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EEEB443 Control & Drives

EEEB443 Control & Drives. Dynamic Model of Induction Machine By Dr. Ungku Anisa Ungku Amirulddin Department of Electrical Power Engineering College of Engineering. Outline. Introduction Three-phase Dynamic Model Space Phasors of Motor Variables

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EEEB443 Control & Drives

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  1. EEEB443 Control & Drives Dynamic Model of Induction Machine By Dr. UngkuAnisaUngkuAmirulddin Department of Electrical Power Engineering College of Engineering EEEB443 - Control & Drives

  2. Outline • Introduction • Three-phase Dynamic Model • Space Phasors of Motor Variables • Three-phase to Two-phase Transformation (Stationary) • Two-phase (Stationary) Dynamic Model (in dsqs frame) • Two-phase Dynamic Model in Arbitrary Rotating Frame (in dq frame) • Voltage equations • Torque equation • Commonly-used Induction Motor Models • Stationary (Stator) Reference Frame Model • Rotor Reference Frame Model • Synchronously Rotating Reference Frame Model • Equations in Flux Linkages • References EEEB443 - Control & Drives

  3. Introduction • Per phase equivalent circuit model only useful for analysing IM performance in steady-state • all transients neglected during load and frequency variations • used in scalar control drives which do not require good transient response • example: drive systems for fans, blowers, compressors • Dynamic model used to observe dynamic (steady-state and transient) behaviour of IM since: • Considers instantaneous effects of varying: • Voltages and currents • Stator frequency • Torque disturbances • machine is part of the feedback loop elements to control the dynamics of the drive system • High performance drive control schemes are based on dynamic model of IM EEEB443 - Control & Drives

  4. Introduction • Dynamic model – complex due to magnetic coupling between stator phases and rotor phases • Coupling coefficients vary with rotor position and rotor position vary with time • Dynamic behavior of IM can be described by differential equations with time varying coefficients • Complexity of dynamic model can be reduced by employing space vector equations EEEB443 - Control & Drives

  5. Magnetic axis of phase B ibs a b’ c’ Magnetic axis of phase A ias b c Simplified equivalent stator winding ics a’ Magnetic axis of phase C Three-phase Dynamic Model IM consists of three-phase windings spaced at 120 apart • Model windings using simplified equivalent stator winding located on the magnetic axis of each phase. EEEB443 - Control & Drives

  6. stator, b rotor, a r rotor, b stator, a rotor, c stator, c Three-phase Dynamic Model • Similar model is applied to represent the rotor ‘windings’ • Rotor rotates at speed r • Rotor phase ‘a’ winding displaced from stator phase ‘a’ winding by angle r EEEB443 - Control & Drives

  7. Three-phase Dynamic Model • Voltage equation for each stator phase: • Similarly, voltage equation for each rotor phase: • These equations can be written in a compact form. EEEB443 - Control & Drives

  8. Three-phase Dynamic Model • Stator voltage equation (compact form): • Rotor voltage equation (compact form): where: •  abcs= stator flux linkage (flux linking stator windings ) • abcr= rotor flux linkage (flux that links rotor windings) (1) (2) EEEB443 - Control & Drives

  9. Three-phase Dynamic Model • The displacements between 3 phase stator (rotor) windings are non-quadrature (i.e. not 90) • Magnetic coupling exists between the 3 stator (or rotor) phases, i.e. the flux linkage each stator (or rotor) phase is sum of: • fluxes produced by the winding itself • fluxes produced from the other two stator (or rotor) windings • fluxes produced by all three rotor (or stator) windings • Example: Flux linkage for stator phase ‘a’ is sum of: • Fluxes produced by stator phase ‘a’ winding itself • Fluxes produced by stator phase ‘b’ and stator phase ‘c’ • Fluxes produced by rotor phase ‘a’, ‘b’ and ‘c’ EEEB443 - Control & Drives

  10. Three-phase Dynamic Model • In general, the stator flux linkage vector: (3) Flux linking stator winding due to stator currents Flux linking stator winding due to rotor currents EEEB443 - Control & Drives

  11. Three-phase Dynamic Model • In general, the rotor flux linkage vector: (4) Flux linking rotor winding due to rotor currents Flux linking rotor winding due to stator currents EEEB443 - Control & Drives

  12. Three-phase Dynamic Model • Self inductances in (3) and (4) consists of magnetising inductance and leakage inductance: • For stator: • For rotor: • Due to symmetry in windings, mutual inductances between stator phases in (3)(and rotor phases in (4)) can be written in terms of magnetising inductances: Stator leakage inductances Rotor leakage inductances Note: Subsrcipt ‘s’ is replaced with ‘r’ for rotor phase leakage inductances, currents and flux linkage EEEB443 - Control & Drives

  13. Three-phase Dynamic Model • Mutual inductances between the stator and rotor windings depends on rotor position r: EEEB443 - Control & Drives

  14. Three-phase Dynamic Model • Equations (1) – (4): • completely describe dynamic characteristics of 3-phase IM • consists of 6 equations (3 for stator and 3 for rotor), i.e. large number of equations • all equations are coupled to one another • Magnetic coupling complicates dynamic model in 3-phase! • Better to develop model based on space phasors: • reduces number of equations • eliminates magnetic coupling between phases EEEB443 - Control & Drives

  15. Space Phasors of Motor Variables • If xa, xb, and xc are the 3-phase IM quantities, whereby: • The space phasorin the 3-phase system is obtained from the vectorial sum of the 3-phase quantities, i.e.: (5) • x is called the space phasor or complex space vector , where a = ej2/3 EEEB443 - Control & Drives

  16. Three-phase to Two-phase Transformation (Stationary) • Any three-phase machine can be represented by an equivalent two-phase machine using Park’s transformation Two-phase equivalent Three-phase easier way to obtain dynamics of IM There is magnetic coupling between phases There is NO magnetic couplingbetween phases (due to 90 angle between phases) EEEB443 - Control & Drives

  17. stator, qs rotor,  r rotating rotor,  r stator, ds Two-phase equivalent Three-phase to Two-phase Transformation (Stationary) • Dynamic model of IM usually obtained using the two-phase equivalent machine stator, b rotating r rotor, b rotor, a r stator, a Three-phase rotor, c stator, c EEEB443 - Control & Drives

  18. Three-phase to Two-phase Transformation (Stationary) • All 3-phase system quantities have to be transformed to 2-phase system quantities • Equivalence between the two systems is based on the equality of mmf produced and current magnitudes, i.e.: • MMF produced by 2-phase system = MMF of 3-phase system • current magnitude of 2-phase system = current of 3-phase system • The use of space phasors enables the transformations from 3-phase to 2-phase system. EEEB443 - Control & Drives

  19. Three-phase to Two-phase Transformation (Stationary) • In the stationary 2-phase system, the space phasor is defined as : (6) • The space phasor in the 2-phase system must equal that in the 3-phase system. • Hence, by comparing (5) and (6): EEEB443 - Control & Drives

  20. Three-phase to Two-phase Transformation (Stationary) • The abc  dsqs transformation is given by: (7) • Under balanced conditions, the zero-sequence component adds to zero, i.e.: Zero-sequence components, which may or may not be present. EEEB443 - Control & Drives

  21. Three-phase to Two-phase Transformation (Stationary) • Assuming balanced conditions, the abc  dsqs transformation: (8) • The inverse transform (dsqs abc transformation) is given by: (9) where: (10) EEEB443 - Control & Drives

  22. Three-phase to Two-phase Transformation (Stationary) • Transformation equations (8) and (9) apply to all 3-phase quantities of the IM (i.e. voltages, current and flux linkages) • Transformation matrices (Tabc and Tabc-1) given by (10) causes the space phasor magnitude to be equal to peak value of the phase quantities, i.e.: • This is one of many abc dsqs and dsqsabc transformation matrices in literature, eg.: • space phasor magnitude to be equal to 1.5 times peak value of the phase quantities ( ) • space phasor magnitude to be equal to rms value of the phase quantities ( ) EEEB443 - Control & Drives

  23. SPACE VECTORS Space vector representation of the mmf distribution in an AC machine created by balanced positive-sequence three-phase sinusoidal currents. Each of the ABC (RGB) space vectors pulsates along its respective axis. The resultant vector (in black), of 1.5 magnitude, rotates at the excitation frequency. Source: http://www.ece.umn.edu/users/riaz/animations/listanimations.html EEEB443 - Control & Drives

  24. SPACE VECTORS This animation shows the motion of space vectors for the case of a balanced three- phase sinusoidal signal: fA = cos(ωt), fB = cos(ωt-α), fC = cos(ωt+α) where α = 2π/3. The corresponding space vector is obtained from fR = (fA + γ fB + γ2fC) = ejωt where γ = ejα. Source: http://www.ece.umn.edu/users/riaz/animations/listanimations.html EEEB443 - Control & Drives

  25. Example 1 - Space Phasor & 3-2 phase transformation • An induction motor has the following parameters: EEEB443 - Control & Drives

  26. Example 1 - Space Phasor & 3-2 phase transformation (contd.) • Assuming that the motor is operating under rated conditions, with stator and rotor current phasors of: Calculate the values the following stator current values at time t = 0: • ias, ibs and ics(3-phase stator phase currents) • isds and isqs(2-phase stator phase currents) • iar, ibr and icr(3-phase rotor phase currents) • irds and irqs(2-phase rotor phase currents) • Show that the magnitude of isds and isqsis equal to the peak stator phase current EEEB443 - Control & Drives

  27. stator, qs rotor,  r rotor,  rotating r stator, ds Two-phase (Stationary) Dynamic Model (in dsqs frame) • From the three-phase dynamic model (eq. 1 and 2): • Applying the transformation given by (8): Note: dsqs – stator equivalent two-phase winding  - rotor equivalent two-phase winding EEEB443 - Control & Drives

  28. stator, qs rotor,  r rotor,  rotating r stator, ds Two-phase (Stationary) Dynamic Model (in dsqs frame) • The rotor winding rotates at a speed r • Hence, need to transform the rotor quantities from the  to the stationary dsqs frame.  qs xr r   Bring all rotor and stator quantities to be on the same axis! r ds EEEB443 - Control & Drives

  29. xrqs xrds Two-phase (Stationary) Dynamic Model -   dsqs frame transform qs  • On the  frame: • On the dsqs frame: • The angle between the two frames is r xr r  xr  xr r ds qs  xr r   r ds EEEB443 - Control & Drives

  30. xrqs xrds Two-phase (Stationary) Dynamic Model -   dsqs frame transform qs  • Therefore: • More elegantly : (full derivation   dsqs frame transform) xr r  xr  xr r r qs  ds (11) xr r  (12)  r ds EEEB443 - Control & Drives

  31. Two-phase (Stationary) Dynamic Model (in dsqs frame) • Two-phase dynamic model: • To transform rotor quantities from the   dsqs frame, from equation (12): Substituting these into the rotor voltage equation above.. Expressed in rotating frame EEEB443 - Control & Drives

  32. Two-phase (Stationary) Dynamic Model (in dsqs frame) • Hence: • Therefore, the two-phase dynamic model in the stationary dsqs frame: Expressed in rotating frame Expressed in stationary frame (13) (14) EEEB443 - Control & Drives

  33. Two-phase (Stationary) Dynamic Model (in dsqs frame) • The flux linkages in (13) and (14) are given by: where and • Note that equations (13)-(16) each consists of two equations. • One from equating real quantities • One from equating imaginary quantities • Final dynamic equations in the stationary dsqs frame is given by substituting (15) and (16) into (13) and (14) and separating the real and imaginary equations. (15) (16) EEEB443 - Control & Drives

  34. Two-phase (Stationary) Dynamic Model (in dsqs frame) • Final dynamic equations in the stationary dsqs frame: • Lm = mutual inductance • Lr’ = rotor self inductances referred to stator • Rr’ = rotor resistance referred to stator • Ls = stator self inductance • vrd, vrq, ird, irq are the rotor voltages and currents referred to stator • S = derivative operator (17) EEEB443 - Control & Drives

  35. Example 2 – Dynamic Model of Induction Motor in dsqs frame • The induction motor from the Example 1 has the following additional parameters: EEEB443 - Control & Drives

  36. Example 2 - Dynamic Model of Induction Motor in dsqs frame (contd.) • Using the values of stator and rotor currents obtained in the Example 1 , calculate the stator flux s and rotor flux r vectors at time t = 0. • Given that Calculate the torque produced by the motor using: • Stator flux s and stator current Is vector • Rotor flux r and stator current Is vector • Stator current  Is and rotor current Ir vector EEEB443 - Control & Drives

  37. SPACE VECTOR DECOMPOSITION • Space vectors under balanced sinusoidal conditions, appears as constant amplitude vectors rotating at the excitation frequency (2f). • In the stationary dsqs (αβ in the diagram) frame: • dsqs components are • time varying sinusoidal • signals at stator frequency • (2f) Source: http://www.ece.umn.edu/users/riaz/animations/listanimations.html EEEB443 - Control & Drives

  38. SPACE VECTOR DECOMPOSITION If we want the Induction Motor to behave like a DC motor, the two-phase components must be constant values. This can be achieved by having a two-phase frame that rotates together with the space vector. A rotating dq frame! Source: http://www.ece.umn.edu/users/riaz/animations/listanimations.html EEEB443 - Control & Drives

  39. Two-phase Dynamic Model in Arbitrary Rotating Frame (in dq frame) • The dq referance frame is rotated at an arbitrary speed g • On the dsqs frame: • On the dq frame: • The angle between the two frames is g where: qs q xqs x g xds d g ds EEEB443 - Control & Drives

  40. Two-phase Dynamic Modelin Arbitrary Rotating Frame(dsqs  dq frame transform) • Therefore: • More elegantly : (full derivation dsqs  dq frame transform) xqs g qs q (18) xd xq x g xds d (19) g ds EEEB443 - Control & Drives

  41. Two-phase Dynamic Model in Arbitrary Rotating Frame (in dq frame) • Hence, space vectors in the stationary dsqs frame will have to be transformed into the rotating dq frame using: • Equation (20) will have to be employed onto equations (13) –(16) to obtain the IM dynamic model in the rotating dq frame. xqs (20) qs q xds x g d g ds EEEB443 - Control & Drives

  42. Two-phase Dynamic Model in Arbitrary Rotating Frame (in dq frame) • Subst. (20) into (13), stator voltage equation: • Subst. (20) into (14), rotor voltage equation: Expressed in stationary frame Expressed in arbitrary rotating frame (21) Expressed in stationary frame Expressed in arbitrary rotating frame (22) EEEB443 - Control & Drives

  43. Two-phase Dynamic Model in Arbitrary Rotating Frame (in dq frame) • Subst. (20) into the flux linkages equations of (15) and (16): • Stator flux linkage: • Rotor flux linkage: • Airgap flux linkage: where and (23) (24) (25) EEEB443 - Control & Drives

  44. Two-phase Dynamic Model in Arbitrary Rotating Frame (in dq frame) • Note that equations (21)-(25) each consists of two equations. • One from equating real quantities • One from equating imaginary quantities • Final dynamic equations in the arbitrary rotating dq frame is given by substituting (23)-(24) into the (21)-(22) and separating the real and imaginary equations. • Final dynamic equations in the arbitrary dq frame: (26) EEEB443 - Control & Drives

  45. Two-phase Dynamic Model in Arbitrary Rotating Frame (in dq frame) • Torque equation in the arbitrary dq frame: • Product of voltage and current conjugate space vectors: • It can be shown that for ias + ibs + ics = 0, • Input power to the IM: EEEB443 - Control & Drives

  46. Two-phase Dynamic Model in Arbitrary Rotating Frame (in dq frame) • Torque equation in the arbitrary dq frame: • Input power to the IM: • If and then: EEEB443 - Control & Drives

  47. Two-phase Dynamic Model in Arbitrary Rotating Frame (in dq frame) • Torque equation in the arbitrary dq frame: • The IM equation given by (26) can be written as: Note that the matrices: [R] = consists of resistive elements [L] = consists of coefficients of the derivative operator S [G] = consists of coefficients of the electrical rotor speed r [F] = consists of coefficients of the reference frame speed g EEEB443 - Control & Drives

  48. Power Losses in winding resistance Power associated with g – upon expansion gives zero Rate of change of stored magnetic energy Mechanical power Two-phase Dynamic Model in Arbitrary Rotating Frame (in dq frame) • Torque equation in the arbitrary dq frame: • Hence, the input power is given by: EEEB443 - Control & Drives

  49. Two-phase Dynamic Model in Arbitrary Rotating Frame (in dq frame) • Torque equation in the arbitrary dq frame: • The mechanical power is most important: • By observing equation (26), [G] consists of terms associated with r : EEEB443 - Control & Drives

  50. Two-phase Dynamic Model in Arbitrary Rotating Frame (in dq frame) • Torque equation in the arbitrary dq frame: • Therefore, mechanical power: EEEB443 - Control & Drives

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