Lecture Electromechanical energy conversion: Electromechanical – energy – conversion principles - Nguyễn Công Phương

This chapter presents the following content: Forces and torques in magnetic field systems, energy balance, energy in singly – excited magnetic field systems, determination of magnetic force and torque from energy and coenergy, multiply – excited magnetic field systems, forces and torques in systems with permanent magnets, dynamic equations, analytical techniques.

Nguyễn Công Phương ELECTROMECHANICAL ENERGY CONVERSION Electromechanical – Energy – Conversion Principles 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 Electromechanical – Energy – Conversion Principles Forces and Torques in Magnetic Field Systems Energy Balance Energy in Singly – Excited Magnetic Field Systems Determination of Magnetic Force and Torque from Energy and Coenergy Multiply – Excited Magnetic Field Systems Forces and Torques in Systems with Permanent Magnets Dynamic Equations Analytical Techniques sites.google.com/site/ncpdhbkhn Forces and Torques in Magnetic Field Systems (1) F  q( E  v  B ) v F  qE B F  q( v  B ) Fv   ( v  B) F J  v Fv  J  B sites.google.com/site/ncpdhbkhn Ex Forces and Torques in Magnetic Field Systems (2) B yˆ ˆ A nonmagnetic rotor contains a single – turn coil, it is in a uniform magnetic field The rotor is of radius R and of length l Find the θ – directed torque as a function of α? Fv  J  B  F  S  1( J  B)  I xˆ I  IB Fin  IB0l sin  Fout  IB0l sin  T  (2 R )  F  RIB0l sin  (Nm) sites.google.com/site/ncpdhbkhn Forces and Torques in Magnetic Field Systems (3)   f fld i Lossless magnetic energy storage system , e Electrical terminal i1  v  Winding resistance x   Mechanical terminal Magnetic core  x e f fld Movable magnetic plunger  Lossless winding sites.google.com/site/ncpdhbkhn Forces and Torques in Magnetic Field Systems (4)   f fld i , e Lossless magnetic energy storage system Electrical terminal x   Mechanical terminal ei  dW fld dt d e dt  f fld dx dt  dW fld  id   f fld dx sites.google.com/site/ncpdhbkhn Electromechanical – Energy – Conversion Principles Forces and Torques in Magnetic Field Systems Energy Balance Energy in Singly – Excited Magnetic Field Systems Determination of Magnetic Force and Torque from Energy and Coenergy Multiply – Excited Magnetic Field Systems Forces and Torques in Systems with Permanent Magnets Dynamic Equations Analytical Techniques sites.google.com/site/ncpdhbkhn Energy Balance Energy is neither created or destroyed, it is merely changed in form dWelectrical  eidt  dWmechanical  dW field sites.google.com/site/ncpdhbkhn Electromechanical – Energy – Conversion Principles Forces and Torques in Magnetic Field Systems Energy Balance Energy in Singly – Excited Magnetic Field Systems Determination of Magnetic Force and Torque from Energy and Coenergy Multiply – Excited Magnetic Field Systems Forces and Torques in Systems with Permanent Magnets Dynamic Equations Analytical Techniques sites.google.com/site/ncpdhbkhn 10 Forces and Torques in Systems with Permanent Magnets (4) S 1 Hm H m d  Fe   d Fe  External 1  Bm S   R S ( H m  H c ) magnetic circuit F       R S   H c  e  d   Bm  R ( H m  H c ) S 2 ( Ni ) equiv Ni  Fe  H  d Fe  Bm  R H m External magnetic circuit   BS   R HS  ( Ni ) equiv Fe     R S    d   d  1   If ( Ni ) equiv   H cd sites.google.com/site/ncpdhbkhn 35 a) Find the x – directed force on the plunger when the current in the excitation winding is zero and x = mm? b) Find the current in the excitation winding required to reduce the plunger force to zero? ( Ni ) equiv   H cd x g0 W   ( Ni ) equiv sites.google.com/site/ncpdhbkhn – ( Ni ) equiv    W fld Rm  Rx  R0 + N12 L Rtotal d + Li1 i1 N1 d d x g0 Rm   ; Rx  ; R0   R S  RWD 0Wg D 0WD   W fld Depth D  Wg W Rm – Ex Forces and Torques in Systems with Permanent Magnets (5) N1i1 Rx Rg 36 a) Find the x – directed force on the plunger when the current in the excitation winding is zero? b) Find the current in the excitation winding required to reduce the plunger force to zero? ( Ni ) equiv   W fld Rm  Rx  R0 x  iequiv const ( Ni ) x ( Ni ) equiv sites.google.com/site/ncpdhbkhn + N1 + ( Ni ) equiv   equiv 0Wg D ( Rm  Rx  R0 ) g0 W equiv ( Ni ) ( Ni ) equiv  N1i1   i1  d N1 dRx ( Rm  Rx  R0 ) dx   i1 – f fld   W fld Depth D  Wg W Rm – Ex Forces and Torques in Systems with Permanent Magnets (6) N1i1 Rx Rg 37 Electromechanical – Energy – Conversion Principles Forces and Torques in Magnetic Field Systems Energy Balance Energy in Singly – Excited Magnetic Field Systems Determination of Magnetic Force and Torque from Energy and Coenergy Multiply – Excited Magnetic Field Systems Forces and Torques in Systems with Permanent Magnets Dynamic Equations Analytical Techniques sites.google.com/site/ncpdhbkhn 38 Dynamic Equationsx (1) K i v0 + R  , e  Electromechanical – energy – conversion system B f fld M f0 d v0  Ri  dt   L( x )i di dL( x ) dx  v0  Ri  L( x )  i dt dx dt sites.google.com/site/ncpdhbkhn 39 – Dynamic Equationsx (2) K i v0  + R , e  Electromechanical – energy – conversion system M – f K   K ( x  x0 ) fD  B B f fld f0 dx dt  f fld d 2x fM  M dt dx d 2x  K ( x  x0 )  B  M  f  dt dt f fld  f K  f D  f M  f  sites.google.com/site/ncpdhbkhn 40 Dynamic Equationsx (3) K i v0 + R  , e  Electromechanical – energy – conversion system B f fld M f0 di dL( x ) dx   v0 (t )  Ri  L( x ) dt  i dx dt  dx d  f (t )   K ( x  x )  B  M x  f ( x, i ) fld  dt dt sites.google.com/site/ncpdhbkhn 41 – Dynamic Equations (4) l0 Ex Spring, K Extract the dynamic equations of motion of the electromechanical system? R l1 Coil Length of flux path in the direction of field  (area of flux path perpendicular to field) Rg  g 0 dx Rg  g 0 da R  Rg  Rg  a x g 0 d g ax 1 1      x a  0 da x N 0 daN x x L( x )    L , R g ax ax 0 daN L  g sites.google.com/site/ncpdhbkhn                  h a g d Applied force, ft Cylindrical steel plunger, M 42 Dynamic Equations (5) l0 Ex Spring, K Extract the dynamic equations of motion of the electromechanical system? x L ( x )  L , ax f fld  e  (i , x ) W fld x i  const l1 0 daN L  g i dL i aL   dx ( a  x ) d ( Li ) di dL di dL dx  L i  L i dt dt dt dt dx dt x di dx  L  L a  x dt ( a  x ) dt sites.google.com/site/ncpdhbkhn Coil a x                  h a g d Applied force, ft Cylindrical steel plunger, M 43 Dynamic Equations (6) l0 Ex Spring, K Extract the dynamic equations of motion of the electromechanical system? i dL i aL f fld   dx ( a  x ) x di dx e  L  L a  x dt ( a  x ) dt di dL( x ) dx  v ( t )  Ri  L ( x )  i  dt dx dt  dx d  f (t )   K ( x  x )  B  M x  f ( x, i ) fld  dt dt x di a dx    v ( t )  Ri  L  L i  a  x dt (a  x ) dt   2 aL  f (t )   K ( x  l )  B dx  M d x  i  dt dt 2 ( a  x ) sites.google.com/site/ncpdhbkhn l1 Coil a x                  h a g d Applied force, ft Cylindrical steel plunger, M 44 Electromechanical – Energy – Conversion Principles Forces and Torques in Magnetic Field Systems Energy Balance Energy in Singly – Excited Magnetic Field Systems Determination of Magnetic Force and Torque from Energy and Coenergy Multiply – Excited Magnetic Field Systems Forces and Torques in Systems with Permanent Magnets Dynamic Equations Analytical Techniques sites.google.com/site/ncpdhbkhn 45 Analytical Techniques (1)  v (0)  V  If  x di a dx   Ri  L , L i  a  x dt ( a  x ) dt  i  x di a dx    v ( t )  Ri  L  L i  a  x dt ( a  x ) dt   2 dx d x i aL  f (t )   K ( x  l )  B  M   dt dt 2 ( a  x ) V R f 0 If  M  dx a V   B  L   K ( x  l0 )  f ( x )  dt ( a  x )  R  t X B dx f ( x) sites.google.com/site/ncpdhbkhn 46 Analytical Techniques (2)  v (0)  V  If  x di a dx   Ri  L , L i  a  x dt ( a  x ) dt  i  x di a dx    v ( t )  Ri  L  L i  a  x dt ( a  x ) dt   2 dx d x i aL  f (t )   K ( x  l )  B  M   dt dt 2 ( a  x ) V R f 0 If  B  d 2x a V   M  L   K ( x  l0 )  f ( x )  dt (a  x )  R  dx  v( x)   dt M  x sites.google.com/site/ncpdhbkhn B dx f ( x) 47 Analytical Techniques (3) If di dx  0, 0 dt dt x di a dx    v ( t )  Ri  L  L i  a  x dt ( a  x ) dt   2 dx d x i aL  f (t )   K ( x  l )  B  M   dt dt 2 ( a  x )  V0  RI    LaI 02  ( a  l )  K ( X  l0 )  f t  If i  I  i, f t  f t  f , vt  V0  v, x  X  x L( X  x) di La ( I  i ) dx    V0  v  R ( I  i )      a  X  x dt ( a  X  x ) dt  0  2    L a ( I  i ) dx d x    K ( X  x  l )  B  M  ft0  f  0 2  ( a  X  x) dt dt sites.google.com/site/ncpdhbkhn 48 Analytical Techniques (4) L( X  x) di La ( I  i ) dx    V0  v  R ( I  i )      a  X  x dt ( a  X  x ) dt  0  2    L a ( I  i ) dx d x    K ( X  x  l )  B  M  ft0  f  0 2  ( a  X  x) dt dt LX di LaI dx    v  Ri    a  X dt ( a  X ) dt 0   2       L aI dx d x L aI 0  i  B  M  K  x  f  3  ( a  X ) dt dt (a  X )   sites.google.com/site/ncpdhbkhn 49 ... Two – Phase Motors IX Speed and Torque Control sites.google.com/site/ncpdhbkhn Electromechanical – Energy – Conversion Principles Forces and Torques in Magnetic Field Systems Energy Balance Energy. .. sites.google.com/site/ncpdhbkhn Electromechanical – Energy – Conversion Principles Forces and Torques in Magnetic Field Systems Energy Balance Energy in Singly – Excited Magnetic Field Systems... sites.google.com/site/ncpdhbkhn 12 Electromechanical – Energy – Conversion Principles Forces and Torques in Magnetic Field Systems Energy Balance Energy in Singly – Excited Magnetic Field Systems