Induction MotorsEquations, Performance, Electrical Equivalent Circuits

Induction Motor by Bullet Points• Stator generates rotating, sinusoidal B-Field: • This field induces current in the rotor cage loops at • The stator B-Field at each rotor wire is such that • Torque pushes in direction of field rotation! (That’s it!!)• Rotor currents generate triangular B-field rotating in the air gap at slip

speed relative to rotor, so at line rate in reference frame!• Rotor field reduces field in stator and line current increases to maintain

the stator winding voltage and the gap magnetic field• Increased line current supplies the mechanical energy and the joule

heating of the rotor.

302 cos( )B B t r

ur $

S L Rf f f

Ri B ur ur $

0 60 120 180 240 300 360-0.2

0

0.2

0.4

0.6

0.8

1

Relative Torque vs Slip Angle

Single Rotor Coil - Resistive Current (Net Power)Reactive Torque (No Net Power)Total Torque - Single TurnTwo Loops at Right Angle

Slip Angle (deg.)

Rela

tive

Torq

ue

0 60 120 180 240 300 360-3

0

3

6

9

12

15

18

21

24

B-Field of Rotor vs. Rotor Angle as Function of Slip Cycle Angle

0 deg.45 deg.90 deg.135 deg.180 deg.225 deg.270 deg.315 deg.

Angle around rotor from one loop (deg.)

Rela

tive

B-Fi

eld

Stre

ngth

(a.u

.) O

ffse

t by

3 fo

r Ea

ch S

lip A

ngle

0 60 120 180 240 300 360-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Shape and Rotation of the Rotor Generated B-Field in the Gap

0 deg.

180 deg.

Angle arount the rotor relative to one turn of the two rotor turns (deg.)

Rela

tive

Mag

netic

Fie

ld S

tren

gth

(a.u

.)

0 60 120 180 240 300 360-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Shape and Rotation of the Rotor Generated B-Field in the Gap

0 deg.45 deg.90 deg.180 deg.

Angle arount the rotor relative to one turn of the two rotor turns (deg.)

Rela

tive

Mag

netic

Fie

ld S

tren

gth

(a.u

.)

0 60 120 180 240 300 360-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Rotor Induced Triangle Wave in Stator Current Showing Waveshape withHarmonic Multiples of 3 Removed and Compared to Sinusoid

Triangle WaveFourier to 27thNo multiples of 3Single Sine

Cycle Angle (deg.)

Rela

tive

Curr

ent (

a.u.

)

Formal Transformer Analogy

• Mutual inductance stator to rotor is time dependent

• The A, B, C voltages are the line voltages • The rotor voltages are • Given rotor frequency, calculate currents and power, subtract rotor and

winding heat to get mechanical power.• Ugh!

2 23 3

4 43 3

2 43 31

2

cos( ) sin( )

cos( ) sin( )

cos( ) sin( )

cos( ) cos( ) cos( ) 00

sin( ) sin(0

A P M M MR R MR R

B M P M MR R MR R

M M P MR R MR RC

MR R MR R MR R RR

MR R MR RR

v L L L L t L t

v L L L L t L td

L L L L t L tvdtL t L t L t Lv

L t Lv

1 1

2 43 3 2 2

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

) sin( ) 0 0 0 0 0

A AW

B BW

C CW

RR R

MR R R RR R

i iR

i iR

i iR

Ri i

t L t L Ri i

R R Rv R i

Simple Per-Phase Transformer Model

• Know that power flow is constant at constant speed (No torque variation!)• Per-phase model with constant impedance that is a function of rotor

speed• Use basic single-phase transformer model with turns ratio and

secondary impedance dependent on rotor speed• Stator field is zero-slip model

Electrical Equivalent Circuit of Stator Alone• Applies when rotor is turning at zero slip• Derive from locked rotor and zero slip conditions• Accounts for wire loss and stator core loss• Leakage inductance usually larger than for a simple transformer because of air gap and slot shape

Electrical Equivalent Circuit with Ideal Transformer

• S is the “slip” or SLIP

NoLoad

fS

f

Electrical Equivalent Circuit Referred to the Stator

• Basis for calculating efficiency, start inrush, etc.• Mechanical energy is loss in ; all else is heat(1 ) RS R

S

Things Left Out!

• Inrush current• No-load mechanical drag from cooling, bearing friction, etc.• Design tradeoffs with cost• Still to go: single phase operation• Government efficiency regulation• Recently – March 2015 – applies to motors to ¼ HP