Synchronous Machine Modeling


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1 ECE 53 Session ; Page / Fall 07 Synchronous Machine Moeling Reference θ Quarature Axis B C Direct Axis Q G F D A F G Q A D C B Transient Moel for a Synchronous Machine Generator Convention
2 ECE 53 Session ; Page / Fall 07 Notation f can represent i, v, λ. α = r rotating reference frame fixe on the rotor s stationary reference frame fixe on the stator β = s stator quantity r rotor quantity a APhase quantity b BPhase quantity c Cphase quantity irectaxis quantity q quaratureaxis quantity F fiel quantity D irectaxis amper wining quantity Q quaratureaxis ampe wining quantity g ey current wining quantity l leakage quantity m mutual quantity f α β f abc = f a f b f c ; f r q0 = f r f r q f r 0 Inuctances Stator Flux Linkage Equations: λ abc = L s i abc + L sr i FDgQ
3 ECE 53 Session ; Page 3/ Fall 07 Rotor Flux Linkage Equations: λ FDgQ = L T sr i abc+ L r i FDgQ Combine Stator an Rotor Flux Linkage Equations: [ λabc λ FDgQ ] = [ Ls L sr L T sr L r ][ iabc i FDgQ ] λ a λ b λ c λ F λ D λ g λ Q = L aa L ab L ac L af L ad L ag L aq L ba L bb L bc L bf L bd L bg L bq L ca L cb L cc L cf L cd L cg L cq L Fa L Fb L Fc L FF L FD L Fg L FQ L Da L Db L Dc L DF L DD L Dg L DQ L ga L gb L gc L gf L gd L gg L gq L Qa L Qb L Qc L QF L QD L Qg L QQ i a i b i c i F i D i g i Q where
4 ECE 53 Session ; Page 4/ Fall 07 Stator Selfinuctances L aa = L s + L m cosθ Stator Mutualinuctances L bb = L s + L m cos(θ π/3) L cc = L s + L m cos(θ + π/3) L ab = L ba = M s L m cos(θ + π/6) Rotor Selfinuctances L bc = L cb = M s L m cos(θ π/) L ca = L ac = M s L m cos(θ + 5π/6) L F L D L g Rotor Mutualinuctances L Q L FD = L DF = M r Statorrotor Mutualinuctances L gq = L Qg = M rq L Fg = L gf = L Dg = L Dg = 0 L FQ = L QF = L DQ = L DQ = 0 L af = L Fa = M F cosθ L bf = L Fb = M F cos(θ π/3) L cf = L Fc = M F cos(θ + π/3) L ad = L Da = M D cosθ L bd = L Db = M D cos(θ π/3) L cd = L Dc = M D cos(θ + π/3)
5 ECE 53 Session ; Page 5/ Fall 07 L ag = L ga = M g cosθ L bg = L gb = M g cos(θ π/3) L cg = L gc = M g cos(θ + π/3) L aq = L Qa = M Q sinθ L bq = L Qb = M Q sin(θ π/3) L cq = L Qc = M Q sin(θ + π/3)
6 ECE 53 Session ; Page 6/ Fall 07 Park s Transformation f r oq = R(θ r)p(0)f abc where θ r = ω r + π + δ Coorinate axis transformation P(0) := , abc oq Transformation to rotating reference frame 0 0 R(θ r ) := 0 cosθ r sin(θ r ), 0q s 0q r, Rotation 0 sinθ r cos(θ r ) Combine into one step P(θ r )=R(θ r )P(0) f r 0q = P(θ r)f abc f r 0 f r f r q = 3 cosθ r cos(θ r π 3 ) cos(θ r+ π 3 ) sinθ r sin(θ r π 3 ) sin(θ r+ π 3 ) f a f b f c
7 ECE 53 Session ; Page 7/ Fall 07 P(θ r ) := 3 cosθ r cos(θ r π 3 ) cos(θ r+ π 3 ) sinθ r sin(θ r π 3 ) sin(θ r+ π 3 ) P (θ r )=P T (θ r ) P (θ r )= 3 cosθ r sinθ r cos(θ r π 3 ) sin(θ r π 3 ) cos(θ r + π 3 ) sin(θ r+ π 3 ) P(θ r ) P (θ r ) t = ω := ωx 0 ω 0
8 ECE 53 Session ; Page 8/ Fall 07 Synchronous Machine Equations. Stator Voltage Equations: (Note: p = /t) v abcs = r s i abcs pλ abcs v 0qs = r s i 0qs pλ 0qs ωxλ 0qs. Rotor Voltage Equation: v FDgQr = R r i FDgQr pλ FDgQr 3. Stator Flux Linkage Equations: λ abcs = L s i abcs + L sr i FDgQr λ oqs = L s i oqs+ L sr i FDgQr 4. Rotor Flux Linkage Equations: λ FDgQr = L T sri abcs + L r i FDgQr λ FDgQr = L T sr i abcs+ L r i FDgQr
9 ECE 53 Session ; Page 9/ Fall 07 λ 0s λ s λ qs λ Fr λ Dr λ gr λ Qr = L L 0 km F km D L q 0 0 km g km Q 0 km F 0 L F M D km D 0 M D L D km g 0 0 L g M Q 0 0 km Q 0 0 M Q L Q i 0s i s i qs i Fr i Dr i gr i Qr k= 3 T E = i λ q i q λ p 3φ (t)=i 0 v 0 + i v + i q v q E a = ω 0M F i F e jδ = E a δ E a = ω 0M F i F
10 ECE 53 Session ; Page 0/ Fall 07 Synchronous Machine Parameters X X q X X q q X X 0 r sc r sac r f r r T 0 T T T a irect axis reactance quarature axis reactance irect axis transient reactance quarature axis transient reactance irect axis subtransient reactance quarature axis subtransient reactance negative sequence reactance zero sequence reactance stator c resistance stator ac resistance fiel resistance referre to the stator negative sequence resistance irect axis opencircuit transient timeconstant irect axis shortcircuit transient timeconstant irect axis shortcircuit subtransient timeconstant armature shortcircuit (.c.) timeconstant
11 ECE 53 Session ; Page / Fall 07 Machine Turbo Hyro Synchronous Synchronous Constant Generator Generator Conensor Motor X X q X X q q X X r sc r sac r T T T T a
12 ECE 53 Session ; Page / Fall 07 Three Phase Short Circuit of a Synchronous Machine i as (t) [ ( ) Ẽ a X + X X e t Ẽ a ( Ẽ a ( T +( + X )e t q Ta sin(α) X X )e t q Ta sin(ω e t+ α) ] )e t T sin(ω e t+ α) Name Magnitue Frequency T Steay E a X Funamental Transient E a ( X X ) Funamental T ( ) Subtransient E a X X Funamental T Asymmetrical E a ( ) + X sin(α) Zero q T a Secon Harmonic E a ( ) X q Double Funamental T a
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