___________________________________________________________________________________________

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ELECTRICAL MACHINES

2006

621.313 31 .., .., .. 31 ELECTRICAL MACHINES: . : -

, 2006. 176 .

ISBN , ,

. . 3-

621.313

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, 2006

ISBN : , 2006

3

PREFACE Electrical Engineering is a dynamic profession, which provides the expertise to meet technical challenges facing the nation. Electrical Engineering concerns generation, supply, distribution, application and their automation. The electrical engineer is often a key figure in different industries. Electrical machine is the main type of converter of mechanical energy into electric, and electric into mechanical, one, as well as one form of electric energy into another one, different in voltage, current and sometimes in frequency. It has played, during the entire historic period of electrical engineering development, a leading role, which has mapped out progress in different fields and, particularly, in the branch termed heavy-current engineering. Continuous improvements in the design of electrical machines have made many new practical applications possible and have become strong impulses for further progress and more diverse use of electric energy. This account for fact that electrical machine was given great attention to by scientists and engineers and that electrical machines attained technical perfection of design so soon. This textbook is intended for studying the course Electrical Machines for students, who go through the Bachelor Degree Program in Electrical Engineering. Students study this course in the fifth semester. The course is based on the higher mathematics, physics, engineering graphics knowledge, mechanics and measurement. The textbook is intended mainly for students, who have already taken courses TEE 201, 202 Electric Circuit Theory, INCABE 202 Electrical Engineering Materials. All important concepts of magnetism, electricity and electromagnetic conversion theory are explained. The mathematical language is as simple as possible. The textbook is based on the classical series of the textbooks on Electrical Machines by A.I. Voldek, M.P. Kostenko and L.M. Piotrovsky, B.F. Tokarev, M.M. Katsman and it consists of the following topics:

1. Transformers. 2. Induction Machines. 3. Synchronous Machines. 4. Direct Current Machines.

The topic "Transformers" includes the following questions: elements of construction; basic voltage equations; schemes and group of transformers winding coupling; distribution of load between transformers and etc. The topic "Induction Machines" includes the following questions: elements of construction; rotating magnetic field, voltage equations of induction motor; energetic diagrams of active and reactive power, induction motor torques, starting and regulation of rotation frequency three-phase induction motor and etc.

4

The topic '' Synchronous Machines' includes information about construction and basic principle of a synchronous machine operation, magnetic field of excitation winding, reaction of armature, voltage vector diagrams of synchronous generators, synchronous motors and compensators. The authors welcome yours suggestions for improvements of future editions of this textbook. The topic '' Direct Current Machines' includes information about basic elements of D.C. Machine construction and principle of their action; the process of commutation in D.C. Machines; characteristics of direct current generators and motors. The authors welcome your suggestions for improvements of future editions of this textbook.

1. TRANSFORMERS

Transformer is a static electromagnetic device with two (or more)

inductively linked windings intended for transforming one (primary) an

alternating current system to another (secondary) one by means of

electromagnetic induction. Power transformers are widely used in

electrotechnical installations as well as in power transmission systems which

change only the values of alternating voltage and current (Fig. 1.1). While

studying the given section attention is focused on general-purpose power

transformers.

1 - pole core, 2, 3 - windings, 4 - case, 5 - cooling pipes, 6 - voltage switch handle, 7, 8 - terminals, 9 expander.

Fig. 1.1

5

1.1. Design and Operation Principle of Transformers

A single-phase transformer consists of a pole core and two windings. One

winding called primary is cut in to alternating current supply at voltage .

Load is cut in to another winding called secondary. Primary and secondary

windings of power transformer are not electrically linked and power is

transmitted from one winding to another by electromagnetic way.

1U

Zload

Transformer operation is based on electromagnetic induction principle. When

cutting in primary winding to a. c. supply at frequency f alternating current

flows in the turns of this winding producing alternating magnetic flux in the

pole core. Being closed in the pole core this flux is linked with both windings

and induces self-induction e.m.f. in primary winding

1i

1

( )dtdwe 111 = , mutual induction e.m.f. in secondary winding is ( )dtdwe 122 = , where is turns number in primary and secondary windings. 21 , ww

When cutting in the load to the terminals of secondary winding current

is produced under the effect of e.m.f. in the winding turns and voltage is

induced across the terminals of secondary winding. Step-up transformer shows

and step-down transformer offers

Zload 2i

2e U 2

12 UU > 12 UU < . The second alternating magnetic flux 2 is produced when current flows

across the turns of secondary winding. Direction of this flux depends on the

character of transformer load and may be in opposition or concordant to the flux

of primary winding. Besides, the fact that current appears in secondary winding

causes current change in primary winding but resultant magnetic flux in the

pole core is not changed and depends only on magnitude and the rate of primary

winding voltage. Thus, one may assume that joint flux

equals flux . 1Modern power transformers are of similar design circuit consisting of 4

main systems, i.e. 1. closed magnetic system - pole core, 2. electrical system - 6

two or more windings, 3. cooling system - air, oil, water or combined systems,

4. mechanical system providing mechanical durability of the construction and

possibility of transformer transportation.

The pole core is intended to increase inductive coupling between the

windings. It forms magnetic circuit along which resultant magnetic flux of

transformer is closed. The pole core is made of iron laminations, which are

isolated from one another by a very thin coat of varnish or oxide on one side of

each lamination. Such pole core construction makes it possible to reduce eddy

currents induced by alternating magnetic flux and to minimize energy losses in

the transformer.

Power transformers are produced with pole cores of three types, i.e. core-

type, shell-type and shell-core-type constructions.

A single-phase transformer of core-type construction [Fig. 1.2(a)] consists of

four areas, they are two limbs ( L ) and two yokes ( Y ). A limb is considered to

be an area of the pole core which is enclosed by turns per coil, an yoke being an

area of the pole core connecting limbs and closed pole core.

(a) (b) (c)

Fig. 1.2

In a single-phase two-winding transformer of core-type construction each

of two windings consists of two parts situated on two limbs connected either in

series or in parallel. Such winding arrangement brings to inductive linking

increase.

7

Cross-sectional area of the limb is step-shaped circle inscribed. An yoke is of

cross-sectional area with less number of steps and four angles which are beyond

8

the circle. Yoke cross-section is larger than that of the limb that allows, in

particular, to improve the parameters of no-load transformer.

In a single-phase transformer of shell-type pole core [Fig. 1.2(b)] there is one

limb and two yokes which partly cover windings like a shell from diametrically

situated sides. Magnetic flux in the limb of such pole core is twice larger than

that in the yokes, therefore each yoke possesses twice less cross-section than

that of the limb. In single-phase transformer pole core of shell-core type

construction [Fig. 1.2(c)] there are two limbs and two yokes as is the case with

core-type transformer and two more lateral yokes as in shell-type transformer.

Such pole core construction requires larger amount of electric steel but makes it

possible to reduce pole core height that is important for transformer

transportation by railway.

Three-phase transformer pole core of core-type construction [Fig. 1.3(a)]

consists of three limbs and two yokes located in one plane if the pole core is flat.

In spatial pole core the limbs are located in different planes. Flat pole core of

core-type construction is not quite symmetrical as pole core length for the mean

phase is somewhat shorter than for the marginal ones. However, it does not

influence.

The pole core of shell-type three-phase transformer [Fig. 1.3(b) may be

schematically represented by three single-phase shell-type pole cores which are

superimposed. The mean phase of such a transformer has reverse switching

relative to marginal phases. In this case fluxes are geometrically added in

contacting areas of the next phases of the pole core instead of being subtracted

that allows to reduce the cross-section of these pole core segments.

In three-phase transformer pole core of shell-core type construction

[Fig. 1.3(c)] there are three limbs and two yokes like in a core-type transformer

and two more lateral yokes like in a shell-type transformer. Advantages and

drawbacks of such pole core design are similar to the like single-phase

transformer design.

For three-phase voltage conversion one can use not only a three-phase

transformer with any type of pole core mentioned above but three single-phase

transformers as well. Such device is called three-phase transformer bank.

(a) (b) (c)

Fig.1.3

Three-phase transformers with the pole core common for all phases are often

used. They are more compact and cheaper. Transformer bank is used in case of

transportation problems and for decreasing stand-by power in case of emergency

repair.

Transformer windings are important elements owing to two reasons, i.e.

1. The cost of materials used for manufacturing makes up about a half of

transformer cost.

2. Transformer durability often depends on winding durability.

In two-winding transformers the winding to which ceiling voltage is applied

is called high-voltage ( HV ) winding and the winding with very low voltage is

called low-voltage ( LV ) winding.

9

10

According to the pole core arrangement on the limb the windings are

classified as concentric and sandwich winding constructions. Concentric

windings are designed in the form of hollow cylinders placed concentrically on

the limbs. LV winding is placed closer to the limb as less isolation distance is

required and HV winding is placed outside.

Sandwich (disc) windings are made in the form of separate HV and LV

sections (discs), which are sandwiched on the limb. They are used only in

special-purpose transformers.

According to engineering design the windings are classified as

1. Cylindrical single- or multilayer windings made of rectangular or round

wire

2. Spiral simplex and multiple windings of rectangular wire

3. Continuous disc windings made of rectangular wire

4. Windings made of foil.

Single- and two-layer cylindrical windings of rectangular wire are used as

LV windings at nominal current up to 800A. The turns of each layer are wound

closer to each other in a spiral manner. Interlayer isolation is made by two layers

of electroisolating 0.5 mm cardboard or by the channel.

Multilayer cylindrical windings made of rectangular wire are used as HV

windings (up to 35 kW). These windings are used in 110 kW transformers and

above.

Spiral simplex and multiple windings are used as LV windings at the current

over 300 A. Turns are wound in the form of one or several movements spiral.

Channels are made between turns and parallel branches.

Continuous-disc windings consist of disk coils connected in series and

wound in continuous spiral without breaking wire between separate coils. The

coils are separated by the channel. They are used as HV and LV windings.

1.2. Basic Transformer Equations

It can be supposed that resultant alternating magnetic flux in the transformer pole core is sinusoidal time function.

Whereas instantaneous e.m.f. value induced in the primary winding equals

( ) ( )2sincos max1max111 === twtwdtdwe where f 2= .

By analogy for the secondary winding this leads to the following

( )2sinmax22 = twe Thus, e.m.f. and lag resulting flux 1e 2e in phase through an angle 2 .

Effective e.m.f. value may be written as

,44.42 max1

max11 fw

EE == ,

max22 44.4 fwE =

E.m.f. ratio of HV and LV windings is called transformation ratio

21

21

21

UU

ww

EEk ==

Currents and in transformer windings besides resultant magnetic

flux induce magnetic leakage fluxes

1I 2I

1 and 2 (Fig. 1.4). Each flux is linked with the turns of only inherent winding and induces e.m.f. leakage in it.

Effective e.m.f. leakage values are proportional to the currents in the

corresponding windings

,xIjE ;xIjE 2 22111 == &&&

11

where are inductive leakage reactances of primary and secondary

windings accordingly. The sign minus in this expression points to leakage e.m.f.

reactance.

21 , xx

Fig. 1.4

If the primary transformer winding with ohmic resistance cut into the

voltage main, voltage equation is

1r

1U

( ) 111111 rIxIjEU ++= &&&&

In power transformer inductive and active voltage drop is not significant,

therefore one can assume that ( )11 EU && . For secondary transformer winding the voltage drop at the load equals

terminal voltage of secondary winding and voltage equation results is

load2222222 ZIrIxIjEU == &&&&&

where is ohmic resistance of secondary winding. 2r

If a transformer runs at primary winding cut into the voltage main and

broken secondary winding we deal with no-load duty. Current in primary

winding under these conditions is called no-load duty.

1U

0I

12

Magnetomotive force (m.m.f.) 10 wI produced by this current induces magnetic flux in transformer pole core with the amplitude

10max 2 RwI = where is pole core magnetic resistance. mR

When secondary winding is closed to load current develops. As

for the primary winding the current increases up to the value . Now magnetic

flux in pole core develops under the effect of two m.m.f. and .

loadZ 2I

1I

11 wI & 22 wI &Thus, it may be considered that resultant magnetic flux value at stable

voltage does not practically depend on transformer load if its value does not

exceed the nominal value. The considered approach leads to the following

transformer m.m.f. equation

1U

221110 wIwIwI += &&& 221110 wIwIwI += &&&

and transformer currents equation is ( )2012112210 ,/ IIIIIwwIII +=+=+= &&&&&&&& , where is secondary winding current brought to the primary

winding turn number.

1222 / wwII = &&

13

1.3. Transformer Equivalent Circuit

Parameters of primary and secondary transformer windings differ

resulting in marked transformation ratio that makes difficult plotting vector

diagrams.

14

2

This problem is eliminated by bringing secondary parameters and the load

to the form of primary winding, they are converted per turn number of primary

winding . As a result, instead of a real transformer with transformation ratio

we get the equivalent transformer with

1w

k w w= 1 / 121 == wwk , where . Such transformer is called the idealized transformer. 12 ww =

Secondary parameters referred above should not influence energetic

transformer values, i.e. all voltages and phase shifts in secondary winding

remain the same as in a real transformer. As a result, turns number of secondary

winding changes into kwwww == 2122 // times and as a consequence

22 EkE && = , . 22 UkU && =If electromagnetic voltages of real and idealized transformer secondary

winding are equal then the expression for the secondary winding current is

obtained

kIIIEkIEIE / , 22222222 &&&&&&&& === Referring to the losses equality in the secondary windings effective

resistance of both real and idealized transformers the expression for idealized

effective resistance of a secondary winding is obtained

( ) ( ) 222222222222 ,/ krrrkIrIrI === &&& Referred inductive leakage reactance of secondary winding is determined

from the equality condition of secondary winding reactive power of real and

idealized transformers

( ) ( ) 222222222222 ,/ kxxxkIxIxI === &&& .

Referred impedance of transformer secondary winding 2

2222 kZxjrZ =+=

Voltage equations for idealized transformer may be written as ( ) 111111 rIxIjEU ++= &&&& , 222222 rIxIjEU = &&&&

Currents equation is ( )201 III += &&& These equations show analytical relation between transformer parameters

in the range between no-load and nominal duties.

Lets consider transformer equivalent circuit [Fig. 1.5(a)]. This diagram shows

Fig. 1.5

that ohmic and inductive resistances are conventionally taken out by convention

from the corresponding windings and are energized in series. 15

As 1=k in the idealized transformer then is obtained. As a result points A and a, X and x in the diagram are of similar potentials, that makes its

possible to connect them electrically and to obtain T-shape electric equivalent

circuit of the idealized transformer [Fig. 1.5(b)]. Magnetic linking between the

windings is substituted for electric linking in this equivalent circuit.

21 EE = &&

T-shaped electric equivalent circuit of the idealized transformer makes

investigation of electromagnetic processes and transformer calculations easier.

The circuit is a complex of three branches. The first branch contains impedance

and current . The second branch (magnetizing) contains

impedance and current , where , are the parameters of a

magnetizing branch. The third branch contains impedances of secondary

winding , load

111 jxrZ += 1I&

mmm jxrZ += 0I& mr mx

222 xjrZ += loadloadload xjrZ = and current 2I&. All the parameters of electric equivalent circuit but loadZ are constant and may be determined either by calculation or experimentally (no-load and short-circuit

duties ).

1.4. No-Load Duty

No-load duty is considered to be transformer duty at closed secondary

winding , . =loadZ 02 =IVoltage and current equations take the form ( ) 101011 rIxIjEU ++= &&&& ;

220 EU = && ; . 01 II && =Magnetic flux 1 in the transformer is alternating one, therefore the pole

core is being steadily remagnetized, there arise magnetic losses from hysteresis

and eddy currents induced by alternating magnetic flux in iron laminations.

Open-circuit current is of two components, namely, active component

owing magnetic losses and reactive one showing magnetizing current

0I& aI0&

I0&

16

20

200 a III +=

Fig. 1.6

Active component of open-circuit current usually is not significant, it does not

exceed 10% of the current and therefore it does not significantly influence

open-circuit current.

0I

As net power of transformer while running

under no-load conditions equals zero active

power consumed under this conditions is

spent for magnetic losses in the pole core

and electric losses in primary winding .

0P

mP

120 rI

Taking into consideration the fact that open

circuit current does not usually exceed 2-

10% of primary winding nominal current

electrical losses can be neglected and

magnetic losses in the iron core can be

0I

nomI1

Fig. 1.7 considered to be open-circuit losses.

Electric equivalent circuit and transformer vector diagram are shown in Fig. 1.6

and Fig. 1.7.

17

Angle through which vector of magnetic flux & 1 lags behind from current is called magnetic loss angle. This angle increases with the growth of open-

0I&

circuit current active component i.e. with the growth of magnetic losses in

the transformer core.

aI0&

1.5. Short-Circuit Duty

Short circuit is the transformer duty at short-circuited secondary winding

, . 0load =Z 02 =UUnder operating conditions when nominal voltage is applied short

circuit is considered to be emergency duty and a serious hazard to the

transformer. Only steady short-circuit current exceeds the nominal current 10-20

times.

nomU1

Short-circuit duty is not a hazard to the transformer as step-down voltage

is supplied to the primary winding, in so doing currents in both windings being

equal to nominal currents.

This step-down voltage is called nominal short-circuit voltage and is

usually expressed as a percentage of nominal voltage

( ) %1051001scsc == nomUUu

As we have found before the resultant magnetic flux in the transformer pole core

is approximately proportional to primary winding voltage. Consequently, at

short-circuit duty resultant magnetic flux in

the pole core is small, magnetizing current

is required to induce it and it may be

neglected, therefore equivalent circuit does

not posses magnetizing branch.

Equations of voltages and currents take the form

( ) ( ) sc1sc111211211sc ZIxIjrIxxIjrrIU scscscscsc &&&&&& =+=+++= , scsc II 21 = && ,

18

where is transformer impedance under

short circuit conditions,

scZ

scsc , xr are active and reactive components of

resistance . scZ

Electric equivalent circuit and vector

diagram are shown in Fig. 1.8 and Fig. 1.9.

Rectangular triangle AOB is called short-circuit triangle, its legs being active and

reactive components of short circuit

voltage

scaU&

scrU&

scasc12211 )( UrIrIrIOB scscsc &&&& ==+= , scrsc12211 )( UxIjxIjxIjBA scscsc &&&& ==+= .

As at short circuit duty the resultant flux is too small compared with its

value at nominal primary winding voltage pole core magnetic losses may be

neglected. It follows that active power , consumed at this duty is spent for

electric losses in transformer windings

scP

sc2

122

112

1sc rIrIrIP scscsc =+= & .

19

1.6. Transformer Vector Diagrams under Load Conditions

For plotting vector

diagrams of electric

equivalent circuit of the

idealized transformer and

the basic equations of

voltage and currents are

used. Vector diagrams

clearly show relations and

phase shifts between the

currents, e.m.f. and

voltages of the

transformer.

For determination of

phase shift angle between

and one should

know the load character.

At active-inductive load (Fig. 1.10) vector lags in phase through an angle

2 & 2I &

2I & 2 &( ) ( )[ ]n2n212 /tan rrxx ++=

At active-capacitive load (Fig. 1.11) vector advances through an angle 2I & 2 &( ) ( )[ ]n2n212 /tan rrxx +=

At marked capacitive load component the voltage U may be larger than

e.m.f. at open circuit (no-load) duty . Besides, reactive component of

secondary winding is in phase with reactive component of the

open circuit current , showing magnetizing effect on the pole core. It causes

2&

2 &

222 sin= II r &&rI0&

20

primary winding current decrease compared with its value at active-inductive

load when the component shows

demagnetizing effect.

1I&

rI 2&

The above vector diagrams of a loaded

transformer cannot be used for practical

calculations as being complex. By

analogy with short- circuit duty in the

transformers running at the load close to

the nominal one open-circuit current is

neglected and it is considered to be

. 21 II = &&As a result, transformer equivalent

circuit takes a simplified form, it lacks

magnetizing branch. The circuit consists

of connected in series elements r [Fig. 1.12(a)].

sc

21

Simplified vector diagram is plotted according to nominal voltage values of

primary winding nomU 1.

, nominal current of primary winding , power factor nomI1&

nom2cos and short circuit triangle parameters , , . scU& scaU& scrU&Lets explain plotting simplified transformer vector diagram at active-inductive

load [Fig. 1.12(b)]. In an arbitrary way for example, one constructs a current

vector is constructed on Y-axis from its origin. A line is drawn at an

angle , where voltage vector

21 II = && 2 ( )2U & is located on it according to the load

character. One constructs - short-circuit triangle is constructed. The leg

BC being equal to active component of short-circuit voltage is in phase with

vector current. The leg AB being equal to reactive component of short circuit

voltage advances current vector by .

ABC

90o

One shifts the triangle without changing the legs of an angle orientation

so that the vertex C could be found on the line directed at an angle to the current vector until the distance from coordinates origin to the vertex A equals

.

ABC2

nomUU 11 && =Then phase shift angle between the primary winding current and its

voltage as well as vector value

1 1I&

1U& ( )2U & are determined. All vector constructions are carried out at the usual scale.

1.7. External Transformer Characteristics

Current change of transformer load causes the changes of its secondary

voltage and efficiency due to the change of voltage drop and active power losses

in the windings.

Secondary voltage change is usually expressed in percent and is

determined as follows

22

10010020

220

20

220 ==

UUU

UUUU &

&&&

&&&

where are ordinary and referred voltages (e.m.f.) of secondary

winding open circuit at nominal voltage of primary winding, are

ordinary and referred voltages across transformer secondary winding terminals

at primary winding rated voltage.

2020 , UU &&

22 ,UU &&

Using simplified transformer vector diagram the expression for calculation of

secondary voltage change is obtained

( ) ( ) ,%200/sincossincos 22sca2scr22scr2sa ++= UUUUU &&& , where is load factor. nomIIII 22nom22 // == &&&&

10010020

220

20

220 ==

UUU

UUUU &

&&&

&&&

From the given expression it follows that secondary voltage change

depends on amount and character of the load.

Dependences at ( )fU = & const=2cos shown in Fig. 1.13(a) are

23

practically linear as the first addend changes proportionally the load and the

second one being insignificant does not practically influence U value. The second addend is neglected in most cases due to its rather small value

and a simplified formula for U calculation is used ( )2scr2sca sincos += UUU &&&

Dependences ( 2 )fU = at const= are of more complicated form [Fig. 1.13(b)]. At sc2 U 0 U== this leads to the following result at

. The largest

voltage change occurs at

sc0

2 90 UU == sc =2 and is

equal to scUU = max .

24

At ( ) 0 ,90 sc020 == U . Dependence of secondary winding

on load current or on load factor

2U&

2I& at rated voltage and primary winding frequency

under stable load conditions is called external transformer characteristic.

For plotting external characteristic the following formula may be used

( ) ,100/1202 UUU = where [ . ] %U =External characteristics (Fig. 1.14) due to linearity dependence ( )fU = are also linear.

1.8. Transformer Voltage Regulation

Voltages at different sections of energy transmission line where step-

down transformers can be cut in differ from each other and as a rule, from

transformer rated primary voltage. Besides, these voltages change owing to load

changes. Taking into consideration the fact that terminal secondary winding

voltage of the transformer should correspond to State Standard requirements it is

possible to provide these requirements, in particular, by changing transformation

ratio.

HV windings of step-down transformers have regulating shunts with the

help of which one can obtain transformation ratio that differs from the nominal

one may be obtained.

Regulating shunts are designed in each phase either close to zero point or

in the middle of the phase. In the first case three or five branches are made in

each phase, in so doing medium shunt corresponds to rated transformer ratio and

two ( four ) other shunts correspond to transformation ratio that differs by 5% ( and ) from the rated one. In the second case each phase is divided

into two parts and six shunts are formed, that makes its possible to get except for

rated transformation ratio four additional values that differ and

2 5. % 5%

2 5. % 5% from the rated one.

Two kinds of power transformer voltage regulation are provided, i.e.

voltage regulation by switching winding branches without excitation (SWE)

after cutting out all transformer windings and voltage regulation without load

break (LBR), without cutting out transformer windings. Branch switches LBR

compared with SWE are of more complex design because each phase is

provided with special switching devices. LBR equipment is located in the

common tank with active transformer part and its switching is automatized or

done at a distance (from switchboard). Transformers with LBR are usually

intended for voltage regulation within the range of 6-10%. 25

At higher transformer voltages

LBR equipment seems to be too

complex. In this case one uses

voltage regulation is used with of an

injector transformer consisting of ST

transformer connected in series and

regulating autotransformer (RA) with

switching device (SD) (Fig. 1.15).

Transformer secondary winding voltage ST U& is summarized with line voltages and changes it up to the value . The value L1U& UUU &&& = L1L2 U& may be changed by the regulation autotransformer (RA) and U& may be changed through by a pitch regulation switch (PRS). 180o

1.9. Transformer Losses and Efficiency

In the process of electric energy transformation some energy is lost in the

transformer in the terms of electric and magnetic losses.

Electric losses cause heating the transformer windings when electric current

flows across them. Power of electrical losses is proportional to current square

and is equal to the sum of electric losses in primary and secondary

windings

eP

1eP 2eP

22

212

121 )( rImrImPPP eee +=+= where m is phase number in transformer windings.

This expression for transformer electric losses is used only at the stage of

designing. When manufacturing transformer electrical losses are determined by

26

the results of short circuit duty taking the voltage at rated currents in the

windings scnomP

scnome PP = 2 , where is load factor.

As electrical losses depend on transformer losses they are called

alternating.

Magnetic losses occur mainly in the transformer pole core. Magnetic

losses of hysteresis are in direct proportion to pole core frequency of magnetic

reversal, i.e. to a.c. frequency (

mP

fph ). Magnetic losses from eddy currents are proportional to the square of this frequency ( ). Total magnetic losses

are considered to be proportional to current frequency by the power 1.3. The

amount of magnetic losses also depends on magnetic induction square in limbs

and yokes of the pole core. If

2fpec

constU =1 and constf = magnetic losses do not depend on transformer load, they are called constant. For the manufactured

transformer magnetic losses are determined by the results of open-circuit duty,

measuring open-circuit power at rated primary voltage. nomP0

Thus, active power released to primary transformer winding is partly

spent for electrical losses in this winding , for magnetic losses in the pole

core respectively. The remainder is called electromagnetic power and it is

released to secondary winding where it is partly spent for electrical losses in this

winding . Active power released to the load of a three-phase transformer

(net power) may be determined as follows

1P

1ep mP

2ep 2P

== PPpppPP eme 12112 or 22222 coscos3 == nomSIUP

where are total losses in the transformer, += scnomnom PPP 20 nomS is rated transformer power,

27

2I , are linear current and voltage values of secondary winding. 2U

Transformer efficiency is determined as active power ratio of secondary

winding output to active power of primary winding input 2P 1P

( ) === 11

11

2 1 PPPPP

PP ,

scnomnomnom

nom

PPSS

++= 2

02

2

coscos

.

The analysis of the above expression shows that transformer efficiency

depends both on the value ( ) and the character ( 2cos ) of the load. Maximum efficiency value corresponds to the load at which magnetic losses are

equal to electrical losses

scnomnom PP = 20 , i.e. at scnomnom PP /0= .

Transformer efficiency usually is of maximum value at = and decreases slightly at load increase.

0 45 0 65. .

1.10. Diagrams and Connection Groups of Transformer Windings

Marking the initial and final windings is done in the following way. In a

single-phase transformer HV winding is denoted by Latin capital letters (A-

origin, X- end). LV winding is denoted by Latin small letters (a - origin, x

end). When the third winding with medium voltage is available its origin and

end is denoted as Am and Xm accordingly. In three-phase transformer HV

winding is denoted by capital letters (A, B, C origins, X, Y, Z ends). LV

winding is denoted by Latin small letters a, b, c origins, x, y, z ends. It is

common practice to consider phase alternation A, B, C from left to right if the

transformer is examined as viewed from HV tapped winding.

28

In most cases three-phase transformer windings are star- (Y), delta- () or seldom zigzag (Z)-connected. The first two diagrams of three-phase winding

connection are denoted by Russian capital letters , accordingly.

Zero terminals of star- and zigzag-connected three-phase winding are

denoted in HV windings by capital letter O and in LV winding by small

letter o. In so doing index N (Yn, Zn) is added to letter designation of winding

connection diagrams.

While connecting a transformer in parallel with other transformers phase

shift between primary and secondary winding e.m.f. is of prime importance. The

notion connection group of winding to characterize this shift is applied.

Consider the fragment of core-type construction pole core of a single-

phase two-winding transformer (Fig. 1.16). Both windings are wound along the

left spiral line and are of similar wind direction.

In both windings origins A and a are arranged

above and ends X and x below respectively, i.e.

they are marked in a similar way.

E.m.f. induced in the winding is considered to

be positive if it acts from initial to final

windings. In both windings e.m.f. is induced by

the same main magnetic flux. Similar wind

direction and marking makes its possible to hold

that the above mentioned e.m.f. of these

windings acts in a similar direction at every

instant, i.e. coincidentally positive or negative.

E.m.f. AE.

and aE.

are in phase. The angle between e.m.f. vectors of

primary and secondary windings equals zero. Conventional symbol is I/I-0 (zero

group).

29

If the marking in one winding is reversed (Fig. 1.17) or wind direction is

changed opposite in sign e.m.f. will act in the windings at every instant. The

angle between e.m.f. vectors of primary and secondary windings is 180. When determining the connection group of the winding this angle should be divided

into 30. Conventional symbol is I/I-6 (the sixth group). Thus, in single-phase transformers two groups of winding connection

zero and the sixth groups are available.

Lets consider three-phase two-winding transformer with HV and LV

star-connected windings under the following conditions:

1. Windings are of similar wind direction

2. Windings are similarly marked

3. Like winding phases are placed on common limbs.

Firstly, vector diagram for HV winding is

plotted, choosing arbitrarily the direction of the first

phase e.m.f., conforming phase alternation with the

others. When plotting vector diagram for LV winding

it should be remembered that the direction of each

vector depends on vector diagram of HV winding.

Then, all the vectors of phase e.m.f. in pairs

AE.

and aE.

, BE.

and bE.

, CE.

and cE.

as well as all

linear e.m.f. vectors in pairs ABE.

and abE.

, BCE.

and

bcE.

, CAE.

and caE.

are in phase at every instant, i.e. the

angle between them equals zero (Fig. 1.18).

30

In three-phase transformers the group of winding connection is defined by

the angle between like linear e.m.f. In the case considered conventional symbol

is Y/Y-0 ( zero group ).

What will be the result if we change LV winding marking per one pitch

around? E.m.f. vector diagram representation for HV winding remains

unchanged. E.m.f. vector diagram of LV winding will be another. The phase a-x

of LV winding is located on a common limb with phase B-Y of HV winding. As

phases possess similar wind direction and are similarly marked core magnetic

flux induces e.m.f. similar in the direction in these phases. Vector aE.

of LV

winding is represented as being in phase with vector BE.

of HV winding.

The same reasoning is provided concerning vector bE.

and cE.

directions.

As a result e.m.f. vector diagram of LV winding is clockwise displaced 120 compared with the previous vector diagram. The angle between like linear e.m.f.

is determined clockwise from e.m.f. vector of HV winding up to e.m.f. vector of

LV winding. The angle is 120, the fourth group. Conventional symbol is Y/Y- Thus, when changing marking of one winding per one pitch around connection

31

grouping of winding varies to four as linear e.m.f. vectors are clockwise

displaced 120.

Fig.1.19

Similar results may be obtained if HV and LV windings have another but

similar winding connection - diagram - delta.

If connection diagrams of HV and LV windings of a three-phase

transformer are similar one can get six even groups are formed: 0, 4, 8, 6, 10, 2

by changing the marking of one winding.

Consider three-phase two-winding transformer with different connection

diagrams (Fig. 1.20) following the conditions mentioned above. LV winding is

delta-connected. E.m.f. vector diagram of HV winding is plotted as shown

above.

E.m.f. vector diagram of LV winding is a triangle, each side being equal

to phase and linear e.m.f. in magnitude and phase. The angle between like linear

e.m.f. is 330, the eleventh group. The symbol is Y/-11. Marking the change of LV winding per one pitch around marking in

changing connection group of windings to four, it will be the third group. If LV

32

winding marking is changed again per a pitch around connection group of

winding will change to four again, it will be the seventh group.

It is not difficult to confirm that marking change of one winding in a

three-phase transformer of different winding connection diagrams makes its

possible to get six odd groups: 11, 3, 7, 5, 9, 1.

According to Russian State Standard there are transformers with the

following connection diagrams and connection groups of windings for using:

1. Y/Yn-0

2. /Yn-11 3. Y/-11 4. Yn/-11 5. Y/Zn-11.

In zigzag-connected circuit each winding phase is divided into two parts

which are placed on different limbs (one part is placed on the main limb, the

second one is arranged on the limb of the neighbouring phase in the order of

alternation). In so doing the second half of each phase is switched on in

opposition to the first half. This makes its possible to get phase e.m.f. 3 times

higher than in the matched switching.

33

However, at matched switching of phase halves e.m.f. of each phase is

1.15 times less than when phase halves are placed on one limb. Therefore wind

wire consumption in zigzag connection increases 15%. This connection is used

only when non-balanced phase load with zero currents is available.

1.11. Parallel Transformer Operation

Parallel operation of two or several transformers is operation at parallel

connection of both primary and

secondary windings. In parallel

connection like terminals of

transformer windings are

connected to the same conductor

(Fig. 1.21) in the mains.

Parallel operation of

transformers instead of one

transformer of total voltage is recommended owing to the following reasoning:

1. to provide regular power supply of consumers in case of emergency

when one of the transformers is under repair,

2. to provide transformer operation with high performance indices

(efficiency, cos2) changing the number of transformers under optimum load conditions,

To distribute the load between parallel transformers proportionally to their

nominal voltages three conditions should be fulfilled.

Firstly, primary and secondary voltages of transformers should be equal

accordingly, i.e. transformers should be of equal transformation ratios

( ). K=== 321 kkkSecondly, transformers should be of the same connection group of

windings. 34

Thirdly, rated short-circuit voltage of transformers should be equal

( K=== 321 scscsc UUU ). When the first condition is not fulfilled even at no-load duty phasing

current Iph develops in parallel transformers. It is due to secondary

e.m.f. difference of the transformers U& (Fig. 1.22)

21 scsch ZZ

UI += &&

where are short-circuit transformer 21 , scsc ZZ

impedances.

When the load is energized phasing current is superimposed

on load current. In transformers with higher secondary e.m.f. ( in step-down

transformers - transformers possessing less transformation ratio) phasing current

is added to load current. The transformer of similar rating but with larger

transformation ratio is underloaded as phasing current is opposed to load

current.

Continuous transformer overload is impermissible as it requires reducing total

load at different transformation ratios. At marked difference of transformation

ratios proper transformer operation is impractible. It makes possible to operate

parallel transformers with unlike transformation ratios if their difference does

not exceed 0.5% geometric mean %5.0100

21

21 =

kkkkk .

When the second condition is not fulfilled secondary linear transformer

voltage is phase-shifted relative to each other. In transformer the voltage

difference in a circuit arises causing marked phasing current. U&Let us consider, for example, energizing two parallel transformers with

equal transformation ratios, one being of zero (Y/Y-0) and another being of the

35

eleventh (Y/-11) connection group of windings. Firstly, linear voltage of the first transformer will be

21U&

3 times higher than linear

voltage of the second transformer. Secondly, the vectors

of these voltages will be phase-shifted relative to each other by

30 (Fig. 1.23)

22U&

2/3 22UOA = as 3/2122 UU = then 2/21UOA = and 22UU = .

Such voltage difference U& results in the phasing current in secondary transformer circuit, the current exceeding

nominal load current 15-20 times, i.e. emergency conditions

occur. The highest U& value appears when energizing parallel transformers with zero or the sixth connection group of

windings ( 22UU = ) as in this case the vector of linear secondary windings is in reverse phase.

When the third condition is not fulfilled neglecting short-circuit currents

parallel transformers [Fig. 1.24(a)] are changed for short-circuit resistances

[Fig. 1.24(b)]. 21 , scsc ZZ

As the currents in parallel branches are inversely related to their

resistances relative voltages (loads) of parallel transformers are inversely related

to their short-circuit voltages as well. As a result, it causes transformer overload

with less U value and underload with high U . sc sc

36

Therefore, State Standard allows parallel transformers energizing at different

short-circuit voltages if their difference does not exceed arithmetic mean 10%

( ) %101005.0 2121 +

=scsc

scscsc UU

UUU

The greater is short-circuit voltage difference the more significant is transformer

difference by voltage. State Standard recommends nominal voltage ratio of

parallel transformers to be not more than 3:1.

Besides, it is necessary to control the order of phase alternation before

energizing three-phase parallel transformers. Phase alternation order should be

similar in all transformers.

aintenance of these

regulations is checked by

transformer phasing (Fig.

1.25). In so doing each

pair of opposite terminals

of a closing switch is

connected by a conductor

(it is not shown) and

voltage is taken with zero

voltmeter between the

remaining pairs of

terminals. If secondary

transformer voltages are

equal and connecting

groups of their windings

are similar zero voltmeter reading is zero if there is similar order of phase

sequence. In this case parallel transformers may be energized. If voltmeter

37

shows some voltage it is necessary to clear out what parallel performance

condition is not fulfilled and eliminate it.

1.12. Non-Balanced Load of Three-Phase Transformers

The reasons of non-balanced load are considered to be uneven distribution

of single-phase receivers by load, emergency conditions that occur at single-

phase, two-phase short circuit or at one phase of wiring line failure.

Non-balance of transformer secondary voltages has a detrimental effect on

both the consumers and the transformer. For example, in a.c. motors permissible

load voltage decreases, durability of filament lamps is reduced at high voltage

and luminous intensity is decreased at low voltage. Overload of separate

transformer phases, excessive phase voltage increase and pole core saturation

occur.

For investigation of transformer operation at non-balanced load the

method of balanced components studied in the course Theoretical fundamentals

of electrotechnics is widely used. While considering three-phase step-down

transformer non-balanced currents of LV may be represented as the sum of three

balanced systems of positive, negative and zero sequence differing by sequence

of current passing through

++=++=++=

021

021

021

cccc

bbbb

aaaa

IIIIIIIIIIII

&&&&&&&&&&&&

(*)

The currents forming positive sequence system reach maximum

successively in phases a, b, c. The currents forming negative sequence system

reach maximum successively in phases a, b, c. Zero sequence currents in all

three phases are of one direction (zero shift).

After coefficients a, a2 are entered into the equations (*) they will be

written as follows

38

++=++=

++=

022

1

0212

021

IIaIaIIIaIaI

IIII

aac

aab

aaa

&&&&&&&

&&&&

(**)

Multiplying any vector into coefficient a does not change its absolute

value, but changes 3/2 its argument, i.e. rotates vector through 120 towards vector rotation.

From (**) currents of positive, negative and zero sequence may be

obtained through non-balanced ones

( )( )( )

++=

++=

++=

cbaS

cbaa

cbaa

IIII

IaIaII

IaIaII

&&&&

&&&&

&&&&

31

3131

22

21

(***)

On the basis of the latter equality in (***) it follows that given the

currents of zero sequence currents sum of three phases is not equal to zero.

The advantage of the method of balanced components includes the fact

that balanced system of each sequence can be transformed regardless of the

systems of other sequences using conventional methods of mathematical and

graphical analysis. However, the method of balanced components suggests

application of superposition method, which is valid only for linear systems.

Therefore, as applied to the transformer one makes assumption taking into

account the lack of pole core iron saturation ( constZ m = ) or neglecting open-circuit current ( =mZ ).

Besides, the transformer at non-balanced load is considered to possess

equal number of secondary and primary winding turns ( 21 ww = ) that does not allow to use reference procedure.

39

At balanced load when transformer phase currents make up a balanced

system one can put down . Substituting these values in

(***) we obtain

acaba IaIIaII &&&& == , , 2

( )( )( ) .01 ;1 as ,0

31

;031

;31

232

242

331

=++==++=

=++=

=++=

aaaIaIaII

IaIaII

IIaIaII

aaaS

aaaa

aaaaa

&&&&

&&&&

&&&&&

Thus, at balanced load there are currents of only positive sequence.

Therefore, all facts considered above regarding balanced load match transformer

operation with positive sequence currents.

What will happen if the position of two terminals of HV windings (for

example B and C) and LV winding (b and c) is interchanged in the transformer

under balanced load condition? Vector alternation of transformer phase currents

will change to reverse, i.e. it corresponds to negative sequence current

alternation. The duty of the transformer and consumers will not change.

Thus, negative sequence currents are converted from one winding to

another as well as positive sequence currents. Transformer operation regarding

positive and negative sequence currents is similar. Above equivalent circuits are

valid both for positive and negative sequence currents, transformer resistance

relative to these sequence currents is similar and equal to short-circuit

resistance . scZ

Currents of zero sequence in star-connected windings may develop only

with zero wire. In delta-connected windings zero sequence currents make up

current flowing across the closed circuit and linear currents as current

differences of adjacent phases do not contain zero sequence currents. Therefore

zero sequence currents in delta-connected winding may develop only as a result

of inducing them by another transformer winding.

40

Zero sequence fluxes are induced by zero sequence currents and therefore

they are in phase in time domain. Let us see how zero sequence fluxes influence

the transformer with different types of pole cores.

In three-phase transformers of shell-type, core-shell-type design and

transformer bank zero sequence fluxes S& are closed across the pole cores. Magnetic resistance for the fluxes S& is slight, therefore even small currents of zero sequence are able to develop large fluxes 000 cba III &&& == S& . If the current

equals short-circuit transformer current magnetic flux equalling

nominal running transformer flux is induced. The similar reasoning refers to

e.m.f. induced by e.m.f. flux .

0aI& S&

S&In a core-type three-phase transformer zero sequence fluxes of all the

phases tend to close from one yoke to another ( for example, in oil transformer

through oil and transformer tank. In this case magnetic resistance for the flux

is rather high and eddy currents are induced in tank walls and losses occur.

Therefore, magnetic flux and induced e.m.f. are small.

S&S&

Physical conditions of transformer operation at non-balanced load.

Case 1. Zero sequence currents are missing. At non-balanced load

voltage drop U in transformer phases is different. If currents of separate phases do not exceed nominal values U is rather small due to small resistance

of the transformer. scZ

Thus, non-balanced transformer load in missing of zero sequence currents

does not distort phase and linear voltage balance at secondary winding

terminals.

As for the case considered primary and secondary currents of positive

sequence in each phase are equal in magnitude and opposite in sign. It is valid

for the currents of negative sequence as well and for the current sum of positive

41

and negative sequences. Therefore the simplifications taken before ( and

neglecting magnetizing current) make it possible to put down, thus

21 ww =

cCbBaA IIIIII &&&&&& === ; ; . As a result, one may state that magnetizing forces and currents of primary

and secondary windings are balanced in each phase and separately in each pole

core area.

Case 2. There are currents of zero sequence. Variant a: currents of zero

sequence develop in both transformer windings. These are transformers with

winding connection Yn/Yn, /Yn. Magnetizing current of zero sequence may be neclected because it contains small total current of sequence and may be written

as:

000000 cbaCBA IIIIII &&&&&& ===== . Thus, magnetizing strength of zero sequence currents of both windings is

mutually balanced in each transformer phase. In such a situation zero sequence

resistance is . Zero components of secondary voltage originate due to

small voltage drop . Therefore, in transformers with winding connection

Yn/Yn, /Yn at non-balanced load phase voltage system is insignificantly distorted.

scS ZZ =0asc IZ &

Variant b: zero sequence currents develop only in one winding. There are

transformers with winding connection Yn/Yn. Zero sequence currents flow only

in secondary winding, they are considered to be magnetizing as they are not

balanced by the currents in primary winding. Zero sequence e.m.f. equals

, where is resistance of magnetizing circuit for zero

sequence currents. E.m.f. may reach high values. For example, in shell-type

transformers, core-shell-type design and transformer banks magnetizing circuit

resistance for zero sequence currents are equal to magnetizing circuit resistance

for positive sequence currents

000 amS IZE && = 0mZSE0

scm ZZ =0 . Therefore, at 42

43

)( na III 05.002.000 = && zero sequence e.m.f. nS UE 0 and the system of phase m.m.f. and voltages is markedly distorted that is unacceptable and dangerous for

single-phase loads. Vector direction depends on the phase of zero sequence

currents and is determined by load conditions.

S

E

&

Zero sequence e.m.f. does not influence linear voltage value, as zero

components disappear in phase voltage differences.

Winding connection Y/Yn in transformers of shell-type, shell-core-type

designs and transformer banks is not used as a rule but if necessary the third

winding is arranged in each phase. It is delta-connected winding. This winding

terminals are not derived outside if this winding is meant only for balancing zero

sequence currents.

In a core-type transformer and winding connection Y/Yn distortion of

phase voltage system with zero sequence currents is less as . scMO ZZ

F and electromagnetic torque directions will change to

the opposite ones as well.

68

The torque will be braking and the machine will run at generator effect

duty and will release active power to the mains. Slip for generator effect duty is

0S

2.6. Voltage Equations of an Induction Motor

There is no electric linking between rotor and stator winding of an

induction motor. There is only magnetic linking and stator winding energy is

conveyed to rotor winding by magnetic field. In this respect an induction motor

is analogous to the two-winding transformer, namely, stator winding is primary

and rotor winding is secondary.

Like in a transformer, in an induction motor there is resultant magnetic

flux linked both with stator and rotor windings and there are two leakage

fluxes as well. is leakage flux of stator winding and

1 2 is leakage flux of

rotor winding.

The amplitude of the resultant magnetic flux m rotating at frequency induces e.m.f. in stationary stator winding, its effective value being

1n

mwfE = 1111 44.4 . Magnetic leakage flux induces leakage e.m.f. in stator winding, the

value is determined by voltage drop in inductive of stator winding

1

111 xIjE = && ,

69

where is inductive leakage reactance of stator winding phase. 1x

Voltage equation of stator winding phase energized at voltage will be

written as follows

1U

11111 rIEEU =++ &&&& , where is voltage drop in pure resistance of stator winding phase . 11 rI 1rFinal equation does not differ from the voltage equation for primary

transformer winding

111111 rIxIjEU ++= &&&& . Resultant magnetic flux outrunning rotating rotor induces e.m.f. in

rotor winding

SEwSfwfE mwmwS === 22212222 44.444.4 where is e.m.f. frequency in the rotating rotor, is

e.m.f. induced in winding phase of a stationary rotor.

Sff = 12 SE2 2E

Magnetic leakage flux induces leakage e.m.f. in rotor winding, the

value of which is determined by the voltage drop in inductive reactance of this

winding

2

SxIjE = 222 && , where is inductive reactance of rotor winding leakage phase of a

stationary rotor.

2x

The voltage equation for rotor winding is

2222 rIEE S =+ &&& , where is pure resistance of rotor winding phase. 2r

Final equation is written

70

0/22222 = SrIxIjE &&& .

2.7. Equations of M.M.F. and Induction Motor Currents

The resultant magnetic flux in an induction motor is produced by joint

action of m.m.f. of stator and rotor windings 1F 2F

( ) mm RFRFF // 021 &&&& =+= , where is magnetic resistance of motor magnetic circuit, is resultant

m.m.f. which is equal to winding m.m.f. of stator at open-circuit duty

mR 0F

pwImF w11010 45.0 = ,

where is open-circuit current in stator winding phase. 0I

M.m.f. of stator and rotor windings per a pole provided motor running

under load conditions are

pwImF w11111 45.0 = ;

wImF w22222 45.0 = ,

where is number of rotor winding phases, is winding coefficient of rotor

winding .

2m 2w

When changing the load on motor shaft the currents in stator and rotor

change as well. The resultant magnetic flux remains unchanged as the

voltage applied to stator winding is invariable (

1I

2I

constU =1 ) and is balanced by stator winding e.m.f. 1E

11 EU && .

71

As e.m.f. is proportional to the resultant magnetic flux it remains invariable at load change

1E

constFFF =+= 210 &&& ,wImwImwIm www 222211111101 45.045.045.0 += &&& .

Dividing this equality into wm w11145.0 we obtain the current equation of an induction motor

21111222210 IIwmwmIII ww +=+= &&&& ,

where 111

22222

w

w

wmwmII =& is rotor current referred to stator winding.

The final current equation of an induction motor is

( )201 III += &&& . From this equation it follows that there are two components in stator

current of an induction motor, i.e. magnetizing ( almost-constant )

component ( ) and alternating component compensating rotor

winding m.m.f. Thus, the rotor winding current exerts the same demagnetizing

action on engine magnetic system as the secondary winding current of a

transformer does.

0I&

opII 0 2I &

2.8. Referred Parameters of Rotor Winding, Vector Diagram

and Equivalent Circuit of an Induction Motor

Rotor winding parameters are brought to the form of stator winding so

that e.m.f. vectors, voltages and current of stator and rotor windings could be

shown in one vector diagram. In so doing rotor winding with phase number ,

with phase turn number and winding coefficient is substituted for the

2m

2w 2w

72

winding with values , , and the powers, and phase shifts of e.m.f.

vectors and rotor currents should be unchanged.

1m 1w 1w

Given stationary rotor referred rotor e.m.f. is e22 kEE = , where ( )221121e / wwEEk ww == is the transformation ratio of an induction motor

voltage under the stationary rotor.

Referred rotor current is ikII /22 = , where is the transformation ratio of induction

motor current.

( ) 2e1222111 // mkmwmwmk wwi ==

Unlike transformers transformation coefficients of voltage and current of

induction motors are not equal ( ikk e ). This is because the phase numbers in windings of stator and short-circuit rotor are not equal ( 21 mm ). Only in phase rotor engines with these coefficients are equal. 21 mm =

Referred resistances of rotor winding phase are

ikkrr = e22 ;

ikkxx = e22 .

There is the specificity of

determination of phase number

and phase turns number .

Each limb of this winding is

considered to be one phase,

therefore phase turns number is

2m 2w

5.02 =w , winding coefficient being 12 =w and phase number equals limbs number in short-

circuit rotor winding, i.e.

22 zm = . Voltage equation of rotor

winding in the referred form is

0/22222 = srIxIjE &&

Fig. 2.8

Value may be written as follows sr /2( ) ssrrrssrsrsr /1/// 222222 +=+= ,

As a result, the voltage equation for rotor winding in the referred form

becomes

( ) ssrIrIxIjE /10 2222222 = &&&& . Hence it follows that induction motor is electrically much like the

transformer running at resistive load.

For induction motors just as for transformers the vector diagram is plotted

by equations of currents and voltages of stator and rotor windings (Fig. 2.8).

73

74

)Phase shift angle between e.m.f. and current is

. Electrical equivalent circuits of an induction motor

correspond to voltage and current equations and to vector diagram as well.

2E& 2I &( 222 / rsxarctg =

In Fig. 2.9(a) T-shape equivalent circuit is shown. Magnetic linking of stator and

rotor windings is substituted for electric linking as it takes place in transformer

equivalent circuit. Pure resistance ( ) ssr /12 may be considered as external alternating resistance cut in to stationary rotor winding. This resistance value is

defined by the slip, i.e. mechanical load on motor shaft.

L-shaped equivalent circuit in which magnetizing circuit is taken out to

input terminals of equivalent circuit is more convenient for practical application.

To keep invariable the value of open-circuit current resistances of stator

winding phases and are turned on in series to this circuit. The obtained

circuit is convenient as it consists of two parallel circuits, namely, magnetizing

one with current and operating circuit with current

0I&

1r 1x

0I& ( )12 cI & . Parameter calculation of L-shaped operating equivalent circuit requires

improvement by introducing coefficient into design formulae such as the ratio

of phase voltage supply circuit and phase e.m.f. of stator winding at ideal

open-circuit duty (

1c

1U

0=s ). As open-circuit current is small at this duty turns out to be not much larger than e.m.f. and coefficient slightly differs from

unity. For motors of 3 kW and above we obtain

1U

1E 1c

02.105.11 =c .

75

Fig. 2.9

76

2.9. Energetic Diagrams of Active and Reactive Power of an

Induction Motor

Energetic diagram of induction motor active power (Fig. 2.10) may be

shown in the following way.

A motor consumes the active power from the mains

11111 cos= IUmP . Some part of this power is lost as electrical losses in pure resistance of

stator winding , another portion is lost in the form of magnetic

stator core losses .

12

111 rImpel =mm rImp = 201

The remaining active power is electromagnetic power released by

magnetic field from stator to rotor

emP

( ) srImsrImppPP melem // 2222222111 === .

Fig. 2.10

Some electromagnetic voltage is lost as electrical losses in pure resistance

of rotor winding . ( ) 222222212 rImrImpel ==

77

The remainder of this power is converted into mechanical power induced

in the rotor

( ) ( ) ( ssrImssrImpPP elemmec /1/1 222222212 === )mec

mecp

adp

.

Some part of mechanical energy P is

lost inside the machine itself in the form of

mechanical losses (for ventilation,

friction in bearings and on the brushes of

slip-ring induction motor if the brushes do

not raise under operating conditions) and

additional losses (due t high

harmonics of winding m.m.f. and stator

and rotor toothing). Fig. 2.11

Net mechanical power on the shaft is ad2 ppPP mecmec = . The sum of motor losses is

ad21 pppppP mecelmel ++++= , = PPP 12 .

Motor efficiency is ( )112 /1/ PPPP == . It is necessary to mention the following important relations emel Psp =2 ,

. They show that for reducing and increasing efficiency it

is required to obtain small slip

( ) emmec PsP = 1 2elps of the motor.

Nominal values of efficiency, slip and power factor of modern general-

purpose induction motors are

95.072.0 =n ; 05.002.0 =ns ; 95.070.0cos =n .

78

Energetic diagram of induction motor reactive power (Fig. 2.11) may be

shown in the following way.

The motor consumes the reactive power from the mains

11111 sinQ = IUm . For leakage flux of stator and rotor winding initiation reactive powers

, are used. 12111 xImq = ( ) 222222212 xImxImq ==

Reactive power spent for motor resultant magnetic flux

is the main portion of reactive power of the motor, which is significantly higher

than in transformers due to the air gap. Large values and significantly

influence power factor of the motor and reduce its value.

mm xIm = 201Q

mQ 0I

2.10. Induction Motor Torques

Electromagnetic torque of an induction motor is produced by current

interaction in rotor winding with rotating magnetic flux and is proportional to

electromagnetic power

( ) ( ) ( )srImspPM elem === 12221121 //// , where pfn /260/2 111 == is angular rotation frequency of magnetic flux.

The above expression shows that electromagnetic torque is proportional to

electrical losses power in rotor winding. From L-shape equivalent circuit the

current in the closed working circuit is

79

)( ) (( ) ( )22112211

1

22

2111

21

2111

112

/

/

xcxsrcr

Uxcxcsrcrc

UcI

+++

=+++

=

Electromagnetic torque formula becomes

( ) ( )[ ]221122111 2211

/2 xcxsrcrsfrUmpM +++= .

Parameter values of an induction motor equivalent circuit at load change

remain practically invariable as well as the voltage in winding phase and

frequency .

1U

1f

Therefore one may conclude that electromagnetic torque at any slip value

is proportional to phase voltage squared (phase rotor current squared). The less

is electromagnetic torque, the larger are such parameters of equivalent circuit as

, , . 1r 1x 2xConsider the dependence (relation) of electromagnetic torque on slip

at , and fixed parameters of equivalent circuit

(Fig. 2.12). This dependence is called mechanical characteristic of an induction

motor.

( )sfM = constU =1 constf =1

Under the slip value 0=s and =s the electromagnetic torque 0=M . Mechanical characteristic exhibits two extrema and maximum induction motor

torque at generator effect duty is slightly larger than at driving duty

( mmmg MM > ).

Critical slip value corresponding to maximum torque is obtained from

the first derivative of the expression for electromagnetic torque, which is

equated to zero

scs

80

( )22112111 / xcxrrscr ++= .

Fig. 2.12

Substituting the expression of critical slip to the formula of

electromagnetic torque we obtain the expression for the maximum

electromagnetic torque

( )[ ]2211211112

11

4 xcxrrcfUmpM m +++= ,

where the sign (+) corresponds to driving and the sign (-) corresponds to

generator effect duty of an induction motor.

Electromagnetic torque reaches maximum value at crss = and further in spite of the increase of the torque reduces as current becomes more

inductive ( ). As noted above the active component of

current determines the value . This active component first increases as

2I 2I ([ // 222 srxarctg = )]

2I 2I increases and then it reduces in spite of the increase of 2I . It should be taken

81

into account that with increase the voltage drop in stator winding increases

and, as a result, e.m.f. and flux

1I

1E somewhat reduce. For general-purpose induction motors it is defined as ( )2111 xcxr +

82

When analysing induction motor operation we shall use mechanical

characteristic shown in Fig. 2.13. When cutting the motor in the

magnetic stator flux possessing no inertia begins rotating at synchronous

frequency and engine rotor under the effect of inertia forces remains

stationary ( ) and slip

( )sfM =

1n

0=n 1=s . Expression of initial starting electromagnetic torque of the motor is

[ ]2' 2112'2111'

22

11

)()(2 xcxrcrfrUmpM S +++++= .

Engine rotor begins rotating under the effect of this torque. In this case the

slip decreases and the torque increases according to characteristic ( )sfM = . Under the critical slip the torque reaches maximum value . At further

rotation frequency increase torque reduces until it reaches the value which is

equal to the sum of opposing torques applied to engine rotor, namely, open-

circuit torque and net torque (

crs m

0M 2M st20 MMMM =+= - static torque ). It should be taken into account that at slips close to unity (starting motor

duty) equivalent circuit parameters significantly change their values. The

reasons are considered to be amplification of magnetic saturation of stator and

rotor teeth layers ( inductive reactance of leakage and decreases ),

current displacement effect in rotor bars (increase of pure resistance and

1x 2x2r 2x

decrease). Calculation of starting characteristics is made by the corresponding

parameters of equivalent circuit.

Static torque is equal to the sum of opposing torques at uniform rotor

rotation ( ). At rated load of the motor steady duty of motor operation

is determined according to mechanical characteristic by point with coordinates

and .

stM

constn =

nMM = nss =

83

Mechanical characteristic analysis shows that a stable induction motor

performance is possible at slips being less than critical ( crss < ), i.e. in the area of mechanical characteristic. It is in this area that the load change on motor

shaft is accompanied by the corresponding change of electromagnetic torque.

When a motor runs at the nominal load there is equality of torques

. If the net load torque increase up to value then the torque

equality is impaired and rotor rotation frequency begins to reduce (slip

increases). It brings to electromagnetic torque increase up to value

(point ) and motor duty becomes stable again. If the motor was

running at nominal load and net load torque decrease up to value occurred

the torque equality is impaired again but rotor rotation frequency begins

increasing (the slip decreases). It brings to electromagnetic torque decrease

up to value (point ). Stable running conditions are restored

again but at another values of and s.

nn MMM 20 += 2

20 MMM +=2M

20 MMM +=

Induction motor operation is unstable at slips . If one obtains

electromagnetic motor torque

crss mMM = and slip crss = then even a slight

increase of load torque brings to electromagnetic torque M decrease. Futher slip

increase follows until it reaches the value 1=s , i.e. until the rotor stops running.

84

2.11. Starting Three-Phase Induction Motors ( IM )

Induction motor starting requirements are the following:

Induction motor should develop a starting torque large enough to

make the rotor rotate and reach nominal frequency.

Starting current should be limited by the value at which motor

damage and normal running duty impairment do not occur.

Starting diagram should be simple, the number and cost of starting

devices should be small.

2.11.1. Starting of Squirrel-Cage Induction Motor

Direct starting. It is the simplest mode of starting. Stator winding is cut in

directly to the mains at nominal voltage (Fig. 2.14). Starting current is

nscscns IrUI 122

11 )74( =+= . Direct starting is possible in case of

powerful mains and starting current of

induction motor does not cause large

voltage drops in the mains (not more than

1015%). Three modes of low-voltage starting.

These modes of starting are used if direct

starting is not available under the condition

of permissible voltage drop in the mains.

Starting torque decrease ( ) is 21s UM considered to be the drawback in this case. Fig. 2.14

85

There for these modes may be realized when starting of an induction

motor at light-running or partial load is possible. This often occurs in powerful

high-voltage motors.

Reactor starting (Fig. 2.15). The first switch SW1 is cut in. Voltage is

applied to stator winding via three-phase reactor R; therefore stator winding is

powered by reduced voltage.

Reactor reactance is

chosen so that voltage in stator

winding phase is not less than

65% of the nominal one.

rx

Having reached stable

rotating frequency the switch

SW2 is cut in. It shunts reactor R

and as a result full line voltage,

which is equal to nominal stator

winding, is applied across the

terminals of stator winding.

Fig. 2.15 Starting current at reactor starting is ( )2211 rscscnsr rUI ++= and it decreases as compared with the current under the direct starting

( )22

22

1

1

scsc

rscsc

sr

s

rr

II

+++= .

The voltage across stator winding terminals decreases at initial stage of

starting the same number of times.

Initial stage of reactor starting decreases as compared with the initial

stage of direct starting

srM

s

86

( )22

22

sr

s

scsc

rscsc

rr

M

+++= times.

In the above relations the changes of the value at starting are not taken

into account. It is not difficult to do if necessary.

sc

Autotransformer starting (Fig. 2.16). At first switches SW1 and SW2 are

cut in and reduced up to ( nU173.055.0 ) voltage is applied to stator winding of induction motor via autotransformer AT.

After a stable rotation frequency is reached the switch SW2 is cut off and

the voltage is applied to stator winding via some winding portion of

autotransformer, the latter working like a reactor in this case. Then switch SW3

is cut in and full line voltage equal to

nominal voltage of stator winding is

applied across the terminals of stator

winding.

If the starting autotransformer decreases

starting voltage of IM k times ( is

transformation ratio of autotransformer),

then the starting current of IM and current

across low-voltage side of

autotransformer also decrease times.

AT ATk

ATk

Starting torque , proportional to

squared voltage across the terminals of

IM stator winding will reduce times.

s

2ATk

Fig. 2.16

Starting current across high-voltage side and supply current decrease

times as well.

2ATk

Thus, at autotransformer starting IM starting torque and starting supply

current reduce a similar number of times. At reactor starting IM starting current

is also starting supply current and starting torque decreases more rapidly

than the starting current. Therefore, at similar values of starting current the

starting torque will be higher at autotransformer starting. In spite of this

advantage of the autotransformer starting over the reactor starting, which is

achieved at the expense of more complicated construction and rise in price of

starting devices, this mode of starting is used seldom compared with reactor one

when reactor starting does not provide necessary starting torque.

s

Starting by star-delta switching (Fig. 2.17). This mode was widely used

at low-voltage IM starting but had

lost its significance at mains

power increase and is used seldom

now.

For its application all the six

terminals of stator winding are

brought out. In so doing line

voltage equals nominal phase

voltage of stator winding. At the

very starting stator winding is

star-connected. When stable

rotation frequency is achieved

winding connection diagram

changes for delta connection by

switching SW. Fig. 2.17 Fig. 2.17

Under this mode of starting

87

88

voltage across stator winding phases is applied 3 times decreased compared

with nominal one, starting torque decreases threefold, starting phase current

decreases 3 times and starting line current decreases threefold. Thus, the

considered mode of starting is equal to autotransformer starting at 3AT =k but commutational overvoltage occurs in stator winding of induction motor at

starting switch.

2.11.2. Slip-Ring Induction Motor Starting

Starting rheostat possessing

several stages as a rule is cut in to rotor

winding circuit. It is calculated for

instantaneous current flows (Fig. 2.18).

Initial starting torque may be

increased up to maximum motor torque

maxs MM = at fixed resistsnce of starting rheostat RRs ( )ms= (Fig. 2.19). Resistance value of starting rheostat

may be determined by equating

critical slip to unity, i.e.

( )mRs

( )( ) ( ) 1221121s21 =+++= xcxrRrs mcr . Fig. 2.18 Referred active phase resistance of starting rheostat is

( ) ( ) 121221121s / crcxcxrR m ++= . Actual resistance of starting rheostat is defined as follows

( ) ( ) ( )21122s / wwmm kwkwRR = .

89

Usually one chooses .

With rotor rotation frequency increase

starting rheostat resistance is reduced

changing one stage for another. Starting

rheostat stages are calculated so that

during switch the torque should be

changed within the chosen range from

up to .

( )mRR ss

maxs,M mins,M

Fig. 2.19

2.12. Regulation of Induction Motor Rotation Frequency

Rotor rotation frequency of IM is ( ) ( )( )spfsnn == 1/601 11 . It follows from this expression that rotor rotation frequency may be regulated by

changing any of three values, namely, slip s, current frequency in stator winding

and pole number of stator winding 2p. 1f

Rotation frequency regulation by slip change occurs only in a loaded

induction motor. Under light running conditions the slip and rotor rotation

frequency remain practically invariable.

Evaluation assessment of any mode of rotation frequency regulation is

made according to the following indices:

possible regulation range,

smooth regulation,

drive efficiency change at regulation.

90

Rotation frequency regulation by changing supplied voltage. Induction

motor torque is proportional to ; therefore mechanical motor characteristics

at voltages less than nominal one (Fig. 2.20) are located under the natural

characteristic.

21U

If static torque is

constant, then the slip of

induction motor increases at

voltage drop in stator winding,

rotor rotation frequency

decreases. Slip regulation using

this mode is possible within the

range

stM

scss

91

. Therefore, resultant electromagnetic torque

of induction motor decreases:

contains reverse component ( reversing field ) which develops torque rM

directed opposition to torque op

r= o

ase supply (2), the latter being asymmetrical limit of three-phase

voltag

single-

phase

s a rule this

rotatio

ostat

construction this rotation frequency regulation may be smooth or stepwise.

p .

Fig. 2.21

Mechanical motor characteristics in this case [Fig. 2.21(a)] are positioned

in th range between symmetrical voltage characteristic (1) and characteristic at

single-ph

e.

Asymmetry regulation of applied voltage is provided by cutting in

regulation autotransformer AT [Fig. 2.21(b)] to one of the phases.

The drawback of this mode of regulation is a narrow regulation range and

efficiency decrease of the motor at asymmetrical voltage increase. A

n frequency regulation is used only in smaller rating motors.

Rotating frequency regulation by changing the pure resistance in rotor

circuit. This regulation of rotation frequency is available only in wound rotor

induction motors. Regulation rheostat similar to starting one but meant for

continuous duty is cut in to rotor circuit. Depending on regulation rhe

92

Mech