DC Motor Reference: Ned Mohan, Electric Machines and Drives, Wiley, 2012. Structure Magnetic Field Produced by Stator Permanent magnets S N f f produced by permanent magnets 0 f f produced by stator winding current 0 Flux Density in the Airgap from DC Field Back-end N 1 N 1' m s1 b1 ia S S 0 f b2 Bf s2 0 0 Torque Production and Commutation Action ia ia ia 1' + N S1 ea - S2 1 1' + - S 0 ea + S2 + S N m - N 0 S2 S1 - 1' 1 + - S 0 m S1 Tem ea Tem 1 Electromagnetic Torque: Tem 2 B f lia r (2 B f lr )ia kT ia e1,1' , i1to1' 0 Induced voltage or back EMF: 0o ea 2 B f lr (2 B f lr ) k E where Torque Constant EMF Constant 270 o ea (t) 360 o Ea (average) 0 kT k E 2 B f lr 180 o 90 o 0o 180 o ea (t) 90 o 270 o 360 o Tem (t) Tem (average) 0 0o 90 o 180 o 270 o 360 o Four Coil Example (1) Four Coil Example (2) ea 4e Ea 2e (average) 0 0o 45o 90 o 180 o 225o 270 o 315o 360o 180 o 225o 270 o 315o 360 o Tem 4Tcond 2Tcond 0 135o Tem (average) 0o 45o 90 o 135o Multi Coil Case Ca : Number of conductors Quadrature axis Electromagnetic Torque: Direct axis Ca B f lr ia Tem Ca B f l r ia kT ia 2 2 Induced voltage or back EMF: Ca B f lr Ca ea B f lr k E 2 2 where Ca B f lr kT k E 2 Torque Constant EMF Constant Equivalent Circuit of DC Motor ia + va + N Tem m S va - _ La + Tem , m ea k E m - Tem kT ia Va1 > Va2 >Va3 >Va4 m,rated Va1 rated Va2 Va3 Va4 TL JL rated Steady State ea k E m va ea + Ra ia + La m Ra JM Basic equations f constant at its rated value T ia em kT d ia dt 1 d m (Tem +TL ) dt J eq Ia Tem ( TL ) kT m = Va - I a Ra kE kT k E Ca B f lr 2 Tem Operating Modes Regenerative Braking: Feeding energy back while braking current and torque direction reversed same polarity of induced emf Operation in reverse direction: polarity of applied voltage reversed Motoring ia >0 Regenerative braking ia <0 Four Quadrant Operation m Regenerative Braking in Forward direction ea + ia - Motoring in Forward direction a f f Tem a m Regenerative Braking in Reverse direction a f f Tem Tem ,m Tem ,m Tem Motoring in Reverse direction ea ia - ea + ia + m a ea ia + Flux‐weakening in Wound Field Machines To Allow Overspeed Operation m m,rated 2 Va rated I a rated f=reduced Va rated I a rated f=rated 1 0 1 Tem Tem,rated Below rated speed, kT maximum to ensure maximum torque/Ampere thereby minimizing resistive losses Above rated speed,Bf reduced to keepVa at its rated value. B f reduced by reducing I f kT and k E reduced; kT , kE B f Since I a is limited to its rated value maximum, Tem reduces. Electronically Commutated Motor Drives Brushless DC “Inside out” machines Electronically commutated armature Feedback Control of DC Motor Feedback Control Objective desired (reference) + signal error error amplifier - P P U Electric Machine Mech Load output Mechanical System Electrical System measured output signal Feedback control makes system insensitive to disturbances and parameter variation Control Objectives Zero steady‐state error Good dynamic response ‐ fast ‐ small overshoot X * (s) + Controller Plant Gc (s) G p (s) E(s) - GOL ( s ) GCL ( s ) X (s) GC ( s )GP ( s ) E (s) GOL ( s ) GC ( s )GP ( s ) X (s) * X ( s ) 1 + GOL ( s ) 1 + GC ( s )GP ( s ) X (s) Definitions 100 Open loop GOL (s) Gc (s)G p (s) Closed loop GCL (s) GOL (s) /(1 + GOL (s)) Crossover frequency fc ,c 0 f c , c -50 -50 -100 Phase of GOL Phase Margin OL | f -( -180o ) c 50 GOL phase margin -180 o -150 -200 10 -2 10 -1 10 10 1 10 2 frequency > 45o for no oscillations 60o preferable Closed loop bandwidth fc desired high for fast response 0 G CL ( j ) -3dB BW Cascaded Control speed* Position controller position* + - torque* Speed controller + + - torque Torque controller torque(current) speed position Torque loop : fastest Speed loop : slower Position loop : slowest Electrical System Mech System speed 1 s position Speed Loop with LP Filter Steps in Designing the Controller Assume system is linear about the steady state operating point; design controller using Linear Control Theory Simulate design under large signal conditions and “tweak” controller as necessary System representation for small signal analysis Assume Steady state system operating point Highest bandwidth at least an order of magnitude lower than switching frequency ; neglect switching frequency components Power Processing Unit (PPU) for DC Motor H Bridge + ea - va(t) can be negative! Switching Power Pole and Its Average Model 1 Averaged Representation of the PPU H Bridge (A) (B) k PWM d(t) or va(t) can be negative! (C) Vd Vˆtri (D) DC Motor and Mechanical Load Basic equations DC motor time constant ea k E m va ea + Ra ia + La Tem kT ia d ia dt 1 d m (Tem +TL ) dt J eq e m Tem ( s ) + TL ( s ) sJ eq La Ra PI Controller Proportional‐Integral (PI) Controller In the torque and speed loops, proportional control without integral control input leads to steady‐state error Cascaded PI Controller PID Controller Design Example Design of the Torque (Current) Loop (1) Simplifying assumptions Neglect TL Interleaved loops redrawn as nested loops Assuming J high enough, inner loop can be ignored Design of the Torque (Current) Loop (2) Design of the Torque (Current) Loop (3) Design of the Speed Loop (1) Assume current loop to be ideal; represent by unity Choose crossover frequency C an order of magnitude lower than CI Choose a reasonable phase margin Design of the Speed Loop (2) Design of the Speed Loop (3) Design of the Position Loop (1) speed loop Assume speed loop to be ideal Proportional gain (k p) alone is adequate due to presence of pure integrator Select the bandwidth of the position loop to be one order of magnitude smaller than that of the speed loop. Design of the Position Loop (2) The code for this example: DCcontrol.m Feed Forward Control Concept Feed Forward Control Compare: Feed Back Control Feed Forward Control Example Feed Forward vs Feed Back Feed Forward in DC Motor Control Enhance the dynamic response Limits on PI Controller Turn-off integrator action when it reaches limit.
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