PLECS 3.7 Online Help

Synchronous Machine (Round Rotor)

Purpose

Smooth air-gap synchronous machine with main-flux saturation

Library

Electrical / Machines

Description

pict

This synchronous machine has one damper winding on the direct axis and two damper windings on 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.

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.

Electrical System

pict
d-axis  
pict
q-axis

Stator flux linkages:

Yd = Llsid + Lm,d (id +i′f + i′k,d)

                 (          )
Yq = Llsiq + Lm,q iq + i′g + 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 inductance L
 m   is not constant but depends on the main flux linkage Y
  m   as illustrated in the Y  ∕i
 m  m   diagram. For flux linkages Y
  m   far below the transition flux Y
  T  , the relationship between flux and current is almost linear and determined by the unsaturated magnetizing inductance L
 m,0  . For large flux linkages the relationship is governed by the saturated magnetizing inductance L
  m,sat  . Y
 T   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.

pict

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:

Te =  3p (iqYd - idYq)
      2

Mechanical System

Mechanical rotor speed ωm  :

       1
ωm  =  J-(Te - Fωm - Tm)

θm  = ωm

Parameters

Most parameters for the Salient Pole Synchronous Machine are also applicable to this round rotor machine. The following parameters are different:

Unsaturated magnetizing inductance
The unsaturated magnetizing inductance L
  m,0  . The value in henries (H) is referred to the stator side.
Saturated magnetizing inductance
The saturated magnetizing inductance Lm,sat  , in H. If no saturation is to be modeled, set Lm,sat = Lm,0  .
Damper resistance
A three-element vector containing the damper winding resistance R′k,d  , R′k,q1   and R′k,q2   of the d-axis and the q-axis. The values in ohms (_O_  ) are referred to the stator side.
Damper leakage inductance
A three-element vector containing the damper winding leakage inductance L′lk,d  , L′lk,q1    and L ′lk,q2    of the d-axis and the q-axis. The values in henries (H) are referred to the stator side.
Initial field/damper current
A two-element vector containing the initial currents  ′
if,0   in the field winding and ′
ik,q1,0   in one of the damper windings in amperes (A), referred to the stator side.

Inputs and Outputs

Same as for the Salient Pole Synchronous Machine.

Probe Signals

Most probe signals for the Salient Pole Synchronous Machine are also available with this machine. Only the following probe signal is different:

Damper currents
The damper currents i′k,d  , i′k,q1   and i′k,q2   in the stationary reference frame in A, referred to the stator side.

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, "Modelling of magnetic saturation in smooth air-gap synchronous machines", IEEE Transactions on Energy Conversion, Vol. 12, No. 2, March 1997.
E. Levi, "Impact of cross-saturation on accuracy of saturated synchronous machine models", IEEE Transactions on Energy Conversion, Vol. 15, No. 2, June 2000.