Bài giảng Electromechanical energy conversion - Chapter VII: Variable - Reluctance Machines and Stepping Motors - Nguyễn Công Phương

NguyễnCôngPhương ELECTROMECHANICAL ENERGY  CONVERSION Variable – Reluctance Machines and Stepping Motors Contents I. Magnetic Circuits and Magnetic Materials II. Electromechanical Energy Conversion Principles III. Introduction to Rotating Machines IV. Synchronous Machines V. Polyphase Induction Machines VI. DC Machines VII.Variable – Reluctance Machines and Stepping Motors VIII.Single and Two – Phase Motors IX. Speed and Torque Control sites.google.com/site/ncpdhbkhn 2 Variable – Reluctance Machines and Stepping Motors 1. Introduction 2. Basics of VRM Analysis 3. Practical VRM Configurations 4. Current Waveforms for Torque Production 5. Nonlinear Analysis 6. Stepping Motors sites.google.com/site/ncpdhbkhn 3 Introduction ‐training‐events/motor‐ tutorial/motor‐principles/switched‐reluctance‐motors:WBT_MOTORSRTUT_WP ‐motor‐drivers.html sites.google.com/site/ncpdhbkhn 4 Variable – Reluctance Machines and Stepping Motors 1. Introduction 2. Basics of VRM Analysis 3. Practical VRM Configurations 4. Current Waveforms for Torque Production 5. Nonlinear Analysis 6. Stepping Motors sites.google.com/site/ncpdhbkhn 5 Basics of VRM Analysis (1) • Two types: singly-salient and doubly- salient. • There are no windings or permanent magnets on their rotors. • Their only source of excitation consists of stator windings. • It means that all the resistive winding losses in the VRM occur on the stator. • Because the stator can typically be cooled much more effectively and easily than the rotor, the result is often a smaller motor for a given rating and frame size. • Both the rotor & stator are constructed of high-permeability magnetic material. • To produce torque, VRMs must be designed such that the stator-winding inductances vary with the position of the rotor • Any number of phases are possible. sites.google.com/site/ncpdhbkhn 6 Basics of VRM Analysis (2) To produce torque, VRMs must be designed such that the stator-winding inductances vary with the position of the rotor sites.google.com/site/ncpdhbkhn 7 Basics of VRM Analysis (3) 11LL11()mm 12 ()i    22LL12()mm 22 ()i o LL22()mm 11 ( 90) Wfld (,12 , m ) Tmech  m ii12, are const 11 WLiLiLii ()22 () () fld2211 m 1 22 m 2 12 m 1 2 22 ii12dL11()mm dL 22 () dL 12 () m Tiimech  12 22ddmm d  m sites.google.com/site/ncpdhbkhn 8 Basics of VRM Analysis (4) 22 ii12dL11()mm dL 22 () dL 12 () m Tiimech 12 22ddmm d  m dL () 12 m  0 dm o LL22()mm 11 ( 90) 22 o ii12dL11()mm dL 11 ( 90) Tmech  22ddmm sites.google.com/site/ncpdhbkhn 9 Basics of VRM Analysis (5) Ex. Given a 4/2 VRM. R = 3.8 cm, α = β = 60o, g = 2.54×10–2 cm, D = 13 cm, the poles of each phase winding are connected in series such that there are 100 turns (50 turns/pole) in each phase winding. The rotor & stator are of infinite magnetic permeability. Neglecting leakage & fringing fluxes, plot the phase-1 inductance as a function of θm? NSN22 RD L 00 max Gg2 10027 (4 10 )( / 3)(3.8 10 2 )(0.13) 0.128H 22.54104 L11()m Lmax m 180o 150o 120o 90o 60o 30o 0 30o 60o 90o 120o 150o 180o sites.google.com/site/ncpdhbkhn 10 Basics of VRM Analysis (6) Ex. 22 o ii12dL11()mm dL 11 ( 90) Tmech  22ddmm L () Lmax 11 m m 180o 150o 120o 90o 60o 30o 0 30o 60o 90o 120o 150o 180o dL11()/mm d o o L / o 180 120 max 60 m 150o 90o 60o 30o 0 30o 90o 120o 150o 180o Torque Tmax1 180o 120o T 60o max 2 m 150o 90o 60o 30o 0 30o 90o 120o 150o 180o 22 Lmax1ILI 1 max1 2 iIi112 ,0 TTmax1, max 2 22  iiI122 0,  sites.google.com/site/ncpdhbkhn 11 Basics of VRM Analysis (7) Torque Tmax1 180o 120o T 60o max 2 m 150o 90o 60o 30o 0 30o 90o 120o 150o 180o 22 Lmax1ILI 1 max1 2 iIi112 ,0 TTmax1, max 2 22  iiI122 0,  • VRM must be designed to avoid the occurrence of rotor positions for which none of the phases can produce torque. • 4/2 machines will always have such positions if they are constructed with uniform, symmetric air gaps. sites.google.com/site/ncpdhbkhn 12 Basics of VRM Analysis (8) • For linear machine iron (no magnetic saturation), finding the torque is simply a matter of: – Finding the stator – phase inductances (self & mutual) as a function of rotor position, – Expressing the coenergy in terms of these inductances, and then, – Calculating the derivative of the coenergy with respect to angular position (holding the phase currents constant when taking the derivative). sites.google.com/site/ncpdhbkhn 13 Basics of VRM Analysis (8) • For nonlinear machine iron (where saturation effects are important): – The coenergy can be found by appropriate integration of the phase flux linkages, – The torque can again be found from the derivative of the coenergy with respect to the angular position of the rotor. sites.google.com/site/ncpdhbkhn 14 Variable – Reluctance Machines and Stepping Motors 1. Introduction 2. Basics of VRM Analysis 3. Practical VRM Configurations 4. Current Waveforms for Torque Production 5. Nonlinear Analysis 6. Stepping Motors sites.google.com/site/ncpdhbkhn 15 Practical VRM Configurations (1) • Criteria: – Low cost, – Constant torque independent of rotor angular position, – A desired operating speed range, – High efficiency, – A large torque – to – mass ratio. • A compromise between the variety of options available to the designer. • A doubly – salient design is often the superior choice because it can generally produce a larger torque for a given frame size. sites.google.com/site/ncpdhbkhn 16 Practical VRM Configurations (2) 22 o idL111()mm idL 211 ( 90) Tmech  22ddmm dL11()m L max L min L max L min 1 dLmmmmax • For a give Lmax & ∆θm, the largest value of Lmax/Lmin will give the largest torque. • Because of its geometry, a doubly – salient structure will typically have a lower Lmin & thus a larger value of Lmax/Lmin  doubly – salient machines are the predominant type of VRM. • The challenge to the VRM designer: to achieve a small value of Lmin. sites.google.com/site/ncpdhbkhn 17 Practical VRM Configurations (3) If the ratio ps/pr (or alternatively pr/ps if pr is larger than ps) is an integer, there will be zero-torque positions. sites.google.com/site/ncpdhbkhn 18 Practical VRM Configurations (4) sites.google.com/site/ncpdhbkhn 19 Practical VRM Configurations (5) sites.google.com/site/ncpdhbkhn 20 Practical VRM Configurations (6) sites.google.com/site/ncpdhbkhn 21 Variable – Reluctance Machines and Stepping Motors 1. Introduction 2. Basics of VRM Analysis 3. Practical VRM Configurations 4. Current Waveforms for Torque Production 5. Nonlinear Analysis 6. Stepping Motors sites.google.com/site/ncpdhbkhn 22 Current Waveforms for Torque Production (1) 2/ p r dL11()m dLpLmr0(2/)(0)  0 dm sites.google.com/site/ncpdhbkhn 23 Current Waveforms for Torque Production (2) 2/ p r dL11()m dLpLmr0(2/)(0)  0 dm 22 o idL111()mm idL 211 ( 90) Tmech  22ddmm 1 Tdmech m 0 2  ii12, are constant To produce a time-averaged torque, the stator currents must vary with rotor position sites.google.com/site/ncpdhbkhn 24 Current Waveforms for Torque Production (3) 22 o idL111()mm idL 211 ( 90) Tmech  22ddmm • To produce a time-averaged torque, the stator currents must vary with rotor position. • Motor operation requires a positive time – average shaft torque. • Braking or generator operation requires negative time – average shaft torque. • Positive torque is produced when a phase is excited at angular positions with positive dL/dθm for that phase. • Negative torque: negative dL/dθm. sites.google.com/site/ncpdhbkhn 25 Current Waveforms for Torque Production (4) sites.google.com/site/ncpdhbkhn 26 Current Waveforms for Torque Production (5) d vRij jjjdt  j  Lijj() m j d vRi [()] L i j jjdt jj m j d di j RLjjjmjjjm[()] iL  () dt dt dLjjm() dm di j RiLjjjjm () ddtm dt sites.google.com/site/ncpdhbkhn 27 Current Waveforms for Ex. 1 Torque Production (6) Given an idealized 4/2 VRM. Assume that it has a winding resistance of R = 1.5 Ω/phase and a leakage inductance Ll = 5 mH in each phase. For a constant rotor speed of 4000 r/min, calculate the phase-1 current as a function of time during the interval a –60° ≤ θm ≤ 0°, assuming that a constant voltage of V0 = 100 V is applied to phase 1 just as dL11(θm)/dθm becomes positive (i.e., at θm = –60° = – π/3 rad). L () Lmax 11 m m 180o 150o 120o 90o 60o 30o 0 30o 60o 90o 120o 150o 180o dL11()mm d di1 vR11 iL 111 ()m ddtm dt 2400  d  4000 rad/s m m 60 3 dt Lmax  LL11(ml ) m   0.122 m  0.018  /3 3 dL() d 400 11 mm0.122 51.1 ddtm 3 sites.google.com/site/ncpdhbkhn 28 Current Waveforms for Ex. 1 Torque Production (7) Given an idealized 4/2 VRM. Assume that it has a winding resistance of R = 1.5 Ω/phase and a leakage inductance Ll = 5 mH in each phase. For a constant rotor speed of 4000 r/min, calculate the phase-1 current as a function of time during the interval a –60° ≤ θm ≤ 0°, assuming that a constant voltage of V0 = 100 V is applied to phase 1 just as dL11(θm)/dθm becomes positive (i.e., at θm = –60° = – π/3 rad). dL11()mm d di1 vR11 iL 111 ()m ddt dt dL() d di m viL11 mm  () 1 1111ddtdt m dL() d m 11 mm51.1 R  1.5 dL () di d() L i ddt 11 m iL() 1111 m dt111m dt dt tt L ()ti () t L (0)(0) i  vdt Vdt 11 1 11 100 1 0 L11()ti 1 () t Vt 0 Vt0 it1() L11()t 100t it1() A L11(mm ) 0.122  0.018 51.1t  0.005 mmtt/ 3  418.9   / 3 sites.google.com/site/ncpdhbkhn 29 Current Waveforms for Ex. 1 Torque Production (8) Given an idealized 4/2 VRM. Assume that it has a winding resistance of R = 1.5 Ω/phase and a leakage inductance Ll = 5 mH in each phase. For a constant rotor speed of 4000 r/min, calculate the phase-1 current as a function of time during the interval a –60° ≤ θm ≤ 0°, assuming that a constant voltage of V0 = 100 V is applied to phase 1 just as dL11(θm)/dθm becomes positive (i.e., at θm = –60° = – π/3 rad). 100t Lttit11(mm ) 0.122 0.018;  mm  / 3 418.9 / 3;1 ( )  A 2 51.1t  0.005 L11()m Lmax 1.5 1 m 0.5 60o 30o 0 30o 60o Phase current (A) 0 0 0.5 1 1.5 2 2.5 Time (s) -3  /3 x 10  0t  0.0025s 0.25 m 418.9 0.2 0.15 3 100 2.5 10 0.1 i (0.0025)  (Nm) Torque 1 51.1 2.5 103  0.005 0.05 0  1.88A 0 0.5 1 1.5 2 2.5 Time (s) -3 x 10 sites.google.com/site/ncpdhbkhn 30 Current Waveforms for Ex. 2 Torque Production (9) Given an idealized 4/2 VRM. Assume that it has a winding resistance of R = 1.5 Ω/phase and a leakage inductance Ll = 5 mH in each phase. For a constant rotor speed of 4000 r/min, calculate the phase-1 current if a negative voltage of –200 V is applied at θm = 0° and maintained until the current reaches zero. L () Lmax 11 m m 180o 150o 120o 90o 60o 30o 0 30o 60o 90o 120o 150o 180o dL11()mm d di1 vR11 iL 111 ()m ddtm dt 2400  d  4000 rad/s m m 60 3 dt Lmax  LL11(ml )  m  0.122 m  0.133  /3 3 dL() d 400 11 mm0.122 51.1 ddtm 3 sites.google.com/site/ncpdhbkhn 31 Current Waveforms for Ex. 2 Torque Production (10) Given an idealized 4/2 VRM. Assume that it has a winding resistance of R = 1.5 Ω/phase and a leakage inductance Ll = 5 mH in each phase. For a constant rotor speed of 4000 r/min, calculate the phase-1 current if a negative voltage of –200 V is applied at θm = 0° and maintained until the current reaches zero. dL11()mm d di1 vR11 iL 111 ()m ddt dt dL() d di m viL11 mm  () 1 1111ddtdt m dL() d m 11 mm51.1 R  1.5 dL () di d() L i ddt 11 m iL() 1111 m dt111m dt dt tt L ()ti () t L (0)(0) i  vdt Vdt 11 1 11 1tt 1 0 00 t L (ti ) ( t ) 0.133 1.88 200 dt 11 1 t 0 0.25 200(t  0.0025) it() 1 Lt 11() 200t 0.75 it1() A L11(mm )  0.122 0.133 51.1t 0.26 mm tt/ 3 418.9   / 3 sites.google.com/site/ncpdhbkhn 32 Current Waveforms for Ex. 1 & 2 Torque Production (11) 2 1.5 1 0.5 Phase current (A) 0 0 0.5 1 1.5 2 2.5 3 3.5 Time (s) -3 x 10 0.3 0.2 0.1 0 -0.1 Torque (Nm) -0.2 0 0.5 1 1.5 2 2.5 3 3.5 Time (s) -3 x 10 sites.google.com/site/ncpdhbkhn 33 Current Waveforms for Torque Production (12) sites.google.com/site/ncpdhbkhn 34 Variable – Reluctance Machines and Stepping Motors 1. Introduction 2. Basics of VRM Analysis 3. Practical VRM Configurations 4. Current Waveforms for Torque Production 5. Nonlinear Analysis 6. Stepping Motors sites.google.com/site/ncpdhbkhn 35 Nonlinear Analysis (1) sites.google.com/site/ncpdhbkhn 36 Nonlinear Analysis (2) • Saturation has 2 important effects: – It limits flux densities for a given current level & thus tends to limit the amount of torque available from the VRM, – It tends to lower the required inverter volt-ampere rating for a given VRM output power & thus tends to make the inverter smaller & less costly. • A well – design VRM system will be based on a trade – off between the 2 effects. sites.google.com/site/ncpdhbkhn 37 Nonlinear Analysis (3) d pivi in dt Net work pdt id  in   Peak energy  max id 0 Inverter volt-ampere rating area(W +W ) = rec net Net output area area(Wnet ) sites.google.com/site/ncpdhbkhn 38 Variable – Reluctance Machines and Stepping Motors 1. Introduction 2. Basics of VRM Analysis 3. Practical VRM Configurations 4. Current Waveforms for Torque Production 5. Nonlinear Analysis 6. Stepping Motors sites.google.com/site/ncpdhbkhn 39 Stepping Motors (1) • When the phases of a VRM are energized sequentially in an appropriate step-wise fashion, the VRM will rotate a specific angle for each step. • Stepping/stepper motors. • Designed to produce a large number of steps per revolution: 50, 100, or 200 steps (7.2o, 3.6o, or 1.8o per step). https://learn.sparkfun.com/tutorials/ • Often used in digital control systems. motors‐and‐selecting‐the‐right‐one • Typical applications: – Paper-feed & print-head-positioning motors in printers. – Drive & head-positioning motors in disk drives & CD/DVD players, – Worktable & tool positioning in controlled machining equipment. sites.google.com/site/ncpdhbkhn 40 Stepping Motors (2) • The angular resolution of a VRM is determined by the number of rotor & stator teeth & can be greatly enhanced by techniques such as castleation. • Have a wide variety of designs & configurations. • The use of permanent magnets in combination with a variable-reluctance geometry can significantly enhance the torque & positional accuracy of a stepping motor. sites.google.com/site/ncpdhbkhn 41 Stepping Motors (3) _Motor_Performance_at_Stepper_Motor_Prices sites.google.com/site/ncpdhbkhn 42 Stepping Motors (4) ‐to‐reverse‐rotation‐direction‐of‐stepper‐motor sites.google.com/site/ncpdhbkhn 43 Stepping Motors (5) press/2012/1186854_6008.html sites.google.com/site/ncpdhbkhn 44