+ −− + ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ −− + cos ( ) cos cos ( ) 2 1 2 2 3 1 2 2 θπ θ π θπ lls A B r LL+− + ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ cos2 2 3 θ π (5.2-8) By a straightforward extension of the work in Chapters 1 and 2 , we can express the self- and mutual inductances of the damper windings. The inductance matrices L sr and L r may then be expressed as L sr skq r skq r sfd r skd r skq r LLLL L=− ⎛ ⎝ 12 1 2 3 cos cos sin sin cos θθθθ θ π ⎜⎜ ⎞ ⎠ ⎟ − ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ − ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ − ⎛ LLL skq r sfd r skd r2 2 3 2 3 2 3 cos sin sin θ π θ π θ π ⎝⎝ ⎜ ⎞ ⎠ ⎟ + ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ + ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ +LLL skq r skq r sfd r12 2 3 2 3 2 cos cos sin θ π θ π θ π 33 2 3 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ + ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ L skd r sin θ π (5.2-9) L r lkq mkq kq kq kq kq lkq mkq lfd mfd fdk LL L LLL LL L = + + + 11 12 12 2 2 00 00 00 dd fdkd lkd mkd LLL00 + ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ (5.2-10) In (5.2-8) , L A > L B and L B is zero for a round rotor machine. Also in (5.2-8) and (5.2- 10) , the leakage inductances are denoted with l in the subscript. The subscripts skq 1, skq 2, sfd , and skd in (5.2-9) denote mutual inductances between stator and rotor windings. The magnetizing inductances are defi ned as LLL mq A B =− 3 2 () (5.2-11) LLL md A B =+ 3 2 () (5.2-12) VOLTAGE EQUATIONS IN MACHINE VARIABLES 147 It can be shown that L N N L skq kq s mq1 1 2 3 = ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ (5.2-13) L N N L skq kq s mq2 2 2 3 = ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ (5.2-14) L N N L sfd fd s md = ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 2 3 (5.2-15) L N N L skd kd s md = ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 2 3 (5.2-16) L N N L mkq kq s mq1 1 2 2 3 = ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ (5.2-17) L N N L mkq kq s mq2 2 2 2 3 = ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ (5.2-18) L N N L mfd fd s md = ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 2 2 3 (5.2-19) L N N L mkd kd s md = ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 2 2 3 (5.2-20) L N N L N N L kq kq kq kq mkq kq kq mkq 12 2 1 1 1 2 2 = ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ (5.2-21) L N N L N N L fdkd kd fd mfd fd kd mkd = ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ (5.2-22) It is convenient to incorporate the following substitute variables, which refer the rotor variables to the stator windings. ′ = ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ i N N i j j s j 2 3 (5.2-23) ′ = ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ v N N v j s j j (5.2-24) ′ = ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ λλ j s j j N N (5.2-25) 148 SYNCHRONOUS MACHINES where j may be kq 1, kq 2, fd , or kd . The fl ux linkages may now be written as l l abcs qdr ssr sr T r abcs qdr ′ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = ′ ′′ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ′ ⎡ ⎣ ⎢ ⎤ ⎦ LL LL i i 2 3 () ⎥⎥ (5.2-26) where L s is defi ned by (5.2-8) and ′ =− ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ L sr mq r mq r md r md r mq r mq LLLL LL cos cos sin sin cos θθθθ θ π 2 3 ccos sin sin cos θ π θ π θ π rmdrmdr mq LL L − ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ − ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ − ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 2 3 2 3 2 3 θθ π θ π θ π θ rmqrmdrmdr LLL+ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ + ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ + ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ + 2 3 2 3 2 3 cos sin sin 22 3 π ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ (5.2-27) ′ = ′ + ′ + ′ + ′ L r lkq mq mq mq lkq mq lfd md md md lkd LL L LLL LL L LL 1 2 00 00 00 00 ++ ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ L md (5.2-28) The voltage equations expressed in terms of machine variables referred to the stator windings are v v rL L LrL i abcs qdr ss sr sr T rr abc pp pp ′ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = + ′ ′′ + ′ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ 2 3 () ss qdr ′ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ i (5.2-29) In (5.2-28) and (5.2-29) ′ = ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ r N N r j s j j 3 2 2 (5.2-30) ′ = ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ L N N L lj s j lj 3 2 2 (5.2-31) where, again, j may be kq 1, kq 2, fd , or kd . STATOR VOLTAGE EQUATIONS IN ARBITRARY REFERENCE-FRAME VARIABLES 149 5.3. TORQUE EQUATION IN MACHINE VARIABLES The energy stored in the coupling fi eld of a synchronous machine may be expressed as W f abcs T s abcs abcs T sr qdr =+ ′′ + ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ′ 1 2 1 2 3 2 () () (iLi iLi i qqdr T r qdr ) ′′ Li (5.3-1) Since the magnetic system is assumed to be linear, W f = W c , the second entry of Table 1.3-1 may be used, keeping in mind that the derivatives in Table 1.3-1 are taken with respect to mechanical rotor position. Using the fact that θ θ rrm P = 2 , the torque is expressed in terms of electrical rotor position as T P e abcs T r s abcs abcs T r sr qdr = ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ∂ ∂ + ∂ ∂ ′′ 2 1 2 () [] () []iLiiLi θθ {{} (5.3-2) In expanded form (5.3-2) becomes T P LL iiiiiiii e md mq as bs cs as bs as cs bs = ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ − −−−−+ 23 1 2 1 2 2 222 () ii i i ii ii cs r bs cs as bs as cs r ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎡ ⎣ ⎢ ⎧ ⎨ ⎩ +−−+ sin ()cos 2 3 2 22 2 22 θ θ ⎤⎤ ⎦ ⎥ + ′ + ′ −− ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ −−Li i i i i i i mq kq kq as bs cs r bs cs () sin( 12 1 2 1 2 3 2 θ ))cos () cos ( θ θ r md fd kd as bs cs r Li i i i i ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ − ′ + ′ −− ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ + 1 2 1 2 3 2 iii bs cs r − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ⎫ ⎬ ⎭ )sin θ (5.3-3) The above expression for torque is positive for motor action. The torque and rotor speed are related by TJ P pT erL = ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ + 2 ω (5.3-4) where J is the inertia expressed in kilogram meters 2 (kg·m 2 ) or Joule seconds 2 (J·s 2 ). Often, the inertia is given as WR 2 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 synchronous machine. 5.4. STATOR VOLTAGE EQUATIONS IN ARBITRARY REFERENCE-FRAME VARIABLES The voltage equations of the stator windings of a synchronous machine can be expressed in the arbitrary reference frame. In particular, by using the results presented in Chapter 150 SYNCHRONOUS MACHINES 3 , the voltage equations for the stator windings may be written in the arbitrary reference frame as [3] vri qd s s qd s dqs qd s p 00 0 =++ ω ll (5.4-1) where ()[ ]l dqs T ds qs =− λλ 0 (5.4-2) The rotor windings of a synchronous machine are asymmetrical; therefore, a change of variables offers no advantage in the analysis of the rotor circuits. Since the rotor variables are not transformed, the rotor voltage equations are expressed only in the rotor reference frame. Hence, from (5.2-2) , with the appropriate turns ratios included and raised index r used to denote the rotor reference frame, the rotor voltage equations are ′ = ′′ + ′ vri qdr r r qdr r qdr r pl (5.4-3) For linear magnetic systems, the fl ux linkage equations may be expressed from (5.2-7) with the transformation of the stator variables to the arbitrary reference frame incorporated l l qd s qdr r ss s ssr sr T sr 0 1 1 2 3 ′ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = ′ ′′ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ − − KL K KL LK L () ()() ⎥⎥ ⎥ ′ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ i i qd s qdr r 0 (5.4-4) It can be shown that all terms of the inductance matrix of (5.4-4) are sinusoidal in nature except ′ L r . For example, by using trigonometric identities given in Appendix A KL ssr mq r mq r md r md r LL L L L ′ = − −−−−−cos( ) cos( ) sin( ) sin( ) θθ θθ θθ θθ mmq r mq r md r md r LL Lsin( ) sin( ) cos( ) cos( ) θθ θθ θθ θθ −− − − ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ 00 0 0 ⎦⎦ ⎥ ⎥ ⎥ (5.4-5) The sinusoidal terms of (5.4-5) are constant, independent of ω and ω r only if ω = ω r . Similarly, K s L s ( K s ) − 1 and (/)( )( )23 1 ′ − LK sr T s are constant only if ω = ω r . Therefore, the position-varying inductances are eliminated from the voltage equations only if the refer- ence frame is fi xed in the rotor. Hence, it would appear that only the rotor reference frame is useful in the analysis of synchronous machines. Although this is essentially the case, there are situations, especially in computer simulations, where it is convenient to express the stator voltage equations in a reference frame other than the one fi xed in the rotor. For these applications, it is necessary to relate the arbitrary reference-frame variables to the variables in the rotor reference frame. This may be accomplished by using (3.10-1) , from which VOLTAGE EQUATIONS IN ROTOR REFERENCE-FRAME VARIABLES 151 fKf qd s rr qd s00 = (5.4-6) From (3.10-7) K r rr rr = −− − −− ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ cos( ) sin( ) sin( ) cos( ) θθ θθ θθ θθ 0 0 001 (5.4-7) Here we must again recall that the arbitrary reference frame does not carry a raised index. 5.5. VOLTAGE EQUATIONS IN ROTOR REFERENCE-FRAME VARIABLES R.H. Park was the fi rst to incorporate a change of variables in the analysis of synchro- nous machines [1] . He transformed the stator variables to the rotor reference frame, which eliminates the position-varying inductances in the voltage equations. Park ’ s equations are obtained from (5.4-1) and (5.4-3) by setting the speed of the arbitrary reference frame equal to the rotor speed ( ω = ω r ). Thus vri qd s r sqds r r dqs r qd s r p 00 0 =+ + ω ll (5.5-1) ′ = ′′ + ′ vri qdr r r qdr r qdr r pl (5.5-2) where ()[ ]l dqs rT ds r qs r =− λλ 0 (5.5-3) For a magnetically linear system, the fl ux linkages may be expressed in the rotor refer- ence frame from (5.4-4) by setting θ = θ r . K s becomes K s r , with θ set equal to θ r in (3.3-4) . Thus, l l qd s r qdr r s r ss r s r sr sr T s r r 0 1 1 2 3 ′ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = ′ ′′ ⎡ − − KL K KL LK L () ()() ⎣⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ′ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ i i qd s r qdr r 0 (5.5-4) Using trigonometric identities from Appendix A, it can be shown that KL K s r ss r ls mq ls md ls LL LL L () − = + + ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ 1 00 00 00 (5.5-5) KL s r sr mq mq md md LL LL ′ = ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ 00 00 0000 (5.5-6) [...]... ⎥ ⎥ ⎥⎢ 2 ⎥ ⎢ ′r ⎥ X d X ′ − X md ⎦ ⎣ψ kd ⎦ fd (5. 5-49) (5. 5 -50 ) (5. 5 -51 ) where 2 Dq = − X mq ( X q − 2 X mq + X kq1 + X kq 2 ) + X q X kq1 X kq 2 ′ ′ ′ ′ (5. 5 -52 ) Dd = − X ( X d − 2 X md + X ′ + X kd ) + X d X ′ X kd ′ fd fd ′ (5. 5 -53 ) 2 md Substituting (5. 5-49)– (5. 5 -51 ) for the currents into the voltage equations (5. 5-22)– (5. 5-26), (5. 5-37), and (5. 5-38) yields ωr p ⎤ ⎡r a 0 0 0 −rs a12 −rs a13 ⎥... = rfd i ′ r + pλ ′ r ′ fd fd (5. 5-13) v ′ = r ′ i ′ + pλ ′ (5. 5-14) r kq 2 r kd r kq 2 kq 2 r kq 2 r kd kd r kd Substituting (5. 5 -5) – (5. 5-7) and (5. 2-28) into (5. 5-4) yields the expressions for the flux linkages In expanded form r r r λqs = Llsiqs + Lmq (iqs + ikq1 + ikq 2 ) ′r ′r (5. 5- 15) r r r λ ds = Llsids + Lmd (ids + i ′ r + ikd ) ′r fd (5. 5-16) λ0 s = Llsi0 s (5. 5-17) r λ kq1 = Llkq1ikq1 + Lmq... ′ ′ ⎥ ⎢ ωb ωb ωb ⎦ ⎣ (5. 5-38) where X q = Xls + X mq (5. 5-39) X d = Xls + X md (5. 5-40) X kq1 = Xlkq1 + X mq ′ ′ (5. 5-41) X kq 2 = Xlkq 2 + X mq ′ ′ (5. 5-42) X ′ = Xlfd + X md ′ fd (5. 5-43) X kd = Xlkd + X md ′ ′ (5. 5-44) The reactances Xq and Xd are generally referred to as q- and d-axis reactances, respectively The flux linkages per second may be expressed from (5. 5-29)– (5. 5- 35) as r ⎡ ψ qs ⎤ ⎡ X q... 152 SYNCHRONOUS MACHINES ⎡ Lmq ⎢L 2 mq (L ′ )T (K r )−1 = ⎢ sr s 3 ⎢ 0 ⎢ ⎣ 0 0⎤ 0⎥ ⎥ 0⎥ ⎥ 0⎦ 0 0 Lmd Lmd (5. 5-7) In expanded form, (5. 5-1) and (5. 5-2) may be written as r r r r vqs = rsiqs + ω r λ ds + pλqs (5. 5-8) r r r r vds = rsids − ω r λqs + pλ ds (5. 5-9) v0 s = rsi0 s + pλ0 s (5. 5-10) vkq1 = rkq1ikq1 + pλ kq1 ′r ′ ′r ′r (5. 5-11) v ′ = r ′ i ′ + pλ ′ (5. 5-12) r v ′fd = rfd i... can therefore be expressed as I as = −8 35 ⋅ 0. 85 8 35 1 − 0. 852 −j kA 3 ⋅ 26 3 ⋅ 26 (5A-4) To determine the real component of Ea, the real components in (5. 9-17) are equated, yielding Es cos δ = Vs + ωe 8 35 1 − 0. 852 26 X d I s sin θ ei (0) = − 1. 457 ⋅ kV ωb 3 3 ⋅ 26 (5A -5) From which Ea = 26 8 35 ⋅ 1 − 0. 852 8 35 ⋅ 0. 85 − 1. 457 ⋅ + j1. 457 ⋅ kV 3 3 ⋅ 26 3 ⋅ 26 (5A-6) For the case in which rated power... ikq 2 ) ′r ′r (5. 5-29) ψ = X i + X md (i + i ′ + i ′ ) (5. 5-30) ψ 0 s = Xlsi0 s (5. 5-31) r ds r ls ds r ds r fd r kd ψ ′ = X ′ i ′ + X mq (i + i ′ + i ′ ) (5. 5-32) r ψ kq 2 = Xlkq 2ikq 2 + X mq (iqs + ikq1 + ikq 2 ) ′r ′ ′r ′r ′r (5. 5-33) r kq1 r lkq1 kq1 r qs r kq1 r kq 2 ψ ′ = X ′ i ′ + X md (i + i ′ + i ′ ) (5. 5-34) r ψ kd = Xlkd ikd + X md (ids + i ′ r + ikd ) ′r ′ ′r ′r fd (5. 5- 35) r fd r lfd fd... vfdr − rs + v0s i0s L ls − Figure 5. 5-1 Equivalent circuits of a three-phase synchronous machine with the reference frame fixed in rotor: Park’s equations 154 SYNCHRONOUS MACHINES vkq1 = rkq1ikq1 + ′r ′ ′r p ψ kq1 ′r ωb (5. 5- 25) vkq 2 = rkq 2ikq 2 + ′r ′ ′r p ψ kq 2 ′r ωb (5. 5-26) r v ′fd = rfd i ′ r + ′ fd p r ψ ′fd ωb (5. 5-27) vkd = rkd ikd + ′r ′ ′r p ψ kd ′r ωb (5. 5-28) where ωb is the base electrical... voltage Thus, 167 ANALYSIS OF STEADY-STATE OPERATION 8 35 1 − 0. 852 kA 3 ⋅ 26 (5A-10) ωe 8 35 1 − 0. 852 26 X q I s sin θ ei (0) = + 1. 457 ⋅ kV ωb 3 3 ⋅ 26 (5A-11) I s sin θ ei (0) = and Es cos δ = Vs + Thus Ea = 26 8 35 1 − 0. 852 8 35 ⋅ 0. 85 + 1. 457 ⋅ + j1. 457 ⋅ kV 3 3 ⋅ 26 3 ⋅ 26 (5A-12) The phasor diagrams for the three load conditions are shown in Figure 5A-1 The amplitude of the stator phase current is... ⎦ ⎣ (5. 5 -54 ) In (5. 5 -54 ), aij and bij are the elements of the 3 × 3 matrices given in (5. 5-49) and (5. 550 ), respectively 157 TORQUE EQUATIONS IN SUBSTITUTE VARIABLES 5. 6 TORQUE EQUATIONS IN SUBSTITUTE VARIABLES The expression for the positive electromagnetic torque for motor action in terms of rotor reference-frame variables may be obtained by substituting the equation of transformation into (5. 3-2)... θ ev (0) (5. 9-24) where These voltages may be expressed in the rotor reference frame by replacing θ with θr in (3.6 -5) and (3.6-6) r vqs = 2 vs cos(θ ev − θ r ) (5. 9- 25) v = − 2 vs sin(θ ev − θ r ) (5. 9-26) r ds If the rotor angle from (5. 7-1) is substituted into (5. 9- 25) and (5. 9-26), we obtain r vqs = 2 vs cos δ (5. 9-27) r vds = 2 vs sin δ (5. 9-28) The only restriction on (5. 9-27) and (5. 9-28) is . + ′ + ′ + ′′ 2 2() (5. 5 -53 ) Substituting (5. 5-49)– (5. 5 -51 ) for the currents into the voltage equations (5. 5-22)– (5. 5-26) , (5. 5-37) , and (5. 5-38) yields . ′ = ′′ + ′ vrip fd r fd fd r fd r λ (5. 5-13) ′ = ′′ + ′ vrip kd r kd kd r kd r λ (5. 5-14) Substituting (5. 5 -5) – (5. 5-7) and (5. 2-28) into (5. 5-4) yields the expressions