Synchronous Machine (Salient Pole)
Purpose
Salient pole synchronous machine with main-flux saturation
Library
Electrical / Machines
Description
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
Stator flux linkages:
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 and
are not constant but depend on the main flux linkage
as illustrated in
the
diagram.
is assumed to be constant at all saturation levels. The equivalent magnetizing flux
in an isotropic machine is defined as
For flux linkages far below the transition flux
, the relationship between
flux and current is almost linear and determined by the unsaturated magnetizing
inductance
. For large flux linkages the relationship is governed by the
saturated magnetizing inductance
.
defines the knee of the transition
between unsaturated and saturated main flux inductance. The tightness of the
transition is defined with the form factor
. If you do not have detailed
information about the saturation characteristic of your machine,
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:
Mechanical System
Mechanical rotor speed :
Parameters
- Model
- Implementation in the rotor reference frame or as a voltage behind reactance.
- Stator resistance
- Armature or stator winding resistance
in ohms (
).
- Stator leakage inductance
- Armature or stator leakage inductance
in henries (H).
- Unsaturated magnetizing inductance
- A two-element vector containing the
unsaturated stator magnetizing inductance
and
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
along the d-axis, in H. If no saturation is to be modeled, set
.
- Magnetizing flux at saturation transition
- Transition flux linkage
, in Vs, defining the knee between unsaturated and saturated main flux inductance.
- Tightness of saturation transition
- Form factor
defining the tightness of the transition between unsaturated and saturated main flux inductance. The default is 1.
- Field resistance
- d-axis field winding resistance
in ohms (
), referred to the stator side.
- Field leakage inductance
- d-axis field winding leakage inductance
in henries (H), referred to the stator side.
- Damper resistance
- A two-element vector containing the damper winding
resistance
and
of the d-axis and the q-axis. The values in ohms (
) are referred to the stator side.
- Damper leakage inductance
- A two-element vector containing the damper
winding leakage inductance
and
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
in
.
- Friction coefficient
- Viscous friction
in Nms.
- Number of pole pairs
- Number of pole pairs
.
- Initial rotor speed
- Initial mechanical speed
in radians per second (
).
- Initial rotor position
- Initial mechanical rotor angle
in radians. If
is an integer multiple of
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
and
of phase a and b in amperes (A).
- Initial field current
- Initial current
in the field winding in amperes (A).
- Initial stator flux
- A two-element vector containing the initial stator flux
and
in the rotor reference frame in Vs.
Probe Signals
- Stator phase currents
- The three-phase stator winding currents
,
and
, in A. Currents flowing into the machine are considered positive.
- Field currents
- The excitation current
in A.
- Damper currents
- The damper currents
and
in the stationary reference frame, in A.
- Stator flux (dq)
- The stator flux linkages
and
in the stationary reference frame in Vs.
- Magnetizing flux (dq)
- The magnetizing flux linkages
and
in the stationary reference frame in Vs.
- Rotational speed
- The rotational speed
of the rotor in radians per second (
).
- Rotor position
- The mechanical rotor angle
in radians.
- Electrical torque
- The electrical torque
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.