1 3상 동기전동기 DQ-모델 IPMSM DQ-모델 한용수, Ph. D. 국민대학교, 전자공학부 조교수 InTelligent Energy Conversion (ITEC) Lab. Kookmin University Types of PMSM 2 ◆PMSM : Permanent Magnet Synchronous Machine ➢ Symmetric structure : Surface mounted permanent magnet synchronous motor (SMPMSM) ✓ Only magnetic torque ➢ Asymmetric structure : Interior permanent magnet synchronous motor (IPMSM) ✓ Both magnetic and reluctance torques ✓ Suitable for fast rotating operation because of inserted PM S S N N N S Axis N S N S Axis S N N N S S ➢ Synchronous Machine ✓ Synchronous machine : 𝝎𝒆 = 𝝎𝒓 ➔ e = r ✓ Asynchronous machine : 𝜔𝑒 ≠ 𝜔𝑟 , 𝜔𝑒 = 𝜔𝑟 + 𝜔𝑠𝑙𝑖𝑝 ➔ 𝒆 ≠ 𝒓 InTelligent Energy Conversion (ITEC) Lab. Kookmin University d-q Modeling of IPMSM 3 ◆Target model : Interior Permanent Magnet Synchronous Motor (IPMSM) ➢ d-q-axis inductances on the rotor reference frame ✓ Including the permanent magnet in the flux path ✓ Permanent magnet has much lower permeability than core. 𝐿𝑑 < 𝐿𝑞 ➢ Inductance variation according to the rotor angle ✓ Average (mutual) inductance component ➔ 𝐿𝐴 q ✓ Pulsating (mutual) inductance component ➔ 𝐿𝐵 d InTelligent Energy Conversion (ITEC) Lab. Kookmin University d-q Modeling of IPMSM 4 ◆Pole number (pole pair * 2) of motor ➢ In electric motor, pole number refers to the number of pairs of magnetic poles (N, S-poles) per one rotation. ✓ 2-pole(1-pole pair) motor : 1*N-pole, 1*S-pole ✓ 4-pole(2-pole pair) motor : 2*N-pole, 2*S-pole q qr qs r a d c qs b r c a dr a b b c ds θrm =θr ds rm = r / 2 c b c b a a a 2pole (1pole-pair) PM motor InTelligent Energy Conversion (ITEC) Lab. c b 4pole (2pole-pair) PM motor Kookmin University d-q Modeling of IPMSM 5 ◆Electrical frequency and Mechanical frequency ➢ In 2-pole motor, 𝜔𝑒 = 𝜔𝑟 = 𝜔𝑟𝑚 (electrical frequency = mechanical frequency) ➢ In 4-pole motor, 𝜔𝑒 = 𝜔𝑟 = 2𝜔𝑟𝑚 (electrical frequency = 2*mechanical frequency) ➢ In 2n-pole (n-polepair) motor, 𝜔𝑒 = 𝜔𝑟 = 𝑛𝜔𝑟𝑚 (electrical frequency = n*mechanical frequency) ➢ As the number of pole increases, the torque output increases. ➢ As the number of pole increases, the Back-EMF increases. ➔ Speed capability decreases. ➢ Example : 𝜔𝑒 = 2*pi*60 q Pole pairs Mechanical speed Rated Torque 2 3600 rpm T1 4 1800 rpm 2T1 6 1200 rpm 3T1 8 900 rpm 4T1 InTelligent Energy Conversion (ITEC) Lab. qr qs r qs b a dr c c a dr a b b c ds θrm =θr ds rm = r / 2 c b c b a a a c b Kookmin University d-q Modeling of IPMSM 6 ◆Stator voltage equation 𝑑 𝑣𝑎𝑏𝑐𝑠 = 𝑅𝑠 𝑖𝑎𝑏𝑐𝑠 + 𝜆𝑎𝑏𝑐𝑠 , 𝑑𝑡 𝑣𝑎𝑠 𝜆𝑎𝑠 𝑖𝑎𝑠 𝑣𝑎𝑏𝑐𝑠 = 𝑣𝑏𝑠 , 𝑖𝑎𝑏𝑐𝑠 = 𝑖𝑏𝑠 , 𝜆𝑎𝑏𝑐𝑠 = 𝜆𝑏𝑠 𝑣𝑐𝑠 𝑖𝑐𝑠 𝜆𝑐𝑠 ◆Stator inductance matrix & Stator flux equation ➢ A-b-c phase windings are coupled each other. 𝐿𝑎𝑠,𝑎𝑠 𝐋𝐬 = 𝐿𝑎𝑠,𝑏𝑠 𝐿𝑎𝑠,𝑐𝑠 𝐿𝑏𝑠,𝑎𝑠 𝐿𝑏𝑠,𝑏𝑠 𝐿𝑏𝑠,𝑐𝑠 𝐿𝑐𝑠,𝑎𝑠 𝐿𝑐𝑠,𝑏𝑠 , 𝐿𝑐𝑠,𝑐𝑠 𝐿𝑎𝑠,𝑎𝑠 𝜆𝑎𝑠 𝜆𝑎𝑏𝑐𝑠 = 𝜆𝑏𝑠 = 𝐋𝐬 𝑖𝑎𝑏𝑐𝑠 + 𝜆𝑎𝑏𝑐𝑚 = 𝐿𝑎𝑠,𝑏𝑠 𝐿𝑎𝑠,𝑐𝑠 𝜆𝑐𝑠 InTelligent Energy Conversion (ITEC) Lab. Magnetic flux component that links from rotor PM to stator windings 𝐿𝑏𝑠,𝑎𝑠 𝐿𝑏𝑠,𝑏𝑠 𝐿𝑏𝑠,𝑐𝑠 𝐿𝑐𝑠,𝑎𝑠 𝐿𝑐𝑠,𝑏𝑠 𝐿𝑐𝑠,𝑐𝑠 𝑖𝑎𝑠 𝑖𝑏𝑠 + 𝜆𝑎𝑏𝑐𝑚 𝑖𝑐𝑠 Kookmin University d-q Modeling of IPMSM 7 ◆Stator inductance equations in detail qs qr ➢ Stator inductance varies according to the rotor angle a dr c b ds θrm =θr 𝐿𝑎𝑠,𝑎𝑠 𝐋𝐬 = 𝐿𝑎𝑠,𝑏𝑠 𝐿𝑎𝑠,𝑐𝑠 𝐿𝑏𝑠,𝑎𝑠 𝐿𝑏𝑠,𝑏𝑠 𝐿𝑏𝑠,𝑐𝑠 𝐿ℓ𝑠 + 𝐿𝐴 + 𝐿𝐵 cos 2 𝜃𝑟 1 2𝜋 = − 𝐿𝐴 + 𝐿𝐵 cos 2𝜃𝑟 − 2 3 1 2𝜋 − 𝐿𝐴 + 𝐿𝐵 cos 2𝜃𝑟 + 2 3 c b 𝐿𝑐𝑠,𝑎𝑠 𝐿𝑐𝑠,𝑏𝑠 𝐿𝑐𝑠,𝑐𝑠 a 1 2𝜋 − 𝐿𝐴 + 𝐿𝐵 cos 2𝜃𝑟 − 2 3 2𝜋 𝐿ℓ𝑠 + 𝐿𝐴 + 𝐿𝐵 cos 2𝜃𝑟 + 3 1 − 𝐿𝐴 + 𝐿𝐵 cos 2 𝜃𝑟 2 InTelligent Energy Conversion (ITEC) Lab. 1 2𝜋 − 𝐿𝐴 + 𝐿𝐵 cos 2𝜃𝑟 + 2 3 1 − 𝐿𝐴 + 𝐿𝐵 cos 2 𝜃𝑟 2 2𝜋 𝐿ℓ𝑠 + 𝐿𝐴 + 𝐿𝐵 cos 2𝜃𝑟 − 3 Kookmin University d-q Modeling of IPMSM 8 ◆Stator flux equation with complex space vector q qs r a Flux of PM λas 𝜆𝑎𝑏𝑐𝑠 = λbs λcs LB + 2 cos 𝜃𝑟 2𝜋 cos 𝜃𝑟 − 3 = 𝐋𝐬 𝑖𝑎𝑏𝑐𝑠 + 𝜆𝑓 2𝜋 cos 𝜃𝑟 + 3 𝑒 𝑗2𝜃𝑟 a2 𝑒 𝑗2𝜃𝑟 a𝑒 j2𝜃r a2 𝑒 𝑗2𝜃𝑟 a𝑒 𝑗2𝜃𝑟 𝑒 j2𝜃r a𝑒 𝑗2𝜃𝑟 𝑒 𝑗2𝜃𝑟 a2 𝑒 𝑗2𝜃𝑟 𝐿ℓ𝑠 + 𝐿𝐴 = 1 − 𝐿𝐴 2 1 − 𝐿𝐴 2 1 − 𝐿𝐴 2 𝐿ℓ𝑠 + 𝐿𝐴 − 𝑒 −𝑗2𝜃𝑟 a𝑒 −𝑗2𝜃𝑟 a2 𝑒 −𝑗2𝜃𝑟 + a𝑒 −𝑗2𝜃𝑟 a2 𝑒 −𝑗2𝜃𝑟 𝑒 −j2𝜃r a2 𝑒 −𝑗2𝜃𝑟 𝑒 −𝑗2𝜃𝑟 a𝑒 −𝑗2𝜃𝑟 1 𝐿 2 𝐴 1 − 𝐿𝐴 2 1 − 𝐿𝐴 2 c ias ibs ics b ds θrm =θr c b 𝐿ℓ𝑠 + 𝐿𝐴 𝑖𝑎𝑠 𝑖𝑏𝑠 + 𝜆𝑓 𝑖𝑐𝑠 dr a 𝑒 𝑗𝜃𝑟 e−𝑗𝜃𝑟 a2 𝑒 𝑗𝜃𝑟 + ae−𝑗𝜃𝑟 a𝑒 𝑗𝜃𝑟 a2 e−𝑗𝜃𝑟 2 𝜆𝑎𝑏𝑐𝑠 = λ𝑎𝑠 + aλ𝑏𝑠 + a2 λ𝑐𝑠 3 3 = 𝐿ℓ𝑠 + 𝐿𝐴 2 2 3 2 2 𝑖𝑎𝑠 + a𝑖𝑏𝑠 + a 𝑖𝑐𝑠 + 𝐿𝐵 𝑖𝑎𝑠 + a2 𝑖𝑏𝑠 + a𝑖𝑐𝑠 𝑒 𝑗2𝜃𝑟 + 𝜆𝑓 𝑒 𝑗𝜃𝑟 3 2 3 InTelligent Energy Conversion (ITEC) Lab. Kookmin University d-q Modeling of IPMSM 9 ◆Stator voltage equation in the rotor reference frame 𝑑 𝑑 𝜆𝑎𝑏𝑐𝑠 → 𝑣𝑎𝑏𝑐𝑠 𝑒 −𝑗𝜃𝑟 = 𝑅𝑠 𝑖𝑎𝑏𝑐𝑠 𝑒 −𝑗𝜃𝑟 + 𝑒 −𝑗𝜃𝑟 𝜆𝑎𝑏𝑐𝑠 𝑑𝑡 𝑑𝑡 3 3 ∗ 𝜆𝑒𝑑𝑞𝑠 = 𝜆𝑎𝑏𝑐𝑠 𝑒 −𝑗𝜃𝑟 = 𝐿ℓ𝑠 + 𝐿𝐴 𝑖𝑎𝑏𝑐𝑠 𝑒 −𝑗𝜃𝑟 + 𝐿𝐵 𝑖𝑎𝑏𝑐𝑠 𝑒 𝑗𝜃𝑟 + 𝜆𝑓 𝑒 𝑗𝜃𝑟 2 2 𝑣𝑎𝑏𝑐𝑠 = 𝑅𝑠 𝑖𝑎𝑏𝑐𝑠 + * : conjugate operator 𝑒 𝑣𝑎𝑏𝑐𝑠 𝑒 −𝑗𝜃𝑟 = 𝑣𝑑𝑞𝑠 , 𝑑 𝑒 𝑒 −𝑗𝜃 𝑟 𝑣𝑑𝑞𝑠 = 𝑅𝑠 𝑖𝑑𝑞𝑠 + 𝑒 𝑑𝑡 𝑒 = 𝑅𝑠 𝑖𝑑𝑞𝑠 + 𝜆𝑒𝑑𝑞𝑠 𝑒 𝑗𝜃𝑟 𝑑 𝑒 𝑒 −𝑗𝜃 𝑗𝜃 𝑟 𝑟 = 𝑅𝑠 𝑖𝑑𝑞𝑠 + 𝑒 𝑒 𝜆𝑑𝑞𝑠 + 𝑗𝜔𝑟 𝜆𝑒𝑑𝑞𝑠 𝑒 −𝑗𝜃𝑟 𝑒 𝑗𝜃𝑟 𝑑 𝑒 𝜆𝑑𝑞𝑠 + 𝑗𝜔𝑟 𝜆𝑒𝑑𝑞𝑠 𝑑𝑡 𝜆𝑎𝑏𝑐𝑠 = 𝜆𝑒𝑑𝑞𝑠 𝑒 𝑗𝜃𝑟 3 𝜆𝑒𝑑𝑞𝑠 = 𝐿ℓ𝑠 + 𝐿𝐴 2 𝑒 𝑖𝑑𝑞𝑠 + 𝑑𝑡 3 ∗ 𝑒 𝐿𝐵 𝑖𝑑𝑞𝑠 + 𝜆𝑓 2 InTelligent Energy Conversion (ITEC) Lab. Kookmin University d-q Modeling of IPMSM 10 ◆Definitions of d- and q- axis inductances 𝜆𝑒𝑑𝑞𝑠 = 3 𝐿ℓ𝑠 + 𝐿𝐴 2 3 ∗ 𝑒 𝑒 𝑖𝑑𝑞𝑠 + 𝐿𝐵 𝑖𝑑𝑞𝑠 + 𝜆𝑓 𝜆𝑒𝑑𝑞𝑠 = 𝐿𝑚𝑑 + 𝐿𝑚𝑞 𝐿ℓ𝑠 + 2 2 𝑟 𝑖𝑑𝑞𝑠 + 𝐿𝑚𝑑 − 𝐿𝑚𝑞 2 𝑒 𝑖𝑑𝑞𝑠 ∗ + 𝜆𝑓 3 𝐿 − 𝐿𝐵 𝐿𝑑 = 𝐿𝑚𝑑 + 𝐿𝑙𝑠 2 𝐴 → 𝐿 =𝐿 +𝐿 3 𝑞 𝑚𝑞 𝑙𝑠 𝐿𝑚𝑞 ≡ 𝐿𝐴 + 𝐿𝐵 2 𝐿𝑚𝑑 ≡ 𝜆𝑒𝑑𝑞𝑠 = 𝜆𝑒𝑑𝑠 𝐿𝑑 = 0 𝜆𝑒𝑞𝑠 𝑒 0 𝑖𝑑𝑠 𝜆𝑓 + 𝑒 𝐿𝑞 𝑖𝑞𝑠 0 InTelligent Energy Conversion (ITEC) Lab. Kookmin University d-q Modeling of IPMSM 11 ◆Final d- and q- axis voltage equations on the rotor reference frame 𝜆𝑒𝑑𝑞𝑠 = 𝜆𝑒𝑑𝑠 𝐿𝑑 = 0 𝜆𝑒𝑞𝑠 𝑒 𝑑 𝑒 0 𝑖𝑑𝑠 𝜆𝑓 𝑒 𝑒 𝑒 + → 𝑣 = 𝑅 𝑖 + 𝜆 + 𝑗𝜔 𝜆 𝑒 𝑠 𝑟 𝑑𝑞𝑠 𝑑𝑞𝑠 𝑑𝑞𝑠 𝑑𝑞𝑠 𝐿𝑞 𝑖𝑞𝑠 𝑑𝑡 0 𝑑 𝑒 𝑒 𝑖𝑑𝑠 − 𝜔𝑟 𝐿𝑞 𝑖𝑞𝑠 𝑑𝑡 𝑑 𝑒 𝑒 𝑒 𝑒 𝑣𝑞𝑠 = 𝑅𝑠 𝑖𝑞𝑠 + 𝐿𝑞 𝑖𝑞𝑠 + 𝜔𝑟 𝐿𝑑 𝑖𝑑𝑠 + 𝜔𝑟 𝜆𝑓 𝑑𝑡 𝑒 𝑒 𝑣𝑑𝑠 = 𝑅𝑠 𝑖𝑑𝑠 + 𝐿𝑑 Lls Lmd v dse if ◆d- and q- axis equivalent circuits of PMSM ➢ On synchronous reference frame (e) ➢ On rotor reference frame (r) ➢ Virtual current for PM flux 𝜆𝑓 𝑖𝑓 = 𝐿𝑚𝑑 InTelligent Energy Conversion (ITEC) Lab. Lls v eqs L mq Kookmin University d-q Modeling of IPMSM 12 ◆Why should we use the d-q modeling on the synchronous reference frame? ➢ On the stationary reference frame… ✓ All variables are AC values. ✓ Controller is designed based on the AC values ➔ PI controller X ✓ As speed increases, the frequency of all variable also increases. ➔ not applicable in digital system @ high speed operation ✓ D-q axes can not be decoupled each other. ✓ And so on…. ➢ On the synchronous reference frame ✓ All variables are DC values. ✓ Controller is designed based on the DC values ➔ PI controller O ✓ As speed increases, the frequency of all variable is fixed to DC. ✓ D-q axes can be decoupled each other. ✓ And so on…. 𝑑 𝑒 𝑒 𝑒 𝑒 𝑣𝑑𝑠 = 𝑅𝑠 𝑖𝑑𝑠 + 𝐿𝑑 𝑖𝑑𝑠 − 𝜔𝑟 𝐿𝑞 𝑖𝑞𝑠 𝑑𝑡 𝑑 𝑒 𝑒 𝑒 = 𝑅 𝑖𝑒 + 𝐿 𝑣𝑞𝑠 𝑖𝑞𝑠 + 𝜔𝑟 𝐿𝑑 𝑖𝑑𝑠 + 𝜆𝑓 𝑠 𝑞𝑠 𝑞 𝑑𝑡 InTelligent Energy Conversion (ITEC) Lab. Kookmin University d-q Modeling of SPMSM 13 q ◆In the derivation of IPMSM d-q modeling… ➢ 𝐿𝐵 = 0 ➢ 𝐿𝑑 = 𝐿 𝑞 = 𝐿𝑠 d ◆Voltage equation on the rotor reference frame 𝜆𝑒𝑑𝑞𝑠 = 𝜆𝑒𝑑𝑠 𝐿𝑠 = 0 𝜆𝑒𝑞𝑠 𝑒 0 𝑖𝑑𝑠 𝜆𝑓 + 𝑒 𝐿𝑠 𝑖𝑞𝑠 0 𝑑 𝑒 𝑒 𝑒 𝑒 𝑣𝑑𝑠 = 𝑅𝑠 𝑖𝑑𝑠 + 𝐿𝑠 𝑖𝑑𝑠 − 𝜔𝑟 𝐿𝑠 𝑖𝑞𝑠 𝑑𝑡 𝑒 = 𝑅 𝑖𝑒 + 𝐿 𝑣𝑞𝑠 𝑠 𝑞𝑠 𝑠 Rs idse InTelligent Energy Conversion (ITEC) Lab. -ωrλqse vdse Rs 𝑑 𝑒 𝑒 𝑖𝑞𝑠 + 𝜔𝑟 𝐿𝑠 𝑖𝑑𝑠 + 𝜔𝑟 𝜆𝑓 𝑑𝑡 𝜆𝑓 𝑖𝑓 = 𝐿𝑚 Lls iqse vqse Lm if Lls ωrλdse Lm Kookmin University HW-2 : d-q Modeling of IPMSM 14 ◆IPMSM 전동기 동기 좌표계 전압 방정식 유도하기 ➢ 수기로 작성 ➢ 단순히 강의자료에 있는 수식만 배껴서 작성하는 경우 인정하지 않음. ➢ 중간에 생략된 수식 유도까지 포함해서 작성할 것 ➢ 상/중/하로 평가 ➢ 다음주 수업 시간전에 A4 용지로 제출 𝑑 𝑣𝑎𝑏𝑐𝑠 = 𝑅𝑠 𝑖𝑎𝑏𝑐𝑠 + 𝜆𝑎𝑏𝑐𝑠 , 𝑑𝑡 𝑣𝑎𝑠 𝜆𝑎𝑠 𝑖𝑎𝑠 𝑣𝑎𝑏𝑐𝑠 = 𝑣𝑏𝑠 , 𝑖𝑎𝑏𝑐𝑠 = 𝑖𝑏𝑠 , 𝜆𝑎𝑏𝑐𝑠 = 𝜆𝑏𝑠 𝑣𝑐𝑠 𝑖𝑐𝑠 𝜆𝑐𝑠 𝑑 𝑒 𝑒 𝑖𝑑𝑠 − 𝜔𝑟 𝐿𝑞 𝑖𝑞𝑠 𝑑𝑡 𝑑 𝑒 𝑒 𝑒 𝑒 𝑣𝑞𝑠 = 𝑅𝑠 𝑖𝑞𝑠 + 𝐿𝑞 𝑖𝑞𝑠 + 𝜔𝑟 𝐿𝑑 𝑖𝑑𝑠 + 𝜔𝑟 𝜆𝑓 𝑑𝑡 𝑒 𝑒 𝑣𝑑𝑠 = 𝑅𝑠 𝑖𝑑𝑠 + 𝐿𝑑 InTelligent Energy Conversion (ITEC) Lab. Kookmin University Torque of IPMSM 15 ◆Input power calculation ➢ In three-phase system, − 1 1 − 𝑇𝑑𝑞−𝑡𝑜−𝑎𝑏𝑐 = 2 1 − 2 0 3 2 3 − 2 𝑃𝑖𝑛 = 𝑣𝑎𝑠 𝑖𝑎𝑠 + 𝑣𝑏𝑠 𝑖𝑏𝑠 + 𝑣𝑐𝑠 𝑖𝑐𝑠 𝑠 𝑣𝑑𝑠 𝑣𝑎𝑠 1 𝑠 3 𝑠 𝑣𝑠 𝑣 + 𝑣 𝑣𝑏𝑠 = 𝑇𝑑𝑞−𝑡𝑜−𝑎𝑏𝑐 𝑑𝑠 𝑑𝑠 = 𝑠 2 2 𝑞𝑠 , 𝑣 𝑞𝑠 𝑣𝑐𝑠 1 𝑠 3 𝑠 − 𝑣𝑑𝑠 - 𝑣𝑞𝑠 2 2 𝑠 𝑖𝑑𝑠 𝑖𝑎𝑠 1 𝑠 3 𝑠 𝑖𝑏𝑠 = − 2 𝑖𝑑𝑠 + 2 𝑖𝑞𝑠 𝑖𝑐𝑠 1 𝑠 3 𝑠 − 𝑖𝑑𝑠 − 𝑖𝑞𝑠 2 2 𝑃𝑖𝑛 = 𝑣𝑎𝑠 𝑖𝑎𝑠 + 𝑣𝑏𝑠 𝑖𝑏𝑠 + 𝑣𝑐𝑠 𝑖𝑐𝑠 1 𝑠 𝑠 3 𝑠 𝑠 1 𝑠 3 𝑠 3 𝑠 1 𝑠 1 𝑠 𝑠 3 𝑠 𝑠 1 𝑠 3 𝑠 𝑠 𝑠 = 𝑣𝑑𝑠 𝑖𝑑𝑠 + 𝑣𝑑𝑠 𝑖𝑑𝑠 + 𝑣q𝑠 𝑖q𝑠 + - 𝑣𝑑𝑠 𝑖𝑞𝑠 + 𝑣𝑞𝑠 𝑖𝑑𝑠 + 𝑣𝑑𝑠 𝑖𝑑𝑠 + 𝑣q𝑠 𝑖q𝑠 + 𝑣 𝑖 + 4 4 2 2 2 2 4 4 2 𝑑𝑠 2 𝑞𝑠 𝟑 = 𝒗 𝒊 + 𝒗𝒒𝒔 𝒊𝒒𝒔 𝟐 𝒅𝒔 𝒅𝒔 𝑃𝑖𝑛 = 1 1 − 2 2 3 3 − 2 2 2 1 𝑇𝑎𝑏𝑐−𝑡𝑜−𝑑𝑞𝑠 = 3 0 3 𝑠 1 𝑠 𝑣 𝑖 2 𝑞𝑠 2 𝑑𝑠 3 3 s 3 -j𝜃 s 3 e 𝟑 𝝎 𝝎 * e∗ −j𝜃 s 𝝎 𝑣𝑑𝑠 𝑖𝑑𝑠 + 𝑣𝑞𝑠 𝑖𝑞𝑠 = vdqs ∙ is* = e v ∙ e i = v ∙ i = 𝒗𝒅𝒔 𝒊𝒅𝒔 + 𝒗𝝎 𝒒𝒔 𝒊𝒒𝒔 dqs dqs dqs dqs dqs 2 2 2 2 𝟐 InTelligent Energy Conversion (ITEC) Lab. Kookmin University Torque of IPMSM 16 ◆By energy conservation 𝑃𝑖𝑛 = 3 𝜔 𝜔 𝜔 𝜔 𝑣𝑑𝑠 𝑖𝑑𝑠 + 𝑣𝑞𝑠 𝑖𝑞𝑠 2 = 3 𝑒 𝑒 3 𝑒 𝑒 𝑣𝑑𝑠 𝑖𝑑𝑠 + 𝑣𝑞𝑠 𝑖𝑞𝑠 = 2 2 = 3 𝑑 𝑒 𝑒 2 𝑒 𝑑 𝑒 𝑒 2 𝑒 𝑅𝑠 𝑖𝑑𝑠 + 𝑖𝑞𝑠 + 𝐿𝑑 𝑖𝑑𝑠 𝑖𝑑𝑠 + 𝐿𝑞 𝑖𝑞𝑠 𝑖 + 𝜔𝑟 2 𝑑𝑡 𝑑𝑡 𝑞𝑠 𝑃𝑐𝑜𝑝𝑝𝑒𝑟 = 𝑑 𝑒 𝑑 𝑒 𝑒 𝑒 𝑒 𝑒 𝑒 𝑖𝑑𝑠 − 𝜔𝑟 𝐿𝑞 𝑖𝑞𝑠 𝑖𝑑𝑠 + 𝑅𝑠 𝑖𝑞𝑠 + 𝐿𝑞 𝑖𝑞𝑠 + 𝜔𝑟 𝐿𝑑 𝑖𝑑𝑠 + 𝜔𝑟 𝜆𝑓 𝑖𝑞𝑠 𝑑𝑡 𝑑𝑡 𝑒 𝑒 𝑒 𝐿𝑑 -𝐿𝑞 𝑖𝑑𝑠 𝑖𝑞𝑠 + 𝜆𝑓 𝑖𝑞𝑠 3 𝑒 2 𝑒 2 𝑅𝑠 𝑖𝑑𝑠 + 𝑖𝑞𝑠 2 𝑃𝑣𝑎𝑟_𝑖𝑛𝑑 = 𝑃𝑚 = 𝑒 𝑅𝑠 𝑖𝑑𝑠 + 𝐿𝑑 3 𝜔 2 𝑟 3 𝑑 𝑒 𝑒 𝑑 𝑒 𝑒 𝐿𝑑 𝑖𝑑𝑠 𝑖𝑑𝑠 + 𝐿𝑞 𝑖𝑞𝑠 𝑖 2 𝑑𝑡 𝑑𝑡 𝑞𝑠 𝑑 𝑒 𝑒 𝑖𝑑𝑠 − 𝜔𝑟 𝐿𝑞 𝑖𝑞𝑠 𝑑𝑡 𝑑 𝑒 𝑒 𝑒 𝑒 𝑣𝑞𝑠 = 𝑅𝑠 𝑖𝑞𝑠 + 𝐿𝑞 𝑖𝑞𝑠 + 𝜔𝑟 𝐿𝑑 𝑖𝑑𝑠 + 𝜔𝑟 𝜆𝑓 𝑑𝑡 𝑒 𝑒 𝑣𝑑𝑠 = 𝑅𝑠 𝑖𝑑𝑠 + 𝐿𝑑 → 𝑃𝑣𝑎𝑟_𝑖𝑛𝑑 = 0 @ 𝑠𝑡𝑒𝑎𝑑𝑦 𝑠𝑡𝑎𝑡𝑒 𝑒 𝑒 𝑒 𝐿𝑑 − 𝐿𝑞 𝑖𝑑𝑠 𝑖𝑞𝑠 + 𝜆𝑓 𝑖𝑞𝑠 𝑃𝑚 = 𝑇𝑒 *𝜔𝑟𝑚 = 𝑇𝑒 ∗ 𝜔𝑟 /𝑝𝑝 𝑻𝒆 = 𝟑 𝒑𝒑 𝟐 𝑳𝒅 − 𝑳𝒒 𝒊𝒆𝒅𝒔 𝒊𝒆𝒒𝒔 + 𝝀𝒇 𝒊𝒆𝒒𝒔 InTelligent Energy Conversion (ITEC) Lab. Kookmin University Torque of IPMSM 17 ◆Two torque components ➢ Torque of IPMSM = Magnetic torque + Reluctance torque 𝑻𝒆 = 𝟑 𝒑𝒑 𝟐 qr 𝑳𝒅 − 𝑳𝒒 𝒊𝒆𝒅𝒔 𝒊𝒆𝒒𝒔 + 𝝀𝒇 𝒊𝒆𝒒𝒔 3 𝑒 ➢ Magnetic torque : 𝑇𝑒_m = 𝑝𝑝𝜆𝑓 𝑖𝑞𝑠 2 qs b c a b c ds rm = r / 2 c b 3 𝑒 𝑒 ➢ Reluctance torque : 𝑇𝑒_𝑟 = 𝑝𝑝 𝐿𝑑 − 𝐿𝑞 𝑖𝑑𝑠 𝑖𝑞𝑠 2 dr a a a c b ➢ Torque can be determined by the currents on the synchronous reference frame. ✓ Current & Torque are DC values. ✓ Constant torque control ➔ constant current (on synchronous reference frame) control InTelligent Energy Conversion (ITEC) Lab. Kookmin University HW-3 : Implementation of IPMSM in Simulation 18 ◆Modeling an IPMSM Motor in Simulink ➢ Basic framework : ➢ 제출 내용 : ✓ 작성한 Simulink 파일은 추후에 제어기 구현 파트에서 이어서 활용할 예정 ✓ 구성한 Simulink 파일의 블록을 적절히 캡쳐하여 구성된 블록도를 설명 • • 설명은 줄글 보다는 수식으로만 간단히 진행 예시 : abc ➔ dq 좌표 변환 (수식) + (블록 캡쳐) ✓ A4 1~2 장 이내로 작성하고, 출력물 제출 (다음주 수업시간 전까지) InTelligent Energy Conversion (ITEC) Lab. Kookmin University
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