❯❱ ❯ P❱

♣rt♠♥t♦ ♥♥r ♥str

♦ ♦tt♦rt♦ r ♥ ♥♥r ♥str

rr♠ ♥ ♥♥r ♥r ttr

♦ ❳❳❳

♥♥♦t Prt rr♥t ♦♥tr♦ ♦r

②♥r♦♥♦s t♥ ♥s

rtt♦r ♦

♠♦ Pr♦ P♦♦ ♦♦♠♦

♦♦r♥t♦r ♥r③③♦

Pr♦ ♦rt♦ rr

♣rs♦r

♠♦ Pr♦ r♦ ♦♦♥♥

♦tt♦r♥♦ ♥ ù

tt♦r

♦♥t♥ts

Pr

②♥r♦♥♦s t♥ ♥

♥tr♦t♦♥

♥r ♠♦ ♦ t ♦② tr ♣s ♠♥

♥r② qt♦♥s

qt♦♥ t♦t strt♦♥

♥r ♠♦ ♦ t ♦② t♦ ♣s ♠♥

♥r② qt♦♥s

♦ t♦t strt♦♥

②♥r♦♥♦s t♥ ♥

♥r② qt♦♥s

♦ t♦t strt♦♥

♥ t s♥s♦② strt ♥♥s

♦♥s♦♥s

Prt rr♥t ♦♥tr♦

♥tr♦t♦♥

Prt rr♥t ♦♥tr♦

♦st ♥t♦♥

♦s P

t ♦ ♣r♠tr ♠s♠t ♥ rt♦♥

♦r P

♦♥s♦♥

♦♥t♥ts

♦♥ st♥t♦♥ ♥ ♦r Prt rr♥t ♦♥tr♦

♥tr♦t♦♥

♥②ss ♦ st♥t♦♥ ts

Pr♦♣♦s ♥tst♥t♦♥ ♠t♦

q♥ ♥tt♦♥

♦ t♦♥

♠tt♦♥s ♥ ♥r♥t st♥t♦♥

trt♣

♦♥s♦♥s

♦ Prt ②strss rr♥t ♦♥tr♦

♥tr♦t♦♥

Pr♦♣♦s ♦♥tr♦ s♠

♣ ♦♦♣

♣rt♥ ♥

P ♦

❱♦t ♦♦♣

♠♣♠♥tt♦♥ ♥ sts

♦♥s♦♥s

♦♥s♦♥s

tr ♦rs

♦r♣②

♥♦♠♥ts

Pr

s Pr srs t ♠♦tt♦♥ ♥ t ♠♥ ♦♥trt♦♥s ♦ t tss ♦♥t♥ts ♦ ♣tr ♦ t tss r r② s♠♠r③ ♥② st ♦ ♣t♦♥s♦ t t♦r s r♣♦rt

r♦♥

♥ r♥t s t s ♦ ♣♦r ♦♥rtrs s ♦♠ r② ♣♦♣r ♥ t ♦tr rs r ♦♥tr♦ t♥qs ♥ ♣r♦♣♦s ♦r ♣♦r ♦♥rtrs ♥r② ②r t ♦♥♦♥ rsr ♥ t ②s ♠♦r ♣♦r ♠r♦♣r♦ss♦rs t♦ ♥ ♣r♦r♠♥ s♦t♦♥s s♣t ts s♥ t ♦t♣t ♦ t ♦rrsr ♦t♥ rsts ♥ ♦♠♣① ♥ r② ♣♣ s♦t♦♥s ♦tr stst♥qs s s ♥r ♥ ②strs②s ♦♥tr♦ t ♣st ♠♦t♦♥ r stt ♠♥ ♦ ♥ rt ♥♠r ♦ ♥str ♣♣t♦♥s rs♦♥s ♦ trrs♣ ♥ ♦♥ ♦♥sr♥ t rtrsts ♦ ts ♠t♦s ♦♥ ♦♥ ss♠♣t② ♦ ♦♠♣r♥s♦♥ ♥ ♠♣♠♥tt♦♥ ♥ ♦♥ t ♦tr s♥t② ♦♦♣r♦r♠♥ ♥ r♦st♥ss t♦ ts r♥t trs s♣t tr s st ①t♥s r♦♦♠ ♦r ♠♣r♦♠♥ts t s ♥♦t ♣♥ss t♦ ♣r♦♣♦s s♦t♦♥s tt ♥ ttrt♦r ♣♦♣ ♦r♥ ♥ ♥str② ♥ t♦ ♦♠♣t t ♥ ♣♦ss② t♦ r♣ trt♦♥♠t♦s

sr② ♦♥tr♦ ♦rt♠ ♦r tr rs s t♦ s♠♣ ♥ s② ♥rst♥ ss t s t♦ st ♦r rt♠ ♣♣t♦♥s ♦st♥ss ♥rt② ②♦♥ tt ♣r♦r♠♥ t♦ r♥t s♥ t ♥tr ♦ tr♥t ♣♣t♦♥s ♦♠ ♣♣♥s ♥ t♦♠♦t

♥ ts ♣rs♣t Prt ♦♥tr♦ ♦ r♣rs♥t ♥t t♦ ♥tr♦ ♠♣r♦♠♥ts ♥ ♥s ♥ t ♦r♠♥t♦♥ ♥str ♣♣t♦♥s Prt ♦♥tr♦s ss ♦ ♦♥tr♦rs tt ss t ♠♦ ♦ t s②st♠ ♦r t ♣rt♦♥ ♦ ttr ♦r ♦ t ♦♥tr♦ rs s ♥♦r♠t♦♥ s s ② t rt♦r ♥♦rr t♦ ♦t♥ t ♦♣t♠ tt♦♥ ♦r♥ t♦ ♣r♥ ♦♣t♠③t♦♥ rtr♦♥r♣rs♥t ② ♦st ♥t♦♥ s ♦♥tr♦ tq s s ♦♥ ♦♥♣ts tt r①tr♠② s♠♣ ♥ ♥tt ♥ ss ♣♥♥ ♦♥ t t②♣ ♦ ♣rt ♦♥tr♦t ♠♣♠♥tt♦♥ ♥ s♦ s♠♣ ♥ ♣rtr ♥t ♦♥tr♦ t ♦s ♦♥sr♥ t srt ♥tr ♦ t ♣♦r ♦♥rtr ♥ rsts ♥ ♥ ①tr♠② s♠♣♠♣♠♥tt♦♥ ②♦♥ s♠♣t② ♦tr ♥ts ♥ r♦♥③ rst t

Pr

♣rt ♦♥tr♦ t s ♣♦ss t♦ ♦ t s strtr ♦t♥♥ r② sttr♥s♥t rs♣♦♥s ss ♥♦♥♥rts ♥ ♥ ♥ t ♠♦ ♦♥ t♥ ♦ ♥r③♥ t ♠♦ ♦r ♥ ♦♣rt♥ ♣♦♥t ♥ ♠♣r♦♥ t ♦♣rt♦♥ ♦t s②st♠ ♦r ♦♥t♦♥s ♥② t s ♣♦ss t♦ ♥ ♠tt♦♥s ♦ t rs♥ s♥♥ t rt♦r

♠ ♦ ts tss s t♦ st② Prt ♦♥tr♦ ♣♣ t♦ t rr♥t ♦♥tr♦♦ s②♥r♦♥♦s ♠♥s ♥②s♥ ♥ rss♥ s♦♠ ♦♣♥ rsr t♦♣s rr♥ts ♥ ♦ ♦♥tr♦ ♥ ♣rtr t♦ ♠♥ s♣ts r st ♥♠② t ♥ ♦ ♣rs ♥♦ ♦ t ♠♥ ♠♦ ♥ t ♣♦sst② t♦ r s②♥r♦♥♦s ♠♥ ♦♥ t ①♠♠ ♦rq ♣r ♠♣r t ① ❲♥♥ ♥ t ①♠♠♦rq ♣r ❱♦t ♦♣rt♦♥s

♣r♦r♠♥ r strt② rt t♦ t r② ♦ t ♠♦ s ♦r t♣rt♦♥ ♥ s ♦ ♣r♠trs ♠s♠t ♦r rt♦♥ rtr t♥ ♦tr ♠♦ ♥qs t ♣rt♦♥ ♦ t s♥ ♦rs♥♥ ♦ t ♦r ♦r♦ t r rst ♣rt ♦ ts ♦r s ♦♠♠t t♦ st② ts s♣t ♥②s♥ tts ♦ ♠s♠ts ♥ rt♦♥s ♦s♥ ♥ ♣rtr ♦♥ t tr♠♥t ts ♦r♦♥ strt♦♥ ♥♦ ♠♦r s♦t♦♥ s ♣rs♥t t♦ ♦r♦♠ t ♠tt♦♥s♥ ② ♥ ♥qt ♠♦

♥ ♦rr t♦ ② ①♣♦t t rtrsts ♦ t r ssr♥ t ♦st♣♦r ♦sss ♥ r② ♦r♥ ♦♥t♦♥ ♣r♦♣r ♦♥tr♦ ♦rt♠ s t♦ s♥ t s♦♥ ♣rt ♦ ts ♦r ♣rt rt♦r t♦ tr t ♠♦st sttrt♦r② ♣♥♥ ♦♥ t ♠♥ ♦♣rt♦♥ s ♣rs♥t ♣r♦♣♦s s ♦♠♥t♦♥ ♦ ♣rt ♦♥tr♦ ♥ ②strss ♦♥tr♦ s♥ ts ♠ s t♦ ♣ t rr♥trr♦r t♥ rt♥ ②strss ♥

s st② s rr ♦t ♦♥sr♥ Prt rr♥t ♦♥tr♦ ♦r ②♥r♦♥♦st♥ ♠♥s s ♥ ♦ ♠♥ s ♦ rt ♥trst t♦ t t tt ttrs ♣♦r ♥st② s♣r♦r rt② ♥② ♥ t s ♦st t t♦ t s♥ ♦ ♣r♠♥♥t ♠♥ts ♥ rts ♥ t r♦t♦r ss s♥ ts♥♥t r♦♥ strt♦♥ ts ♦♥tr♦ r♣rs♥ts ♥ ♥ ♣rtr ♦r ♣rt♦♥tr♦ s♠s ♥ ♦r ts rs♦♥ t s ♣rt s st②

t♥ ♦ t tss

rtr t ♦♥t♥ts ♦ t ♣tr ♦ t tss r r② sr

♣tr ♠♦ ♦ t s②♥r♦♥♦s rt♥ ♠♥ s r ♥ sr tr qt♦♥s r rst ♦t♥ ♦♥sr♥ ♦② tr ♣s♠♥ tt s t ♠♦st ♥r r♣rs♥tt♦♥ ♦r ♥ tr ♠♥ ♠♦ s ♦♣ t t ♦♥② ss♠♣t♦♥ ♦ ♥♦ ②strss ♥ ♥♦ ② rr♥ts tr tt t s ♦ ♥rt② t ♥♦ r♦♥ strt♦♥ ts sst tr ♣s s②st♠ s tr tr♥s♦r♠ ♥ t q♥t t♦ ♣s♠♥ ♥② s♣ r♣rs♥tt♦♥ ♦r t rt♥ ♠♥ s ♦t♥② ♠♣♦s♥ ♣♣r♦♣rt ♦♥t♦♥s ♦♥ t r♦t♦r rr♥ts

♣tr Prt rr♥t ♦♥tr♦ s ♣rs♥t strt♥ t ♥ ♦r ♦t ♠♥ trs tr tt t s②st♠ s sr ♦♥sr♥ t s♣

Pr

s ♦ s②♥r♦♥♦s rt♥ ♠♥ ♣♣t♦♥s ♦♠ ♦♥srt♦♥s ♦♥t ♦ ♦ t ♦st ♥t♦♥ r ♣r♦ t♦ ♦ ♠♣r♦♣r ♦♣rt♦♥s ♥♦♥tr♦♥ ♥ ♥s♦tr♦♣ ♠♥ r♠♥r ♦ t ♣tr srs ♦s ♥ ♦r ♣rt♦♥ ♦r♠r s ♣rs♥t ♥ ♥②s ♥♥t t ♦ ♣r♠trs ♠s♠t ♥ ♣rtr t ts ♦ r♦♥ strt♦♥♦♥ t ♣rt rt♦r r ♣rs♥t ♦r ♣rt♦♥ s ♥tr♦t♥ t ♠rts ♦ ts t♥q ss t s♦ ♦t t♦rst♥t♦♥ s ♥②③

♣tr ♦r ♣rt s♠s sr r♦♠ ♦t t♦r st♥t♦♥ ttr ♦rs ♥ ♦ rr♥t rt♦♥ s ♥♦t ♣t ♦r sr st♣ss ♣♥♦♠♥♦♥ s t♦ ♥ ♠♣rs rr♥t ♣rt♦♥ ♥ ts ts ♣rsst♥stts ♥ ♦ss ♦ ♦♥tr♦ ♦ ♦r ♥ ts ♣tr ♥♦ ♥tst♥t♦♥s♦t♦♥ s ♣rs♥t ♣r♦♣♦s ♠t♦ ♦s ♦♥t♥♦s ♣t t♦t♦r♥ ♥♦♥ ♦♣t♠ ♦t t♦rs

♣tr ♣rt ♦♥tr♦ s♠ ♦r t s♣ ♦♣rt♦♥ ♦ t s②♥r♦♥♦srt♥ ♠♥ s ♣rs♥t ♦rt♠ ♦♥s t♦ ♦t ♣rt ♦♥tr♦♥ ②strss ♦♥tr♦ s♥ ts ♠ s ♣♥ t rr♥t rr♦r t♥ t♦r♥♥ ♣r♦♣♦s s♠ s t♦ ♦♥tr♦ t ♠♥ ♦♥ t ①♠♠♦rq ♣r ♠♣r t ①❲♥♥ ♥ t ①♠♠ ♦rq ♣r ❱♦ttrt♦rs r♥t ♦♣rt♦♥s r ♦t♥ ♥tr♦♥ t ♦♣rt♥ ♥ ♦s ♦ ♦♣rt♥ ♣♦♥ts ♦t t♥ t P ♥ t P❱tr♥st♦♥ t♥ t r♥t ♦ s ♦t♥ ② ♠♥ ♦ ♥ ♥♥♦t ♦t♦♦♣ tt ♦s ♥♦t rqr t ♥♦ ♦ t s s♣

st ♦ ♣t♦♥s

r ♣rts ♦ ts P tss ♥ ♣rs♥t ② t t♦r r♥ s P♦rs ♥ ♥tr♥t♦♥ ♦♥r♥s ♥ ♦r♥s st ♦ ♣t♦♥s s r r♣♦rt

❼ ♦rs ù ♥ ♦♦♥♥ ♦st rr♥t ♦♥tr♦ s♦♥ Pr♦♣♦rt♦♥♥tr srrs ♦r Pr♠♥♥t ♥t ②♥r♦♥♦s ♥s r♥st♦♥s ♦♥ ♥str② ♣♣t♦♥s ♥ Prss

❼ ♦rs ù ♥ ♦♦♥♥ t♠ str♥ ♦♠♣♥st♦♥ ♦rt♠ ♦r t rr♥t ♦♥tr♦ ♦ P rs ♥r②♦♥rs♦♥ ♦♥rss ♥ ①♣♦st♦♥ ♥♥♥t t♦r

❼ ❱ ♥③♦♥ ù ♥ ♦♦♥♥ ♥ t ❱♦t ♦♥tr♦ ♦♦♣ ♦r ♣ ①❲♥♥ ♥ P ②♥r♦♥♦s ♦t♦r rs ♥r②♦♥rs♦♥ ♦♥rss ♥ ①♣♦st♦♥ ♥♥♥t t♦r

❼ ♦s♦ ù ♥ ♦♦♥♥ ♦♥ ♦r③♦♥ st♠t♦r ♦r t♣ ♥ ♦t♦r P♦st♦♥ ♦ ♥s♦rss P r ②♠♣♦s♠ ♦♥♥s♦rss ♦♥tr♦ ♦r tr rs t♥ t② ♣t♠r

Pr

❼ ù P♦t♦ ♥ ♦♦♥♥♦r Prt rr♥t ♦♥tr♦♦r ②♥ r s ♦♥ ♥ t ♣t ♦ ♠sr rr♥t rs♣♦♥ss ②♠♣♦s♠ ♦♥ Prt ♦♥tr♦ ♦ tr rs ♥ P♦r tr♦♥s P Ps♥ ③ ♣ ♣t♠r

❼ ❱ ♥③♦♥ ù ♥ ♦♦♥♥ ♥ ♦♥tr♦ strt② ♦r ♥② s♣ r♥ s②♥r♦♥♦s rt♥ ♠♦t♦r rs ♥tr♥t♦♥ tr ♥s ♥ rs ♦♥r♥ ♠ ♣♣ ♦

❼ ù ♦r♥♥ ♦♦♥♥ ♥ st♦ st ♥ ♦r ♠t♥ rt② ♦ s♥t r♦t♦r tr ♣r♦♣s♦♥ ♠♥s t s♥ ♣r♠♥♥t♠♥t s②♥r♦♥♦s ♠♥ r ♥ tr ②st♠s ♥ r♥s♣♦rtt♦♥♦ ♥♦ ♣♣ ♦ tst

❼ ♦rs ù ♥ ♦♦♥♥ r♦st rr♥t ♦♥tr♦ s♦♥ ♣r♦♣♦rt♦♥♥tr ♦srrs ♦r ♣r♠♥♥t ♠♥t s②♥r♦♥♦s ♠♥s ♥r② ♦♥rs♦♥ ♦♥rss ♥ ①♣♦st♦♥ ❲ ♣♣ ♦

❼ ù ♦r♥♥ ♦♦♥♥ ♥ st♦ ♦ Prt ②strss rr♥t ♦♥tr♦ ♦r s♣ ♦♣rt♦♥ ♦ ②♥r♦♥♦s t♥ ♠♥ r ♥ ♥♥ ♦♥r♥ ♦ t ♥str tr♦♥s ♦t② ♦r♥ ♣♣ ♦

❼ Pr♥t ù ♥ r ♥ ♦♥r②s♥s ♦♣♥ t♥q ♦r ss♥s♥ ♥trt ♠trs ②♠♣♦s♠ ♦♥ ♥s♦rss ♦♥tr♦ ♦r tr rs ♣♣ ♦

❼ ♦r♥♥ ù ♦♦♥♥ ♥ ♥ ♥ ♥trt trtrtr♥t♦r s ♦♥ ♥s♦rss ②♥r♦♥♦s t♥ ♥ r ❱ P♦r ♥ Pr♦♣s♦♥ ♦♥r♥ ❱PP ♦♥tr ♣♣ ♦ ❱PP

❼ ù ♦r♥♥ ♦♦♥♥ ♥ st♦ ♥ P ♦t♦r rrss s P ♦t♦r r ♦r ① st ♥ ♦ ♥t ♦t♦r Pr♦♣s♦♥ ♥s ❱ P♦r ♥ Pr♦♣s♦♥ ♦♥r♥ ❱PP♦♥tr ♣♣ ♦ ❱PP

❼ ♦r♥♥ ù ♦♦♥♥ ♥ st♦ tst ♥ ♦r ②r♣r♦♣s♦♥ tr♥ rsr ♥ ♦♣♠♥t ♥tr♥t♦♥ tr ❱ ♦♥r♥ ❱ ♦r♥ ♣♣ ♦ ❱

Chapter 1

②♥r♦♥♦s t♥ ♥

♥②s♥ ♥ ♦♣♥ ♦♥tr♦ s♠s rqrs ♣ ♥rst♥♥ ♦ t ♣♥t♣②ss s ♣tr ♥s t ♥②ss ♦ t tr ♠♦ ♦ s②♥r♦♥♦srt♥ ♠♥ ♥ t s t ♥♠♥t ss ♦ ts ♦r

♥tr♦t♦♥

♥ ts ♣tr t ♠♦ ♦ ②♥r♦♥♦s t♥ ♥ ②♥ s r♥ sr ♦r t s ♦ ♥rt② ♥ ♥rs ♠♦ ♦ s②♥r♦♥♦s ♠♥s r t rst t s ♦t♥ ② t qt♦♥s ♦ ♦② tr ♣s ♠♥♣rt♦r③ ♦r♥② ttr s t ♠♦st ♥r r♣rs♥tt♦♥ ♦r ♥ tr♠♥ s♥ t s tr ♣s ♥♥ ♦♥ ♦t t stt♦r ♥ t r♦t♦r ♦r♥t♦ ts ♠♦ t ♠♥ s sr ② ♠♥ ♦ qt♦♥s rt♥ t stt♦r ♥r♦t♦r ♦ts t t rr♥ts ♥ ①s ♠♦ s ♦♣ t t ♦♥②ss♠♣t♦♥ ♦ ♥♦ ②strss ♥ ♥♦ ② rr♥ts ss t tr qt♦♥s rrst r ♦♥sr♥ ♥r rt♦♥s♣ t♥ ① ♥ rr♥ts ♥ t♥ ②♥tr♦♥ t ②♣♦tss ♦ ♥rt② r♦♥ strt♦♥ s ♥t

tr ♣s s②st♠ s tr tr♥s♦r♠ ♥ t t♦ ♣s s②st♠ ♦t♥♥ tqt♦♥s ♦ t ♦② t♦ ♣s ♠♥ s♠ ♦♥srt♦♥s ♠ ♦r ttr ♣s ♠♥ r ①t♥ ♦r t t♦ ♣s s②st♠ strt♥ t ♥r♠♦ ♦t♥ t ♥♦ ss♠♣t♦♥s ♦♥ t ♠♥t ♦r ♦ t ♠♥ ♥t♥ ♥tr♦♥ t ②♣♦tss ♦ ♥rt②

♥② s♣ r♣rs♥tt♦♥ ♦r rt♥ ♠♥ s ♦t♥ ② ♠♣♦s♥♣♣r♦♣rt ♦♥t♦♥s ♦♥ t r♦t♦r rr♥ts ♥ t ♠♦ s ♦♣ strt♥r♦♠ t ♠♦st ♥r r♣rs♥tt♦♥ tr tt t ②♣♦tss ♦ ♥♦ r♦♥ strt♦♥ s♥tr♦ ♥ t♥ t qt♦♥s r r ♦♥sr♥ s♥s♦ strt♦♥ ♦t ♥♥ t t r♣

②♥r♦♥♦s t♥ ♥

♥r ♠♦ ♦ t ♦② tr ♣s ♠♥

isa

sa

sb

sc

ra

rb

rc

isc

isb

ira

irb

irc

m

r ♦② tr ♣s ♠♥ r♣rs♥tt♦♥

♥r ♠♥ ♦♥sr ♥ ts st♦♥ ♣rs♥ts tr ♣s ♥♥ ♦♥♦t t stt♦r ♥ t r♦t♦r s s♦♥ ♥ ♠♥ ♣♦st♦♥ ♦ t r♦t♦rϑm s ♥♥t t t ♥ t♥ ♥♥ sa ♥ ra ♥ s ♦ ♠♥s t♠t♣ ♣♦s t ♣♦ ♣r ♥♠r p > 1 t rt♦♥s♣ t♥ t tr♥ t ♠♥ ♣♦st♦♥ ♥ ①♣rss s ϑ = pϑm ss t ♠♥s♣ ♦ t r♦t♦r s stt t ωm ♥ t tr s♣ s ω = pωm trqt♦♥s rt♥ t ♦ts u t rr♥ts i ♥ t ① ♥s λ r

us = Rsis +ddtλs

ur = Rrir +ddtλr

r us = [usa usb usc]t is = [isa isb isc]

t ♥ λs = [λsa λsb λsc]t r t stt♦r

♦t rr♥t ♥ ① t♦rs rs♣t② stt♦r rsst♥ ♠tr① s ♥ sRs = Rs · I3×3 r I3×3 s t ♥tt② ♠tr① r♦t♦r q♥ts r s② ♥s♥ ssr♣t r

st ♦ s① qt♦♥s r♣rs♥ts t tr ♥ ♦ ♣s ♦t t t tr♠♥s s t s♠ ♦ t rsst ♦t r♦♣ ♥ t t♠ rt♦♥♦ t ① ♦♣ t t ♦♥sr ♥♥ tt♦r ♥ r♦t♦r qt♦♥s ♥ ♥ ♥ ♦♥ ①♣rss♦♥ s

u = Ri+d

dtλ

r u = [us ur]t i = [is ir]

t λ = [λs λr]t r t♦rs t s③ 6 × 1 ss t

rsst♥ ♠tr① R ♥ rtt♥ s ♦ ♠tr①

R =

[

Rs 03×3

03×3 Rr

]

r 03×3 s t ♥ ♠tr①

♥ t ♠♦ s ♦♣ ♥r t ss♠♣t♦♥ ♦ ♥♦ ②strss ♥ ② rr♥ts t rt♦♥s♣ tt ♥s t rr♥t ♥ t ① ♥ s ♥♦ ♥

♥r ♠♦ ♦ t ♦② tr ♣s ♠♥

♣rtr

λ = λ(is, ir, ϑ) =

[

λs(is, ir, ϑ)

λr(is, ir, ϑ)

]

♥

i = i(λs,λr, ϑ) =

[

is(λs,λr, ϑ)

ir(λs,λr, ϑ)

]

♦r rt♥ ♣♦st♦♥ ϑ t rt♦♥s♣ t♥ t ① ♥ t rr♥t ♥ r♣rs♥t ② t ♠♥t③t♦♥ r qtt② r♣♦rt ♥

0.5

Current

Flu

x

wm

wm'

r ♥t③t♦♥ r

① t ♣♦st♦♥ ϑ t ♠♥t ♥r② wm ♥ ♦♥r② w′

m r ♥ s

wm(λ, ϑ) =

∫ λ

0i(λ, ϑ)dλ

w′

m(i, ϑ) =

∫ i

0λ(i, ϑ)di

♦♠tr ♠♥♥ ♦ ts t♦ q♥tts s r♣rs♥t ♥ ♥ trt♦♥s♣ t♥ t♠ s

itλ = wm + w

′

m

♥ t ① ♥ λ s ♥t♦♥ ♦ t rr♥ts i ♥ t ♣♦st♦♥ ϑ λ = λ(i, ϑ) t tr qt♦♥ ♥ rrtt♥ s♣tt♥ t ① rt♦♥sr♥ t ♣rt rts t rs♣t t♦ t rr♥t ♥ t ♣♦st♦♥ t rsts

u = Ri+ ℓd

dti+

∂

∂ϑλd

dtϑ

r ℓ s t 6× 6 r♥t ♥t♥ ♠tr① ttr r♣rs♥ts t s♦♣♦ t ♠♥t③t♦♥ r t t ① rr♥t ♥ t ♥ ①♣t② rtt♥ s

ℓ = ℓ(i, ϑ) =

∂λsa

∂isa. . . ∂λsa

∂irc

∂λrc

∂isa. . . ∂λrc

∂irc

②♥r♦♥♦s t♥ ♥

♥r② qt♦♥s

♥r② ♥ ♥ ♦t♥ ② ♠t♣②♥ ② idt t ♥ ①♣rss s

itudt︸ ︷︷ ︸

dwin

= itRidt︸ ︷︷ ︸

dwJ

+ itdλ︸︷︷︸

dwm+dwem

r dwin s t ♥♣t tr ♥r② dwJ s t ♦ ♦ss ♦♠♣♦♥♥t dwm s trt♦♥ ♦ t ♠♥t ♥r② st♦r ♥ t s②st♠ ♥ dwem s ♥r② ♦♥rtr♦♠ tr t♦ ♠♥ ❯♥r t ss♠♣t♦♥ ♦ ♥♦ ♠♥ ♦sss t ttr♦rrs♣♦♥s t♦ t ♦t♣t ♠♥ ♦r t♦ ttr tr♠s ♦ ♥ s② ♥t strt♥ r♦♠ ♥ ♠t♣②♥ t ② idt s s♦♥ ♥

itudt︸ ︷︷ ︸

dwin

= itRidt︸ ︷︷ ︸

dwJ

+ itℓdi︸︷︷︸

dwm

+ it∂λ

∂ϑdϑ

︸ ︷︷ ︸

dwem

♦♠♣r♥ ♥ ♥ ♠♣♦s♥ dwem = τdϑm t s ♣♦ss t♦ rt

itdλ = dwm + τdϑm

r τ s t ♥st♥t♥♦s t♦rq

♥ t ♠♥t ♥r② s ♥t♦♥ ♦ t stt♦r ♥ r♦t♦r rr♥t ②♦♥ ttt ♣♦st♦♥ t ♥♥ts♠ rt♦♥ dwm ♥ ①♣rss s

dwm =∂wm

∂λdλ+

∂wm

∂ϑdϑ =

∂wm

∂λsadλsa + · · ·+

∂wm

∂λradλra + · · ·+

∂wm

∂ϑdϑ

♦♠♥♥ ♥ t s ♣♦ss t♦ rt

itdλ− τdϑm =

∂wm

∂λdλ+

∂wm

∂ϑdϑ

♥ t stt rs r ♥♣♥♥t q♥tts ♥ s♣rt ♥ ts t♦♦♠♣♦♥♥t ♥ ♥t s

i(λ, ϑ) =∂wm

∂λ

τ(λ, ϑ) = −p∂wm

∂ϑ

r t ♦r♠r r♣rs♥ts t ♥♥ rr♥t ♥ t ttr t t♦rq ①♣rss♦♥s ♥t♦♥ ♦ t ♠♥t ♥r②

t♦rq ♥ s♦ ①♣rss s ♥t♦♥ ♦ t rr♥t i ♥ t ♣♦st♦♥ ϑs ♥ ♦♥ ② ♦♥sr♥ t ♠♥t ♦♥r② ♥ ♥ ♣rtr

w′

m(i, ϑ) = itλ− wm(λ, ϑ)

♥ ①♣rss♦♥ ♦r t t♦rq ♥ ♦t♥ ② r♥tt♥ ♥♦♠♥♥ t t rst s r♣♦rt ♥

τ(i, ϑ) = p∂w

′

m

∂ϑ

♥r ♠♦ ♦ t ♦② tr ♣s ♠♥

qt♦♥ t♦t strt♦♥

0 1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

1

Current [A]F

lux

[Vs]

w/i iron saturationw/o iron saturation

r ♥t③t♦♥ r ♥ s ♦ ♥rt②

♠♦ ♦♣ ♥ t♦♥ ♦♥srs ♥r ♦② tr ♣s♠♥ qt♦♥s r ♥♦ s♣ ♦r ♥r ♠♥ t r♦♥ strt♦♥s ♥t ♠♥t③t♦♥ r s qtt② r♣♦rt ♥ ♥ ts st ① ♥ ①♣rss s

λ(i, ϑ) = L(ϑ)i

r L s t 6 × 6 ♥t♥ ♠tr① tt ♣♥s ♦♥ t ♣♦st♦♥ ♥ ♥♦t ♦♥ trr♥ts t ♥ ①♣rss s

L =

[

Ls(ϑ) Msr(ϑ)

Mrs(ϑ) Lr(ϑ)

]

♥ ts s♠trs r

Ls(ϑ) =

Ls,a(ϑ) Ms,ab(ϑ) Ms,ac(ϑ)

Ms,ba(ϑ) Ls,b(ϑ) Ms,bc(ϑ)

Ms,ca(ϑ) Ms,cb(ϑ) Ls,c(ϑ)

Msr(ϑ) = Mtrs(ϑ) =

Msr,aa(ϑ) Msr,ab(ϑ) Msr,ac(ϑ)

Msr,ba(ϑ) Msr,bb(ϑ) Msr,bc(ϑ)

Msr,ca(ϑ) Msr,cb(ϑ) Msr,cc(ϑ)

Lr(ϑ) =

Lr,a(ϑ) Mr,ab(ϑ) Mr,ac(ϑ)

Mr,ba(ϑ) Lr,b(ϑ) Mr,bc(ϑ)

Mr,ca(ϑ) Mr,cb(ϑ) Lr,c(ϑ)

r s ♥t♥s r stt t L(ϑ) M(ϑ) s s ♦r t ♠t ♥t♥ t♥ t♦ ♥♥s

t s ♦rt t♥ tt Ls(ϑ) Lr(ϑ) ♥ Msr(ϑ) r r♥t ♠tr① ♥ r♦ t♦r s st ♦♥ ♠♥t t♦ t rt t rs♣t t t ♣r♥t r♦t♦r s ♥ ①♠♣

❼ Ms,ab(ϑ) Ms,ba(ϑ)

❼ Ls,b(ϑ) = Ls,a(ϑ− 2/3π) ♥ Ls,c(ϑ) = Ls,a(ϑ− 4/3π)

②♥r♦♥♦s t♥ ♥

stt♦r ♥ r♦t♦r ① ♥ ♥ ①♣rss s

λs = Lsis +Msrir

λr = Mrsis + Lrir

♥ s ♦ ♥rt② t ♠♥t ♥r② ♥ ♦♥r② r q ♥ t♦♥ ♦♥sr♥ wm = 1/2itλ t ♣♦ss t♦ rt

τ(i, ϑ) = p∂

∂ϑw

′

m(i, ϑ) = p∂

∂ϑwm(λ, ϑ)

=1

2pit

∂

∂ϑλ =

1

2pit

d

dϑLi

t♦rq ♥ s♣tt ♥

τ(i, ϑ) =1

2pits

[d

dϑLs(ϑ)

]

is

︸ ︷︷ ︸

τr

+ pits

[d

dϑMsr(ϑ)

]

ir

︸ ︷︷ ︸

τed

+1

2pitr

[d

dϑLrϑ

]

ir

︸ ︷︷ ︸

τc

r τr s t rt♥ t♦rq τed t tr♦②♥♠ t♦rq ♥ τc s t ♦♥t♦rq

rt♥ t♦rq τr s rt t♦ t rt♦♥s ♦ t stt♦r ♥t♥ ♠tr①t t r♦t♦r ♣♦st♦♥ tt ♦rs r② t♠ t r♦t♦r s ♥ ♥s♦tr♦♣ strtr t♦ ♣♦ s♥s ♦r ① rrrs t ♥ s♥ s

τr =1

2p∂λt

s(is, 0, ϑ)

∂ϑis

=1

2pits

∂

∂ϑλs(is, 0, ϑ)

= p∂

∂ϑwm(is, 0, ϑ)

r λs(is, 0, ϑ) s t stt♦r ① ♥ t♦ t stt♦r rr♥t ♦♥② ir = 0

tr♦②♥♠ t♦rq s rt t♦ t rt♦♥ ♦ t ♠t ♥t♥st♥ t stt♦r ♥ t r♦t♦r ♥♥s t t r♦t♦r ♣♦st♦♥ ♠♥ rs♦♥♦r ts rt♦♥ s t ♠♦♠♥t ♦ t r♦t♦r t rs♣t t♦ t stt♦r t ♥ ①♣rss s

τed(is, ir, ϑ) = pits∂Msr(ϑ)

∂ϑir

= p∂λt

sr(0, ir, ϑ)

∂ϑis = pits

∂

∂ϑλtsr(0, ir, ϑ)

= p∂λt

rs(is, 0, ϑ)

∂ϑir = pitr

∂

∂ϑλrs(is, 0, ϑ)

r λsr(0, ir, ϑ) s t stt♦r ① ♥ t♦ t r♦t♦r rr♥t ♦♥② is = 0♥ λrs(is, 0, ϑ) s t r♦t♦r ① ♥ t♦ t stt♦r rr♥t ♦♥② ir = 0

♥r ♠♦ ♦ t ♦② t♦ ♣s ♠♥

♦♥ t♦rq s t♦ stt♦r ♥s♦tr♦♣s ♠♥② s ② t s♦ts ♦♣♥♥t ♥ ①♣rsss s

τcog(ir, ϑ) =1

2pitr

∂Lr(ϑ)

∂ϑir

=1

2p∂λt

r(0, ir, ϑ)

∂ϑir

=1

2pitr

∂λtr(0, ir, ϑ)

∂ϑ

= p∂wm(0, ir, ϑ)

∂ϑ

r λr(0, ir, ϑ) s t stt♦r ① ♥ t♦ t r♦t♦r rr♥t ♦♥② is = 0

♥r ♠♦ ♦ t ♦② t♦ ♣s ♠♥

iryirz

ry

isy

isz

sy

rz

sz

r ♦② t♦ ♣s ♠♥ r♣rs♥tt♦♥

tr♣s ♥♥ ♦♥ t stt♦r ♥ ♦♥ t r♦t♦r ♥ st ♦♥sr♥♥ q♥t t♦ ♣s s②st♠ yz r♣rs♥t ♥ r ♦t ♥♥s rtr♥s♦r♠ ♥ t q♥t t♦ ♣s s②st♠

❼ tr♥s♦r♠t♦♥ s ①♣rss ♦r ♥r tr♣s ♥♥ abc r♦tt♥t ♥ ♥r s♣ ωabc ♣♦st♦♥ ♦ t ♣s a t rs♣t t♦ tsss ①s s ϑabc

❼ t♦♣s rr♥ r♠ yz s r♦tt♥ t ♥ ♥r s♣ ♦ ωyz ♣♦st♦♥ ♦ t ♣s y t rs♣t t♦ t sss ①s s ϑyz

❼ t ♦♠♦♣♦r ♦♠♣♦♥♥t s ss♠ t♦ ③r♦

rt♦♥s♣ t♥ t tr♣s abc ♥ t t♦♣s yz s②st♠s ♥ ①♣rss s

fyz = Tfabc =⇒ fabc = T−1

fyz

r f s s t♦ r♣rs♥t ♥r q♥tt② t ♦t t rr♥t ♥ t① ssr♣ts abc ♥ yz rr t♦ t tr♣s ♥ t t♦♣s s②st♠

②♥r♦♥♦s t♥ ♥

rs♣t② T s t tr♥s♦r♠t♦♥ ♠tr① t s③ 2 × 3 ♥ T−1 s t ♥rs

♠tr① ② ♥ ①♣rss s

T(∆ϑ) =

[

cos[∆ϑ] cos[∆ϑ− 2/3π] cos[∆ϑ+ 2/3π]

−sin[∆ϑ] −sin[∆ϑ− 2/3π] −sin[∆ϑ+ 2/3π]

]

T−1(∆ϑ) =

cos[∆ϑ] −sin[∆ϑ]

cos[∆ϑ− 2/3π] −sin[∆ϑ− 2/3π]

cos[∆ϑ+ 2/3π] −sin[∆ϑ+ 2/3π]

r ∆ϑ = ϑyz −ϑabc ♥ t ♦♦♥ t ♣♥♥② ♦♥ ∆ϑ s ♦♠tt t s ♦rtt♥ tt t ♥rs ♠tr① ♥ ♦♥ ② r♥tr♦♥ t tr r♦ ♥ t♠tr① T ♦♥sr♥ t ♦♠♦♣♦r ♦♠♣♦♥♥t ♥ ts ② t trs♦r♠t♦♥♠tr① s sqr 3 × 3 ♠tr① ♥ ts ♥rs ♥ rtt♥ s ♥ r t♦♠♦♣♦r ♦♠♣♦♥♥t s ♥ ♦♠tt tr t ♥rs♦♥

trt♥ r♦♠ t qt♦♥ ♦ t tr♣s ♥♥ ♥ s♥ t ♣♦sst♦ rt

uabc = Riabc +d

dtλabc

T−1

uyz = RT−1

iyz +d

dt

(T

−1λyz

)

T−1

uyz = RT−1

iyz +T−1 d

dtλyz +

d

dt

(T

−1)λyz

rt ♦ t tr♥s♦r♠t♦♥ ♠tr① s ♦t♥ ② ♥ ♠♥t②♠♥trt♦♥

d

dt

(T

−1)= (ωyz − ωabc)

−sin[∆ϑ] −cos[∆ϑ]

−sin[∆ϑ− 2/3π] −cos[∆ϑ− 2/3π]

−sin[∆ϑ+ 2/3π] −cos[∆ϑ+ 2/3π]

= (ωyz − ωabc)T−1

J

r

J =

[

0 −1−1 0

]

♦♠♥♥ ♥ t s ♣♦ss t♦ rt

T−1

uyz = RT−1

iyz +T−1 d

dtλyz +T

−1Jλyz(ωyz − ωabc)

uyz = Riyz +d

dtλyz + Jλyz(ωyz − ωabc)

t s ♦rt ♠♥t♦♥♥ tt s ♦r ♦t t stt♦r ♥ t r♦t♦r ②♥ r♥tt ② ♠♣♦s♥ t s♣ ♦ t t♦ s②st♠s ♥♠② ωabc = 0 ♥ωabc = ω ♦r t stt♦r ♥ t r♦t♦r rs♣t②

♥ t ♥②ss ♦ t s②♥r♦♥♦s ♠♥ t♦ ♣rtr tr♥s♦r♠t♦♥s r ①t♥s② s ♥♠② t r ♥ t Pr tr♥s♦r♠t♦♥s rst s ♦t♥

♥r ♠♦ ♦ t ♦② t♦ ♣s ♠♥

♦♥sr♥ t♦♣s stt♦♥r② ♥♥ ♥♠ αβ t ωyz = 0 ♥ ts s t①s r♣rs♥t♥ t ♦ α s ♥ t t ♣s a tr♥s♦r♠t♦♥ ♠tr① ♥ rrtt♥ ♦♥sr♥ ϑyz = ϑabc ♥ ωyz = ωabc = 0 ♥ t qt♦♥s r

us,αβ = Ris,αβ + ddtλs,αβ

ur,αβ = Rir,αβ + ddtλr,αβ − ωJλr,αβ

s♦♥ s tr♥s♦r♠t♦♥ s ♦t♥ ♦♥sr♥ t♦♣s ♥♥ ♥♠dq s②♥r♦♥♦s② r♦tt♥ t t r♦t♦r ♥ ts s t tr♥s♦r♠t♦♥ ♠tr① ♥ ♦t♥ t ωyz = ω ♥ t qt♦♥s r

us,dq = Ris,dq +ddtλs,dq + ωJλs,dq

ur,dq = Rir,dq +ddtλr,dq

♦ r ♠♦r ♥r ①♣rss♦♥ tt ♥s ♦t t stt♦r ♥ t r♦t♦rqt♦♥s t tr♥s♦r♠t♦♥ ♠tr① P t s③ 4× 6 s ♥ s

P =

[

T(ϑyz) 02×3

02×3 T(ϑyz − ϑ)

]

♥ ts ♥rs

P−1 =

[

T−1(ϑyz) 03×2

03×2 T−1(ϑyz − ϑ)

]

tt r ♦t♥ ♦♥sr♥ t ♣♦st♦♥ ♦ t stt♦r ♥♥ ϑabc = 0 s r♣rs♥t♥ ♥rs♦♥ ♦ P ♥ ♦♥ ② r♥tr♦♥ t ♦♠♦♣♦r ♦♠♣♦♥♥t ♥ ts ② t ♠tr① s sqr t s③ 6 × 6 tr t ♥rs♦♥ t♦♠♦♣♦r ♦♠♣♦♥♥t s ♦♠tt ♥ ♥ ts ② ♥ rrtt♥ s

uyz = Riyz +d

dtλyz +Gλyz

r

G =

0 −ωyz 0 0

ωyz 0 0 0

0 0 0 −(ωyz − ω)

0 0 ωyz − ω 0

♥r② qt♦♥s

♥r② ♥ ♦r t♦ ♣s ♠♥ ♥ ♦t♥ ♠t♣②♥ ②iyzdt

ityzuyzdt︸ ︷︷ ︸

dwin

= ityzRiyzdt︸ ︷︷ ︸

dwJ

+ ityzdλyz + i

tyzGλyzdt

︸ ︷︷ ︸

dwm+dwem

rt♦♥ ♦ t st♦r ♥r② ♥ t ♠♥ ♥r② ♦t♣t ♥ rtt♥ s

3

2ityzdλyz + i

tyzGλyzdt = dwm + τdϑm

②♥r♦♥♦s t♥ ♥

tr♦♠♥t t♦rq τ ♥ ①♣rss strt♥ r♦♠ t ♥r② ♥ s

τ =1

ωm[ωyz(λsyisz − λszisy) + (ωyz − ω)(λryirz − λrziry)]+pityz

∂λyz

∂ϑ−p d

dϑwm

r ωm s t ♠♥ s♣ ♦ t r♦t♦r ♥ s ωm = ω/p

ss ♦♥sr♥ t rt♦♥s♣ t♥ ♠♥t ♥r② ♥ ♦♥r② wm +w

′

m = itλ t qt♦♥ ♦♠s

τ =1

ωm[ωyz(λsyisz − λszisy) + (ωyz − ω)(λryirz − λrziry)] + p

∂

∂ϑw

′

m

♦ t♦t strt♦♥

s♠ ♦♥srt♦♥s ♣rs♥t ♥ t♦♥ ♦r t ♥strt tr ♣s ♠♥ ♥ ♦♥ ♦r t t♦ ♣s q♥t ♠♥ ♥ ♣rtr t rt♦♥s♣ ♥♥ t ① ♥ t rr♥t s st ♥ t t♦ ♣s s②st♠ ♥ ♦t♥ ② ♠♥ ♦ t tr♥s♦r♠t♦♥ ♠tr① P s ♦♦s

λ(i, ϑ) = L(ϑ)i

P−1

λyz(i, ϑ) = L(ϑ)P−1iyz

λyz(i, ϑ) = PL(ϑ)P−1

︸ ︷︷ ︸

Lyz(ϑ)

iyz

♥ t ♥t♥ ♠tr① ♥ ①♣rss s

Lyz = PL(ϑ)P−1

=

[

T(ϑyz) 02×3

02×3 T(ϑyz − ϑ)

][

Ls(ϑ) Msr(ϑ)

Mrs(ϑ) Lr(ϑ)

][

T−1(ϑyz) 03×2

03×2 T−1(ϑyz − ϑ)

]

=

[

T(ϑyz)Ls(ϑ)T−1(ϑyz) T(ϑyz)Msr(ϑ)T

−1(ϑyz − ϑ)

T(ϑyz − ϑ)Mrs(ϑ)T−1(ϑyz) T(ϑyz − ϑ)Lr(ϑ)T

−1(ϑyz − ϑ)

]

=

[

Ls,yz Msr,yz

Mrs,yz Lr,yz

]

rLs,yz = T(ϑyz)Ls(ϑ)T

−1(ϑyz)

Msr,yz = Mtrs,yz = T(ϑyz)Msr(ϑ)T

−1(ϑyz − ϑ)

= T(ϑyz − ϑ)Mrs(ϑ)T−1(ϑyz)

Lr,yz = T(ϑyz − ϑ)Lr(ϑ)T−1(ϑyz − ϑ)

t s ♦rt ♠♥t♦♥♥ tt ♥ r t♦t② ♥r ♥ r ♦rr② ♦ ♦ t rr♥ r♠ ♠tr①s ♥ r ♥♦t ①♣t② ①♣rsss♥ tr ♦♠♣t ♦r♠ ♦s ♥♦t ♣r♦ s ♥♦r♠t♦♥s ♣rs ①♣rss♦♥ ♥ ♦♥sr♥ t r ♥ Pr tr♥s♦r♠t♦♥s

♥r ♠♦ ♦ t ♦② t♦ ♣s ♠♥

t r tr♥s♦r♠t♦♥ s ♣r♦r♠ ωyz = 0 t q♥t ♥t♥♠tr① s

Lαβ =

[

Ls,αβ Msr,αβ

Mrs,αβ Lr,αβ

]

r

Ls,αβ =

[

Ls,α Ls,αβ

Ls,βα Ls,β

]

Msr,αβ = Mrs,αβ =

[

Msr,αα Msr,αβ

Msr,βα Msr,ββ

]

Lr,αβ =

[

Lr,α Lr,αβ

Lr,βα Lr,β

]

♥

Ls,α =1

6(4Ls,a + Ls,b + Ls,c − 4Ms,ab − 4Ms,ac + 2Ms,bc)

Ls,αβ = Ls,βα =−Ls,b + Ls,c + 2Ms,ab − 2Ms,ac

2√3

Ls,β =1

2(Ls,b + Ls,c − 2Ms,bc)

Msr,αα =1

6(4Msr,aa +Msr,bb +Msr,cc − 4Msr,ab − 4Msr,ac + 2Msr,bc)

Msr,αβ = Msr,βα =−Msr,bb +Msr,cc + 2Msr,ab − 2Msr,ac

2√3

Msr,ββ =1

2(Msr,bb +Msr,cc − 2Msr,bc)

Lr,α =1

6(4Lr,a + Lr,b + Lr,c − 4Mr,ab − 4Mr,ac + 2Mr,bc)

Lr,αβ = Lr,βα =−Lr,b + Lr,c + 2Mr,ab − 2Mr,ac

2√3

Lr,β =1

2(Lr,b + Lr,c − 2Mr,bc)

♥ ts rr♥ r♠ t t♦rq ♥ rrtt♥ s

τ(iαβ , ϑ) =1

2pits,αβ

[d

dϑLs,αβ(ϑ)

]

is,αβ

︸ ︷︷ ︸

τr

+ pits,αβ

[d

dϑMsr,αβ(ϑ)

]

ir,αβ

︸ ︷︷ ︸

τed

+1

2pitr,αβ

[d

d(ϑ)Lr,αβϑ

]

ir,αβ

︸ ︷︷ ︸

τc

r♥t② t Pr tr♥s♦r♠t♦♥ s ♣r♦r♠ t ♥t♥ ♠tr① ♥ rtt♥ s

Ldq =

[

Ls,dq Msr,dq

Mrs,dq Lr,dq

]

②♥r♦♥♦s t♥ ♥

trt♥ r♦♠ ♥ ♦♥sr♥ ωyz = ω t t♦rq ♥ rrtt♥ s

τ =1

ωm[ω(λsdisq − λsqisd)] + p

∂

∂ϑw

′

m

τ = p(λsdisq − λsqisd) + p∂

∂ϑw

′

m

❯♥r t ss♠♣t♦♥ ♦ ♥rt② t ♥r② ♥ t ♦♥r② r q ♥

∂

∂ϑw

′

m =∂

∂ϑwm =

1

2it ∂

∂ϑλ =

1

2it ∂

∂ϑLi

♦♥r② ♥ rtt♥ s

∂

∂ϑw

′

m =1

2its,dq

d

dϑLs,dqis,dq + i

ts,dq

d

dϑMsr,dqir,dq +

1

2ir,dq

d

dtLr,dqir,dq

Ptt♥ t♦tr ♥ t s ♣♦ss t♦ ♦t♥

τ = p(λsdisq − λsqisd)

+1

2pits,dq

d

dϑLs,dqis,dq + pits,dq

d

dϑMsr,dqir,dq +

1

2pir,dq

d

dtLr,dqir,dq

②♥r♦♥♦s t♥ ♥

♠♦ t♦ sr rt♥ ♠♥ s ♦t♥ stt♥ t ①tt♦♥ rr♥ts♦ t r♦t♦r q t♦ ③r♦ t ♦♦s tt ♥ t tr ♣s s②st♠ t rr♥ts r

ir =[

0 0 0]t

♠r② ♥ t q♥t t♦ ♣s ♠♥

ir,yz =[

0 0]t

♦s② t tr qt♦♥ ♥ s ♦ ②♥ ♥ ♥ ♥r ♦r r②♠♥ t♦t r♦t♦r ♥♥s ♦♥② ♥s t stt♦r q♥tts

us = Rsis +d

dtλs

♥ ♦♥② ♦♥ ♥♥ s ♣rs♥t t ♦♥ ♦♥ t stt♦r t ssr♣t s s ♦♠tt♥ t♦r ♥♦tt♦♥ rtr

s♠ rt♦♥s♣ t♥ ① ♥ rr♥t ♥ ①♣rss s ♥ t s ♦♦② tr ♣s ♠♥

λ = λ(i, ϑ)

♥i = i(λ, ϑ)

②♥r♦♥♦s t♥ ♥

♦♠♥♥ t tr qt♦♥ t t rt♦♥s♣ ♥♥ rr♥t ♥ ① ts♠ ①♣rss♦♥ ♥ ♦t♥ ♦♥② r♥ s tt ♥ s ♦ ②♥t r♥t ♥t♥ ♠tr① s s③ 3× 3 ♥ ♥ ①♣rss s

ℓ = ℓ(i, ϑ) =

∂λsa

∂isa. . . ∂λsa

∂isc

∂λsc

∂isa. . . ∂λsc

∂isc

♥r② qt♦♥s

rsts ♦t♥ t t s ♦ ♦② tr ♣s ♠♥ ♥ s②①t♥ t♦ t s ♦ ②♥ ♥r② ♥ s ♦t♥ ♠t♣②♥ t trqt♦♥ ② idt rst s

♥ t s♠ ② t s ♥ ♦♥ ♥ t s ♣♦ss t♦ rt

itdλ = dwm + τdϑm

r♥t② s t♦ ♠♦ s♥ t ♠♥t ♥r② s ♥t♦♥ ♦ t♣♦st♦♥ ♥ t stt♦r rr♥t ♦♥②

dwm =∂wm

∂λdλ+

∂wm

∂ϑdϑ =

∂wm

∂λsadλsa + · · ·+

∂wm

∂λscdλsc + · · ·+

∂wm

∂ϑdϑ

♦♠♥♥ ♥ t s ♣♦ss t♦ rt

itdλ− τdϑm =

∂wm

∂λdλ+

∂wm

∂ϑdϑ

tt ♥ s♣rt ♥ ts t♦ ♦♠♣♦♥♥t

i(λ, ϑ) =∂wm

∂λ

τ(λ, ϑ) = −p∂wm

∂ϑ

r t ♦r♠r r♣rs♥ts t ♥♥ rr♥t ♥ t ttr t t♦rq ①♣rss♦♥

t♦rq ♥ s♦ ①♣rss s ♥t♦♥ ♦ t rr♥t i ♥ t ♣♦st♦♥ ϑ♦t♥♥

♦ t♦t strt♦♥

♥ ts st♦♥ t ♠♦ ♦ ②♥ s ♦♣ ♦♥sr♥ ♥r ♠♥t rs ♥ rsts r ♦t♥ s ♥ ①t♥s♦♥ ♦ t ♦♥srt♦♥s ♠ ♦rt ♦② tr ♣s ♠♥

♥ s ♦ ♥♦ strt♦♥ t ① ♥ ①♣rss s r L s t 3 × 3♥t♥ ♠tr① tt ♥ rtt♥ s

λ(i, ϑ) = L(ϑ)i

②♥r♦♥♦s t♥ ♥

r L s t 3× 3 ♥t♥ ♠tr① tt ♥ rtt♥ s

L =

Ls,a(ϑ) Ms,ab(ϑ) Ms,ac(ϑ)

Ms,ba(ϑ) Ls,b(ϑ) Ms,bc(ϑ)

Ms,ca(ϑ) Ms,cb(ϑ) Ls,c(ϑ)

♥ s ♦ ♥rt② t ♠♥t ♥r② ♥ ♦♥r② r q t♦rq ♥ ♥ rtt♥ s ♥ t s ♦rt t♥ tt s♥♥t r♥ ♥ ♦♥ t♥ t ♥r ♦② ♠♥ ♥ t ②♥ ♥ s♣tt♥ tt♦rq ♥ ts ♦♠♣♦♥♥ts ♥ t s ♥ s♦♥ tt t t♦rq ♦ ♦② tr ♣s ♠♥ ♥s t rt♥ t tr♦②♥♠ ♥ t ♦♥t♦rq ♥ t ♦♥trr② t t♦rq ♣r♦ ② t ②♥ s ♦♥② ♦♥ tr♠ ♥♠②t rt♥ t♦rq tt ♥ ①♣rss s

τ(i, ϑ) =1

2pit[d

dϑL(ϑ)

]

i

︸ ︷︷ ︸

τr

tr♦②♥♠ t♦rq s ♣r♦ ② t rt♦♥ ♦ t ♠t ♥t♥st♥ t stt♦r ♥ t r♦t♦r ♥♥s s tr♠ s ♣rs♥t ♥♦t ♦♥② ♥ ♠♥st r♦t♦r ♥♥ t ♥ ♣r♠♥♥t ♠♥t ♠♥s s ♥ ts s t t♦rqs ♣r♦ ② t ♥trt♦♥ ♦ t ♠♥t ① ♥ t t stt♦r ♥♥s s♦♠♣♦♥♥t ♦ t♦rq s ♥ ♥ s ♦ ②♥ s♥ t r♦t♦r s ♥♦ ♣r♠♥♥t ♠♥t♥tr ♥♥s

♦♥ t♦rq s s ② t ♥trt♦♥ t♥ t r♦t♦r ① ♥ ♥t stt♦r ♥s♦tr♦♣s ♠♥② s ② s♦ts ♦♣♥♥ s tr♠ s ♣rs♥t ♥ ♦t♦② ♥ ♣r♠♥♥t ♠♥t ♠♥s ♥ t ♦♥trr② ②♥ ♦ ♥♦t tst♦rq ♦♠♣♦♥♥t

♦♥② ♦♠♣♦♥♥t ♥ ♥ t t♦rq ①♣rss♦♥ ♦ t ②♥ s trt♥ t♦rq ttr s rt t♦ t rt♦♥ ♦ t ♥t♥ ♠tr① tt r♦t♦r ♣♦st♦♥ ♥ t s ♣rs♥t ♥ r② ♠♥ ♣rs♥t♥ ♥ ♥s♦tr♦♣ r♦t♦r♥ ② ♣♦ s♥s ♦r ① rrrs t ♥ ①♣rss s

♥ t s♥s♦② strt ♥♥s

♥ ♠♥s t stt♦r ♥♥ s s♥ t♦ ♥♥ t ♥♠♥t ♦♠♣♦♥♥t♦ t r♣ ① strt♦♥ ② t ♥♥ s s♥s♦ strt♦♥r♦♥ t r♣ t rst♥ ① ♥ s s♥s♦ t t r♦t♦r ♣♦st♦♥ ♥ tsst♦♥ t qt♦♥s ♦ t ②♥ r ♥②③ ♥r t ss♠♣t♦♥ ♦ s♥s♦strt♦♥ ♦ t stt♦r ♥♥s

❯♥r t ss♠♣t♦♥ ♦ strt ♥♥s t 3 × 3 ♥t♥ ♠tr① ♦ ttr ♣s ♠♥ ♥ ①♣rss s

L =2

3

LΣ + L∆♦s2ϑ −LΣ

2 + L∆♦s2ϑ2 −LΣ

2 + L∆♦s2ϑ1

−LΣ

2 + L∆♦s2ϑ2 LΣ + L∆♦s2ϑ1 −LΣ

2 + L∆♦s2ϑ

−LΣ

2 + L∆♦s2ϑ1 −LΣ

2 + L∆♦s2ϑ LΣ + L∆♦s2ϑ2

②♥r♦♥♦s t♥ ♥

r ϑ1 = ϑ− 2/3π ♥ ϑ2 = ϑ− 4/3π LΣ = (Ld + Lq)/2 ♥ L∆ = (Ld − Lq)/2

rr♥t ♥ ss♠ ② t ♥r ①♣rss♦♥

ia = IM cos(ϑ+π

2− γ)

ib = IM cos(ϑ1 +π

2− γ)

ic = IM cos(ϑ2 +π

2− γ)

tr ♣s ① ♥s ♥ ♦t♥ strt♥ r♦♠ ♥ sλ = Li ♦t♥♥

λa = IM [−LΣ sin(ϑ− γ) + L∆ sin(ϑ+ γ)]

λb = IM [−LΣ sin(ϑ1 − γ) + L∆ sin(ϑ1 + γ)]

λc = IM [−LΣ sin(ϑ2 − γ) + L∆ sin(ϑ2 + γ)]

rst s ♦♥sst♥t t t ss♠♣t♦♥s tt t ① ♥s ♥ srs s♥s♦ ♥t♦♥

① qt♦♥ ♥ s♠r② r ♦r t q♥t t♦ ♣s ♠♥ ♥ ♣rtr ② ♣♣②♥ t r tr♥s♦r♠t♦♥ t♦ t s ♣♦ss t♦①♣rss t ♥t♥ ♠tr① ♥ t stt♦♥r② rr♥ r♠ ♦t♥♥

L =

[

LΣ + L∆ cos 2θ L∆ sin 2θ

L∆ sin 2θ LΣ − L∆ cos 2θ

]

s♠ tr♥s♦r♠t♦♥ ♥ ♣♣ t♦ ♦t♥♥

iα = −IM sin(θ − γ)

iβ = IM cos(θ − γ)

① ♥ ♥ ♦t♥ strt♥ r♦♠ ♥ s λ = Li ♦t♥♥

λα = IM [−Lq cos(γ) sin(θ) + Ld sin(γ) cos(θ)]

λβ = IM [Lq cos(γ) cos(θ) + Ld sin(γ) sin(θ)]

♠r② t qt♦♥ ♦ t t♦ ♣s ♠♥ ♥ r ♣♣②♥ tPr tr♥♦r♠t♦♥ ①♣rss♥ t q♥tts ♥ t rr♥ r♠ r♦tt♥ s②♥r♦♥♦s② t t r♦t♦r

♥t♥ ♠tr① ♦t♥ t ts tr♥s♦r♠t♦♥ s

Ldq =

[

Ld 0

0 Lq

]

♥ t q rr♥ r♠ t rr♥ts ♥ ①♣rss s

id = IM sin(γ)

iq = IM cos(γ)

②♥r♦♥♦s t♥ ♥

♥② r♣t♥ t s♠ t♦♥ ♦♥ ♦r t tr ♣s s②st♠ ♥ ♦r tt♦ ♣s ♠♥ ♥ t stt♦♥r② rr♥ r♠ t ① ♥ ♥ ①♣rsss t ♣r♦t t♥ ♥

λd = IMLd sin(γ)

λq = IMLq cos(γ)

t s ♦rt ♥♦t♥ tt ♥ ts s t s♥s♦ ♥♥ strt♦♥ tt r♣ t s ♥t♥s r ♦♥st♥t ♥ t ♠t ♥t♥s Ldq = Lqd r♥ ♦r ts rs♦♥ strt♥ r♦♠ t t♦rq ♥ ①♣rss s

τ = p(λdiq − λqid)

♦♥s♦♥s

♥ ts ♣tr t ♠♦ ♦ t s②♥r♦♥♦s rt♥ ♠♥ s ♥ ♦♣rst t qt♦♥s ♥ r ♦♥sr♥ t ♠♦st ♥r s ♦ tr ♣str ♠♥ t ♦② tr ♣s ♠♥ s ♠♥ ♣rs♥ts♥♥s ♦♥ ♦t t stt♦r ♥ t r♦t♦r t s ♦rt t♥ tt ts ♠♦♦ r♣rs♥t t ♦r ♦ r② ♥ ♦ s②♥r♦♥♦s ♠♥ ♥♥ ♣r♠♥♥t ♠♥t ♥ rt♥ ♠♥s ♠♦ s ♥ ♦♣ t t ♦♥②ss♠♣t♦♥ ♦ ♥♦ ②strss ♥ ♥♦ ② rr♥ts tr tt t qt♦♥s ♥♦t♥ ♥tr♦♥ t ②♣♦tss ♦ ♥rt② ♥t♥ r♦♥ strt♦♥

tr ♣s s②st♠ s ♥ tr♥s♦r♠ ♥t♦ ♦② t♦ ♣s ♠♥ s♠ ♦♥srt♦♥s ♠ ♦r t tr ♣s ♠♥ ♥ r♣t s♣②♥ t qt♦♥ ♦r t t♦ ♣s s②st♠ ♥ t ♠♦ s ♥ rst ♦♣t ♥♦ rstrt♦♥ ♦♥ t ♠♥t ♦r ♦ t ♠♥ ♥ tr ♥tr♦♥ t②♣♦tss ♦ ♥♦ r♦♥ strt♦♥

♥② t rsts ♦t♥ ♦r t ♦② ♠♥s ♦t t tr ♣s♥ t t♦ ♣s ♥ s t♦ ♦♣ t ♠♦ ♦ t ②♥ ② ♠♣♦s♥s♣ ♦♥t♦♥ ♦♥ t r♦t♦r rr♥ts ♥ ♣rtr t♥ ♥ r♦t♦r rr♥ts ♥ t♠♦ ♦ t ♦② ♠♥ t s ♣♦ss t♦ r♣rs♥t s②♥r♦♥♦s rt♥♠♥ qt♦♥s ♥ ♦♣ ♦♥sr♥ t ♠♦st ♥r s t♥♥tr♦♥ t ②♣♦tss ♦ ♥rt② ♥ ♥② ♦♥sr♥ s♥s♦ strt♦♥♦ t ♥♥s

Chapter 2

Prt rr♥t ♦♥tr♦

Prt rr♥t ♦♥tr♦ s♠s str♦♥② r② ♦♥ t ♥♦ ♦ t ♣♥t ♠♦ r② ♦ t ♦s rr♥t ♣rt♦♥ ♦ t ② ♣r♠trs rt♦♥♦r ♠s♠t ♥♦♥ ts ♥ ♦tr ♠♦ ♥qs ♥ ②♥r♦♥♦s t♥♥ ts t ♦ ♣rtr② rt s♥ ts ♥r♥t ♥t♥s r♦♥ strt♦♥ss t rt♦♥ ♦ t ♥t♥s ♥ r♥ ♥t② ♦r ♣rt♦♥s ♥ ♣rs♥t s ♥ t s♦t♦♥ t♦ ♦r♦♠ t ♠♦ ♥qs ♦♥tr♦s♠s s ♦♥ ts ♦ ♥♦t rqr ♠♦ ♦r t ♣rt♦♥ s♥ t② r② ♦♥ tst♦r rr♥t rt♦♥s ♦t♥ ② ♠sr♠♥ts s ♥♥♦t ♦♥tr♦ tqr♥ts r♦st♥ss t♦ ♣r♠tr rt♦♥s ♥ ♠s♠ts ♥ t ♦tr ♥ rq♥t ♣t ♦ t st♦r rr♥t rt♦♥s s ♠♥t♦r② t♦ ♦ t s♦ ♦t t♦r st♥t♦♥ tt ts t ♣r♦r♠♥ ♥ t♦ t ♦ss ♦ t ♦♥tr♦♥ ts ♣tr Prt rr♥t ♦♥tr♦ s♠s r ♥②s ♥ ♣rs♥t ♦s♥♦♥ ♦t ♦s ♥ ♦r ♣rt♦♥

♥tr♦t♦♥

♥ t♦ ts s♠♣t② ♦ ♦♠♣r♥s♦♥ ♥ ♠♣♠♥tt♦♥ ts ♠♣r♦ ②♥♠s♥ st rs♣♦♥s Prt ♦♥tr♦ P s ♥ ② ♥stt ♥ ♠♣♠♥t s ♥ ♦ ♦♥tr♦ s t♦ ♣rt t tr ♣♥t ♦r ♥ ss ts♣t② t♦ st t ♦♥tr♦ t♦♥ ♦ ♦ t ttr s s ♦♥ t t♦♥ ♦ ♦st ♥t♦♥ ♥ t s ♦ ♣♦r tr♦♥s ♥ tr rs Pss♠s ♥ ♦t♥ s t♦ ♦♥tr♦ t rr♥t ♦♥ ♥ t ♦ ♥ tr ♠♦t♦r ♦♥♥t t♦ t ♣♦r ♦♥rtr ❬❪ s ♥ ♦ s♠s r rrr sPrt rr♥t ♦♥tr♦ P ♥ ♥ ss ♥ t♦ t♦rs ♣♥♥ ♦♥♦ t st♥ s♥s ♦r tt♥ t ♠♥ ♦ts r ♥rt rst♣♣r♦ ♥♠ ♦♥① ♦♥tr♦ t ♣r♦s ♥② ♦t t♦r ♦t♥ t ♦♠♥t♦♥ tr♦ ♣s t ♠♦t♦♥ st ♦ t♦ ♦r tr s t♦rs ♦t ♥rtr s♦♥ ♥t ♦♥tr♦ t ①♣♦ts t srt ♥tr♦ t ♣♦r ♦♥rtr ♥ ♦s ♥♦t rqr ♠♦t♦r s♥ t ss ♦♥② t st♦rs

♣rt♦♥ ♦♠♠♦♥② rs ♦♥ t ♣♥t ♠♦ ♥ rs t♦ ♦s P

Prt rr♥t ♦♥tr♦

P t♦ ts ♥tr ts ♣r♦r♠♥ r② ♣♥s ♦♥ t r② ♦ t♠♦ t ♦ ♣r♠tr ♠s♠t ♥ t rr♥t ♣rt♦♥ ♦ P ♦r t♦ ♦♥rtr r ♣② ♥stt ♥ ❬❪ ♥ ts ♦r t s s♦♥ tt t♣rt♦♥ rr♦r ♣♥s ♥♦t ♦♥② ♦♥ t ♥rt♥t② ♦ t ♠♦ ♣r♠trs t s♦♦♥ t ♥st♥t♥♦s s ♦ ♦ rr♥t ♥ ♦t♣t ♦t ♦ t ♥rtr st s ♠♦r sr ♥ ♣rs♥ ♦ ♥t♥ ♠s♠ts ♥ ♥ ♣rtr ♥ t♣rt♦♥ ♠♦ ♦rst♠ts t r ♦♥sr♥ ♥ tr ♠♦t♦r s t♣♥t t ♦ ♠♥t♦♥ stt♦♥ ♦ ♦r t♦ r♦♥ strt♦♥ ♥ t ♦ ♣rtr② rt ♥ ②♥ rs♦♥ s tt t♦ t ♦♠tr② ♥ t r♥ ♦♣rt♦♥ ♣t② t ♥t♥s r str♦♥② ♣♥♥t ♦♥ t rr♥ts ♥t② r② ♥ r♥ ♥tr♦♥ s♥♥t rr♦rs ♥ t ♣rt♦♥ ttr♥ tr♥ ♠♣s tr♦rt♦♥ ♦ t r ♣r♦r♠♥

r♦st♥ss ♦ P s ♥ rss ♥ r♥t ②s s s ② ♠♥♦ str♥ stt ♦srr ♦r ♦♥♥ ♣r♠tr st♠t♦♥ ♥ ❬❪ ♥ ❬❪ ♥ ♦srrs s♠ s s t♦ ♠♣r♦ t ♣r♦r♠♥ ♦ P s♠ ♦r ♥ ♥t♦♥♠♥ ♥ t ♦r♠r ♥rr ♦srr s s t♦ ♦♠♣♥st ♦r t ♥♥♦s ②♥♠s t ttr ①♣♦ts s♦♥ ♦rr s♥ ♠♦ ♦srr ♦tts ♦rs ♦♠♣r t ♣r♦r♠♥ t trt♦♥ P s♠ ♥r tt ♦ ♣r♠tr ♠s♠t s♦♥ t t♥ss ♦ t ♦srrs ♥ ❬❪ r♦st ♣rt rr♥t ♦♥tr♦ s♠ ♦r Pr♠♥♥t ♥t ②♥r♦♥♦s ♥P s ♣rs♥t ♥ ♣rtr t ♣r♦r♠♥ ♦ t ♦♥tr♦r r ♠♣r♦ ♥st ♣r♠tr rt♦♥s ② ♠♥ ♦ ♥ ♥tr tr♠ ♥ ❬❪ ♥ ♣tP ♦r P s ♣rs♥t r t ♥t♥s r st♠t ♦♥♥ ② ♠♥ ♦♥ ♣t ♦srr ♥ r s ♥ t ♠♦ ♣rt♦♥ ♥ ♥ t rr♥t rr♥♥rt♦♥ s P♣r ❬❪ ♥②ss t t ♦ ♣r♠tr rt♦♥s ♥ P♦r P ♥ tr tt ♣r♦♣♦ss ♥ ♥tt♦♥ ♣r♦ss s ♦♥ st sqr♠♥♠③t♦♥ t♦ ♥♥ t ♣r♦r♠♥ ♦ t s♠

♥ t s♦t♦♥ rrr s ♦r P P t♦ ♦r♦♠ t ♦♥sq♥s ♦ t ♠♦ ♥qs s ♥ ♣r♦♣♦s ♥ ❬❪ ♥ t♥ ♠♣r♦ ♥ ❬❪ ♥ts ♦rs t rr♥t rt♦♥s rt t♦ rt♥ ♦t t♦r r r♦r ♥st♦r ♥ ♦♦♣ ❯ rr♥t ♣rt♦♥ s tr ♦t♥ ② ss♥t ❯ ♥ t ♣♦st♦♥ rt t♦ t ♦♥sr ♦t t♦r rsts s♦♥♥ ❬❪ t tt r rr♥t ♣rt♦♥ ♥ ♦♥② t ❯ srq♥t② ♣t ♦♥ t ♣♥♦♠♥♦♥ ♦t t♦rs st♥t♦♥ s♠♣ ♥tst♥t♦♥ s♦t♦♥ ♣r♦♣♦s ♥ ❬❪ ♦♥ssts ♥ ♠♦♥t♦r♥ t ♦t ♦t t♦rs ♥ tr ♦♥ ♦ t♠ ♦r♦♠s ① trs♦ ♣♣②♥t t♦t t s ♦ ♠♣♠♥tt♦♥ ts s♦t♦♥ s s♦♠ s♥♥t rsrst② t ♣r♦♣♦s ♥tst♥t♦♥ ♠t♦ ♠♣s t ♣♣t♦♥ ♦ ♥♦♥ ♦♣t♠ t♦rs s♥ t② r ♥♦t t♥ r♦♠ t ♦st ♥t♦♥ ♠♥♠③t♦♥ ss t ♦♦ t trs♦ s ♠ ♦♥ t s ♦ ♠♣r ♦♥srt♦♥s t ♦ ♥♦t st ♦r r② ♦r♥ ♦♥t♦♥ ♥ t r♣rs♥ts tr♦ t♥ t rt②♦ t ♣rt♦♥ ♥ t ♦♠♣♥ t t ♦st ♥t♦♥ ♠♥♠③t♦♥

Prt rr♥t ♦♥tr♦

SynRMCost Function

Minimization ~

UDC

ikPrediction

at k+1

iabck

22

i z,k2|k16

Prediction

at k+216

i*

2 2

uk

ik1|kuz

r Prt rr♥t ♦♥tr♦ s♠

Prt rr♥t ♦♥tr♦

♥r r♣rs♥tt♦♥ ♦ t P s♠ s r♣♦rt ♥ ♥P s t♦ ♣rt t tr ♦r ♦ t s②st♠ ♥ t♦ ♦♦s t ♥①t ♦♥tr♦t♦♥ ♦r♥ t♦ rt♥ ♦♣t♠ rtr♦♥ ♥ ♣rtr ♥ P t ♦♦s♥♦t t♦r s t ♦♥ ♠♦♥ t t ♣♦ss ♥rtr stts u

z = [uzd uzq ]t t

z ∈ Q ♥ Q = 0, . . . , 7 tt ♠♥♠③ t ♦st ♥t♦♥ ttr s t♦ r② s♥ s♥ t ♦r ♦ t ♦ s②st♠ s s ♦♥ t r♥t♦t ♥t♦♥s ♥ s ♥ trtr s ♥②s ♥ ♥ ts♦r t ♦♥sr ♦st ♥t♦♥ s

♠♥z∈Q J =[

i∗d − iz,k2|kd

]2+[

i∗q − iz,k2|kq

]2

t st♣ t rr♥t s♠♣s ik = [ikd ikq ]

t ♥ t st ♦t t♦r ♣♣

uk = [ukd ukq ]

t r s t♦ ♣rt t ♥①t s ♦ rr♥t ik1|k = [i

k1|kd i

k1|kq ]t

♣rt♦♥ t ♥st♥t k1 = k + 1 ♥ ♥r② ①♣rss s

ik1|kd = ikd +∆ikd

ik1|kq = ikq +∆ikq=⇒ i

k1|k = ik +∆i

k

r ∆ik = [∆ikd ∆ikq ]

t s t rr♥t rt♦♥ t♥ st♣s k ♥ k1 ♣rtrr♥ts ik1|k t♦tr t t t ♥rtr stts uz t z ∈ Q r s ♥ tr♥t♦ st♠t t t ♣♦ss rr♥ts iz,k2|k = [i

z,k2|kd i

z,k2|kq ]t t st♣ k2 = k + 2

ttr r ♦♠♣t s

iz,k2|kd = ik1d +∆iz,k1d

iz,k2|kq = ik1q +∆iz,k1q

=⇒ iz,k2|k = i

k1 +∆iz,k1

s ♦♠♣t t r s t♦ t t ♦st ♥t♦♥ ♥ t♦ ♥t st ♦t t♦r tt ♣♣ t t ♥①t st♣ t♦ st♣ ♣rt♦♥s rqr t♦ ♦♠♣♥st ♦r t ♥r♥t ② ♥ t t ♠♣♠♥tt♦♥ ♦ tP s♠ ❬❪ s♥♥t r♥ t♥ P ♥ P st♥s ♥♦ t rr♥t rt♦♥s ∆i

k ♥ ∆iz,k1 ♥ ♥ r t s

sr ♥ t ♥①t st♦♥s

Prt rr♥t ♦♥tr♦

♦st ♥t♦♥

♦ ♦ t ♦st ♥t♦♥ r♣rs♥ts r s♣t ♥ t s♥ ♦ ♦♥tr♦s♠ r♥t ♦st ♥t♦♥s ♥ s ♥ trtr ♥ ❬❪ ♦st ♥t♦♥s ♦♥ t s♦t rr♦r s ♠♣♦② ♥ t abc tr ♣s s②st♠ ♦r ♥ ♥tr♦rPr♠♥♥t ♥t P ♠♥s ♥ ❬❪ t s♠ s ♠♣♦② ♥ t αβ stt♦♥r②rr♥ r♠ ♦r ②♥ s♠ s s ♥ ❬❪ ♦r r♠♦♥t Pr♠♥♥t♥t P ♠♥ ♥ ❬❪ t ♦st ♥t♦♥ s s ♦♥ t rr♥t sqr rr♦r♦r ♥ P ♠♥ t r ♦ t ♣r♦♣r ♦ ♦ t ♦st ♥t♦♥ ♥ ♦♥ ♥ ❬❪ r r♥t s♦t♦♥s r ♥②s ♥ s ♦♥srt♦♥s r ♥♦♥ t t♥ t♦rs ♥ t r ♦♥str♥ts rr♥t ♠ts ♥ ♥ t♦st ♥t♦♥

♥ trt♦♥ r♥t ♦♥tr♦ s♠s t q rr♥ts r ♦♥tr♦② t♦ ♥♣♥♥t rt♦rs s r♥ts tt ♥♦ ♠tt♦♥s r ♥♦♦t t rr♥s r ♣rs② tr ♥ ♥r ②♥♠s ♦r♥ t♦ t ♥t ♦ t t♦ ♦♦♣s P rs ♦♥ t ♠♥♠③t♦♥ ♦ t ♦♥② ♦st ♥t♦♥ tts t♦ ♥ ♦t t rr♥t rr♦rs ♦♥sr♥ t ②♥ ♣♣t♦♥ t r♠r ♥s♦tr♦♣② ♦ rst ♥ t♦♦ r♥t ♦r ♥ t t♦ ①s ♥ ♣rtr♥ ♠♣r♦♣r ♦ ♦ t ♦st ♥t♦♥ ♦ rst ♥ s ♦♥tr♦ t♦♥ ♥ t♣r♦rt② t♦ ♦♥ ♦ t♦ ①s tt ♦ ♣rtr② tr♠♥t r♥ t ♠♥♦♣rt♦♥ ♦♠ ♦♥srt♦♥s ♦♥ ts s♣t r r♣♦rt ♥ ts st♦♥

rst ♦st ♥t♦♥ ♦♥sr s t ♦♥ s ♦♥ t s♦t rr♦r tt ♥ rtt♥ s

♠♥z∈Q J = |i∗d − iz,k2|kd |+ |i∗q − iz,k2|kq |

♦♥sr♥ t s ♣♦♥t s t rr♥ t s♦ rs ♦ ts ♦st ♥t♦♥r ♣♦tt ♥ t s ♥s

♥ ♦rr t♦ r② ts stt② t P s ♦♥ s ♥ tst ♥ s ♦st ②♥♠s ♥♦♥ rr♥t st♣s ♥ ♣rtr t t ♠♥ t st♥st s♣ st♣ s ♥ s rr♥ t♦ t s♣ rt♦r s ♦♥sq♥ ts ♦t♣tstrts t♦ t ♠①♠♠ rr♥t ♠♣t t s ♣♦♥t r♣rs♥ts trr♥t rr♥ rsts ♦ t s♠t♦♥ r r♣♦rt ♥ ♥ ♣rtrt q rr♥t ♦r♠s t t♦rq ♥ t ♦t t♦r ♥①s r s♦♥ ♥ t s ♦rt t♥ tt ♥ t ♥t st t rt♦r ♦♥② ts t♦♥② t ①s rr♦r s ♥ s♥ s♥ t st♣ t ♦t t♦r t ♥①4 s ♦s♥ ♥ ϑ = 0t ①s rr♥t r♣② rs t ♥t rr♥ t q①s s st ③r♦ r r♣rs♥tt♦♥ s ♥ ② t ♣♦♥t ♦ ♦r trt♦r②♥ t q ♣♥ s♦ ♥ ♥ ♥ t rt rr♥t rs trr♥ i∗d = −1/

√2 ❬♣❪ t ♣rt rt♦r strts ♦♦s♥ t♦rs t♦

♥rs t qrtr rr♥t ♥①s 2 ♥ 3 ♦♥tr♦ t♦♥ s r②t♦♦ t rr♥t t t ♦r ♥t♥ s rst② ♣s t♦rs ts rr♥♥ ♦♥② tr t rr♦r ♦ t s♦♥ rr♥t s ♠♥♠③ ♦tr t stt t t♦rq ♣r♦ ② t ♠♥ r♠♥s ③r♦ ♦♥ ♦ t t♦ rr♥t s♥ ♥ ts ♥ tr♥ ♠♣s s♦r s♣ rs♣♦♥s

♦♥srt♦♥s ♠ s♦ r ♥ rs♠ s ♦♦s t♦ t s♥♥tr♦t♦r ♥s♦tr♦♣② t ♦♥tr♦ t♦♥ ♣rs♥ts s ♥ ♣rtr t stst ② t♦r t rr♥t rr♦r s t♦ st t ♣♦♥t ♦ ♦r rst ♦♥ t ①s t t ♦r

Prt rr♥t ♦♥tr♦

0 100 200−1

−0.5

0

d−ax

is c

urre

nt

[p.u.]

ActualRef

0 100 2000

0.5

1

q−ax

is c

urre

nt

0 100 2000

0.5

1

Time step [−]

Tor

que

0 20 400

1

2

3

4

Time step [−]

Vol

tage

vec

tor

inde

x [p

.u.]

♦♥tr♦ q♥tts

−1 −0.8 −0.6 −0.4 −0.2 00

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.2

0.2

0.2

0.4

0.4

0.4

0.4

0.40.6

0.6

0.6

0.6

0.6

0.8

0.8

0.8

0.8

0.81

1

1.2

d−axis current [p.u]

q−ax

is c

urre

nt [p

.u]

q rr♥t ♣♥

r ♦st ♥t♦♥ s ♦♥ s♦t rr♦r

♥t♥ s ♦♥sq♥ ♦♥② ♦♥ ♦ t t♦ rr♥ts r♣② rs t rr♥t s ♦rt ♠♥t♦♥♥ tt ts ♥sr ♦r s ♦t♥ ♥♦t ♦♥② t t strt♣♦ t ♠♥ s ♥ t rsts ♦ t ♥ t♦s ♦♣rt♦♥s ♥♦♥ rr♥t rr♦rs s t s ♣rtr② ♥t ♥ t ①s s ♥ t ♦♥ ♦t s① t ♦t t♦rs ♦ t ♥rtr ♥ ts stt♦♥ t ♣rt rt♦r♦♦ss t ♥ t♦r ♦r ts ♦♣♣♦st ♣♥♥ ♦♥ t s♥ ♦ t rr♥t t♦st② r② t ①s rr♥t ♣♥ ♦♥st♥t qrtr rr♥t

rst t♦ ♦r♦♠ ts ♥sr ♦r s t♦ t t t♦ rr♦r ♦♠♣♦♥♥ts ♥ t ♦st ♥t♦♥ ♥ ♣rtr s♥ ♥ r ♦r t q①s ♦ ♠♦r ♠♣♦rt♥ t♦ ts rr♦r ♦♥ tt t rr♥t r♠♥s t♦♦ st♥t r♦♠ trr♥ ♥ t s s ♥ ② t ♥s♦tr♦♣② rs♦♥ tt♠♣t s t♦ tt rr♦rs ♥ t ♦st ♥t♦♥ t t s♥② rt♦ ♦t♥♥

♠♥z∈Q J = |i∗d − iz,k2|kd |+ Lq

Ld|i∗q − iz,k2|kq |

♠♣ ♦ t ttr s r♣♦rt ♥ t s ♥s rsts ♦ tP s ♦♥ r r♣♦rt ♥ ♦r ♦t♥ t t ♥ ♦st♥t♦♥ s ♦♠♣t② r♥t t ♥ s♥ tt ♥ t ♥t st t ♣rt♦♥tr♦r ♦t♣ts t ♦t t♦r t ♥① 3 ttr ♣ss ♦t t rr♥tst♦rs tr rr♥ ♦♥ t t ♦t♥ s♥ s r② s♦♥ ♥ t♦t t rt rr♥t s ♥ssr② str t ♣♦♥t ♦ ♦r ♦s ♥♦t♠♦ ♦♥ t ①s s ①♣t t t♦rq rs♣♦♥s s str t strts rs♥r♦♠ t rst st♣

♥♦tr ♦st ♥t♦♥ s s ♦♥ t sqr rr♦r ♥ t ♥ rtt♥ s

♠♥z∈Q J =[

i∗d − iz,k2|kd

]2+[

i∗q − iz,k2|kq

]2

r♣♦rts t s♦rs ♦ t ♦♥sr ♦st ♥t♦♥s ♦ s♦♦ tt t ♣♦♥t ♦ ♦r ts t♦♦ r r♦♠ t rr♥ ♥ t rr♦rs r sqr♥ t ♦st ♥t♦♥ s s♦♦♥ s ♦♥ ♦ t t♦ rr♥ts ♠♦ r♦♠ ts rr♥ ts tr♠♦♠s ♠ r t♥ t ♦tr ♦♥ ♦r♥ t ♦st ♥t♦♥ t♦ r t s

Prt rr♥t ♦♥tr♦

0 100 200−1

−0.5

0

d−ax

is c

urre

nt

[p.u.]

ActualRef

0 100 2000

0.5

1q−

axis

cur

rent

0 100 2000

0.5

1

Time step [−]

Tor

que

0 20 400

1

2

3

Time step [−]V

olta

ge v

ecto

r in

dex

[p.u

.]

♦♥tr♦ q♥tts

−1 −0.8 −0.6 −0.4 −0.2 00

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.50.5

0.5

0.50.5

1

1

1

1

1

1

1

1.5

1.5

1.5

1.52

2

2

2

2.5

2.5

d−axis current [p.u]

q−ax

is c

urre

nt [p

.u]

q rr♥t ♣♥

r ♦st ♥t♦♥ s ♦♥ t s♦t rr♦r

♦st ♥t♦♥ ♥ rrtt♥ s

♠♥z∈Q J =[

i∗d − iz,k2|kd

]

·[

i∗d − iz,k2|kd

]

+[

i∗q − iz,k2|kq

]

·[

i∗q − iz,k2|kq

]

= wd(εd) ·[

i∗d − iz,k2|kd

]

︸ ︷︷ ︸

εd

+wq(εq) ·[

i∗q − iz,k2|kq

]

︸ ︷︷ ︸

εq

♥ t ♥ s♥ s t s♠ ♦ t rr♥t rr♦rs t r ts wd(εd)♥ wq(εq) ♣♥♥ ♦♥ t rr♦rs εdεq t♠ss t ts r ts q t♦ trr♦rs

rsts ♦t♥ t r r♣♦rt ♥ s ♦r t ♦st ♥t♦♥t t s♦t rr♦r t rst st♣s r ♦♠♠t t♦ r t ①s rr♦r ♦ ♦ts t ♣rt rt♦r ♦♦ss t ♦t t♦r t ♥① 4 tt ♦s tstst rs ♦ id ♣♥ t qrtr rr♥t t ③r♦ ♦♠♣rs♦♥ t s♦s tt ts ♦♣rt♦♥ sts ♦r ♦r ♥♠r ♦ st♣s s♥ t sqrrr♦r rs♦♥ s tt s s♦♦♥ s t rr♦r ♦ t ①s ♦♠s ♦r t♥ rt♥trs♦ t q①s rr♦r rsts t ♦♠♥♥t ♦♠♣♦♥♥t ♥ t ♦st ♥t♦♥ s ♦♥sq♥ tr t s♦rt ♥t st t ♦♦s♥ ♦t t♦r s t ♦♥ t♥① 3 ❲t t ♦st ♥t♦♥ s tr♦r ♣♦ss t♦ s♦rt♥ t s ♦♥tr♦t♦♥ t ♥ q①s rr♥t ♥ t♦rq

0 100 200−1

−0.5

0

d−ax

is c

urre

nt

[p.u.]

ActualRef

0 100 2000

0.5

1

q−ax

is c

urre

nt

0 100 2000

0.5

1

Time step [−]

Tor

que

0 20 400

1

2

3

4

Time step [−]

Vol

tage

vec

tor

inde

x [p

.u.]

♦♥tr♦ q♥tts

−1 −0.8 −0.6 −0.4 −0.2 00

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

10.1

0.1

0.1

0.1

0.1

0.2

0.2

0.2

0.2

0.3

0.3

0.30.3

0.4

0.4

0.4

0.4

0.4

0.5

0.5

0.5

0.50.5 0.6

0.6

0.7

0.8

0.9

d−axis current [p.u]

q−ax

is c

urre

nt [p

.u]

q rr♥t ♣♥

r ♦st ♥t♦♥ s ♦♥ sqr rr♦r

st ♦st ♥t♦♥ t ♥ ts st♦♥ s ♦t♥ ♦♠♥♥ t ♥

Prt rr♥t ♦♥tr♦

ts ♦ s♥ t s♥② rt♦ s t ♥ t sqr rr♦r rst♥ ♦st♥t♦♥ s

♠♥z∈Q J =[

i∗d − iz,k2|kd

]2+

Lq

Ld

[

i∗q − iz,k2|kq

]2

s♦s t rsts ♦t♥ t t ♥ s♥ tt t ♣r♦r♠♥r ♠♣r♦ t rs♣t t♦ t♦s ♦ ♥ r t s♠ ♦

0 100 200−1

−0.5

0

d−ax

is c

urre

nt

[p.u.]

ActualRef

0 100 2000

0.5

1

q−ax

is c

urre

nt

0 100 2000

0.5

1

Time step [−]

Tor

que

0 20 400

1

2

3

Time step [−]

Vol

tage

vec

tor

inde

x [p

.u.]

♦♥tr♦ q♥tts

−1 −0.8 −0.6 −0.4 −0.2 00

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

10.2

0.2

0.2

0.20.2

0.4

0.4

0.4

0.40.4

0.6

0.6

0.60.6

0.60.8

0.80.8

1

111

1.21.21.2 1.41.41.4

1.6

1.6

1.61.8

2

d−axis current [p.u]q−

axis

cur

rent

[p.u

]

q rr♥t ♣♥

r ♦st ♥t♦♥ s ♦♥ t sqr rr♦r

rsts ♥ t ♦♥srt♦♥s ♣rs♥t ♥ ts st♦♥ t tt rt♥♦st ♥t♦♥ ♦ ♥♦t st ♦r r② ♦♣rt♦♥ ♦ t ②♥ ♥ ♣rtr tr♦t♦r ♥s♦tr♦♣② ♦ ♥tr♦ s ♥ t ♦♥tr♦ t♦♥ tt ♥ tr♥ ♦ tt ♣r♦r♠♥ r♥ ②♥♠s t s ♦rt ♠♥t♦♥♥ tt t ♦♥sr ♦st♥t♦♥s ①t ♦♠♣r st② stt ♣r♦r♠♥

♦ ♦ ♥tr♦♥ t s♥② rt♦ s t ♥ t ♦st ♥t♦♥ s♦♥ ♣r♦r♠♥ ♥ ♣rtr ♥r strt ②♥♠s ♥ ♦t ①s t ♦tt♦r tt ♣ss ♦t t rr♥t t♦rs tr rr♥ s ♦s♥ ♠♥ r ♦ ts t♥q s tt t ♥♦ ♦ t s♥② rt♦ s rqr t♥t♥s t♦ ♣rs② ♥♦♥ ss t♦ t s♥♥t r♦♥ strt♦♥tr s r ♥♦t ♦♥st♥t t ♣♥ ♦♥ t rr♥ts ♦r ts rs♦♥ ♣rt♦♠♣♥st♦♥ ♦ rqr t ♣rs ♠♥t rtr③t♦♥ ♦ t ♠♥ ♥r② ♦r♥ ♦♥t♦♥ ♥ s ♦♥ ts r ts s♦ s

♥ ts ♣rt ♦ t ♦r t ♦st ♥t♦♥ s ♦♥ t sqr rr♦r s ♥♦♥sr ②♥♠ tst s♦ tt t ♦s s♥t② ♦♦ ♣r♦r♠♥ rtr♠♦r t ♦s ♥♦t rqr t ♥♦ ♦ t ♠♥ ♣r♠trs s r♥tsr♦st♥ss ♥ts ♣r♠tr ♥rt♥t② ♥ rt♦♥s ♥ ♦♥sr♥ t ♣r♣♦s♦ ts ♣rt ♦ t ♦r t r♣rs♥ts t ♠♦st st ♦

Prt rr♥t ♦♥tr♦

♦s P

t♥ t t ♦ r♦ss strt♦♥ ♥ ♦♥sr♥ s♥s♦ strt♦♥ ♦ tstt♦r ♥♥s s ♣tr t q♥t ♠♦ ♦ t ②♥ ♥ ①♣rss♥ t s②♥r♦♥♦s r♦tt♥ rr♥ r♠ s

∆ik = Ai

k +Buk

A =

[

−R/ℓd · Ts ωLq/ℓd · Ts

−ωLd/ℓq · Ts −R/ℓq · Ts

]

B =

[

Ts/ℓd 0

0 Ts/ℓq

]

r R st♥s ♦r t stt♦r ♥♥ rsst♥ Ld(ikd) ♥ Lq(i

kq ) r♣rs♥t t

♣♣r♥t ♥t♥s ℓd(ikd) ♥ ℓq(ikq ) r t r♥t ♥t♥s ♥ ω s t

tr s♣ ♦r ♦♠♣t♥ss t rr♥t ♣♥♥s r ♦♠tt ♥

trt♥ r♦♠ t rr♥t s♠♣s t ♥st♥t k ♠♦ s ♣rt♦♥ ♦t rr♥ts t t ♥①t st♣ ♥ ♦t♥ ♦♥sr♥ ♥ ②♥ ♠♦ ♥ t ♥t♥s r ♦♥st♥t ♥ ℓd = Ld ♥ ℓq = Lq s ♦s ♦t♥♥t ①♣rss♦♥ ♦ t rr♥t rt♦♥s ∆i

k t♥ t♦ ♦♥st s♠♣s s

∆ik = Ami

k +Bmuk

Am =

[

−R/Ld · Ts ωLq/Ld · Ts

−ωLd/Lq · Ts −R/Lq · Ts

]

Bm =

[

Ts/Ld 0

0 Ts/Lq

]

s ♥r③ ♠♦ s ♦t♥ s ♥ trtr t♦ ♣rt t rr♥t ♦ s②♥r♦♥♦s♠♥s t s ♦rt ♠♥t♦♥♥ tt t ②♣♦ts②s ♦ ♥♦ r♦♥ strt♦♥ s ♠♦rst ♦r P ♠♥s r t t ♦ r♦♥ strt♦♥ s ss s♥♥t t♥ ♥P ♦r ②♥

♥ t rr♥ts ik1|k r ♦t♥ t t② r s t♦ t t

♣♦ss ♥①t rr♥t rt♦♥s ∆iz,k1 t z ∈ Q s

∆iz,k1 = Ami

k1|k +Bmuz

♥ tr♥ ♦s ♣rt♥ t rr♥ts iz,k2|k t ttr r s ♥

t t♦♥ ♦ t ♦st ♥t♦♥

r♥ t r♥t ♠♥ ♦♣rt♦♥s ♠s♠t t♥ t r ♥ t①♣t ♣r♠trs s s t♥ r♥t ♥ ♣♣r♥t ♥t♥s ♦♦r ss ②♥ ♣rs♥ts ♠♥t rtrsts tt r str♦♥② ♥♦♥ ♥r♠♥ t ♠♦ s ♦r t ♣rt♦♥s ♥ ♥st

Prt rr♥t ♦♥tr♦

t ♦ ♣r♠tr ♠s♠t ♥ rt♦♥

♥ ts st♦♥ t ts ♥ ② ♣r♠tr ♠s♠t ♥ rt♦♥ r r② ♥stt t♦ t ♥♦♥♥r ♥tr ♦ P t s ♥♦t ♣♦ss t♦ ♥②s ts ♦r ♥ tr♠s ♦ stt② t trt♦♥ ♠t♦s s ♦r ♥r ♦♥tr♦ s♠s t r♦♦t ♦s ♥ t ♦s ♦♦♣ ♥②ss ♥ t ♦♥sq♥s ♦ ♠♦♣r♠tr ♠s♠t s ♥ ♠♣r② t ② ♥②s♥ t ♣r♦r♠♥ ♦t P ♥r r♥t t♥ ♥rt♥ts s sts ♥ rr♦t ♥ ♣♣t♦♥s ♥♦♥ rr♥t ♣rt♦♥ ♥ ♦♥tr♦ t tr♣s t♦♥rtrs ❬ ❪ ♠t♣s rs ❬❪ ♥ t r♦♥t ♥ ♣♣t♦♥s ❬❪ ♦♠♦ t ♦ t ♦rs ♦♥ tt t ♥rt♥t② ♥ t ♦ rsst♥ s ss r♥t ♦r ♥ ♥♥ ♦♥ t P ♦♣rt♦♥ ♥ ♦tr ♦rs t ♥♥ ♦ rsst♥ ♥ ♥t♥ ♠s♠t s ♥ st s♣rt② ♥t②♥ t♦r♥t ♦r rsst♥ rt♦♥s r rs♣♦♥s ♦r st②stt rr♦rs ♥rt♥ts ♦♥ t ♦ t ♥t♥ r rt t♦ rr♦r ♥ ♦t st②stt♥ ②♥♠s ♣rt rr♥ts ♠t♠t ♥②ss ♦♥ t t ♦ ♣r♠tr♠s♠t ♥ t rr♥t ♣rt♦♥ s ♥stt ♥ ❬❪ ♣rt♦♥ rr♦r s ♠♦♥strt t♦ ♠♦r sr ♥ t t ♥t♥ s ♦r t♥ t ①♣tt♦♥s t ♥ ♦rst♠t ♠♦ ♥t♥ ss t r ♣r♦r♠♥ r♦rs♥ ② ♥ ♥rs r♣♣ ♥ t ♦♥tr♦ rr♥ts

♦r ♣rs♥t ♥ ❬❪ ♦♥srs rsst♥ ♥ ♥t♥ ♠s♠t t♥t s s ♦r t ♣rt♦♥ ♥ t t ♦♥ ♥ s ♦ ♥ ♦ ♦♠♦♥srt♦♥s t♦ ♦♥ t♦ ①t♥ t rsts ♦t♥ ♥ ❬❪ t♦ t s ♦ ②♥♥ ♣rtr ts ♥ ♦ ♠♥s ①ts ♥ ♥r♥t s♥♥t r♦♥ strt♦♥tt ♥ ♥t s t ♠♥ rs♣♦♥s ♦r ♠♦ ♥rs ❲tr r♦♥strt♦♥ ♦rs t t s ♥♦t ♦♥② rt♦♥ ♥ t ♥t♥ s t t♠♥ ♠♦ ts s r♥t Prs② t ①t ♠♦ ♦ t strt ♠♥♥t♥ r♦ss strt♦♥ s t ♦♥ r♣♦rt ♥ ♥ s ♦ r♦♥ ♥rt②t ①t ♠♦ s

t s ♦rt t♥ t ♣rt♦♥ rr♦r ♥ s ♥r ♠♦ s s ♦r trr♥t ♣rt♦♥ t♦ ♦♥tr♦ strt ②♥ s ♠t♦ s ♦t♥ s ♦r ts s♦ ♠♣♠♥tt♦♥ ♥ s t ♦s ♥♦t rqr t ♣rs ♠♥t rtr③t♦♥♦ t ♠♥ rr♦r ♥ ♦♠♣t s t r♥ ♦ ♥ ♦t♥♥

ε = ∆A ik +∆Bu

k

∆A = A−Am =

R(ℓd−Ld

ℓdLd

)

ωme

(LdLq−Lqℓd

ℓdLd

)

ωme

(Ldℓq−LdLq

ℓqLq

)

R(ℓq−Lq

ℓqLq

)

Ts

∆B = B−Bm =

(Ld−ℓdℓdLd

)

0

0(Lq−ℓq

ℓqLq

)

Ts

r ℓ ♥ L st♥ ♦r t t r♥t ♥ ♣♣r♥t ♥t♥ L s s♦r t ①♣t ♦ t ♣♣r♥t ♥t♥ s ♥ t ♥r ♠♦ ♦r t rr♥t ♣rt♦♥ rr♦r ①♣rss ♥ ♥s ♦t t ♦♠♣♦♥♥t ♥ ②t r♥ t♥ t ♥r ♥ t strt ♠♦ ℓ 6= L ♥ t ♦♠♣♦♥t♥t

Prt rr♥t ♦♥tr♦

♥ ② t ♣♣r♥t ♥t♥ ♠s♠t L 6= L rsts ♦t♥ ♥ ❬❪ r st ♥ s ♦ ②♥ ♦♥sr♥ s♠ rt♦♥ ♥②ss ♥ ♣rtr t ♦♥srt♦♥ss ♠ ♥ ❬❪ ♥ ①t♥ ♥r③♥ t ♠♦ ♦t ♦r♥ ♣♦♥t ♦♥sr♥ ♥r ♠♥ t ♥t♥ q t♦ t r♥t ♥t♥s♥ t ♦♥sr ♦♣rt♥ ♣♦♥t

ts ♦ r♦♥ strt♦♥ r r ♥ r t ♣r♦r♠♥ ♦ ♥ ♥ strt ♠♥ r s♦♥ ♥ ♦t ss t ♥rs ♠♦ ss t♦ ♣rt t rr♥ts tst s ♣r♦r♠ t ♦ r♦t♦r ♥ ♣r♦♥ t♦rr♥t st♣s ①♣t ♥rs r♣♣ s ♥t ♥

0 1000 2000−0.8

−0.6

−0.4

−0.2

0

d−ax

is c

urre

nt [p

.u.]

Actual Ref

0 1000 20000

0.5

1

q−ax

is c

urre

nt [p

.u.]

−1 −0.5 00

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.1

0.1

0.1

0.2

0.2

0.2

0.3

0.3

0.3

0.4

0.4

0.4

0.5

0.5

0.6

0.6

0.7

0.7

0.8

0.8

0.9

0.9

1

1

d−axis current [p.u]

q−ax

is c

urre

nt [p

.u]

♠♥

0 1000 2000−1

−0.5

0

d−ax

is c

urre

nt [p

.u.]

Actual Ref

0 1000 20000

0.5

1

q−ax

is c

urre

nt [p

.u.]

−1 −0.5 00

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.1

0.1

0.1

0.2

0.2

0.2

0.3

0.3

0.3

0.4

0.4

0.4

0.5

0.5

0.6

0.6

0.7

0.7

0.8

0.8

0.9

0.9

1

1

d−axis current [p.u]

q−ax

is c

urre

nt [p

.u]

trt ♠♥

r ♦♠♣rs♦♥ t♥ ♥r ♥ strt ♠♥

t s r r♦♠ tt s♥ ♥rs ♠♦ t♦ ♦♥tr♦ strt ♠♥♠♣s ♥ st♦rt rr♥t ♦r♠ rs♦♥ s tt s♥ t ①♣t♦r s t♦t② r♥t r♦♠ t t ♦♥ t ♣rt rt♦r ♦s ♥♦t ♦r♣r♦♣r② s ♦♥sq♥ t ♦t t♦rs ♦s♥ ② t ♦st ♥t♦♥ ♠♥♠③t♦♥r r♥t r♦♠ t s s♦s t ♦ t rr♥t ♣rt♦♥r② ♥ t ♦t t♦r st♦♥ s ♥ s♥ t ♣rt♦♥ ♥ tsqr ♠rrs s ♣rs ♥ r ♥ s ♦ ♠♥ ♥ t ♦tr ♥t s r tt t ♣rt♦♥ t t strt ♠♥ s ♥♦t ♣rs ♥ ♣rtrt rr♥t rt♦♥s r ②s ♥rst♠t s ♦♥sq♥ t ♦t t♦rst♦♥ s r♥t ♥ t t♦ ss ♥ t ♦♥sr ①♠♣ t t ♠♥t ♣rt rt♦r ♦♦ss ♥①s 3 ♥ 5 t♦ r ♠♦r ♥t s trt rr♥t ♦r♠r ♦s ♥ ♥rs t ttr rs ♦ t q①srr♥t ♥ stt ♥① 0 s s t♦ ♥rs ♠♦r ♣♦st s t ①s s♦② r♥ t qrtr rr♥t ♦♥sr♥ t strt ♠♥ t♦♦♣♣♦st t♦rs 1 ♥ 4 r s t♦ ♥rs ♥ r t rt rr♥t rs♣t② r♥ t qrtr rr♥t ❱t♦rs 2 ♥ 3 ♦s ♥rs♥ t q①s

s s♦♥ ♥ t sq♥ ♦ ♦t t♦rs s ♦♠♣t② r♥t ♥ tt♦ ss s ♠♣s r♥t ♥♠r ♦ st♥ s♦s t ♥♠r ♦st♥ t st♣ ♥ t t♦t ♥♠r t ♥ s♥ tt t st♥ ♣ttr♥t t ♠♥ ♥♦ ♦♥② ♦♥ ♥ t ♦♥trr② ♥ s ♦ strt♠♥ t st♣ ♦r s ♥ ♥♦ s ♦♥sq♥ t t♦t ♥♠r♦ ♦♠♠tt♦♥s s r ♥ t s ♦ strt ♠♥ t s ♦rt ♠♥t♦♥♥ ttt ttr ♦♥srt♦♥ s ♥♦t ♥r t s ♥♦t sr tt s♥ t ♥♦♠♥ ♠♦ ♥

Prt rr♥t ♦♥tr♦

300 305 310 315 320 325−0.72

−0.7

d−ax

is [p

.u.]

300 305 310 315 320 3250.69

0.7

0.71

q−ax

is [p

.u.]

300 305 310 315 320 325012345

Vol

tage

vec

tor

Time step [−]

♠♥

300 305 310 315 320 325

−0.8

−0.7

−0.6

d−ax

is [p

.u.]

300 305 310 315 320 325

0.65

0.7

0.75

q−ax

is [p

.u.]

300 305 310 315 320 3250

1

2

3

4

Vol

tage

vec

tor

Time step [−]

trt ♠♥

r ♦♠♣rs♦♥ t♥ ♥r ♥ strt ♠♥

t ♣rt♦♥ ♦♥tr♦♥ strt ♠♥ ♦s ♥♦t ♠♣② ♥ r ♥♠r ♦♦♠♠tt♦♥s ♥ ♥r t st♥ ♣ttr♥ ♦t♥ t ♥ ①t ♠♦ ♥ t♥ ♠♣rs ♦♥ s r♥t tr tt t t♦t ♥♠r ♦ ♦♠♠tt♦♥s ♣♥s ♦♥t ♥ ♦ ♦♣rt♦♥ ♦♥ t ♦st ♥t♦♥ tt ♦ ♥ tr♠ t♦ ♣♥③ t♥♠r ♦ st♥s ♥ ♦tr t♦rs

0 500 1000 1500 20000

1

2

3Commutations

Eac

h st

ep [−

]

0 500 1000 1500 20000

500

1000

1500

Time step [−]

Tot

al [−

]

♠♥

0 500 1000 1500 20000

1

2

3Commutations

Eac

h st

ep [−

]

0 500 1000 1500 20000

1000

2000

3000

Time step [−]

Tot

al [−

]

trt ♠♥

r ♦♠♣rs♦♥ t♥ ♥r ♥ strt ♠♥

♦r P

♦♥♣t ♦ P s ♥ rst ♣rs♥t ♥ ❬❪ ♥ t♥ ♠♣r♦ ♥ ❬❪ ♠ ♦ ts ♦♥tr♦ s♠ s t♦ r t ♥♦ ♦ ♣r♠trs rqr ② P♥ t♦ ♠♣r♦ t ♣r♦r♠♥ ♦ rr♥t tr♥ ♥ ♥r t t ♦ str♦♥ r♦♥strt♦♥ ♥ ♦ ♦rs ♥r t ts ♦ ♦tr ♣♥♦♠♥ t♥ t ♣rt♦♥

s ♦ P ♥ sr s ♦♦s tr ♦t t♦r s♦t♣t r♦♠ t ♦st ♥t♦♥ ♠♥♠③t♦♥ t rst♥ rr♥t rt♦♥s ♥ t t♦①s r r♦r ♥ st♦r ♥ t♦ r♥t ❯s s r♣rs♥t ♥ ♥ts ② t s ♦ ∆izd ♥ t s ♦ ∆izq r ♥ t♦ t ♦rrs♣♦♥♥♥rtr stt uz t z ∈ Q rr♥t ♣rt♦♥s ♥ r s② ♦t♥② ss♥ t s st♦r ♥ t ❯s ♦rrs♣♦♥♥ t♦ t ♦♥sr ♦tt♦r s ♥ t s ♦ ∆i

z r ♦t♥ r♦♠ ♠sr♠♥ts t②

Prt rr♥t ♦♥tr♦

♥r♥t② ♦♥t♥ t ♥♦r♠t♦♥ ♦t t r ♦r ♦ t ♠♥ ♥ ♦♥t♦♥s♥r t♦ t t ♦♥s

u1 u

4

iΔ1

iΔ4

iΔ1

iΔ4

t♦r♥ rr♥t tt♦♥

iΔ1

iΔ4

iΔ7

iΔ6

iΔ5

iΔ3

iΔ2

iΔ0

♦r ♣rt♦♥

r ♣rs♥tt♦♥ ♦ t ♦r ♣rt♦♥

t♦ t s ♣r♥♣ s t s♠ r♥t ♠♣♠♥tt♦♥ s s ♥ ❬❪ ♥♥ ❬❪ ♦r♠r s r♣rs♥t ♥ r♥ s♠♣ t♠ t♦ rr♥t♠sr♠♥t r ♣r♦r♠ ♥ ♦rr t♦ ♦t♥ t ♦rrt r♥ s♥ ♥ t♦♦ t rr♥t s♣s ♥ ② t sts t rst rr♥t tt♦♥ s ♥s♦r t t st♥ stt s ♣r♦r♠ t s♦♥ tt♦♥ s ② ② ① t♠ ♣r♦ tr t t st♥ s ♣r♦r♠ s ♥ s♥ ♥ t strt♥ ♣♦♥t ♦ t st♥ ♥tr s ♥ ♥ tt ♦ t ♦rrs♣♦♥♥s♠♣♥ ♥tr ② ① ♠♦♥t ♦ t♠ s ♦♥sq♥ t stt♦r rr♥t t t♥ ♦ t t st♥ ♥tr ♠sr ♥ t t s♠♣♥ ♥trs rst t rr♥t r♥ ♦rrs♣♦♥♥ t♦ t t st♥ ♥tr ♥♥♦t t ♥ t t s♠♣♥ ♥tr ♥ ♦tr ♦rs t rr♥t r♥♦rrs♣♦♥♥ t♦ t ♣r♦s st♥ stt ♥ t ♥ t ♣rs♥t s♠♣♥♥tr

Current sampling

Ts

rr♥t r♥ tt♦♥ t♥q

1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 2 3 4 5 6

ia(t) iak-2,1

iak-2,2

iak-1,1

iak-1,2

iak,1

iak,2

uk-3 uk-2 uk-1 uk

1: Read current and rotor position

2: Output trigger signals of new switching state

3: Read current

4: Calculate and update current difference

5: Estimate future current difference

6: Compute current command

7: Predict stator current, calculate cost and choose next switching state

♠ ♦♣rt♦♥s

r P rr♥t rt♦♥ tt♦♥ ♥ ♦♣rt♦♥ s ♥ ❬❪

rr♥t tt♦♥ t♥q r♣rs♥t ♥ rqrs t♦ s♠♣s t ♥tr ♦ ♦ t tt♦♥ ♦ s♣s t s♦♥ rrr♥t ♠sr♠♥t s② ♦ ① ♥tr ♦ t♠ tr t ♥ ♦♥t♥ ♠♦ s ♣♣ ♦rs♥ tr r ♥♦ ♥r ♥s rr♥ t ♦ ♦ t t♠ ② t r②♣♥s ♦♥ ♠♣r ①♣r♥ ♥ t ♥rtr rtrsts ♠♣r♦♣r ♦ ♦t ttr ♦ rst ♥ ♣r♠♥♥t ♠s t♦ t r s②st♠ ♦ ♥② rs♦ tsss s♥ rr♥t tt♦♥ ♠t♦ s ♥ s ♥ ❬❪ ♥ s♠♣♥ ♣r♦t rr♥t tt♦♥ s ♣r♦r♠ ♠♠t② ♦r ♦♥t♥ ♠♦ s ♣♣ s

Prt rr♥t ♦♥tr♦

s♦♥ ♥

Current sampling

Ts

rr♥t r♥ tt♦♥ t♥q

1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6

ia(t) iak

iak1|k

iak2|k

uk-1 uk uk+1 uk+2

1: Measure stator current

2: Output new switching state

3: Calculate and update old current variation

4: Estimate current variation

5: Compute current command

6: Compute current command

7: Execute current prediction, calculate cost and choose next switching

state

♠ ♦♣rt♦♥s

r P rr♥t rt♦♥ tt♦♥ ♥ ♦♣rt♦♥ s ♥ ❬❪

❲tr ♥ ♥rtr stt s ♥♦t s ♦r sr s♠♣♥ ♥trs t rtrr♥t rt♦♥s r ♥♦t ♣t ♦r ♦♥ t♠ ♥ ❬❪ t t♦rs rr t♦ ts sss t st♥t♦♥ ♦ rr♥t rt♦♥ ♣t ♥ t② ♣r♦♣♦s s♠♣ s♦t♦♥ t♦rss t ♥ ♣rtr ♦t t♦r s ♥♦t ♥ ♣♣ ♦r ♥♠r Nold ♦♦♥st st♣s ts t♦r ♦r t t ♥①t ♥tr ♥ ts ② t ♦t♣t♦ t ♦st ♥t♦♥ ♠♥♠③t♦♥ s sr ♥ t ♠♥ t ♥♦♥♦♣t♠♦t t♦r ♦♠ ♦♥srt♦♥s ♦♥ ts s♣t ♥ tr ♥ ts st♦♥

t♥ss ♦ t ♠♦r ♣rt♦♥ s r② s♦♥ ♥ rP ♥ t P ♣r♦♣♦s ♥ ❬❪ r ♦♠♣r ♦♥sr♥ t strt ♠♦♦♥ t ②♥ rsts r ♦t♥ t trs♦ Nold = 200 ♣♦ts s♦t rsts ♦t♥ t s♣ st♣ tst t rr♥ ♦ 100 ❬r♣♠❪ strt♥ r♦♠st♥st s r② sr s t P ♣rs♥ts ♥ ♥rsrr♥t r♣♣ t♦ t ♠♦ ♥q② ♥ t ♦♥trr② s♥ t P ①♣♦tst rr♥t ♠sr♠♥ts t ♥♦r♠t♦♥ ♦♥ t ♠♦ r ♦t♥ r♦♠ tt ♠♥ t ①ts ttr rr♥ tr♥ ♥ ♦ r♣♣ s ♥ t s♦ ♥♦♥ strt ♠♥ s♠ ♦♥srt♦♥s ♥ ♠ ♦♥ t st♥ t ♠♦r ♣rt♦♥ ♣rs♥ts r♥t st♥ ♣ttr♥ tt ♥ ts tst ♦s♥ ♦r ♥♠r ♦ ♦♠♠tt♦♥s s s♦♥ ♥

0 200 400 600 800 1000−1

−0.5

0

d−ax

is c

urre

nt [p

.u.]

MB MB−ref MF MF−ref

0 200 400 600 800 10000

0.5

1

q−ax

is c

urre

nt [p

.u.]

Time step [−]

rr♥t ♦r♠s

0 200 400 600 800 10000

200

400

600

800

1000

1200Commutations

Time step [−]

t♥

r ♦♠♣rs♦♥ t♥ P ♥ P t Nold = 200

s ♥ s♥ ♥ t rr♥t ♦r♠s ♦ t P ♣rs♥t s♦♠♥t s♣s s r ♥ ② t s♠♣ ♥tst♥t♦♥ s♦t♦♥ ♣r♦♣♦s ♥ ❬❪

Prt rr♥t ♦♥tr♦

♥ r♣♦rt t rr♥t ♦r♠s ♦t♥ t Nold = 100♥ Nold = 50 rs♣t②

0 200 400 600 800 1000−0.8

−0.6

−0.4

−0.2

0d−

axis

cur

rent

[p.u

.]

0 200 400 600 800 10000

0.5

1

q−ax

is c

urre

nt [p

.u.]

Time step [−]

Nold = 100

0 200 400 600 800 1000−0.8

−0.6

−0.4

−0.2

0

d−ax

is c

urre

nt [p

.u.]

0 200 400 600 800 10000

0.5

1

q−ax

is c

urre

nt [p

.u.]

Time step [−]

Nold = 50

r st t r♥t s ♦ t trs♦ Nold

♥tr♦♥ ♥♦♥♦♣t♠ ♦t t♦rs ♦s ♦♥ t ♦t t♦r st♥t♦♥ t t t s♠ t♠ ss s♣s ♥ t rr♥t ♦r♠ ♠♣t ♦ts s♣s ♣♥s ♦♥ sr s♣ts t r♦t♦r ♣♦st♦♥ ♥ t s ♦ ♥t♥ ♦r t s ♦ rt② t rsts ♦t♥ t Nold = 50 r ♥②s ♥ t♥ ♣rtr t♠ ♥tr ♦ 100 st♣s s ♦♥sr ♥ ♣♦st♦♥ ♦t q①s t t ♥♥♥ ♦ ts ♥tr s s♦♥ ♥ r♥ t ♥♦r♠♦♣rt♦♥ t ♣rt rt♦r ♦t♣ts t rsts ♦ t ♦st ♥t♦♥ ♠♥♠③t♦♥Prs② ♦♥sr♥ t r♦t♦r ♣♦st♦♥ t tr ♦♣t♠ t♦rs

❼ ♥① ♦s s♠ rt♦♥ ♦ t rt rr♥t ♥ ♥rss t q①srr♥t

❼ ♥① rss t ①s rr♥t ♦t♥♥ ♥t rt♦♥ ♦ tqrtr rr♥t

❼ ♥① t s ♦s♥ t♦ ♦t♥ s♠ rt♦♥s t♦rs t ♦r♥ ♦ ♦t trr♥ts s ♥ t ♣♦♥t ♦ ♦r s ♦s t♦ t rr♥

ttr ♥ t rt rr♥t rt♦♥s r ♣♦♥t t r ♦♦r ♥ ♥ t s♦♥ ♣♦t ♦ ♥ ♥ ♥ t♦♥ t♦ t ♦st ♥t♦♥♠♥♠③t♦♥ t ♦rt♠ ♥tr♦s r♥t ♥①s t♦ r♥t rq♥t ❯♣t rst ♣♦t ♦ s♦s t ♦ t t t♦rs t♠ t♦r s ♦♥ t♦ ♦ ♥ t s ♥♦t ♥ ♣♣ ♦r 50 ♦♥st st♣s t♦rt♠ ♣♣s t t ♥ s♥ tt ♥ ts s t st♥t t♦rs r 1 2 ♥ 3r♥ t ♦♦r tr t ♣♣t♦♥ ♦ ♦♥ ♦ ts tr ♥♦♥ ♦♣t♠ t♦rs ♥♦tr ♥♦♥♦♣t♠ t♦r s ♣♣ t♦ r♥ t rr♥ts t♦ t rr♥s t♦r ♥① 5 ♥ ts ①♠♣ t ♦♦r s ♥♦t ♦r ② t ♥tst♥t♦♥♦rt♠ s♥ t ♦s ♥♦t r t trs♦ Nold t t s ♦s♥ ② t ♦st ♥t♦♥t♦ r♥ t ♣♦♥t ♦ ♦r t♦ t rr♥ ♥ ♦tr ♦rs ts ♦t t♦r r♣rs♥ts ♥ ♦♣t♠ ♦ ♥ t ♠♦ stt♦♥ tr t ♣♣t♦♥ ♦ ♦ t♦rst t s ♥♦t t♥ s ♦♣t♠ r♥ t ♥♦r♠ ♦♣rt♦♥ s ①♠♣ s♦s ttr♥ t st② stt ♦♣rt♦♥ t ♥tst♥t♦♥ ♠t♦ ♣r♦♣♦s ♥ ❬❪ ♦rs

Prt rr♥t ♦♥tr♦

760 780 800 820 840 860−0.4

−0.35

−0.3

−0.25

−0.2

d−ax

is c

urre

nt [p

.u.]

760 780 800 820 840 860

0.32

0.34

0.36

0.38

0.4

q−ax

is c

urre

nt [p

.u.]

Time step [−]

rr♥t ♦r♠s

760 780 800 820 840 8600

20

40

60

Age

[−]

0 1 2 3 4 5 6 7

760 780 800 820 840 8600

2

4

6

Vol

tage

vec

tor

inde

x [−

]

Time step [−]

♥ ♦t t♦r ♥①

r t ♦ t P ♦♣rt♦♥

4

5 6

0 1

23

d

q

dq-axes

stagnantnew after stag.optimal

r t ♦ t P ♦♣rt♦♥

♦t t♦rs tt r ♥♦t ♦s♥ ② t ♦st ♥t♦♥ ♠♥♠③t♦♥ ♥ tt ♦ ♥ ♥ ♦♣♣♦st♦♥ t t ♦♣t♠ t♦rs

♦ ♦ Nold s s ♦♥ ♠♣r ♦♥srt♦♥s ♥ t r♣rs♥ts tr♦ t♥ t ♦♠♣♥ t t ♦st ♥t♦♥ ♠♥♠③t♦♥ ♥ t ❯ ♣t♥rq♥② s rt② ♥ t t ♣rt♦♥ r② ♥ ♥ tr♥ t t stt② ♦ t ♦♥tr♦ s♠ ♦r t trs♦ t ♠♦r rq♥t t ♣t ♥t r t ♥♠r ♦ ♥♦♥ ♦♣t♠ t♦rs ♣♣t♦♥s r t trs♦t r t ♦r♥ t t ♦st ♥t♦♥ ♦t♣t t t ♦r t ♣rt♦♥r② s ♦♥srt♦♥s r qtt② r♣♦rt ♥ t ♣♦t ♦

s♠♥ t ♦♥srt♦♥s ♥ t rsts s♦♥ s♦ r t t ♣♦ss t♦ ttt t ♥tst♥t♦♥ s♦t♦♥ ♣r♦♣♦s ♥ ❬❪ s ♠♥ r t ♣♣t♦♥ ♦ ♥♦♥ ♦♣t♠ ♦t t♦rs ♥ r② t♠ tt ♥ ♥rtr stt s♦♥ t♦ ♦ t ♦rt♠ ♣♣s t sr♥ t ♦t♣t ♦ t ♦st ♥t♦♥♠♥♠③t♦♥ ss t ♦ ♦ Nold r♣rs♥ts tr ♦ t♥ t qt② ♦t ♣rt♦♥ ♥ t ♦♠♣♥ t t ♦st ♥t♦♥ ♠♥♠③t♦♥ ♥ ♦tr ♦rs s♠ ♦ Nold ♠♣s rq♥t t ♣t ♥ r② ♦s st♥t♦♥ ♦t♥② ♣♣②♥ ♥♠r ♦ ♥♦♥ ♦♣t♠ ♦ts ♥ t ♦tr ♥ ♦♦s♥ r ♦ Nold ♦s ♣rsr♥ t ♦♣t♠ ♦♥tr♦ t t rr♥t ♣rt♦♥♦ sr② ♦rs♥ ♥② Nold s ♦♥ s ♦♥ ♠♣r ♦srt♦♥s ♥ts ♦ ♥♦t r♥t t stt② t r② ♦r♥ ♦♥t♦♥

Prt rr♥t ♦♥tr♦

Nold1

Optimality

Update frequency,

prediction accuracy,

stability

r tt r♣rs♥tt♦♥ ♦ t ♦ ♦ Nold

♦♥s♦♥

♥ ts st♦♥ t ♥r Prt rr♥t ♦♥tr♦ s sr ♥ ♥②s ♦♠♦♥srt♦♥s ♦t t ♦ ♦ t ♣r♦♣r ♦st ♥t♦♥ ♥ s ♦ s♥t ♠♥s ♥ ♣r♦ ♥ ♣rtr t t ♥ ② t r♦t♦r ♥s♦tr♦♣② ♥♥②s r♥ tr♥s♥t ♦♣rt♦♥s ♥♦♥ rr♥t st♣s t ♠♥ strt♣t s ♥ s♦♥ tt t ♦st ♥t♦♥ s ♦♥ t s♦t rr♦r ♦ rst ♥ s ♦r ♥ t ♣r♦rt② t♦ t ①s t t ♦r ♥t♥ rst stt r♥ t rst st♣s t rr♥t ♦♥ t ①s t t ♦r ♥t♥ st②rs ts rr♥ t ♦tr ♦♥ r♠♥s ③r♦ ♦tr t s tt t t♦rq♥ ♥ r♥ ts ♥tr rst♥ ♥ s♦r s♣ rs♣♦♥s t s ♥ s♦♥tt ♥tr♦♥ t♥ t♦r q t♦ t s♥② rt♦ ♦s ♦r♦♠♥ ts♦r ❲t ts s♦t♦♥ t s s ♦♠♣♥st ♥ t rr♦r ♦♥ t t♦ ①sr r ♥ ♥ ② ♠♥ r ♦ ts s♦t♦♥ s tt t rqrst ♥♦ ♦ t s♥② rt♦ t s ♦ t ♥t♥ ♥ ②♥ tss r str♦♥② ♣♥♥t ♦♥ t rr♥t t♦ r♦♥ strt♦♥ ♦r ts rs♦♥s♥ ♦♥st♥t t♥ t♦r ♦♥sr♥ t ♥♦♠♥ ♣♣r♥t ♥t♥s ♦♥♦t st ♦♠♣♥st♦♥ r♥ t ♠♥ ♦♣rt♦♥s rsts ♦t♥t t ♦st ♥t♦♥ s ♦♥ t sqr rr♦r s♦ s♥t② ♥♦r ♦♥ ♦t ①s ss t ♦s ♥♦t rqr t ♥♦ ♦ t ♠♥♥t♥s ♦r ts rs♦♥s ts ♦st ♥t♦♥ s ♥ ♦s♥ ♦r ts ♣rt ♦ t♦r

♦♣rt♦♥ ♦ t ♣rt rt♦r s ♥ sr ♦♥sr♥ t t♦st♣s ♣rt♦♥ rqr t♦ ♦♠♣♥st ♦r t ②s ♥tr♦ ② t t ♠♣♠♥tt♦♥ tr tt ♦t t ♦s ♥ t ♦r ♣rt♦♥ ♥♣rs♥t ♥ t ♦r♠r s t t ♦ ♣r♠tr ♠s♠t ♥ rt♦♥ ♥♥stt ♥ ♣rtr ♥ ②♥ ♣♣t♦♥s r♦♥ strt♦♥ s ♥ r♦♥ss t ♠♥ rs♣♦♥s ♦r ♠♦ ♥qs

Chapter 3

♦♥ st♥t♦♥ ♥ ♦r

Prt rr♥t ♦♥tr♦

t♦ ts ♥tr ♦r ♣rt♦♥ s ♥r♥t② r♦st t♦ ♣r♠tr rt♦♥s♥ t rr♥t rt♦♥s s ♦r t ♣rt♦♥ r rt② ♦t♥ r♦♠ ♠sr♠♥ts t ♣rt rt♦r ♣rs② ♥♦s t t ♠♦ r s♣t ♦r t♣r♦♣r ♦♣rt♦♥ ♦ t r s t ❯ ♣t ♥ ♣rtr ♣rs ♣rt♦♥ rqrs tt t ttr s ♣r♦r♠ rq♥t② ♦trs t s st♦r ♥ t ❯ r♥♦t r ♥②♠♦r ♥ t② ♦ t♦ sr ♦rs♥♥ ♦ t r ♣r♦r♠♥♥ ♥ t ♦rst s t ♦ss ♦ ♦♥tr♦ ❲tr ♦t t♦r s ♥♦t ♣♣ ♦rsr ♥trs t rt rr♥t rt♦♥ s ♥♦t ♣t ♥ t ♣rt♦♥ s♦♥ ts s ①tr♠② r♥t r♦♠ t t ♦r s ♣♥♦♠♥♦♥ s rrr s st♥t♦♥ ♦ t ♦t t♦r ♥ ts ♣tr ♥ ♥②ss ♦ t ts ♦t ♦t t♦r st♥t♦♥ s ♣r♦ tr tt ♥♦ ♥t st♥t♦♥ ♠t♦ s♣r♦♣♦s ♥ tst

♥tr♦t♦♥

P s ♥ ♦♣ ♥ ❬❪ ♥ ❬❪ ♥ ♠♣♠♥t ♦r t rr♥t ♦♥tr♦♦ ♥ P ♥ ②♥ rs♣t② s ♦rs s♦ t rt ♥ts ♥② t ♦r ♣rt♦♥ ♥ t ♦♥sq♥t ♥♥ ♣r♦r♠♥ ♦ t r rsts ♠♦♥trt t st② ♥ t t♥ss ♦ ts ♥ ♦ ♦♥tr♦s♦♥ tt P s ♣r♦♠s♥ t♥♦♦② t♦ r♥t ♥s t♦ ts ♥r♥t r♦st♥ss t♦ ♣r♠tr rt♦♥s ts s♣r ♦ ♣r♦ rt ♥ts ♥♥str ♣♣t♦♥s rst ♦ t ♦ ♣♣t② ♦ ♣rt ♦♥tr♦ ♦ rt② ①t♥ s♥ t s rr♥t② ♠t ② t ♠♥♥ ♥ ♦ t ♣rs♥♦ ♦ t ♠♦ ❲t♦t t trt♦♥ ♣rt s♠s ♦s tr t♥ss ♥ ♦♠ ♥st ♠♥ ♦tr sts ♠t♦s ♥r ♦♥tr♦t rst ♦ ♥ ♥str② ♥ ♥ ♣rtr ♥ rs ♣♣t♦♥s rtr♠♦r ts♥ ♦ ♥② rqr♠♥ts ♦♥ t ♥♦ ♦ t ♠♥ ♣r♠trs s sr♦tst♥♥ ts t rsts ♥ ♥ s ♦ ♠♣♠♥tt♦♥ s♥ t ♦♥② rqrs rr♥t♠sr♠♥ts ♥ t ♦s ♦♠♣① ♥ t♠ ♦♥s♠♥ ♦♠♣tt♦♥s s♥

♦♥ st♥t♦♥ ♥ ♦r Prt rr♥t ♦♥tr♦

s ♥ ♦rtss ♥♦ tr② t♥♥ ♦ ♥s ♥ trs♦s s ♦tr trt♦♥ ♦♥tr♦s♠s ♣r♦♣♦rt♦♥♥tr rt♦rs ♦srrs ♥ st♠t♦rs t s♥ tr♥ r♣rs♥ts ♥ ♥ ♠♦r ttrt tr t ♣♦sst② t♦ s ♥ t♦♣♠♥t ♦ ♥ ♥rs r ♠♥s t ♦♠♣t② r♥t rtrsts ♦ ♦♥tr♦ t♦t ♥♥ t ♦♥tr♦ ♦rt♠ ♥ t♦t ♦r♦ss♦♠♠ss♦♥♥ ♦r ♥tt♦♥ ♣r♦rs

s♣t t rt ♣♦t♥tts tr P s ♥ ♣rs♥t ♥ ❬❪ ♦♥② ♦♥♥♦tr ♦r ♣r♦♣♦s ② t s♠ t♦rs s ♥ ♣s ♦♥ t s♠ t♦♣r② rtr rsr s ♥ t♦ ♦♣ ts ♥ ♦ ♦♥tr♦ ♥ t♦ ♠ t ♦♠♣tt

rst r s♣t tt s t♦ rss s t s♦ ♦t t♦rst♥t♦♥ s ♣♥♦♠♥♦♥ ♦rs ♥ ♦ rr♥t rt♦♥ s ♥♦t ♣t♦r sr st♣s t rt ♦t t♦r s ♥♦t ♣♣ ♥ tt t♠ ♥tr♥ t rr♥t rs♣♦♥s ♥r rt♥ ♥rtr ♦t♣t r♠t② ♥s ♥♦t s♥ ♥ ♠♣t t t ♠♥ stt ♥ ♥rq♥t ♣t ♦ t ❯ss ♦♠♣t② r♦♥ ♣rt♦♥s tt r t ♠♥ ♥ ♥①♣t ♦♣rt♦♥s st♥t♦♥ s ♥♦t ♣r♥t ♦r r② s♦ t ♣r♦r♠♥ r r② ♦rs♥♥ ♥ ♣ t t t♦t ♦ss ♦ ♦♥tr♦

❲ ts s♥♥t ss s ♥♦t ♥ ♠♥t♦♥ ♥ ❬❪ t s ♥ rsst s♠♣ s♦t♦♥ ♥ ❬❪ s r② sr ♥ ♣tr ts ♠t♦ ♣rs♥tss♦♠ ♥t rs t ♦rs ♥♦♥ ♦♣t♠ t♦rs ♥ t rqrs t t♥♥ ♦ trs♦ tt s ♠ t ♠♣r ♦♥srt♦♥s ♥ tt ♦ ♥♦t st ♦rr② ♦♣rt♦♥

♥ ts ♣tr ♥♦ ♥tst♥t♦♥ s♦t♦♥ s ♣rs♥t t ①♣♦ts t rt♦♥s♣ t♥ t rr♥t rt♦♥s ♥ ② t t ♥rtr stts s ♦♥ ts tss t tr ♠♦st r♥t s st♦r ♥ t ❯ t♦ r♦♥strt t ♦tr ♦r tt♦ ♥ t♦rs s t s♠ rr♥t rt♦♥ ♥ ♦♥t♥♦s ♣t

♥②ss ♦ st♥t♦♥ ts

P s♠s ♣rs♥t ♥ ❬❪ ♥ ❬❪ r② ♦♥ t rr♥t rt♦♥s st♦r ♥t ❯ ♥ ♣rtr r② t♠ tt ♦t t♦r s ♣♣ t rt rr♥trt♦♥ s ♣t s ①♣♥ ♥ t ♥①t st♦♥s t rr♥t rt♦♥ ♥ ② rt♥ ♦t t♦r ♣♥s ♦♥ t ♠♥ stt ♦♥ t rr♥t ♦♥ t♣♦st♦♥ ♥ ♦♥ t s♣ ♥ ♣rtr ♦t ts ♠♣t ♥ s♥ ♣♥ ♦♥ tttr s ♠♥s tt r② t♠ ♦t t♦r s ♥♦t ♦s♥ ② t ♦st ♥t♦♥♦r ♥♠r ♦ st♣s t rt rr♥t rt♦♥ s ♥♦t ♣t ♥ t ♦♥srt♠ ♥tr ♥ ts s t s ♦♥t♥ ♥ t ❯ ♦ s♥♥t②r♥t r♦♠ t t ♦♥s r② t ♦♥r t t♠ ♥tr ♥ t ♣ts ♥♦t ♣r♦r♠ t r t rr♦r t♥ t ❯ ♥ t t rs♣♦♥s ♦t ♠♥ ♥t ts ♥ ② ts rr♦rs ♦♥ t ♦♥tr♦ ♣♥ ♦♥ t♠♥ ♦♣rt♦♥

♦r t s ♦ rt② ♥ ①♠♣ ♦ t ts ♦ st♥t♦♥ s ♣r♦ ♥ s♠t♦♥ s rr ♦t ♣r♦♥ s♣ st♣ t t strt♣ t ♣rt♦♥s s ♦♥ t ♥♦♠♥ ♠♦ ♥ tr 25 st♣s t ♠♦r ♣rt♦♥ s strt

♥②ss ♦ st♥t♦♥ ts

♠♦s strt♣ ♦s ♥ t ❯ t ♦rrt s ♦ rr♥t rt♦♥tr tt t P s t♦ tr t rr♥t ♥ t s♣ rr♥s t ♦rst♣ k = 2000 t ♦♥tr♦ s ♦st t rr♥t rr♦rs strt ♥rs♥ ♥ t s♣r② rs ③r♦ rst ♣♦t ♦ s♦s tt tr ♥st♥t k = 1000t ♣rt rt♦r strt ♦♦s♥ ♦♥② ♥①s 0 5 ♥ 6 s ♠♥s tt ♦♥② ttr rt rr♥t rt♦♥s r ♣t ♥ t ❯ t ♦tr r r♠♥s♦♥st♥t ♥ q t♦ t st ♣t t t ♥ ♦♥② ♥① 0 ♥ 6 r ♦♣t r♦♠t ♦st ♥t♦♥ ♠♥♠③t♦♥ ♥ t ♦♥tr♦ s ♥♦t t♦ r♥ t rr♥s rs♦♥ ♥ ♦♥ ♦♥sr♥ t s ♦ t rr♥t rt♦♥s ♥ t t♦ ❯♣♦tt ♥ t ♥ s♥ tt tr rt♥ ♥st♥t ♥ ♦t t ❯ t♠♦st ♦ t rr♥t rt♦♥s t s♠ s♥ ♥ ♣rtr ♥ t ❯ ♦ t rt①s t s r ♣♦st t t ♦♥ rt t♦ ♥① 6 ♠r② ♥ t q①s t st♦r s r ♣♦st t ♥①s 0 ♥ 1 ♦r ts rs♦♥ t ♣rtrt♦r ss t s st♦r ♥ t ❯ t♦ t t ♦st ♥t♦♥ ♥ t♦ rt rr♥t rr♦rs t ♥♦ ♥ ♣rtr t ♦♦ss ♥① 6 t♦ ♣r♦ ♥trt♦♥ ♦ t rt rr♥t ♥ 0 t♦ ♣r♦ ♥t rt♦♥ ♦♥ t q①s t ♦tr r♦♥ s st♦r ♥ t ❯ s② t ♦st ♥t♦♥ t♦♥ ♠♥♠♣♦ss ♦r t ♣rt rt♦r t♦ ♦♦s r♥t ♥①

0 1000 2000 3000 4000 5000−200

0

200

Spe

ed [r

pm]

0 1000 2000 3000 4000 5000−1

−0.5

0

d−ax

is [p

.u.]

0 1000 2000 3000 4000 50000

0.5

1

q−ax

is [p

.u.]

Time step [−]

♣ ♥ rr♥ts

01234567

Vol

tage

inde

x [−

]

−0.2

0

0.2

d−ax

is L

UT

[p.u

.]

0 1 2 3 4 5 6 7

0 1000 2000 3000 4000 5000

−0.1

0

0.1

Time step [−]

q−ax

is L

UT

[p.u

.]

❱♦t t♦r ♥①

r ①♠♣ ♦ t stt♦♥ ts

①♠♣ ♣r♦ ♥ ts st♦♥ ♦ t♦ qtt ①♣♥t♦♥ ♦ t♣♥♦♠♥♦♥ ♦ st♥t♦♥ t s ♥ s♦♥ t ♦r ♦ t ♣rt rt♦r♥ ♣rs♥ ♦ st♥♥t t♦r ♥ t s st♦r ♥ t ❯ r ♥♦t r ts ♥ r tt t rr♥t rt♦♥s r ♥♦t r② ♣t t ♦st ♥t♦♥t♦♥ s ♠sr♣rs♥t ♥ t♦ r♦♥ ♦ ♦ t ♦♣t♠ ♦t t♦r♥ t ♦rt♠ ♥trs ts ♦♥t♦♥ t ♦ ♠♣♦ss ♦r ♠ t♦ r♥ t♦♥tr♦ ♦ t ♠♥

♥r ♥②ss ♦♥ t ♣♥♦♠♥♦♥ ♦ st♥t♦♥ s ♥♦t ♣♦ss s♥ ts ts♣♥ ♦♥ ♠tt ♦ r♥t s♣ts s s t ♥ ♦ tst t ♦st ♥t♦♥t

♦♥ st♥t♦♥ ♥ ♦r Prt rr♥t ♦♥tr♦

Pr♦♣♦s ♥tst♥t♦♥ ♠t♦

♣r♦♣♦s ♣t♥ ♠t♦ s sr ♥ ts st♦♥ ♠ s t♦ ♥ trt♦♥s♣ t♥ t rr♥t rt♦♥s ♥ ② t t ♥rtr stts ♥ t♦①♣♦t t ♦r t ❯ ♣t ♥ ts ② t ♦ ♣♦ss t♦ s t ♠♦st r♥ts st♦r ♥ t t t♦ r♦♥strt t ♦trs s ♠t♦ s ①♣t t♦ ♣r♦ ♦♥t♥♦s ❯ ♣t ♥ tr♥ ♠♥s ♥ ♣rs ♥ r rr♥t♣rt♦♥ t♦t ♦r♥ ♥♦♥ ♦♣t♠ ♦t t♦rs

♦♥sr♥ t srt ♠♦ ♦ t ②♥ t rr♥t rt♦♥ ♥ t qrr♥ r♠ s ② t ③t ♦t t♦r ♥ ①♣rss s

∆iz,k = δi

∅ + δiz

r t t♦t rr♥t rt♦♥ ∆iz,k s ①♣rss s t s♠ ♦ t♦ r♥t t♦rs

♥♠② δi∅ = [δi∅d δi∅q ] ♥ δiz = [δizd δizq ] ttr ♥ ①♣rss s

δi∅ = Ai

k

δiz = Bu

z t z ∈ Q

r t ♦r♠r r♣rs♥ts t rr♥t rt♦♥ s ② t ♥ stt ♦ t ♥rtru∅ = u

z = [0 0] t z = 0, 7 ♥ ♦tr ♦rs ts s t ♦t♦♥ ♦ t s②st♠ t♥ ♥♣t ♦t t ♥tr rs♣♦♥s s♦♥ tr♠ δiz s ♥ ② t t♦t t♦r u

z t z ∈ Q ♥ t ♥ s♥ s t ♦r rs♣♦♥s ♦s②∆i

∅ = δi∅

t s ♦rt ♥♦t♥ tt δi∅ ♣♥s ♦♥ t t rr♥ts ♥ s♣ δi∅ =f(ik, ω) ss t s♦♥ tr♠ δiz ♣♥s ♦♥ t rr♥ts tt t t ♥t♥ ♦♥ t rt ♥ qrtr ♦ts tt ♥ t t ♥① ③ ♥t tr r♦t♦r ♣♦st♦♥ ϑ δiz = f(ik, z, ϑ) ♥ ♦tr ♦rs ♥ ♦tt♦r tt ♠♥s ①♥ z t rt rr♥t rt♦♥ ∆i rs t t rr♥tt s♣ ♥ t ♣♦st♦♥ ♥ ♣rtr t s ♣♦ss t♦ ①♣rss t rr♥t rt♦♥s② s♣tt♥ t♠ ♥ t t♦ q ♦♠♣♦♥♥ts ♦t♥♥

∆izd = δi∅d(ik, ω) + δizd(i

k) · ♦s(ϑ− ϑz)

∆izq = δi∅q(ik, ω) + δizq(i

k) · s♥(ϑ− ϑz)

r ϑz s t ♣♦st♦♥ ♦ t t ♦t t♦rs ♥ t αβ ♣♥ t st② stt t ♦♥st♥t rr♥t i

k ♥ s♣ ω t rt♦♥s ♥ r♣rs♥t st s♠ ♦ ♦♥st♥t ♦st ♥ ♥ ♦st♥ tr♠ t ♦♥st♥t ♠♣t r♣♦rts ts stt♦♥

t s ♦rt t♥ tt t s st♦r ♥ t ❯ ♥ r♦rsr t♠ ♥trs ♦r t r♥t ♠♥ stt t② r ♥♦t r♥②♠♦r ♥ t② ♦ str♦♥② r r♦♠ t ♦rrt s ♥ ts s t ♦st♥t♦♥ t♦♥ ♥ ♥ tr♥ t ♦ ♦ t ♥①t ♦t t♦r s s ♦♥ r♦♥s ♥ t♦ ♥ ♥♦rrt ♠♥ ♦♣rt♦♥

♣r♦♣♦s ♣t♥ ♠t♦ s s ♦♥ t ♦♦♥ r♠rs t st♣ ① t rr♥ts t s♣ ♥ t ♣♦st♦♥ t ♥ stt ♦♠♣♦♥♥t δi∅ st s♠ ♦r t t♦rs ∆i

z ss t♦ t r♦t♦r ♥s♦tr♦♣② t t

Pr♦♣♦s ♥tst♥t♦♥ ♠t♦

0 100 200 300−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5d−axis

Electrical angle θ [°]

Current variations [A]

0 100 200 300−0.15

−0.1

−0.05

0.05

0.1q−axis

Electrical angle θ [°]

zid

∅id z

iq

z

iq0

r rr♥t rt♦♥s t st② stt

t♦rs δiz r ♣ ♦♥ ♥ ♣s ♥ t q ♣♥ s t♥ t♦ ♦♠♥ tr♠sr t♦rs ∆i

z ♥ ①♣♦t♥ r♠rs ♥ t♦ r♦♥strt t ♦trs

t s ♦rt ♠♥t♦♥♥ tt ♥ ♦ ♦♥sr♥ t t♦rs ∆iz t t s♠

♥st♥t ♦r t s♠ rr♥tss♣ ♥ ♣♦st♦♥ r♥ t r r ♦♣rt♦♥t rr♥t rt♦♥s r ♦t♥ t r♥t ♥st♥ts ♥tr♦♥ ♥ ♣♣r♦①♠t♦♥♦r ts rs♦♥ st ♦rt♠ ♦r t sq♥ ♥tt♦♥ s t♦ s♥♥ ♦rr t♦ r♥t rq♥t ♣t ♦ t ❯ s ①♣♥ ♥

qtt r♣rs♥tt♦♥ ♦ r♠rs ♥ s ♣r♦ ♥

2,k12

3,k13

5,k1 6,k1

6

1

1,k114

4,k1

6

5

z,k1

z

z∅z Q∈

i

k

δδδ δ

δ δ Δ

Δ

∅Δ =

∅δ

k

Δδδii

i i

i i

i

i

ii

iii

i

i i i

i

i

i

i

r ❱t♦r r♣rs♥tt♦♥

r♥t sq♥s s =< s1, s2, s3 > ♦♠♣♦s ② tr ♠sr t♦rs ♥ s t♦ ♦r ♦♠♣t ❯ ♣t st♠t♥ t ♦tr ♦r ♦r s ♥t ♦♦♥ t sq♥s r ①♣rss ♦♥sr♥ z = 1 s t ♠♥♠♠ ♥① ♦t tr♣ts rsts ♥ s② ①t♥ t♦ z ∈ Q ② ♠♥ ♦ ♣r♦♣r ♥①st♥ ♥ r♦tt♦♥

♦♠♥♥ ♠①♠♠ ♦ tr rr♥t rt♦♥s ♥ ② t t♦rs t s ♣♦ss t♦ ♦t♥ t rt t♦ t ♥ t♦r ♥ tr r♥t ②s s rt♦♥s♣s r s ♥ ♦♥ ♦ t ♣♦ss sq♥s t♦ r♦♥strt t ♠ss♥s strt♥ r♦♠ t ♠♦st r♥t rr♥t rt♦♥s ♥ ♣rtr t ♥ t♦r♥ ♦t♥ ♦♠♥♥ t♦ ♦♣♣♦st t♦rs ①♠♣ ♥ tr♦♥stt t♦rs ①♠♣ ♥ ♦r tr t♦r t ♣s st ♦2/3π ①♠♣ ♥

♦♥ st♥t♦♥ ♥ ♦r Prt rr♥t ♦♥tr♦

i (2),k1δi2i (3),k1δi3

δi5i (5),k1 i (6),k1

δi6 Δi1

i (1),k1δi1δi4

i (4),k1

Δi4

i k

δi∅δi1 δi4+ =[00]+Δi1Δi4

=∅

Δi2

♦ ♦♣♣♦st t♦rs

δi4i (4),k1

δi5i (5),k1 i (6),k1

δi6i (1),k1δi1

Δi2

Δi1

δi∅Δi3 δi2δi3

i (3),k1 i (2),k1

i k

δi1 δi3+ =[00]δi2- +Δi1Δi3

=∅

Δi -Δi2

r ♦♥st t♦rs

i (2),k1δi2δi4

i (4),k1

i (6),k1

δi6 i (1),k1δi1Δi5

Δi1

δi∅Δi3δi3

i (3),k1

i k

δi5i (5),k1

δi1 δi3+ =[00]δi5+

=∅

Δi+Δi1Δi3Δi5+

3

r t♦rs t 2/3π♣s st

r ♦♠♥t♦♥ ♦ δiz rst♥ ♥ t ♥ t♦r [0 0]

q♥

♦♠♣♦s ② tr ♦♥st t t♦rs s =< 1, 2, 3 > s s♦♥ ♥ t①♠♣ ♦ ❯s ♥ ♣t t

∆i∅ = ∆i

1 +∆i3 −∆i

2

∆i4 = 2∆i

∅ −∆i1

∆i5 = 2∆i

∅ −∆i2

∆i6 = 2∆i

∅ −∆i3

rst t rt♦♥ ♦ t ♥ t♦r s ♦♥ ♦♠♥♥ t tr ♦♥st t♦rss ♥ tr tt ② ♥rt♥ t rt♦♥s♣ t♥ t♦ ♦♣♣♦sttt t♦rs ♥ t ♥ t♦r s ♥ t ①♠♣ ♦ t s ♣♦ss t♦ ♥t rr♥t rt♦♥s rt t♦ t tr ♦tr t t♦rs

q♥

♦♠♣♦s ② tr t t♦rs t t♦ ♦♣♣♦st s =< 1, 2, 4 > s s♦♥ ♥t ①♠♣ ♦ ❯s r ♣t t

∆i∅ =

1

2(∆i

1 +∆i4)

∆i3 = ∆i

2 +∆i4 −∆i

∅

∆i5 = 2∆i

∅ −∆i2

∆i6 = 2∆i

∅ −∆i3

rs♣♦♥s t♦ t ♥ t♦r s ♦t♥ ♦♠♥♥ t t♦ ♦♣♣♦st t♦rss ♥ tr tt t t t♦r ♦♥st t♦ t♦ t t♦rs ♥ ts s z = 3 ♥ ♦♥ ② ♥rt♥ t rt♦♥s♣ ①♣rss ♥ ♦t t♦ ♠ss♥ s ♥♠② 5 ♥ 6 r ♦♠♣t ①♣♦t♥ trt♦♥s♣ t♥ t♦ ♦♣♣♦st t♦rs

Pr♦♣♦s ♥tst♥t♦♥ ♠t♦

q♥

♦♠♣♦s ② t♦ t ♦♥st t♦rs ♥ ♥ t♦r s =< 1, 2, 0 > ss♦♥ ♥ t ①♠♣ ♦ ❯s ♥ ♣t t

∆i4 = 2∆i

∅ −∆i1

∆i5 = 2∆i

∅ −∆i2

∆i3 = ∆i

2 +∆i4 −∆i

∅

∆i6 = 2∆i

∅ −∆i3

rst t t♦rs ♥ ♦♣♣♦st♦♥ t t t♦ t t♦rs ♥ ♦♥♥rt♥ t rt♦♥s♣ ①♣rss ♥ ♥ ♦♠♥♥ ♦♥ ♦ t ♥♦♥ z = 2 ♥ ♦♥ r♦♥strt z = 4 t t ♥ t♦r t s ♣♦ss t♦ ♥t t♦r ♦♥st t♦ t ♦♥sr t♦ t t♦rs st s ♦♥①♣♦t♥ t rt♦♥s♣ ♥

q♥

♦♠♣♦s ② t♦ t t♦rs t ♣s st ♦ 2/3π ♥ ♥ t♦r s =< 1, 3, 0 > s s♦♥ ♥ t ①♠♣ ♦ ❯s ♥ ♣t t

∆i2 = ∆i

1 +∆i3 −∆i

∅

∆i4 = 2∆i

∅ −∆i1

∆i6 = 2∆i

∅ −∆i3

∆i5 = 2∆i

∅ −∆i2

tr s r s t♦ ♥ t t♦r t♥ t t♦ ♥♦♥ ttr tt s♠r② t♦ q♥ t ♥①t st♣ ♦♥ssts ♥ ♥♥ t ♠ss♥ t♦rs② ♠♥ ♦ t rt♦♥s♣ tt ♥s t♦ ♦♣♣♦st t♦rs ♥ t ♥

q♥

♦♠♣♦s ② tr t t♦rs t ♠t ♣s st ♦ 2/3π s =< 1, 3, 5 >s s♦♥ ♥ t ①♠♣ ♦ ❯s ♥ ♣t t

∆i∅ =

1

3(∆i

1 +∆i3 +∆i

5)

∆i2 = 2∆i

∅ −∆i5

∆i4 = 2∆i

∅ −∆i1

∆i6 = 2∆i

∅ −∆i3

rst st♣ ♦s ♥♥ t ♥ t♦r ♦♠♣♦♥♥t t t ①♣rss♦♥ ♥ tr tt t ♦tr tr t♦rs r ♦♥ s t ♦♣♣♦st ♦ ♦t ♠sr♥ r♦♥strt s

t ♦tr sq♥s tt ♦ ♥♦t strt t ♠♥t 1 ♥ ♦♥♥t t tsq♥s r♣rs♥t ♥ ♥ ♣rtr ts ♥ ♦♥ ② s♦rt♥ ♥ s♥♥

♦♥ st♥t♦♥ ♥ ♦r Prt rr♥t ♦♥tr♦

4

5 6

0

7

1

23

q♥

5 6

0

7

3

4 1

2

q♥

3

4

5 6

7

1

2

0

q♥

2

4

5 6

7

1

3

0

q♥

0

2

4

6

7

1

3

5

q♥

r ❱♦t t♦r sq♥s

SEQUENCE

IDENTIFICATIONSUBSET

SELECTION

LUT

UPDATE

idx

r q♥ ♥tt♦♥ ♦rt

♦rr t ♠♥ts ♦ t ♥r sq♥ t ts ♣♦♥t t rst ♠♥t s[1] s t♠♥♠♠ ♥① ♥ ♥ t sq♥ tr tt t ♠♥ts ♦ r strt② t q♥tt② s[1] − 1 ♥ ts ② ♦r ♣♦ss tr♣ts ♥ sq♥ s♦t♥ ttr strts t ♥① 1 ♥ ♣s t s♠ s♣t s♣♠♥tst♥ t tr t♦rs ♦ t ♦r♥ sq♥ t s ♦rt ♠♥t♦♥♥ tt ttr♣ts r② s ♠♥♠♥ ♠♥t q t♦ 1 t♥ t strt♦♥ s ♥♦ t s♥s[1]− 1 = 0

q♥ ♥tt♦♥

♣r♦♣r ♦rt♠ ♦r t sq♥ ♥tt♦♥ s ♥ ♦♣ ♦♥sr♥ t♦♦♥ rqr♠♥ts rst t s t♦ r♦♥③ tr t ♦t t♦rs ♦t♥② t ♦st ♥t♦♥ ♠♥♠③t♦♥ ♦r♠ st sq♥ ss t s t♦ r♦rt ♦rr ♥ t t♦rs r ♦s♥ s ♠♣s tt ♦♥sr♥ t ss♥ ♥ ♥ ♦♥② ♦♥ ♥ t♦r r♥t sq♥s r t♦ ♣tt ❯ ♥② t ♦rt♠ ♠st st ♦r r t♠ ♣♣t♦♥s

♦r s sr ♥ ♦rt♠ tr♣ts r stt s ♥ t ♣r♦sst♦♥ t s =< s1, s2, s3 >∈ S ♥ ♦r ♦ t♠ fs s ♥tr♦ ♥♣t ♦t t♦r ♥① ① s ♦♠♣r t s[fs] ♥ fs s ♣t ♦r♥②♥s ♥ ❲♥ tr♣t s ♦♠♣t ♥ ts tr ♠♥ts r ♦♥t ❯ ♣t s ①t ♥s ♥ tr t ♣t s ♣r♦r♠ t rst♠♥t ♦ t ♥t sq♥ s[1] t ♦st ♦t t♦r s sr t♦♦t♥ r ♣t r♥ t ♠♥ ♦♣rt♦♥ ♥ st A ♥s t ♥①s tt t st♣ ♥ s t♦ ♦r♠ st sq♥ t♦s t♦rstt r② ♥ ♦s♥ ② t ♦st ♥t♦♥ ♠♠♥♠③t♦♥ ♥ ♥ tt ♥♦t ♥ sr tr t ❯ ♣t

♦ r t ♦♠♣tt♦♥ r♥ t ♦rt♠ s t♦ st t♦ ♦ tt♦♥ ♦ t r♥t tr♣ts t st♣ ♦r ts rs♦♥t sq♥♥tt♦♥ s ♠ ♦♥sr♥ ♦♥② t sq♥s s ∈ Sidx,j ⊆ S r t ssts ♥ s Si,j ≡ s ∈ S|∃k, s[k] = i∧ s[1] = j s ♠♥s r♥ t rsr t♦ t sq♥s ♥♥ ① ♥ strt♥ t q s s tt sq♥s t tr ts rst ♠♥t s ① ♥ ① 6∈ A ♦r ts rst ♠♥t s q 6= ①♥ q ∈ A ♥

q♥ ♥tt♦♥

♦rt♠ q♥ ♥tt♦♥ ♥♣t ① s t rr♥t ♦t t♦r ♥① Sidx,q s t sst ♦♥t♥♥ ①

♥ strt♥ t q ∈ Q A s t sst ♥♥ t ♥①s tt ♥ st♦ ♦r♠ tr♣t fs∀s

♦r q ∈ Q ♦

♦r s ∈ Sidx,q ♦ q ∈ A ∧ q 6= idx ∨ ① 6∈ t♥ ① == s[fs] t♥ fs ← fs + 1

A ← A∪ ① ♦r q ∈ Q ♦

♦r s ∈ Sidx,q ♦ fs == 3 t♥

♥t♦♥ ❯ ❯P A ← A\s[1]

s♠t r♣rs♥tt♦♥ ♦ t ♦rt♠ s r♣♦rt ♥ s ①♣♥t ♦♥ssts ♦ tr ♠♥ ♣rts rst sst st♦♥ ♦s r♥ t t♦t♥♠r ♦ sq♥ t♦♥s ♦♥sr♥ t ♥ ♥♣t ♥① ① ♦r ♦ t♦rt♠ s ♥ t t ♦ sq♥ ♥tt♦♥ ♥ t ♥①s r♦♠♣r t t sq♥s ♥ t rt s r ♣t ♥② tr sq♥ s ♥t t ❯ r ♣t ♦r♥②

♥ ①♠♣ ♦ t ♦rt♠ ♦♣rt♦♥ s r♣♦rt ♥ ♥ t t t♦t♣t ♦ t ♦st ♥t♦♥ ♠♥♠③t♦♥ r st st♣ ② st♣ ♥ t rt t♦♣rt♦♥ ♦ t ♦rt♠ r r② sr ♥ t ①♠♣ t t rst st♣ t♥♣t ♥① s 1 ♦rt♠ ♣t t ♦ t sq♥s strt♥ t t tt s♦♥ st♣ t ♥♣t s ♥① 4 s ♦ t sq♥s strt♥ t ts ♥①r ♣t ss t sq♥s strt♥ t 1 ♥ t 4 ♥ t s♦♥ ♣♦st♦♥r s ♣t t t ♥①t st♣ t ♥♣t s 0 sq♥s strt♥ t ts♥① ♥ t♦s strt♥ t 4 ♥ ♥ 0 ♥ t s♦♥ ♣♦st♦♥ r ♣t ♥s =< 1, 4, 0 > s ♥♦t ♣♦ss sq♥ t s ♥♦t ♣♦ss t♦ ♣t t ❯ t tsst♣ t t st st♣ t ♥♣t ♥① s 2 t ts ♣♦♥t t sq♥ s =< 4, 0, 2 >s ♥t ♥ t s ♦r t ❯ ♣t rst ♠♥t ♦ t sq♥♥♠② 4 s t♥ sr s ♠♥s tt ts ♥① s ♥♦t ♥②♠♦r ♦rt ❯ ♣t s♠♣ ①♠♣ r♣rs♥t ♥ ♠♦strt t ♦r

1

1 4

1 4 0

1 4 0 2

update: s=<1,-,->

update: s=<1,4,->, s=<4,-,->

update: s=<4,0,->, s=<0,-,->

update: s=<4,0,2>, s=<0,2,->s=<2,-,->

identify: s=<4,0,2>

discard: s[1]=4

r ①♠♣ ♦ t sq♥ ♥tt♦♥

♦ t ♦rt♠ t s ♦rt t♥ tt tr t rst ♠♥t ♦ t ♥tsq♥ s sr 4 ♥ t ①♠♣ tr st ♦ ♦r ♥①s 1 ♥ t①♠♣ s ♦ r♣rs♥t ♥ ss s♥ t ❯ ♣t ♦ ♣r♦r♠ s♥

♦♥ st♥t♦♥ ♥ ♦r Prt rr♥t ♦♥tr♦

♦ ♥① s♣t t sr♥ ♠t♦ ♠♣♦② ♥ t ♦tr ♥ sr♥ t♦r s ♦ rqr t ♥♦ ♦ t ♦ t ♥①s ♦r t st t ♦rr♥ t ♥①s r ♣t ♥s t st A s s♦t♦♥ s ♥♦t ♥ ♠♣♠♥ts t ♦ ♠ t ♦rt♠ ♠♦r ♦♠♣① ♦♥ ♥ts t rqr♠♥t ♦♣♣t② ♥ r t♠ ♣♣t♦♥s ss t ♥②ss ♦ t rsts ♦t♥r♥ t ♣r♠♥r② tsts ♦♥r♠ tt t ♠♣♠♥t s♦t♦♥ s s♥t②♥t ♥ tt t ♦s ♥♦t sr r♦♠ t ♦ ♠♥t♦♥ ss

s♦s t ①s ❯ t♦t ♥ t t ♣r♦♣♦s ♣t♥ ♠t♦ s♣s st♣ tst s ♥ ♣r♦r♠ ♥ t s♠t♦♥ s♦ ♥s r ♥②t②♦♠♣t ♥ ♦♠♣r t t s ♦t♥ t t rr♥t rt♦♥ ♠sr♠♥ts s ♥s t s ♦rt t♥ tt t♦t ♥② ♠t♦ t♦ ♣r♥t♥ ♦ st♥t♦♥ t ♦ss ♦ ♦♥tr♦ ♦rs s t ♥ r♦♥s st r ♥s r ♥♦t s♥s♦s ♥②♠♦r t t ♥ ♦ t tst s ♣♣♥s s♥ trr♥ts ♦ ♥♦t tr t rr♥s ♦♠♠♥ ② t s♣ rt♦r t♦ ♣r♦ trqr t♦rq ♥ ts stt♦♥ t s♣ r② rs ♥t t ♦♠s ③r♦♥ t ♦♥trr② t♥s t♦ t ♣r♦♣♦s ♥tst♥t♦♥ t♥q t ❯ r ♦♥t♥♦s② ♣t ♦t♥♥ ♣rs ♥ r ♣rt♦♥ ♥ tr♥ r♥tst ♣r♦♣r ♦♣rt♦♥ ♦ t ♠♥

−0.02

0

0.02

∆ i q0 [p

.u]

−0.05

0

0.05

∆ i q1 [p

.u]

−0.05

0

0.05

∆ i q2 [p

.u]

−0.05

0

0.05

∆ i q3 [p

.u]

−0.05

0

0.05

∆ i q4 [p

.u]

−0.05

0

0.05

∆ i q5 [p

.u]

0 500 1000−0.05

0

0.05

∆ i q6 [p

.u]

Time step [−]0 500 1000

−0.02

0

0.02

∆ i q7 [p

.u]

Time step [−]

Correct Actual

❯ st♥t♦♥

0

0.01

0.02

∆ i q0 [p

.u]

−0.05

0

0.05

∆ i q1 [p

.u]

−0.05

0

0.05

∆ i q2 [p

.u]

−0.05

0

0.05

∆ i q3 [p

.u]

−0.05

0

0.05

∆ i q4 [p

.u]

−0.05

0

0.05

∆ i q5 [p

.u]

0 500 1000−0.05

0

0.05

∆ i q6 [p

.u]

Time step [−]0 500 1000

0

0.01

0.02

∆ i q7 [p

.u]

Time step [−]

Correct Actual

❯♣t ❯

r t♥ss ♦ t ♣r♦♣♦s ♦rt♠

♦ t♦♥

SynRMMASTER

MACHINE

r ①♣r♠♥t st♣

♣r♦♣♦s s♠ s ♥ ♠♣♠♥t ♦♥ t ①♣r♠♥t st♣ ♦ t s♠♣♥ t♠ ♦ Ts = 200[µs] ♥ t ♠♥ t ♦ t ②♥ r r♣♦rt♥ st② stt ♣r♦r♠♥ ♥ t ♥ ♦♠♣r t

♦ t♦♥

♦t♦r t

P♦ ♣r ♥♠r p 2

Ps rsst♥ R 4.5 Ω

rt ♥t♥ Ld 60 mH

rtr ♥t♥ Lq 190 mH

♦♠♥ rr♥t IN 10 Apk

trt♦♥ P s♠ s♦s t rsts ♦t♥ t ♦♥st♥t s♣n = 100 ❬r♣♠❪ ♥ ♦r r♥t ♦s rr♥t rr♥s r ♥rt ♦♥sr♥♥ ♥strt ♠♥ ss♠♥ ♥r ①♠♠ ♦rq Pr ♠♣rP trt♦r② ♦s② t♦ r♦♥ strt♦♥t t P rs r♦♠t ♦♥ ♥ ♦r ts rs♦♥ t ♠♥ ♦♣rt♦♥ s ♥♦t ♦♣t♠ ♦r tr♥t t trt♦r② rqrs t ♣rs ♥♦ ♦ t ♠♥ ♠♦ t♦t ♦ ♦ s♥ t P s ♥t rs ♠♦r ♦ ♦sss ♥r ♠①♠♠ t♦rq ♣t② t ♦s ♣♥ t s♠ ♦♠♣t② r r♦♠t ♠♦

rst tst s ♥ ♣r♦r♠ t♦ r② t t♥ss ♦ t ♣r♦♣♦s P♥ ♣rtr tr♥st♦♥ t♥ t ♦s ♥ t ♦r ♣rt♦♥ s♥ ♦♥ r♥ t st② stt ♦♣rt♦♥ ♦ t ♠♥ rsts r r♣♦rt♥ r t ♥st♥t ♥ t tr♥st♦♥ s ①t ♥ s② r♦♥sst♣ k = 500 t s ♦rt t♥ tt tr t tr♥st♦♥ t r♣♣ s r♥ ♦t ①s t s ♠♦r ♥t ♥ t qrtr ①s r r♦♥ strt♦♥ss r♠r ♠s♠t t♥ t♥ t t ♥ t ♥♦♠♥ ♠♦ s♦r t ♣rt♦♥ r♥ t P st t ❯ r ♣t t t ♣r♦♣♦s♥tst♥t♦♥ ♠t♦ t t② r ♥♦t s ♦r t ♣rt♦♥

0 250 500 750 1000−0.55

−0.5

−0.45

−0.4

−0.35Load 0.4 [p.u.]

d−ax

is c

urre

nt [p

.u.]

0 250 500 750 10000.35

0.4

0.45

0.5

0.55

q−ax

is c

urre

nt [p

.u.]

Time step [−]

r r♥st♦♥ t♥ P ♥ P

s♦s t q rr♥t ♦r♠s ♦ t P ♥ t ♣r♦♣♦s P t ♥ s♥ tt t ♦ ♦ t ♣r♦r♠♥ ♦ t t♦ s♠s r ♦♠♣r♥ tr♠s ♦ r♣♣ ♥ ts stt♦♥ t ①s ♣rs♥ts ♥ r r♣♣ t♦ t r♦t♦r♥s♦tr♦♣② Lq > Ld t r ♦s r♦♥ strt♦♥ s ♠♦r ♥t♥s ♥ t tsr ♠♥② ♥ t qrtr ①s s ♦♥sq♥ t ♠♦ s ♦r t ♣rt♦♥s s♥♥t② r♥t ♥ t q①s t ①s s ss st t♦ r♦♥ strt♦♥ st ♣♦t ♦ s♦s tt r♦♥ ♠♦ s ♥ t ♣rt♦♥ ss

♦♥ st♥t♦♥ ♥ ♦r Prt rr♥t ♦♥tr♦

♥ ♥rs ♥ t rr♥t r♣♣ ♦♥r♠♥ t ♥②ss ♣rs♥t ♥ ❬❪ s ts r ♥ t qrtr ①s r t ♥t♥ s ♦rst♠t t ♦ ♦st rr♥t r♣♣ s r ♥ t ①s ♣rs♥t♥ t ♦r ♥t♥ ♥♠② t①s ♣♥♥ ♦♥ t str♥t ♦ t r♦♥ stt♦♥ ♥ ♥ tr♥ ♦♥ t ♣r♠tr♠s♠t t rr♥t ♦r♠s ♦ ♣rs♥t t ♦♣♣♦st ♦r t r♣♣s r ♥ t ♦tr ①s s ♥ s♥ t P ♦s ♥♦t ♣rs♥t t s♠t ♥ t rr♥t r♣♣ ♦s ♥♦t ♥ t t ♦

0 500 1000−0.12

−0.1

d−axis current [p.u]

mL=

0.1

[p.u

.]

0 500 10000.08

0.1

0.12

q−axis current [p.u]

0 500 1000

−0.3

−0.2

mL=

0.4

[p.u

.]

0 500 10000.2

0.3

0 500 1000

−0.5

−0.4

−0.3

mL=

0.7

[p.u

.]

0 500 10000.3

0.4

0.5

0 500 1000

−0.7

−0.6

−0.5

mL=

1 [p

.u.]

Time step [−]0 500 1000

0.5

0.6

0.7

Time step [−]

rr♥t ♦r♠s

d−axis q−axis0

0.5

1

1.5

2x 10

−3 Load 0.1 [p.u.]

d−axis q−axis0

0.02

0.04

0.06Load 0.4 [p.u.]

d−axis q−axis0

0.05

0.1

0.15

0.2Load 0.7 [p.u.]

d−axis q−axis0

0.05

0.1

0.15

0.2Load 1 [p.u.]

MB MF

rtr♦♥

r ①♣r♠♥t rsts t 100 ❬r♣♠❪ ♥ r♥t ♦s

t ♦ t ♣rt♦♥ r② s s♦♥ ♥ r t r s♦ ♥♥ t s ♥ r♣rs♥t t t ♥ t ♣rt rr♥t t ♥ s♥tt t ♣rt s s rt s♥ t ♣rt② r♣s t ♦r♠ ♦ t♠sr rr♥t t ♦♥ st♣ ♥ ♥ ♣rt♦♥ r② ♦ t t♦ s♠ss ♦♠♣r ♥ ② ♠♥ ♦ t ♥tr qr rr♦r rtr♦♥ t ♥ s♥ tt ♦r t P t r t ♦ t r t ♣rt♦♥ rr♦r s② t ♠s♠t ♥ t ♠♦ ss t q①s rr♦r s r t♥ t ①s ♦♥ P rr♦rs r ♦♥st♥t ♥ ♦ ♥♦t ♥ ♥ t ♦r tsts t♥ss♦ t ♠♦ r ♣rt♦♥ s ♣rtr② ♥t ♥ t st rsts r② s♦tt t ♣rt♦♥ s r ♥ r② ♦r♥ ♦♥t♦♥

540 545 550 555−0.12

−0.1

−0.08

−0.06d−axis current [p.u]

Load

0.1

[p.u

]

540 545 550 5550.095

0.1

0.105q−axis current [p.u]

540 545 550 555−0.3

−0.28

−0.26

−0.24

Load

0.4

[p.u

]

Time step [−]540 545 550 555

0.26

0.27

0.28

0.29

0.3

Time step [−]

♦ ♦

540 545 550 555−0.48

−0.46

−0.44

−0.42d−axis current [p.u]

Load

0.7

[p.u

]

540 545 550 5550.4

0.45

0.5q−axis current [p.u]

540 545 550 555−0.68

−0.66

−0.64

−0.62

−0.6

−0.58

Load

1 [p

.u]

Time step [−]540 545 550 555

0.62

0.64

0.66

0.68

0.7

Time step [−]

♦

r t ♦ t ♣rt♦♥ r②

♥ s♦ t ♦t♣t ♦ t ♦st ♥t♦♥ ♠♥♠③t♦♥ ♥t ♥♠r ♦ st♥ ♣r st♣ ♦r ♦t t P ♥ t P t ♥ ♥♦t tt st♥ ♣ttr♥ ♦ t t♦ s♠s ♥ t tst t ♦ 0.1[p.u.] s r②

♠tt♦♥s ♥ ♥r♥t st♥t♦♥

s♠r s ♦♥r♠s tt t ♥♦♠♥ ♠♦ s ♦r t ♣rt♦♥ ♥ t Ps s♥t② ♣rs ♥ ts ♦r♥ ♦♥t♦♥ r♥t② t r ♦s t ♦st♥t♦♥ ♦♦ss r♥t ♦t t♦rs ♥ t t♦ s♠s ♥ ♣rtr t P♦s ♥♦t ♣rs♥t ♥s t♥ ♦♣♣♦st t♦rs ♥♦ tr♥st♦♥s tt rqrs ♦♠♠tt♦♥s r ♥♦

0 500 10000

2

4

6MBPCC

Load

0.1

[p.u

]

0 500 10000

2

4

6MFPCC

0 500 10000

2

4

6

Load

0.4

[p.u

]

Time step [−]0 500 1000

0

2

4

6

Time step [−]

♦ ♦

0 500 10000

2

4

6MBPCC

Load

0.1

[p.u

]

0 500 10000

2

4

6MFPCC

0 500 10000

2

4

6

Load

0.4

[p.u

]

Time step [−]0 500 1000

0

2

4

6

Time step [−]

♦

r ♦st ♥t♦♥ ♦♣t

0 500 10000

1

2

3MBPCC

Load

0.1

[p.u

]

0 500 10000

1

2

3MFPCC

0 500 10000

1

2

3

Load

0.4

[p.u

]

Time step [−]0 500 1000

0

1

2

3

Time step [−]

♦ ♦

0 500 10000

1

2

3MBPCC

Load

0.7

[p.u

]

0 500 10000

1

2

3MFPCC

0 500 10000

1

2

3

Load

1 [p

.u]

Time step [−]0 500 1000

0

1

2

3

Time step [−]

♦

r ♠r ♦ ♦♠♠tt♦♥ ♣r st♣

s♦s t ❯ ♦r t tsts t ♦ ♥ t ♥ s♥ tt t ♣t s rq♥t ♥ t ♥ ♦s ♦t♥♥ s♠♦♦t rr♥t rt♦♥ ♦r♠s

♠tt♦♥s ♥ ♥r♥t st♥t♦♥

rsts s♦♥ ♥ t ♣r♦s st♦♥ t t st② ♦ t ♣r♦♣♦sss t ♦♠♣rs♦♥ t t P ♦♥r♠ ts t♥ss rtss ts ♦rt ♠♥t♦♥♥ ♥ t ts ♥♦♥tr r♥ t ♠♣♠♥tt♦♥ ♥ r t stt♦♥s tt r♣rs♥t rtt② ♦r t P

rst t ①sts ♦♠♥t♦♥ ♦ tr ♥①s tt ♦s ♥♦t ♦ t ❯ ♣t♥ ♣rtr ♦r t tr♣ts tt ♥ r♣rs♥t s t ♦♥ ♥ t sq♥s ♦♠♣♦s ② t♦ ♦♣♣♦st t♦rs ♥ ♥ t♦r t s ♥♦t ♣♦ss t♦

♦♥ st♥t♦♥ ♥ ♦r Prt rr♥t ♦♥tr♦

0 500 1000 1500−1

−0.5

0

0.5

1d−axis

Load

0.1

[p.u

.]

0 500 1000 1500−0.2

−0.1

0

0.1

0.2q−axis

0 500 1000 1500−1

−0.5

0

0.5

Load

0.4

[p.u

.]

Time step [−]0 500 1000 1500

−0.5

0

0.5

Time step [−]

♦ ♦

0 500 1000 1500

−1

−0.5

0

0.5

d−axis

Load

0.7

[p.u

.]

0 500 1000 1500

−1

−0.5

0

0.5

q−axis

0 500 1000 1500

−1.5

−1

−0.5

0

0.5

Load

1 [p

.u.]

Time step [−]0 500 1000 1500

−1.5

−1

−0.5

0

0.5

Time step [−]

♦

r rr♥t rt♦♥s ❯

2

5 6

7

3

4 10

r ❯♥st sq♥

r♦♥strt t ♦tr ♥♥♦♥ s ♦r t s ♥ tt ts stt♦♥ t②r♣rs♥t rtt② r♥ t P ♦♣rt♦♥ rs♦♥ ♥ qtt②①♣♥ s ♦♦s ♣rt rt♦r ♥r♥t② r♥s t ♣♦♥t ♦ ♦r s♦s s ♣♦ss t♦ t rr♥ ♥ ♦♥ t rs t t t♥s t♦ r♠♥ ♦s t♦ t ts ② ♥ tt t♦ ♦♣♣♦st t♦rs tt s ♦♣♣♦st rs♣♦♥ss r tr♥t②♥ ♥q② t♦tr t t ♥ t♦r ♦s♥ s t ♦♣t♠

s♥♥t rtt② ♦♥r♥s stt♦♥s ♥ t st♥t♦♥ s ♥♦ ♣r♦♣♦s ♥tst♥t♦♥ s♦t♦♥ s s ♦♥ t ♥tt♦♥ ♦ sq♥s ♦♠♣♦s ② tr♣ts ♦ ♦t t♦rs ♥①s s ♦♥sq♥ ♥♦ sq♥s r♥t t ♣r♦♣♦s ♠t♦ ♥♥♦t ♦r rsts s♦♥ ♥ ♦♥r♠tt r♥ t ♥♦r♠ ♠♥ ♦♣rt♦♥ s♥t ♥♠r ♦ sq♥s r ♥tt♦ r♥t t st ♦♣rt♦♥ ♦ t r ♥ t ♦tr ♥ ♣rtr ♦♣rt♦♥s s s st tr♥s♥ts ♥♦♥ rr♥t rr♦rs ♦ r♣rs♥t ♥ ♦st t♦t ♣r♦♣r ♦♣rt♦♥ ♦ t P Prs② ♥ ♣rs♥ ♦ rr♥t st♣s t s♠♦t t♦r ♦ ♦s♥ ♦r ♥♠r N ♦ ♥tr r♥ ts t♠ ♦♥② ♦♥ ♦ t ❯ s ♣t t ♦trs r ♥♦t s♥ ♥♦ tr♣ts r ♦r♠ ♦rts rs♦♥ tr N st♣s t t stt ♦ t ♠♥ s rst② ♥ t♦r ♣rt♦♥ t ♠♥ t♦ ♥①♣t ♦♣rt♦♥s r t ♦ N t r t rs ♦ ♥stt② t s ♦rt t♥ tt ts stt♦♥r♣rs♥ts rtt② ♥ ♦r t P ♣r♦♣♦s ♥ ❬❪ s♥ tr ♠t♦ ♦♣♣② s① ♥♦♥ ♦♣t♠ t♦rs ♦♥st②

♣rt♦♥ s ② t ♣rt rr♥t rt♦r ♥ t ♣r♦♣♦s Ps s ♦♥ t ♠sr rr♥t rt♦♥s ♥ ts t rr♥t qst♦♥ s t♦ ♣rs ♥ r r♦♠ rq♥② ♥♦ss ♦trs t s st♦r ♥ t ❯♥ t♥ s ♥ t ♣rt♦♥ r ♥♦t r t s t♦ ♥♦t tt r♦♥ s st♦r ♥ t ❯ ts ♦ tr s t♦ ♣t t rr♥t rt♦♥s

♠tt♦♥s ♥ ♥r♥t st♥t♦♥

rt t♦ ♦ ♦t t♦rs rst♥ ♥ s♦♥ ♦ t rr♦r ♥ t ♦ t ts ♦rt ♠♥t♦♥♥ tt t ♣rs♥ ♦ ♦st ♥ t ♠sr♠♥ts ♦s ♥♦t r♣rs♥t rtt② ♦r t ♦♥tr♦ s♠ rs♦♥ s tt r② ♣♦ss ♦♥st♥t ♦st s♠♥t ② t rr♥t strt♦♥ ♥ t ♦tr ♥ rq♥② ♥♦s ♦t t ♦r ♦ t s♠ ♦ ♣r♥t ♥② r♦♥ ♦♣rt♦♥ s ② t ♣rs♥ ♦ ♥♦s ♥ t ♠sr ♦r♠s t ❯ ♥ ♦♣ss tr ss♦t♦♥ ♦s ♦♥ rr♦♥♦s s♣s ♥ t s st♦r ♥ t ❯ t♠♦♥st♥t ♦ t tr s t♦ ♣r♦♣r② s♥ ♥ ♦rr t♦ t t rq♥②♥♦s ♥♦t ♥tr♦♥ ②s ♥ t rr♥t rt♦♥ ♦r♠s s r② ①♣♥ ♥ t str♥ ♥ ② t st♥ ♥ ♦ ② ♣r♦♣r②♣♥ t rr♥t s♠♣♥ s ♥ ♦♥ ② st♥ t s②♥r♦♥③t♦♥ ♦ t♥trr♣t sr r♦t♥ ♥ t ♦tr ♥ ♥ t ①♣r♠♥t st♣ ♦ t ♦rt♦r② s t♦ ♠♥s r s♠t♥♦s② ♦♥tr♦ ♥♠② t ②♥♥ t ♠str t♦ ♣♦r ♦♥rtrs r ♥♦t s♦t s♥ t② sr t r♦♥♦♥♥t♦♥ ss t♦ t s♠♣♥ t♠s s ♥ t t♦ r t s♠ ♦r r♠t♣s tr st r s♦♠ st r♥s ♦r ts rs♦♥s t str♥s♥tr♦ ② t ♦♠♠tt♦♥s ♦ t ♥rtr ♦ t ♠str ♠♥ t t rr♥t♠sr♠♥ts ♦ t ②♥ ♥ t s♠ r♥ t♥ tr rq♥s tstr♥ s ♥♦t ① ♥ rt♥ ♣♦st♦♥ ♦ t s♠♣♥ t♠ ♦ t ②♥ t tss r♦♠ t ♥♥♥ t♦ t ♥ t s r tt t s ♥♦t ♣♦ss t♦ ♦ ts ♥♦ ♦♠♦♣♦r str♥ ② s② ♣♥ t rr♥t s♠♣♥s ♥ ♣r♦♣r ♣♦st♦♥s ♠♥s tt ♣r♦② ♥ t str♥ ♦r ♥ ♦rrs♣♦♥♥ t♦ trr♥t s♠♣♥s t tr♣s rr♥ts t ② t t♥ ♥ tr♣ss②st♠ ♦♥② t♦ rr♥ts r ♠sr t tr s ♦t♥ ② strt♥ t♦tr t♦ s s♦♥ ♥ r ia ib ic r t t rr♥t isa i

sb i

sc r t

s♠♣ s ♥ ∆i s t ♦♠♦♣♦r str♥ ts ♠t♦ s ♠♣② tstr♥ ♦♥ t tr ♣s s②st♠ s t♥ tr♥srr t♦ t q♥t t♦♣s s②st♠ ♥ t ♦♥tr♦ s♠ s ♦♥sq♥t② t s t s s♦♥ ♥♦♥sr♥ t r tr♥s♦r♠t♦♥ ♥ r t ♥ s♥ tt ♥ t t♦♣s rr♥ts ♥ tr♠ ♥ ② t ♦♠♦♣♦r str♥

isa = ia +∆i

isb = ib +∆i

isc = −isa − isb = −ia − ib︸ ︷︷ ︸

ic

−2∆i

isα = 23

[isa − 1

2 isb − 1

2 isc

]= 2

3

[ia − 1

2 ib − 12 ic +

32∆i

]

isβ =√33 [isb − isc] =

√33 [ib − ic −∆i]

r♥t② t tr rr♥ts r ♠sr t str♥ s ♥♦t tr♥srr t♦ t t♦ ♣s s②st♠ ♦r ts rs♦♥ ts s♦t♦♥ s ♥ ♦♣t♥ t ♠♣♠♥tt♦♥

isa = ia +∆i

isb = ib +∆i

isc = ic +∆i

isα = 23

[isa − 1

2 isb − 1

2 isc

]= 2

3

[ia − 1

2 ib − 12 ic]

isβ =√33 [isb − isc] =

√33 [ib − ic]

♦♥ st♥t♦♥ ♥ ♦r Prt rr♥t ♦♥tr♦

♥♦tr ♣♦ss s♦t♦♥ tt ♦s ♥ t♦ ♦♥② rr♥t ♠sr♠♥ts ♦♥ ♦♠♦♣♦r str♥ ♥ t t♦ ♣s s②st♠ ♦♥ssts ♥ ♠sr♥ t♦rr♥t r♥s s s♦♥ ♥ ♥ t ♦♥sq♥t t♦ ♣s tr♥s♦r♠t♦♥rsts s ♥

isa − isb = ia − ib

isb − isc = ib − ic

isα = 23 [(i

sa − isb) + 1/2(isb − isc)] =

23

[ia − 1

2 ib − 12 ic]

isβ =√33 [isb − isc] =

√33 [ib − ic]

♥♦tr str♦♥ ♠tt♦♥ ♦ t ♣r♦♣♦s ♠t♦ s r♣rs♥t ② t ♦♠♣tt♦♥ r♥ t♦t t sq♥ ♥tt♦♥ ♦rt♠ ♦s ♥♦t ♥ ♦♠♣①t♦♥s ♦♥② ♦ ♦♠♣rs♦♥s ♥ t♦♥s r ①t t ♥♠r♦ sq♥s tt t♦ t ss ♥ ♥rs ♦ t tr♥r♦♥ t♠ r t♠ ♣♣t♦♥ s ♦♦s② ♦♥str♥ ② t s♠♣♥ t♠ t ♦♠♣t♦rt♠ s t♦ ①t ♥ t♠ ♥tr ♦r t♥ Ts

r♥ t ♠♣♠♥tt♦♥ ♦ t ♦rt♠ ♥ t ①♣r♠♥t st♣ ♥♠r♦ ♦rr♥s ♦r t Ts = 100[µs] ♦t r② tst s t ② ts ♥ ♦t rs♦♥ ♥ ♦♥ ♦♥ ♦ t ♦rt♠ s ♠ st s ①♣♥♥ t ♥♠r ♦ sq♥s tt t♦ t t st♣ ♣♥s ♦♥t ♥♣t ♥① ♥ ♦♥ t ♠♥ts ♦ t st A s ♦♥sq♥ r♥ rt♥tst t s③ ♦ t st sq♥ sst ♦ ♦r ♦ ♥♦t s t ♦rr♥ ♦ tr t♠ ♣♣t♦♥

♥ ♦rr t♦ ♦ ♥t ♦rr♥ stt♦♥s t♦ st ①♣r♠♥ts ♥r♣r♦ rsts t s♠♣♥ t♠ s ♥ ♥rs t♦ Ts = 200[µs] s♦ s s♦♠ s♥t ts ♦♥ t ♣r♦♣r ♦♣rt♦♥ ♦ t ♣r♦♣♦s P ♥♣rtr s ①♣♥ ♥ t rt♦♥s♣ tt ♥s t rr♥t rt♦♥s♥ tt s s t♦ ♣t t ❯ s s ♦♥ t ss♠♣t♦♥ tt t t sr t♥ t t s♠ ♥st♥t ts ②♣♦tss ♦s ♥♦t ♦ t t♦rs ♦ trr♥t rt♦♥s r ♥♦t ♣ ♦♥ ♥ ♣s r♥ t r ♦♣rt♦♥ t sr st♦r ♥ t ❯ t r♥t ♥st♥ts tt ♠♥s ♥tr♦♥ ♥ ♣♣r♦①♠t♦♥♦s② t s♠r t t♠ ♥trs t♥ t rr♥t rt♦♥ ♠sr♠♥tst ss t ♠♣t ♥tr♦ ② t ♣♣r♦①♠t♦♥ t♦ t r t♠ ♦♥str♥t♥ t ♦♥sq♥t ♦ ♦ Ts = 200[µs] ts ♠♣♠♥tt♦♥ ♦ t P ♦s♥♦t ♦ t st♥t♦♥ ♦♥ r♥ rt♥ ♦♣rt♦♥s ♥ ♣rtr t s ♥♦♥ tt st♥t♦♥ ♦ ♦r r♥ s♣ tsts ♦r st ②♥♠s

trt♣

trt♣

t t ♠♥ strt♣ ♥ ♥♦ rr♥t rt♦♥s ♥ ♠sr ②t t ❯r ♠♣t② ♥ ts s t rr♥t ♣rt♦♥ ♥♥♦t ♦t♥ ♥ ❬❪ ♥ ❬❪ tstrt♣ ♦ t P s ♥♦t rss r t♦ r♥t s♦t♦♥s r ♣r♦♣♦s

rst s② s♦t♦♥ ♦ r♣rs♥t ② ♦s strt♣ ♥ ♣rtrr♥ t rst st♣s t ♠♦ ♦ s t♦ ♣rt t rr♥ts tr tr rt♥ ♥♠r ♦ st♣s ♦r tr t ❯ ♥ ♣t ♦r t rst t♠ t♦♥tr♦ ♦ stt t♦ t ♦r ♣rt♦♥ t s ♦rt t♥ tts♥ ① ♥♠r ♦ st♣s s t rtr♦♥ ♦r t tr♥st♦♥ ♦ ♥♦t st ♦r t ♦♣rt♦♥ ♦ t ♠♥ rs♦♥ s tt t t♠ ♥tr ♥ t♠♦s ♣rt♦♥ s s ♦ st r♥ rt♥ tsts ♦♥ t ❯t♦ ♣t ♦r t ♠♦r ♣rt♦♥ s strt t t ♦ ♥♦t s♦ r♥♦tr tsts ♥ ♦tr ♦rs t ♦ ♦ ts trs♦ s ♥♦t ♥q ♥ t s s ♦♥t ①♣r♥ ♦ t s♥r ss t ♠♥ r s tt ts ♠t♦ ♦rqr t ♥♦ ♦ t ♠♥ ♣r♠tr t♦ ♠s♠t t♥ t♠♦ ♥ t t ♣♥t ♦ rs♦♥② ♣t ♦r ♠t ♥♠r ♦ t♠st♣s ♥ ts ♣rt ♦ t ♦r t s ♥ t♦ ♦♠♣t② ① r② s♦t♦♥tt rqrs t ♥♦ ♦ t ♠♦ ♥ ♦♠♣t② ♠♦r ♦♥tr♦s♠ ♦r ts rs♦♥ t ♠♦s strt♣ s ♥♦t ♥ ♠♣♠♥t

st strt♣ ♣r♦r s ♥ ♦♣ ♥ ♠♣♠♥t s t♦♦t♣t t t ♦t t♦rs ♦r t ♠♥ s strt ♥ ts ② t ❯r ♥ t ♠♥ ♥ strt r♥♥♥ s♥ t ♣rt♦♥ s ♦♥ t st♦rrr♥t rt♦♥ ♠sr♠♥ts sq♥ ♦ ♥①s r♥ ts ♣r♦r s t♦♦ tt t rr♥ts ♥rs ♦r rs r♥ ♥r♦s ♦r t ♠♥♦ ♦ ts sq♥ ♦ ♦♣♣♦st t♦rs s ♣r♦ ♦r t s ♦ rt② ♥♠rs♠t♦♥ s ♥ rr ♦t ♥ t rsts r s♦♥ ♥ t ♥ s♥tt t sq♥ s ♦rr t♦ ♣r♦ ♦♣♣♦st t♦rs 1− 4 2− 5 ♥ 3− 6 ♥ts ② t rr♥t rt♦♥s ♦♣♣♦st rt♦♥s ♦♥ ♥r♦s ♦rt ♠♥ s♥ ♦ t rr♥t rt♦♥ ♥r ♦t t♦r ♣♥s ♦♥t ♣♦st♦♥ ♦ t ♠♥ ♥ ts s ϑme = −π/2 ♥ t ①♠♣ ♥① s♥ ♦r 10 st♣s t♦ r② s♦ t rsts t s ♦rt t♥ tt t s♠♣r♦r ♦ ♦♥ ♣♣②♥ ♥① ♦r st ♦♥ st♣

0 10 20 30 40 50 60 7001234567

Vol

tage

inde

x [−

]

0 10 20 30 40 50 60 70−0.5

0

0.5

d−ax

is [p

.u.]

0 10 20 30 40 50 60 70−0.2

0

0.2

q−ax

is [p

.u.]

Time step [−]

r ♠t strt♣ ♣r♦r

♣r♦♣♦s strt♣ ♣r♦r s ♥ ♠♣♠♥t ♥ tst ♥ t rstsr r♣♦rt ♥ t ♦ t strt♣ ♣r♦r r s♦♥ ♥

♦♥ st♥t♦♥ ♥ ♦r Prt rr♥t ♦♥tr♦

t ♥ s♥ tt tr t ttr s ①t s♣ st♣ s ♥ s rr♥ t♦ ts♣ rt♦r ♦♥st♥t ♦ ♦ 0.3[p.u.] s ♣r♦ ② t ♠str ♠♥ ♥st♦ t st ♣r♦r t ♣rt rt♦r s t♦ r♦♥③ t♦r s tst t♦ r t rr♥t rr♦r ♥ ♥ tr♥ t♦ tr t s♣ rr♥

0 500 1000 1500 20000

50

100

Spe

ed [r

pm]

0 500 1000 1500 2000−5

0

5

d−ax

is [p

.u.]

0 500 1000 1500 2000−5

0

5

q−ax

is [p

.u.]

Time step [−]

♣ ♥ rr♥ts

0 10 20 30 40 50 60 7001234567

Vol

tage

inde

x [−

]

0 10 20 30 40 50 60 70−0.5

0

0.5

d−ax

is [p

.u.]

0 10 20 30 40 50 60 70−0.1

0

0.1

q−ax

is [p

.u.]

Time step [−]

t ♦ t strt♣

r trt♣ ♣r♦r

♦♥s♦♥s

♥ ts ♣tr t ts ♦ ♦t t♦r st♥t♦♥ ♦♥ P s♠s ♥t ♥ ①♠♣ ♦ t ♣♣♥s ♥ ♣rs♥ ♦ ♥rq♥t ❯ ♣t s ♥♣r♦

♥♦ ♣t♥ ♠t♦ s ♥ ♣rs♥t s t♥q ♦s r♦♥strt♥ t rr♥t rt♦♥s s♥ ♦♥② t s rt t♦ t tr ♠♦st r♥t ♦tt♦rs ♣r♦♣♦s s♦t♦♥ s s ♦♥ t rt♦♥s♣ ♥♥ t rr♥t rt♦♥s♥ ② t t ♦t t♦rs ttr ♥ ①♣♦t t♦ r♦♥strt t♦r s ♥ t ❯ strt♥ r♦♠ t ♠♦st r♥t st♦r rr♥t rt♦♥s rt♦♥s♣ ♦s ♥r t ss♠♣t♦♥ tt t t s r t♥ t t s♠♥st♥t r♥t② r♥ t ♠♥ ♦♣rt♦♥ t rr♥t rt♦♥s r ♦t♥t r♥t st♣s ♥tr♦♥ ♥ ♣♣r♦①♠t♦♥ tt s ♥ t t♠ t♥t ♦♥sr s s s♥t② s♦rt ♦r ts rs♦♥ st ♦rt♠ ♦r tsq♥ ♥tt♦♥ s ♥ ♦♣ ♥ ♠♣♠♥t ♠ ♦ t ttr st♦ r♦♥③ tr t ♦t ♥①s ♣ttr♥ ♦r♠s st sq♥ ♦r t ❯♣t ♦rt♠ s ♥ ♦♣ ♦♥sr♥ t strt ♦♥str♥t ♥ ②t r t♠ ♣♣t♦♥ ♥ ♣rtr t ♥♠r ♦ ♣♦ss sq♥s tt s t♦ t s r ♣♥♥ ♦♥ t ♥♣t ♦t ♥①

♣r♦♣♦s ♦rt♠ s ♥ t ② ♠♥ ♦ ①♣r♠♥ts rsts ♥ ♦♠♣r t P s ♦♥ t ♥♦♠♥ ♠♦ ♦ t ②♥ t♦ t tt♥ss ♦ t ♣r♦♣♦s s♠ ♥ ♥r t t ♦ ♣r♠tr rt♦♥s ♥♣rtr t tsts ♥ ♣r♦r♠ t r♥t ♦s t♦ t t ♦r♦ t s♠ t str♦♥ r♦♥ strt♦♥ rsts r② s♦ t ♥ts ♦t ♦r ♣rt♦♥ ♥ ♣rtr t ♣rt♦♥ rr♦r s rt② r ♥ s ♦♥sq♥ t rr♥t ♦r♠s ♣rs♥t r r♣♣

♠tt♦♥s ♦ t ♣r♦♣♦s ♦♥tr♦ s♠ ♥ t ♥ ♣rtr

♦♥s♦♥s

t s ♥ t t ♣rs♥ ♦ stt♦♥s ♥ st♥t♦♥ s ♥♦ ♥ ♥♦ sq♥s ♥ ♥t ♥ s ♦r t ❯ ♣t s t②r♣rs♥ts s♥♥t rtt② ss s♥ t ♣rt♦♥ s s ♦♥ t ♠srrr♥t rt♦♥s t ♣rs♥ ♦ ♥♦ss ♦r str♥s ♦ t t ♣r♦r♠♥ ♦t r ♥ t ♠♣♠♥tt♦♥ tr rr♥t ♠sr♠♥ts ♥ ♦♣ss tr ♦rt ❯ ♥ s t♦ ♦ sss rt t♦ rq♥② ♥♦ss ♥ ♦♠♦♣♦r str♥s ♥tr♦ ② t ♣♦r ♦♥rtr ♦ t ♠str ♠♥ ♥♦trs♥t ss s ♥ ② t ♦♠♣tt♦♥ r♥ ♥ ♦rr t♦ ♦ ♦rr♥ tst s♠♣♥ t♠ s ♥ ♦

♥② ♣r♦r ♦r t strt♣ s ♥ ♣r♦♣♦s ♥ ts ♦♥t♦♥ t ❯r ♠♣t② ♠♥♥ tt t ♣rt rt♦r ♥♥♦t t t t ♦ t t♦t t♦rs rs ♥ ② ♠♦s strt♣ ♥ t♥ tr tt st ♣r♦r s ♥ ♠♣♥t ttr ♦s ♥ t❯ ♦r t ♠♥ strt♣

Chapter 4

♦ Prt ②strss rr♥t

♦♥tr♦

♥ ts ♣tr ♦ Prt ②strss rr♥t ♦♥tr♦ s♠ ♦r ②♥r♦♥♦st♥ ♠♥ s ♣rs♥t ♣r♦♣♦s ♦rt♠ ♣rts t ♣♥t ♦r♦♥sr♥ t ♥t st ♦ t ♣♦r ♦♥rtr stts ♥ t ♦♦ss t ♠♦st st♦t t♦r ♥ ♦rr t♦ ♠♥♠③ t ♦st ♥t♦♥ ttr s s♥ t♦ ♠tr♥t ♦s ♣♥♥ ♦♥ t ♦♣rt♥ ♦♥t♦♥s ♦ t ♠♥ ♥ ♣rtr t①♠♠ ♦rq Pr ♠♣r trt♦r② t ① ❲♥♥ r♦♥ ♥ t ①♠♠♦rq Pr ❱♦t ♦♣rt♦♥ r ♦♥sr ♥ ts ② s♣ r♥ ♦♣rt♦♥s ♥ t rr♥t ♥ ♦t ♠ts r ♥r♥t② ♦sr ss trt ♦st ♥t♦♥ ♦t♣ts rt♥ ♦t t♦r t s♠ t♦r s ♣t ♥t t rr♦r♠♣t ♦s ②♦♥ ① trs♦ ♦r ts rs♦♥ t ♣r♦♣♦s s♠ ♦♥st♦ ②strss ♦♥tr♦

♥tr♦t♦♥

♦♥♥t♦♥ rt ♦rq ♦♥tr♦ s t♥q s ♦♥ t ♦♥tr♦ ♦t t♦rq ♥ t stt♦r ① t♦r ♠♦ t♦t ♥♥ rr♥t ♦♦♣s ♥ t♦ss♠s s♣ rt♦r ♣r♦s t t♦ rr♥s ♥ t ♠♥ ♦♥tr♦r ts t♦♣ t♦rq ♥ ① rr♦rs t♥ ♥ ②strss ♥ ❬❪ ❲ ♥♦♥ ♠rts ♦ tss♠s r t s♠♣ ♥ r♦st ♦♥tr♦ strtr ♥ t st t♦rq ②♥♠ r ♥ ♦♥trst t t t♦rq ♥ rr♥t r♣♣s ♥ t r st♥rq♥② ss s♦rt s♠♣♥ t♠ s rqr t♦ ♣ t t♦rq t♥ t②strss r♦♥ ❬❪

♥ rtr ♦♣♠♥t ♦ t ♦♥♥t♦♥ s♠ t st♥ ♦ t ♥rtr s♥♦t s ♦♥ t t♦rq ♥ ① ②strss ♦♥tr♦r t ♦♥ t ♠♥♠③t♦♥ ♦ ♦st♥t♦♥ s r♣rs♥ts ♠rt♦♥ ♦ t♦rs ♦ Prt ♦♥tr♦ P♥ t ♥ ♥t s P ❬❪

r ♦rs ♣rs♥t s♠s ♦r ②♥ s ♦r ♥st♥ ❬❪ s ♠♥t♣♦♦② trs ♥ ♥r♥t s♣ r♥ ♦♣rt♦♥ r♦r ♥ t ♦♥tr♦

♦ Prt ②strss rr♥t ♦♥tr♦

s♠ ♦r ②♥ ♠♥ s♦ ①♣♦t t ♠♥ ♣ts ♥ ♦♣rt♦♥t♦♥s ♥ ♣rtr ♣♥♥ ♦♥ t s♣ ♦ t ♠♥ t ①♠♠ ♦rqPr ♠♣r P t ① ❲♥♥ ❲ ♥ ①♠♠ ♦rq Pr ❱♦tP❱ trt♦rs t♦ tr s♠ ♦r ②♥ t♦ ♦♥tr♦t ♠♥ ♥ t ♦r♠♥t♦♥ r♦♥s s ♣rs♥t ♥ ❬❪ s♠ s s♦♥ t t② rt♦ rt♦♥ ♥tr♦ ♥ ❬❪ ♦s r♥ t t♦rq r♣♣rt♥♥ t ②♥♠ ♣r♦r♠♥ ♦ t ♦♥♥t♦♥ ss ♥ ♣t stt♦r① rt♦♥ ♦rt♠ s ♦♥ t stts ♦ t t② rt♦ s ♣r♦♣♦s ♦r ❲♦♣rt♦♥

♥ ts ♦r ♥ ♥ ♦♥tr♦ s♠ ♦r s♣ ♦♣rt♦♥ ♦ ②♥♠♥s ♣rs♥t ♠♥ s ♦♥tr♦ t♦ rt② tr t P t ❲ ♥ tP❱ trt♦rs ② ♥tr♦♥ t ♦♣rt♥ ♥ s s ♦s ♦ ♦♣rt♥ ♣♦♥ts♦t t♥ P ♥ P❱ trt♦rs t ♦♥s t ♦r♠r ♦r s♣ ♦rt♥ t s s♣ ♦t ♦r t♥ t ♥♦♠♥ ♦♥ t ♦♥s tt ttr ♦r t st s♣ ♣ t♦ t ♠①♠♠ r s♣

r♥t② r♦♠ ♦♥♥t♦♥ s♠s t ♦rt♠ ♦s ♥♦t ♦♥sr tt♦rq ♥ ① rr♦rs ♦r t s♦♥ ♦ t ♠♦st st ♦t t♦r ♥st trr♥t ♠♣t rr♦r ♥ t st♥ r♦♠ t ♦♣rt♥ ♥ r s t♦ ♦♠♣① rr♦r ♥ tr♥ s s t♦ ♦♠♣t ♦st ♥t♦♥ t♦ ♠♥♠③ t♠ t rr♦r ♠♣t ①s ♥ trs♦ rst tr♠ ♦ t ♦♠♣① rr♦rs t ♦♠♣r♥ t rr♥ ♣r♦ ② t s♣ ♦♥tr♦r ♥ t ♠srrr♥ts s♦♥ ♦♠♣♦♥♥t s ♦t♥ t♥ t st♥ t♥ t t♣♦♥t ♦ ♦r ♥ t ♦♣rt♥ ♥

♥rs♥ t s♣ t tr♥st♦♥ t♥ t P ♥ ❲ s ♦t♥ ②r♦tt♥ t ♦♣rt♥ ♥ r♦♠ t P ♣ t♦ t P❱ trt♦r② ② ♠♥ ♦♥ ♥♥♦t ♦t ♦♦♣ ♦♥♥t♦♥ ♦t ♦♦♣ r s ♦♥ t ♥♦ ♦t s s♣ ♥ tr♥ rqrs t ♠♥ ♣r♠trs ♥r r② ♦r♥♦♥t♦♥ str♥ ♣r♠tr ♥rt♥ts ♥ rt♦♥s ♦ t♦ sr♦rs♥♥ ♦ t r ♣r♦r♠♥ ♦r t♦ ♥♦♥ ② t③t♦♥ ♦ t s ❬❪

♥ ♦rr t♦ ♦r♦♠ ts sss ♥ ts ♣tr t s ♣r♦♣♦s ♥ ♣t ♦rt♠ t♦ r t ♠♥ t s♣ r♥ t ♠♥ ① ♦rt♠ ss ♦♥ t t tt ♥ t ♠ ♦♠s rr t♥ t ♦t t♦♥tr♦ strt② ♥ ♥♦t ♥ ♦t t♦r t♦ r♥ t rr♦r ♥s t♠t r♦r t r♣tt ♦t♦♥ ♦ s♣ ♦♥r♥ rtr♦♥ ♥ ss ♥ ♥①

♥ t ♦ ♦ t ♦t t♦r t♦ ♣♣ s s ♦♥ t ♠♥♠③t♦♥ ♦t ♦st ♥t♦♥ ♣♥s ♦♥ t ♦♠♣① rr♦r ♦♠♣♦s ② t rr♥t ♠♣t rr♦r ♥ t st♥ r♦♠ t ♦♣rt♥ ♥ t ♣r♦♣♦s ♦♥tr♦ s♠s rrr s ♦ Prt ②strss rr♥t ♦♥tr♦ P ss t♦♥s t♦ t ss ♦ s♥ t ♣rts t ♠♥ ♦r ♦♥sr♥ t t♣♦ss ♥rtr stts

t s ♦rt t♥ tt t ♣r♦♣♦s ♦♥tr♦ strt② rs ♦♥ t ♥♦♦ t ♠♥ ♣r♠tr ♦r r② ♦r♥ ♦♥t♦♥

Pr♦♣♦s ♦♥tr♦ s♠

Pr♦♣♦s ♦♥tr♦ s♠

♦r ♦ t ♣r♦♣♦s ♦♥tr♦ s♠ s r♣♦rt ♥ s s ♦♠♣♦s② s♣ rt♦r PIn ♦t ♦♦♣ ❱ ♥ t P ♦ ♦ r♣rs♥ts t ♦r ♦ t s♠

n*

n-

+

MP

HCC

w*=0

*PIn

~SynRM

VL

uz

lim+

z

r ♦♥tr♦ s♠

♣ ♦♦♣

♦t♣t ♦ t P s♣ ♦♥tr♦r s ♥ ♥ ♥ t ♥s t rr♥tt♦r ♠♣t |I∗| ♥ t s♥ ♦ t t♦rq ♠♥ m∗ s q♥tt② r♣rs♥tst rr♥t ♠♣t ♦♠♠♥ t♦ tr t s♣ rr♥ n∗ ♠ts ±µlim ♦ t rt♦r ♣♥ ♦♥ t ①tr♥ ♦t ♦♦♣ ♣r♠tr α ♥ r sr ♥

µ∗ = |I∗| · sign(m∗)

µlim = ±α · IN

r♥ t♥ t rr♥ µ∗ ♥ t t µ st♠t r♦♠ t♠sr rr♥ts r♣rs♥ts t rst ♦♠♣♦♥♥t ♦ t ♦♠♣① rr♦r ♥tr♦ ♥

♣rt♥ ♥

♥r ②♥ ♠♦ s r ♦♥sr ♦r t s ♦ s♠♣t② ♥ ts s tP ♥ t P❱ trt♦rs r strt ♥s ♥ t qrr♥t ♣♥ s ♥♥ ♥ rs♣t② r id ≤ 0 ♥ iq > 0 s ♦r ♠♦t♦r ♦♣rt♦♥

iq = ±id

iq = ±Ld

Lq· id

trt♥ r♦♠ ts t ♦♣ ♥ ts ♦r s t♦ ♦t♥ t rr♥ trt♦r② ② r♦tt♥ t P ♦ ♦ ts t ♦♣rt♥ ♥ w s ♥ ♥ ♥ t♦r♥t ②s rst s♣rt② ♦♥srs t st♥ r♦♠ t P ♥t P❱ stt t wMTPA ♥ wMTPV rs♣t② ♥ t♥ t ts t

♦ Prt ②strss rr♥t ♦♥tr♦

s♠ ♦ t t♦ tr♠s s♦♥ s ♦t♥ strt♥ r♦♠ t ♣♦♥t t♦ ♥st♥ ♦r♠ ♥ ♣rtr sL(β) s t s♦♣ ♦ ♥r ♥ tr♦ t ♦r♥t♥ t P ♥ t P❱ ♥ ♦t t ♥t♦♥s t r♦tt♦♥ s ♦t♥② ♠♥ ♦ t ♣r♠tr β s t ♦t♣t ♦ t ♦t ♦♦♣ sr ♥

w = β · wMTPA + (1− β) · wMTPV

wMTPA = i2d − i2q

wMTPV = i2d −(Lq

Ld

)2

i2q

sL(β) =sign(m∗)

β(Lq

Ld− 1)

− Lq

Ld

w(β) =|iq − sL(β)id|√

1 + sL(β)2

♦r♥ t♦ t t♦ r♥t ♥t♦♥s ♦ t ♦♣rt♥ ♥ t ②strss ♥s ♥ r♣rs♥t ♥ t q rr♥t ♣♥ ♥ ♠♥t ♥ t rs r♣rs♥t t ②strss ♥ ♦♠♣t t ♥ rs♣t② t s♦rt ♥♦t♥ tt s♥ t ♥t♦♥ s♣t t ②strss ♥ s r ♥t rr♦r ♣♥ ts s♣ rsts ♦r♠ ♥ t rr♥t ♣♥ ♣♥♥ ♦♥ t ♣♦♥t♦ ♦r s ♦ rst ♥ ♠♣r♦♣r tr ♦ t rr♥t rr♥ ♥ r♥t♦r ♦ ♦t♥ ♥ t t♦ ①s ♣♥♥ ♦♥ t ♣♦♥t ♦ ♦r ♥ t♦♥trr② ♦s ♦t♥♥ rr s♣ ♥ t ♦ ♣♥♦r ts rs♦♥ tttr s ♥ ♦s♥ ♥ ♠♣♠♥t

−1 −0.8 −0.6 −0.4 −0.2 00

0.2

0.4

0.6

0.8

1

d−axis current [p.u.]

q−a

xis

curr

ent [p

.u.]

=0.25

= 0.5

= 0.75

= 1

=1

=

0.7

=

0

r ②strss ♥ ♥ t q rr♥t ♣♥

st♥ t♥ t t ♣♦♥t ♦ ♦r ♥ t ♦♣rt♥ ♥ r♣rs♥ts ts♦♥ tr♠ ♦ t ♦♠♣① rr♦r ♥ ♥

t s ♦rt ♥♦t♥ tt t ♥t♦♥ ♦ t q♥tt② w ♥ s② ①t♥ t♦t s ♦ strt ♠♥ ♣r♦ tt t P ♥ P❱ ♦ r ♥ ♥♣rtr s♦t♦♥ st ♦r ts s ♦ ♠♣♠♥tt♦♥ ♦♥ssts ♥ ♥r③♥ tt trt♦rs ♦ t ♠♥ s qtt② r♣rs♥t ♥ r tP ♥ t P❱ r ♥r③ ♦♥sr♥ t ♥trst♦♥ t♥ t trs ♥ t rr♥t r♠r♥ ♠t ❯s♥ t ♥r③ trt♦rs ♥tr♦s♥ ♣♣r♦①♠t♦♥ ♥ t ♦♥trr② t s ♣♦♥t ♥ t strt♥ ♣♦♥t ♦ t P❱r t s♠ ♦ t t rs rt ♥t ♦ ts s♠♣ s♦t♦♥ s

Pr♦♣♦s ♦♥tr♦ s♠

♥ ♥ tr♠s ♦ s rst t ♦s ♥♦t rqr t ♦♠♣t ♥♦ ♦ t ♠♥♠♥t rtrsts s♥ ♦♥② t♦ ♣♦♥ts r ♥ ♥♠② t ♥trst♦♥s ♦t t rs t t rr♥t r♠r♥ ss t t ♥r③ rst s♠ ♦rt♠ ♣rs♥t ♦r t ♥r s ♥ s t s♠ ♥t♦♥♦ ♦♣rt♥ ♥ s

−1.5 −1 −0.5 00

0.5

1

1.5

d−axis current [p.u.]

q−a

xis

curr

ent [p

.u.]

MTPA

MTPV

Linearized

MTPA

Linearized

MTPV

(IPM axis convention)

r t ♥ ♥r③ trt♦rs

P ♦

♦r ♦ t s♠ s t P ♦ ♦ r♦♠ t ♦♥tr♦ st t ♦rs s ♥♣t µ∗ r♦♠ t s♣ ♦♥tr♦r t rr♥ ♦ t st♥ r♦♠ t♦♣rt♥ ♥ w∗ = 0 ♥ t ♦♥t β r♦♠ t ♦t ♦♦♣ rtr♠♦r trr♥ts ♥ t ♣♦st♦♥ r ♠sr r♦♠ t ♠♥ ♥ ♣r♦ t♦ t P ♦ ♦t♣t s t ♦t t♦r uj s rt② ♣♣ t♦ t ♣♦r♦♥rtr

♠r② t♦ ❬❪ ♥ rr♦r t♦r E = [Eµ Ew]t s ♥tr♦ r Eµ Ew

r t ♥♦r♠③ rr♦r ♦♠♣♦♥♥ts E s t rr♦r ♠♣t ♥ φE ts ♣s

Eµ =µ∗ − µ

INEw =

w∗ − w

I2N

E =√

E2µ + E2

w φE = tan−1

(Ew

Eµ

)

trt ♦ t ♦♥tr♦ s t♦ ♦sr ♣♥ t ♠♦ ♦ t rr♦rt♥ r♠r♥ ♦ rs Emax

E ≤ Emax

t ♥st♥t E rs t ♠t t ♦♥tr♦ ts ♦♥ t ♥ ♦t ♥ ♦rrt♦ r♥ t ♣♦♥t ♦ ♦r ♥s t ♠t r ♥ ♣rtr t ♦t t♦r t t♥①t st♣ s ♦♦s♥ ♠♦♥ ♦♥ ♦ t t ♣♦ss ♥rtr stt u

z t z ∈ Q♥ ♦rr t♦ r t rr♦r E s♦ tt s ♦sr ♥ st ♦tt♦r uk1 s t♦ r♥t rt♦♥ ♦ E t ♦♥r♥ ♦♥t♦♥ s ♦t♥② ♠♣♦s♥ ♥t rt ♦ t rr♦r ♠♣t r t ♦t st♥s ♦rrt

d

dtE ≤ 0

d

dt

√

E2µ + E2

w =EµEµ + EwEw

E≤ 0

♦ Prt ②strss rr♥t ♦♥tr♦

r♥t rtr ♥ ♦♣t ♦r t ♦ ♦ t ♦t s♣t t♦r ♠♦rt♥ ♦♥ ♠ts ② ♦♦ r♥t rqr♠♥ts r♥ t st♥rq♥② ♦r t st♥ ♦sss ♥ ❬❪ t ♦rt♠ ♦♦ss t s♣t t♦r tt♠♥♠③ ♠①♠♠ ♥t t rt ♦ t rr♦r r♥t② ♥ ts ♦rt ♦♥tr♦ strt② ♣rts t trt♦r② ♦ t ♦r♥ ♣♦♥t ♦r t t ♣♦ss♦t t♦rs u

z ♥ t sts t ♦♥ tt ♣s t rr♦r t♥ t ♠t ♦r t♦♥st t♠

P ♦♣rt♦♥ ♥ sr s ♦♦s t ♥st♥t t rr♥ts ik

♥ t s♣ ωk r ♠sr t ♦ts uk r ♥♦♥ r♦♠ t ♣r♦s st♣ q rr♥ts t t ♥①t s♠♣♥ t♠ i

k1|k r ♣rt t ♥ ♣rt rr♥ts r s t♦ t µk1 wk1 ♥ t rr♦rs Ek1

µ Ek1w

Ek1 s t ♣rt rr♦r ♦srs t♥ t ♥①t ♣♣ ♦t t♦rs t s♠ s ♥ t t st♣ trs t rt ♦ t rr♦r s ♦♠♣t♦r t t ♣♦ss ♥rtr stts uz t z ∈ Q ♥♦♥ t ttr t s ♣♦sst♦ st♠t t rr♦r trt♦rs ♥ t t♠ ∆T z ♥ t ♣♦♥t st② t♥t ♠t Emax ♦s♥ ♦t t♦r s t ♦♥ tt ♥ts t ♠①♠♠ ♦ ∆T z ♥ t tt t ♥st♥t r♦r t ♦st ♥t♦♥ f t♦ ♠♥♠③ s r♣♦rt ♥

f =1

∆T z

♦r t s ♦ rt② t ♦♣rt♦♥ ♦ t ♦rt♠ r r♣rs♥t ♥

μk1

E

< E max

: min fz

= 1/ ΔTz

μ*SPEED LOOP OP. LINE

w *

α β

uk ik

ikΔ = A mik + B m uk

ik1|k = ik+ ikΔ

ik1|k

w k1

wEk1 μE

k1

k1

uk1 = uk uz

uk1

E k1

r ♣rs♥tt♦♥ ♦ t P ♦♣rt♦♥

♦ ∆T z ♥ ♦♠♣t s ♦♦s t E t rr♦r t ♥st♥t k♥ s ♥ Prt♥ t rr♦r t t ♥①t ♥st♥t t s ♣♦ss t♦♦♠♣t t rt E

z = [Eµ Ew]t ①♣rss s ♥ ♠♣t Ez s t

s♣ νz ② t ♦♣rt♥ ♣♦♥t ♠♦s ♥ t rr♦r ♣♥ ♦♥ t rt♦♥ ♦E

z

r♣rs♥ts t ♣♦♥t ♦ ♦r ♥ ts trt♦r② ♥ t rr♦r ♣♥ ss♠♥E = Emax r ϕz s t ♥ t♥ t rr♦r ♥ ts rt ♥ γz s ts♦♠♣♠♥tr② s ①♣rss ♥

Ez =√

(Ezµ)

2 + (Ezw)

2 φEz = tan−1

(

Ezw

Ezµ

)

Pr♦♣♦s ♦♥tr♦ s♠

E

E

wE

E

E

Ez

z

z

Ej

Cz

r rr♦r ♣♥

ϕz = π − γz and γz = φE + φEz

ss♠♥ t s♣ νj ♦♥st♥t t ♣♦♥t ♠♦s ♦♥ t ♦r Cj r♠♥♥t♥ t r♠r♥ ♦r t♠ ∆T j ♦♠♣t ♥

∆T z =Cz

vz=

2Emax · cos(ϕz)

Ez

❱♦t ♦♦♣

②♦♥ t s s♣ t ♠ ♦♠s rr t♥ t s ♥ ts♦♥t♦♥ t P srs ♦r t st ♦t t♦r t♦ st t rr♦r t♥t r♠r♥ t ♥♦ s ♦r ♥ s t♦ st♥s tr ts ♥ssr② t♦ t ① ♦ t ♠♥

♦♥t♦♥ r♣♦rt ♥ s ♥♦t st t♦ s s rtr♦♥ t♦ r♦ttt ♦♣rt♥ ♥ t♦ ♦sr♥ t ttr ♦♥t♦♥ ♠♥s ♥♥ t♦r ttr t rr♦r ♠♣t t ♦s ♥♦t ♠♣② t ♦♥r♥ t♦ t r♠r♥♥ ♦tr ♦rs ♦sr♥ t ♠♣t rt♦♥ ♦♥② s ♥♦t s♥t ♥ t rr♦rrt♦♥ s t♦ ♦♥sr s

s♦s t ♥r s ♥ t rr♦r E s ♦ts t ♠t Emax st ♦t t♦r s t♦ r t rr♦r ♠♣t ♥ t♦ st t ♣♦♥tt♦rs t ♠t r s r♣rs♥ts ♦♥str♥t ♦♥ t ♦ t ♥ γz ♥♥ ♥ t ♥ ①♣rss s ♥ s ♥t♦♥ ♦ ϕz

lim = sin−1(Emax/E) ttr ♠ts t s r ♦r t rr♦r t t ♥①t ♥st♥t Ek1 ♥ t ①♠♣♣t ♥ t rr♦r t s s♠r ♠♣t t t trt♦r② ♦s♥♦t r♦ss t r♠r♥ ♦♥t♦♥ s ♦sr s ♥♦t

γz ∈ π ± ϕzlim

cos(γz) < −cos(ϕzlim)

ξz = EµEzµ + EwE

zw + Ez ·

√

E2 − E2max < 0

♦ Prt ②strss rr♥t ♦♥tr♦

E lim

wE

E

z

z

EzTs

Ek1

r ①♠♣ ♦ t rr♦r s r

♦♥t♦♥ s ②s ♦sr ♥ t ♠♥ s ♦ t s s♣ t ♦s ♥♦t ♦ ♥ t ♠ rs t ♠①♠♠ ♦t t♦ tst r♣rs♥ts t rtr♦♥ t♦ r♦♥③ tr t s ♥ssr② t♦ t ①♦ t ♠♥ ♥ t ♥ s t♦ r t ♠♥ ♦♥ t ❲ ♥ t P❱trt♦rs ♥ ♣rtr t ♦rt♠ ♠♦♥t♦rs t ♦ ξz ♥ t rts t♦♣rt♦♥ ♦ t ♠♥ t♥ ♦♥ t ♣r♠tr η ♦r♥② ♦ ♥ sr s ♦♦s tr s t st ♦♥ ♦t t♦r ♦sr♥ ξz < 0ts ♥t s r♦r ♥rs♥ t ♦ t ♦♥tr cntη ≥ 0 s♦♥ ♦♥trcntcycle s ♣t t r② trt♦♥ tr Ncycle st♣s cntη = 0 t ♠♥s ttr♥ t ♦♥sr ♥tr t P ♥r ♦♥ st ♦t t♦r ♥t ♣r♠tr η s r ♦ q♥tt② ∆η ♥ t ♦♥trr② cntη = Ncycle t ♠♥stt t r② ♥st♥t t st ♦♥ ♦t t♦r uz ♦sr ts s t st ♦rt♠ trs ♥rs♥ t ♦ η tr Ncycle st♣s t ♦♥trs cntη ♥cntcycle r ③r♦ ♥ t ♣r♦ss s r♣t

t s ♦rt ♠♥t♦♥♥ tt t ♣r♦r♠♥ ♦ t ♦rt♠ str♦♥② r② ♦♥ t♣r♦♣r t♥♥ ♦ t ♣r♠trs Ncycle ♥ ∆η

♦rt♠ ♦ rt s r♣♦rt ♥ t s ♦rt t♥ tt η ∈[−1; 1] t ♣♣r ♦♥ ♦ t ♥tr ♠♣s t P ♦♣rt♦♥ t ♦r♠t s ♦r t ♠♥ ♠①♠♠ s♣

trt♥ r♦♠ t ♦ η t♦ r♥t ♦♥ts r ♥tr♦ rst♥♠② α s s t♦ ♠t t s♣ ♦♥tr♦r ♦t♣t µ∗ t♥ t r♥ ±α · IN s♣r♦s② sr ♥

s♦♥ ♦t♣t β s s t♦ r♦tt t ♦♣rt♥ ♥ r♦♠ t P t♦ tP❱ trt♦r② s sr ♥

♦r t s ♦ rt② t ♦r ♦ t ♦t ♦♦♣ ♥ t s ♦ t tr♣r♠tr η α ♥ β r r♣♦rt ♥ ♦r r♥t ♣♦♥ts ♦ ♦r ♥♠② tP t ❲ r♦♥ ♥ t P❱ trt♦r② ♥ t q rr♥t ♣♥

♠♣♠♥tt♦♥ ♥ sts

z = 0:7

< 0z

0flag = 1flag =

cnt + flagcnt=

cnt = Ncyclecycle

cnt = Ncycle

+= -=

cnt = 0

=

cntcycle cntcycle= + 1

=

1

-1

y

y

y

y

n

n

n

n

cnt = 0cnt = 0cycle

>0

= 1=

αβ = 1 + = 0

αβy n

η η

η η η

η

η

η η η η η η η η η η

η

η

η

ηη

r ❱ ♦rt♠ ♦ rt

−1 −0.8 −0.6 −0.4 −0.2 00

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.1

0.1

0.1

0.10.1

0.2

0.2

0.2

0.2

0.3

0.3

0.3

0.3

0.4

0.4

0.4

0.4

0.5

0.5

0.5

0.6

0.6

0.6

0.7

0.7

0.7

0.8

0.8

0.8

0.9

0.9

1

1

d−axis current [p.u]

q−a

xis

cu

rre

nt [p

.u]

Current limit

MTPA

MTPV

MTPAη = 1

= 1 = 1

FWη = (1; 0]

= 1 = η

η =(0; -1]= 1+η = 0

MTPV

BASE POINT

MTPV POINT

r ❱♦t ♦♦♣ ♦♣rt♦♥

♠♣♠♥tt♦♥ ♥ sts

P strt② s ♥ ♠♣♠♥t ♥ ①♣r♠♥t② tst s♦st ①♣r♠♥t st♣ ♠♥ t ♦ t ②♥ ♥ t stt♥ ♣r♠trsr r♣♦rt ♥ ♠♣t ♦ t ②strss ♥ s ♥ ① t♦Emax = 2.5e−2

rst tst s ♥ rr ♦t t♦ r② t rr♥t ♦♥tr♦ strt② ♦r t♦♣rt ♦♥t♦♥s ♥ ♣rtr t ②♥ s ♥ r t t ♠str ♠♦t♦rt s♣ r♠♣ strt♥ r♦♠ ③r♦ t♦ s ②♦♥ t s s♣ P s♣rt♦r ♦ t ②♥ ♠♥ s ♥ rt② strt t♦ t ♠①♠♠ µ∗ = α · IN ♥ ts ② t ♣♦♥t ♦ ♦r strt r♦♠ t s ♣♦♥t ♥ ♠♦♦♥ t ♠①♠♠ rr♥t r♠r♥ r♥ ❲ ♥ ♦♥ t P❱ trt♦r②♥ t stt♥ ♦ ts ①♣r♠♥t t ♦♥tr♦ strt② ♥ tst t t ♠♥♦r♥ t ts ♠ts

rsts r r♣♦rt ♥ r t q①s rr♥ts ♥ t ❱ ♣r♠tr r s♦♥ r t t② ♦ t ♦♥tr♦ strt② ♥ ♦♥r♠ s♦♥ tt t

♦ Prt ②strss rr♥t ♦♥tr♦

s t♦ r t ♠♥ ♥ t ♦ s♣ r♥ r♥t ♦♣rt♦♥s ♥♠②P❲ ♥ P❱ r r② ♥t

0 5 10 15 20−1

−0.5

0

d−axis current [p.u.]

0 5 10 15 200

0.5

0 5 10 15 20−1

0

Time [s]

z

αβ

1

q−axis current [p.u.]

1

VL parameters [−]

rr♥t rs♣♦♥s ♥ ❱ ♣r♠trs

−1 −0.8 −0.6 −0.4 −0.2 00

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.1

0.1

0.1

0.10.1

0.2

0.2

0.20.2

0.3

0.3

0.3

0.3

0.4

0.4

0.4

0.4

0.5

0.5

0.5

0.6

0.6

0.6

0.7

0.7

0.7

0.8

0.8

0.8

0.9

0.9

1

1

d−axis current [p.u.]

q−ax

is c

urre

nt [p

.u.]

♦♠♣t r♥ ♦♣rt♦♥ ♥ qrr♥t ♣♥

r rst tst ♦♥tr♦ strt② t♦♥

rst tst ♦ r♥ t t② ♦ t ♠♦ t t ♦s ♥♦t t ts②st♠ ♥ ts ♦♠♣t♥ss s♥ t P s♣ rt♦r t♦♥ s ♥ ♥t♥t♦♥②① ♣r ♥②ss s rqr t♦ ♥rst♥ t r ♣r♦r♠♥ ♦ t s②st♠♥ ♥ ♣rtr t ♥trt♦♥ t♥ t s♣ ♥ t ♦t ♦♦♣s s♦♥♦t s ♦st♦♥s ♥ ♥stts

r♣♦rts t rsts ♦t♥ ♣r♦♥ ♦♥st♥t s♣ rr♥ t♦ t②♥ ♥ ts s ♦ t s s♣ ♣r♦♥ ♥ ♥rs♥ r♥ t♦rqt t ♠str ♠♦t♦r ♠♣s t ♠♦♠♥t ♦ t ♣♦♥t ♦ ♦r ♦♥ t Ptrt♦r② r♥ t ♦ t♦rq r♠♣ t q rr♥ts ♦s② tr t Ptrt♦r② ♥ t t ❱ ♦t♣ts η = α = β = 1

0 0.5 1 1.5 2 2.5 30

0.5

1Torque [p.u]

0 0.5 1 1.5 2 2.5 3−1

−0.5

0d−axis current [p.u]

0 0.5 1 1.5 2 2.5 30

0.5

1q−axis current [p.u]

Time [s]

♦rq rr♥ ♥ rr♥t rs♣♦♥s

−1 −0.8 −0.6 −0.4 −0.2 00

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.1

0.1

0.1

0.10.1

0.2

0.2

0.20.2

0.3

0.3

0.3

0.3

0.4

0.4

0.4

0.4

0.5

0.5

0.5

0.6

0.6

0.6

0.7

0.7

0.7

0.8

0.8

0.8

0.9

0.9

1

1

d−axis current [p.u]

q−ax

is c

urre

nt [p

.u]

P trt♦r② ♥ qrr♥t♣♥

r sts ♦ t t♦rq r♠♣ tst

♥② t ♦♥tr♦ s♠ s ♥ tst t s♣ st♣ ♥t s ♦ t s s♣ t ♥ rr♥ s ②♦♥ t rsts ♦ ts①♣r♠♥t r r♣♦rt ♥ t ♥ s♥ tt t ♣r♠tr η ♥ β strtrs♥ t ♥st♥t t = 1[s] s♥ t r♦tt♦♥ ♦ t ♦♣rt♥ ♥ ts s♦♣ sr② r ♥t t ♦♥s t t P❱ ♥ β = 0 tr ts ♠♦♠♥tη ♦s ♦♥ rs♥ t♦ ♥t s ♣r♦♦♥ t rt♦♥ ♦ α ttr s♥♦ t ♦♥ t rr♥ts s♥ t② r ♦ t α · IN ♥ tr♦r t t

♠♣♠♥tt♦♥ ♥ sts

rr♥t ♠♣t ♦s ♥♦t r t ♠t r

0 0.5 1 1.5 2 2.5 3−1

−0.5

0

d−axis current [p.u.]

0 0.5 1 1.5 2 2.5 30

0.5

1

q−axis current [p.u.]

0 0.5 1 1.5 2 2.5 3−1

0

1

VL−parameter [−]

Time [s]

ηαβ

rr♥t ♥ ❱ ♣r♠tr

−1 −0.8 −0.6 −0.4 −0.2 00

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.1

0.1

0.1

0.10.1

0.2

0.2

0.20.2

0.3

0.3

0.3

0.3

0.4

0.4

0.4

0.4

0.5

0.5

0.5

0.6

0.6

0.6

0.7

0.7

0.7

0.8

0.8

0.8

0.9

0.9

1

1

d−axis current [p.u]

q−ax

is c

urre

nt [p

.u]

P t♦ P❱ tr♥st♦♥ ♥t qrr♥t ♣♥

r sts ♦ t s♣ st♣ tst

st♥ rq♥② ♦ t ♣r♦♣♦s ♦♥tr♦ s♠ s ♥ t ♥♦♠♣r t tt ♦ trt♦♥ ♣ ❱t♦r ♦t♦♥ ❱ strt② r♥t tsts ♥ ①t t♦ r② t ♦r ♦ t s♠ ♥ r② ♦r♥♦♥t♦♥ rt♦ t♥ t ♥♠r ♦ ♦♠♠tt♦♥s ♦ t t♦ s♠s ♥♥ s ♥ s s ♥① ♦ ♦♠♣rs♦♥ t s ♥ r tt t rt♦ s♥ t r♥ Nsw = 4 − 5.5 ♠♥♥ tt t ♣r♦♣♦s s♠ 4 t♦ 5.5 t♠s♦r ♥♠r ♦ ♦♠♠tt♦♥s

Nsw =NSVM

NMPHCC

s♦s t ①♣r♠♥t tst rr ♦t t♦ t t ♥♠r ♦ ♦♠♠tt♦♥ ♥ t P r♦♥ s♦s t s♣ t rr♥ts ♥ t ♣♦♥t ♦♦r ♥ t q ♣♥ s♦s t ♥♠r ♦ st♥ ♦♥ ♣s ♥t t♦t t ♥ ♥♦t tt t ♦♠♠tt♦♥s r q② strt ♦♥ t tr♣ss

0 0.5 10

100

200

300

Time [s]

Sp

ee

d [rp

m]

0 0.5 1−1

0

1

Time [s]

Cu

rre

nt [p

.u]

−1 −0.5 00

0.2

0.4

0.6

0.8

1

0.1

0.1

0.1

0.2

0.2

0.2

0.3

0.3

0.3

0.4

0.4

0.4

0.5

0.5

0.6

0.6

0.7

0.7

0.8

0.8

0.9

0.9

1

1

d−axis current [p.u.]

q−a

xis

cu

rre

nt [p

.u.]

id

iq

P tst

0 0.5 10

0.2

0.4

0.6

0.8

1

Nmax

=x60000

Timex[s]

Sw

itch

ing

xCo

un

ter

SVM

MP−HCC

0 0.5 10

0.2

0.4

0.6

0.8

1

Singlexphase:xNmax

=x20000

Timex[s]

Sw

itch

ing

xCo

un

ter SVM

MP−HCC

♠r ♦ st♥

r ♦♠♣rs♦♥ t♥ ❱ ♥ P

t s ♦rt t♥ tt t rsts r♣♦rt ♥ ts st♦♥ s♦ r♠r rr♥t r♣♣ rs♦♥ ♦ t ttr s t♦♦ st t ♣rt♦♥ s ♠①♣♦t♥ t ♥♦♠♥ ♠♦ ♦ t ②♥ s ①♣♥ ♥ ts ss ♥

♦ Prt ②strss rr♥t ♦♥tr♦

♥rs r♣♣ ♥ t rr♥t ♦r♠s ss ts s ♥ ②strss s ♦♥tr♦tt ♥r♥t② ♣rs♥ts ♥ r r♣♣ ♦♠♣r t♦ ♦♠♠♦♥ rr♥t ♦♥tr♦ ♦♥srt♦♥s ♠ ♦♥ t ♥♠r ♦ st♥ t tt t ♣r♦♣♦s s♠ ♦r ♥♠r ♦ ♦♠♠tt♦♥s ♦r ts rs♦♥ t rr♥t r♣♣ ♦ ♠tt ② ♥rs♥ t rq♥② t♦t ①♥ t ♥♠r ♦ ♦♠♠tt♦♥s♦t♥ t trt♦♥ ❱ ♦♥tr♦ strt②

♦r ♦ t s②st♠ t Ts = 30[µs] s ♥ t t s♠t♦♥ rsts r r♣♦rt ♥ t ♥ s♥ tt t r♣♣ s s♥♥t②r r♦♠ 30.8% t♦ 14.8% ss t ♥♠r ♦ ♦♠♠tt♦♥s s st ♦r t♥♥ ❱ s♥ t rt♦ s Nsw = 3.3 ♥② t s♦ ♥♦t r♦♠ t st ♣♦t♦ tt t ♠♣t ♦ t rr♦r E s rst② rs ♥ t ♠♦st②r♠♥s ♦ t ♠t Emax ♦r ts rs♦♥ t ②strss ♥ ♠♣t ♥ rs♦♥② r

0 0.05 0.1 0.15 0.2 0.25 0.3−0.6

−0.5

−0.4

d−ax

is [p

.u.]

0 0.05 0.1 0.15 0.2 0.25 0.30.4

0.5

0.6

q−ax

is [p

.u.]

0 0.05 0.1 0.15 0.2 0.25 0.30

0.05

0.1

Err

or [−

]

Time [s]

rr♥ts ♥ rr♦r

−1 −0.8 −0.6 −0.4 −0.2 00

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.1

0.1

0.1

0.10.1

0.2

0.2

0.20.2

0.3

0.3

0.3

0.3

0.4

0.4

0.4

0.40.

5

0.5

0.50.

6

0.6

0.6

0.7

0.7

0.7

0.8

0.8

0.8

0.9

0.9

1

1

q−ax

is c

urre

nt [p

.u]

d−axis current [p.u]

P♦♥t ♦ ♦r ♥ t q rr♥t♣♥

r ♦♠♣rs♦♥ t♥ Ts = 100[µs] ♥ Ts = 30[µs]

s♠ tst s ♥ r♣t t ②strss ♥ rsts r s♦♥♥ r ♦♠♣rs♦♥ t s ♦ Ts = 100[µs] s r♣♦rt ♥ t②strss ♥ ♦s rtr rs ♦ t rr♥t r♣♣ ♥ ts tst ts s8.3% t s ♦rt ♠♥t♦♥♥ tt t ♥♠r ♦ ♦♠♠♥tt♦♥ ♥ ts tst sst ♦r t♥ ❱ ♥ Nsw = 1.5

0 0.05 0.1 0.15 0.2 0.25 0.3−0.6

−0.5

−0.4

d−ax

is [p

.u.]

0 0.05 0.1 0.15 0.2 0.25 0.30.4

0.5

0.6

q−ax

is [p

.u.]

0 0.05 0.1 0.15 0.2 0.25 0.30

0.05

0.1

Err

or [−

]

Time [s]

rr♥ts ♥ rr♦r

−1 −0.8 −0.6 −0.4 −0.2 00

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.1

0.1

0.1

0.10.1

0.2

0.2

0.20.2

0.3

0.3

0.3

0.3

0.4

0.4

0.4

0.4

0.5

0.5

0.5

0.6

0.6

0.6

0.7

0.7

0.7

0.8

0.8

0.8

0.9

0.9

1

1

q−ax

is c

urre

nt [p

.u]

d−axis current [p.u]

P♦♥t ♦ ♦r ♥ t q rr♥t♣♥

r st t ②strss ♥

♦♥s♦♥s

♦♥s♦♥s

♥ ts ♣tr ♥ ♥ ♦♥tr♦ s♠ ♦r s♣ ♦♣rt♦♥ ♦ ②♥ ♠♥s ♣rs♥t s♠ ♦♥s t♦ t ss ♦ ♦ Prt ♦♥tr♦ s♥ t♦rt♠ s s ♦♥ t ♠♥♠③t♦♥ ♦ t ♦st ♥t♦♥ s♦t♦♥ ♦t♥ ②ts ♣r♦ss s t ♦♥ tt ♦s ♣♥ t rr♦r t♥ t ②strss ♥ ♦r t♦♥st t♠

♠♦ s ♦♣ ♦♥sr♥ ♥r ②♥ ♥ ♦t t ♣rt♦♥ ♥ ♥t rr♥ ♥rt♦♥ ♥ t ♥t♦♥ ♦ t ♦♣rt♥ ♥ rtss s♠♣ ♠t♦ t♦ ♥ t ts ♦ r♦♥ strt♦♥ s ♥ ♣r♦♣♦s

♣r♦♣♦s ♦♥tr♦ s♠ ♦s t ①♣♦tt♦♥ ♦ t s②♥r♦♥♦s rt♥ ♠♥ ♣♦t♥ts ♥ ♣rtr t s ♥ s♦♥ t t♥ss ♦ t♦rt♠ ♥ ♦♣rt♥ ♣♦♥ts ♥r P ❲ ♥ P❱ trt♦rs s♣ ♣ts r ② r♦tt♥ t ♦♣rt♥ ♥ ♥ r♥ t P s♣♦♥tr♦r ♠ts ② ♠♥ ♦ ♥ ♥♥♦t ♦t ♦♦♣ ttr rs ♥♦r♠t♦♥♦♥ t rr♦r stts r♦♠ t ♦♥tr♦ strt② t ♥r♥t② r♦♥③s ♥ t ♦t♠t s r ♥ t ♦♣rts ♦r♥②

♥② ♦♠♣rs♦♥ t trt♦♥ ❱ strt② s ♥ ♠ ♥ ♣rtrt ♥♠r ♦ st♥ ♦ t t♦ s♠s s ♥ t ♥ t ♣r♦♣♦ss♠ ♦♥s t♦ ②strss ♦♥tr♦ ① t s♠♣♥ t♠ t ♥♠r ♦ ♦♠♠tt♦♥s s s♥♥t② ♦r t♥ ♥ ❱ s ♦ t♦ rs t s♠♣ t♠ t♦r t rr♥t r♣♣ ttr s s t ♠♦s ♣rt♦♥ tt rs♦♥ t ♥♦♠♥ ♠♦ ♦ t ②♥ ss ②strss ♦♥tr♦ s♠s ♦♠♠♦♥②♣rs♥ts ♥ ♥rs r♣♣

♦♥tr♦ s♠ s ♥ sr ♥ t ♥ r ② ♠♥ ♦ ①♣r♠♥ts rsts ♦t♥ ♦♥ t tst ♥ s♦ ts st② ♥ ts t♥ss ♥♣rtr ♦♦ ♣r♦r♠♥ ♥ t ♥ t ♦ s♣ r♥ ♥ ♥t tr♥st♦♥ t♥ r♥t ♦♣rt♦♥ ♦ t ♠♥

♦♥s♦♥s

♥ t rst ♣rt ♦ ts ♦r t ♠♦ ♦ t s②♥r♦♥♦s rt♥ ♠♥ s ♥♦♣ tr qt♦♥s ♥ ♦t♥ ♦♥sr♥ t ♠♦st ♥rs ♦ tr ♣s tr ♠♥ t ♦② tr ♣s ♠♥ t s♦rt t♥ tt ts ♠♦ ♦ r♣rs♥t t ♦r ♦ r② ♥ ♦ s②♥r♦♥♦s ♠♥ ♥♥ ♣r♠♥♥t ♠♥t ♥ rt♥ ♠♥s ♠♦s ♥ ♦♣ t t ♦♥② ss♠♣t♦♥ ♦ ♥♦ ②strss ♥ ♥♦ ② rr♥tstr tt t qt♦♥s ♥ ♦t♥ ♥tr♦♥ t ②♣♦tss ♦ ♥rt② ♥t♥ r♦♥ strt♦♥ tr ♣s ♠♦ s ♥ t♥ tr♥s♦r♠ ♥ tq♥t t♦ ♣s s②st♠ ♥tr♦♥ ♥r rr♥ r♠ qt♦♥s ♥ ♥② ♦t♥ ♦♥sr♥ stt♦♥r② ♥ s②♥r♦♥♦s② r♦tt♥ rr♥r♠

rsts ♦t♥ ♦r t ♦② ♠♥s ♦t t tr ♣s ♥ t t♦♣s ♥ s t♦ ♦♣ t ♠♦ ♦ t ②♥r♦♥♦s t♥ ♠♥② ♠♣♦s♥ s♣ ♦♥t♦♥ ♦♥ t r♦t♦r rr♥ts ♥ ♣rtr t♥ ♥ r♦t♦rrr♥ts ♥ t ♠♦ ♦ t ♦② ♠♥ t s ♣♦ss t♦ r♣rs♥t ts ♥ ♦♠♥ ♥②ss r♣♦rt ♥ ts ♣tr r t ss ♦ ts ♦r ♥ r str♦♦t t tss

Prt rr♥t ♦♥tr♦ s ♥ sr ♥ ♥②s ♥ t s♦♥ ♣tr♦♠ ♦♥srt♦♥s ♦t t ♦ ♦ t ♣r♦♣r ♦st ♥t♦♥ ♥ s ♦ s♥t♠♥s ♥ ♣r♦ ♦♥sr♥ t s ♦r ♥ ♣rs♥ ♦ r♦t♦r♥s♦tr♦♣② t s ♥ s♦♥ tt t ♦st ♥t♦♥ s ♦♥ t s♦t rr♦r ♦♣rs♥t ♥♦♥ ♦♣t♠ ♦r r♥ ♣rtr ♦♣rt♦♥s rst♥ ♥ ♦ ②♥♠s♥ t ①s t t ♦r ♥t♥ t s ♥ s♦♥ tt ♥tr♦♥ t♥t♦r q t♦ t s♥② rt♦ ♦s ♦r♦♠♥ ts ♦r ❲t ts s♦t♦♥t s s ♦♠♣♥st ♥ t rr♦r ♦♥ t t♦ ①s r r ♥ ♥ ② ♠♥ r ♦ ts s♦t♦♥ s tt t rqrs t ♥♦ ♦ t s♥②rt♦ t s ♦ t ♥t♥ t s ♥ s♦♥ tt s♥ ♦st ♥t♦♥s ♦♥ t sqr rr♦r ♦s ♦t♥♥ s♥t② ♥ ♦r ♦♥ ♦t①s t♦t ♥② rqr♠♥ts ♦♥ t ♥♦ ♦ t ♠♥ ♣r♠trs ♦r tsrs♦♥s ts ♦st ♥t♦♥ s ♥ ♦s♥ ♥ ts ♦r

♦♣rt♦♥ ♦ t ♣rt rt♦r s ♥ sr ♦♥sr♥ t t♦st♣ ♣rt♦♥ rqr t♦ ♦♠♣♥st ♦r t ②s ♥tr♦ ② t t ♠♣♠♥tt♦♥ tr tt ♦t t ♦s ♥ t ♦r ♣rt♦♥ ♥

♦♥s♦♥s

♣rs♥t ♥ t ♦r♠r s t t ♦ ♣r♠tr ♠s♠t ♥ rt♦♥ ♥♥stt ♥ ♣rtr ♥ ②♥r♦♥♦s t♥ ♣♣t♦♥s r♦♥ strt♦♥ s♥ r♦♥s s t ♠♥ rs♣♦♥s ♦r ♠♦ ♥qs

♦r ♣rt♦♥ s ♥ rtr ♥②s ♥ ♣tr tr ♣r♦♥ s♦♠♦♥srt♦♥s ♦♥ t tr♠♥t ts ♦ ♦t t♦r st♥t♦♥ ♦ ♦r♦♠♥stts ♥ t ♦ss ♦ ♦♥tr♦ ♥ ② ♥ ♥rq♥t ❯ ♣t ♥♦ ♣t♥ ♠t♦ s ♥ ♣rs♥t s t♥q ♦s r♦♥strt♥ t rr♥trt♦♥s s♥ ♦♥② t s rt t♦ t tr ♠♦st r♥t ♦t t♦rs rt♦♥s♣ t♥ t t rr♥t rt♦♥s s ♥ s t♦ ♥t② ♣t t❯ st ♦rt♠ ♦r t sq♥ ♥tt♦♥ s ♥ ♦♣ ♥ ♠♣♠♥t ♠ ♦ t ttr s t♦ r♦♥③ tr t ♦t ♥①s ♣ttr♥ ♦r♠s st sq♥ ♦r t ❯ ♣t ♦rt♠ s ♥ ♦♣ ♦♥sr♥t strt ♦♥str♥t ♥ ② t r t♠ ♣♣t♦♥ ♥ ♣rtr t ♥♠r ♦♣♦ss sq♥s tt s t♦ t s r ♣♥♥ ♦♥ t ♥♣t ♦t♥①

♣r♦♣♦s ♦rt♠ s ♥ t ② ♠♥ ♦ ①♣r♠♥ts ♥ t rsts ♥ ♦♠♣r t t♦s ♦ trt♦♥ ♦s ♣rt ♦♥tr♦ t♦t t t♥ss ♦ t ♣r♦♣♦s s♠ ♥ ♥r t t ♦ ♣r♠trrt♦♥s ♥ ♣rtr t tsts ♥ ♣r♦r♠ t r♥t ♦s t♦ tt ♦r ♦ t s♠ t str♦♥ r♦♥ strt♦♥ rsts r② s♦♥t ♥ts ♦ t ♦r ♣rt♦♥ ♥ ♣rtr t ♣rt♦♥ rr♦r s rt②r ♥ s ♦♥sq♥ t rr♥t ♦r♠s ♣rs♥t r r♣♣

♠tt♦♥s ♥ t rtts ♦ t ♣r♦♣♦s ♦♥tr♦ s♠ ♥ t ♥ ♣rtr t ♣rs♥ ♦ stt♦♥s ♥ st♥t♦♥ s ♥♦t s♥stt② t♦ ♥♦ss ♥ str♥s ♥ t ♥rs tr♥r♦♥ t♠ ♦ ts♣rtr ♠♣♠♥tt♦♥ r t ♠♥ sss

♣r♦r ♦r t strt♣ s ♥ ♣r♦♣♦s ♥ ts ♦♥t♦♥ t ❯ r♠♣t② ♠♥♥ tt t ♣rt rt♦r ♥♥♦t t t t ♦ t t♦t t♦rs rs ♥ ② ♠♦s strt♣ ♥ t♥ tr tt st ♣r♦r s ♥ ♠♣♠♥t

st ♣rt ♦ ts ♦r s ♦♠♠t t♦ t ♦♣♠♥t ♦ ♥ ♥ ♦♥tr♦s♠ ♦r s♣ ♦♣rt♦♥ ♦ ②♥r♦♥♦s t♥ ♠♥ s♠♦♥s t♦ ♦t ♦ Prt ♥ ②strss ♦♥tr♦ s♥ t ♦rt♠ s s♦♥ t ♠♥♠③t♦♥ ♦ t ♦st ♥t♦♥ ♥ t rst ♦ ts ♣r♦ss s t ♦♥ tt♦s ♣♥ t rr♥t rr♦r t♥ ① trs♦ ♦r t ♦♥st t♠

♠♦ s ♥ ♦♣ ♦♥sr♥ ♥r ♠♥ ♥ ♦t t ♣rt♦♥♥ ♥ t rr♥ ♥rt♦♥ ♥ t ♥t♦♥ ♦ t ♦♣rt♥ ♥ rtss s♠♣ ♠t♦ t♦ ♥ t ts ♦ r♦♥ strt♦♥ s ♥ ♣r♦♣♦s

t s ♥ s♦♥ tt t ♣r♦♣♦s ♦♥tr♦ s♠ ♦s t ①♣♦tt♦♥ ♦t ②♥r♦♥♦s t♥ ♠♥ ♣♦t♥ts ♥ ♦♣rt♥ ♣♦♥ts ♥r P❲ ♥ P❱ trt♦rs s♣ ♣ts ♥ ② r♦tt♥ t ♦♣rt♥ ♥ ♥ r♥ t P s♣ ♦♥tr♦r ♠ts ② ♠♥ ♦ ♥♥♥♦t ♦t ♦♦♣ ttr rs ♥♦r♠t♦♥ ♦♥ t rr♦r stts r♦♠ t♦♥tr♦ strt② t ♥r♥t② r♦♥③s ♥ t ♦t ♠t s r ♥ t ♦♣rts ♦r♥②

♦♥s♦♥s

♦♠♣rs♦♥ t trt♦♥ ♣ ❱t♦r ♦t♦♥ strt② s ♥ ♦♥♥ ♣rtr t ♥♠r ♦ st♥ ♦ t t♦ s♠s s ♥ t ♥ t♣r♦♣♦s s♠ ♦♥s t♦ ②strss ♦♥tr♦ ① t s♠♣♥ t♠ t ♥♠r ♦♦♠♠tt♦♥s s s♥♥t② ♦r ♦♥sq♥t② t s ♦ t♦ rs t s♠♣t♠ t♦ r t rr♥t r♣♣ ttr s s ② t♦ s♣ts ②strsss s♠s ♥r♥t② ♣rs♥t ♥rs r♣♣ ♥ ss t rr♥t ♣rt♦♥s ♦t♥ s♥ t ♥♦♠♥ ♠♦ tt ♦s ♥♦t ♥ t ts ♦ r♦♥ strt♦♥

tr ♦rs

s tss r♣♦rts ♣rt ♦ t ♦r ♦♥ r♥ t tr ②rs ♦ P sts♥ ♥♦ t t ♥ ♦ ts ♣t r♠② tt tr st r r♦♦♠ ♦r♠♣r♦♠♥ts ♥ rtr ♦♣♠♥ts ②♦♥ tt ♥ ♦♣♥ t♦♣s s r s♦♠tr ♦rs tt ♣rs♦♥② ♦ t♦ ♦ ♠② P s ♥♦t ♠t ♥ t♠

♦r ♥tr♦s rt ♥ts ♥ tr♠s ♦ r② ♦ t ♣rt♦♥ ♥t ♦tr ♥ ts t♥q srs r♦♠ ♦t t♦r st♥t♦♥ tt ♦ t♦♥stt② ts ♣♥♦♠♥♦♥ s ♥♦t t♥ ♥t♦ ♦♥t ♥ ♣r♥t ♥ ts ♦r ♥♦ ♠t♦ t♦ r♥t rq♥t ♥ ♣rs rr♥t rt♦♥s ❯ ♣ts ♥ ♣rs♥t rtss s r② ♠♥t♦♥ s♦♠ rt stt♦♥s ♥ st♥t♦♥ s ♥♦ ♦ ♦r r♥ t ♠♥ ♦♣rt♦♥ r♥t② ♦s ♦♥tr♦ trs ♥ ②s st ♦♣rt♦♥ ♥ ♥r t ts♦ rs♦♥ ♣r♠trs ♠s♠t ♦r ts rs♦♥ t ♦ ♥trst♥ t♦ s ♣rt♦♥ s ♦♥ ♠♦ ♥ t ♣r♠trs r ♥t r♥ t ♠♥♦♣rt♦♥ ♥ ♣rtr t s♠ ♠t♦ s ♦r t ❯ ♣t ♦ ♣tt♦ ♦♣ st ♣r♠tr ♥tt♦♥ ♦rt♠ ♦t♣t ♦ s t♦ t ♦s ♣rt♦♥

♥s t♦ ♠t♦ ♣r♦♣♦s ♥ ts ♦r t ❯ ♦♥t♥s ♣rs ♥♦r♠t♦♥s♦t t t stt ♦ t ♠♥ ♦r ts rs♦♥ t s ♦ t rr♥t rt♦♥s st♦r ♥ t ❯ ♦ s t♦ r♦♥③ tr t ♦t ♠t s ♥r s r♥t tr ♦♠♥ t ♦tr s♦t♦♥s ♦♣ ♥ ts ♦r♦ s t♦ ♦♣ ♦t ♦♦♣ ♦r ① ❲♥♥ ♥ ①♠♠ ♦rq ♣r❱♦t ♦♣rt♦♥s

♦r♣②

❬❪ P ♦rts P ③♠r♦s ♥♥ ♦ ♥ ♦r③Prt ♦♥tr♦ ♥ ♣♦r tr♦♥s ♥ rs r♥st♦♥s ♦♥ ♥str tr♦♥s ♦ ♥♦ ♣♣

❬❪ ❨♦♥ Pr③ ♥ ♦r③ ♥②ss ♦ ♥t♦♥tr♦st ♠♦♣rt rr♥t ♦♥tr♦ t ♠♦ ♣r♠tr ♠s♠t ♥ tr♣s ♥rtr r♥st♦♥s ♦♥ ♥str tr♦♥s ♦ ♥♦ ♣♣ ②

❬❪ ❲♥ ❳ ♥ ❨ ❨ ❲♥ ♥ ❳ ♦st ♣rt rr♥t ♦♥tr♦t ♦♥♥ str♥ st♠t♦♥ ♦r ♥t♦♥ ♠♥ rs r♥st♦♥s ♦♥ P♦r tr♦♥s ♦ ♥♦ ♣♣ ♥

❬❪ ❲♥ ❩ ♦♥ ❨ ❨ ❲♥ ♥ ❳ ttrr♦rss t ♣rt rr♥t ♦♥tr♦ s♥ s♦♥♦rr s♥♠♦ str♥ ♦srr ♦r ♥t♦♥ ♠♥ rs r♥st♦♥s ♦♥ P♦r tr♦♥s ♦ PP ♥♦ ♣♣

❬❪ rr ❯ ②s ♥ ♥ r♦st ♣rt rr♥t ♦♥tr♦r♦r ♣♠s♠ rs r♥st♦♥s ♦♥ ♥str tr♦♥s ♦ ♥♦ ♣♣ ♥

❬❪ ❩ ♥ ♥ ♥ ♣t ♥t♦♥tr♦st ♠♦ ♣rt rr♥t♦♥tr♦ ♦r ♣♠s♠ rs t ♥t♥ rt♦♥ tr P♦r ♣♣t♦♥s ♦ ♥♦ ♣♣

❬❪ ♠ t♦♥♥ ♦♥♠ss♦♥ ♥ ♥ ♦s♥ ♥②ss♦ t ♠♣t ♦ ♦♥♥ ♥tt♦♥ ♦♥ ♠♦ ♣rt rr♥t ♦♥tr♦ ♣♣ t♦♣r♠♥♥t ♠♥t s②♥r♦♥♦s ♠♦t♦rs tr P♦r ♣♣t♦♥s ♦ ♥♦ ♣♣

❬❪ ♥ t ❨ ♥ s♦ ♦r ♣rtrr♥t ♦♥tr♦ ♦r ♥tr♦r ♣r♠♥♥t♠♥t s②♥r♦♥♦s ♠♦t♦r rs s ♦♥rr♥t r♥ tt♦♥ t♥q r♥st♦♥s ♦♥ ♥str tr♦♥s ♦ ♥♦ ♣♣

♦r♣②

❬❪ ♥ t ❨ ❨ ♥ ❨ ♠♣r♦ ♠♦r ♣rtrr♥t ♦♥tr♦ ♦r s②♥r♦♥♦s rt♥ ♠♦t♦r rs r♥st♦♥s ♦♥♥str tr♦♥s ♦ ♥♦ ♣♣ ♥

❬❪ P ♦rts ♦r③ ♥ ♦rs ② ♦♠♣♥st♦♥ ♥ ♠♦ ♣rt rr♥t ♦♥tr♦ ♦ tr♣s ♥rtr r♥st♦♥s ♦♥ ♥strtr♦♥s ♦ ♥♦ ♣♣

❬❪ ♠ r s③ ♥ ♦r③ ♦st♥ss ♠♣r♦♠♥t ♦ ♣rt rr♥t ♦♥tr♦ s♥ ♣rt♦♥ rr♦r ♦rrt♦♥ ♦r ♣r♠♥♥t♠♥t s②♥r♦♥♦s ♠♥s r♥st♦♥s ♦♥ ♥str tr♦♥s♦ ♥♦ ♣♣ ♥

❬❪ ♦r③ ♥ P ♦rts Prt ♦♥tr♦ ♦ P♦r ♦♥rtrs ♥ trrs ♦♥ ❲② ♦♥s t

❬❪ ♦r③ P♦♥tt P ♦rr P ③♥ P ♦rts ♥ ❯ ♠♠♥♥ Prt rr♥t ♦♥tr♦ ♦ ♦t s♦r ♥rtr r♥st♦♥s♦♥ ♥str tr♦♥s ♦ ♥♦ ♣♣

❬❪ ♦♦ rrr♦ r ♦r ♥ ♥stt② t♦ tr♣r♠tr rt♦♥s ♦ ♣rt rr♥t ♦♥tr♦ ♥ ♠t♣s rs ♥ t ♥♥ ♦♥r♥ ♦ t ♥str tr♦♥s ♦t② ♦ ♣♣

❬❪ ❯ ♦♦♥ ♥ Pr Prt♦♥tr♦s rt ♣♦r ♦♥tr♦ t ♥ ♣t ♣r♠tr ♥tt♦♥ t♥q ♦r ♠♣r♦ ♣r♦r♠♥ r♥st♦♥s ♦♥ P♦r tr♦♥s ♦ ♥♦ ♣♣ ♦

❬❪ ♥ P ③♠r♦s rt t♦rq ♦♥tr♦ ♦ ♣♠ ♥rtr ♠♦t♦rs sr② r♥st♦♥s ♦♥ ♥str tr♦♥s ♦ ♥♦ ♣♣

❬❪ ♦rs♣♦r ♥ Ps ♥♦rss ♣rt rt t♦rq ♦♥tr♦ ♦rs②♥r♦♥♦s rt♥ ♠♥s t r② ♦ ♥ ③r♦ s♣ r♥st♦♥s♦♥ ♥str tr♦♥s ♦ ♥♦ ♣♣

❬❪ rt♥ ❱r♥ ♥ rs♥ Prt rt t♦rq ♦♥tr♦ ♦r ①♥ t♦rq r♣♣ rt♦♥ r♥st♦♥s ♦♥ ♥str tr♦♥s ♦ ♥♦ ♣♣ ♥

❬❪ ♦r③ ♥♥ s♣♥♦③ r♥♦ ♥ ♦s ♣r♦r♠♥ ♦♥tr♦ strts ♦r tr rs ♥ ①♣r♠♥tssss♠♥t r♥st♦♥s ♦♥ ♥str tr♦♥s ♦ ♥♦ ♣♣

❬❪ ②r P♣♦t♦ ♥ ♦rr ♦ ♣rt rt t♦rq ♦♥tr♦ ♣rt ♦♥♣t ♦rt♠ ♥ ♥②ss r♥st♦♥s ♦♥ ♥str tr♦♥s ♦ ♥♦ ♣♣ ♥

♦r♣②

❬❪ r♥ P ♦rts ❨③ ♥ ♦r③ Prt t♦rq ♦♥tr♦ ♦♥t♦♥ ♠♥s s ♦♥ stts♣ ♠♦s r♥st♦♥s ♦♥ ♥strtr♦♥s ♦ ♥♦ ♣♣ ♥

❬❪ ❲ ❳ ❳ ❲♥ ❲♥ ❲ ❳ ♥♥ r♥ ♥ ♦r♥③♥t♦♥tr♦st ♠♦ ♣rt t♦rq ♦♥tr♦ t t s♦t♦♥ ♦r♣♠s♠ rs r♥st♦♥s ♦♥ ♥str tr♦♥s ♦ ♥♦ ♣♣ ♣t

❬❪ Pr♥ ♥ ♦♦♥♥ ♦ ♣rt rt t♦rq ♦♥tr♦ t ♥t♦♥tr♦ st ♦r ♣♠s♠ r s②st♠s ♣rt ①♠♠ t♦rq ♣r ♠♣r ♦♣rt♦♥ r♥st♦♥s ♦♥ ♥str ♥♦r♠ts ♦ ♥♦ ♣♣ ♦

❬❪ ♦ ♣rt rt t♦rq ♦♥tr♦ t ♥t ♦♥tr♦ st ♦r ♣♠s♠ rs②st♠s ♣rt ♥♥ ♦♣rt♦♥ r♥st♦♥s ♦♥ ♥str♥♦r♠ts ♦ ♥♦ ♣♣ ②

❬❪ ♥ ♥ ② ②♥♠ t♦rq ♦♥tr♦ ♦ s②♥r♦♥♦s rt♥♠♦t♦r r♥st♦♥s ♦♥ P♦r tr♦♥s ♦ ♥♦ ♣♣

❬❪ ❳ ❩♥ ♥ ♦♦ r♦st ♥♥ ♦rt♠ s ♦♥t② rt♦ rt♦♥ ♦r rt t♦rq ♦♥tr♦ s②♥r♦♥♦s rt♥ ♠♦t♦r r♥st♦♥s ♦♥ tr♦♥s ♦ ♥♦ ♣♣ ♣r

❬❪ ❨ ♥ ❩ ❩ ♥ rt t♦rq ♦♥tr♦ ♦ ♣r♠♥♥t♠♥t s②♥r♦♥♦s ♠♥ rs t s♠♣ t② rt♦ rt♦r r♥st♦♥s♦♥ ♥str tr♦♥s ♦ ♥♦ ♣♣ t

❬❪ ♦♦♥♥ ♥ ♦♥ t ♦r♠t♦♥ ♦ t t strt② ♦r ♦♥r♥ ♥ stt② ♥②ss t ♣♠ ♠♦t♦r r s st② ♥ ♥tr♥t♦♥ ②♠♣♦s♠ ♦♥ ♥s♦rss ♦♥tr♦ ♦r tr rs ♥ Prt ♦♥tr♦ ♦ tr rs ♥ P♦r tr♦♥s P t ♣♣

♥♦♠♥ts

s ss s rtt♥ ♥ ❳♥ tr♦♥ rs♦♥ s t tt♣♣rsr♥♣t