Machine Fundamental Electromagnetics

# Machine Fundamental Electromagnetics - Synchronous Machine...

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Unformatted text preview: Synchronous Machine Equations March 7, 2011 SOME BACKGROUND MAGNETIC CIRCUITS Magnetic Circuit for High Permeability Core Rc = Lc = = 1 Pc N2 Rc F = NI = = lc µAc N 2 µAc lc lc µAc φ = Rc φ Faraday’s Law V = − dΨ dt SYNCHRONOUS MACHINES Electrical to mechanical conversions . p = # of poles ωs = pωm 2 for constant speed Pm = ω m T m Voltage Equations - abc axes in Fundamental Parameters (armature equations for terminal voltage) ea = d dt Ψa − Ra ia eb = d dt Ψb − Ra ib ec = d dt Ψc − Ra ic (and the rotor equations with a single damper winding) 1 ef d = d dt Ψf d + Rf d if d 0= d dt Ψ1d + R1d i1d 0= d dt Ψ1q + R1q i1q Flux Equations abc axes in Fundamental Parameters (just one armature phase shown for illustration, showing self and mutual inductances) (constant component and varying with angle θ, 2θ due to symmetry - max linkage twice for every rotation) ￿ ￿ Ψa = −ia [Laa0 + Laa2 cos 2θ] +ib Lab0 + Laa2 cos(2θ + π ) + . . . 3 ￿ ￿ ic Lab0 + Laa2 cos(2θ − π ) + i1d La1d cos θ − i1q La1q sin θ 3 (and the rotor equations) ￿ ￿ ￿ ￿ ￿￿ π π Ψf d = Lf f d if d + Lf kd ikd − Laf d ia cos θ + ib cos θ − 23 + ic cos θ + 23 ￿ ￿ ￿ ￿ ￿￿ π π Ψ1d = Lf 1d if d + L11d i1d − La1d ia cos θ + ib cos θ − 23 + ic cos θ + 23 ￿ ￿ ￿ ￿ ￿￿ π π Ψ1q = L11q i1q + La1q ia sin θ + ib sin θ − 23 + ic sin θ + 23 Park’s Transformation ￿ ￿ ￿ ￿ π π cos (θ) cos ￿ − 23 ￿ cos ￿ + 23 ￿ θ θ π π − sin θ + 23 P = 2 − sin (θ) − sin θ − 23 3 P −1 1 2 cos (θ) ￿ ￿ π = cos ￿θ − 23 ￿ 2π cos θ + 3 1 2 and − sin (θ) ￿ ￿ π − sin ￿θ − 23 ￿ 2π − sin θ + 3 1 2 1 1 1 i dq0 = P iabc and iabc = P −1 idq0 edq0 = P eabc and eabc = P −1 edq0 Ψdq0 = P Ψabc and Ψabc = P −1 Ψdq0 Partial Calculation using Park’s Transformation for Ψd with id terms only (no damper windings) Ψd = cos(θ)Ψa + cos(θ − 2π 3 )Ψb + cos(θ + 2π 3 )Ψc cos(θ)Ψa = cos(θ)(−ia￿[Laa0 + Laa2 cos 2θ] ￿ ￿ ￿ +ib Lab0 + Laa2 cos(2θ + π ) + ic Lab0 + Laa2 cos(2θ − π ) + . . .) 3 3 2 ￿ ￿ 2π π − 2π ￿cos(θ − 3 )Ψb = cos(θ2π ￿3 )(ia Lab0 + Laa2 cos(2θ + 3 ) −ib Laa0 + Laa2 cos(2θ − 3 ) + ic [Lab0 + Laa2 cos(2θ − π )] + . . .) ￿ ￿ π π cos(θ + 23 )Ψc = cos(θ + 23 )(ia Lab0 + Laa2 cos(2θ − π ) 3 ￿ ￿ π +ib [Lab0 + Laa2 cos(2θ − π )] − ic Laa0 + Laa2 cos(2θ + 23 ) + . . .) (summing these will give the d-axis ﬂux and we can rearrange in terms of the inductances) Laa0 [−ia cos(θ) −ib cos(θ − 2π 3) − ic cos(θ + 2π 3 )] = −Laa0 id π π π Lab0 [ia (cos(θ − 23 ) + cos(θ + 23 )) + ib (cos(θ) + cos(θ + 23 )) + ic (cos(θ) + 2π 2π π cos(θ − 3 ))] = Lab0 [−ia cos(θ) −ib cos(θ − 3 ) − ic cos(θ + 23 )] = −Lab0 id π π Laa2 [ia (− cos(θ) cos(2θ) + cos(θ − 23 ) cos(2θ + π ) + cos(θ + 23 ) cos(2θ − π )) + 3 3 π 2π 2π 2π ib (cos(θ) cos(2θ + 3 ) − cos(θ − 3 ) cos(2θ − 3 ) + cos(θ + 3 ) cos(2θ − π )) + . . . π π π π ic (cos(θ) cos(2θ − 23 ) + cos(θ − 23 ) cos(2θ − π ) + cos(θ + 23 ) cos(2θ + 23 )) = 1 π . . .Laa2 [ 2 ia (− cos(θ) − cos(3θ) + cos(θ + π ) + cos(3θ − 3 ) + cos(θ − π ) + π π cos(3θ + π )) + 1 ib (−3 cos(θ − 23 )) + 1 ic (−3 cos(θ + 23 )) = − 3 Laa2 id 3 2 2 2 Flux Equations 0dq axes in Fundamental Parameters Deﬁne Ld = Laa0 + Labo + 3 Laa2 2 Lq = Laa0 + Labo − 3 Laa2 2 L0 = Laa0 − 2Labo then Ψd = −Ld id + Laf d if d + La1d i1d Ψq = −Lq iq + La1q i1q Ψ0 = −L0 i0 Ψf d = Lf f d if d + Lf 1d i1d − 3 Laf d id 2 Ψ1d = L11d i1d + Lf 1d if d − 3 La1d id 2 Ψ1q = L11q i1q − 3 La1q iq 2 Voltage Equations - 0dq axes in Fundamental Parameters Note: d dt Ψabc d = P −1 dt Ψdq0 + 3 d −1 Ψdq0 dt P and with edq0 = P eabc edq0 now if ωs = Ψd d d = −Ridq0 + dt Ψdq0 + P dt P −1 Ψq Ψ0 assuming near synchronous speed (i.e., θ(t) = ωs t + θ0 ) − sin (θ) ￿ − cos (θ) ￿ 0 ￿ ￿ π π d −1 = ωs − sin ￿θ − 23 ￿ − cos ￿θ − 23 ￿ 0 dt P 2π 2π − sin θ + 3 − cos θ + 3 0 d dt θ d P dt ￿ −1 = ￿ P π − sin (θ) ￿ cos ￿ + 23 ￿ θ ￿ π π − sin θ + 23 − sin ￿θ − 23 ￿ 2π 1 − sin θ + 3 2 ￿ ￿ π cos (θ) cos ￿ − 23 ￿ θ 2 − sin (θ) − sin θ − 2π 3 3 ωs 1 2 1 2 d P dt P −1 0 = ωs 1 0 −1 0 0 0 0 0 and substituting for Ψabc and d dt Ψabc ed = d dt Ψd − Ψq ωs − Ra id eq = d dt Ψq + Ψd ωs − Ra iq e0 = ef d = d dt Ψ0 d dt Ψf d − Ra i0 + Rf d if d 0= d dt Ψ1d + R1d i1d 0= d dt Ψ1q + R1q i1q 4 − cos (θ) ￿ ￿ π − cos ￿θ − 23 ￿ 2π − cos θ + 3 0 0 0 ...
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## This note was uploaded on 07/14/2011 for the course ECE 522 taught by Professor Tomsovic during the Summer '10 term at University of Florida.

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