# Squirrel Cage Induction Machine

## Description

Squirrel Cage Induction Machine

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

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]