1 / 22 Lecture 1: Induction Motor ELEC-E8402 Control of Electric Drives and Power Converters (5 ECTS) Marko Hinkkanen Aalto University School of Electrical Engineering Spring 2016
2 / 22 Learning Outcomes After this lecture and exercises you will be able to: Transform the T-model parameters to the inverse-γ-model parameters (no need to remember the scaling equations, however) Express the dynamic inverse-γ model in stator coordinates Later in this course, only the inverse-γ model will be used.
3 / 22 Outline Introduction Review: Space Vectors Dynamic T Model Dynamic Inverse-Γ Model
3-Phase Induction Motor: Structure Figure: http://www.ctiautomation.net/about-motors.htm. See also: http://www.youtube.com/watch?v=n8luotqkxlk 4 / 22
5 / 22 Steady-State T-Equivalent Circuit Rotor angular speed ω m (in electrical rad/s) i s R s jω s L sσ jω s L rσ i r Supply angular frequency ω s i m Slip angular frequency u s jω s ψ s jω s L m jω s ψ r R r ω r /ω s ω r = ω s ω m
6 / 22 3-Phase Induction Motor Most common motor in industrial applications Advantages Robust, inexpensive, durable Can be started by direct connection to the mains (unlike other standard AC motors) Low-performance control with a frequency converter is very simple (open-loop scalar control aka volts-per-hertz control) Disadvantages High-performance control is complicated Less efficient than synchronous machines (but more efficient than DC machines)
Operating Principle Magnetised from the stator If the motor rotates synchronously, no currents are induced in the rotor Once the motor is loaded mechanically, the rotor starts lagging behind the synchronous rotation of the flux Currents are induced in the short-circuited rotor winding Torque counteracting the lagging effect is produced Animation: https://en.wikipedia.org/wiki/induction motor#/media/file:asynchronmotor animation.gif 7 / 22
8 / 22 Outline Introduction Review: Space Vectors Dynamic T Model Dynamic Inverse-Γ Model
Magnetic Axes in the Complex Plane e j2π/3 e j0 e j0 Phase a e j4π/3 All 3 phases Windings are sinusoidally distributed along the air gap 9 / 22
Space-Vector Transformation Space vector is a complex variable (signal) i s s = 2 3 (i a + i b e j2π/3 + i c e j4π/3) where i a, i b, and i c are arbitrarily varying instantaneous phase variables Superscript s marks stator coordinates Same transformation applies for voltages and flux linkages Space vector does not include the zero-sequence component (not a problem since the stator winding is delta-connected or the star point is not connected) Peak-value scaling of space vectors will be used in this course. Furthermore, we will use the subscript s to refer to stator quantities, e.g., the stator current vector i s and the stator voltage vector u s. The subscript r refers to rotor quantities. 10 / 22
e j2π/3 β e j2π/3 i c e j4π/3 i s s 3 2 is s i c i s s i b e j2π/3 i a α i b i a e j0 e j4π/3 e j4π/3 i a = Re { i s } s i s s = 2 ( ia + i b e j2π/3 + i c e j4π/3) i b = Re { e j2π/3 i s } s 3 i c = Re { e j4π/3 i s } s 11 / 22
12 / 22 Representation in Component and Polar Forms Component form Polar form i s s = i α + ji β i s s i α β i s s = i s e jθ i = i s cos(θ i ) +j i }{{} s sin(θ i ) }{{} i α Generally, both the magnitude i s and the angle θ i may vary arbitrarily in time i β Positive sequence in steady state: i s = 2I is constant and θ i = ω m t + φ ji β θ i α
Physical Interpretation: Sinusoidal Distribution in Space i s s i α β 3-phase winding creates the current and the mmf, which are sinusoidally distributed along the air gap Space vector represents the instantaneous magnitude and angle of the sinusoidal distribution in space Magnitude and the angle can vary freely in time ji β α Rotating current distribution produced by the 3-phase stator winding 13 / 22
14 / 22 Outline Introduction Review: Space Vectors Dynamic T Model Dynamic Inverse-Γ Model
15 / 22 Stator Winding Stator phase voltages β u a = R s i a + dψ a u b = R s i b + dψ b u c = R s i c + dψ c Corresponding stator voltage space vector α u s s = R si s s + dψs s In the following, space vector equations will be directly given
16 / 22 Rotor Winding Short-circuited rotor winding is modelled similarly as the stator winding y β ω m x u r r = R ri r r + dψr r = 0 ϑ m Superscript r refers to rotor coordinates Coordinates transformations α i r r = e jϑm i s r ψ r r = e jϑm ψ s r leads to 0 = R r i s r + dψs r jω m ψ s r ω m = dϑ m
17 / 22 Flux Linkages Stator flux linkage Rotor flux linkage ψ s s = L sσi s s + L m i s m β i s s ψ s r = L rσi s r + L m i s m Magnetising current α i s m = i s s + i s r Figure illustrates the currents and flux paths in a no-load condition (where i s r = 0) Airgap flux Stator leakage flux
18 / 22 Dynamic T-Model Flux linkages ψ s s = L si s s + L m i s r ψ s r = L mi s s + L r i s r i s s R s L sσ L rσ R r i s r Stator inductance L s = L sσ + L m u s s dψ s s i s m L m dψ s r jω m ψ s r Rotor inductance L r = L rσ + L m
19 / 22 Outline Introduction Review: Space Vectors Dynamic T Model Dynamic Inverse-Γ Model
20 / 22 Γ Model and Inverse-Γ Model T model can be transformed into simpler models with no loss of information Rotor variables are scaled i s r = γi s R ψ s r = ψs R /γ Scaling factor γ can be chosen arbitrarily Scaled rotor voltage equation 0 = (γ 2 R r )i s R + dψs R jω m ψ s R Stator voltage equation is not affected
Γ Model and Inverse-Γ Model Scaled flux equations ψ s s = L si s s + (γl m )i s R ψ s R = (γl m)i s s + (γ 2 L r )i s R Two reasonable selections for γ γ = L s /L m Γ model γ = L m /L r inverse-γ model Both these selections decrease the number of model parameters from 5 to 4 Inverse-Γ model is convenient for control purposes For details about model transformations, see: G. R. Slemon, Modelling of induction machines for electric drives, IEEE Trans. Ind. Applicat., vol. 25, no. 6, 1989. 21 / 22
Dynamic Inverse-Γ Model Scaled parameters L M = γl m L σ = L sσ + γl rσ i s s R s L σ R R i s R R R = γ 2 R r i s M where γ = L m /L r Flux linkages u s s dψ s s L M dψ s R jω m ψ s R ψ s s = L σi s s + ψ s R ψ s R = L M(i s s + i s R ) 22 / 22