Appendix A Field-Oriented Control

A.1 Introduction Chronologically, Field-Oriented Control (F.O.C.) was the first vector control method developed for controlling induction motors. The principle of this method was proposed in the early 1970s by F. Blaschke of Siemens, who used physical analysis to show that the two components of the stator current space vector projected along two rectangular axes, to be defined later, called the direct and transverse axes, play the same roles as the field and armature currents in a DC motor. The direct axis was found to be oriented along the axis of the magnetic rotor field, !e:lm2' (see Chap.7, Sec7.2.2, Eq.7.21), which is why this approach has been called Field Orientation. In the following paragraphs we shall present a more direct approach to Field-Oriented Control and explain how to implement it. However, first we must introduce some preliminary investigations before taking up discussion of the Field Orientation Principle.

A.2 Preliminary Investigation Imagine a set of symmetrical three-phase rotor windings revolving inside a magnetic field. The magnitude and orientation of the field may be time­ dependent. However, we assume that the field has a plane of symmetry containing the rotor axis. Therefore, the cross section of the field by any plane perpendicular to the shaft will have a common axis of symmetry at any given time. Let the electrical angle of the axis of the field with respect to rotor phase "a" axis be A. The flux linkages of the various rotor phases would then be

"'a = ",(t)COS(A) (A.1), '"b = ",(t) COS(A - 21r / 3) (A.2), "'c = ",(t)cos(A-4n /3) (A.3). 206 Field-Oriented Control

The rotor flux linkage space vector is defined as

3 . lJI = -lJI(t)e J). (AA). _2 2

Notice that this vector has the same direction as the rotor field axis of symmetry. The space vector of the induced emf is obtained as

_2 __ dlJl 3 dlJl(t) e j ). + jCO':!!.2 (A5), g2 =Tt- 2 dt in which the rotor angular velocity, co, is defined as

CO=dA (A6). dt

Two induced electromotive forces may be identified: transformational emf, given by the first term in Eq.(A5) and rotational emf, given by the second term in Eq.(A5). Note that the rotational emf is characterized by a flux linkage multiplied by the factor jco. If we assume that the current space vector i2 flows in the rotor windings, two output powers may be defined: the transformation power, which enters into the circuit's balance of magnetic energy exchange, defined as

= - dlJl(t) P•. ~rs Re{ dt e !2*} (A.7). and the rotation or mechanical power, defined as

p.~c =- ~ coRe{j~2!~}= ~ coIm{~2!~} (A8).

The output electromagnetic torque is thus

m T=-=-pP « 2 Im {lJI b.*} (A9), n 3 - 2 in which n = CO / p is the mechanical angular velocity and p the number of pair-poles. We shall now analyze the principle of the F.O.c. in the light of this introduction Appendix A 207

A.3 Field-Oriented Control The rotational emf appears in the rotor loop of the general equivalent circuit shown in Fig.2.4, Chap.2, Sec2.6, and is defined as

~2q = jW(Lo!1 + L2!2) (A. 10), which shows that the rotor windings revolve within a magnetic field defined in the stator frame of reference as

'i!..lm2 =Lo!1 +L2!2 = Lo[tl +(1+<12)!J=Lo!/l (A.ll).

The direction of the vector ':!!.lm2 in the complex plane, or equivalently that of !IJ ' defines the axis of symmetry of this field, also called the direct axis (Fig.A.l). Perpendicular to the direct axis is the transverse axis. The direct and transverse axes play major roles in the Field Orientation approach.

Let the two components of the current space vectors, !I' and !2, along the direct and transverse axes be (ild , ilq ) and (i2d , i2q), respectively. According to Eq.(A.ll), we have the following relations (see also Fig.A.l(a) and (b)):

ild +(l+<12)i2d =ilJ (A. 12),

i lq +(1+<12)i2q =0 (A. 13). where ilJ is the magnitude of !IJ' The torque is thus given by

2 { 2 L I {. 2pLo .. T=-pIm "'I 2!z.o} =-p 0 m !1J!z.o} 1IJ/lq (A. 14). 3 - m 3 3(1+<12)

Eq.(A.l4) shows that the torque is proportional to the product of a field current, ilJ and an annature current, iJ q , as in a DC motor. One complementary fact is that the magnitude of the field current, i lJ , can itself be controlled by the direct component, ild, of the stator current space vector. This may be confirmed by writing the equation for the rotor loop, which can easily be transformed into

di Try ~+(l- jwr2 )i" =il (A.l5). - dt -~ - 20S Field-Oriented Control

The time-constant of the current II' is thus 'l"2, (see Chap.l, Sec. 1.3, Eq.1.l6). Let the currents II' and II be respectively written as (See Fig.AI)

. -' p,p !1'- l l'e (A.l6).

II = (ild + jhq )e P'p (AI7).

We can obtain the components of the two members of Eq.(A.15) along the direct and transverse axes by multiplying them by e-jJ.p, respectively taking the real and imaginary parts of it, resulting in

q-axis q-axis

d-axis

12(1 + 0" 2) Rea/axis !2(1 +0"2)

(a) (b)

(Fig.A.i) Components o/the currents over the direct and transverse axes (a) general case; (b) stabilized il'

dil' . . 'l"2 --+11' =/ld (AIS), dt

dAI' i lq --=W+-- (A.l9). dt i JJ 'l"2

Eq.(AIS) shows that the magnitude of II' is controlled by ild. If the direct current, hd, is maintained constant for an interval of time equal or superior to 5'l"2, iJJ = ild results. In such case the expression of the torque given by Eq.(AI4) becomes: Appendix A 209

L T- 2p o .. (A.20). 3(1+0"2) 'ld'lq

Another result is that according to Eq.(A.12), when the relation iJI =ild is achieved, the rotor-current space vector, !2, maintains the direction of the transverse axis (see also Fig.AI (b)). We have already seen that this condition is achieved in the permanent regime (see Chap.3, Sec.3.2, Eq.3.18, with fL2 = 0 ).

Thus, given the reference magnetlSlng energy (flux), W"I, the required reference direct current, Ildre!, that must be maintained constant in order to achieve this flux may be determined (for example, the magnetising energy may be maintained constant, equal to its steady-state rated value). In addition, given the reference torque, T"f, the required transverse current, Ilqre! , may be calculated from Eq.(A.20).

In light of what has already been discussed, one may maintain the direct component, ild, of the stator current constant, in order to set a constant flux in the magnetic circuit of the motor, and rapidly change the transverse component of this current, ilq , thereby achieving high dynamic torque control. This constitutes the principle of the Field-Oriented Control.

Implementation of F.O.C. requires that the current vector !/1 (or, equivalently, 'llm2) be known at any given time. Direct and indirect methods can be used in order to achieve this task:

• The first direct method consists in measuring, by means of some appropriate flux-sensing device, the magnetising field, 'lImO' as well as the stator current, i I . The rotor field is then determined as

L (A.21). V'_1m2 = (1+0"2)11'_lmO -0"2 O!1

This method may be quite accurate. However, for purposes of reliability, the use of a sensor inside the motor is not desirable and this approach often is not used .

• The vector 'lImO may be determined by measuring the current !I as well as its time derivative (see Chap.?, Sec.?2.2). This direct method does not require the use of a sensor inside the motor and may be accurately achieved, provided that the rotor resistance, R2, is correctly evaluated at any given time. 210 Field-Oriented Control

• One indirect method may be used by integrating Eq.(A.15) using the current II as the driving input signal. This method may be inaccurate, since integration of Eq.(A.15) is difficult to achieve. In addition, it requires that 't'2, that is, the rotor resistance parameter, be correctly evaluated at any given time .

• Another indirect method consists in integrating Eq.(7.32), Chap.7, Sec.7.2.3, using the current II' and the voltage fll' as the driving input signals. Having measured the stator current space vector, i. , the rotor flux is then determined as

VI = (1 +0" 2)VI LO!I (A.22). _1m2 -Iml -..!!....1-0"

This method may be inaccurate, since accurate integration of Eq.(7.32) would be difficult. Meanwhile, it should be pointed out that this method is rather independent of the value of the rotor resistance.

Computation of the driving voltage, fll' required to achieve the reference torque and the reference energy in a voltage-source inverter-fed induction motor under Field-Oriented Control is more involved. This computation may be conducted as follows: The required driving voltage, fll' is calculated using the equation for the stator loop of the general equivalent circuit. After necessary simplifications, this equation reads

. Lo d!1l d!1 !!I =RI!I +---+O"(I+O"I)Lo- (A.23). 1+0"2 dt dt

Taking Eq.(A.15) into account, Eq.(A.23) becomes

L di !!I=RI!.+ 0 [!I-(l-jCO't'2)!Il]+O"(l+0"1)Lo -=.!.. (A.24). (1+0"2)'t'2 dt

We can compute fll from Eq.(A.24), provided the required current derivative, dII / dt, has been previously calculated. Thus, we must once again calculate the acceleration of the charges within the stator phases, confirming that the approach is a variant of the Computed Charges Acceleration Method. However, we only need to determine the stator current charge acceleration; the charge acceleration of the rotor current need not be known. Field co-ordinates thus appear as a convenient reference frame in Appendix A 211

which resolution of the linear equations established in Chap.7, Sec.7.2.1, Eqs.(7.14) to (7.17) may be simplified.

Taking into account Eqs.(A.17) and (A.l9), the stator current derivative reads

d · d' di i ! I (lid . Iq) jA~ + '(r., + _lq_)I' -= --+J--e J"'. _I (A.25). dt dt dt II' 'r 2

The unknown variables, diId / dt and dilq / dt, that is, the charges acceleration of the direct and transverse currents are calculated as

di = lldret-ild ld (A.26), dt - I1t dilq llqret-ilq (A.27), dt I1t in which ivt and iIq are the actual values of the direct and transverse currents; lldre! and llqre! are the direct and transverse reference currents, and I1t is the time-interval within which the reference energy and reference torque must be attained.

Bringing Eqs.(A.26) and (A.27) into Eq.(A.25), and the result of Eq.(A.25) into Eq.(A.24), the required driving voltage, H-I' is determined. The voltage H-I remains to be achieved by space vector modulation and applied across the motor terminals. Appendix B Direct Torque Control

B.1 Introduction Direct Torque Control (D.T.C.) is a vector control method that was developed in the late 1980s by Asea Brown Bovery (ABB) inspired by the work of M. Depenbrock. In this method, the major role is assigned to the stator field, ~Iml' which is firmly related to the driving input voltage, HI (see Chap.7, Sec.7.2.3, Eq.7.32). The rotor field, VI_1m2 ,serves as the reference base for positioning the stator field, VI_Iml . The magnitude of this latter vector is controlled in order to set the magnetic flux level in the machine. The position angle of VI I with respect to VI is controlled so as _1m _1m2 to achieve the required torque. The following paragraphs present the Direct Torque Control Principle: B.2 Direct Torque Control Principle The relative positions of the stator and rotor fields, VII I' VI ,as well as - m _1m2 the stator current space vector, ii, at an arbitrary given time are represented in Fig.B.l. These three vectors (therefore also i2) may be determined by direct measurement or by indirect methods involving various flux linkages (see Appendix A, Sec.A.3). The motor's electrical angular velocity, ro, may be easily determined as well. The output torque is obtained as

T-~ 1-0" Im{ .} (B.l). - 3 O"L P !.lm2 !.Iml o

Moreover, increments of the stator and rotor fields, fiVl_Iml and fiVl_1m 2 during the time-interval f1t are calculated as (also see Eqs.(7.32) and (7.33), Chap.7, Sec.7.2.3) fi!.lml =(!!I-RIUfit (B.2), fiVl = (R2 i 2 jroVl )f1t (B.3). _1m2 - + _1m2 214 Direct Torque Control

1:i 3 1:i 2

1m

". *11 ~ f1. 6 ,,' .. d!lm2

! lm2

...... -= :. Re

(Fig.B.]) Stator-current, stator-field and rotor-field space vectors

Note that the increment d~lm2 does not depend on the driving voltage, !!I' whereas d~lml does.

Assuming that the reference torque and flux are provided, it is possible to calculate what the increment d!lml of the stator flux linkage, ~Iml' would have to be at a given time in order to meet these reference values within the time-interval dt, assuming that the rotor flux linkage, ~lm2' is known. The vector ~lm2 itself and its increment within the time-interval & may be calculated at any given time, using Eq.(B.3) and Eq.7.33, Chap.7, Sec.7.2.3. Note that the increment dV' over the time-interval dt represents some -1m) charge acceleration, which confirms that this method is also a variant of the more general Computed Charges Acceleration Method. Finally, the required applied voltage, !!I' is detennined according to Eq.(B.2).

A simplified version of this method consists in assuming that during time­ interval & the rotor flux increment is nil; that is, flY' = At the same _1",2 o. time, the appropriate voltage space vector is applied across the phase windings during time-step & in order to move toward the reference torque and flux. If, for example, referring to Fig.B.l, the angle of vector ~Iml with respect to V' must increase and its length must decrease over the next _1m2 time-step, dt, the voltage vector f1.4' or 115 , or a modulated vector fabricated using these two generating vectors, may be applied to the phases during this time. At the end of the time-step, the three complex vectors ~Iml ' Appendix B 215

lfI 0' and II are re-determined by measurement and the control process is _Im_ reiterated. After several such reiterations the actual torque and energy closely approach their reference values. Note that due to the simplified hypothesis concerning the zero-increment of the rotor field within each time­ step, the reference targets may not be precisely attained, since this method amounts to a kind of "shot-in-the-dark" approach. However, if ru is sufficiently short, the flux and torque values achieved should not fall too far from their targets and would be distributed within a hysteresis band above and below their reference values. Appendix C Double-Cage Induction Motors

C.1 Introduction In this appendix we shall first establish the general equivalent circuit of a short-circuited double-cage induction machine. Next, we shall give the steady-state equivalent circuit of such a machine fed from fixed frequency and voltage mainlines. Equivalent circuits of double-cage induction machines may be established exactly in the same way as was proceeded for deriving the equivalent circuit of single-cage machines. The difference is that in the latter case, as we already know, a two-loop equivalent circuit is obtained, whereas a three-loop circuit results in the case of double-cage machines. Therefore, transients phenomena in the latter category of machines are governed by three time constants, whereas only two time constants are present in the former case. The steady state three-loop equivalent circuit may be used to study the operation of class B and class C squirrel-cage induction motors widely used in industrial applications.

C.2 Voltage Equations in Double-Cage Induction Motors We shall designate the currents within the three-phase stator windings, A, B and C, by iA, iB and ie, respectively. The current within the first three­ phase rotor-cage windings, a, band c, will be designated by ia , ib , ie' and those of the second rotor-cage windings, n, ~ and y by ia , ifJ and i y • The parameters of the stator phases are designated by R1 , Lal ; those of the first rotor cage by Rz,I' LaZ,I' and those of the second cage by R Z,2' Laz,2' The magnetizing inductance is labeled Lo' The various stator and rotor current space vectors are defined as usual by:

i = i + i ej21r/3 +' j41r/3 _I A B Ice (C.l),

i = i +i ejZ1r/3 +' j41r/3 _2,1 a b lee (C.2),

i = i + i e jZ1r 13 +' j41r 13 _2,2 a fJ lye (C.3), 218 Double-Cage Induction Motors

After the necessary transformations, the voltage equations of the stator phase "A" is written as (see also Chap2, Sec.2.5, Eqs.(2.63) to (2.65) )

d' d uA=RliA+ LUI~ +LO-[(iA+ ia + ia)] (C.4). dt dt

Similar equations for stator phases Band C are

· L di B L d r(. . .)] uB=R IIB+ ul--+ o-~IB+lb+l/l (C.5), dt dt

d' d uc=Rlic+ LUI--.!..£. +Lo-[(ic+ ic + ir )] (C.6). dt dt

We multiply Eq.(C.4) by eo = 1, Eq.(C.5) by ej21f/3 and Eq.(C.6) by ej41f/3. Summing up these three equations member-to-member, we obtain

. d~1 d~m ~I = RI!I + LUI -;it + Lo dt (C.7),

where the magnetizing current, ~m ' is defined as:

~m = ~I + ~2.1 + ~2.2 (C.8).

Similar equations for the rotor phases in the rotor frame of reference can be written as (see also Chap2, Sec.2.5, Eqs.(2.69) )

., dL.I dtm O=R 1 21 +L_21---+LO (C.9), 21.-. v'dt dt

, d" di 22 !...m R ., +L ---'-+Lo-- (C. 10). o = 2.2 !...2.2 u2.2 dt dt

In the rotor frame of reference, the magnetizing current, tm' may be expressed as: tm = L + tv + L.2 (C.11 ). AppendixC 219

C.3 General Equivalent Circuit of Double-Cage Induction Motors The voltage equations of the rotor (Eqs.C.9 and C.l 0) can be expressed in the stator frame of reference by multiplying its members by e jA , resulting in

d" d" O - R Ie., p. + L e--jA !..2.1 + L e--jA !..m (C.l2), 2.1_2.1 0-2.1 dt 0 dt d" d" 0= R i' e jA + L e jA !..2.2 + L e jA !..m (C.l3), 2.2_2.2 0'2.2 dt 0 dt

It is then easy to check the following general derivative relation

jA jA d! d(!.e ) • d). (' jA) e -=---j-Ie (C.14), dt dt dt- where i is an arbitrary complex current. Transforming Eqs.(C.12) and (C.l3) according to the relation (C.l4) we obtain:

_ R . db di ~2q.1 - 2.1 !2.1 + La2.1 --+ Lo ~ (C.l5), dt dt _ R . d!2.2 di ~2q.2 - 2.2 !2.2 + La2.2 --+ Lo ~ (C.16). dt dt

Lal RI

i.

III Lo

~2q.1 iJ

(Fig, C. J) General eqUivalent circuit ofa double-cage rotor induction motor 220 Double-Cage Induction Motors

The respective current-dependent rotor voltage sources read:

~2q.1 = jW(L,,2.1!2.1 + Lo!.,) (c. 17),

~2q.2 = jw (L"2.2 !2.2 + Lo! .. ) (C.18).

The equivalent circuit shown in Fig.C.1 can represent the three voltage­ equations (C.7), (C.l5) and (C.16), which may be put in the matrix form as u=Zi+L di (C.19), dt with:

!I (C.20), i { !2.1 1 (C.21), u{~'1 _2.2

R, o

Z= [ - jwLo R2•1 - jwL2,1 -ojwLo ) (C.22), -jwLo - jwLo R2.2 - jwL2.2

LI Lo Lo ) L= [ Lo L 2,1 Lo (C.23), Lo Lo L2,2

in which L 2•1 and L 2•2 are defined as

L2.1 = Lo + L,,2.1 (C.24), L2.2 = Lo + L"2.2 (C.25).

Let:

A =-L-1Z (C.26), B=L-I (C.27).

Using the above notations, Eq.(C.19) is transformed into Appendix C 221

di = Ai + Bu (C.28). dt

If we assume that matrices Z and L , thus A and B , are time-independent, that is, the angular velocity and all other parameters are constant within the interval of time [0 t], the current vector may be calculated as:

i(t) = e/Aio + fe(l-AlA B U(A)dA (C.29), o

where io specifies the vector column of the initial conditions of the system. One must note that among the three initial-condition values: (!l) 1=0' (~2,1) 1=0 and (!2,2)/=0' the stator current, (!1)/=0' can be measured directly. Instead, the rotor currents, (~2.1 )/=0 and (~2.2 )/=0' cannot be measured. However, they may be determined indirectly, using the value of the stator current (!l )/=0' as well as the values of its first and second derivatives, (d~l / dt)/=o and (d 2!1 / dt 2)/=0' Actually, using Eq.(C.28), we may write:

(i)/=O = io (C.30)

(~: )/=0 = Aio + Buo (C.31),

d 2 i 2. du (dt 2 )/=0 = A 10 + ABuo + B(Tt)/=o (C.32).

This calculation is especially easy to perform under zero sequence applied voltage (u == 0; du / dt == 0). In this case we obtain:

(i)/=o = io (C.33)

(-di) 1=0 = Aio (C.34), dt d 2 • 2. (-I) 1=0 = A 10 (C.35). dt 2

The three components of the vector column i o , that is, (!l) 1=0' (!2,1) 1=0 and (!2.2) 1=0' may be calculated by resolving the system of linear equations obtained, using the elements of the first line of Eqs.(C.33), (C.34) and (C.35).

The instant value of electromagnetic torque is given by 222 Double-Cage Induction Motors

T =--I 2 R ee{.* I +e I.*} =- 2 L I mil { .. * +11..• } (C.36), n 3 _2q.'_2.' _2q.2 _2,2 3 P 0 _'_2,2 _'_2,'

The magnetizing energy is still calculated as

W I L ·2 mag = 3" O'm (C.37).

Torque control of double-cage induction motors using the Computed Charge Acceleration Method may be easily implemented. However, in this case six unknown variables: (X" y,), (X2." Y2,I ) and (X2,2' Yz.2) must be introduced in order to characterize the charge accelerations of the stator- and rotor­ current space vectors. Then, a system of six linear equations with the above mentioned unknown variables must be resolved (see Chap.7, Sec.7.2.1, Eqs.(7.14) to (7.17». Finally, the required applied voltage is obtained using Eq.(C.7).

C.4 Steady-State Equivalent Circuit of Double-Cage Induction Motors Similar to the steady-state voltage equations of single-cage induction motors derived in Chap.3, Sec.3.3, Eqs.(3.14) and (3.15), the steady-state voltage equations of double-cage machines are written as

Y.., = R,!..., + jW,L",L + jw,LO!...m (C.38), !i2q., = R2.'!...2.I + jW,L"2.'!...2,, + jw,LO!...m (C.39), !i2q,2 = R2,2!...2,2 + jW,L,,2.2!...2.2 + jw,Lo!...m (C.4D).

R, jW,L", l2.2

Lm R2.I v, S jW,L,,2,' L

(Fig.C.2) Steady-state equivalent circuit ofa double-cage rotor induction motor Appendix C 223

It is easy to demonstrate that we obtain:

1 f..2Q.\ = (1- - )R2.\ L.\ (C.4l), S 1 f..2Q.2 = (1- - )R2•2 L.2 (C.42), S where "s" stands for the natural slip. The steady state equivalent circuit is thus the one shown in Fig.C.2.

Torque and magnetizing energy are respectively calculated as

T = 3pLo Im{.L£~.2 + .L£;.\ } (C.43), and

3 2 Wmag = '2 LOlm (C.44).

Industrial line-fed, double-cage induction motors are designed so as to achieve:

OJ\L,,2.2 » OJ\L,,2.\ » R2,l » R2•2 (C.45), and

R 2.\ » R2•2 » SOJ\L,,2.2 » SOJ\L,,2.\ (C.46).

Therefore, at start-up (s=l) the rotor current would principally flow in the branch having the lowest reactance and higher resistance. This provides a significant starting torque and acceptable power factor. However, at rated regime (s:=(W3), the rotor current principally flows in the branch having the lowest resistance and higher reactance, which guarantees a good efficiency and high power factor. Appendix 0 Transient Analysis in Single-Phase Induction Motors

0.1 Introduction While transient phenomena in three-phase induction machines have been relatively well studied, technical papers dealing with the analysis of transient phenomena in single-phase machines are quite rare. We know that the steady-state operation of single-phase machines may be treated as a special case of a certain symmetrical three-phase machine with one phase disconnected (see ChapA). In this appendix we shall see that analysis of transient phenomena in single-phase machines may be conducted in a similar way, that is, as a special case of transient phenomena taking place in the corresponding symmetrical three-phase machine with one of its phases disconnected.

0.2 Transients in Three-Phase Induction Motors We shall begin to write in a new form the voltage equations (Chap.5, Sec.5.3, Eqs.(5.l) and (5.2», which govern the behavior of three-phase induction machines under general conditions. These equations written in their new form are suitable to be applied to the case of single-phase machines. For ease of reference they are rewritten below

. di di U =R I + T --=.L + T --..d.. (D.l), -I 1-1 L.oj dt '-0 dt .. .. d!1 d!z (D.2). O=-JwLo'-1+ (Rz- JwLz)'z+L -o -+ & L z-- &

Let the various stator and rotor complex voltages and currents be written as

~I= U A + u B e}Z1C/3 + u c e}41C/3 = u 1r + jU lim (D.3),

!I= iA + iBeiZ1r/3 + ic e}41r/3 = i1r + jilim (DA), 226 Transients in Single-Phase Induction Motors

!2= i. + ibej21r/3 + icej41t13 = i 2r + ji21m (D.5).

Separating the real and imaginary parts of Eqs.(D.1) and (D.2), it is easy to verify that we obtain the following matrix equation: u= Z·1+ L -di (D.6), dt with

ilr ] i u ~ •• (D.7) { ~Ii" (D.8), {""] '2r i2inl

(ilr),=o di di lim / dt (D.9), i = [ (~lim ) ,=0 I(D.lO), dt f"/~ di / dt 2r I • (12r )'=0 di 2im / dt (i2m )'=0

[R.o RI0 00 0] 0 (D.11), Z = 0 wLo R2 WL2 -(f)Lo 0 -(f)L2 R2

o 4 LI 0 Lo0] (D.12). L~[~Lo o L2 0 0 Lo 0 L2

Eq.(D.10) represents the current vector i at time" t = 0 ", that is, it specifies the initial conditions of the system. Furthermore, let:

A =-L-1Z (D. 13), B=L-I (D. 14), Appendix D 227

where L-I is the inverse of the matrix L. Using the above notations, Eq.(D.6) is transformed into

di = Ai +Bu (D.l5). dt

If we assume that matrices Z and L, thus A and B , are time-independent, the current vector within the time-interval [0 t] may be calculated as: , i(t) = e'Ai o + feu-AlA BU(A)dA (D.16), o

0.3 Transients in Single-Phase Induction Motors Transient currents in single-phase machines may be computed in a similar way. Indeed, an induction machine may be considered to operate as a three­ phase machine with one phase (i.e. phase A) disconnected. The only new fact is that in this case the stator current is a pure imaginary number and reads: !l = jJ3 iL = jilim ' where iL is the line current. In addition, while the real part of the applied voltage, u 1r ' remains unknown, its imaginary part, U 1im ' is related to the single-phase line voltage, U CB = U c - u B = u L ' as:

U +U +U lim = Im{u A B ej31r/2 C ej41r/3}_- --uJ3 (D.l7). 2 L

Disregarding the real part of the Eq.(D.l), we obtain three linear differential equations with three unknown variables: ilim = J3iL' i2r and i2im . We may still write these equations in the matrix form, as given by Eq.(D.6), where:

2 (D.l8), uft ]

J3iL] i { i2r (D.19),

i2im 228 Transients in Single-Phase Induction Motors

.{J3 diL / dtj -d. di 2r / dt (D.20), dt di. / dt 2""

13 (iL)I=O j i 0 = [ (i2JI=O (D.21).

(i2im ) 1=01

[R. 0 Z = wLO R2 :L, J (D.22), o -roL2 R2

0 L= [L.0 L2 L.0 J (D.23). LO 0 L2

We shall define again matrices A and B by Eqs.(D.l3) and (D.14). The solution of the system is still given by Eq.(D.l6).

Once the rotor and stator currents are calculated, the transient torque is obtained as

T 2 L {..* } 213 L" ="'3 P olm !1!2 = -3- P OlLI2r (D.24).

It is clear that the instant torque value in single-phase induction machines may not be controlled. In fact, the product i L i 2r cannot be forced to maintain permanently a constant value, using a single set of driving voltage. However, a given mean torque may be achieved when the motor is fed through a single­ phase, harmonic (sinusoidal) voltage-source of appropriate magnitude, U L , and angular frequency, WI' The latter values depend on the rotor speed. The mean torque is unavoidably accompanied by a pulsating torque of angular frequency 2wl , the mean value of which is nil (see Chap.4, Sec4.3). Appendix E Synchronous Motors Torque Control

E.1 Introduction The Computed Charges Acceleration Method (C.C.A.M.), as described earlier in Chap.7, Sec.7.2.1, can be successfully applied in order to control torque in synchronous motors. However, implementation of this technique in this case is much easier than for induction motors. The essential reason for this difference is that synchronous motors can be modeled by a single loop equivalent circuit, whereas the equivalent circuit of induction motors, as we know, possesses two loops. Actually, the rotor windings in synchronous motors are fed through a separated DC source and the amount of current flowing in them is maintained at a fixed level. Indeed, these rotor windings are often replaced by a set of permanent magnets. In addition, the problem of rotor resistance adjustment does not arise in the case of synchronous motor. Moreover, sensor-less speed evaluation in synchronous motors is easier to achieve than for induction motors. Torque control of a synchronous motor using the Computed Charges Acceleration Method requires only that the stator-current space vector and angular position of the rotor be measured at regular, short intervals.

In a first step we shall consider the case of permanent-magnet synchronous motors, without field winding and damper cage. These category of machines are widely used as electrical actuators in robotics and other industrial applications. In a second step we will establish the operating equations of synchronous machines including the effects of both field winding and damper cage. This category of machines includes large alternators and motors used in power generating plants.

We shall begin by establishing the operating equations for permanent-magnet synchronous motors and draw the general equivalent circuit for the case of a cylindrical-rotor machine. In a second step, we shall discuss the case of a salient-pole machine. We shall then treat the application of the Computed Charges Acceleration Method to controlling the torque of a synchronous motor in the two cases. Finally, we shall present some simulations 230 Synchronous Motors Torque Control

concerning torque control of an industrial synchronous motor, the characteristics of which are presented in the corresponding section. E.2 Operating Equations of Cylindrical-Rotor Synchronous Motors The schematic of a 2-pole three-phase synchronous motor is represented in Fig.E.I. For ease of representation, the rotor windings are replaced in this figure by permanent magnets. As in the case of induction motors, the stator phase winding of a synchronous motor is characterized by its resistance, R" its principal inductance, L p , and its leakage inductance, La, . The magnetizing inductance is still defined as: Lo = 3Lp /2. In cylindrical-rotor synchronous motors the mutual inductances between the various stator phases are LAB = LBC = LCA = -Lp /2 . The so called synchronous inductance, L" is defined as L, = Lo + La, . Notations for stator phase currents and phase voltages are the same as those used for induction motors. Phase windings are assumed to have an isolated neutral point. Thus we have

iA +iB +ic =0 (E.1).

I Map."'" ro'.'

I / T.,

shaft

(Fig.E.l) Schematic ofa 2-po/e synchronous machine AppendixE 231

The voltage equation of the phase A is written as

. diA diA diB die uA = RI'A + LUI -+Lp -+LAB -+L(;A -+eA (E.2), dt dt dt dt

where eA is the back-emf induced by the magnetic field of the rotor in stator phase A. Similar equations for phases Band C may be written. Taking into account Eqs.(2.49) and (2.50), Chap.2, Sec.2.3, these three equations can be simplified into

· L diA uA = RI'A + 1-+eA (E.3), dt · L diB uB= RI'B+ 1-+eB (E.4), dt · L die ue = RI'e+ I-+ee (E.5), dt

Multiplying Eq.(E.3) by eo = I, Eq.(E.4) by e j1.1C 13 , Eq.(E.5) by ej4tr/3, and adding the three equations member-by-member, we obtain the voltage equation of a synchronous motor in its complex form, that is, having the voltage and current space vectors as variables. This equation reads

. d!1 ~I =RI!I +LI dt+~1 (E.6).

In Eq.(E.6) the last term, ~I' represents the back-emf space vector generated by the rotor magnetic field. Assuming that this field has sinusoidal distribution within the air gap, one can write

e A = -(J)lf/ 21 sin(J.) (E.7), eB = -(J)lf/ 21 sin(J. - 2rr / 3) (E.8), ee =-(J)lf/21 sin(J.-4rr /3) (E.9), where (Olf/21 is the peak back-electromotive force per phase. The back-emf is proportional to the rotor angular velocity, since the magnitude of the rotor field is assumed to be constant. Note also that J., in Eq.(E.7), represents the rotor electrical angular position, which must be determined by measurement 232 Synchronous Motors Torque Control

or evaluated by indirect methods; /iJ is the instant electrical angular velocity, and is related to A as:

dA -=/iJ (E. 10). dt

The value of 1/hf can easily be determined by opening the stator-phase windings and measuring the voltage induced across their terminals, assuming that the rotor is driven by an auxiliary prime mover with constant angular velocity.

Taking into account Eqs.(E.7), (E.8) and (E.9), ~l reads

3 . J'A. e /iJ1JI e /iJ1JI (E.II), _I = -2 } 2f = } -2f

According to Eq.(E.6), the general equivalent circuit of a cylindrical-rotor synchronous motor is the one drawn in Fig.E.2. The torque is given as

T= ~ ~Re{~lf:} (E.12), in which Q=/iJ/p is the mechanical angular velocity of the rotor. Assuming that the current space vector is written in the form il = ile jA1 , taking Eq.(E.ll) into account, the expression of the torque becomes

T = -PlJI2fil sin(A-A1 ) (E.l3).

RI

!!I jOJ'I' -2f

i l

(Fig.E.2) General eqUivalent circuit ofa cylindrical-rotor synchronous machine AppendixE 233

E.3 Torque Control of Cylindrical-Rotor Synchronous Motors The problem of torque control can be stated as follows: Given the stator current space vector, i, = iIeiJ.J, the angular position, A, and the angular velocity, w, at a given instant, what voltage space vector, !!,' should be applied to the stator terminals so as to attain the value of the reference torque, T"f , over a given interval of time, llt?

Let the acceleration of the charges of the stator current be

d!, dt = (x+ jy)eill (E. 14).

This leads to:

di, x=- (E.l5), dt . dA, . y=/,-=/,W, (E.l6), dt where all is the instantaneous value of the angular frequency of the stator­ current space vector. After the time-interval llt, the magnitude and angle of the stator current, i" would have changed by xllt and yllt / h, respectively. However, assuming that the angular velocity, w, remains unchanged during this interval of time, the rotor position angle, A, would have changed by wtl.t. Consequently, according to Eq.C.13, the reference torque is reached within the time-interval llt if one can write

Tref =-Pl/f2f(i1 +X!lt)Sin[A-A1 +(w- ~)llt] (E.l7).

Eq.(E.l7) contains two unknown variables: x and y. Therefore, some complementary condition must be imposed in order to obtain a unique set of solutions. Let this condition be set as follows: The torque reference, T"f, must be reached with the minimum value of the stator current, iI, that is, with minimum copper loss. For this, the sinus of the angle it refers to in Eq.(E.l7) must be equated to unity. This leads to 234 Synchronous Motors Torque Control

Y Tr A-AI +(w--)M= --+2kTr (E.lS), iI 2

Tref =Pl/f2f(i l +xM) (E.l9).

From Eqs.(E.S7) and (E.l9), x and y are calculated as:

T x=(~-il)1 At (E.20), Pl/f2f (A-AI)+Tr 12-2ktr ]. y =[ +W II (E.2l). M

Note that in Eq.(E.2l), the relative integer, k, must be appropriately chosen so as to achieve: -2tr::;; (A - AI) + Tr 12 - 2kTr ::;; 2Tr. Next, according to Eqs.(E.6) and (E.ll), the required voltage, HI' is calculated as

-R· L( .)jA\ 3. jA ~I - I!I + I x+ lye +"2lWl/f2fe (E.22).

E.4 Examples of Simulation Torque control of a three-phase, 2-pole, 5kW, 220V/phase, 2,400rpm synchronous motor using C.C.A.M. is simulated in this section. The main parameters ofthis motor are: l/f2f =0.90Wb, RI =0.350 and LI =0.Ol6H . The DC link voltage is U 0 =450V. The total momentum of inertia reflected to the shaft is J r =0.05kg .m2. The load torque is assumed to be given as: T/ = O.OSw. The condition of minimum copper losses is assumed for these simulations .

• The first simulation (Sim.C.l) shows, from left-to-right and top-to-bottom, torque, phase current, motor velocity, and phase voltage, assuming that the motor starts from standstill with a constant motoring torque of 20Nm. After SOOms, a braking torque of -20Nm is applied and maintained so that the rotor velocity changes direction. As may be verified, the required motoring and braking torques are established almost instantly. Naturally, such an abrupt change in applied torque may be mechanically undesirable and should be avoided. AppendixE 235

40, 20 , -c( E 20 -f"------:g 10 I Z I m I m _1 ______::I- - o ------I U 0 ::I - ~ I . I m --20 ------I :I .10 ~ .c , n. -40' -20 0 1 2 0 1 2 time(s) time(s) 2000, 400 -~ 1000~--~----\~------~ m ~ 2001------~------_ .,IILli. I D E 0 ------~------>, ~ r: '~j .,m 0 ... ·1 000 ------L------.. .cn. -2000' I J -200 0 1 2 0 1 2 time(s) time(s)

(Sim.C.J) Variations in relevant variables when the cylindrical-rotor synchronous motor starts sunder C. C.A.M. from standstill with a reference torque of Tref = 20Nm and, after BOOms, the reference torque is abruptly changed to T ref = -20Nm 236 Synchronous Motors Torque Control

40 1 20 , -$ -1"------C I CD I -.... ______LI ______.... :::a I U 0 CD• ro at -10 .c Q. -40 1 I 1 -20 0 1 2 0 1 2 time(s) time(s) 2000 1 - 200 1 000 ~ --- -/-- - - -\ ------~ 100 ~at at :l1:li 0 E 0 ------}------>, 0 ~ '~j CD .... -1 000 r------L------:I -100 .c Q. -2000 1 I 1 -200 0 1 2 0 1 2 time(s) time(s)

(Sim.C.2) Variations in relevant variables when the cylindrical-rotor synchronous motor starts under C.C.A.M. from standstill with a reference torque progressively increasing to Tref = 20Nm and, after 8ooms, the reference torque progressively decreasing to T"f = -20Nm . Appendix E 237

• In the second simulation (Sim.C.2) the motoring torque rise-time and the breaking torque fall-time are progressively controlled so that they are mechanically tolerable for the motor shaft. E.S Operating Equations for Salient-Pole Synchronous Motors Salient-pole synchronous motors may be analyzed as cylindrical-rotor machines, except for the mathematical treatments, which take longer to complete. In the first approximation, the principal phase inductances of a salient-pole machine may be expressed as

LpA =M[l+acos(2A)] (E.23), LpB = M[l +aCOS(2A+ 2Jr 13)] (E.24), Lpc = M [1+ a cos(2A + 4n 1 3)] (E.25), where 0 s: a < 1 is a constant real number that accounts for periodic oscillations of the principal inductance around its mean value, M, as a function of the rotor angular position, A. In addition, the mutual inductance between phase-pairs is approximately expressed as

LAB = LBA = M[ -~+acOS(2A+4n 13)] (E.26),

LBC =LCB =M[-~+acOS(2A)] (E.27),

LCA = LAC =M[-~+acOS(2A+2Jr/3)] (E.28).

The voltage equations for phases A, Band C read

d· d UA = RliA +L"I 2+-(LpA iA +LBAiB +LCAic)+eA (E.29), dt dt · L diB d (L· L· L·) UB = RlIB + "I -+- ABIA + pAIB + CBIc +eB (E.30), dt dt . L dic d (L . L . L·) Uc = R Ilc + "I -+- ACIA + BCIB + pclc +ec (E.3I). dt dt 238 Synchronous Motors Torque Control

We multiply Eq.(E.29) by ejO =1, Eq.(E.30) by ej21t13, and Eq.(E.3I) by e j41t 13, and we add the three equations thus obtained. Designating: Lo = 3M /2 and L. = Lo + La. ' we obtain the following voltage equation:

. d!. d u. = R.I. +L.-+e. +-('" ) (E.32), -- dt - dt-· where !!:. is given by Eq.(E.l2) and !e:1 is expressed as

'" = ~aM t e j (2A) (E.33). _I 2 -.

Thus we can write

d (E.34), -dt ("'.)=~'- +e_r where the transformational and rotational emf, !!:, and !!:r' are respectively given as

3 d" e =-aM 2 e j (2.l) (E.35), -, 2 dt

e = 3J'waM" j(2.l) _r !.e (E.36).

According to Eqs.(E.32), (E.35), and (E.36), the equivalent circuit of a salient-pole synchronous motor is shown in Fig.E3.

RI

~. ~r

~. ~, i l

(Fig.E3) General eqUivalent circuit ofa salient-pole synchronous machine AppendixE 239

E.6 Torque Control of Salient-Pole Synchronous Motors The reluctance torque, that is, the torque due to pole saliency, is obtained as

T , =~3.Re{eQ3 _,_I t} (E.37). where Q = OJ / P is the mechanical rotor angular velocity. Let the stator current be expressed as !I = ile P.I. We thus obtain

T, =-2paMi~ sin 2(1\.-1\.1) (E.38).

Total torque is the sum of the synchronous torque given by Eq.(E.l3) and the reluctance torque given by Eq.(E.38), that is

T = -PlJI2fil sin(I\.-I\.I)-2paMiI2 sin 2(1\.-A,) (E.39).

Given the actual value of the torque, T, our goal now is to determine the voltage space vector, Itl' to be applied to the motor terminals in order to attain the reference torque, T"f, in a given time, I1t. Let the required stator charges acceleration be written as

d!1 dt = (x+ jy)e jAI (EAO).

We will first determine the value of y so that the reference torque is attained for minimum cooper loss, that is, the minimum value for il. For this we equate to zero the derivative of torque with respect to angle 0 = I\. -1\.1. It is easy to verify that if il is kept constant, the total torque, given by Eq.(E.39), reaches its peak value for o· = -arcos(-f3 +~O.5+ f32 )+2k1t' (EAl), where f3 is defined as

lJIzf (EA2). f3 = 8aMii 240 Synchronous Motors Torque Control

Thus, we may first determine y in Eq.(E.40) for which the reference torque is obtained with minimum cooper loss. For this we must realize

(A+~t)-(AI + y~t)=D* -2kn (E.43). II

We therefore obtain

(A-AI)-D* -2ktc] . y= [ W+ II (E.44). &

The above condition being fulfilled, a second condition to be satisfied is

T"f =-P'I'2f(i1 +x&)sin(D*)-2paM(i1 +~t)2 sin(2D*) (E.45).

Neglecting terms in &2, we obtain

x = _ TUf + pil'l' 2f sin 15 * + 2paMil2 sin 215 * (E.46). p('I' 2f sin 15 + 4aMil sin 215 *)&

Finally, according to Eqs.(E.32), (E.35), and (E.36), we obtain the expression for the required voltage, H.I' as

_ R . T( • ) jAI 3. jA ~I = I~I + .... X + lY e + "21W'I'2/e (E.47), + :iMa(x - jy)ej(2A-AI) + 3jwaM~ej(n-AI) 2

It can be readily verified that for a = 0, we obtain the voltage expression already established in Eq.(E.22) for the case of a cylindrical-rotor machine. Note also that if we let '1'2/ = 0, that is, if the rotor field is absent, we are dealing with the case of a synchronous reluctance motor. As an illustration, consider the case of a 1.5kW, 2,400rpm, 2-salient-pole, ferrite-type permanent magnet synchronous motor with a saliency factor of a = 0.30. The other parameters of this motor are: '1'2/ = 0.25Wb, RI = 0.250, and M = 0.OO8H. The DC link voltage is U 0 = 150V. The total momentum of inertia reflected to the shaft is J r = 0.0Ikg.m 2 • The load torque is assumed to be given as T/ = 0.03w . Appendix E 241

10 20 total -< -~------:g 10 E 5r------I - ID Z relue. I I -~ -: 0 ------t:------u 0 e- ID. ______I _;ylll;. _____ {!!. -5 :I -10 .II: D.. _10 1 1 -20 0 1 2 0 1 2 time(s) time(s) 4000, 100 -~ ID 50 til a. III -=D E o~------~------~ >, 0 - ID :I -50 ~ -4000 1 1 -100 0 1 2 0 1 2 time(s) time(s)

(Sim.C.3) Variations in relevant variables when a salient-pale-rotor synchronous motor starts under C.C.A.M. from standstill with a reference torque progressively increasing to T"f == 7 Nm and, after BOOms, the reference torque progressively decreasing to T"f == -7 Nm . 242 Synchronous Motors Torque Control

• Sim.C3 shows variations in relevant variables when the given salient-pole synchronous motor starts under C.C.A.M. from standstill with a reference torque progressively increasing to T"f = 7 Nm and, after 800ms, the reference torque progressively decreasing to T"f = -7 Nm . As can be verified, for a total torque of 7Nm in this regime, 2Nm are provided as reluctance torque due to pole-saliency; the remaining 5Nm are supplied as synchronous torque due to the rotor field. Recall that for this case study, the reluctance torque constitutes a significant part of the total torque.

E.7 Operating Equations for Synchronous Machines with Damper Cages and Field Winding Until now we have been dealing with the simple case of unsaturated, permanent-magnet synchronous motors currently used in industrial applications. Saturation and armature magnetic-reaction can legitimately be neglected in this case. We shall end this appendix by giving the general equations for synchronous machines with damper cages, under saturated regime, and taking into account the effect of the armature magnetic-reaction.

Magrretized rotor acting as damper cage

/ 'T,

Shaft

I L 3-pluue stator winding

(Fig.E.4) Schematic 0/ a 2-pole synchornous machine with damper cage andfield winding AppendixE 243

The basic equations will be given for the motoring case, however they may be used for generators, just by inverting the conventional positive directions of the various voltages and currents of their equivalent circuits. As far as transient phenomena in synchronous machines are concerned, the circuit layout of an actual 2-pole synchronous machine may be approximately represented as shown in Fig.EA. The angle of the rotor axis with the phase "A" axis is designated by A. According to this model there are two system of coupled symmetrical windings on the rotor:

• The solid rotor fabricated in hard steel often plays the role of the damper cage in cylindrical-rotor synchronous machines. On the other hand, salient­ pole synchronous machines are provided with special "damper bars", regularly distributed around the rotor. These bars are short-circuited at their two ends and play exactly the same role as the short-circuited cages in induction machines. The various phases of this circuit will be designated by "a'" "f3'" and "y'''. Damper cages play a major role in the transient behavior of a synchronous machine. The relevant parameters of the damper cage may be determined by direct measurements, with the field winding being kept open. They are designated as R2.1' L"2.1' and Lo.

• The short-circuited single-phase field winding in a synchronous machine plays a significant role in transient phenomena as a second damper cage. One should recall that the field winding is a single-phase circuit, which we shall replace by an equivalent three-phase symmetrical windings, similar to that of the stator windings, with one phase disconnected. The various phases of this circuit will be designated by "a''', "b'" and "c'''. The phase "a'" has its axis perpendicular to that of the rotor and is assumed to be disconnected. Thus the electric angle between the stator phase "A" axis and the rotor phase" a'" axis is (J = A + 1r / 2. The relevant parameters of this fictitious three-phase field winding are designated as R2.2, L"2.2' and Lo' Note that Lo is the common magnetizing inductance of the various stator and equivalent rotor three-phase windings. Also recall that a DC source of appropriate constant voltage, U ;/0' is present between the phases "b'" and "c' ". This source provides the required DC field current. The rotor field winding is characterized by: I) The amount of its rated excitation field, If/;/o' 2) Its rated copper losses, p;/o' 3) The amount of its rated leakage field, If/:2/0' These quantities must previously be determined by direct measurements. While If/;/o and p;/o can be easily measured, determination of If/:2/0 is somewhat more difficult. The replacement of the actual field winding by an equivalent three-phase windings with its phase "a'" 244 Synchronous Motors Torque Control

disconnected is valid if the latter windings may generate the same three quantities: "';/0' p;/o, and "':uo' For this, the parameters R2.2 and L"2.2' as well as the voltage U;/o must verify the following relations:

, -L (--U;/o) (E.48), '"2/0 - 0 2R2•2

u;/o) = 2L"2.2 ( 2R (E.49), "",,2/0 2.2 U;/o )2 p' = 2R2• (2R (E.50). ,,2/0 2 2.2

From Eqs.(E.48), (E.49) and (E.50), R2.2, L"2.2 and U;/o may be determined. In the following, we shall study the transient behavior of both cylindrical-rotor and salient-pole synchronous machines.

E.7.1 Cylindrical-Rotor Synchronous Machines Since there are two windings on the rotor and only one on the stator, it is more advantageous to derive the operating equations and the equivalent circuit of the machine in the rotor frame of reference. In reference to the case of a double-cage induction motor studied in Appendix C, the stator voltage equation of a synchronous motor in the stator frame of reference may be written as: (also see Chap.2, Sec.2.5)

. d!1 d!m ~I = RI!I + L"I -;]I + Lo ~ (E.51), with im =il +i2.1 +h2 (E.52).

We multiply Eq.(E.51) by e-j8 , with (} =A.+1t'/2. After the required transformations we may write Eq .(E.51) in the rotor frame of reference as:

d" d" d" u' = R i' +L --.!::.!.+L L2.1 +L L2.2 +e' (E.53), -I I-I I dt 0 dt 0 dt -I AppendixE 245

The current-dependent voltage source ii, in Eq.(E.53), is written as

e' - 'WIII' -I -J Lim (E.54),

1fI' where W = d')../ dt ; the stator field _1m ,in Eq.(E.54), is defined by:

1fI'lm = LIf.1 + Lof.2.1 + LOf.Z.2 (E.54').

The field current L2.2 may be written as the sum of two components: LzIO and L2.2' The first component, L2/0' is provided by the DC source; the second, L2.2' is some induced current. The stator field 1fI'lm may thus be written as

1fI'lm = LoL2/0 +(LILI +LoL2.1 +LoL2.2) (E.55), in which the first term represents the excitation field and the second (within the brackets) the magnetic field (magnetic reaction) due to the load.

The voltage equations of the damper and field-winding cages in the rotor frame of reference are respectively written as:

di' di' di' O=R i' +L --=!....+L ~L ~ (E.56), 2,1_2,1 0 dt 2,1 dt 0 dt , " df.1 dL,1 dL,2 ~2 2 = R22!:....2 Z + Lo-- +Lo --==-t L22 --- (E.57), , " dt dt 'dt with

L2,1 = Lo + La2.1 (E.58), L2,2 = Lo + Lu2.2 (E.59).

The general equivalent circuit of a cylindrical-rotor synchronous machine is almost similar to that of a double-cage induction motor. This equivalent circuit is shown in the rotor frame of reference in Fig.E.5. However, as we can see, there is a major difference between this equivalent circuit and that of a double-cage induction motor. Indeed, a current-dependent voltage source, !£z.z' is present in the rotor circuit of the field winding of the synchronous machine. The important fact here is that because of the disconnected phase 246 Synchronous Motors Torque Control

"a'" the field current space vector, (.2.2' is a pure imaginary number. In fact, it may be easily demonstrated that we may write:

L.2 = j.J3 i;, (E.60), where i;, is the instant field current. On the other hand, since we have u;. =u;. + U ;'0 ' and u;. + u;. + u;. =0 (because of the symmetry of the field windings), the current-dependent voltage space vector, i2.2' is written as

, 3, . .[iU' lL2.2 = "2 ua' - JT 2,0 (E.61), where u:. is the voltage of the disconnected phase " a' ". We shall designated the real part of all variables in the rotor frame of reference by the index "cr', and their imaginary part by the index "q". Taking the real and imaginary parts of Eqs.(E.53), (E.56) and (E.57) we obtain a system of six linear equations with six unknown variables. These variables are: i:d, i:q, i;.ld' i;.lq' i;.2d and i;.2q' However, we know that i;.2d = O. Therefore the number of unknown variables may be reduced from six to five. In addition, we may disregard the equation related to the unknown voltage U;.2d' (the real part of Eq.E.57). We thus obtain five differential equations with five unknown variables. The above system of five equations may be written in the matrix form as

LUI ., RI L2.2

., , Lm LI Lo , ILl , !!...2,2 ., LI

(F;g.E.5) Equivalent circuit ofa cylindrical-rotor synchronous motor with cIamper cage andfield winding. AppendixE 247

, , di' u = Zi +L_l (E.62), dt

with: , ., Uld di;d 1 dt , 'id., Ulq 'lq di;q 1 dt ., di' U' ~ 0 (E.63), i' '2,ld (E.64), - di;,ld 1 dt I (E.65), ., dt 0 '2,lq di;"q 1 dt ., - .,f3u;/o 12 '2,2q di;,2q 1 dt

R, -roLl 0 -roLo -roLo roLl R, roLo 0 0 Z =1 0 0 R2,I 0 0 (E.66),

0 0 0 R2,I 0 0 0 0 0 R2,2

LI 0 Lo 0 0 0 LI 0 Lo Lo L =1 Lo 0 L2,I 0 0 (E.67). 0 Lo 0 L2,I Lo 0 Lo 0 Lo L2,2

In a saturated regime the magnetizing inductance, Lo' would depend on the total amount of the magnetizing current. Variations in the inductance ratio, Lo (i) ILo (i = 0), versus an arbitrary current, i, can be determined by the curve plotted in Sim.l.l, Chap. I, Sec. 1.3. As we see, the non-linear equations for saturated synchronous machines with both damper and fieId­ winding cages, and taking into account the armature magnetic reaction, have a rather complicated structure. Nevertheless the analysis may be dealt with using numerical methods. In order to simplify the analysis of transient phenomena in electric machines, it is often assumed that the magnetic circuit remains unsaturated. This is an acceptable assumption for synchronous machines, which usually have quite a large air gap. In this case, all the 248 Synchronous Motors Torque Control

various inductances would remain constant. With the above-mentioned assumption, the following matrices are defined:

A =-L-1Z (E.68), B=L-I (E.69).

Now, Eq.(E.62) can be transformed into:

d" , __I = Ai' +Bu (E.70). dt

If we assume that matrices Z and L , thus A and B , are time-independent, that is, the angular velocity and all other parameters are constant within the interval of time [0 t], the current vector may be calculated as:

I j'(t) = e'Aj~ + f eu-.t)A B u'C".)dA (E.71), o where i; specifies the initial condition in the rotor frame of reference. The electromagnetic torque is obtained as

I 2 {,.,.} T= n 3 Re f-ILI (E.72), where n = 0) / P represents the mechanical angular velocity of the shaft; p is the number of pair poles. The electromagnetic torque has thus two components: a synchronous or field component, T5)'11, and an asynchronous component, Tas).", respectively expressed as:

T 2 L I {., .,.} syn =3 porn L2JoLI (E.73),

Tosyn = ~ pLo Im{ Ci2.1 + L2,2 )r; } (E.74),

The mechanical equation of the system reads as:

J, dO) =T-Tl. (E.75), pdt AppendixE 249

where TL is the load torque and J r the total momentum of inertia reflected to the shaft. Eqs. (E.71) to (E.75) govern the electrical and mechanical behavior of the drive under general conditions. Note that in a permanent regime the rotor damper loop becomes open, and sole remain the stator and the field winding loops.

In the following examples, using three case-studies, we shall illustrate how this theory may be applied to predict the transient behavior of a cylindrical­ rotor synchronous generator:

• In this first case it is assumed that the rotor of a synchronous generator is driven at a constant angular velocity (J), while the stator windings remain open (unloaded generator). Initially, the field current is nil. Then, the DC source is connected to the field winding. The field current rises and ultimately reaches its permanent value. The goal is to study how the voltage at the generator terminals builds up.

For this case we have ild = 0 and ilq = O. Thus the number of voltage equations may be reduced from five to three. The various elements of Eq.(E.62) read:

f u 0 , (E.76), if ~~.Iq (E.77), ~ d~~.Iq II~) dt (E.78), f- .J3u 2/0 12 ) r12.2q ) ., f"d12.2q I dt

0 o 0) Z= [R"0 R2.I (E.79), L= [L"0 L2.1 La (E.80). 0 0 :J 0 Lo L2.2 We again define matrices A and B by Eqs.(E.68) and (E.69), and use Eq.(E.71) with the initial conditions: i'(to) = O. The output voltage is then: (also see Eqs.(E.54), (E.54') and (E.60), with L = 0)

, , '.1 (., .,) '.1" , .•1" ~I = ~I = J{J)LO !..2.1 + !..2.2 = J{J)LO!..2.1 - WLOl 2,2q (E.8l).

When the transient currents have vanished (L,I ~ 0), and the field current has reached its permanent value (i;,2q ~ -.J3u;/o 1(2R2,2)) ' we obtain: 250 Synchronous Motors Torque Control

, _ 13 wLo U~fO (E,82), ~I - 2 R2,2

• The second case-study deals with an excited generator the shaft of which is driven at the electrical angular velocity w. The stator phases are initially open (unloaded generator). Then, a sudden three-phase short-circuit occurs at the machine terminals. The goal is to study the rise of the transient phase currents. Actually, transient currents expressed in the rotor frame of reference may be calculated analytically using Eq.(E.71), with the initial conditions given as: i;d = i;q = i;,ld = i~,lq = 0, and i;,2q = -13U ~fO /(2R 2,2)' In addition, in Eq.(E.63) one must set U;d = U;q = 0 .

• In the third case-study it is assumed that under the same condition a sudden two-phase short-circuit occurs. More precisely, it is assumed that phases "B" and "C' are short circuited at their two terminals, while the phase "A" remains open (disconnected). It is more difficult to treat mathematically the case of an asymmetric, two-phase short-circuit than the symmetric, three­ phase short-circuit studied above.

We shall begin by observing that in the stator frame of reference the stator voltage space vector, !!I' is a pure real number (because of the windings symmetry, we have: UA + UB + Uc = 0), whereas the current space vector, ii' is a pure imaginary number. Indeed we may write:

3 !!1='2 UA (E.83),

il = j.,fjilL (E.84), where UA represents the voltage of the disconnected phase "A", and ilL the current flowing in the phase "B". Recall that the current flowing in the phase "C' is -ilL' Assuming that the rotor angular position is fJ =A +Jr / 2, the stator voltage and current in the rotor frame of reference are obtained as:

, _ 3 -j8 _, • I ~1-2uAe -Uld+JUlq (E.85),

L = jJ3iIL e-j8 = i;d + ji;q (E.86).

Thus we may readily deduce the two following equations: AppendixE 251

, • {'j , {'j Uld sm U+ Ulq cOSu = 0 (E.87),

" ., ('j lId = Zlq tan u (E.88).

Since d~1 / dt is also a pure imaginary number, the two components of the stator current derivative in the rotor frame of reference are related as:

., di; dZ ld _ --q- tan e (E.89). Tt- dt

Using Eq.(E.87) the number of the available differential equations may be reduced from five to four. Moreover, using Eqs.(E.88) and (E.89) the number of the unknown variables may also be reduced from five to four. We will consider the following variables: i;q' i~.ld' i~.lq and i~.2q' The above system of four equations may be written in the matrix form as expressed in Eq.(E.62). It is straightforward to verify by direct calculation that the various elements of this matrix equation are written as:

o ] {i;q"., ] {di;q d" / dt/ d u' { (E.90), j' ~~.Id (E.91),!!:!..- ~~.Id t I (E.92), o Z2.lq dt dZ 2.lq / dt

- J3U ~fO / 2 i;.2q di;.2q / dt

OJLo cos 2 e - OJ Lo sin e cos e W4 OO R2•1 cose 0 - '9] (E.93), 0 R2•1 r zf~ 0 0 R2.2

Lo sin e cos e Lo cos 2 e L. 00;' LI sin e L2.1 cose 0 9] L= [ ~ (E.94). Lo 0 L2.1 Lo

Lo 0 Lo L2•2

We observe that the coefficients of the linear equations thus obtained depend on time via the rotor position angle, e. However, the system may be resolved by numerical methods, using the Range-Kutta algorithm. 252 Synchronous Motors Torque Control

E.7.2 Salient-Pole Synchronous Machines The case of salient-pole synchronous machines may be treated as that of the cylindrical-rotor machines, except for the mathematical developments, which take longer to complete. In the following, we shall limit our analysis to the first harmonic (fundamental) of the various voltage and current waveforms. In a first step both the damper cage and field windings may be transferred to the stator frame of reference. In the latter frame they would appear to a stator based observer as two stationary three-phase windings, having their axes aligned with those of the stator windings. Furthermore, for such an observer the various principal inductances of the stator phases, "A" , "R" and "e', as well as those of the rotor phases, "a" , "b", "e", and "a" , "f3 ", "r", would depend on the rotor position and may be approximately expressed as:

Lp(A.a.a) = M[l +acos(29)] (E.95),

Lp(B.b.p) = M[l +acos(29 +2nI3)] (E.96),

Lp(c.c.r) = M[l + acos(28 + 4n 13)] (E.97), where 0 ~ a < 1 is a constant real number that accounts for periodic oscillations of the principal inductance around its mean value, M, as a function of the rotor angular position, 9 =..1, +n 1 2. In addition, the mutual inductances between the various phase-pairs would be approximately:

L(A.a.a)(B.b.p) = L(B.b.p)(A.a.a) = M[ - ~ + a cos(29 + 4n 13)] (E.98),

L(B.b.p)(c.c.r) = L(c.c.r)(B.b.p) = M[ - ~ + a COS(29)] (E.99),

L(c.c.r)(A.a.a) = L(A.a.a)(c.c.r) = M[ - ~ + a cos(29 + 2n 13)] (E.lOO).

In the following we shall designate: Lo =3M 12 and (J) = d9 1 dt. Thus the voltage equations in the stator frame of reference may be written as:

. d!1 d!m dl/fm u l = Rill +Lul--+Lo--+-- (E.lOl), -- dt dt dt with im and '1!..m defined respectively as: AppendixE 253

im =i. +h. +i2.2 (E. 102),

j29 l/f_m = aLO_m i' e (E. 103).

Next, we multiply Eq.(E.S3) by e-j9 (9 =A +n /2). After the required transformations we may write this equation in the rotor frame of reference as:

, of dr.. dr..., , , !£. = R.!... + Lu. dt + Lo dt +~. + ~ReI + ~mo8 (E.I04), where the current-dependent voltages i., ~Rel and ~ma8 are defined as: i. = jm(L.r.. + LOr.2 .• + LoL.2) (E.105), ~Rel = jmaLo(LJ (E. 106),

I d(LY ~""'8 =aL0dt (E. 107).

For a rotor-based observer the various principal inductances of the stator phases, "A'" , "B'" and" C' ", as well as those of the rotor phases, "a' " , "b'" and "c' '', and "a' " , "{3''', "r''', would not depend on the rotor position, since the stator has a cylindrical structure. Furthermore, in this frame of reference, the principal inductances of the various phases may be approximately expressed as:

Lp(A'.a',a') = M [I + a] (E. lOS),

Lp(B', b', P') = M[I + a cos(2n / 3)] (E.l09),

Lp(C', c', r') = M[I + a cos(4n / 3)] (E.lIO),

In addition, in the same frame of reference the mutual inductances between the various phase-pairs would be approximately:

L(A',a',a')(B',b',P') = L(B',b',P')(A',a',a') = M[- ~ + a cos(4n / 3)] (E.lIl),

L(B',b',P')(C',c',r') = L(c',C',r')(B',b',P') = M[ - ~ + a] (E. I 12),

L(c',C',r')(A',a',a') = L(A',a'.a')(c'.c',r') = M[- ~ + a cos(2n /3)] (E.I13). 2S4 Synchronous Motors Torque Control

Thus, in the rotor frame of reference, the voltage equations of the damper and field-winding cages read respectively:

di' di' ·' L _2.1 L _m , O = R21 / 21+ -21---+ 0--+10 -"8 (E.114), • - • v. dt dt _"N' di' di' u' =R i' +L ~+L ---=.!!!....+e' (E.llS). _2.2 2.2_2.2 a2.2 dt 0 dt _mag

In the rotor frame of reference, the general equivalent circuit for a salient­ pole synchronous machine may be established as shown in Fig.E.6. This equivalent circuit corresponds to Eqs.(E.I 04), (E.l14) and (E.lIS).

Taking the real and imaginary parts of Eqs.(E.l 04), (E. 1 14) and (E. 1 IS), and disregarding the equation related to the unknown voltage U;.2d' we obtain five differential equations with five unknown variables. These variables are: i;d' i;q' i;.ld' i;.,q and i;.2q' Also recall that we have i;.2d = O. The above system is written in the matrix form as

d" U , = Z"1+ L - I (E. 1 16), dt

., La' !...2.2

R2•1 , !!....I

, !£.I

, ., , ~ma8 , !...Z.I !Ll.2 ~Rel ., !...I

(Ftg.E6) Equivalent circuit ofa saJient-pole synchronous machine with damper cage and field winding AppendixE 255

with: , ., .U1d 'id di;d / dt , of ulq 'Iq di;q / dt ., ., di' u' (E.l17), I (E. I IS), (E.119), o 2 ld di;.ld / dt o '., • dt 2 lq di;.lq / dt , '., • U2.2q '2.2q di;.2q / dt

R, -(j)L,+ aaLo o -(j)L,,+ aaL. -(j)L,,+ aaL. (E. 120), fl)L,+aaL. R, fl)L,,+aaL. o o

Z=I 0 o R2., o o

o o o R,., o o o o o R,.,

L,iaL" o L"+aL,, o o

o L,-aL" o L,,- aL" L,,- aL" (E.l21). L =I L"+aL,, o L,.,+aL" o o

o L,,-a L" o L,.,- aL" L"-aL,,

o L"-aL,, o L"-aL,, L,.,-aL"

Recall that L2.1 and L2.2 are given by Eqs.(E.5S) and (E.59). Furthennore, U;.2q ,in Eq.(E.117), is calculated as:

U;.29 = -../3u;/o /2 (E. 122),

It can be easily verified that by setting a = 0 we retrieve the voltage equations of the cylindrical-rotor synchronous machines. As usual, Eq.(E.l16), as a system of linear differential equations with constant coefficients, may be integrated using the integral fonnula given by Eq.(E.71). References

Books

A. Fitzgerald, Ch. Kingsley: Electric Machinery, McGraw-Hill, N.Y., 1961.

Ph. Alger: The Nature of Induction Machines, Gordon and Breach, N.Y., 1965.

W. Leonard: Control of Electrical Drives, Springer-Verlag, Berlin, 1985.

P.c. Krause: Analysis of Electrical Machinery, McGraw-Hill, 1985.

P.K. Kovacs: Transient Phenomena in Electrical Machines, Elsevier, 1986.

B.K. Bose: Power Electronics and AC Drives, Prentice Hall, 1987.

P. Vas: Vector Control ofAC Machines, Oxford, Clarendon Press, 1990.

D.W. Novotny, T.A. Lipo: Vector Control and Dynamics of AC Drives, Oxford, Clarendon Press, 1996.

H. Wayne Beaty, J.L. Kirtley, Jr.: Electric Motor, McGraw-Hill, 1998.

S. E. Lyshievski, Electromechanical Systems, Electric Machines, and Applied Mechatronics, CRC-Press, 1999.

J. Bonal, G. Seguier. Variable Speed Electric Drives, Editions TEC & Doc 1999, Paris.

General Papers

F. Blaschke: The Principle of Field Orientation as Applied to the New Transvector Closed-Loop Control System for Rotating-Field Machines, Siemens Review, Erlangen, 1972, No.5, pp. 217- 220

M. Depenbrock: Direct Self Control (D.S.C). of Inverter-Fed Induction Machine, IEEE Trans. On Power Electron. PE-3 (1988), no.4, pp. 420-429. 258 References

D. Zinger, F. Profumo, T.A. Lipo, D.W. Novotny: A Direct Field Oriented Control for Induction Motor Drives Using Tapped Motor Windings, Proc. Of Power Electronics Specialists Conference (1988), PP. 855-861

T.A. Lipo: Recent Progress in the Development of Solid-state AC Drives, IEEE Trans. On Power Electronics, (1989) pp. 105-117.

J. Holtz: Pulsewidth Modulation, a Survey, IEEE Trans. On Industrial Electronics, Vol.39, No.5, Dec. 1992, pp. 410-420.

B.K. Bose: Power Electronics and Motion Control Technology States and Recent Trends. Rec. IEEElPESC. (1992), pp. 3-10.

D. Casadei, G. Grandi, G. Serra, A. Tani: Switching Strategies in Direct Torque Control of Induction Machines, Proc. ICEM (1994), Vol.2, pp. 204- 209.

P. Tiitinen, P. Pohjalainen, J. Lalu: The Next Generation Motor Control Method: Direct Torque Control (D. T. C.), EPE Europ. Power Electron. Vol.5, no.l, pp. 14-18, March 1995.

B. Arnin: Proposal of a New Method for Controlling the Torque and the Flux of a Voltage Source Inverter-Fed Induction Motor, ETEP, Europ. Trans. on Electrical Power, Vol. 8, No.1, pp. 65-68, January/February 1998.

B. Amin: Spectral Analysis of Modulated Periodic Waveforms Produced by Voltage-Source Inverters. EERR, Electrical Engineering Research Report, Nr.4, pp.14-25, January 1998, Naples, Italy. Subject Index

Current harmonics: 126, 144, 149, 150. AC current: 109, 110. Current transfer ratio: 41,43,81. Active power: 34,40,91,92,150. Current-source inverters: 54, 110, 151, Air gap energy: 41, 160, 17l. 152,153. Air gap length: 7. Current-source line-fed induction motors: Air-gap flux: 4,36,38, 166. 54,57. Amplifier: 164. Damper cage: 242,243,245,247. Angular frequency: 34, 57, 59, 63, 64, Damper bars: 243. 67,78,82,105,117, 161, 183, 22l. DC current: 109. Aperture ratio: 1I6. DC link voltage: 222, 230. Armature current: 205, 208. De-energized motor: 104, 105, 107, 192, Auto-sequentially commuted current­ 203. source inverters: 155. Deep-bar rotor: 5l. Auxiliary DC motor (prime mover): 39, Delta-connected load: 104. 9l. Direct axis: 205,207. Auxiliary windings: 78. Direct Torque Control: 159, 170,213. Back-electromotive force (bemf): 25,219, Distributed winding: 4, 5. 230. Dominant time-constant: 192. Back-iron: 2. Double-cage rotor: 51, 217, 219, 222, 223. Backward admittance (impedance): 60, Driving input current (voltage): 159, 170, 62 .. 213. Backward driving input: 60. Emf of rotation: 206,207,226. Basic model (of the induction motor): 13, Emf of transformation: 206, 226. 14. Electrical angle: 20, 24, 205. Biased linear system: 62. Electrical angular velocity: 27, 36, 77, Bloc diagram: 168. 185,213,219. Braking torque: 45, 55, 80, 180, 185. Electromagnetic compatibility: 126. Capacitive load: 153, 154. Electromagnetic time constant: 97, 98, Capacitor-motor: 78. 101, 102, 185, 19l. Carrier signal: 123. Electromagnetic torque: 39, 96. Carter coefficient: 8. Electromotive forces (emf): 6, 25, 206, Ceiling voltage: 110. 219. Centrifugal switch: 78. Electronic converter: 95, 96. Characteristic equation: 100, 165. Elementary conductor: 3. Characteristic tests: 38,9l. Elliptic trajectory: 63. Class B and C induction motors: 51. End-rings: 2. Closed-loop phase-controlled rectifier: Fall-time (of the controlled torque): 225. 15l. Fault-sensitive motor: 105. Coil-wound induction motor: 39,5l. Faulty condition: 33, 67,95, 104. Commutation: 123, 155. Ferrite-core transformer: 164. Complementary state (of switches): Ill. Fictitious rotor current: 25. Complex power: 150. Field current: 208. Computed Charges Acceleration Method: Field Oriented Control: 159, 170, 205, 159,161,162,168,211,215,217. 207. Concentrated windings: 4,5,7, 13. flux pulsation: 2. Converter-fed drives: 96. flux-density: 7,8,36,37,38. Convolution: 100. flux-sensing device: 170,209. Copper loss: 6,29,52,221,222. Forward admittance (impedance): 60, Current displacement: 200. Forward sequence voltage: 34. 260 Subject Index

Fourier analysis: 117. Magnetically coupled windings: 13, 22. Fractional horsepower motor: 77. Magnetising current: 19, 24, 35, 42, 63, Fractional-pitch section: 4. 81,160,166, 171. Frequency domain analysis: 96, 102. Magnetising energy: 22, 41, 46, 54, 64, Front -end rectifier: 151. 81,85. Frozen power switch: 105, 108. Magnetising field: 163, 170,209 .. Full-aperture mode: 116. Magnetising inductance: 2, 10, ll, 24, Full-load regime: 47,50,58. 36,40,45,92,166,171,200,218. Full-pitch section: 4. Magnetising power: 30, 53. Full-wave mode: 154. Magnetomotive force (mmf): 5, 14. Fundamental component: 5. Major (middle, minor) 6-pulse pattern of a General equivalent circuit: 13, 17, 28, 60, three-level inverter: 114, 115. 81,83,96,207,210,217,220. Mean aperture ratio: 116 Generating vectors (for modulation): Mechanical angular velocity: 27, 36, 104, 122,215. 207,220. Hannonic losses: 5. Mechanical losses: 30, 53. Hannonics: 5, 111, 126. Minimum copper loss condition: 221, High dynamic control: 98, 158, 160,209. 22.2. High-side switching block: 110. Modulated waveform: 126,143, 144. Hysteresis band: 227. Modulation index: 117,128,139. Ideal current-source inverter: 151. Motoring torque: 46, 55, 80, 180, 185, Inductive load: 118, 120, 123, 124, 154, 187,191,192,199,222,225. 155. Mutual inductance: 13, 23, 225. Inertial wheel: 155. Natural pulse pattern (of inverters): 112, Initial condition generator: 102. 113, 115, 116, 118, 119, 120. Input admittance (impedance): 39, 40, 41, Negative (positive) slip: 46,48,59. 42,60,62,92,103,104,181. Neutral point: 14, 15, 16, 62, 68, 118, Input power: 150. 124,218. Integral- horsepower motor: 77. No-load impedance: 39,91. Inverter's leg: 11 0, 111. No-load test: 39, 11. Inverter-fed induction motor: 51,54, 102, Non-zero vectors: 112, 113, 153. 105,108,126,154,159,210. Normalized current: 10,44,45,46. Iron losses: 30, 31, 53, 171. Normalized input admittance(impedance): Iron-loss conductance: 36, 39, 53. 42, 43, 70, 83, 84. Isolated neutral point: 14, 15, 16, 62, 68, Normalized input impedance (impedance): 218. 42,42,74,171. Laminations: 2. Normalized slip: 43,44,45,46,47, 51, Laplace transform: 102,103. 55,57,59, 69,86, 182. Leakage magnetic flux: 6, 22. Normalized torque: 46, 52, 72, 86. Linear circuit: 14, 63, 81, 102. ON/OFF state switches:96, 112, 123, 155. Linear differential equation: 98. On-line (measurements): 199. Linear regime: 14,96. Open-loop-rotor stator input-inductance: Line-fed induction motor: 33,44,47,49, 11. 51,54,57,58,59,64,93. Open-loop-rotor stator time-constant: II, Line-to-line voltage: 87. Open-loop-stator rotor input-inductance: Load torque: 50, 95, 97, 104, 170, 185, 11. 222,230.. Open-loop-stator rotor time-constant: 11. Locked-rotor impedance: 40, 92. Operating equations: 11, 40, 43, 82, 96, Low (high) dynamic control technique: 159,180,217,218,225,230. 158, 160,209. Optimum phase angle of the driving Low-side switching block: 110. voltage: 183. Magnetic pattern: 4, 5, 6 Output power: 29, 150. Magnetic power: 29, 30, 52, 53, 201. Output terminals: 110. Magnetic saturation: 8. Subject Index 261

Output torque: 46,67,77,79,105, 183, Shading coil: 78. 213. Short-time overload: 33 Overheating: 72, 126, 171, 191. Sine-triangle modulation: 109. Overload: 33,87,96. Single-phase equivalent circuit: 87, 91, Parasitic effect: 164. 92. Parasitic torque: 2, 5. Single-phase induction motor: 62, 77, 78, Passive windings: 6. 80, 83, 85, 91. Peak torque: 46,49,50,55,57,87. Single-phase mainlines: 85,87. Perturbing load: 95. Single-phase winding: 22, 77. Phasor: 35,36, 127, 130. Six-step waveform: 125. Pole: 4. Slip: 36,43,46,47,49,50. Position control: 186. Slipping bags: 51. Power balance: 150,201. Slots: 2, 3, 4, 5, 8. Power electronics: 33,109. Space harmonics: 5. Power factor: 44,47,48,50,51,56,57, Space vector modulation: 109, 118, 120, 58,85. 121, 123, 143, 154, 165, 168,211.. Power switches: 96, 105, 108, Ill, 123, Space vector: 13. ISS, 164, 168. Spectrum (of a periodic waveform): 121, Principal inductance: 1, 2, 3, 4, 6, 8, 9, 133,144. 10,23,218,225. Speed regulation: 109,203. Quasi-regular 12-pulse pattern: 115. Split-bank capacitor: Ill. Quasi-square waveforms: 126. Split-phase induction motors: 78. Rated regime (value): 38,39,47,54,91, Squirrel-cage rotor: 51. 95, 17l, 199,200. Stair-like magnetic pattern: 5. Real-time analysis: 98, 100. Star configuration: 104. Rectangular magnetic pattern: 5, 15. Star-connected load (phases): 62,87, 104, Reference targets: 162,171,215. 105,170. Reflected momentum of inertia: 186. Star-delta starter: 104, 106. Regular waveforms: 117, 123. Start-up (variables at): 33,51, 57, 58, 78, Resistive loads: 153, 154. 80,104, 105, 106. Response time: 158,192. State variables: 14, 102, 158, 169. Revolving field: 38,79. Static converter: 109. Rise-time (of the controlled torque): 225. Stationary windings: 21,22,38. Rotating winding: 21. Stator loop: 42, 157, 198. Rotor (stator, total) coefficient of Stator resistance: 1,42, 157,210. dispersion: 11, 43. Steady-state equivalent circuit: 36, 158. Rotor bars: 5,6, 13,51,230. Steady-state regime: 10, 34, 43, 67, 80, Rotor field: 163, 170,205,206,209,223, 87, 104, 180. 214,215,219,228,230. Steady-state-based control strategy: 158. Rotor (stator) leakage inductance: 22, 29, Step-by-step transient analysis: 100, 102. 30,40,41,45,52,53,93,163. Step function (Stp): 127. Rotor loop: 36, 41, 81, 103, 157, 207, Stored magnetic energy: 25. 208. Superposition theorem: 63,81,83. Rotor (stator) resistance: 1, 40, 41, 51, Surge current: 51, 105. 92,163,157,199,2011,202,210,217. Switching blocks: 123. Rotor (stator) yoke: 2. Switching frequency: 123,124. Salient poles: 78. Synchronous inductance: 218. Sampling function (Sa): 128. Synchronous motor: 217, 227. Saturation coefficient: 8. Tachometer: 202. Scalar control: 158, 159, 168, 180, 191. Temporary-capacitor-motors: 78. Scanning modes: 116,117,118,133. Thevenin equivalent circuit: 102, 103. Sections (of windings): 3,4. Thevenin impedance: 103. Sensor-less speed evaluation: 217. Thevenin inherent voltage source: 103. Shaded-pole motor:: 78. 262 SUbject Index

Three-phase induction motors: 2, 13, 34, Transverse axis: 207,209 .. 67,77,80,157,159. Two-terminal-connected mode: 153, 155. Three-phase mainlines: 87. Two-level inverter: 111,112, 113, 123. Three-phase transformer: 22. Undesirable oscillations: 166. Three-phase windings: 6, 13, 14, 21. Uniform waveforms: 121, 131, 138, 143. Three-level inverter: Ill, 112, 113, 115, Unbalance voltages: 62, 63, 65, 66, 67, 138,143. 71. Three-terminal-connected mode: 112, 113, Unit step function: 127. 153. Asymmetrical phases: 67, 158, 159. Threshold speed (in scalar control) 186, Vector control: 158, 159. 191, 199. Voltage equations: 25, 27, 36, 96, 170, Thyristors: 155. 219,225. Time-constant (mechanical thermal, Voltage space vector: 14, 26, 34, 59, 62, electromagnetic): 97, 98, 102, 184, 185, 84, 112, 118, 124, 151, 162, 170, 182, 191. 215,221,227. Time domain (analysis): 96, 100, 104. Voltage spikes: 154. Time of transition (of pulses): 116. Voltage-source inverter: 44, 100, 110, Torque: 24. 112, 118, 121, 151,210. Total harmonic distortion: 117. Windage losses: 30, 53. Transient phenomena: 33,88, 100, 104. Zero-sequence voltage: 115, 116, 120, Transient regime: 95, 96,100, 157,230. 143,163,164,165,199,201,202,203. Transient-regime-based control strategy: 158.