Electromechanical Systems

Asinchronous (induction) machines

• Types of machines with alternating current

• Types of induction machines with alternating current

• Components of asinchronous (induction) machines, (squirel cage and slip-ring induction machines)

• How does it works!

• Mathematical model

• Equivalent circuit

• Vector (phasor diagram)

Literature

1. R. Wolf: Osnove električnih strojeva, Školska knjiga, Zagreb, 1991. (72-95, 107-117, dijelovi 181-220), in Croatian

2. B. Jurković: Elektromotorni pogoni, Školska knjiga, Zagreb, 1985. (Statička stanja elektromotornih pogona s asinkronim motorima, str.49-62), in Croatian

3. D. Ban: Mirna, pulzirajuća i okretna magnetska polja, predavanja (pogledati dodatnu literaturu na web stranicama), in Croatian

Electrical machines- types

Stator with 3 phase windingRotor, squirel cage or slip-ring type

Stator with electromagnet or permanent magnet

Rotor with winding, (armature winding)

Stator with winding

Rotor with permanent magnets

Stator with winding on the pole

Iron rotor; different reluctance in different axces !

Asynchronous machine

Synchronous machine

DC current machine

Reluctant machine

ASINCHRONOUS (INDUCTION) MACHINES

Induction machine (IM)

Stator with three symetrical (balanced) distributed phases , a, b. c (windings)

air gap

Rotor winding

Stator windings

Fig.1.Cross section of IM a), Spatial stator winding distribution, b)

ASINCHRONOUS MACHINES – industrial construction

Fig.2. Two types of induction motors – industrial products

Squirel cage Induction machine (motor), IM

- squirrel cage construction, the rotor winding consists of a number of rotor bars, short-cut by rings from both rotor side, see figures below

ASINCHRONOUS induction machines

barsbars

ring

ring

ring

Fig. 3. Squirrel cage rotor of induction motor, rings and bars a), squirrel cage rotor industrial product b).

ring

a)

b)

Slip-ring asynchronous (induction, IM) machine

stator is identical as squirrel cage induction motor

rotor has clasical winding, not a bars

usualy 3 windings (phases) on the rotor

rotor winding ends connected to the stationary rings, see figure below

ASINCHRONOUS induction machines

rings

resistors

Fig. 4. Stator and rotor connections of a slip-ring a), squirrel cage rotor industrial product b).

StatorSliced iron, slices electrically isolated from conductors (windings) placed in slots. There are 3 isolated balanced phase (windings), spaced with 120° (for 2-pole machine). 3-phase symmetrical stators winding is supplied by 3-phase symmetrical voltage supply 120°

RotorSliced iron, slices electrically isolated from rotor conductors (windings), placed in rotor. Rotor winding is usually 3-phase, in “star” connection. The ends of 3-phase winding are short connected altogether from one side in one point. Three others ends of windings are usually connected , to three slip rings, see Fig. 4. Those rings are connected then on stator connection box. For squirrel cage type rotor, conductors are made from cooper (Cu) or aluminium (Al).

Air gapIt must be as small as possible, taking into account bearings specifications, as well as a mechanical stress. Smaller air gap resulting in small magnetizing current needed for magnetic field. That field is important for effective electromechanical conversion.

Physical concept of IM

• Three phase (3f) IM motor supplied from stator side by symmetrical 3f voltage supply, results with SYMMETRICAL ROTATING FIELD. This

field rotate with synchronous speed s (1)

• Rotational field “is cutting” rotor conductors by relative speed s-

(slip, (2), inducing in conductors (windings) voltage E2=s·E20 , (3)

• In short connected rotor winding (squirrel cage rotor) induced voltage (3) will generate current, which will together with rotational field produce tangentional force on the rotor, ie. torque.

• Developed torque will accelerate rotor, and after reaching desired speed, (steady state), rotor speed will be close to the synchronous speed, (1)

p→number of pole pairs (see explanation at the end)

s

ss

slip

%100%

s

ss

slip (%)(2)(1)

p

fs

2 f

pn ss

60

260

Synchronous speed

Rotor voltage – dependence of slip

• When rotor is blocked (s=1, speed=0), rotational field induce in rotor winding voltage E20 , see Fig.5.

• When rotor start to move, relative speed is changing, as well as relative speed between rotational (stator) field against rotor, and voltage E2 is changing according (3)

• When the relative speed is zero, ie. s=0, there is no voltage in rotor winding, no current, nor force, no torque!! It means that motor cannot work when s=0. Conclusion is that motor can work only when different speed between rotor and rotational speed exist!!! This phenomena define term ASINCHRONOUS MACHINE.

202 EsE (3)

Fig.5. Rotor voltage vs rotor speed

• Rotor voltage and current frequencies are depending of relative speed between rotor and rotational (stator) field. i.e. slip. Those variables have frequency determined by relative speed between rotor and rotational

(stator) field.

Rotor current frequency vs slip

n

1

0 ns

f2f1

12 60fs

nnpf s

s

ss

%100%

s

ss

p

fs

2 f

pn ss

60

260

Reminder !!!!

Rotor speed– vs. slip

rotor rotates with synchronous speed s = 0rotor blocked , zero speed s=1rotor rotates faster than rotational speed s < 0rotor rotates opposite than rotational field speed s > 1

-111 60

1 o/min, rpm,min60

sn s fn s

p

The sam units are used for the synchronous speed ns

Number of pole pairs- Explanation

• The term “1 pair poles” defines the region in the stator of machine where three windings (phases) are simetrically spaced inside stator slots. It is said that the angle between axces of the phases are 120geometricly , Fig.1. a)

• In the a) this space is 360, in b) it is 180 geometricly.

• For one supply stator voltage period, rotating field always passing 1 pair poles space!!!. That means, for one cycle T, rotating field will pass in case a) 360, but in case b) only half space, i.e. 180 geometricly

• Conclusion 1: rotating field speed in case a) is 2 times larger than in case b)

• Conclusion 2. In the machine with p-pole pairs, rotating field will pass in one T cycle 360/p parts of machine stator space.

a) 1par polova c) 2 para polovab) 1par polova

• Physical process with one pole pairs machine doesn’t changed increasing the number of poles. In that case, all analysis can be performed on one pair poles machines.

• In this case the term electrical angle (el), is defined and it is identical to

the geometric angle (g) for 2-pole machine, p=1.

• Generally, for the case of p- pair poles machine, relation between electrical and geometric angle is

el gp (4)

Number of pole pairs- Explanation

INDUCTION MACHINE – HOW DOES IT WORK

Fig.6. Animation of PULSATING field

Initial position of pulsating field is maximal field (maximal current) (maximal sinusoid) the circles are “maximal red”, vector is maximal right oriented.

When the field is zero, vector is in the middle of circle (point!), "circles are red”, current in conductors is zero.

Next position is maximum fields in another (left) side, vector is maximal and on the left, circles are red (maximal negative current)

Fig.7. Animation of SYMMETRICAL ROTATIONAL field (black) and

PULSATING fields of each phase (red, green, blue)

Thru each of 3 winding SYMMETRICAL Y spaced in stators slot (namot A, B i C) flow one of the 3f currents, (delayed each other in120°).

The picture shows that each of the fields are PULSATING, only the amount is changing in one position.

Resulting field is ROTATIONAL field, (BLACK), the sum of pulsating fields of all 3 phases, with maximal amount 50%,greater than maximum of one phase pulsating field.

INDUCTION MACHINE – HOW DOES IT WORK

INDUCTION MACHINE – HOW DOES IT WORK

Fig.8. Animation of ROTATIONAL field (black) and PULSATING fields of each of the 3 phase of IM

• The principle of work is based on the force (i.e. torque) generation

• Torque is result of rotational field and rotor current . Rotor voltage is induced by rotational stator field

• Questions: Why rotor cannot reach the speed of rotational field? How rotor could reach the speed of rotational field? Explain!

Fig. 9. Rotational field speed (ns), rotor speed (n), and rotor

speed relative to rotational field speed (ns -n)

INDUCTION MACHINE – HOW DOES IT WORK

• One phase equivalent induction machine circuit

1R 1X2X

s

R22E1EU

1I2I

)( 1111 jXRIUE

202 EsE

1

2120 f

fEE

Induction machine – equivalent circuits

E1,I1 - induced stator voltage and current

U, U1 - stator voltage (supply voltage)

R1 - stator winding (coil) resistance

R2 - rotor winding resistance

X1 - stator leakage reactance

X2 - rotor leakage reactance

E2 - inducied rotor voltage,

E20 - induced rotor voltage, (rotor locked,

stator connected to suply voltage, U)

f1 - stator voltage frequency,

f2 - rotor voltage frequency,

N1, N2- stator and rotor number of coils

Fig.10. Equivalent circuit per phase of induction motor with rotor parameters relative to the stator side

• Recalculation of rotor’s parameters to the stator side with parameter (k)

2

1 1

2 2

n

n

N fk

N f

Induction machine – equivalent circuits

(5)

Magnetic field generated from primary side and coupled with secondary side and magnetic field generated from secondary side and coupled with primary side are the main (coupled) magnetic field (12 or 21). Magnetic field which couple only primary winding is leakage field 1. Magnetic field which couple only secondary winding is leakage field 2

.

Primary winding

secondarywinding

Main path

Leackage path

Explanation of the main and leakadge path - transformer

Fig.11.

mI , ,m m

2I1I

s

RI 2

2

2 1 2j I L 1E

1E11 RI

1 1 1j I L 1U

2

Induction machine_ vector-dijagram with k=1

Fig.12. Vector diagram of induction machine

Rotor current and leakage reactance

• Rotor current is defined by induced voltage E2 and rotor impedance Z2:

• In standstil E2 = E20 , see (3)

• This formalism can be applied on leakage reactance, X2σ,, so,

• X2σ0 is leakage reactance in standstil, n=0.

• Leakage reactance is defined for 50Hz (standstill), and influence of the frequency f2 can be involved multiplying by slip s.

• For s=0, rotor current is I2(s)=0 (SYNCRONISM !!!)

(6) 2

22

2

20

22

22

20

2

22

/ XsR

E

sXR

Es

Z

EI

2 1 2 2 0 2 02 ,X s f L s X X

)( 1111 jXRIUE

Assumption: Magnetic (rotating) field in the air gap induce in stator winding voltage e1, defined by

1 1

de N

dt

)sin( tNe 11 ;11 fkE

For small slip and small current (load) it can be wrote:

1 fkU

neglect

Electromagnetic torque-dependence of a voltage and frequency

111 44.4 NfE

How torque is changing by stator voltage and frequency?

1 1 1 1 1

1 1 1

1 1

1 1 1 1

( 2 ) (7)

(8)

(9)

U E I R j fL

E k f

E U

k f k f

Electromagnetic torque Mem can be expressed as

Electromagnetic torque - derivation

pr

prmem

ss

s

sΦ

L

NpMM

2

43 2

2

21

2

1

12

2

11

1

2

21

43

f

Uk

fk

U

L

NpM pr

2

1

1

f

UM

(10)(11)

Torque speed characteristics-derivation

• Primary impedance Z1=R1+jXσ1 is neglected in equivalent circuits.

• It shoud be emphasized that motor torque in each working point is proportional to the square of the motor voltage

s

XsR

EkP

s

PsPM

s

okr

s

okr

m

m 1

1

1

22

2

220

• Machine torque dependence of voltage supply can be described using energy balance,

2( )M f U

• Detailed derivation can be found in course textual material on the web pages

(12)

(13)

Machine torque characteristics-Kloss equation

n

n

n

s

s

s

sM

M

max

max

max

2

• Kloss equation describes general torque-speed characteristics of induction machine.

• Functionally, Kloss equation involving two working points: arbitrary working point and working point with maximal slip.

• In the example below, developed torque at maximal and nominal (rated) torque are used for calculation

2

2max X

Rs

Which simplification is used in Kloss-equation?

(14) (15)

Torque vs speed characterestics of IM (It doesn’t worth for motors less than 1kW)!!

important 3 points:• s= 1, n=0 - standstil torque, Mk

• s= sn, n= nn - rated (nominal) torque, Mn

• s= smax, n= nmax - maximal torque, Mmax (Mpr)

M

0 nsnn

Mn

Mk

Mmax

01

n

s snsmax

nmax

Fig.13. Motor torque vs speed induction motor (IM) characteristics

• Derivation for torque (1) – (4) has been done with assumption that recalculation factor , see (5), is k=1

12

22

11

n

n

fN

fN

• From (10)–(13) it can be seen quadratic relation between torque and magnetic field (voltage).

• Expression (14), represent simplified Kloss-equation and can be used for slip-ring motors and squirrel-cage motors without skin effect in rotor slots. If the skin effect is present, Kloss equation (14) can be used only in the region of the small slip.

Electromagnetic torque-dependence of a voltage and frequency

Fig.14. Simulation results given from mathematical model

0 200 400 600 800 1000 1200 1400 16000

2

4

6

8

10

12

14x 10

Ele

ctric

al a

nd m

echa

nica

l pow

er[k

W]

Ulazna (električna) snaga

Izlazna (mehanička) snaga

speed[rpm]

600 800 1000 1200 1400 16000

400

500

600

speed [rpm]

Ele

ctro

mag

netic

torq

ue [N

m]

300

200

100

0 200 400 0 200 400 600 800 1000 1200 1400 16000

50

100

150

200

250

300

350

Sta

tor

curr

ent[A

]

speed [rpm]

0200 400 600 800 1000 1200 1400 16000

10

20

30

40

50

60

70

80

90

100

speed[rpm]

Effi

cien

cy [%

]

P1 is electrical power (power supply)

P2 is power on the motor shaft (mechanical Power)!!!

Induction machine– energy balance

Fig.15. Energy balance in induction motor

For magnetic field getting, IM taking reactive power

Total power of IM is

Active power (on the motor shaft!) P=P2 .

m1 is the number of phases

Example of motor Data:

3f induction motor, P= 1000 kWVoltage 6000 V, frequency 50 Hznominal speed1485 ,(rpm), cosφ=0,88, =0.8nominal current 115 A

Nominal data- Total, Active and Reactive power of IM

1 1 1 1 1sinQ mU I

1 1 1 1S mU I

1 1 1 1 1cosP mU I

s

s

n ns [%] 100%

n

• The amount of slip is directly indicator of the amount of losses in induction motors (see energy balance).

• It is needed to set working point in the way that slip must be very low.

• Nominal slip is usually between 0.1 i 5 %. Low power machine (up to cca 1kW), has larger slip.

Take into account the problem of

overheating .High losses means high

heating, conductor’s isolation getting

badly, it is possible dielectric breakdown!

Take into account the problem of

overheating .High losses means high

heating, conductor’s isolation getting

badly, it is possible dielectric breakdown!

Induction motors - Slip and Losses

Working range of induction squirel cage motor

s = 0 n = ns unloaded machines = 0.01 n = 0.99 ns working region of large machines (over 100kW)s = 0.04 n = 0.96 ns working region of medium and small machines s = 1 n = 0 blocked rotors > 1 revers current braking, plugings < 0 generatory braking

Magnetic field rotation

Rotor rotation

Generator brakingPluging Motoring

Fig.16. 4-quadrant operation

END