Squirrel Cage Induction Machine

Symbol

Description

A three-phase Squirrel Cage Induction Machine (IM). In this model, the parameter slip dependency can be modeled using a piecewise linear approximation. Often two-axis rectangular coordinate is employed to represent the machine dynamics. When the coordinate is fixed on the stator, it is called being in a stationary reference frame (αβ reference frame). Meanwhile, when the coordinate rotates in the synchronized speed with the excitation from the stator, it is called being in a synchronous reference frame (dq reference frame). All equations in this document are also expressed in the stationary reference frame.

Electrical model and equations

vd_vq_circuits

$v_{s\alpha} = R_s i_{s\alpha} + \frac{d }{dt} \phi_{s\alpha}$

$v_{s\beta} = R_s i_{s\beta} + \frac{d }{dt}\phi_{s\beta}$

$0 = R_r i_{r\alpha} + \frac{d }{dt} \phi_{r\alpha} - \omega \phi_{r\beta}$

$0 = R_r i_{r\beta} + \frac{d }{dt} \phi_{r\beta} - \omega \phi_{r\alpha}$

Where v_sαis α-axis stator voltage [V], v_sβ is β-axis stator voltage[V], i_sα is α-axis stator current[A], i_sβ is β-axis stator current[A], i_rα is α-axis rotor current[A], i_rβ is β-axis rotor current[A], R_s is Stator resistance[Ω], R_r is Rotor resistance[Ω], ω is Rotor speed in electrical angle[rad / sec], ϕ_sα is α-axis stator flux[Wb], ϕ_sβ is β-axis stator flux[Wb], ϕ_rα is α-axis rotor flux[Wb], ϕ_rβ is β-axis rotor flux[Wb],

The magnetic flux in induction motors can be represented using the inductance as follows. $$ \phi_{s\alpha} = L_s i_{s\alpha} + L_m i_{r\alpha} $$

$\phi_{s\beta} = L_s i_{s\beta} + L_m i_{r\beta}$

$\phi_{r\alpha} = L_r i_{r\alpha} + L_m i_{s\alpha}$

$\phi_{r\beta} = L_r i_{r\beta} + L_m i_{s\beta}$

Where L_s is Stator self-inductance[H], L_r Rotor self-inductance[H], L_m is Mutual inductance across from the stator to the rotor[H]

Electromechanical equations

Electro-magnetic torque:

$T_e = 1.5 * N_{ pp}*(i_{s\beta} * i_{r\alpha} - i_{s\alpha} * i_{r\beta} )$

Mechanical rotational speed $\Omega$ :

$J \frac{d\Omega}{ dt} = T_e - B \Omega$

Library

Electrical > Motors

Pins

Name	Description
Pin_A	Phase A (Electrical)
Pin_B	Phase B (Electrical)
Pin_C	Phase C (Electrical)
Pin_R	Rotor (Rotational Mechanical)
Pin_Angle	Rotor Angle in radians (Control)

Parameters

Name	Description
Rs	Stator Winding Resistance [Ohm]
Ls	Stator winding leakage inductance [H]
Rr	Rotor resistance [Ohm]
Lr	Rotor leakage inductance [H]
Lm	Magnetizing inductance [H]
J	Rotor Inertia [kg.m²]
B	Rotor Friction Coefficient [N.m/(rad/s)]
InitialCurrent	Initial Current Vector [IA IB IC]
NPP	Number of pole pairs