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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:
- Magnetizing the machine through flux control.
- Torque control.
- 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
R_{s} : Stator resistance
R_{r} : Rotor resistance
L_{s} : Stator self-inductance
L_{r} : Rotor self-inductance
M_{sr} : Stator-Rotor mutual inductance
s : Dispersion coefficient,
T_{s} : Stator electrical time constant, T_{s} = L_{s} / R_{s}
T_{r} :Rotor electrical time constant, T_{r} = L_{r} / R_{r}
DC machine characteristics
P_{un} : Nominal power.
V_{an} : Armature nominal voltage.
V_{fn} : Field nominal voltage.
I_{fn} : Field nominal current.
N_{n} : Nominal speed in RPM.
Inverter characteristics
U_{in} : Nominal line to line output voltage.
U_{dcn} : DC nominal input voltage.
f_{s} : Switching frequency.
Other symbols
f : Stator voltages frequency
ω : Stator voltages angular frequency.
Ω : Rotor angular mechanical speed.
ω_{e} : Rotor angular electrical speed, ω_{e} = PΩ
T_{em} : Electromagnetic torque.
f_{v} : Viscous friction factor.
J : Total inertia (IM and mechanical load).
R_{L} : Load resistance connected to the DC generator.
T_{L} : Total load torque.
T_{L0} : Load torque due to the mechanical load only, proportional to the speed, T_{L0} = k_{L}.W
T_{f} : Viscous friction torque, T_{f} = f_{v}.W
_{m0} : Mechanical time constant at no load, _{m0} = J / f_{v}
_{mL} : Mechanical time constant at load, _{mL} = J / (f_{v} + k_{L})
: 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).
ω_{mr} : Angular speed of the rotating reference frame d-q,
i_{mr} : Rotor magnetizing current.
v_{sα}, v_{sβ}, i_{sα}, i_{sβ} : Stator voltages and currents expressed in the stationary reference frame.
v_{sd}, v_{sq}, i_{sd}, i_{sq} : Stator voltages and currents expressed in the rotating reference frame.
v_{sdc}, v_{sqc} : Decoupling stator voltages expressed in the rotating reference frame.
v_{sdref}, v_{sqref} : Reference stator voltages expressed in the rotating reference frame.
v_{sα}_{ref}, v_{sβ}_{ref} : Reference stator voltages expressed in the stationary reference frame.
t_{TCL} : Desired time constant of the torque closed loop.
t_{Tr }: 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 | 3-phase |
Voltage LL RMS | 460 V |
Maximum Current | 24 A |
Power | 13.5 kW |
Table 1: Three-Phase Two-Level Inverter Nameplate Ratings
Nominal Power | P_{un} | 16 HP |
Armature Nominal Voltage | V_{an} | 460 V |
Field Nominal Voltage | V_{fn} | 460 V |
Field Nominal Current | I_{fn} | 1.11 A |
Nominal Speed | N_{n} | 1746 rpm |
Table 2: DCM Nameplate Ratings
Note
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.
Parameters | Designation | Value | Unit |
---|---|---|---|
IM Stator resistance | R_{s} | 0.6837 | Ω |
IM Stator self-inductance | L_{s} | 0.1528 | H |
IM Rotor resistance referred to the stator | R_{r} | 0.451 | Ω |
IM Rotor self-inductance referred to the stator | L_{r} | 0.1528 | H |
IM Stator-rotor mutual inductance | M_{sr} | 0.1486 | H |
IM Dispersion coefficient | 0.0536 | ||
IM Pole pairs | P | 2 | |
IM base speed | N_{b} | 1800 | rpm |
IM Nominal torque | T_{n} | 54.7 | N.m |
IM Rotor nominal flux in dq axis | Φ_{rdn} | 1.22 | Wb |
IM Rotor nominal magnetizing current in dq axis | i_{mrn} | 8.21 | A |
IM Viscous friction factor | f_{v} | 0.0135 | kg.m^{2}.s^{-1} |
Total inertia (IM and DC machine) | J | 0.31 | kg.m^{2} |
IM Apparent nominal power | S_{IMn} | 12700 | VA |
IM Nominal Line-to-line voltage | U_{IMn} | 460 | V |
IM Nominal current | I_{IMn} | 16 | A |
IM Nominal Speed | N_{IMn} | 1746 | rpm |
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:
Where,
i_{mr ref} = i_{mr}* = Φ_{rd}* / M_{sr} (Φ_{rd}* = Φ_{rd ref }is the flux reference)
The flux response time t_{Φr} at 95 % of the final value, is given by:
t_{Φr} = 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 Φ_{rd ref} is computed by the following equation:
Where,
- Φ_{rd} n is the nominal rotor flux in the rotating reference frame d-q.
- Ω_{b} is the base speed.
- Ω_{M} is the maximum speed authorized for a limited duration as defined by the manufacturer with Ω_{b} < Ω_{M}
Torque Control
The torque control is performed using the following equations:
The open loop torque transfer function H_{TOL}(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 H_{TCL}(s), can be written as:
With T_{em}* = T_{emref} : 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 T_{r0}(t) is proportional to the speed, T_{r0}(t) = k_{L}.Ω(t), the open loop speed transfer function H_{ΩOL}(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 T_{L0} 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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