EE 410/510: Electromechanical Systems Chapter 5 Chapter 5. Induction Machines Fundamental Analysis ayssand dcontrol o of Induction Motors Two phase induction motors Lagrange Eqns. (optional) 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 1 All figures taken from primary textbook unless otherwise cited.
AC Motors: Induction Machines 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 For a squirrel cage motor: Note: Each of the two sets of inductive circuits are orthogonal. This means that the mutual inductance between l ss and l rr is zero.
AC Motors: Induction Machines circuit resistance: Magnetic flux through each inductor Mutual inductance between coupled circuits Mutual inductance of circuits that are orthogonal in space
AC Motors: Induction Machines Cyclic nature of the rotating ti system in r on inductance Magnetic flux through each inductor can then be written as
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 An alternate derivation using the relations previously described is provided below
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 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 These equations can be rewritten in terms of differentials of current vs. time
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, L sr, provides electromagnetic torque. recalling
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 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
Torque Speed Characteristics and Control of Induction Motors As we have observed, the electromagnetic torque generated by induction motors is a function of both the stator and rotor currents, as well as rotor displacement 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 One defines dfi the synchronous angular velocity of induction machines as Note that the electrical angular velocity of an induction machine will always be less than or equal to the synchronous angular velocity. 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 One can generate a steady state response in terms of electrical angular velocity vs. the electromagnetic torque generated 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 For steady state operation (and neglecting friction) Acceleration of the motor to steady state from zero requires T estart > T L0 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
Torque Speed Characteristics and Control of Induction Motors Most industrial motors are either type A or type B which have a normal starting torque and a low slip. 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 20 times the slip of types A, B, and C. Two additional motor classes, E, and F, have low starting torque, but high leakage inductances G = generator leading to high h slip values M = motor B = breaking
2 vs. 3 Phase Induction Motor Torque 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 The equation for a Three phase motor is: Effective operation can be achieved using
Motor Control and Operation Voltage control is achieved by changing the magnitude of the applied voltages circuits in the 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 One can further minimize losses by regulating the applied voltage as the frequency is 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 Control of both the voltage and the frequency provide a multi variable control scheme in which to further improve performance optimization
Example 1: Torque Speed Characteristics Calculate the torque speed characteristic for a 4 pole induction motor. r s = 24.5 ohm X s = 10 ohm X r = 40 ohm X m = 25 ohm u max = 110 V f max f 1 f 2 f 3 = 60 Hz = 20 Hz = 40 Hz = 60 Hz f = 60 Hz f = 40 Hz X = magnetizing reactance f = 20 Hz
Example 2: Analysis the Performance of a Two Phase Induction Motor 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. By adding the concept of torque speed characteristic performance, describe the following A and D class motors Using: We find that: Zero load and applied load conditions are examined using T L =0, and T L = 5 N m. Transient dynamics are developed using the following Simulink program based on our previously developed dseries of ODEs used to describe the system.
Motor Control and Operation Control of both the voltage and the frequency provide a multi variable control scheme in which to further improve performance optimization
3 Phase Induction Motor Equations
3 Phase Induction Motor Example
20 30 minutes SIMULINK Discussions
Pulse Width Modulation Pulse width modulation is the primary means for induction motor control. 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 required for operation Positive or negative voltage outputs from the switch inverter allow for direct phase matched control of the feedback current in the system, thereby reducing the time required to achieve the ideal operating condition 3-phase hard switch inverter
Hard vs. Soft Switch Inverter 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 Square wave voltage source inverters are often used Known as six step inverters Each switch is closed for 180 o of the pulse and closed for the other half Each set of switches is offset by 60 o to provide three phase operation
Voltage Control
Thyristor Inversion: Current Control Current driven inverter with rectified voltage source that is then polled to both 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 the phase of the system to control the slip Result: the DC linked inductor smoothens the current out in the system