Induction Machine with Saturation

Purpose

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

Library

Electrical / Machines

Description

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

Fig. 229 d-axis

Fig. 230 q-axis

The rotor flux is defined as:

\[\Psi_\mathrm{r,d} = L'_\mathrm{lr} \, i'_\mathrm{r,d} + L_\mathrm{m} \, ( i_\mathrm{s,d} + i'_\mathrm{r,d} )\]

\[\Psi_\mathrm{r,q} = L'_\mathrm{l,r} \, i'_\mathrm{r,q} + L_\mathrm{m} \, ( i_\mathrm{s,q} + i'_\mathrm{r,q} )\]

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 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 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. 231 Magnetic saturation characteristic

For flux linkages far below the transition flux \(\Psi_\mathrm{T}\), the relationship between flux and current is almost linear and is 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. 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 > Subsystem menu or the block’s context menu. If you want to make changes, you must first choose Break library link and then Unprotect, both from the same menu.

Electro-Mechanical System

Electromagnetic torque:

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

Mechanical System

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

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

\[\omega = p \, \omega_\mathrm{m}\]

Mechanical rotor angle \(\theta_\mathrm{m}\):

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

\[\theta_\mathrm{e} = p \, \theta_\mathrm{m}\]

Parameters

Model

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

Stator resistance

Stator winding resistance \(R_\mathrm{s}\) in ohms \((\Omega)\).

Stator leakage inductance

Stator leakage inductance \(L_\mathrm{ls}\) in henries \((\mathrm{H})\).

Rotor resistance

Rotor winding resistance \(R_\mathrm{r}'\) in ohms \((\Omega)\), referred to the stator side.

Rotor leakage inductance

Rotor leakage inductance \(L_\mathrm{lr}'\) in henries \((\mathrm{H})\), referred to the stator side.

Unsaturated magnetizing inductance

Unsaturated main flux inductance \(L_\mathrm{m,0}\), in henries \((\mathrm{H})\), referred to the stator side.

Saturated magnetizing inductance

Saturated main flux inductance \(L_\mathrm{m,sat}\) in henries \((\mathrm{H})\), referred to the stator side. If you do not want to model saturation, set \(L_\mathrm{m,sat} = L_\mathrm{m,0}\).

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.

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 rotor 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 stator windings are aligned with the rotor windings at simulation start.

Initial stator currents

A two-element vector containing the initial stator currents \(i_\mathrm{s,a,0}\) and \(i_\mathrm{s,b,0}\) of phases a and b in amperes \((\mathrm{A})\).

Initial stator flux

A two-element vector containing the initial stator flux \(\Psi_\mathrm{s,d,0}\) and \(\Psi_\mathrm{s,q,0}\) in the stationary reference frame in \((\mathrm{Vs})\).

Probe Signals

Stator phase currents

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

Rotor phase currents

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

Stator flux (dq)

The stator flux linkages \(\Psi_\mathrm{s,d}\) and \(\Psi_\mathrm{s,q}\) 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 \((\mathrm{Vs})\).

Rotor flux (dq)

The rotor flux linkages \(\Psi_\mathrm{r,d}'\) and \(\Psi_\mathrm{r,q}'\) in the stationary reference frame in \((\mathrm{Vs})\), referred to the stator side.

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 \((\mathrm{Nm})\).

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, “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.