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Transcript of Electrical Machines
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___________________________________________________________________________________________
.. , .. , ..
ELECTRICAL MACHINES
2006
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621.313 31 .., .., .. 31 ELECTRICAL MACHINES: . : -
, 2006. 176 .
ISBN , ,
. . 3-
621.313
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, .
, ..
, 2006
ISBN : , 2006
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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.
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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.
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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
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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
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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
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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.
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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 .
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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
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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
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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
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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 +
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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 =
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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.
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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
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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
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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
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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 = .
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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.
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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
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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
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Mech