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Induction Machine

Purpose

Non-saturable induction machine with slip-ring rotor

Library

Electrical / Machines

Description

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This model of a slip-ring induction machine can only be used with the continuous state-space method. If you want to use the discrete state-space method or if you need to take saturation into account, please use the Induction Machine with Saturation.

The machine model is based on a stationary reference frame (Clarke transformation). A sophisticated implementation of the Clarke transformation facilitates the connection of external inductances in series with the stator windings. However, external inductors cannot be connected to the rotor windings due to the current sources in the model. In this case, external inductors must be included in the leakage inductance of the rotor.

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 windings is 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

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The rotor flux is computed as

Yr,d = L ′lri′r,d + Lm (is,d + i′r,d)

                (        )
Yr,q = L′lri′r,q + Lm is,q + i′r,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   1√3-    vr,bc

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

⌊    ⌋    ⌊           ⌋
  is,a     |  1    √0  |  [ i  ]
⌈ is,b ⌉  = ⌈ - 12   -32√- ⌉ ⋅  si,d
  is,c       - 12  - 23-     s,q

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

[     ]     [            (     ) ] [      ]
  v′r,d   =  2  cosθ  - cos(θ- 2π3)  ⋅  v′r,ab
  v′r,q      3   sinθ  - sin  θ- 2π3      v′r,bc

⌊     ⌋    ⌊                        ⌋
  i′r,a           c(osθ  )    (sin θ  )   [  ′  ]
⌈ i′r,b ⌉ =  ⌈ cos(θ+ 2π3)  sin(θ + 23π) ⌉⋅  ir,′d
  i′r,c        cos θ- 2π3   sin θ - 23π      ir,q

Electro-Mechanical System

Electromagnetic torque:

Te =  3 pLm (is,q i′  - is,di′ )
      2         r,d      r,q

Mechanical System

Mechanical rotor speed ωm  :

       1
ωm  =  J-(Te - Fωm - Tm)

ω  = pωm

Mechanical rotor angle θm  :

θm  = ωm

θ = pθ
      m

Parameters

Stator resistance
Stator winding resistance Rs   in ohms (_O_  ).
Stator leakage inductance
Stator leakage inductance Lls   in henries (H).
Rotor resistance
Rotor winding resistance R′
 r    in ohms (_O_  ), referred to the stator side.
Rotor leakage inductance
Rotor leakage inductance  ′
Llr   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 ωm,0   in s- 1  .
Initial rotor position
Initial mechanical rotor angle θm,0   in radians. If θm,0   is an integer multiple of 2π∕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 phases a and b in amperes (A).
Initial stator flux
A two-element vector containing the initial stator flux Y′
 s,d,0   and Y′
 s,q,0   in the stationary reference frame in Vs.

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 3 signals:

(1) Rotational speed
The rotational speed ωm   of the rotor in radians per second (s-1  ).
(2) Rotor position
The mechanical rotor angle θ
 m   in radians.
(3) Electrical torque
The electrical torque Te   of the machine in Nm.

Probe Signals

Stator phase currents
The three-phase stator winding currents is,a  , is,b    and is,c  , in A. Currents flowing into the machine are considered positive.
Rotor phase currents
The three-phase rotor winding currents i′r,a  , i′r,b    and i′s,c   in A, referred to the stator side. Currents flowing into the machine are considered positive.
Stator flux (dq)
The stator flux linkages Y
 s,d   and Y
 s,q   in the stationary reference frame in Vs:

Y   = L  i  + L  (i  + i′ )
  s,d    ls s,d   m   s,d   r,d

Ys,q = Llsis,q + Lm (is,q + i′ )
                       r,q

Magnetizing flux (dq)
The magnetizing flux linkages Ym,d   and Ym,q   in the stationary reference frame in Vs:

Ym,d = Lm (is,d + i′r,d)

          (       )
Ym,q = Lm  is,q + i′r,q

Rotor flux (dq)
The rotor flux linkages Y′
 r,d   and Y′
 r,q   in the stationary reference frame in Vs.
Rotational speed
The rotational speed ωm   of the rotor in radians per second ( -1
s  ).
Rotor position
The mechanical rotor angle θm   in radians.
Electrical torque
The electrical torque T
 e   of the machine in Nm.