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Synchronous Machine (Salient Pole)

Purpose

Salient pole synchronous machine with main-flux saturation

Library

Electrical / Machines

Description

pict

This synchronous machine has one damper winding each on the direct and the quadrature axis of the rotor. Main flux saturation is modeled 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 winding and the positive pole of the field winding are marked with a dot.

Electrical System

pict

pict

Stator flux linkages:

Yd = Llsid + Lm,d (id +i′f + i′k,d)

                (       )
Yq = Llsiq + Lm,q iq + i′k,q

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

Rotor Reference Frame Using Park's transformation, the 3-phase circuit equations in physical variables are transformed to the dq rotor reference frame. This results in constant coefficients in the stator and rotor equations making the model numerically efficient. However, interfacing the dq model 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.

Voltage behind Reactance This formulation allows for 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 rotor reference frame.

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.

pict

In this machine model, the anisotropic factor

    ∘ ------------  ∘---------
m =   L    ∕L     ≡  L   ∕L    = const.
       m,q,0  m,d,0      m,q   m,d

is assumed to be constant at all saturation levels. The equivalent magnetizing flux Y
  m   in an isotropic machine is defined as

      ∘ -2------2----2-
Ym  =   Ym,d + Y m,q∕m .

For flux linkages Ym   far below the transition flux YT  , the relationship between flux and current is almost linear and 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. For rotating reference frame formulation, the stator currents, the field current and the main flux linkage are chosen as state variables. With this choice of state variables, the representation of dynamic cross-saturation could be neglected without affecting the performance of the machine. The computation of the time derivative of the main flux inductance was not required.

Electro-Mechanical System

Electromagnetic torque:

      3
Te =  2 p (iqYd - idYq)

Mechanical System

Mechanical rotor speed ω
 m  :

       1-
ωm  =  J (Te - Fωm - Tm)

θm  = ωm

Parameters

Model
Implementation in the rotor reference frame or as a voltage behind reactance.
Stator resistance
Armature or stator winding resistance Rs   in ohms (_O_  ).
Stator leakage inductance
Armature or stator leakage inductance Lls   in henries (H).
Unsaturated magnetizing inductance
A two-element vector containing the unsaturated stator magnetizing inductance Lm,d,0   and Lm,q,0   of the d-axis and the q-axis. The values in henries (H) are referred to the stator side.
Saturated magnetizing inductance
The saturated stator magnetizing inductance Lm,d,sat   along the d-axis, in H. If no saturation is to be modeled, set Lm,d,sat = Lm,d,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.
Field resistance
d-axis field winding resistance   ′
R f   in ohms (_O_  ), referred to the stator side.
Field leakage inductance
d-axis field winding leakage inductance L′lf   in henries (H), referred to the stator side.
Damper resistance
A two-element vector containing the damper winding resistance R ′k,d    and R′k,q    of the d-axis and the q-axis. The values in ohms (_O_  ) are referred to the stator side.
Damper leakage inductance
A two-element vector containing the damper winding leakage inductance  ′
Llk,d   and  ′
Llk,q   of the d-axis and the q-axis. The values in henries (H) are referred to the stator side.
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 speed ωm,0   in radians per second (s- 1  ).
Initial rotor position
Initial mechanical rotor angle θm,0   in radians. If θm,0   is an integer multiple of 2π ∕p   the d-axis is aligned with phase a of the stator windings at simulation start.
Initial stator currents
A two-element vector containing the initial stator currents ia,0   and ib,0   of phase a and b in amperes (A).
Initial field current
Initial current i′f,0    in the field winding in amperes (A), referred to the stator side.
Initial stator flux
A two-element vector containing the initial stator flux Y ′d,0   and Y′q,0   in the rotor 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 (s-1  ).
(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 ia  , ib   and ic  , in A. Currents flowing into the machine are considered positive.
Field currents
The excitation current i′
 f   in A, referred to the stator side.
Damper currents
The damper currents  ′
ik,d   and  ′
ik,q   in the stationary reference frame, in A.
Stator flux (dq)
The stator flux linkages Yd   and Yq   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.
Rotational speed
The rotational speed ω
 m   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, "Saturation modelling in D-Q axis models of salient pole synchronous machines", IEEE Transactions on Energy Conversion, Vol. 14, No. 1, March 1999.
E. Levi, "Impact of cross-saturation on accuracy of saturated synchronous machine models", IEEE Transactions on Energy Conversion, Vol. 15, No. 2, June 2000.