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

Fig. 239 d-axis

Fig. 240 q-axis

Stator flux linkages:

\[\Psi_\mathrm{d} = L_\mathrm{ls} \, i_\mathrm{d} + L_\mathrm{m,d} \, ( i_\mathrm{d} + i'_\mathrm{f} + i'_\mathrm{k,d} )\]

\[\Psi_\mathrm{q} = L_\mathrm{ls} \, i_\mathrm{q} + L_\mathrm{m,q} \, ( i_\mathrm{q} + i'_\mathrm{k,q} )\]

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 \(L_\mathrm{m,d}\) and \(L_\mathrm{m,q}\) are not constant but depend on the main flux linkage \(\Psi_\mathrm{m}\) as illustrated in the \(\Psi_\mathrm{m}/i_\mathrm{m}\) diagram.

Fig. 241 Magnetic saturation characteristic

In this machine model, the anisotropic factor

\[m = \sqrt{L_\mathrm{m,q,0}/L_\mathrm{m,d,0}} \equiv \sqrt{L_\mathrm{m,q}/L_\mathrm{m,d}} = \text{const.}\]

is assumed to be constant at all saturation levels. The equivalent magnetizing flux \(\Psi_\mathrm{m}\) in an isotropic machine is defined as

\[\Psi_\mathrm{m} = \sqrt{\Psi_\mathrm{m,d}^2 + \Psi_\mathrm{m,q}^2/m^2}\]

For flux linkages \(\Psi_\mathrm{m}\) far below the transition flux \(\Psi_\mathrm{T}\), the relationship between flux and current is almost linear and determined by the unsaturated magnetizing inductance \(L_\mathrm{m,0}\). For large flux linkages the relationship is governed by the saturated magnetizing inductance \(L_\mathrm{m,sat}\). \(\Psi_\mathrm{T}\) defines the knee of the transition between unsaturated and saturated main flux inductance. The tightness of the transition is defined with the form factor \(f_\mathrm{T}\). If you do not have detailed information about the saturation characteristic of your machine, \(f_\mathrm{T} = 1\) 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:

\[T_\mathrm{e} = \frac{3}{2} \, p ( i_\mathrm{q} \, \Psi_\mathrm{d} - i_\mathrm{d} \, \Psi_\mathrm{q} )\]

Mechanical System

Mechanical rotor speed \(\omega_\mathrm{m}\):

\[\dot{\omega}_\mathrm{m} = \frac{1}{J} ( T_\mathrm{e} - F \omega_\mathrm{m} - T_\mathrm{m} )\]

\[\dot{\theta}_\mathrm{m} = \omega_\mathrm{m}\]

Parameters

Model

Implementation in the rotor reference frame or as a voltage behind reactance.

Stator resistance

Armature or stator winding resistance \(R_\mathrm{s}\) in ohms \((\Omega)\).

Stator leakage inductance

Armature or stator leakage inductance \(L_\mathrm{ls}\) in henries \((\mathrm{H})\).

Unsaturated magnetizing inductance

A two-element vector containing the unsaturated stator magnetizing inductance \(L_\mathrm{m,d,0}\) and \(L_\mathrm{m,q,0}\) of the d-axis and the q-axis. The values in henries \((\mathrm{H})\) are referred to the stator side.

Saturated magnetizing inductance

The saturated stator magnetizing inductance \(L_\mathrm{m,d,sat}\) along the d-axis, in henries \((\mathrm{H})\). If no saturation is to be modeled, set \(L_\mathrm{m,d,sat} = L_\mathrm{m,d,0}\).

Magnetizing flux at saturation transition

Transition flux linkage \(\Psi_\mathrm{T}\), in \((\mathrm{Vs})\), defining the knee between unsaturated and saturated main flux inductance.

Tightness of saturation transition

Form factor \(f_\mathrm{T}\) defining the tightness of the transition between unsaturated and saturated main flux inductance. The default is 1.

Field resistance

d-axis field winding resistance \(R'_\mathrm{f}\) in ohms \((\Omega)\), referred to the stator side.

Field leakage inductance

d-axis field winding leakage inductance \(L'_\mathrm{lf}\) in henries \((\mathrm{H})\), referred to the stator side.

Damper resistance

A two-element vector containing the damper winding resistance \(R'_\mathrm{k,d}\) and \(R'_\mathrm{k,q}\) of the d-axis and the q-axis. The values in ohms \((\Omega)\) are referred to the stator side.

Damper leakage inductance

A two-element vector containing the damper winding leakage inductance \(L'_\mathrm{lk,d}\) and \(L'_\mathrm{lk,q}\) of the d-axis and the q-axis. The values in henries \((\mathrm{H})\) are referred to the stator side.

Inertia

Combined rotor and load inertia \(J\) in \((\mathrm{Nms}^2)\).

Friction coefficient

Viscous friction \(F\) in \((\mathrm{Nms})\).

Number of pole pairs

Number of pole pairs \(p\).

Initial rotor speed

Initial mechanical speed \(\omega_\mathrm{m,0}\) in radians per second \((\frac{\mathrm{rad}}{\mathrm{s}})\).

Initial rotor position

Initial mechanical rotor angle \(\theta_\mathrm{m,0}\) in radians. If \(\theta_\mathrm{m,0}\) is an integer multiple of \(2\pi/p\), 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 \(i_\mathrm{a,0}\) and \(i_\mathrm{b,0}\) of phase a and b in amperes \((\mathrm{A})\).

Initial field current

Initial current \(i_\mathrm{f,0}\) in the field winding in amperes \((\mathrm{A})\).

Initial stator flux

A two-element vector containing the initial stator flux \(\Psi'_\mathrm{d,0}\) and \(\Psi'_{q,0}\) in the rotor reference frame in weber \((\mathrm{Vs})\).

Probe Signals

Stator phase currents

The three-phase stator winding currents \(i_\mathrm{a}\), \(i_\mathrm{b}\) and \(i_\mathrm{c}\), in amperes \((\mathrm{A})\). Currents flowing into the machine are considered positive.

Field currents

The excitation current \(i_\mathrm{f}\) in amperes \((\mathrm{A})\).

Damper currents

The damper currents \(i'_\mathrm{k,d}\) and \(i'_\mathrm{k,q}\) in the stationary reference frame, in amperes \((\mathrm{A})\).

Stator flux (dq)

The stator flux linkages \(\Psi_\mathrm{d}\) and \(\Psi_\mathrm{q}\) in the stationary reference frame in \((\mathrm{Vs})\).

Magnetizing flux (dq)

The magnetizing flux linkages \(\Psi_\mathrm{m,d}\) and \(\Psi_\mathrm{m,q}\) in the stationary reference frame in \((\mathrm{Vs})\).

Rotational speed

The rotational speed \(\omega_\mathrm{m}\) of the rotor in radians per second \((\frac{\mathrm{rad}}{\mathrm{s}})\).

Rotor position

The mechanical rotor angle \(\theta_\mathrm{m}\) in radians.

Electrical torque

The electrical torque \(T_\mathrm{e}\) of the machine in newton-meters \((\mathrm{Nm})\).

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.