PLECS 1.3 Online Help

Induction Machine

Purpose

Asynchronous machine.

Library

Machines

Description

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The library contains two different models for the Induction Machines. Mathematically, both models are identical. They differ only in the way they can be used.

The standard model of the Induction Machine can only be used with the continuous state-space method. The model is based on a three-phase system instead of the usual reference frame. This facilitates the connection of external inductances in series with the stator windings. However, external inductors cannot be connected to the rotor due to the current sources in the model. In this case, external inductors must be included in the leakage inductance of the rotor.

The discretizable model can be used with both the continuous and the discrete state-space method. This model is based on machine equations in the stationary reference frame (Clarke transformation). To facilitate the discretization of the model the Integrator block is used which allows non-linear feedback in the circuit. All external inductors must be included in the leakage inductances.

The machines operates as a motor or generator; the sign of the mechanical torque determines the mode of operation (positive for motoring, negative for generating). 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: Standard Model

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2     3        2                       3 2              3
  vx;a      2                + 2-3        23-      2v0r;ab + v0r;bc
4 vx;b 5 =  - cos4       2-3             + 23- 5  4   v0r;bc    vr0;ab  5 +
  vx;c      9        + 2-3       2-3                 2v0r;bc   v0r;ab

       2                    3
         is;c   is;b  i0sr;c   i0sr;b
+  1p--!4 is;a   is;c  i0sr;a   i0sr;c 5     0 Lm
    3    is;b   is;a  i0sr;b   i0sr;a     Llr + Lm

                                      2  0s  3
 i0        2                   2--     + 2--      ir;a
ir;0a   =  3 cos   + 2--       3        32--    4 i0sr;b 5
 r;b                3             3      i0sr;c

Stator and rotor currents transformed into the stationary reference frame:

is;d = is;a

      1
is;q = p3 (is;a + 2is;b)

i0 = i0  cos      p1-  i0 + 2 i0    sin
 r;d   r;a         3  r;a    r;b

                1
i0r;q = i0r;a sin  + p-- i0r;a + 2i0r;b cos
                3

In the preceding equations, is the electrical rotor position.

Electrical System: Discretizable Model

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The rotor flux is computed as

'r;d = L0lri0r;d + Lm is;d + i0r;d


's;q = L0lri0r;q + Lm is;q + i0r;q

The three-phase voltages vs;ab and vs;bc at the stator terminals are transformed into dq quantities:


vs;d   =   23  13       vs;ab
vs;q       0  1p3-    vr;bc

Likewise, the stator currents in the stationary reference frame are transformed back into three-phase currents:

2     3   2           3
  is;a     6   1   p0  7     i
4 is;b 5 = 4    12   -32p- 5     si;d
  is;c          12     -32-     s;q

Similar equations apply to the voltages and currents at the rotor terminals with being the electrical rotor position:


v0r;d  =  2   cos       cos       2-3        v0r;ab
v0r;q     3   sin       sin      2-3       vr0;bc

2     3    2                        3
  i0r;a           cos            sin               0
4 i0r;b 5 =  4 cos   +  2-3    sin     + 23-   5    ir0;d
  i0r;c        cos      2-3   sin       23-      ir;q

Electro-Mechanical System

Electromagnetic torque:

Te =  3p Lm   is;q i0     is;d i0
      2         r;d      r;q

Mechanical System

Mechanical rotor speed !m:

       1
!m  = J- (Te    F!m    Tm)

!  = p!m

Mechanical rotor angle m:

m  = !m

= p
   m

Parameters and Dialog Box

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Stator resistance
Stator winding resistance Rs in ohms (_O_).
Stator leakage inductance
Stator leakage inductance Lls in henries (H).
Rotor resistance
Rotor winding resistance Rr0 in ohms (_O_), referred to the stator side.
Rotor leakage inductance
Rotor leakage inductance Llr0 in henries (H), referred to the stator side.
Magnetizing inductance
Magnetizing inductance Lm in henries (H), 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 rotor speed !m;0 in s1.
Initial rotor position
Initial mechanical rotor angle m;0 in radians. If m;0 is an integer multiple of 2=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 is;a;0 and is;b;0 of phase a and b in amperes (A).
Initial rotor currents
A two-element vector containing the initial currents ir;a;00 and ir;b;00 in phase a and b of the rotor windings in amperes (A), referred to the stator side.

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 5 signals:

(1) Rotor speed
The rotational speed !m of the rotor in radians per second (s1).
(2) Rotor position
The mechanical rotor angle m in radians.
(3) Electrical torque
The electrical torque Te of the machine in Nm.
(4-5) Stator flux
The stator flux linkages 's;d and 's;q in Vs:


's;d = Llsis;d + Lm is;d +i0r;d

                         0
's;q = Llsis;q + Lm is;q + ir;q