Chapter 4 permanent magnet ac machines

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

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

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

where

(fabcs)T =[f as f bs f cs] (4.2-2)

Us= diag[r s r s r s] (4.2-3) The fl ux linkages may be written as

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

1 2

3

(4.2-5)

VOLTAGE AND TORQUE EQUATIONS IN MACHINE VARIABLES 123

bs-axis

as-axis q-axis

Sensor

The fl ux linkage l′m may be expressed as

where λm is the amplitude of the fl ux linkages established by the permanent magnet as ′

viewed from the stator phase windings In other words, pλ would be the open-circuit m′

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

32

⎭⎪ (4.2-9)

where L and L are

VOLTAGE AND TORQUE EQUATIONS IN ROTOR REFERENCE-FRAME VARIABLES 125

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 λm′ based

on the geometry, material properties, and stator winding confi guration are provided in Chapter 15

4.3 VOLTAGE AND TORQUE EQUATIONS IN ROTOR

r dqs r

qd s r p

lqd s r

010 (4.3-3)

To be consistent with our previous notation, we have added the superscript r to λm′ In expanded form, we have

r

s qs r

r ds r qs r

r

s ds r

r qs r ds r

λqs r

q qs r

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 pλm′ =r 0 , we can write

v qs r =(r s+pL i q)qs r +ωr L i d ds r +ω λ r m′r (4.3-10)

v ds r =(r s+pL i d)ds r −ωr L i q qs r (4.3-11)

v0s=(r s+pL i ls)0s (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

r qs r ds r ds r

ANALYSIS OF STEADY-STATE OPERATION 127

Substituting (4.3-7) and (4.3-8) into (4.3-15) yields

T e= ⎛⎝⎜ ⎞⎠⎟⎛⎝⎜P⎞⎠⎟ ′m r i qs r + L d−L i i q qs r ds r

3

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 vqs r and vds r Recall that

d

dt

r r

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

Transforming (4.3-18)–(4.3-20) to the rotor reference frame yields

v qs v r

v ds v r

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

V qs r =r I s qs r +ωr L I d ds r +ω λ r m′r (4.4-1)

V ds r =r I s ds r −ωr L I q qs r (4.4-2) where uppercase letters denote steady-state (constant) quantities Assuming no demag-netization, λm′r is always constant The steady-state torque is expressed from (4.3-16) as

r qs r

r ds r

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

j 2 I as =I ds r +jI qs r (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

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 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

V a=r I a a+ωr k v (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, I r a a2 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

Let us now return to the permanent-magnet ac machine From our earlier discussion,

we are aware that the values of Vqs r

and Vds r are determined by the fi ring of the drive

inverter When the inverter is switched so that ϕ v = 0, V qs V

r s

case, (4.4-2) may be solved for Ids r in terms of Iqs r

r m r v

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 ωr2L L q d

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 ωr2L L q d

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 Ids r

and substitute these expressions into the expression for T e (4.4-3) ,

we obtain the following expression

e

s m r

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 ωr2L2s

is neglected, then (4.5-5) yields a straight line T e versus ω r characteristic for a constant Vqs r

Thus, if ωr2L2s

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

BRUSHLESS DC MOTOR 131

minimum torque Let us take the derivative of (4.5-5) with respect to ω r and set the result to zero and solve for ω r Thus, zero slope of the torque versus speed character- istics for L q = L d , V qs r = 2 , and V V s ds

This is shown in Figure 4.5-2 , where L md = 0.6 L mq for the machine considered in Figure

4.5-1 The machine parameters for the characteristics shown in Figure 4.5-1 are

r s = 3.4 Ω , L ls = 1.1 mH, and L mq = L md = 11 mH, thus L q = L d = L s = 12.1 mH The device

is a four-pole machine, and when it is driven at 1000 r/min, the open-circuit to-winding voltage is sinusoidal with a peak-to-peak value of 60 V From this, λm′r is calculated to be 0.0827 V·s (The reader should verify this calculation.)

It is instructive to observe the machine variables during free acceleration and step

changes in load torque with ϕ v = 0 In this case, J = ×5 10− 4 ⋅ 2

kg m , which represents the inertia of the machine and connected mechanical load The dynamic performance

is shown for applied stator phase voltages that are sinusoidal and stepped, as would be the case if a typical six-step voltage source inverter were used to supply the machine The free acceleration characteristics with sinusoidal phase voltages are shown

in Figure 4.5-3 The applied stator phase voltages are of the form given by (4.3-18)–

(4.3-20) with v s = 11.25 V The phase voltage v as , phase current i as , q -axis voltage v qs r,

q -axis current i qs r , d -axis current i ds r , electromagnetic torque T e , and rotor speed ω r in electrical rad/s, are plotted It is clear that since ϕ v = 0, v ds r = 0 The device is a four-

pole machine, thus 200 electrical rad/s is 955 r/min A plot of T e versus ω r is shown in

Figure 4.5-2 Same as Figure 4.5-1 with L = 0.6 L

T e