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 A-Phase quantity b B-Phase quantity c C-phase quantity irect-axis quantity q quarature-axis quantity F fiel quantity D irect-axis amper wining quantity Q quarature-axis 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 Self-inuctances L aa = L s + L m cosθ Stator Mutual-inuctances 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 Self-inuctances 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 Mutual-inuctances L Q L FD = L DF = M r Stator-rotor Mutual-inuctances 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 open-circuit transient time-constant irect axis short-circuit transient time-constant irect axis short-circuit subtransient time-constant armature short-circuit (.c.) time-constant

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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