ECE 8830 Electric Drives

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ECE 8830 - Electric Drives
Topic 5 Dynamic Simulation of
Induction Motor Spring 2004
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Stationary Reference Frame Modeling of the
Induction Motor

• We now consider how the model of the induction
  motor that we have developed can be used to
  simulate the dynamic performance of the induction
  motor.
• We will consider the model of the motor in the
  stationary reference frame.

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Stationary Reference Frame Modeling of the
Induction Motor (contd)

• Consider a 3?, P-pole, symmetrical induction
  motor in the stationary reference frame with
  windings as shown below

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Stationary Reference Frame Modeling of the
Induction Motor (contd)

• Consider first the input voltages for the
  given neutral connections of the stator and rotor
  windings shown.
• The three applied voltages to the stator
  terminals vag, vbg, and vcg need not be balanced
  or sinusoidal. In general, we can write

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Stationary Reference Frame Modeling of the
Induction Motor (contd)

• Therefore,
• In simulation, the voltage vsg can be
  determined from the flow of phase currents into
  the neutral connection by
• where Rsg and Lsg are the resistance and
  inductance between the two neutral points. Of
  course, if s and g are shorted, vsg0.

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Stationary Reference Frame Modeling of the
Induction Motor (contd)

• Now consider transformation of stator abc
  phase voltages to qd0 stationary voltages.
• With the q-axis aligned with the stator
  a-phase axis, the following equations apply

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Stationary Reference Frame Modeling of the
Induction Motor (contd)

• Transformation of the abc rotor winding
  voltages to the qd0 stationary reference frame
  can be done in two steps.
• First transform the referred rotor abc phase
  voltages to a qd0 reference frame attached to the
  rotor with the q-axis aligned to the axis of the
  rotors a-phase winding.
• In the second step, transform the qd0 rotor
  quantities to the stationary qd0 stator reference
  frame.

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Stationary Reference Frame Modeling of the
Induction Motor (contd)

• Step 1 -gt
• where vrn voltage between points r and n and
  the primes indicate voltages referred to the
  stator side.

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Stationary Reference Frame Modeling of the
Induction Motor (contd)

• Step 2 -gt
• where ?r(t) rotor angle
• at time t, ?r(0) rotor angle
• at time t0, and ?r(t)
• angular velocity of rotor.

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Stationary Reference Frame Modeling of the
Induction Motor (contd)

• The qd0 voltages at both the stator and rotor
  terminals, referred to the same stationary qd0
  reference frame, can be used as inputs along with
  the load torque to obtain the qd0 currents in the
  stationary reference frame. These can then be
  transformed to obtain the phase currents in the
  stator and rotor windings.

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Stationary Reference Frame Modeling of the
Induction Motor (contd)

• The inverse transformation to obtain the stator
  abc phase currents from the qd0 currents is given
  by

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Stationary Reference Frame Modeling of the
Induction Motor (contd)

• The abc rotor currents are obtained by a
  two-step inverse transformation process. Step 1
  transforms the stationary qd0 currents back to
  the qd frame attached to the rotor. Step 2
  resolves the qd rotor currents back to the abc
  rotor phase currents.

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Stationary Reference Frame Modeling of the
Induction Motor (contd)

• Step 1 -gt
• Step 2 -gt

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Stationary Reference Frame Modeling of the
Induction Motor (contd)

• The model equations can be rearranged into the
  form of equations (6.112) to (6.117) in Ongs
  book (provided in separate handout).

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Stationary Reference Frame Modeling of the
Induction Motor (contd)

• The torque equation is
• 
  (eq. 6.118)
• The equation of motion of the rotor is given
  by
• where Tmech is the externally-applied
  mechanical torque in the direction of the rotor
  speed and Tdamp is the damping torque in the
  opposite direction of rotation.

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Stationary Reference Frame Modeling of the
Induction Motor (contd)

• Normalized to the base (or rated speed) of the
  rotor ?b is given by
• (eq. 6.120)

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Stationary Reference Frame Modeling of the
Induction Motor (contd)

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Stationary Reference Frame Modeling of the
Induction Motor (contd)

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Stationary Reference Frame Modeling of the
Induction Motor (contd)
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Saturation of Mutual Flux

• See Ong text.

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Linearized Model

• Solving the nonlinear equations by numerical
  integration allows visualization of the dynamic
  performance of a motor. However, in designing a
  control system, we would like to use linear
  control techniques. For this application we need
  to develop a linearized model of the induction
  motor.

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Linearized Model (contd)

• To develop a linearized model for the
  induction motor, we select an operating point and
  perturb the system with small perturbations over
  a linear regime.

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Linearized Model (contd)

• The general form of the behavior of the
  induction motor may be described by the function
• f( , x, u, y) 0
• where x is a vector of state variables
• ( ) u is the vector of
  input variables ( ) and y is the
  vector of desired outputs, such as
• .

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Linearized Model (contd)

• When a small perturbation ? is applied to each
  of the components of the x, u, and y variables,
  the perturbed variables will satisfy the
  equation
• where the 0 subscript denotes the steady state
  value about which the perturbation is applied.

f( xxx0?x , x0 ?x , u0?u, y0?y) 0
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Linearized Model (contd)

• In steady state,
• xxx0
• Neglecting higher order terms and regrouping
  some of the terms in the earlier equations, the
  linear equations including perturbations can be
  re-written as

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Linearized Model (contd)

• See Ong text to learn how to use
  Matlab/Simulink to solve for the A B C D
  matrix.
• See handout from Krishnans book for more
  detailed description of small signal analysis of
  induction motor.