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ECE 8830 - Electric Drives

Topic 3 Induction Motor Modeling -

Steady State Spring 2004

Introduction

- Induction machines are the most widely used of

all electric motors. They offer the following

attractive features - Generally easy to build and cheaper than

corresponding dc or synchronous motors - Rugged and require little maintenance
- Offer reasonable asynchronous performance
- A manageable torque-speed curve
- Stable operation under load
- Generally satisfactory efficiency
- Range in size from few Watts to several MW

Introduction (contd)

- Some disadvantages of induction motors are
- Speeds not as easily controlled as dc motors
- Draw large starting currents, typically 6-8 x

their full load values - Operate with a poor lagging power factor when

lightly loaded

Introduction (contd)

- New designs for high performance induction

machines, such as in high speed motors for gas

compressors, will be required to have new

characteristics from existing machines, it is

important to have a good fundamental

understanding of these types of machines. - Goal To develop a simple model for the

induction machine that is useful for control and

simulation.

Structure of an Induction Machine

- Two types of induction machine
- Wound rotor or squirrel cage rotor

Rotating Magnetic Field and Slip

- We previously showed that a balanced set of

three-phase currents flowing in a set of

symmetrically placed, three-phase stator windings

produces a rotating mmf given by - eq. (6.1) Ong, eq. (2.9) Bose
- where ?ae is the electrical angle measured

from the a-phase axis and ?e is the angular speed

of the stator mmf in electrical radians/second.

Rotating Magnetic Field and Slip (contd)

Rotating Magnetic Field and Slip (contd)

- In mechanical radians/sec. the synchronous

speed is related to the electrical speed by - If the rotor is rotating at an angular speed

?rm the slip speed is simply equal to ?sm - ?rm.

The slip,s, is the normalized slip speed and is

given by

Torque Production

- The torque produced by an induction motor may

be derived and expressed by the following

equation (see ref. 1 in Bose) - where P of poles
- l axial length of motor
- r radius of motor
- Bp peak air-gap flux density
- Fp peak value of rotor mmf
- and

Per-Phase Equivalent Circuit Model

- A per-phase transformer-like equivalent

circuit is shown below

Per-Phase Equivalent Circuit Model (contd)

- Synchronously rotating air gap flux wave

generates a counter emf Vm. This in turn is

converted to a slip voltage in the rotor phase,

Vr nsVm, where nrotorstator turns ratio, and

snormalized slip. - Stator terminal voltage, Vs Vm VRs VLls
- where VRsvoltage drop across stator

resistance (Rs) and VLlsvoltage drop across

stator leakage inductance (Lls).

Per-Phase Equivalent Circuit Model (contd)

- Excitation current, I0 Ic Im
- where Ic is core loss current (Vm/Rm)
- and Im is magnetizing current (Vm/ )
- Rotor-induced voltage, Vr VRr VLl
- where VRr voltage drop across rotor resistance
- and VLl voltage drop across rotor leakage
- inductance
- The induced voltage in the rotor leads to a rotor

- current Ir at slip frequency ?sl.

Per-Phase Equivalent Circuit Model (contd)

- The stator current, IS I0 Ir
- where Ir is the rotor-reflected current

induced in the stator.

I0

Per-Phase Equivalent Circuit Model (contd)

Per-Phase Equivalent Circuit Model (contd)

Per-Phase Equivalent Circuit Model (contd)

- Torque expression can be written as
- where peak value of air gap flux
- linkage/pole
- and peak value of rotor current

Per-Phase Equivalent Circuit Model Power

Expressions

- Input Power where cos? is input PF
- Stator copper loss
- Rotor copper loss
- Core loss
- Power across air gap
- Output power
- Shaft Power where PFw is friction and

windage power loss

Per-Phase Equivalent Circuit Model Torque

Expression

- The torque can be expressed as
- where is the rotor
- mechanical speed (radians/sec.)

Per-Phase Equivalent Circuit Model Torque

Expression (contd)

- Using a little algebra (see Bose) it can
- be shown that the torque may be further
- expressed as
- where .
- This torque expression is similar to that for
- a dc motor, where Im magnetizing
- component of stator current and Ia
- armature component of stator current.

Simplified Per-Phase Equivalent Circuit

- A simplified circuit dropping Rm and shifting

Lm to the input is applicable to integral

horsepower machines. - Performance of this equivalent circuit is

typically within 5.

Simplified Per-Phase Equivalent Circuit Model

(contd)

- The current Ir in this circuit is given by
- The torque of the motor using this circuit
- is given by

Example of Calculating Efficiency of an Induction

Motor

- Example 5.1 Krishnan

Flowchart for Computing Steady State Performance

of Induction Motor

Ref R. Krishnan, Electric Motor Drives

Torque-Speed Curve of Induction Motor

- The torque-speed curve as a function of slip

can be calculated from the equation two slides

back.

Torque-Speed Curve of Induction Motor (contd)

- Three regions in torque-speed curve
- 1) Plugging (braking) region (1ltslt2)
- Rotor rotates opposite to direction of air

gap flux. Can happen, for example, if stator

supply phase sequence reversed while rotor is

moving. - 2) Motoring region (0ltslt1)
- Te0 at s0. As s increases (speed decreases),

Te increases until max. torque (breakdown torque)

is reached. Beyond this point, Te decreases with

increasing s.

Torque-Speed Curve of Induction Motor (contd)

- 3) Regenerating Region (slt0)
- Here the induction machine acts as a

generator. Rotor moves faster than air gap flux

resulting in negative slip.

Torque-Speed Curve of Induction Motor (contd)

Ref R. Krishnan, Electric Motor Drives

Performance Characteristics of Induction Motor

Ref R. Krishnan, Electric Motor Drives

Starting Torque of Induction Motor

- The starting torque of an induction motor is

given by substituting for s1 and is given by

Starting Torque of Induction Motor (contd)

- This torque can be enhanced for line start

motors (ones started directly with full line

voltage) by increasing the rotor resistance. This

can be achieved by connecting external resistors

in the case of slip ring rotors. However, with

squirrel cage rotors where the rotor is shorted,

deep bar or double-cage rotors can be used to

increase starting torque.

Characterizing Induction Motors

- One way to characterize an induction motor is

with the No-load/blocked rotor tests which yield

the per-phase equivalent circuit model shown

earlier (see figure below).

iar

ias

vas

M

Characterizing Induction Motors (contd)

- We can characterize an induction motor with

the variables Rs, Lls, M, Llr, Rr determined

through lab tests using balanced 3? excitation.

This circuit described the impedance perceived

per phase on a line-neutral connected machine.

Everything in the dashed box is a rotor quantity

that has been referred to the stator by the

ideal transformer in the machine model. From now

on, assume that Llr, Rr and iar are referred to

the stator.

Characterizing Induction Motors (contd)

- No-Load Test (s0)
- Equivalent circuit

Ref R. Krishnan, Electric Motor Drives

Characterizing Induction Motors (contd)

- No-Load Test (s0) yields
- In sinusoidal steady state, ignoring

resistances - But -ias (ibsics)
- ?
- From transformer model

Las

gt

Characterizing Induction Motors (contd)

- Blocked rotor test (s1) yields estimates of Lls

and Llr. Equivalent circuit at standstill is

shown below

Ref R. Krishnan, Electric Motor Drives

Characterizing Induction Motors (contd)

- Ohmmeter/Power loss tests give Rs
- and Rr.
- So, with Llr, Rr and all irs understood as
- referred rotor quantities, the stator-side
- tests identify all the model parameters for
- the induction motor.

Example of Determining Induction Motor Model

Parameters

- Example 5.2 Krishnan

NEMA Classification of Induction Motors

- The National Electrical Manufacturers

Association (NEMA) has classified induction

motors based on their torque-slip

characteristics. (see text for details)

Circuit Model of a Three-Phase Induction Machine

(State-Space Approach)

Voltage Equations

- Stator Voltage Equations

Voltage Equations (contd)

- Rotor Voltage Equations

Flux Linkage Equations

Model of Induction Motor

- To build up our simulation equation, we could

just differentiate each expression for ?, e.g. - But since Lsr depends on position,
- which will generally be a function of time,

the trig. terms will lead to a mess! - Parks transform to the rescue!

first row of matrix