L1:1436-459 Advanced Control and Automation Control of AC servo motors • 3-phase permanent magnet synchronous motors – “brushless DC” motor • trapezoidal back-EMF profile • rectangular pulse current profile • requires only 3 Hall-effect position sensors for electronic commutation – AC servo motor • sinusoidal back-EMF profile • balanced sinusoidal current profile • requires precise motor position measurement (resolver or encoder) L1:2 Recall DC servo motor http://www.servomag.com/flash/motor_types/brush_motor.swf L1:3 Permanent magnet AC servo motor L1:4 Control of brushless DC motor Source: P. Krause, O. Wasynczuk, S. Sudhoff, Analysis of Electric Machinery and Drive Systems, Wiley (2002) L1:5 Electrical torque production Tmωm = ea ia + eb ib + ec ic • Back-EMF ea = ka (θ e )ωm , eb = kb (θ e )ωm , ec = kc (θ e )ωm , • For balanced windings and 3-phase supply, current and back-EMF waveform shapes are identical, but displaced by 120ºE (electrical degrees) • Hence, motor torque is Tm (θ e ) = ka (θ e )ia (θ e ) + ka (θ e − 23π )ia (θ e − 23π ) + ka (θ e + 23π )ia (θ e + 23π ) • Two ‘standard’ ways of producing constant torque: – Trapezoidal back-EMF and square wave current – Sinusoidal back-EMF and sinusoidal current • Shape of back-EMF profile depends on geometry of magnets and windings L1:6 Brushless DC waveforms and torque production Kp Ip T ωm = ea ia + ebib + ec ic T (θ e ) = 2 K p I p • Three Hall-effect sensors provide commutation switching points – ‘six-step’ drive Source: D. Hanselman, Brushless Permanent Magnet Motor Design, 2nd ed, 2003 L1:7 http://www.servomag.com/flash/4-pole/smi-motor007.htm L1:8 AC servomotor waveforms and torque production Tm (θ e ) = ka (θ e )ia (θ e ) + ka (θ e − 23π )ia (θ e − 23π ) + ka (θ e + 23π )ia (θ e + 23π ) • Sinusoidal back-EMF: ka (θ e ) = K p cos(θ e ) • Sinusoidal phase current: ia (θ e ) = I p cos(θ e ) • Hence Tm (θ e ) = K p I p ⎡⎣cos2 θ e + cos2 (θ e − 23π ) + cos2 (θ e + 23π ) ⎤⎦ 3 i.e., Tm (θ e ) = K p I p 2 • Need precise measurement of rotor position to generate phase current with correct phase; i.e., encoder or resolver L1:9 Mechanical and electrical position Coil 2 of Coil 1 of phase a phase a 4 pole motor ia (θ e ) = I p cos(ωe t ) P=4 90 ω 10 e t = 120 o E 120 60 8 • Balanced sinusoidally-distributed phase windings 6 ωe t = 60 o E supplied with balanced 3-phase currents 150 30 4 generate a spatially-sinusoidal MMF which rotates at angular speed ωm = (2/P)ωe radM/s 2 (mechanical radians), where ωe is frequency of 180 0 ωe t = 0 phase currents, P is number of motor poles: N ⎛ 3⎞ ⎛ P ⎞ MMFs = s I p ⎜ ⎟ cos ⎜ ωe t − φs ⎟ 210 330 P ⎝2⎠ ⎝ 2 ⎠ where φs = stator angular coordinate (radM) 240 300 Ref: P. Krause, O. Wasynczuk, S. Sudhoff, Analysis of Electric Machinery and Drive Systems, Wiley (2002) 270 L1:10 Dynamics of AC servo motor • KVL for phases: di abc v abc = R s i abc + L s + e abc dt v abc = [ va vb vc ] , i abc = [ia ic ] T T ib R s = R ph I 3×3 ⎡ Ll + Lph − 12 Lph − 12 Lph ⎤ ⎢ ⎥ L s = ⎢ − 12 Lph Ll + Lph − 12 Lph ⎥ All this results in ⎢⎣ − 12 Lph − 12 Lph Ll + Lph ⎥⎦ a very complex, Ll , Lph = leakage and magnetising inductances nonlinear of coils expression for the ⎡ cos θ r ⎤ motor torque: e abc 2 ⎢ 2π ⎥ (sinusoidal = ωr K p cos(θ r − 3 ) back-EMF) P ⎢ ⎥ Tm = Tm(ia, ib, ic, θr) ⎢⎣cos(θ r + 23π ) ⎥⎦ L1:11 Park transformation to qd0 variables • The equations are simplified by a transformation of variables from the machine frame abc to a quadrature-direct-zero qd0 reference frame rotating with the rotor, at speed ωr = (P/2)ωm radE/s • Transformation for voltage, current, flux linkage or charge variables: ⎡ vq ⎤ ⎡cos θ r cos (θ r − 23π ) cos (θ r + 23π ) ⎤ ⎡va ⎤ ⎢v ⎥ = 2 ⎢ sin θ sin θ − 2π 2π ⎥ ⎢ ⎥ v = K (θ ) v ⎢ d⎥ 3⎢ r ( r 3) sin ( r 3 ) ⎥ ⎢ b ⎥ qd 0 s r abc θ + v ⎢⎣ v0 ⎥⎦ ⎢⎣ 12 1 2 1 2 ⎦⎥ ⎢⎣ vc ⎥⎦ ⎡va ⎤ ⎡ cos θ r sin θ r 1⎤ ⎡ vq ⎤ ⎢ v ⎥ = ⎢ cos θ − 2π sin θ − 2π 1⎥ ⎢v ⎥ ⎢ b⎥ ⎢ ( r 3 ) ( r 3 ) ⎥⎢ d⎥ v abc = K s (θ r ) v qd 0 −1 ⎢⎣ vc ⎥⎦ ⎢⎣cos (θ r + 23π ) sin (θ r + 23π ) 1⎦⎥ ⎢⎣ v0 ⎥⎦ • Then, total power: Pabc = va ia + vbib + vc ic = ( vq iq + vd id + 2v0i0 ) 3 = Pqd 0 2 L1:12 Equations of motion in transformed variables • KVL diq vq = R ph iq + ( Ll + Lph ) + ωr ( Ll + Lph ) id + K pωr P dt 2 vd = R ph id + ( Ll + Lph ) − ωr ( Ll + Lph ) iq ⎛P⎞ did K pωm = K p ⎜ ⎟ ωr dt ⎝2⎠ di0 v0 = R ph i0 + Ll dt 3 iq is the ‘torque producing’ • Motor torque Tm = K p iq 2 component of the stator currents • Mechanical dynamics J ω m = Tm − Bωm − Tl i.e., ⎛2⎞ ⎛2⎞ J ⎜ ⎟ ωr = Tm − B ⎜ ⎟ ωr − Tl ⎝P⎠ ⎝P⎠ L1:13 Balanced 3-phase set • Phase voltages are controlled to have a frequency equal to the rotor speed (in radE/s): ωe = ωr ⎡ va ⎤ ⎡ cos(ωr t + φv ) ⎤ v abc = ⎢ vb ⎥ = Vs ⎢ cos(ωr t − 23π + φv ) ⎥ ⎢ ⎥ ⎢ ⎥ ⎢⎣ vc ⎦⎥ ⎢⎣cos(ωr t + 23π + φv ) ⎥⎦ • Quadrature and direct voltages and currents ⎡ vq ⎤ ⎡ cos φv ⎤ v qd 0 = K s (θ r ) v abc = ⎢vd ⎥ = Vs ⎢ − sin φv ⎥ ⎢ ⎥ ⎢ ⎥ ⎣⎢ v0 ⎦⎥ ⎢⎣ 0 ⎥⎦ • To maximise torque production and minimise losses, φv = 0 vq = Vs , vd = 0 L1:14 Electromechanical model of AC servo motor Source: S. Lyshevski, Electromechanica Systems, Electirc Machines, and Applied Mechatronics, CRC Press (2000) L1:15 Control based on quadrature and direct currents Source: SIEMENS ducumentation for SIMODRIVE 611U
Report "Modelado matemático de un servomotor trifásico"