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Squirrel-cage Induction Motor - Introduction

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This courseware allows the students to test virtually (using real-time simulation) the operation of a squirrel-cage induction motor, henceforth designated IM, when a speed control is required.
The mechanical load of the IM consists of a direct current machine, henceforth designated DCM, operating as a generator and feeding a variable load resistance (see figure 1).

Figure 1: Basic Configuration of the Simulated Circuit for the IM Drive

 Two main control strategies are implemented for the squirrel-cage induction motor: the well-known indirect field-oriented control (IFOC) and the V/F control.

Indirect Field Oriented Control (IFOC)

The IFOC method can be defined as follows:

  • Use of the Concordia transformation which transforms three-phase quantities (a, b, c) into two-phase quantities (, ) in a stationary reference frame.
  • Use of the Park transformation, which transforms the two-phase quantities (, ) in a rotating reference frame.
  • Write the equations of the IM in the rotating reference frame.
  • Choose the d-axis so that it is aligned with the rotor flux.
    This goal is achieved by ensuring that a set of equations is verified.
  • The speed control of the IM required the three following steps:
    1. Magnetizing the machine through flux control.
    2. Torque control.
    3. Speed control.

Practical Implementation of the IFOC for IM Drive

First, let define the list of symbols and parameters that are going to be used throughout this document.

IM parameters

Rs : Stator resistance

Rr : Rotor resistance

Ls : Stator self-inductance

Lr : Rotor self-inductance

Msr : Stator-Rotor mutual inductance

s : Dispersion coefficient,

Ts : Stator electrical time constant, Ts = Ls / Rs

Tr :Rotor electrical time constant, Tr = Lr / Rr

DC machine characteristics

Pun : Nominal power.

Van : Armature nominal voltage.

Vfn : Field nominal voltage.

Ifn : Field nominal current.

Nn : Nominal speed in RPM.

Inverter characteristics

Uin : Nominal line to line output voltage.

Udcn : DC nominal input voltage.

fs : Switching frequency.

Other symbols

f : Stator voltages frequency

ω : Stator voltages angular frequency.

Ω : Rotor angular mechanical speed.

ωe : Rotor angular electrical speed, ωe = PΩ

Tem : Electromagnetic torque.

fv : Viscous friction factor.

J : Total inertia (IM and mechanical load).

RL : Load resistance connected to the DC generator.

TL : Total load torque.

TL0 : Load torque due to the mechanical load only, proportional to the speed, TL0 = kL.W

Tf : Viscous friction torque, Tf = fv.W

m0 : Mechanical time constant at no load, m0 = J / fv

mL : Mechanical time constant at load, mL = J / (fv + kL)

 : Angle between the d-axis of the rotating reference frame and the a-axis of the stationary reference frame (e.g., Phase “a” of the stator).

ωmr : Angular speed of the rotating reference frame d-q,

imr : Rotor magnetizing current.

v, v, i, i : Stator voltages and currents expressed in the stationary reference frame.

vsd, vsq, isd, isq : Stator voltages and currents expressed in the rotating reference frame.

vsdc, vsqc : Decoupling stator voltages expressed in the rotating reference frame.

vsdref, vsqref : Reference stator voltages expressed in the rotating reference frame.

vref, vref : Reference stator voltages expressed in the stationary reference frame.

tTCL : Desired time constant of the torque closed loop.

tTr : Torque response time.

ΦCL : Time constant of the regulated flux.

The values of the parameters previously described are summarized in tables 1, 2, and 3.
It should be noted that a three-phase, two-level inverter has been considered for the IM drive.

DC Input

Voltage DC

1000 V

Maximum Current

13.5 A

AC Output


Voltage LL RMS

460 V

Maximum Current

24 A


13.5 kW

Table 1: Three-Phase Two-Level Inverter Nameplate Ratings


Nominal Power


16 HP

Armature Nominal Voltage


460 V

Field Nominal Voltage


460 V

Field Nominal Current


1.11 A

Nominal Speed


1746 rpm

Table 2: DCM Nameplate Ratings


The values of the DC machine nameplate ratings in table 2 are given for motor operation.
When the DC machine is operating as a generation, some of these values may change.






IM Stator resistance




IM Stator self-inductance




IM Rotor resistance referred to the stator




IM Rotor self-inductance referred to the stator




IM Stator-rotor mutual inductance




IM Dispersion coefficient


IM Pole pairs



IM base speed




IM Nominal torque




IM Rotor nominal flux in dq axis




IM Rotor nominal magnetizing current in dq axis




IM Viscous friction factor




Total inertia (IM and DC machine)




IM Apparent nominal power




IM Nominal Line-to-line voltage




IM Nominal current




IM Nominal Speed




Table 3: IM Parameters Values and Nameplate Ratings

The general IFOC scheme used to implement the speed control of the IM is shown in figure 2.

Figure 2: General Scheme of the IFOC

 In what follows, the three main steps of the IFOC are discussed.

Flux Control

The flux control is performed using the following equations:


imr ref = imr* = Φrd* / Msr             (Φrd* = Φrd ref is the flux reference)

The flux response time tΦr at 95 % of the final value, is given by:

tΦr = 3.ΦCL     

The block diagram of the flux loop is presented in figure 3.

Figure 3: Flux Control Loop

The curve of the flux reference versus speed is given in figure 4.

Figure 4: Flux Reference Computation

The flux reference Φrd ref is computed by the following equation:


  • Φrd n is the nominal rotor flux in the rotating reference frame d-q.
  • Ωb is the base speed.
  • ΩM is the maximum speed authorized for a limited duration as defined by the manufacturer with Ωb < ΩM

Torque Control

The torque control is performed using the following equations:

The open loop torque transfer function HTOL(s) is given by the following equation,

To obtain a fast closed loop response for the torque, we usually seek a second order response.
Since the torque is an electrical quantity while the speed is a mechanical quantity, we will assume, for the sake of simplification of the speed control loop that the transfer function of the torque closed loop HTCL(s), can be written as:

With Tem* = Temref : Electromagnetic torque reference.

The block diagram of the torque loop is shown in figure 5.

Figure 5: Torque Control Loop

Speed Control

The speed control is performed using the equations below:

Assuming that the load torque Tr0(t) is proportional to the speed, Tr0(t) = kL.Ω(t), the open loop speed transfer function HΩOL(s) is given by the following equation:

The mechanical time constant at load mL is given as:

To make the speed control independent of the load torque TL0 which will then be considered as a disturbance, we calculate its parameters using the following transfer function:

Where m0 is the mechanical time constant at no load:

The simplified speed control loop scheme is given in figure 6.

Figure 6: Simplified Speed Control Loop

V/F Control

V/F is abbreviated from voltage/frequency.
V/F control is an induction motor control method which ensures the output voltage proportional with the frequency, so it maintains a constant motor flux, preventing weak magnetic and magnetic saturation phenomenon from happening.

Control principle: V/F control principle is to produce a circuit called voltage-controller oscillator with oscillator frequency.
It is a voltage-dependent capacitance, when subjected to a change in voltage, its capacity will change, and then the change in capacity will cause changes in the oscillation frequency, resulting in variable frequency.
This controlled frequency is used to control the frequency of the output voltage, in order to achieve speed changes of the controlled electric motors.

Applications: Asynchronous electric motor torque is a result of the interaction of flux and rotor flux.
At a rated frequency, if the voltage is set to a certain value and only reduce the frequency, then there will be large magnetic flux and magnetic circuit saturation (severely, it will burn motor).
Therefore, the frequency and voltage must be changed proportionally.
When changing the frequency, we should control the output voltage of AC drive, in order to keep constant flux and avoid weak magnetic and magnetic saturation phenomenon.
This control method is commonly applied to fans and pumps.

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