ECE 8830 Electric Drives

1
ECE 8830 - Electric Drives
Topic 3 Induction Motor Modeling -
Steady State Spring 2004
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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 motors
• Rugged and require little maintenance
• Offer reasonable asynchronous performance
• A manageable torque-speed curve
• Stable operation under load
• Generally satisfactory efficiency
• Range in size from few Watts to several MW

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Introduction (contd)

• Some disadvantages of induction motors are
• Speeds not as easily controlled as dc motors
• Draw large starting currents, typically 6-8 x
  their full load values
• Operate with a poor lagging power factor when
  lightly loaded

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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.

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Structure of an Induction Machine

• Two types of induction machine
• Wound rotor or squirrel cage rotor

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Rotating Magnetic Field and Slip

• We previously showed that a balanced set of
  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.

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Rotating Magnetic Field and Slip (contd)

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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

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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 mmf
• and

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Per-Phase Equivalent Circuit Model

• A per-phase transformer-like equivalent
  circuit is shown below

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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 nrotorstator turns ratio, and
  snormalized slip.
• Stator terminal voltage, Vs Vm VRs VLls
• where VRsvoltage drop across stator
  resistance (Rs) and VLlsvoltage drop across
  stator leakage inductance (Lls).

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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 leakage
• inductance
• The induced voltage in the rotor leads to a rotor
  
• current Ir at slip frequency ?sl.

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Per-Phase Equivalent Circuit Model (contd)

• The stator current, IS I0 Ir
• where Ir is the rotor-reflected current
  induced in the stator.

I0
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Per-Phase Equivalent Circuit Model (contd)
15
Per-Phase Equivalent Circuit Model (contd)

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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 current

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Per-Phase Equivalent Circuit Model Power
Expressions

• Input 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 loss

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Per-Phase Equivalent Circuit Model Torque
Expression

• The torque can be expressed as
• where is the rotor
• mechanical speed (radians/sec.)

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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.

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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.

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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

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Example of Calculating Efficiency of an Induction
Motor

• Example 5.1 Krishnan

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Flowchart for Computing Steady State Performance
of Induction Motor

Ref R. Krishnan, Electric Motor Drives
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Torque-Speed Curve of Induction Motor

• The torque-speed curve as a function of slip
  can be calculated from the equation two slides
  back.

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Torque-Speed Curve of Induction Motor (contd)

• Three regions in torque-speed curve
• 1) Plugging (braking) region (1
• Rotor rotates opposite to direction of air
  gap flux. Can happen, for example, if stator
  supply phase sequence reversed while rotor is
  moving.
• 2) Motoring region (0
• Te0 at s0. As s increases (speed decreases),
  Te increases until max. torque (breakdown torque)
  is reached. Beyond this point, Te decreases with
  increasing s.

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Torque-Speed Curve of Induction Motor (contd)

• 3) Regenerating Region (s
• Here the induction machine acts as a
  generator. Rotor moves faster than air gap flux
  resulting in negative slip.

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Torque-Speed Curve of Induction Motor (contd)

Ref R. Krishnan, Electric Motor Drives
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Performance Characteristics of Induction Motor

Ref R. Krishnan, Electric Motor Drives
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Starting Torque of Induction Motor

• The starting torque of an induction motor is
  given by substituting for s1 and is given by

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Starting Torque of Induction Motor (contd)

• This torque can be enhanced for line start
  motors (ones started directly with full line
  voltage) by increasing the rotor resistance. This
  can be achieved by connecting external resistors
  in the case of slip ring rotors. However, with
  squirrel cage rotors where the rotor is shorted,
  deep bar or double-cage rotors can be used to
  increase starting torque.

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Characterizing Induction Motors

• One way to characterize an induction motor is
  with the No-load/blocked rotor tests which yield
  the per-phase equivalent circuit model shown
  earlier (see figure below).

iar
ias
vas
M
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Characterizing Induction Motors (contd)

• We can characterize an induction motor with
  the variables Rs, Lls, M, Llr, Rr determined
  through lab tests using balanced 3? excitation.
  This circuit described the impedance perceived
  per phase on a line-neutral connected machine.
  Everything in the dashed box is a rotor quantity
  that has been referred to the stator by the
  ideal transformer in the machine model. From now
  on, assume that Llr, Rr and iar are referred to
  the stator.

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Characterizing Induction Motors (contd)

• No-Load Test (s0)
• Equivalent circuit

Ref R. Krishnan, Electric Motor Drives
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Characterizing Induction Motors (contd)

• No-Load Test (s0) yields
• In sinusoidal steady state, ignoring
  resistances
• But -ias (ibsics)
• ?
• From transformer model

Las

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Characterizing Induction Motors (contd)

• Blocked rotor test (s1) yields estimates of Lls
  and Llr. Equivalent circuit at standstill is
  shown below

Ref R. Krishnan, Electric Motor Drives
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Characterizing Induction Motors (contd)

• Ohmmeter/Power loss tests give Rs
• and Rr.
• So, with Llr, Rr and all irs understood as
• referred rotor quantities, the stator-side
• tests identify all the model parameters for
• the induction motor.

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Example of Determining Induction Motor Model
Parameters

• Example 5.2 Krishnan

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NEMA Classification of Induction Motors

• The National Electrical Manufacturers
  Association (NEMA) has classified induction
  motors based on their torque-slip
  characteristics. (see text for details)

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Circuit Model of a Three-Phase Induction Machine
(State-Space Approach)

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Voltage Equations

• Stator Voltage Equations

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Voltage Equations (contd)

• Rotor Voltage Equations

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Flux Linkage Equations

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Model of Induction Motor

• To build up our simulation equation, we could
  just differentiate each expression for ?, e.g.
• But since Lsr depends on position,
• which will generally be a function of time,
  the trig. terms will lead to a mess!
• Parks transform to the rescue!

first row of matrix