Derivation of the VOC method

Ske ch o an idea
Ca alin A ghi
No embe 4, 2014
Suppose we s a wi h he ollowing model o a 3-phase in e e in α-β ame:
˙x1=u1−w1
˙x2=u2−w2(1)
Whe e x ep esen s he ou pu capaci o ol ages, u he con e e side cu -
en s cha ging he wo 1 Fa ad capaci o s and w he g id side cu en s.
We plan o s abilize he se :
γ=x:px2
1+x2
2=ρ0
We use he ollowing coo dina e ans o ma ion:
 =px2
1+x2
2
φ= a c an(x2
x1)(2)
By aking he ime de i a i es o (2) we ge :
˙ =1
2
2x1˙x1+2x2˙x2
√x2
1+x2
2
=x1(u1−w1)+x2(u2−w2)
=1
x1x2u1
u2−1
p
˙
φ=1
1+ x2
2
x2
1
x1˙x2−x2˙x1
x2
1
=x1(u2−w2)−x2(u1−w1)
2=1
2−x2x1u1
u2−1
2q
(3)
Whe e p and q a e he ins an aneous ac i e and eac i e powe s:
p=x1w1+x2w2
q=x1w2−x2w1(4)
We ew i e eq. (3) as:
˙
˙
φ=1
2x1 x2
−x2x1u1
u2−1
p
1
2q(5)
We use he ollowing eedback ans o ma ion (in oking a ans e se eed-
back linea iza ion a gumen o simply o esemble a ha monic oscilla o ):
u1
u2=x1−x2
x2x1b
ω(6)
Eq. (5) becomes:
1
˙
˙
φ=1
2x1 x2
−x2x1x1−x2
x2x1b
ω−1
p
1
2q=1
2(x2
1+x2
2) 0
0 (x2
1+x2
2)b
ω−1
p
1
2q
We end up wi h he ollowing sys em:
˙
˙
φ= 0
0 1b
ω−1
p
1
2q(7)
By le ing ˙
φ=ω+˙
θand se ing ω=cons ., we ge he ollowing phase
dynamics:
˙
θ=−
1
2q
By le ing b=b01− 2
ρ2
0, whe e ρ0is he adius o he ci cle-se o be
s abilized, we ge he ollowing ampli ude dynamics:
˙ =b0 1− 2
ρ2
0−
1
p
In ac one could pick any unc ion b( ) ha is ze o a ρ0, s ic ly posi i e
o < ρ0and s ic ly nega i e o > ρ0as b is he eal pa o he complex
poles o he ha monic oscilla o :
˙x1
˙x2=b−ω
ω b x1
x2(8)
The ske ch shows ha i ins ead o s a ing wi h a Liena d oscilla o , one
s a s wi h a 2 s a e in eg a o , by applying a ce ain coo dina e and eedback
ans o ma ion one is able o highligh eedback ha u ns a gi en se in o an
a ac i e limi cycle. Mo eo e , o a pa icula choice o eedback, he closed
loop sys em may ake he shape o a Van de Pol oscilla o .
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