Varable Structure Control ~ Motor Control Harry G. Kwatny Department of Mechancal Engneerng & Mechancs Drexel Unversty
Outlne Moels of ac Motors Synchronous motors Inucton motors Brushless c motors VS Control formulaton Ajustable spee synchronous motor Poston control of brushless c motor
Nomenclature (3 phase synchronous motor) θ rotor angle n nertal frame ω rotor angular velocty v,, q fel wnng voltage, current, charge q = t f f f f f v,, q, j = 1,2,3 stator wnng voltage, current, charge τ M M J R j j j, j ff ( q), mechancal torque (loa) nerta matrx n neral frame nerta matrx n rotatng (Blonel) frame mechancal rotatng nerta sspaton matrx,, j = 1,2,3 stator nuctances - self ( = j) an mutual ( j) fel wnng self nuctance f, = 1,2,3 fel/stator wnng mutual nuctance
Lagrange Equatons L L = Q, L q, q = q, q U q t q q = θ 1 2 3 f q q q q q q = ω 1 2 3 f q generalze coornates L Lagrangan ( ) ( ) ( ) 1 ( ) = 2 M ( ) U( q) = knetc energy, q, q q q q U potental energy, 0 Q generalze force Q= D qwhere ( ) 1 D potental functon Dqq, = 2 qrq + τ v1 v2 v3 vf q
Synchronous Motor J 0 0 0 0 0 L1 L3 L3 L5cosθ 2π 0 L3 L1 L3 L5cos θ 3 M ( q) = 2π 0 L3 L3 L1 L5cos θ + 3 2π 2π 0 L5cosθ L5cos θ L5cos θ + ff 3 3 2π 2π = L1, j = L3, f 1 = L5cos θ, f 2 = L5cos θ, f 3 = L5cos θ + 3 3 R= ag rrrr ( 0,,,, )
ransform to Rotatng Frame,,,, v, v, v v, v, v 1 2 3 q 0 1 2 3 q 0 2π 2π cosθ cos θ cos θ + 3 3 1 2 2 2 π π q = B 2, B= snθ sn θ sn θ + 3 3 3 0 3 1 1 1 2 2 2 J 0 0 0 0 ω 0 0 Lff 0 0 ω τ 0 Ls 0 0 L f 0 r ωl 0 0 s v 0 0 Ls 0 0 q= L f f ω L s r 0 0 q+ v q t 0 0 0 L0 0 0 0 0 0 r 0 0 v0 0 Lf 0 0 L f f 0 0 0 0 r f f vf 3 Ls : = L1+ L3, L0 = L1 2 L3, Lf = Lff 2
Synchronous ~ spee control We want to esgn a spee controller that mantans a esre spee, ω 1 1) ω ω exponentally: y1 = ω+ λ( ω ω) = { Lff q τ} + λ( ω ω), λ > 0 J 2) zero -axs current: y = 2 3 0 0 () 3) balance operaton: y χ, χ v t t 4) constant fel current: y4 = f Proceure: - reuce to regular form, - check zero ynamcs - esgn slng surfaces, - esgn reachng controller, - smulate = = f t
Synchronous ~ spee control Now, we have the system n the form: ( ) ( ) x = f x + G x v [ ω θ χ] ( ) [ ] Reuce to normal form 1 2 3 4 1 2 3 4 ( ) ( ) r = 1, r = 1, r = 1, r = 1 z x = h x an m zeroynamcs = 3 In fact zero ynamcs are: r ω = λω, 0 = L B x = vb = v vq v0 vf y = h x y = y y y y 0 0, θ = ω Note that slng ynamcs are the zero ynamcs! ( ) he stanar choce for K k = 1, = 1,2,3,4 s z = z
Synchronous ~ spee control Now, set up V = sqs= zqz V = 2zQz ( ) ( ) Furthermore, snce z = h x h x z = x = H( x) f ( x) + H( x) G( x) v x z = H x f x + H x G x B x v ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) { } V = 2zQz= 2h xqh x f x + H x G x B x v ( *) * ( ) ( ) v = Vmax, sgn s, s x = HGB Qh x, B
Synchronous ~ spee control
Synchronous ~ spee control
Inucton Motor he nucton motor has two fel ω ω τ wnngs. hey are v close crcuts so that q q the fel currents are vq M = C + 0 nuce as the cols t 0 v0 move through the f1 f1 0 stator fel. f 2 f 2 0 J 0 0 0 0 0 0 Lff 2 Lff 1 0 0 0 0 Ls 0 0 Lf 0 Lff 2 r ωl 0 0 0 s 0 0 Ls 0 0 Lf Lff 1 ωls r 0 0 0 M =, C = 0 0 0 L0 0 0 0 0 0 r 0 0 0 Lf 0 0 Lf 0 0 0 0 0 rf 0 0 0 Lf 0 0 Lf 0 0 0 0 0 rf = L ( ) f f q f 1 2
Inucton Motor Nomenclature ω rotor angular velocty v f fel wnng voltage v, =1,2,3 stator wnng voltages v a, a=,q,0 stator Blonel voltages f, =1,2 fel wnng currents,, =1,2,3 stator wnng currents a, a=,q,0 stator Blonel currents τ mechancal torque (loa) electrcal torque J mechancal rotatng nerta r,r f stator an fel wnng resstances L s stator & q axs nuctances L 0 stator zero sequence axs nuctance L f fel wnng self nuctance fel/stator mutual nuctance L f
Inucton Motor Spee Control Agan, we wsh to regulate spee ω to the esre value ω. Defne χψ, : χ = v0 ψ = rotor flux magntue, Choose ψ = ψ + ψ, ψ = L + L, ψ = L + L 2 2 1 2 1 f f f 2 f q f f ( q f1 f2 ) ( ω q f1 f2 χ) ( q f f ) y = h ω,,,,,, χ : = ω ω 1 1 0 y = h,,,,,, : = χ 2 2 0 1 2 r = 2, r = 1, r = 2 1 2 3 z ( 1 ω ω0 ) z L + L τ 2 f f 2 f f 1 q z χ 3 = y3 = h3 ω,,, 0,,, χ : = ψ ψ 1 2 z ( 4 ψ ψ0) z rψ rψ 5 f 1 f1 f 2 f 2
2-m Zero Dynamcs {(Ψ 1,Ψ 2 ) Ψ 1 2 +Ψ2 2 = Ψ0 2 } 0 zero ynamcs manfol R S 1
Brushless c Motor In the brushless c motor, the fel cols are replace by permanent magnets; thereby removng one coornate: [ θ 1 2 3] [ ω ] q= q q q q = 1 2 3 3 J 0 0 0 ω 0 0 2 k 0 e ω τ 0 Ls 0 0 0 r ωls 0 v = + 0 0 Ls 0 t q 3 2ke ωls r 0 q vq 0 0 0 L0 0 0 0 0 r 0 v0 = 3 2 k e q
Brushless c ~2 t ω 1 0 ( ( ) ) q τ 0 θ J ω 1 v ( r Lsqω ) L L 3 s = s +, ( ) = k 2 1 v q e q ( ( 32 ) ω) q q rq + Ls ke Ls L 0 s v 0 1 r L 0 0 L0
Brushless c ~ poston control We want to esgn a poston controller that mantans a esre poston, θ 1 3 0 ( ) 1) θ θ exponentally: y = ω+ λ θ θ, λ > 0 2) constant stator current: y = + I or zero -axs: y = 3) balance operaton: y = 2 2 2 2 q s 2 Proceure: - reuce to regular form, - check zero ynamcs - esgn slng surfaces, - esgn reachng controller, - smulate
Brushless c ~ poston control ( θω,,,, ): ( ) q ω λθ θ y = h = + 1 1 0 ( θω ) y = h,,,, : = + I 2 2 2 2 2 q 0 q s ( θω ) y = h,,,, : = 3 3 q 0 0 he zero ynamcs are: θ = λ θ θ ( ) r = 1, r = 1, r = 1 1 2 3 z 1 2 3 0 ( ) z = + I z = ω + λ θ θ = 2 2 2 q s