a ADSP-21990: Reference Frame Conversions

Transcription

1 a ADSP-21990: Reference Frae Conversons a ADSP-21990: Reference Frae Conversons Analog Devces Inc., January 2002 Page 1 of 14

2 a ADSP-21990: Reference Frae Conversons Table of Contents SUMMARY INTRODUCTION TO REFERENCE FRAME THEORY Overvew Three-phase to two-phase transforaton (Clarke transforaton) Vector rotaton (Park transforaton) Practcal pleentaton of the transforatons THE REFERENCE FRAME CONVERSION ROUTINES Usng the converson routnes Forats of nputs and outputs Usage of the DSP regsters The progra code: refrae.dsp Access to the lbrary: the header fle: reffrae.h SOFTWARE EXAMPLE: TESTING THE CONVERSION ROUTINES The an progra: an.dsp Man.h Exaple output IMPLEMENTATION OF THE FUNCTIONS. ERROR! BOOKMARK NOT DEFINED. Analog Devces Inc., January 2002 Page 2 of 14

3 a ADSP-21990: Reference Frae Conversons Suary The ntroducton of reference fraes n the analyss of electrcal achnes has turned out not only to be useful n ther analyss but also has provded a powerful tool for the pleentaton of sophstcated control technques. Ths applcaton note gves an ntroducton to the theory of the ost coonly used reference fraes and provdes routnes that allow for easy converson aongst the. They are pleented n a lbrary-lke odule for edate and ntutve applcaton. 1 Introducton to reference frae theory 1.1 Overvew As the applcaton of ac achnes has contnued to ncrease over ths century, new technques have been developed to ad n ther analyss. Much of the analyss has been carred out for the treatent of the wellknown nducton achne. The sgnfcant breakthrough n the analyss of three-phase ac achnes was the developent of reference frae theory. Usng these technques, t s possble to transfor the phase varable achne descrpton to another reference frae. By judcous choce of the reference frae, t proves possble to splfy consderably the coplexty of the atheatcal achne odel. Whle these technques were ntally developed for the analyss and sulaton of ac achnes, they are now nvaluable tools n the dgtal control of such achnes. As dgtal control technques are extended to the control of the currents, torque and flux of such achnes, the need for copact, accurate achne odels s obvous. Fortunately, the developed theory of reference fraes s equally applcable to the synchronous achnes, such as the Peranent Magnet Synchronous Machne (PMSM). Ths achne s soetes known as the snusodal brushless achnes or the brushless ac achne and s very popular as a hgh-perforance servo drve due to ts superor torque-to-weght rato and ts hgh dynac capablty. It s a three-phase synchronous ac achne wth peranent-agnet rotor exctaton and s desgned to have a snusodal torque-poston characterstc. The a of ths secton s to ntroduce the essental concepts of reference frae theory and to ntroduce the space vector notaton that s used to wrte copact atheatcal descrptons of ac achnes. Over the years, any dfferent reference fraes have been proposed for the analyss of ac achnes. The ost coonly used ones are the so-called statonary reference frae and the rotor reference frae. 1.2 Three-phase to two-phase transforaton (Clarke transforaton) Three-phase ac achnes are conventonally odeled usng phase varable notaton. However, for a threephase, star-connected achne, the phase quanttes are not ndependent varables so that: v + + v + + v = 0 = 0 ϕ + ϕ + ϕ = 0 where s, v s and ϕ s denote stator phase currents, voltages and flux lnkages, respectvely. As a result of ths redundancy n the phase varable representaton t s possble to transfor the syste to an equvalent two-phase representaton. The transforaton fro three-phase to two-phase quanttes s wrtten n atrx for as: (1) Analog Devces Inc., January 2002 Page 3 of 14

4 a ADSP-21990: Reference Frae Conversons sβ () t cos( γ) cos( γ) () t = sn( γ) sn( 2γ) where γ =2π/3. The transforaton s equally vald for the voltages and flux lnkages. The stator current space vector s defned as the coplex quantty: and t s possble to wrte (2) ore copactly as: s s s () t () t () t (2) () t = α() t + j β () t (3) 2 [ ] 2 s() t = () t + a() t + a () t (4) 3 where a s a vector operator that produces a vector rotaton of γ = 2π/3 and s defned as: ( j ) ( ) j ( ) a = exp γ = cos γ + sn γ (5) The choce of the constant n the transforatons of (2) and (4) s soewhat arbtrary. Here, the value of 2/3 s adopted. Its an advantage s that agntudes are preserved across the transforaton. Therefore, snusodal phase currents wth a peak agntude of I produce a current space vector wth a peak agntude of I. For ths transforaton, the nverse relatonshp s wrtten: 1 = cos cos 0 ( γ ) sn( γ ) ( 2γ ) sn( 2γ ) As the a of any dgtal current control schee s to control current apltude, t s felt that ths for s ore sutable for control purposes. Transforaton (2) and ts counterpart (6) are denoted hereafter as the Forward Clarke Transforaton and the Reverse Clarke Transforaton, respectvely. The space vector ay be vewed n the coplex plane as shown n Fgure 1. The conventonal agnetc axes of the three achne phases are separated by γ = 2π/3. The real or axs of the new two-axs coordnate syste s arbtrarly chosen to concde wth the axs. Obvously, the agnary or jsβ axs les n quadrature wth the axs. The current space vector s shown at an arbtrary locaton n the coplex plane. The phase currents ay be obtaned by projectng the current space vector onto the respectve phase axs. sβ (6) Analog Devces Inc., January 2002 Page 4 of 14

5 a ADSP-21990: Reference Frae Conversons jsβ Axs γ s Axs Axs Fgure 1: Relatonshp of stator current space vector and stator phase currents. Consder the case of a balanced set of three-phase stator currents: = I sn ( θ ϕ) ( θ γ ϕ) ( θ 2γ ϕ) = I sn = I sn (7) where I s the agntude of the phase currents, θ = ωt s the angular poston n radans and ϕ s the phase angle. Usng (2) the currents of (7) ay be transfored to the equvalent two-phase representaton to gve: sβ ( θ ϕ) ( θ ϕ) = I sn = I cos (8) so that the current space vector of such a syste ay be wrtten as: ( θ ϕ) ji cos( θ ϕ) s = I sn = ji e j( θ ϕ) (9) whch descrbes a crcular trajectory n the space vector plane. Therefore, a balanced three-phase syste n phase varables transfors to a crcular locus n the equvalent two-axs representaton. The radus of the crcle s the peak agntude of the phase quanttes. The crcular locus s descrbed at a rate equal to the angular frequency of the phase quanttes. 1.3 Vector rotaton (Park transforaton) Analog Devces Inc., January 2002 Page 5 of 14

6 a ADSP-21990: Reference Frae Conversons The stator current, voltage and flux lnkage space vectors are coplex quanttes defned n a reference frae whose real axs s fxed to the agnetc axs of stator wndng. However, the correspondng quanttes defned for the rotor crcut of a three-phase ac achne are slarly stated n a reference frae fxed to the rotor. In the analyss of electrcal achnes, t s generally necessary to adopt a coon reference frae for both the rotor and the stator. For ths reason, a second transforaton, known as a vector rotaton, s forulated that rotates space vector quanttes through a known angle. In the space vector dagra of Fgure 1, the axes of the space vector plane are statonary. Meanwhle the space vectors of the current, voltage and flux lnkages rotate about these axes at a rate equal to the angular frequences of the correspondng phase quanttes. If nstead a new reference frae s defned where the axes are ade to rotate at the sae rate as angular frequency of the phase quanttes, statonary current, voltage and flux lnkage space vector result. Consder applyng the vector rotaton through an angle θ: ν θ = e j (10) to the current space vector of (3). The current space vector n ths new reference frae s gven by: whch ay also be wrtten n atrx for as: sd sq θ j dq = sd + j sq = s e cos = sn ( θ) sn( θ) ( θ) cos( θ) (11) The real coponent of the current space vector n ths new reference frae s the drect axs coponent whle the agnary coponent s called the quadrature axs coponent. The relatonshp of the real and agnary coponents of the current space vector n the orgnal statonary two-axs reference frae and the new rotatng reference frae s shown n Fgure 2. Clearly, fro the vewpont of the statonary (αβ) frae, both the current space vector and the drect and quadrature axes are rotatng at a speed ω. However, when vewed fro the rotatng reference frae, the current space vector s statonary. The real axs of the rotatng reference frae s located at an angle θ fro the real axs of the statonary reference frae. The elnaton of poston dependency fro the achne electrcal varables s the an advantage of a vector rotaton. Ths transforaton wll be referred to as the Reverse Park Transforaton. Rotatng Iagnary Axs jsq Statonary Iagnary Axs jsβ sβ s ω sβ Rotatng Real Axs sd (12) sd sq θ Statonary Real Axs Fgure 2: Relatonshp of current space vector coponents n statonary and rotatng reference fraes. Analog Devces Inc., January 2002 Page 6 of 14

7 a ADSP-21990: Reference Frae Conversons The nverse vector rotaton, to transfor fro a rotatng to a statonary reference frae, ay be wrtten: or n atrx for as: sβ s cos = sn = dq e jθ ( θ) sn( θ) ( θ) cos( θ) sd sq (13) (14) Ths rotaton s coonly called the Forward Park Transforaton. Consder applcaton of the vector rotaton of (10) to the current space vector of (9) derved for the balanced set of stator currents: ϕ j dq = ji e (15) so that: sd sq = I = I cos sn ( ϕ) ( ϕ) whch are ndependent of the nstantaneous angular poston of the phase quanttes. Therefore, a balanced three-phase syste n phase varables ay be transfored to an equvalent two-axs representaton that s ndependent of the angular poston by applyng a three-phase to two-phase transforaton followed by a vector rotaton by the angular poston of the phase quanttes. 1.4 Practcal pleentaton of the transforatons (16) In ths secton, soe consderatons are ade about the overall coputatonal effcency of the transforatons. The Forward Clarke Transforaton gven by (2) ay be reduced nto a coputatonally ore effcent for by explctly substtutng the (constant) cosne and sne values and applyng the frst equaton of (1). It s easly seen that these actons lead to: sβ = ( ( )) 1 t 3 The Reverse Clarke Transforaton gven by (6) s pleented by explctly substtutng the (constant) cosne and sne values and applyng the frst equaton of (1): = sβ 3 2 s β (17) (18) Analog Devces Inc., January 2002 Page 7 of 14

8 a ADSP-21990: Reference Frae Conversons Unlke the Clarke transforatons, both Park transforatons requre the real-te coputaton of the transforng atrx tself. Ths entals the evaluaton of one sne and one cosne functon for each. However, the forward and reverse atrces are utually transposed. A ore effcent usage of these transforaton s therefore to calculate the sne and cosne functon once and to buld the two atrces by reversng the sgn of the sne where approprate. Ths wll be explaned ore n detal n the next secton. At last, t ay be noted that the two transforaton steps of (2) and (12) can be cobned nto a sngle transforaton that ay be wrtten as: sd sq = 2 3 ( θ) ( θ γ) ( θ 2γ) ( θ) ( θ γ) ( θ 2γ) cos cos cos sn sn sn Ths transforaton converts the phase quanttes to the rotatng reference frae n a sngle step. However, t s often ore desrable and coputatonally ore effcent to apply the transforaton n two steps as descrbed prevously. In ths way t s necessary to copute only two snusodal functons of poston unlke the sx requred for (19). (19) 2 The Reference Frae Converson routnes 2.1 Usng the converson routnes The routnes are developed as an easy-to-use lbrary, whch has to be lnked to the user s applcaton. The lbrary conssts of two fles. The fle reffrae.dsp contans the assebly code for the subroutnes. The user sply has to nclude the header fle reffrae.h, whch provdes functon-lke calls to the routnes. The exaple fle n the followng secton wll deonstrate the usage of all the converson routnes. The followng table reassues the set of acros defned n ths lbrary. Table 1 Ipleented routnes Transforaton Operaton Usage Intalsaton reffrae_set_dag_regsters_for_transforatons; Clarke Forward Clarke reffrae_forward_clarke(v abc, V αβ ); Reverse Clarke reffrae_reverse _Clarke(V αβ, V abc ); Park Method 1 Calculate Sne & Cosne reffrae_calc_sncos(ϕ, V sncos ); Reverse Park reffrae_reverse_park_sncos(v αβ, V dq, V sncos ); Forward Park reffrae_forward_park_sncos (V dq, V αβ, V sncos ); Park Method 2 Reverse Park reffrae_reverse_park_angle(v αβ, V dq, ϕ); Forward Park reffrae_forward_park_angle (V dq, V αβ, ϕ); where V abc s a three-eleent vector contanng the three phase quanttes, V αβ conssts of two eleents holdng the coponents of the stator quantty vector, V dq represents the two coponents of the rotor Analog Devces Inc., January 2002 Page 8 of 14

9 a ADSP-21990: Reference Frae Conversons quanttes and ϕ s the angle of rotaton. V SnCos holds the value for sn(ϕ) and cos(ϕ) for teporary storage. The followng exaple wll clarfy ts usage, whle secton 2.2 descrbes the forat of the quanttes. As already entoned n secton 1.4, there are two ways of pleentaton for the Park transforaton. The three routnes lsted for Method 1 n Table 1 should be used whenever both the forward and reverse transforaton are requred snce the trgonoetrc functons are evaluated only once. Ths s coputatonally ore effcent at the expense of two addtonal eory locatons for V SnCos. Method 2 calculates the sne and cosne functons every te the Park routne s called, but requres no eory locatons for the. Ths s recoended whenever only one transforaton s requred. 2.2 Forats of nputs and outputs The pleentaton of the routnes s such that values for angles are expected to be n the usual scaled 1.15 forat. Therefore, +1 (0x7FFF) corresponds to +π radans or 180 degrees, and 1 (0x8000) to -π radans or -180 degrees. Ths apples to the functons reffrae_calc_sncos( ), reffrae_forward_park_angle( ) and reffrae_reverse_park_angle( ). The other nputs are vectors. That eans that the values passed to the routnes are addresses of contguous eory locatons. Typcally, they are represented by labels, such as Valphabeta. The coponents of the vectors are nubers n the scaled 1.15 forat. However, havng the sngle coponents n ths forat does not necessarly ental that the nput s n ths forat. As a sple exaple, havng Vα and Vβ both equal to 1 leads to a vector n the statonary frae wth agntude 2 > 1. Obvously, f ths vector s the nput to the nverse Clarke transforaton, t corresponds to deand a three-phase syste of apltude bgger than one. It s up to the user to ensure that the nputs are eanngful and adequately scaled. 2.3 Usage of the DSP regsters The acros lsted n Table 1 are based on three subroutnes that are reported n the followng table. Also, an overvew s gven on the usage of the core DSP regsters. It ay be noted that the DAG regsters M0, M1, M2 ust be set to 1 and L0, L1, L2 ust be set 0 and that they are not odfed by the converson routnes. Furtherore, the trgonoetrc routnes requre that M5 and L5 be set to 1 and 0 respectvely. The already entoned call to reffrae_set_dag_regsters_for_transforatons prepares these regsters for the converson routnes. It now becoes clear that ths routne s necessary only once f none of the above-entoned regsters are odfed n another part of the user s code. However, for the sake of effcency, t ay be ore approprate to use sngle nstructons to restore eventually odfed regsters rather than a call to ths acro that requres 8 nstructons. Table 2 Usage of DSP core regsters for the subroutnes Subroutne Input Output 1 Modfed regsters Other regsters reffrae_forward_clark I1 = Input I0 = Output N/A MX0, MY0, MR, AR, I0, I1 M0, M1 = 1, L0, L1 = 0 reffrae_reverse_clark I1 = Input I2 = Output N/A MX0, MY0, MX1, MY1, MR, I1, I2 M1, M2 = 1, L1, L2 = 0 reffrae_rotate_vector I0 = Input I1 = Output y0 = cos(angle) N/A MX0, MX1, MR, I0, I1 I0, I1 M0, M1= 1, L0, L1 0 1 The output values are stored n the output vector n Data Meory. No DSP core regster s used. Analog Devces Inc., January 2002 Page 9 of 14

10 a ADSP-21990: Reference Frae Conversons y1 = sn(angle) When usng the functon lke acros, the odfed regsters are those reported below. The ajor dfference to Table 2 s seen n the Park routnes. Ths s because the routne reffrae_rotate_vector only perfors a vector rotaton wth the transforng atrx as nput, whereas the overall Park transforaton requres the coputaton of a sne and a cosne functon. Table 3 Usage of DSP core regsters for the Macros Usage Modfed regsters reffrae_set_dag_regsters_for_transforatons; M0, M1, M2, M5 = 1, L0, L1, L2, L5 = 0 reffrae_forward_clarke(v abc, V αβ ); reffrae_reverse _Clarke(V αβ, V abc ); reffrae_calc_sncos(ϕ, V sncos ); reffrae_reverse_park_sncos(v αβ, V dq, V sncos ); reffrae_forward_park_sncos (V dq, V αβ, V sncos ); reffrae_reverse_park_angle(v αβ, V dq, ϕ); reffrae_forward_park_angle (V dq, V αβ, ϕ); MX0, MY0, MR, AR, I0, I1 MX0, MY0, MX1, MY1, MR, I1, I2 AX0, AY0, AX1, AY1, AR, AF, MY0, MX1, MY1, MR, SR, I0, I6, M6, L6 AX0, AR, MX0, MX1, MY0, MY1, MR, I0, I1, I2 MX0, MX1, MY0, MY1, MR, I0, I1, I2 AX0, AY0, AX1, AY1, AR, AF, MX0, MX1, MY0, MY1, MR, SR, I0, I1, I6, M6, L6 AX0, AY0, AX1, AY1, AR, AF, MX0, MX1, MY0, MY1, MR, SR, I0, I1, I6, M6, L6 2.4 The progra code: reffrae.dsp The code contaned n the fle reffrae.dsp defnes the three routnes that are lsted n Table 2. The routne reffrae_rotate_vector perfors a vector rotaton and s the core routne for both the reverse and forward Park transforaton. In the code, equaton (17) of the forward Clarke transforaton s pleented. Note that the order n whch the sngle operatons are executed s changed. The reason for t s that the dfference Vb-Vc n a three phase syste ay assue values that are bgger than one ( V V 3 ) and therefore ay not be b c represented n 1.15 forat. Ths s avoded by pre-ultplyng both coponents wth the constant before subtractng the. Also, before subtractng the two values, the saturaton ode of the ALU has to be set n order to avod overflow fro postve to negatve values and vce-versa. However, ths s only necessary for nput apltude equal to one. The two nstructons for enablng and dsablng the saturaton ode could be reoved f the nput syste has agntude less than one. The reverse Clarke transforaton of equaton (18) s also pleented. Slarly to the case of the forward transforaton, overflows of the MAC need to be avoded, for nput agntudes equal to one. Agan, the correspondng SAT MR nstructons could be reoved f ths never occurs. As already entoned the reffrae_rotate_vector routne s used by both the forward and reverse Park transforatons. It sply ultples the nput vector by the atrx of equaton (14) whose eleents are provded as nputs as well. Refer to the descrpton of the header fle n the next secton for ore detals on the evaluaton of ths atrx. Once agan, the sae consderatons as for the reverse Clarke Analog Devces Inc., January 2002 Page 10 of 14

11 a ADSP-21990: Reference Frae Conversons transforaton regardng saturaton apply. For ore nforaton regardng the progra code, please refer to the coents n the reffrae.dsp fle. 2.5 Access to the lbrary: the header fle: reffrae.h The lbrary ay be accessed by ncludng the header fle reffrae.h n the applcaton code, as was already explaned n the secton 2.1. The header fle s ntended to provde functon-lke calls to the routnes presented n the prevous secton. It defnes the calls shown n Table 1. The fle s ostly selfexplanng. Ths fle, after ncludng the trgonoetrc lbrary and the core routnes, defnes the ntalsaton routne Set_DAG_regsters_for_transforatons and the acros for the forward and reverse Clarke transforatons. The only coent s that, snce the nput and output vectors are ponted to by ndex regsters of the DM-DAG, they ust be defned n data eory. It s worth addng a few coents about the evaluaton of the rotaton atrx n the dfferent fors of the Park Transforaton. The sne and cosne values are calculated and the vector rotaton routne s called. For the reverse transforaton, the only acton that s requred s to copleent the value of the angle. Note that, snce the negate nstructon ar = -angle, produces the copleent except for an nput of -1 (0x8000). If ths occurs, the twos-copleent s 0x8000 wth the overflow flag set to one. The condtonal nstructon edately after t detects ths case and loads AR wth the correct value of +1 (0x7FFF). Method 1 of the Park transforaton s pleented by the reffrae_calc_sncos, reffrae_forward_park_sncos and the reffrae_reverse_park_sncos acros. The dfference s that the atrx for the reverse transforaton s not obtaned by recalculatng the trgonoetrc functons wth the copleented angle, but by akng use of the know syetres of these functons. Snce sn(-x)=sn(x) and cos(-x)=cos(x), the resultng atrx s obtaned by nvertng the sgn of only the (prevously stored) sne value. Agan, wth a condtonal nstructon, the specal case of 1 s correctly handled. For ore nforaton regardng the header fle, please refer to the coents n the reffrae.h fle. 3 Software Exaple: Testng the Converson Routnes 3.1 The an progra: an.dsp The exaple deonstrates how to use the routnes. All t does s to generate a three-phase sne wave syste. The phase coponents are then transfored nto the statonary (αβ-) and rotor (dq) frae. After that, the nverse transforatons are executed. The applcaton has been adapted fro a prevous note 2, hence ths secton wll only explan the few and ntutve odfcatons to that applcaton. The fle an.dsp contans the ntalsaton and PWM Sync and Trp nterrupt servce routnes. Frst, the general systes constants and PWM confguraton constants (an.h- see the next secton) are ncluded. Also ncluded are the PWM lbrary, trgonoetrc lbrary and the reference converson lbrary. Next, all the vectors for the transforaton routnes are declared. The ntalsaton of PWM block s executed and the nterrupts are enabled. The progra then enters a loop, whch just wats for nterrupts. The nterrupt servce routne sply shows how to ake use of the reference frae converson routnes. After pleentng the sne-wave generaton (see the applcaton note entoned above for detals) the three phase voltages are stored nto a vector (Vabc) for use wth the transforaton routnes. Then Vabc s converted nto the statonary reference frae (Valphabeta). Method 1 s then executed to transfor Valphabeta nto the dq-frae (Vdq) and back nto Valphabeta_rev by calculatng the sne and cosne of the angle once and nvokng respectvely the reverse and forward Park transforaton. Method 2 s also 2 AN : Generaton of Three-Phase Sne-Wave PWM Analog Devces Inc., January 2002 Page 11 of 14

12 a ADSP-21990: Reference Frae Conversons shown, although the related code s coented. The user ay play wth the by sply reovng the coent drectve //. At last, the statonary vector s transfored back nto the orgnal 3-phase descrpton Vabc_rev. Secton 3.3 wll show the outputs that are produced by ths exaple. For ore nforaton on the an progra, please refer to coents n the an.dsp fle. 3.2 Man.h The confguraton of the PWM block requres soe constants, whch are defned n the an.h fle. The use of ths fle s to defne general syste paraeters and constants, such as the crystal clock [khz], core clock [khz] and the perpheral clock [khz] (also denoted as H clk ). Besdes, any constants requred by the lbrary fles wll be defned here. In ths case, the constants needed by the PWM lbrary, such as the PWM swtchng frequency [Hz], the dead te [nsec] and the PWM sync pulse te [nsec] wll be ncluded n ths fle. Two fles, the ADSP h and the acro.h are also ncluded n the an.h fle. The ADSP h defnes the perpheral regsters of the ADSP whle the acro.h defnes the ost coonly used acro. 3.3 Exaple output The graphcal representaton of the vector varables Vabc, Valphabeta, Vdq are shown n the followng fgures. The orgnal three-phase syste s descrbed as follows: V V V A B C ( ωt) 2π ( ωt ) sn = V sn 3 ( ) sn 4 π ωt 3 ( 20 ) where ω and V denote the angular speed and the (scaled) apltude. Note that t s n the for of equaton (7) wth ϕ equal to zero. Fgure 3 shows the produced wavefors versus the angle (-π to +π). V c V b V a ωt Fgure 3 Vector varable Vabc (the saw-tooth s the angle representaton rangng fro -π to +π) Applyng the forward Clarke transforaton (17) to ths partcular syste leads to: V V α β sn = V cos ( ωt) ( ) ωt ( 21 ) descrbng a crcular locus n the αβ-plane. Ths s shown n the next fgure. Analog Devces Inc., January 2002 Page 12 of 14

13 a ADSP-21990: Reference Frae Conversons V α V β Fgure 4 Vector varable Valphabeta versus theta (left) and n the αβ-plane (rght) The next step conssts of applyng the reverse Park transforaton of (12) to (21). It s easly seen that the resultng dq-vector s constant: V V d q 0 = V 1 Fgure 5 represents ths vector graphcally n the te doan. Note that n the dq-plane the vector s reduced to one pont. ( 22) V d V q Fgure 5 Vector varable Vdq versus theta Analog Devces Inc., January 2002 Page 13 of 14

14 a ADSP-21990: Reference Frae Conversons The vectors Valphabeta_rev and Vabc_rev, whch represent the back-transfored quanttes, obvously are, apart fro eventual roundng errors, dentcal to Valphabeta and Vabc, respectvely. As an exaple, the followng fgure shows a coparson between Va and Va_rev both n the te doan and n xyrepresentaton. V a =V a_rev ωt Fgure 6 Va and Va_rev versus theta (left) and n xy representaton (rght) Analog Devces Inc., January 2002 Page 14 of 14