A.1 Appendix a Field-Oriented Control

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A.1 Appendix a Field-Oriented Control 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).
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