Download EE 410/510: Electromechanical Systems Chapter 5

EE 410/510: Electromechanical Systems
Chapter 5
• Chapter 5. Induction Machines
•
•
Fundamental Analysis and Control of Induction u da e ta
a ys s a d Co t o o duct o
Motors
• Two phase induction motors
• Lagrange Eqns. (optional)
Torque speed characteristics and control
• Torque speed characteristics and control
• Three phase induction motors in machine variables
Simulation and Analysis of Induction Motors in MATLAB
Note: We will be skipping multiple sections of this chapter in attempt to provide a clear introduction to the material and allow us to move onto other equally important topics
5/21/2010
All figures taken from primary textbook unless otherwise cited.
1
AC Motors: Induction Machines
AC Motors: Induction Machines
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Three main torque and energy conservation mechanisms for electromagnetic motion devices
– Induction: Electromagnetic torque is the result of time varying electromagnetic fields present due to time varying voltage or motion or the rotor w.r.t. the stator
– Syncrhonous: Torque results because of the interaction of a time varying field generated in the stator windings and a stationary field established by the windings or magnets in the motor
– Variable reluctance: Torque produced to minimize the reluctance of the electromagnetic system. Thus the torque is created in attempt to align the minimum reluctance path of the rotor with the time varying rotating air gap.
Two Phase Induction Motor
Two Phase Induction Motor
For a squirrel – cage motor:
Note: Each of the two sets of inductive circuits
are orthogonal. This means that the mutual
inductance between lss and lrr is zero.
AC Motors: Induction Machines
AC Motors: Induction Machines
circuit resistance:
Magnetic
M
i flflux
through each
inductor
Mutual inductance between coupled circuits
Mutual inductance of circuits that are orthogonal in space
AC Motors: Induction Machines
AC Motors: Induction Machines
C li nature
Cyclic
t
off th
the rotating
t ti system
t
in
i r on inductance
i d t
Magnetic flux through each inductor can then be written as
AC Motors: Induction Machines
AC Motors: Induction Machines
One can then write the equation for magnetic flux in matrix form:
Where each of the inductance terms is composed of a 2x2 matrix:
N is the number of turns in the inductor, and R is the reluctance
AC Motors: Induction Machines
AC Motors: Induction Machines
AC Motors: Induction Machines
AC Motors: Induction Machines
AC Motors: Induction Machines
AC Motors: Induction Machines
AC Motors: Induction Machines
AC Motors: Induction Machines
An alternate derivation using the relations previously described is provided below
AC Motors: Induction Machines
AC Motors: Induction Machines
These equations can be rewritten in terms of differentials of current vs. time
AC Motors: Induction Machines
AC Motors: Induction Machines
AC Motors: Induction Machines
AC Motors: Induction Machines
Now that we have differential equations for current, we can write the mech. ODE
The mechanical rotational velocity and rotation angle is
equal to two times electrical angular component divided by
the number of magnetic poles in the system
AC Motors: Induction Machines
AC Motors: Induction Machines
These equations can be rewritten in terms of differentials of current vs. time
AC Motors: Induction Machines
AC Motors: Induction Machines
Next we need to develop the equation for the electromagnetic torque in the system
where
The self inductance terms as well as the leakage inductances are not functions of angular
displacement. Thus only the mutual inductance term, Lsr , provides electromagnetic
torque.
recalling
AC Motors: Induction Machines
AC Motors: Induction Machines
Thus one can now write the mechanical equations of motion for a two phase AC motor
Complete set of ODE’s governing a two phase motor
AC Motors: Induction Machines
AC Motors: Induction Machines
Governing ODEs written in matrix form. Note that these are HIGHLY nonlinear
Remember that for squirrel cage
motors (very common devices) that
2 Phase Induction Motor
2 Phase Induction Motor
Torque Speed Characteristics and Control of Induction Motors
f d i
•
As we have observed, the electromagnetic torque generated by induction motors is a f
function of both the stator and rotor currents, as well as rotor displacement
i
fb h h
d
ll
di l
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Control of induction motors is achieved by changing the frequency and magnitude of the Control
of induction motors is achieved by changing the frequency and magnitude of the
voltages supplied to the phase windings.
Remember to use the voltage rating of the stator windings as the maximum allowable applied voltage, or risk device failure due to resistive heating of the stator windings
O d fi
One defines the synchronous angular velocity of induction machines as
th
h
l
l it f i d ti
hi
•
•
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Note that the electrical angular velocity of an induction machine will always be less than or equal to the synchronous angular velocity. q
y
g
y
–
Equal values for electronic and synchronous angular velocities are only achieved under zero load and zero friction conditions
Torque Speed Characteristics and Control of Induction Motors
f d i
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One can generate a steady state response in terms of electrical angular velocity vs. the electromagnetic torque generated l
i
d
Using this approach, industrial standards have been developed to classify induction machines into four distinct classes: A,B,C,D
Each class is defined by its “slip” which provides an efficiency ratio for the electrical to y
p
p
y
mechanical angular velocity
Torque Speed Characteristics and Control of Induction Motors
f d i
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For steady state operation (and neglecting friction)
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Acceleration of the motor to steady state from zero requires Testart > TL0
The motor then accelerates until rc is reached at the maximum electromechanical torque
The torque is then decreased back to TL as the speed of the motor increases to r
Note that e requires the torque on the system to go to zero implying no load or friction forces and that this value is slightly higher than r
forces, and that this value is slightly higher than 
Torque Speed Characteristics and Control of Induction Motors
f d i
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Most industrial motors are either type A or type B which have a normal starting torque and a l
low slip.
li
Type C motors have two rotors and thus require higher starting torques. Slip is generally greater in this class as well.
Type D motors have a high rotor resistance and approximately 10 –
yp
g
pp
y
20 times the slip of types p
yp
A, B, and C.
Two additional motor classes, E, and F, have low starting torque, but high leakage inductances
G = generator
l di t hi h li
leading to high slip values
l
M = motor
t
B = breaking
2 vs. 3 Phase Induction Motor Torque
2 vs. 3 Phase Induction Motor Torque
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We have derived the electromagnetic torque for a two‐phase motor as:
•
One can guarantee balanced operation of two‐phase induction motors using either:
or
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The equation for a Three‐phase motor is:
q
p
•
Effective operation can be achieved using
Motor Control and Operation
Motor Control and Operation
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Voltage control is achieved by changing the magnitude of the applied voltages circuits in the stator.
stator
However voltage control reduces the effective start torque of the system and prevents adequate control of type A, B, and C motors
Thus frequency control is more widely used for standard motor operation. In frequency control, the voltage is stepped between zero and the phase voltage desired. The frequency (and effective duty cycle) of the system is controlled using the concept of slip and synchronous angular velocity, where the control frequency is given by =2f.
Motor Control and Operation
Motor Control and Operation
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One can further minimize losses by regulating the applied voltage as the frequency is changed.
changed
As one can see in the figures below, the voltage is decreased linearly while reducing the frequency
A constant volts per hertz control can is achieved by maintaining the following experimental relationship
Or one can vary the system performance slightly by relating Motor Control and Operation
Motor Control and Operation
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Control of both the voltage and the frequency provide a multi‐variable control scheme in which which
to further improve performance optimization
Example 1: Torque Speed Characteristics
Example 1: Torque Speed Characteristics
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Calculate the torque‐speed characteristic for a 4 pole induction motor.
rs = 24.5
24 5 ohm
h
Xs = 10 ohm
X’r = 40 ohm
Xm = 25 ohm
umax = 110 V
fmax
f1
f2
f3
= 60 Hz
= 20 Hz
= 40 Hz
= 60 Hz
X = magnetizing reactance
f = 60 Hz
f = 40 Hz
f = 20 Hz
Example 2: Analysis the Performance of a Two‐Phase Induction Motor
h
d i
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Assume a motor operated at 115 Volts, and 60Hz has a 4 pole design. Use the differential equations previously derived to describe the dynamics of the motor
equations previously derived to describe the dynamics of the motor. By adding the concept of torque‐speed characteristic performance, describe the following A and D class motors
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Using:
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We find that: •
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Zero load and applied load conditions are examined using TL =0, and TL = 5 N‐m.
Transient dynamics are developed using the following Simulink program based on our previously developed series of ODEs used to describe the system.
i l d l
d i
f ODE
d d
ib h
Motor Control and Operation
Motor Control and Operation
•
Control of both the voltage and the frequency provide a multi‐variable control scheme in which which
to further improve performance optimization
3 Phase Induction Motor Equations
3 Phase Induction Motor Equations
3 Phase Induction Motor Example
3 Phase Induction Motor Example
SIMULINK Discussions
SIMULINK Discussions
• 20
20‐30
30 minutes
minutes
Pulse Width Modulation
Pulse Width Modulation
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Pulse width modulation is the primary means for induction motor control.
Uses an inverter to supply fixed step voltages based
Uses an inverter to supply fixed step voltages based over the current motor phases
The additional applied voltage sums with the individual 3‐phase voltages to provide discrete Fourier transforms that provide inductor with the phase
transforms that provide inductor with the phase required for operation
Positive or negative voltage outputs from the switch inverter allow for direct phase matched control of the f db k
feedback current in the system, thereby reducing the i h
h b
d i
h
time required to achieve the ideal operating condition
3-phase hard switch inverter
Hard vs. Soft Switch Inverter
Hard vs. Soft Switch Inverter
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Hard switching provides digital type pulsing
li
Soft witching using a capacitivly coupled diode yields rounds the top of the pulse to provide a smoother transfer function.
3-phase soft switch inverter
Six Step Inverters
Six Step Inverters
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Square wave voltage source i
inverters are often used
f
d
Known as six step inverters
Each switch is closed for 180o of the pulse and closed for the other p
half
Each set of switches is offset by 60o to provide three phase operation
Voltage Control
Thyristor Inversion: Current Control
Thyristor Inversion: Current Control
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Current driven inverter with rectified voltage source that is then polled to both l
h i h
ll d b h
positive and negative terms to provide three current values: ‐i, 0, I
Each value is 60 degrees out of phase providing rotation dictating the rotation of the motor
Application of current control from an internal voltage source helps modulate g
p
the phase of the system to control the slip
Result: the DC linked inductor smoothens the current out in the system
the current out in the system