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Induction Machine with Saturation

Purpose

Induction machine with slip-ring rotor and main flux saturation.

Description

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The Induction Machine with Saturation models main flux saturation by means of a continuous function.

The machine operates as a motor or generator; if the mechanical torque has the same sign as the rotational speed the machine is operating in motor mode, otherwise in generator mode. All electrical variables and parameters are viewed from the stator side. In the component icon, phase a of the stator and rotor winding is marked with a dot.

Electrical System:

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The rotor flux is defined as

Yr,d = L ′lri′r,d + Lm (is,d + i′r,d)

                 (       )
Yr,q = L′lri′r,q +Lm   is,q + i′r,q  .

The machine model offers two different implementations of the electrical system: a traditional stationary reference frame and a voltage-behind-reactance formulation.

Stationary Reference Frame This implementation is based on machine equations in the stationary reference frame (Clarke transformation). Constant coefficients in the stator and rotor equations make the model numerically efficient. However, interfacing the reference frame with the external 3-phase network may be difficult. Since the coordinate transformations are based on voltage-controlled current sources inductors and naturally commutated devices such as diode rectifiers may not be directly connected to the stator terminals. In these cases, fictitious RC snubbers are required to create the necessary voltages across the terminals. The implementation can be used with both the continuous and the discrete state-space method.

Voltage behind Reactance This formulation allows direct interfacing of arbitrary external networks with the 3-phase stator terminals. The rotor dynamics are expressed using explicit state-variable equations while the stator branch equations are described in circuit form. However, due to the resulting time-varying inductance matrices, this implementation is numerically less efficient than the traditional reference frame. Currently, it can only be used with the continuous state-space method.

In both implementations, the value of the main flux inductances L
  m,d   and L
  m,q   are not constant but depend on the main flux linkage Y
  m   as illustrated in the Y ∕i
 m  m   diagram.

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For flux linkages far below the transition flux YT  , the relationship between flux and current is almost linear and is determined by the unsaturated magnetizing inductance Lm,0  . For large flux linkages the relationship is governed by the saturated magnetizing inductance Lm,sat  . YT   defines the knee of the transition between unsaturated and saturated main flux inductance. The tightness of the transition is defined with the form factor fT  . If you do not have detailed information about the saturation characteristic of your machine, fT = 1   is a good starting value. The function

plsaturation(Lm0, Lmsat, PsiT, fT)

plots the main flux vs. current curve and the magnetizing inductance vs. current curve for the parameters specified.

The model accounts for steady-state cross-saturation, i.e. the steady-state magnetizing inductances along the d-axis and q-axis are functions of the currents in both axes. In the implementation, the stator currents and the main flux linkage are chosen as state variables. With this type of model, the representation of dynamic cross-saturation can be neglected without affecting the machine's performance. The computation of the time derivative of the main flux inductance is not required.

In order to inspect the implementation, please select the component in your circuit and choose Look under mask from the Edit menu. If you want to make changes, you must first choose Break library link and then Unprotect, both from the Edit menu.

Electro-Mechanical System

Electromagnetic torque:

      3
Te =  2 p (is,qYs,d - is,dYs,q)

Mechanical System

Mechanical rotor speed ωm  :

ωm  =  1-(Te - Fωm - Tm)
       J

ω  = pωm

Mechanical rotor angle θm  :

θm  = ωm

θ = pθm

Parameters

Model
Implementation in the stationary reference frame or as a voltage behind reactance.
Stator resistance
Stator winding resistance Rs   in ohms (_O_  ).
Stator leakage inductance
Stator leakage inductance Lls   in henries (H).
Rotor resistance
Rotor winding resistance R′r    in ohms (_O_  ), referred to the stator side.
Rotor leakage inductance
Rotor leakage inductance L′lr   in henries (H), referred to the stator side.
Unsaturated magnetizing inductance
Unsaturated main flux inductance Lm,0  , in henries (H), referred to the stator side.
Saturated magnetizing inductance
Saturated main flux inductance Lm,sat    in henries (H), referred to the stator side. If you do not want to model saturation, set Lm,sat = Lm,0  .
Magnetizing flux at saturation transition
Transition flux linkage YT  , in Vs, defining the knee between unsaturated and saturated main flux inductance.
Tightness of saturation transition
Form factor fT   defining the tightness of the transition between unsaturated and saturated main flux inductance. The default is 1.
Inertia
Combined rotor and load inertia J   in Nms2  .
Friction coefficient
Viscous friction F   in Nms.
Number of pole pairs
Number of pole pairs p  .
Initial rotor speed
Initial mechanical rotor speed ωm,0   in s- 1  .
Initial rotor position
Initial mechanical rotor angle θm,0   in radians. If θm,0   is an integer multiple of 2π∕p   the stator windings are aligned with the rotor windings at simulation start.
Initial stator currents
A two-element vector containing the initial stator currents is,a,0   and is,b,0   of phases a and b in amperes (A).
Initial stator flux
A two-element vector containing the initial stator flux Ys,d,0   and Ys,q,0   in the stationary reference frame in Vs.

Inputs and Outputs

Mechanical torque
The input signal Tm   represents the mechanical torque at the rotor shaft, in Nm.

The output vector “m” contains the following 3 signals:

(1) Rotational speed
The rotational speed ωm   of the rotor in radians per second ( -1
s  ).
(2) Rotor position
The mechanical rotor angle θm   in radians.
(3) Electrical torque
The electrical torque Te   of the machine in Nm.

Probe Signals

Stator phase currents
The three-phase stator winding currents is,a  , is,b    and is,c  , in A. Currents flowing into the machine are considered positive.
Rotor phase currents
The three-phase rotor winding currents i′r,a  , i′r,b    and i′s,c   in A, referred to the stator side. Currents flowing into the machine are considered positive.
Stator flux (dq)
The stator flux linkages Ys,d    and Ys,q    in the stationary reference frame in Vs.
Magnetizing flux (dq)
The magnetizing flux linkages Ym,d    and Ym,q    in the stationary reference frame in Vs.
Rotor flux (dq)
The rotor flux linkages Y′r,d    and Y ′r,q    in the stationary reference frame in Vs, referred to the stator side.
Rotational speed
The rotational speed ωm   of the rotor in radians per second (s-1  ).
Rotor position
The mechanical rotor angle θm   in radians.
Electrical torque
The electrical torque Te   of the machine in Nm.

References

D. C. Aliprantis, O. Wasynczuk, C. D. Rodriguez Valdez, "A voltage-behind-reactance synchronous machine model with saturation and arbitrary rotor network representation", IEEE Transactions on Energy Conversion, Vol. 23, No. 2, June 2008.
K. A. Corzine, B. T. Kuhn, S. D. Sudhoff, H. J. Hegner, "An improved method for incorporating magnetic saturation in the Q-D synchronous machine model", IEEE Transactions on Energy Conversion, Vol. 13, No. 3, Sept. 1998.
E. Levi, "A unified approach to main flux saturation modelling in D-Q axis models of induction machines", IEEE Transactions on Energy Conversion, Vol. 10, No. 3, Sept. 1995.
E. Levi, "Impact of cross-saturation on accuracy of saturated induction machine models", IEEE Transactions on Energy Conversion, Vol. 12, No. 3, Sept. 1997.