AC Modeling and Control of AC Motors Seiji Kondo, Member 1. q q (1) PM (a) N d q Dept. of E&E, Nagaoka Unive

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1 AC Moeling an Control of AC Motors Seiji Kono, Member 1. (1) PM (a) N Dept. of E&E, Nagaoka University of Technology 163 1, Kamitomioka-cho, Nagaoka, Niigata (a) 巻数 N のコイル 1 I θ Nsinθ I Ncosθ (b) コイル巻数の分解 1(a) N 1(b) 1(b) N N cos θ, N N sin θ (1) P P L e P (N ), L e P (N ) () 1(a) N L(θ) L e L e L e cos θ L e sin θ (3a) L e P N, L e P N (3b) P l (3a) (3b) enacmcontrol.tex Nov.85 1

2 N α sinθ N sinθ β N β N α θ' θ I N α cosθ N α θ N N cosθ β (a) 巻数 N のコイル (b) コイル巻数の分解 3 L(θ) L cos θ L sin θ (4a) β i s v s L (P l P )N, L (P l P )N (4b) L(θ) i rβ α θ P P L(θ) L ( L ) (5) v rβ vrα i rα θ v s i s θ 1 (a) N α N α N α cos θ, N α N α sin θ (6) N β N β N β cos θ, N β N β sin θ (7) N α N β M P N α N β cos θ cos θ (8) N α N β M P N α N β sin θ sin θ (9) M(θ) M M P N α N β cos θ cos θ P N α N β sin θ sin θ (1) N α N β N θ θ π/ 4 (4b) P P M(θ) (1) (a) θ N β 3 N (1) θ N β N M α (θ) P N N α cos θ (13) 4 s s [ ] s (P l P )N L s (P l P )N r a P N N α cos θ P N N α sin θ P N N β sin θ P N N β cos θ r b M(θ) 1 (P P )N sin θ 1 (L L ) sin θ (11) (3b) L L (4b) S.Kono Note, FileName: enacmcontrol.tex Nov.8, 5

3 r α P N N α cos θ P N N α sin θ (P l P cos θ P sin θ)n α 1 (P P )N α N β sin θ r β P N N β sin θ P N N β cos θ 1 (P P )N α N β sin θ (P l P sin θ P cos θ)nβ (14) (14) s s s s r α r α s (14) (14) N N N s N α N β N r L s P Ns, L s P Ns L r P Nr, L r P Nr M P N s N r, M P N s N r l s P l Ns, l r P l N r (15) (14) s s [ ] s l s L s L s l s L s r α M cos θ M sin θ M sin θ M cos θ r β l r L r L r r α M cos θ M sin θ L r L r L r L r r β M sin θ M cos θ L r L r l r L r L r sin θ sin θ L r L r cos θ cos θ (16) (15) P P M M M, L s L s L s, L r L r L r (17) (16) s s r α r β [ ] s l s L s M cos θ M sin θ L s l s L s M sin θ M cos θ (18) r α M cos θ M sin θ l r L r r β M sin θ M cos θ l r L r (l s L s )[I ] M[C(θ)] M[C(θ)] T (l r L r )[I ] (19) [C(θ)] cos θ sin θ 3 sin θ cos θ, [I ] 1 1 () 4 ( 6(b) αβ ) 1 s r α θ θ θ θ (1) θ 3 s α, s β, r, r (16) [ L ] s α s β r r s α l s L s L s L s L s cos θ L s L s sin θ M cos θ M sin θ (1) () αβ αβ (1) () enacmcontrol.tex Nov.85 3

4 s β L s L s sin θ l s L s L s L s L s M sin θ M cos θ cos θ i v v v i u i β v β i α v w v u v α r r M cos θ M sin θ M sin θ M cos θ l r L r l r L r () i w (a) 5 (b) (14) (16), (18), () 4 *** 4.14 *** P P [ L ] s a s b s c r a r b r c s a s b s c l s L s L s cos(π/3) L s cos(π/3) L s cos(π/3) l s L s L s cos(π/3) L s cos(π/3) L s cos(π/3) l s L s M cos θ θ π 3 θ π 3 θ π 3 M cos θ θ π 3 θ π 3 θ π 3 M cos θ r a r b r c M cos θ θ π 3 θ π 3 θ π 3 M cos θ θ π 3 θ π 3 θ π 3 M cos θ l r L r L r cos(π/3) L r cos(π/3) L r cos(π/3) l r L r L r cos(π/3) L r cos(π/3) L r cos(π/3) l r L r (3) l s[i 3 ] L s [C ()] M[C (θ)] M[C (θ)] T l r [I 3 ] L r [C ()] (4) cos θ cos ( ) θ π 3 cos ( ) θ π 3 [C (θ)] cos ( ) θ π 3 cos θ cos ( ) θ π 3 cos ( ) θ π 3 cos ( ) (5) θ π 3 cos θ (1) p (u, v, w) (α, β) u α (u, v, w) (α, β) u v w α β 3 3 1/ 1 1/ 1/ 3/ 1/ 1/ α (6a) 3/ β 1/ 1/ 1/ u 1 1/ 1/ v (6b) 3/ 3/ w R s R r (4) (1) 4..3 D-8 p.73, D-16 p.18-4 S.Kono Note, FileName: enacmcontrol.tex Nov.8, 5

5 v su v sv v sw v ru v rv v rw [ Z 3 ] i su i sv i sw i ru i rv i rw [ ] Z3 R s[i 3 ] R r [I 3 ] t l s[i 3 ] L s [C ()] M[C (θ)] M[C (θ)] T l r [I 3 ] L r [C ()] [C (θ)] (5) (7a) (7b) (6a) v u v v v w i u i v i w [C 3] [C 3] v v α v β i i α i β (8a) (8b) s r [C 3 ] (6a) 1/ 1 [C 3 ] 1/ 1/ 3/ 3 1/ 1/ (9) 3/ (8a)(8b) (7a) v s v s v s v r v rα v rβ [Z ] i s i s i s i r i rα i rβ (3) [Z ] 6 6 [Z ] [C 1 3] [C 3 ] [Z 3 ] [C 3] [C 3 ] R s[i 3 ] R r [I 3 ] t l s[i 3 ] l r [I 3 ] L s 3 [I ] M [C(θ)] t M [C T (θ)] L r (31) [I ] 3 sα sβ s s (18) a [C(θ)] () (31) (18) L <> L <3> 3/, L <> s L <> r 3 L<3> s 3 L<3> r M <> 3 M<3> (3a) (3b) (3c), l <> s l <3> s, l r <> l r <3> R <> s R <3> s, R <> r R <3> r 33 (3) (3e) (6b) i i u α i 1 1/ 1/ β 3 3/ 3/ i v (i u i v ) 3 1 i u i (33) v i v i 3 α i β 1 i u i (34) w (6a) (31) 1 4 3/ D-8 p.74 enacmcontrol.tex Nov.85 5

6 β i s v s β isβ v sβ [L 11 ] l s L s l s L (38b) s i rβ v rβ v rα i rα θ α i s i r v r v r i r θ ' θ i sα α [L 1 ] M cos θ M sin θ M sin θ M cos θ (38c) (a) ~ v s v sα (b) ~{ [L ] L r L r1 cos θ L r1 sin θ L r1 sin θ L r L r1 cos θ (38) v uv v vw v wu v u v v v w v v v w v u 1/ 1 1/ v 1/ 3/ 1/ 1/ v α 3/ v β 1/ 1/ 3/ 1/ 1/ v 3/ 1/ v α 1 v β 3/ 3/ v 3 3/ v α (35) 3/ [ v v α v β ] T [ vα v β ] T v α /3 1/ 6 1/ v uv (36) v β 4. v vw 41 6 αβ s r 6(a) 6(b) 411 v β 6(a) [L] T [ ψs ψ s ψ rα ψ rβ ] T [L] [ is i s i rα i rβ ] T (37) [L] (16) [L] [L 11] [L 1 ] [L 1 ] T [L ] (38a) L r l r L r L r, L r1 L r L r (38e) (38a)-(38e) v s R s v s R s v rα R r R r v rβ i s i s i rα i rβ [L] t i s i s i rα i rβ (39) 6(a) 41 6(b) 6(a) (b) (38a)-(38e) [L] 1 s r θ θ 3 [ ψsα ψ sβ ψ r ψ r ] T [L ] [ i sα i sβ i r i r ] T (4) 6(b) [L ] () [L ] [L ] [L 1 ]T [L 1 ] [L 11 ] (41a) [L ] L s L s1 cos θ L s1 sin θ L s1 sin θ L s L s1 cos θ (41b) [L 1 ]T M cos θ M sin θ M sin θ M cos θ (41c) [L 11 ] l r L r l r L (41) r L s l s L s L s, L s1 L s L s (1) p.8 (4.) (41e) 6 S.Kono Note, FileName: enacmcontrol.tex Nov.8, 5

7 (41a) (38a) (41a)-(41e) v sα R s v sβ R s v r R r v r R r i sα i sβ i r i r [L ] t i sα i sβ i r i r (4) 6(b) 4 (39) [v] [R][i] t [[L][i]] [R][i] [L] [i] t θ [L] [i] (43) θ [i] T P i P i [i] T [v] [i] T [R][i] [i] T [L] [i] t θ[i] T [L] [i] (44) θ ( ) 1 t [i]t [L][i] 1 ( [i] T [L][i] [i] T [L] [i] [i] T [L] [i] ) t t t [i] T [L] [i] 1 t [i]t θ [L] [i] (45) θ 1, 3 (38a) [L] 3 (45) (44) [i] T [L][i]/t P i [i] T [v] [i] T [R][i] t θ 1 ( ) 1 [i]t [L][i] [L] [i]t [i] (46) θ (46) 1 3 θ T [i] T P i θ θ T T 1 [L] [i]t [i] (47) θ p θ M T M θ M θ/p, T M pt (48) (4) [L] [L ]θ θ 43 (47) (18) θ [L] θ sin θ cos θ M cos θ sin θ sin θ cos θ cos θ sin θ (49) (47) T T 1 [L] [i]t θ [i] 1 [ ] is i s i rα i rβ M sin θ cos θ cos θ sin θ sin θ cos θ cos θ sin θ i rα ( Mi s sin θ Mi s cos θ) i rβ (Mi s cos θ Mi s sin θ) i s i s i rα i rβ i rα ψ rβ i rβ ψ rα (5) ψ rβ M i s sin θ M i s cos θ (51a) ψ rα M i s cos θ M i s sin θ (51b) ψ rβ S rβ ψ rα S rα (16) θ (46) 3 1/ D-13 pp enacmcontrol.tex Nov.85 7

8 [L] θ s s r α r β s s r α M sin θ M cos θ M sin θ M cos θ (L r L r ) sin θ M cos θ M sin θ (L r L r ) sin θ r β M cos θ M sin θ (5) (L r L r ) cos θ (L r L r ) sin θ (47) T T i rα ( M i s sin θ M i s cos θ ) i rβ ( M i s cos θ M i s sin θ ) (L r L r )i rα i rβ cos θ L r L r ( ) i rα i rβ sin θ (53) 1 (51a) (51b) T i rα ψ rβ i rβ ψ rα (L r L r )i rα i rβ cos θ L r L r ( ) i rα i rβ sin θ (54) 1 34 ***3.1.3 *** (a) θ ω αβ ω ω αβ 7(a) α θ (, ) (α, β) C(θ) α β (55a) α β C( θ) (55b) C(θ) cos θ sin θ sin θ cos θ (55c) (1) p.56 β α (a) j θ 7 β α (b) αj θ' θ 7(b) αβ 7(b) (55a)-(55c) θ θ 5 1 7(a) x j x αβ α jβ x exp(jθ)x αβ (cos θ j sin θ)(α jβ) (56) exp(jθ) θ 1 No.3 (55a) (56) (55a) (56) x t t ( exp(jθ)xαβ ) exp(jθ)( t j θ)x αβ (57) 53 cossin cos sin x r, x i j -1 x x r jx i (58) 7(a) θ 8 S.Kono Note, FileName: enacmcontrol.tex Nov.8, 5

9 x x r jx i y r jy i x r y r x i y i (59a) (x r jx i ) ± (y r jy i ) (x r ± y r ) j(x i ± y i ) (59b) (x r jx i ) (y r jy i ) (x r y r x i y i ) j(x r y i x i y r ) (59c) (x r jx i ) (y r jy i ) (x r y r x i y i )/Y j(x i y r x r y i )/Y (59) Y y r y i x r j x r j j (x r, x i ) (x r, x i ) (y r, y i ) x r y r x i y i (6a) (x r, x i ) ± (y r, y i ) (x r ± y r, x i ± y i ) (6b) (x r, x i ) (y r, y i ) (x r y r x i y i, x r y i x i y r ) (6c) (x r, x i ) (y r, y i ) ( x ry r x i y i Y Y y r y i, x iy r x r y i ) (6) Y (x r, x i ) x r (x r, ) (, 1) j (6c) (, 1) (, 1) (-1, ) j j -1 x r y r???????? (61) x i y i 1!! (x r jx i ) (y r jy i ) x r y r x i y i j(x r y i x i y r ) x r x i x i x y r r y x ry r x i y i i x r y i x i y (6) r 1 x r x i x r x i x r x i X cos θ sin θ sin θ cos θ (63a) x i X x r x r x i, ( ) θ xi tan 1 (63b) x r X θ x r x i 53 jθ e jθ exp(jθ) cos(θ) j sin(θ) (64) (63a) θ exp(jθ) cos θ sin θ sin θ cos θ exp(jθ) (65) e x e y e xy (66) (3) n n 1, n4 (3) p.4 enacmcontrol.tex Nov.85 9

10 e jα e ±jβ e j(α±β) cos(α ± β) j sin(α ± β) (67a) (cos α j sin a)(cos β ± j sin β) (67b) (cos α cos β sin α sin β) j (sin α cos β ± cos α sin β) (67c) (67a) (67c) 533 x r Re(x r jx i ), x i Im(x r jx i ) (68) Re(), Im() x r x x x i ( j) x x ( j) jx i (x r jx i ) (x r jx i ) ( j) (x r jx i ) (x r jx i ) x r (69a) (69b) j Re(), Im() Re(), Im() cos cos sin Re() cos Re() (x ± y) x ± y (7a) (xy) x y (7b) ( ) x x y y (7c) x x x (7) Im(xx ) (7e) x r x i x i x r x r x i a b c (71) (71) A 1 1 1, A 1 1, (7a) A 3 1 1, A (7b) 1 A 1 A 3 1, A 1 A 3 1 A A 4 1, A A 4 1 (73a) (73b) a,, c, (38c), (38), (41b), (41c) 1 S.Kono Note, FileName: enacmcontrol.tex Nov.8, 5

11 1 No. (A jb)(x jy) 1 A B x (Ax By) j(bx Ay) B A y Ax By Bx Ay exp(jθ) cos θ sin θ 3 exp(jθ) cos θ sin θ sin θ cos θ 4 j conj() j conj() 1 1 ( 1) ( ) 1 ( 3) A 1 1 1, A 1 1, A 3 1 1, A A 1 A 3 1, A 1 A 3 1, A A 4 1, A A 4 1 b (71) A 1 A A 3 A 4 (71) (7a)(7b) A (74a) A 1 1 exp(jπ/) j (74b) A conj() (74c) A j conj() (74) 1 54 (41a)-(41e) [ ] [ ] [ L ] T L L 1 [ ] [ ] L 1 L (75a) 11 (38a) [ L ] L s L s1 cos θ L s1 sin θ L s1 sin θ L s L s1 cos θ (75b) [ L 1 ] T M cos θ M sin θ M sin θ M cos θ (75c) [ L ] 11 l r L r l r L (75) r L s l s L s L s, L s1 L s L s (75e) v sα v sβ v r v r R s R s R r R r i sα i sβ i r i r [ L ] t i sα i sβ i r i r (76) [L ] (75b) (74a)-(74) [L ] L sa 1 L s1 (cos θ A 3 sin θ A 4 ) (77a) L s L s1 ( cos θ conj() sin θ j conj() ) L s L s1 exp(jθ )conj() (77b) (75c) (73b)(74a)-(74) [L 1 ]T M cos θ A 1 A 3 M cos θ A 1 A 3 M sin θ A A 4 M sin θ A A 4 M M ( cos θ A 1 sin θ ) A M M (cos θ A 3 sin θ A 4 ) (78a) M M ( cos θ sin θ j ) M M (cos θ sin θ j)conj() M M exp(jθ ) M M exp(jθ )conj() M exp(jθ ) M 1 exp(jθ )conj() (78b) enacmcontrol.tex Nov.85 11

12 [L 1 ] M cos θ A 1 A 3 M cos θ A 1 A 3 M sin θ A A 4 M sin θ A A 4 M M ( cos θ A 1 sin θ ) A M M (cos θ A 3 sin θ A 4 ) (78c) M M ( cos θ sin θ j ) M M (cos θ sin θ j)conj() M M exp( jθ ) M M exp(jθ )conj() M exp( jθ ) M 1 exp(jθ )conj() (78) M M M, M 1 M M (78e) (75) (74a)-(74) [L 11 ] (l r L r ) A 1 A 3 (l r L r ) A 1 A 3 (l r L r L r )A 1 L r L r L r L r1 conj() A 3 (79a) (79b) [L L s L s1 exp(jθ )conj() ] M exp( jθ ) M 1 exp(jθ )conj() M exp(jθ ) M 1 exp(jθ )conj() L r L r1 conj() (8) conj() (b) (4) (41a)- (41e) D-14 pp.-5 M M M L ss l s L s l s L s L rr l r L r l r L r 3 v r v r 4 6(b) γ, δ (4) v sα v sβ R s DL ss R s DL ss DM cos θ DM sin θ DM sin θ DM cos θ DM cos θ DM sin θ DM sin θ DM cos θ R r DL rr R r DL rr i sα i sβ i rγ i rδ D /t (81) θ (81) (81) v sαβ R s DL ss DM exp(jθ ) DM exp( jθ ) R r DL i sαβ rr i (8) rγδ 6(b) α θ (8) i sαβ exp(jθ ) exp ( j(θ θ ) ) i s (83) i rγδ i r i v v s (D R s (D j θ )L ss j( θ θ ) ) M (D j θ )M R r ( D j( θ θ ) ) L i s (84) rr i r θ θ α 1 S.Kono Note, FileName: enacmcontrol.tex Nov.8, 5

13 θ (84) v s R s DL ss θ L ss v s θ L ss R s DL ss DM ( θ θ )M ( θ θ )M DM DM θ M i s θ M DM i s R r DL rr ( θ θ (85) )L rr i r ( θ θ )L rr R r DL rr i r (46) (84) i s i s i s i s Current CTL i s i s v s, s θ - - flux position calc. uvw uvw λ r v u, v, w i u, v, w INV - Current CTL v s, s - uvw v u, v, w IM PS INV T M(i s i r i s i r ) MIm(i r i s) (86) Im * 6 (84) i s i s θˆ - phase calc. λˆr uvw v u, v, w flux observer i u, v, w IM PS λ r Mi s L rr i r (87) (87) (84) 9 (D j( θ θ ))Mi s R r i r (D j( θ θ ))L rr i r R r i r (D j( θ θ ))λ r (88) R rm i s R r λ r (D j( θ θ ))λ r (89) L rr L rr (88) (89) (87) i r i s (89) t λ R r θ θ r λ L rr r ( θ θ ) R r λ r λ R rm r L i s rr i (9) s L rr (9) (86) (87) ( ) 1 T MIm (λ r L Mi s )i s rr M Im ( λ r L i ) s rr M L rr (λ r i s λ r i s ) (91) Im(i i) i s i s - - is i s i s τ L rr / R r 1 τ 1 τ P 1 Current CTL is ωsl 63 v s, s θ ω1 t ω uvw uvw v u, v, w i u, v, w ωr INV IM PS λ r λ r λ r λ r (9a) λ r (9b) enacmcontrol.tex Nov.85 13

14 (91) T M L rr λ r i s (93) i s i s (9) 1 (9b) t λ r R r L rr λ r R rm L rr i s (94) T re f λ re f r (93) (94) i re f s i re f s L rr T re f M λ re f r 1 M (1 L rr R r (95) t )λre f r (96) (95) (96) 64 λ r λ r λ re f r λre f r j (9) t λ r R r L rr λ r ( θ θ )λ r R rm L rr i s (97) ( θ θ ) λ r () ( θ θ ) R rm L rr i re f s λ re f r (98) (95), (96) (98) 1 τ L rr /R r, ω r θ, ω sl θ θ (99) λ r (9) (95), (96), (98) t λ o r Ro r M o ˆL rr ˆR r ˆML rr o λ o r λ re f r Ro r Lrr o ( θ θ ) R o r M o [ Lrr o ˆM 1 Ro r /Lrr o ] ˆR r / ˆL rr θ θ λ o r Ro r M o ˆL rr ˆR Ro r r ˆML rr o Lrr o λ o r λ re f r λ re f r (1) o ˆ (1) o ˆ [ ] T 1 R o r /L o rr ± j( θ θ ) (λ o r λ r ) λo r (λ o r ) (λ o r λre f r ) 7. PM (b) (4) 6(b) ψ M i r 3 (4) αβ v sα v R s i sα sβ i [A 1] sβ t i sα i sβ [A ] i sα θ sin θ ψ cos θ (11a) i sβ [A 1 ] L s L s1 cos θ L s1 sin θ L s1 sin θ L s L s1 cos θ (11b) [A ] θ L s1 sin θ cos θ cos θ sin θ (11c) L s l s L s L s, L s1 L s L s * (11) 14 S.Kono Note, FileName: enacmcontrol.tex Nov.8, 5

15 i sαβ v sαβ R s i sαβ L s L s1 exp(jθ ) i sαβ t t θ L s1 exp(jθ )ji sαβ exp(jθ )j θ ψ (1) 6(b) v sαβ exp(jθ )v s, i sαβ exp(jθ )i s (13) i s i s v s R s i s L s L s1 t t j θ L s i s j θ L s1 i s j θ ψ (14) v s v R s i s s i l s L s s l s L s t i s i s θ (l s L s ) l s L s i s θ ψ (15) (15) (46) T (L s L s )i s i s ψi s (16) p (48) 7 i s PM :, 1984 :, file name NoteD14P.oc 5/11/7 (14) 3/4/5 [3/1/14 v.1] ieejsk.cls D-1 pp.5-5 (1) i sαβ i sαβ D-13 pp.4-41 enacmcontrol.tex Nov.85 15

> > <., vs. > x 2 x y = ax 2 + bx + c y = 0 2 ax 2 + bx + c = 0 y = 0 x ( x ) y = ax 2 + bx + c D = b 2 4ac (1) D > 0 x (2) D = 0 x (3

> > <., vs. > x 2 x y = ax 2 + bx + c y = 0 2 ax 2 + bx + c = 0 y = 0 x ( x ) y = ax 2 + bx + c D = b 2 4ac (1) D > 0 x (2) D = 0 x (3 13 2 13.0 2 ( ) ( ) 2 13.1 ( ) ax 2 + bx + c > 0 ( a, b, c ) ( ) 275 > > 2 2 13.3 x 2 x y = ax 2 + bx + c y = 0 2 ax 2 + bx + c = 0 y = 0 x ( x ) y = ax 2 + bx + c D = b 2 4ac (1) D >