Induction Motor Modeling _steady State

Topic 3: Induction Motor Modeling - Steady State Spring 2004 ECE 8830 - Electric Drives Introduction Induction machines are the most widely used of all electric motors. They offer the following attractive features:Generally easy to build and cheaper than corresponding dc or synchronous motorsRugged and require little maintenanceOffer reasonable asynchronous performanceA manageable torque-speed curveStable operation under loadGenerally satisfactory efficiencyRange in size from few Watts to several MW Introduction (contd) Some disadvantages of induction motors are:Speeds not as easily controlled as dc motorsDraw large starting currents, typically 6-8 x their full load valuesOperate with a poor lagging power factor when lightly loaded Introduction (contd) New designs for high performance induction machines, such as in high speed motors for gas compressors, will be required to have new characteristics from existing machines, it is important to have a good fundamental understanding of these types of machines. Goal: To develop a simple model for the induction machine that is useful for control and simulation. Structure of an Induction MachineTwo types of induction machine:Wound rotor or squirrel cage rotor Rotating Magnetic Field and Slip We previously showed that a balanced setof three-phase currents flowing in a set of symmetrically placed, three-phase stator windings produces a rotating mmf given by:

[eq. (6.1) Ong, eq. (2.9) Bose] where ae is the electrical angle measured from the a-phase axis and e is the angular speed of the stator mmf in electrical radians/second.3 4( , ) cos( )2e ea m a eNF t I tP Rotating Magnetic Field and Slip (contd) Rotating Magnetic Field and Slip (contd) In mechanical radians/sec. the synchronous speed is related to the electrical speed by:If the rotor is rotating at an angular speed rm the slip speed is simply equal to sm - rm. The slip,s, is the normalized slip speed and is given by: 2sm eP sm rm e rsm es Torque Production The torque produced by an induction motor may be derived and expressed by the following equation:(see ref. [1] in Bose) where P= # of poles l = axial length of motor r = radius of motor Bp= peak air-gap flux density Fp=peak value of rotor mmfand sin2e p pPT lrB F | ` . ,2r + Per-Phase Equivalent Circuit Model A per-phase transformer-like equivalent circuit is shown below: Per-Phase Equivalent Circuit Model (contd) Synchronously rotating air gap flux wave generates a counter emf Vm. This in turn is converted to a slip voltage in the rotor phase, Vr = nsVm, where n=rotor:stator turns ratio, and s=normalized slip. Stator terminal voltage, Vs = Vm + VRs +VLls where VRs=voltage drop across stator resistance (Rs) and VLls=voltage drop across stator leakage inductance (Lls). Per-Phase Equivalent Circuit Model (contd) Excitation current, I0 = Ic + Im where Ic is core loss current (=Vm/Rm)and Im is magnetizing current (=Vm/) Rotor-induced voltage, Vr = VRr + VLl where VRr = voltage drop across rotor resistance and VLl = voltage drop across rotor leakageinductanceThe induced voltage in the rotor leads to a rotor current Ir at slip frequency sl. e mL Per-Phase Equivalent Circuit Model (contd) The stator current, IS = I0 + Ir where Ir is the rotor-reflected current induced in the stator. I0 Per-Phase Equivalent Circuit Model (contd)2'' 'm mr rrr sl lre lrn sV VI nIRR j Lj Ls + | `+ . ,'2rrRRn'2lrlrLLn Per-Phase Equivalent Circuit Model (contd) Per-Phase Equivalent Circuit Model (contd)Torque expression can be written as:where = peak value of air gap flux linkage/pole and= peak value of rotor current3sin2 2e mrPT I | ` . ,mrI Per-Phase Equivalent Circuit Model: Power ExpressionsInput Power: where cos is input PF

Stator copper loss:Rotor copper loss:Core loss:Power across air gap:Output power:Shaft Power: where PFw is friction and windage power loss3 cosin s sP V I 23ls s sP I R 23lr r rP I R 23 /lc m mP V R 23 /g r rP I R s 23 (1 / )o g lr r rP P P I R s s sh o FwP P P Per-Phase Equivalent Circuit Model: Torque ExpressionThe torque can be expressed as:where is the rotor mechanical speed (radians/sec.)2 23 132o re r r rm m eP R s PT I R Is s | ` . ,2 2(1 )m r esP P | ` | ` . , . , Per-Phase Equivalent Circuit Model: Torque Expression (contd)Using a little algebra (see Bose) it can be shown that the torque may be further expressed as:where . This torque expression is similar to that for a dc motor, where Im = magnetizing component of stator current and Ia = armature component of stator current. 32e mmaPT L I I| ` . ,sina rI I Simplified Per-Phase Equivalent Circuit A simplified circuit dropping Rm and shifting Lm to the input is applicable to integral horsepower machines.

Performance of this equivalent circuit is typically within 5%. Simplified Per-Phase Equivalent Circuit Model (contd)The current Ir in this circuit is given by:The torque of the motor using this circuit is given by:2 2 2( / ) ( )srs r e ls lrVIR R s L L + + +22 2 232 ( / ) ( )s ree s r e ls lrV R PTs R R s L L | ` + + +. , Example of Calculating Efficiency of an Induction MotorExample 5.1 Krishnan Flowchart for Computing Steady State Performance of Induction MotorRef: R. Krishnan, Electric Motor Drives Torque-Speed Curve of Induction MotorThe torque-speed curve as a function of slip can be calculated from the equation two slides back. Torque-Speed Curve of Induction Motor (contd)Three regions in torque-speed curve:1) Plugging (braking) region (1