Chapter 4 permanent magnet ac machines

Analysis of Electric Machinery and Drive Systems Editor(s): Paul Krause, Oleg Wasynczuk, Scott Sudhoff, Steven Pekarek

121 4.1. INTRODUCTION The permanent-magnet ac machine supplied from a controlled voltage or current source inverter is becoming widely used. This is attributed to a relatively high torque density (torque/mass or torque/volume) and ease of control relative to alternative machine architectures. Depending upon the control strategies, the performance of this inverter– machine combination can be made, for example, to (1) emulate the performance of a permanent-magnet dc motor, (2) operate in a maximum torque per ampere mode, (3) provide a “fi eld weakening” technique to increase the speed range for constant power operation, and (4) shift the phase of the stator applied voltages to obtain the maximum possible torque at any given rotor speed. Fortunately, we are able to become quite familiar with the basic operating features of the permanent-magnet ac machine without getting too involved with the actual inverter or the control strategies. In particular, if we assume that the stator variables (voltages and currents) are sinusoidal and balanced with the same angular velocity as the rotor speed, we are able to predict the predominant operating features of all of the above mentioned modes of operation without becoming involved with the actual switching or control of the inverter. Therefore, in this chapter, 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. PERMANENT-MAGNET AC MACHINES 4 122 PERMANENT-MAGNET AC MACHINES we will focus on the performance of the inverter–machine combination assuming that the inverter is designed and controlled appropriately and leave how this is done to Chapter 14 . 4.2. VOLTAGE AND TORQUE EQUATIONS IN MACHINE VARIABLES A two-pole, permanent-magnet ac machine, which is also called a permanent-magnet synchronous machine, is depicted in Figure 4.2-1 . It has three-phase, wye-connected stator windings and a permanent-magnet rotor. The stator windings are identical wind- ings displaced at 120°, each with N s equivalent turns and resistance r s . For our analysis, we will assume that the stator windings are sinusoidally distributed. The three sensors shown in Figure 4.2-1 are Hall effect devices. When the north pole is under a sensor, its output is nonzero; with a south pole under the sensor, its output is zero. During steady-state operation, the stator windings are supplied from an inverter that is switched at a frequency corresponding to the rotor speed. The states of the three sensors are used to determine the switching logic for the inverter. In the actual machine, the sensors are not positioned over the rotor, as shown in Figure 4.2-1 . Instead, they are often placed over a ring that is mounted on the shaft external to the stator windings and magnetized in the same direction as the rotor magnets. We will return to these sensors and the role they play later. The voltage equations in machine variables are vri abcs s abcs abcs p=+l (4.2-1) where ()[ ]f abcs T as bs cs fff= (4.2-2) U s = diag[ ]rrr sss (4.2-3) The fl ux linkages may be written as ll abcs s abcs m =+ ′ Li (4.2-4) where, neglecting mutual leakage terms and assuming that due to the permanent magnet the d -axis reluctance of the rotor is larger than the q -axis reluctance, L s may be written as / 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 (4.2-5) VOLTAGE AND TORQUE EQUATIONS IN MACHINE VARIABLES 123 Figure 4.2-1. Two-pole, three-phase permanent-magnet ac machine. q r r s r s r s N s N s N s n v as v cs i cs i as i bs v bs + + + bs-axis as-axis q-axis cs-axis d-axis as ¢ bs ¢ cs ¢ cs as bs S N w r 1 2 3 SensorSensor Sensor 124 PERMANENT-MAGNET AC MACHINES The fl ux linkage ′ l m may be expressed as ′ = ′ − ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ + ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ l mm r r r λ sin sin sin θ θ π θ π 2 3 2 3 ⎢⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ (4.2-6) where ′ λ m is the amplitude of the fl ux linkages established by the permanent magnet as viewed from the stator phase windings. In other words, p m ′ λ would be the open-circuit voltage induced in each stator phase winding. Damper windings are neglected since the permanent magnets are typically relatively poor electrical conductors, and the eddy currents that fl ow in the nonmagnetic materials securing the magnets are small. Hence, in general large armature currents can be tolerated without signifi cant demagnetization. We have assumed by (4.2-6) that the voltages induced in the stator windings by the permanent magnet are constant amplitude sinusoidal voltages. A derivation of (4.2-5) and (4.2-6) is provided in Chapter 15 . The expression for the electromagnetic torque may be written in machine variables using Τ Ρ e c r W = ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ∂ ∂2 θ (4.2-7) where WW c abcs T s abcs abcs T mpm =+ ′ + 1 2 iLi il (4.2-8) In (4.2-8) , W pm is the energy in the coupling fi eld due to the presence of the permanent magnet. Substituting (4.2-8) into (4.2-7) and neglecting any change in W pm with rotor position, the electromagnetic torque is expressed Τ Ρ e md mq as bs cs as bs as cs bs LL ii ii i= ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ − () −−−−+ 23 1 2 1 2 2 222 LLL ii i i ii ii cs r bs cs as bs as cs r ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎡ ⎣ ⎢ ⎧ ⎨ ⎩ +−−+ sin ()cos 2 3 2 22 2 22 θ θ ⎤⎤ ⎦ ⎥ + ′ −− ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ +− ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ⎫ λθθ masbscs r bscs r iii ii 1 2 1 2 3 2 cos ( )sin ⎬⎬ ⎭ ⎪ (4.2-9) where L mq and L md are VOLTAGE AND TORQUE EQUATIONS IN ROTOR REFERENCE-FRAME VARIABLES 125 LLL mq A B =+ 3 2 () (4.2-10) LLL md A B =− 3 2 () (4.2-11) The above expression for torque is positive for motor action. The torque and speed may be related as TJ P pB P T ermrL = ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ + ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ + 22 ωω (4.2-12) where J is the inertia of the rotor and the connected load is in kg·m 2 . Since we will be concerned primarily with motor action, the torque T L is positive for a torque load. The constant B m is a damping coeffi cient associated with the rotational system of the machine and the mechanical load. It has the units N·m·s per radian of mechanical rota- tion, and it is generally small and often neglected. Derivations of L mq , L md , and ′ λ m based on the geometry, material properties, and stator winding confi guration are provided in Chapter 15 . 4.3. VOLTAGE AND TORQUE EQUATIONS IN ROTOR REFERENCE-FRAME VARIABLES The voltage equations in the rotor reference frame may be written directly from (3.4-3) and (3.4-9) with ω = ω r . vri qd s r sqds r r dqs r qd s r p 00 0 =+ + ω ll (4.3-1) where ()[ ]l dqs rT ds r qs r =− λλ 0 (4.3-2) l qd s r ls mq ls md ls qs r ds r s LL LL L i i i 0 0 00 00 00 = + + ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥⎥ ⎥ ⎥ + ′ ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ λ m r 0 1 0 (4.3-3) To be consistent with our previous notation, we have added the superscript r to ′ λ m . In expanded form, we have vri p qs r sqs r rds r qs r =+ + ωλ λ (4.3-4) vri p ds r sds r rqs r ds r =− + ωλ λ (4.3-5) vrip sss s00 0 =+ λ (4.3-6) 126 PERMANENT-MAGNET AC MACHINES where λ qs r qqs r Li= (4.3-7) λλ ds r dds r m r Li=+ ′ (4.3-8) λ 00slss Li= (4.3-9) where L q = L ls + L mq and L d = L ls + L md . It is readily shown that if mutual leakage between stator windings shown in Chapter 2 is included in (4.2-5) , the form of the q - and d -axis fl ux linkages remains unchanged. Indeed, the only impact will be on the respective leakage terms in L q and L d . Substituting (4.3-7)–(4.3-9) into (4.3-4)–(4.3-6) , and since p m r ′ = λ 0 , we can write v r pL i L i qs r sqqs r rdds r rm r =+ + + ′ () ωωλ (4.3-10) v r pL i L i ds r sdds r rqqs r =+ −() ω (4.3-11) v r pL i ss lss00 =+() (4.3-12) The expression for electromagnetic torque in terms of q and d variables may be obtained by substituting the expressions for the machine currents in terms of q - and d -currents into (4.2-9) . This procedure is quite labor intensive; however, once we have expressed the voltage equations in terms of reference-frame variables, a more direct approach is possible [1] . In particular, the expression for input power is given by (3.3-8) , and the electromagnetic torque multiplied by the rotor mechanical angular velocity is the power output. Thus we have T P vi vi vi erqs r qs r ds r ds r ss 23 2 2 00 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ =++ () ω (4.3-13) Substituting (4.3-4)–(4.3-6) into (4.3-13) gives us T P ri i i i i ersqs r ds r sds r qs r qs r ds r 23 2 2 3 2 22 0 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ =++ ( ) +− () ωλλωω λλ λ r qs r qs r ds r ds r ss ip ip ip+++ () 3 2 2 00 (4.3-14) The fi rst term on the right-hand side of (4.3-14) is the ohmic power loss in the stator windings, and the last term is the change of stored magnetic energy. If we equate the coeffi cients of ω r , we have T P ii eds r qs r qs r ds r = ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ − 3 22 () λλ (4.3-15) ANALYSIS OF STEADY-STATE OPERATION 127 Substituting (4.3-7) and (4.3-8) into (4.3-15) yields T P iLLii em r qs r dqqs r ds r = ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ′ +− 3 22 [()] λ (4.3-16) The electromagnetic torque is positive for motor action. When the machine is supplied from an inverter, it is possible, by controlling the fi ring of the inverter, to change the values of v qs r and v ds r . Recall that d dt r r θ ω = (4.3-17) Mathematically, θ r is obtained by integrating (4.3-17) . In practice, θ r is estimated using Hall sensors or a position observer, or measured directly using an inline position encoder. For purposes of discussion, let us assume that the applied stator voltages are sinusoidal so that vv as s ev = 2cos θ (4.3-18) vv bs s ev =− ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 2 2 3 cos θ π (4.3-19) vv cs s ev =+ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 2 2 3 cos θ π (4.3-20) When the machine is supplied from an inverter, the stator voltages are controlled such that θθφ ev r v =+ (4.3-21) With power electronics, the voltages will generally have a waveform with switching harmonics included. Nevertheless, as a fi rst approximation, (4.3-18)–(4.3-20) may be considered as the fundamental components of these stepped phase voltages. Transforming (4.3-18)–(4.3-20) to the rotor reference frame yields vv qs r sv = 2cos φ (4.3-22) vv ds r sv =− 2sin φ (4.3-23) 4.4. ANALYSIS OF STEADY-STATE OPERATION For steady-state operation with balanced, sinusoidal applied stator voltages, (4.3-10) and (4.3-11) may be written as 128 PERMANENT-MAGNET AC MACHINES VrI LI qs r sqs r rdds r rm r =+ + ′ ωωλ (4.4-1) VrI LI ds r sds r rqqs r =− ω (4.4-2) where uppercase letters denote steady-state (constant) quantities. Assuming no demag- netization, ′ λ m r is always constant. The steady-state torque is expressed from (4.3-16) as T P ILLII em r qs r dqqs r ds r = ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ′ +− 3 22 [()] λ (4.4-3) It is possible to establish a phasor voltage equation from (4.4-1) and (4.4-2) . For steady- state operation, ϕ v is constant and represents the angular displacement between the peak value of the fundamental component of v as and the q -axis fi xed in the rotor. If we refer- ence the phasors to the q -axis and let it be along the positive real axis of the “stationary” phasor diagram, then ϕ v becomes the phase angle of  V as , and we can write  VVe VjV as s j svsv v = =+ φ φφ cos sin (4.4-4) Comparing (4.4-4) with (4.3-22) and (4.3-23) , we see that 2  VVjV as qs r ds r =− (4.4-5) If we write the three-phase currents similar to (4.3-18)–(4.3-20) in terms of θ ei , then (4.3-21) would be in terms of θ ei and ϕ i rather than θ ev and ϕ v and we would arrive at a similar relation for current as (4.4-5) . In particular, 2  IIjI as qs r ds r =− (4.4-6) and jI I jI as ds r qs r 2  =+ (4.4-7) Substituting (4.4-1) and (4.4-2) into (4.4-5) and using (4.4-6) and (4.4-7) yields  VrjLIE as s r q as a =+ +() ω (4.4-8) where  ELLIe ardqds r rm rj =−+ ′ [] 1 2 0 ωωλ () (4.4-9) It is noted that  E a for the permanent-magnet ac machine, (4.4-9) , is along the reference (real) axis. BRUSHLESS DC MOTOR 129 4.5. BRUSHLESS DC MOTOR The permanent-magnet ac machine is often referred to as a “brushless dc motor.” This is not because it has the physical confi guration of a dc machine, but because by appro- priate control of the driving inverter, its terminal characteristics may be made to resemble those of a dc motor. In order to show this, it is necessary to give a brief dis- cussion of the dc machine that will be a review for most; however, a more detailed analysis is given in Chapter 14 . The voltages induced in the armature (rotating) windings of a dc machine are sinusoidal. These induced voltages are full-wave rectifi ed as a result of the windings being mechanically switched by the action of the brushes sliding on the surface of the commutator mounted on the armature. This “dc voltage,” which is often called the counter electromotive force or back voltage, is proportional to the strength of the sta- tionary fi eld, in which the armature windings rotate, and the armature speed. This stationary fi eld is established by either a winding on the stationary member of the machine or a permanent magnet. The steady-state armature voltage equation may be written VrI k aaa rv =+ ω (4.5-1) where V a is the armature terminal voltage, r a is the resistance of the armature windings between brushes, I a is the armature current, ω r is the rotor speed in rad/s, k v is propor- tional to the fi eld strength, and ω r k v is the counter electromotive force. If a fi eld winding is used to establish the stationary fi eld, k v will vary with the winding current; if the fi eld is established by a permanent magnet, k v is constant. In either case, k v has the units of V·s/rad. It is clear from (4.5-1) that the voltage ω r k v is the open-circuit ( I a = 0) armature voltage. The commutation of the armature windings is designed so that the magnetic fi eld established by the current following in the armature windings is stationary and orthogo- nal to the stationary magnetic fi eld established by the fi eld winding or the permanent magnet. With the two fi elds stationary and always in quadrature, the maximum possible torque is produced for any given strength of the magnetic fi elds. We will fi nd that this optimum torque characteristic is the objective of many of the advanced control tech- niques for ac machines. The expression for torque may be obtained by multiplying (4.5-1) by I a and recognizing that V a I a is the power input, Ir aa 2 is the ohmic power loss, and ω r k v I a is the power output. Since the torque times rotor speed is the power output, the electromagnetic torque may be expressed as TkI eva = (4.5-2) Let us now return to the permanent-magnet ac machine. From our earlier discussion, we are aware that the values of V qs r and V ds r are determined by the fi ring of the drive inverter. When the inverter is switched so that ϕ v = 0, VV qs r s = 2 and V ds r = 0 . In this case, (4.4-2) may be solved for I ds r in terms of I qs r . 130 PERMANENT-MAGNET AC MACHINES I L r I ds r rq s qs r v == ω φ for 0 (4.5-3) Substituting (4.5-3) into (4.4-1) yields V rLL r I qs r srqd s qs r rm r v = + ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ + ′ = 22 0 ω ωλ φ for (4.5-4) We now start to see a similarity between the voltage equation for the permanent-magnet ac machine operated in this mode ( ϕ v = 0) and the dc machine. If we neglect ω rqd LL 2 in (4.5-4) , then (4.5-1) and (4.5-4) would be identical in form. Let us note another similarity. If L q = L d , then the expression for the torque given by (4.4-3) is identical in form to (4.5-2) . We now see why the permanent-magnet ac motor is called a brushless dc motor when ϕ v = 0, since the terminal characteristics appear to resemble those of a dc motor. We must be careful, however, since in order for (4.5-1) and (4.5-4) to be identical in form, the term ω rqd LL 2 must be signifi cantly less than r s . Let us see what effects this term has upon the torque versus speed characteristics. To do this, let us fi rst let L q = L d = L s , and if we then solve (4.5-4) for I qs r , and if we take that result along with (4.5-3) for I ds r and substitute these expressions into the expression for T e (4.4-3) , we obtain the following expression T Pr rL V e sm r srs qs r rm r v = ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ′ + − ′ = 3 22 0 222 λ ω ωλ φ ()for (4.5-5) The steady-state, torque-speed characteristics for a brushless dc motor are shown in Figure 4.5-1 . Therein L q = L d and ϕ v = 0; hence, Figure 4.5-1 is a plot of (4.5-5) . If ω rs L 22 is neglected, then (4.5-5) yields a straight line T e versus ω r characteristic for a constant V qs r . Thus, if ω rs L 22 could be neglected, the plot shown in Figure 4.5-1 would be a straight line as in the case of a dc motor. Although the T e versus ω r is approximately linear over the region of motor operation where T e ≥ 0 and ω r ≥ 0, it is not linear over the complete speed range. In fact, we see from Figure 4.5-1 that there appears to be a maximum and Figure 4.5-1. Torque-speed characteristics of a brushless dc motor with L q = L d , ϕ v = 0, VV qs r s = 2 , and V ds r = 0 . T e w r [...]... increased by a factor of 3 by decreasing rs 2.0 60 1.5 40 1.0 fvMT, deg 80 TeM fv = 0.5 20 –200 fvMT fv = 0 1 p 2 –100 100 1 fv = – p 2 0 –0.5 Te, N·m Speed, rad/s 200 –1.0 –1.5 –2.0 fv = p Figure 4. 6-3 Torque-speed characteristics of permanent- magnet ac machine with τs increased by a factor of 3 by increasing Ls  140 PERMANENT- MAGNET AC MACHINES 2π ⎞ ⎛ ibs = 2is cos ⎜ θ r + φi − ⎟ ⎝ 3⎠ (4. 7-2) 2π ⎞ ⎛... 1 fv = – p 2 –1.5 fv = p –2.0 Figure 4. 6-1 Torque-speed characteristics of a permanent- magnet ac machine with Lq = Ld The steady-state, torque-speed characteristics for the permanent- magnet ac machine with the parameters given in Section 4. 5 are shown in Figure 4. 6-1 for Lq = Ld The value of Vs is 11.25 V In order to illustrate the limits of the torque-speed characteristics for the various possible... Figure 4. 5-6 Free-acceleration characteristics of a brushless dc motor supplied from a typical six-step voltage source inverter (ϕv = 0) Compare with Figure 4. 5-3 136 PERMANENT- MAGNET AC MACHINES Te, N·m 1.5 1.0 0.5 0 100 wr, rad/s 200 Figure 4. 5-7 Torque-speed characteristics for free acceleration shown in Figure 4. 5-6 r Vqs = 2Vs cos φv (4. 6 -4) V = − 2Vs sin φv (4. 6-5) r ds wherein ϕv is determined... Power Conditioner— Machine Interaction for Electronically Commutated DC Permanent Magnet Machines, ” IEEE Trans Magn., Vol 17, November 1981, pp 32 84 3286 [3] T.M Jahns, “Torque Production in Permanent- Magnet Synchronous Motor Drives with Rectangular Current Excitations,” IAS Conf Rec., October 1983, pp 47 6 48 7 PROBLEMS 1 Verify (4. 3-3) 2 Write the voltage equations given by (4. 3-10)– (4. 3-12) and the torque... machine, thus 200 electrical rad/s is 955 r/min A plot of Te versus ωr is shown in Te ωr Figure 4. 5-2 Same as Figure 4. 5-1 with Lmd = 0.6Lmq 132 PERMANENT- MAGNET AC MACHINES 20 10 vas, V 0 –10 –20 5 ias, A 0 –5 20 r v qs, V 10 0 0.05 second r i qs, A 5 0 r i ds, A 5 0 1.0 Te, N·m 0.5 0 200 wr, rad/s 100 0 Figure 4. 5-3 Free-acceleration characteristics of a brushless dc motor (ϕv = 0) Figure 4. 5 -4. .. A three-phase permanent- magnet ac machine is operating with I qs = 100 A and r 2 I ds = −10 A The load is a fan with TL = 0.1ω r The parameters of the machine are P = 4, rs = 0.01 Ω, Lq = Ld = 1 mH, λ mr = 0.133 V ⋅ s Determine Vas and the machine ′ efficiency r r r r 14 A six-pole permanent- magnet ac machine has λqs = Lq iqs, λ ds = Ld iqs − λ mr Draw a ′ cross sectional view of the machine that shows... gated to track the commanded currents Provided that the inverter is able to track the commanded currents, the phase currents are of the form ias = 2is cos(θ r + φi ) (4. 7-1) 139 CONTROL OF STATOR CURRENTS 4 Te, N·m 80 fvMT, deg 60 40 3 TeM 2 fv = 1 p 2 1 20 –200 fvMT fv = 0 100 –100 Speed, rad/s 200 1 fv = – p 2 –1 –2 –3 fv = p 4 Figure 4. 6-2 Torque-speed characteristics of permanent- magnet ac machine... MACHINES the load torque is stepped back to 0.1 N·m In these studies, the inertia is 2 × 10 4 kg·m2, which is 40 % of the inertia used in Figure 4. 5-3 and Figure 4. 5 -4 The free-acceleration characteristics with the machine supplied from a typical voltage source inverter are shown in Figure 4. 5-6 with J = 5 × 10 4 kg·m2 so that a direct comparison can be made with Figure 4. 5-3 Although this may seem inappropriate... torques occur may be determined by taking the derivative of (4. 6-6) with respect to ωr and setting the result to zero This yields ω rMT = 1 τ s sin φv − τ v 1 ⎡ ⎤ 2 2 ⎢ − cos φv ± τ τ s + τ v − 2τ sτ v sin φv ⎥ ⎣ ⎦ s It is left to the reader to show that (4. 6- 14) , with ϕv = 0, is (4. 5-6) (4. 6- 14) 138 PERMANENT- MAGNET AC MACHINES 2.0 60 1.5 40 20 –200 Speed, rad/s fvMT Te, N·m fvMT, deg 80 1.0 TeM 0.5... balanced steady-state operation given by (4. 4-1)– (4. 4-3) From (4. 4-2) r r Vds + ω r Lq I qs rs (4. 6-1) rs2 + ω r2 Lq Ld r ω r Ld r I qs + Vds + ω r λ mr ′ rs rs (4. 6-2) r I ds = Substituting (4. 6-1) into (4. 4-1) yields r Vqs = r Let us again set Lq = Ld = Ls, which simplifies our work In this case, Te and I qs differ r only by a constant multiplier Solving (4. 6-2) for I qs yields r I qs = rs r + ω r2 . & Sons, Inc. PERMANENT- MAGNET AC MACHINES 4 122 PERMANENT- MAGNET AC MACHINES we will focus on the performance of the inverter–machine combination. as (4. 4-5) . In particular, 2  IIjI as qs r ds r =− (4. 4-6) and jI I jI as ds r qs r 2  =+ (4. 4-7) Substituting (4. 4-1) and (4. 4-2) into (4. 4-5)