transformer engineering design and practice 1_phần 3 docx

As the transmission voltages areincreased to higher levels in some part of the power system, transformers againplay a key role in interconnection of systems at different voltage levels.T

Transformer Fundamentals

1.1 Perspective

A transformer is a static device that transfers electrical energy from one circuit toanother by electromagnetic induction without the change in frequency Thetransformer, which can link circuits with different voltages, has been instrumental

in enabling universal use of the alternating current system for transmission anddistribution of electrical energy Various components of power system, viz.generators, transmission lines, distribution networks and finally the loads, can beoperated at their most suited voltage levels As the transmission voltages areincreased to higher levels in some part of the power system, transformers againplay a key role in interconnection of systems at different voltage levels.Transformers occupy prominent positions in the power system, being the vitallinks between generating stations and points of utilization

The transformer is an electromagnetic conversion device in which electricalenergy received by primary winding is first converted into magnetic energy which

is reconverted back into a useful electrical energy in other circuits (secondarywinding, tertiary winding, etc.) Thus, the primary and secondary windings are notconnected electrically, but coupled magnetically A transformer is termed as either

a step-up or step-down transformer depending upon whether the secondaryvoltage is higher or lower than the primary voltage, respectively Transformers can

be used to either step-up or step-down voltage depending upon the need andapplication; hence their windings are referred as high-voltage/low-voltage orhigh-tension/low-tension windings in place of primary/secondary windings

Magnetic circuit: Electrical energy transfer between two circuits takes place

through a transformer without the use of moving parts; the transformer thereforehas higher efficiency and low maintenance cost as compared to rotating electrical

machines There are continuous developments and introductions of better grades

of core material The important stages of core material development can besummarized as: non-oriented silicon steel, hot rolled grain oriented silicon steel,cold rolled grain oriented (CRGO) silicon steel, Hi-B, laser scribed andmechanically scribed The last three materials are improved versions of CRGO.Saturation flux density has remained more or less constant around 2.0 Tesla forCRGO; but there is a continuous improvement in watts/kg and volt-amperes/kgcharacteristics in the rolling direction The core material developments arespearheaded by big steel manufacturers, and the transformer designers canoptimize the performance of core by using efficient design and manufacturingtechnologies The core building technology has improved from the non-mitred tomitred and then to the step-lap construction A trend of reduction of transformercore losses in the last few years is the result of a considerable increase in energycosts The better grades of core steel not only reduce the core loss but they alsohelp in reducing the noise level by few decibels Use of amorphous steel fortransformer cores results in substantial core loss reduction (loss is about one-thirdthat of CRGO silicon steel) Since the manufacturing technology of handling thisbrittle material is difficult, its use in transformers is not widespread

Windings: The rectangular paper-covered copper conductor is the most

commonly used conductor for the windings of medium and large powertransformers These conductors can be individual strip conductors, bunchedconductors or continuously transposed cable (CTC) conductors In low voltageside of a distribution transformer, where much fewer turns are involved, the use ofcopper or aluminum foils may find preference To enhance the short circuitwithstand capability, the work hardened copper is commonly used instead of softannealed copper, particularly for higher rating transformers In the case of agenerator transformer having high current rating, the CTC conductor is mostlyused which gives better space factor and reduced eddy losses in windings Whenthe CTC conductor is used in transformers, it is usually of epoxy bonded type toenhance its short circuit strength Another variety of copper conductor oraluminum conductor is with the thermally upgraded insulating paper, which issuitable for hot-spot temperature of about 110°C It is possible to meet the specialoverloading conditions with the help of this insulating paper Moreover, the aging

of winding insulation material will be slowed down comparatively For bettermechanical properties, the epoxy diamond dot paper can be used as an interlayerinsulation for a multi-layer winding High temperature superconductors may findtheir application in power transformers which are expected to be availablecommercially within next few years Their success shall depend on economicviability, ease of manufacture and reliability considerations

Insulation and cooling: Pre-compressed pressboard is used in windings as

opposed to the softer materials used in earlier days The major insulation (betweenwindings, between winding and yoke, etc.) consists of a number of oil ducts

formed by suitably spaced insulating cylinders/barriers Well profiled angle rings,angle caps and other special insulation components are also used.

Mineral oil has traditionally been the most commonly used electrical insulatingmedium and coolant in transformers Studies have proved that oil-barrierinsulation system can be used at the rated voltages greater than 1000 kV A highdielectric strength of oil-impregnated paper and pressboard is the main reason forusing oil as the most important constituent of the transformer insulation system.Manufacturers have used silicon-based liquid for insulation and cooling Due tonon-toxic dielectric and self-extinguishing properties, it is selected as areplacement of Askarel High cost of silicon is an inhibiting factor for itswidespread use Super-biodegradable vegetable seed based oils are also availablefor use in environmentally sensitive locations

There is considerable advancement in the technology of gas immersedtransformers in recent years SF6 gas has excellent dielectric strength and is non-flammable Hence, SF6 transformers find their application in the areas where fire-hazard prevention is of paramount importance Due to lower specific gravity ofSF6 gas, the gas insulated transformer is usually lighter than the oil insulatedtransformer The dielectric strength of SF6 gas is a function of the operatingpressure; the higher the pressure, the higher the dielectric strength However, theheat capacity and thermal time constant of SF6 gas are smaller than that of oil,resulting in reduced overload capacity of SF6 transformers as compared to oil-immersed transformers Environmental concerns, sealing problems, lowercooling capability and present high cost of manufacture are the challenges whichhave to be overcome for the widespread use of SF6 cooled transformers

Dry-type resin cast and resin impregnated transformers use class F or C

insulation High cost of resins and lower heat dissipation capability limit the use ofthese transformers to small ratings The dry-type transformers are primarily usedfor the indoor application in order to minimize fire hazards Nomex paperinsulation, which has temperature withstand capacity of 220°C, is widely used fordry-type transformers The initial cost of a dry-type transformer may be 60 to 70%higher than that of an oil-cooled transformer at current prices, but its overall cost

at the present level of energy rate can be very much comparable to that of the cooled transformer

oil-Design: With the rapid development of digital computers, the designers are freed

from the drudgery of routine calculations Computers are widely used foroptimization of transformer design Within a matter of a few minutes, today’scomputers can work out a number of designs (by varying flux density, corediameter, current density, etc.) and come up with an optimum design The realbenefit due to computers is in the area of analysis Using commercial 2-D/3-Dfield computation software, any kind of engineering analysis (electrostatic,electromagnetic, structural, thermal, etc.) can be performed for optimization andreliability enhancement of transformers

Manufacturing: In manufacturing technology, superior techniques listed below

are used to reduce manufacturing time and at the same time to improve the productquality:

- High degree of automation for slitting/cutting operations to achieve betterdimensional accuracy for the core laminations

- Step-lap joint for core construction to achieve a lower core loss and noise level;top yoke is assembled after lowering windings and insulation at the assemblystage

- Automated winding machines for standard distribution transformers

- Vapour phase drying for effective and fast drying (moisture removal) andcleaning

- Low frequency heating for the drying process of distribution transformers

- Pressurized chambers for windings and insulating parts to protect againstpollution and dirt

- Vertical machines for winding large capacity transformer coils

- Isostatic clamping for accurate sizing of windings

- High frequency brazing for joints in the windings and connections

Accessories: Bushings and tap changer (off-circuit and on-load) are the most

important accessories of a transformer The technology of bushing manufacturehas advanced from the oil impregnated paper (OIP) type to resin impregnatedpaper (RIP) type, both of which use porcelain insulators The silicon rubberbushings are also available for oil-to-air applications Due to high elasticity andstrength of the silicon rubber material, the strength of these bushings againstmechanical stresses and shocks is higher The oil-to-SF6 bushings are used in GIS(gas insulated substation) applications

The service reliability of on load tap changers is of vital importance since thecontinuity of the transformer depends on the performance of tap changer for theentire (expected) life span of 30 to 40 years It is well known that the tap changerfailure is one of the principal causes of failure of transformers Tap changers,particularly on-load tap changers (OLTC), must be inspected at regular intervals

to maintain a high level of operating reliability Particular attention must be givenfor inspecting the diverter switch unit, oil, shafts and motor drive unit Themajority of failures reported in service are due to mechanical problems related tothe drive system, for which improvements in design may be necessary For servicereliability of OLTCs, several monitoring methods have been proposed, whichinclude measurement of contact resistance, monitoring of drive motor torque/current, acoustic measurements, dissolved gas analysis and temperature risemeasurements

Diagnostic techniques: Several on-line and off-line diagnostic tools are available

for monitoring of transformers to provide information about their operatingconditions Cost of these tools should be lower and their performance reliability

should be higher for their widespread use The field experience in some of themonitoring techniques is very much limited A close cooperation betweenmanufacturers and utilities is necessary for developing good monitoring anddiagnostic systems for transformers.

Transformer technology is developing at a tremendous rate The computerizedmethods are replacing the manual working in the design Continuousimprovements in material and manufacturing technologies along with the use ofadvanced computational tools have contributed in making transformers moreefficient, compact and reliable The modern information technology, advanceddiagnostic tools and several emerging trends in transformer applications areexpected to fulfill a number of existing and future requirements of utilities andend-users of transformers

1.2 Applications and Types of Transformers

Before invention of transformers, in initial days of electrical industry, power wasdistributed as direct current at low voltage The voltage drop in lines limited theuse of electricity to only urban areas where consumers were served withdistribution circuits of small length All the electrical equipment had to bedesigned for the same voltage Development of the first transformer around 1885dramatically changed transmission and distribution systems The alternatingcurrent (AC) power generated at a low voltage could be stepped up for thetransmission purpose to higher voltage and lower current, reducing voltage dropsand transmission losses Use of transformers made it possible to transmit thepower economically hundreds of kilometers away from the generating station.Step-down transformers then reduced the voltage at the receiving stations fordistribution of power at various standardized voltage levels for its use by theconsumers Transformers have made AC systems quite flexible because thevarious parts and equipment of the power system can be operated at economicalvoltage levels by use of transformers with suitable voltage ratio A single-linediagram of a typical power system is shown in figure 1.1 The voltage levelsmentioned in the figure are different in different countries depending upon theirsystem design Transformers can be broadly classified, depending upon theirapplication as given below

a Generator transformers: Power generated at a generating station (usually at

a voltage in the range of 11 to 25 kV) is stepped up by a generator transformer to

a higher voltage (220, 345, 400 or 765 kV) for transmission The generatortransformer is one of the most important and critical components of the powersystem It usually has a fairly uniform load Generator transformers are designedwith higher losses since the cost of supplying losses is cheapest at the generatingstation Lower noise level is usually not essential as other equipment in thegenerating station may be much noisier than the transformer

Generator transformers are usually provided with off-circuit tap changer with a

Chapter 1

Figure 1.1 Different types of transformers in a typical power system

small variation in voltage (e.g., ±5%) because the voltage can always becontrolled by field of the generator Generator transformers with OLTC are alsoused for reactive power control of the system They may be provided with acompact unit cooler arrangement for want of space in the generating stations(transformers with unit coolers have only one rating with oil forced and air forcedcooling arrangement) Alternatively, they may also have oil to water heatexchangers for the same reason It may be economical to design the tap winding as

a part of main HV winding and not as a separate winding This may be permissiblesince axial short circuit forces are lower due to a small tapping range Special carehas to be taken while designing high current LV lead termination to avoid any hot-spot in the conducting metallic structural parts in its vicinity The epoxy bondedCTC conductor is commonly used for LV winding to minimize eddy losses andprovide greater short circuit strength Severe overexcitation conditions are takeninto consideration while designing generator transformers

b Unit auxiliary transformers: These are step-down transformers with primary

connected to generator output directly The secondary voltage is of the order of 6.9

kV for supplying to various auxiliary equipment in the generating station

c Station transformers: These transformers are required to supply auxiliary

equipment during setting up of the generating station and subsequently duringeach start-up operation The rating of these transformers is small, and theirprimary is connected to a high voltage transmission line This may result in asmaller conductor size for HV winding, necessitating special measures forincreasing the short circuit strength The split secondary winding arrangement isoften employed to have economical circuit breaker ratings

d Interconnecting transformers: These are normally autotransformers used to

interconnect two grids/systems operating at two different system voltages (say,

400 and 220 kV or 345 and 138 kV) They are normally located in thetransmission system between the generator transformer and receiving endtransformer, and in this case they reduce the transmission voltage (400 or 345 kV)

to the sub-transmission level (220 or 138 kV) In autotransformers, there is noelectrical isolation between primary and secondary windings; some volt-amperesare conductively transformed and remaining are inductively transformed.Autotransformer design becomes more economical as the ratio of secondaryvoltage to primary voltage approaches towards unity These are characterized by awide tapping range and an additional tertiary winding which may be loaded orunloaded Unloaded tertiary acts as a stabilizing winding by providing a path forthe third harmonic currents Synchronous condensers or shunt reactors areconnected to the tertiary winding, if required, for reactive power compensation Inthe case of an unloaded tertiary, adequate conductor area and proper supportingarrangement are provided for withstanding short circuit forces underasymmetrical fault conditions

e Receiving station transformers: These are basically step-down transformers

reducing transmission/sub-transmission voltage to primary feeder level (e.g., 33kV) Some of these may be directly supplying an industrial plant Loads on thesetransformers vary over wider limits, and their losses are expensive The farther thelocation of transformers from the generating station, the higher the cost ofsupplying the losses Automatic tap changing on load is usually necessary, andtapping range is higher to account for wide variation in the voltage A lower noiselevel is desirable if they are close to residential areas

f Distribution transformers: Using distribution transformers, the primary feeder

voltage is reduced to actual utilization voltage (~415 or 460 V) for domestic/industrial use A great variety of transformers fall into this category due to manydifferent arrangements and connections Load on these transformers varieswidely, and they are often overloaded A lower value of no-load loss is desirable toimprove all-day efficiency Hence, the no-load loss is usually capitalized with ahigh rate at the tendering stage Since very little supervision is possible, usersexpect the least maintenance on these transformers The cost of supplying lossesand reactive power is highest for these transformers

Classification of transformers as above is based on their location and broadfunction in the power system Transformers can be further classified as per theirspecific application as given below In this chapter, only main features arehighlighted; details of some of them are discussed in the subsequent chapters

g Phase shifting transformers: These are used to control power flow over

transmission lines by varying the phase angle between input and output voltages

of the transformer Through a proper tap change, the output voltage can be made

to either lead or lag the input voltage The amount of phase shift required directlyaffects the rating and size of the transformer Presently, there are two types ofdesign: single-core and two-core design Single-core design is used for smallphase shifts and lower MVA/voltage ratings Two-core design is normally used forbulk power transfer with large ratings of phase shifting transformers It consists oftwo transformers, one associated with the line terminals and other with the tapchanger

h Earthing or grounding transformers: These are used to get a neutral point that

facilitates grounding and detection of earth faults in an ungrounded part of anetwork (e.g., the delta connected systems) The windings are usually connected

in the zigzag manner, which helps in eliminating third harmonic voltages in thelines These transformers have the advantage that they are not affected by a DCmagnetization

i Transformers for rectifier and inverter circuits: These are otherwise normal

transformers except for the special design and manufacturing features to take intoaccount the harmonic effects Due to extra harmonic losses, operating flux density

in core is kept lower (around 1.6 Tesla) and also conductor dimensions are smallerfor these transformers A proper de-rating factor is applied depending upon themagnitudes of various harmonic components A designer has to adequately checkthe electromagnetic and thermal aspects of design For transformers used withHVDC converters, insulation design is the most challenging design aspect Theinsulation has to be designed for combined AC-DC voltage stresses.

j Furnace duty transformers: These are used to feed the arc or induction

furnaces They are characterized by a low secondary voltage (80 to 1000 V) andhigh current (10 to 60 kA) depending upon the MVA rating Non-magnetic steel isinvariably used for the LV lead termination and tank in the vicinity of LV leads toeliminate hot spots and minimize stray losses High current bus-bars areinterleaved to reduce the leakage reactance For very high current cases, the LVterminals are in the form of U-shaped copper tubes of certain inside and outsidediameters so that they can be cooled by oil/water circulation from inside In manycases, a booster transformer is used along with the main transformer to reduce therating of tap-changers

k Freight loco transformers: These are mounted on the locomotives within the

engine compartment itself The primary voltage collected from an overhead line isstepped down to an appropriate level by these transformers for feeding to therectifiers, whose output DC voltage drives the locomotives The structural designshould be such that it can withstand vibrations Analysis of natural frequencies ofvibration is done to eliminate possibility of resonance

l Hermetically sealed transformers: This construction does not permit any

outside atmospheric air to get into the tank It is completely sealed without anybreathing arrangement, obviating need of periodic filtration and other normalmaintenance These transformers are filled with mineral oil or synthetic liquid as acooling/dielectric medium and sealed completely by having an inert gas, likenitrogen, between the coolant and top tank plate The tank is of welded coverconstruction, eliminating the joint and related leakage problems Here, theexpansion of oil is absorbed by the inert gas layer The tank design should besuitable for pressure buildup at elevated temperatures The cooling is not effective

at the surface of oil, which is at the highest temperature In another type of sealedconstruction, these disadvantages are overcome by deletion of the gas layer Theexpansion of oil is absorbed by the deformation of the cooling system, which can

be an integral part of the tank structure

m Outdoor and indoor transformers: Most of the transformers are of outdoor

duty type, which have to be designed for withstanding atmospheric pollutants.The creepage distance of bushing insulator gets decided according to thepollution level The higher the pollution level, the greater the creepage distancerequired from the live terminal to ground Contrary to the outdoor transformers,

an indoor transformer is designed for installation under a weatherproof roof and/

or in a properly ventilated room Standards define the minimum ventilationrequired for an effective cooling Adequate clearances are kept between thewalls and transformer to eliminate the possibility of higher noise level due toreverberations

There are many more types of transformers having applications in electronics,electric heaters, traction, etc Some applications have significant impact on thedesign of transformers The duty (load) of transformers can be very onerous Forexample, current density in transformers with frequent motor starting duty has to

be lower to take care of high starting current of motors, which can be of the order

of 6 to 8 times the full load current

Shunt and series reactors are very important components of the power system.Design of reactors, which have only one winding, is similar to transformers inmany aspects Their special features are given below

n Shunt Reactors: These are used to compensate the capacitive VARs generated

during low loads and switching operations in extra high voltage transmissionnetworks, thereby maintaining the voltage profile of a transmission line withindesirable limits These are installed at a number of places along the length of theline They can be either permanently connected or switched type Use of shuntreactors under normal operating conditions may result in poor voltage levels andincreased losses Hence, the switched-in types are better since they are connectedonly when the voltage levels are required to be controlled When connected to thetertiary windings of a large transformer, they become cost-effective Voltage drop

in high series reactance between HV and tertiary windings must be accounted forwhen deciding the voltage rating of tertiary connected shunt reactors Shuntreactors can be of core-less (air-core) or gapped-core (magnetic circuit with non-magnetic gaps) design The flux density in the air-core reactor has to be smaller asthe flux path is not well constrained Eddy losses in the winding and stray losses inthe structural conducting parts are higher in this type of reactor In contrast, thegapped-core reactor is more compact due to higher permissible flux density Thegap length can be suitably designed to get a desired reactance value Shuntreactors are usually designed to have a constant impedance characteristics up to1.5 times the rated voltage to minimize the harmonic current generation underover-voltage conditions

o Series Reactors: These reactors are connected in series with generators, feeders

and transmission lines for limiting fault currents under short circuits Seriesreactors should have linear magnetic characteristics under fault conditions Theyare designed to withstand mechanical and thermal effects of short circuits Seriesreactors used in transmission lines have a fully insulated winding since both itsends should be able to withstand the lightning impulse voltages The value ofseries reactance has to be judiciously selected because a higher value reduces thepower transfer capability of the line The smoothing reactors used in HVDC

transmission system, connected between the converter and DC line, smoothen the

connected to a source of sinusoidal voltage of frequency f Hz Primary winding draws a small excitation current, i0 (instantaneous value), from the source to set upthe mutual flux in the core All the flux is assumed to be contained in the core

(no leakage) The windings 1 and 2 have N1 and N2 turns respectively Theinstantaneous value of induced electromotive force in the winding 1 due to themutual flux is

(1.1)Equation 1.1 is as per the circuit viewpoint; there is flux viewpoint also [1], inwhich induced voltage (counter electromotive force) is represented as

The elaborate explanation for both the viewpoints is given in[2] If the winding is assumed to have zero winding resistance,

Figure 1.2 Transformer in no-load condition

Since v1 (instantaneous value of the applied voltage) is sinusoidally varying, theflux must also be sinusoidal in nature varying with frequency f Let

(1.3)

substituting the value of in equation 1.1, we get

(1.4)

The r.m.s value of the induced voltage, E1, is obtained by dividing the peak value

in equation 1.4 by

(1.5)

Equation 1.5 is known as emf equation of a transformer For a given number of

turns and frequency, the flux (and flux density) in a core is entirely determined bythe applied voltage

The voltage induced in winding 2 due to the mutual flux is given by

(1.6)The ratio of two induced voltages can be derived from equations 1.1 and 1.6 as

where a is known as ratio of transformation Similarly, r.m.s value of the induced

voltage in winding 2 is

(1.8)

The exciting current (i0) is only of magnetizing nature (im ) if B-H curve of core

material is assumed without hysteresis and if eddy current losses are neglected

The magnetizing current (im ) is in phase with the mutual flux in the absence of

hysteresis Also, linear magnetic (B-H) characteristics are assumed

Now, if the secondary winding in figure 1.2 is loaded, secondary current is set

up as per Lenz’s law such that the secondary magnetomotive force (mmf), i2N2,

opposes the mutual flux tending to reduce it In an ideal transformer e1=v1,because for a constant value of the applied voltage, induced voltage andcorresponding mutual flux must remain constant This can happen only if the

primary draws more current (i1’) for neutralizing the demagnetizing effect ofsecondary ampere-turns In r.m.s notations,

(1.9)

Thus, the total primary current is a vector sum of the no-load current (i.e.,

magnetizing component, Im , since core losses are neglected) and the load current

(I1’),

(1.10)For an infinite permeability magnetic material, magnetizing current is zero.Equation 1.9 then becomes

Thus, for an ideal transformer when its no-load current is neglected, primaryampere-turns are equal to secondary ampere-turns The same result can also bearrived at by applying Ampere’s law, which states that the magnetomotive forcearound a closed path is given by

(1.12)

where i is the current enclosed by the line integral of the magnetic field intensity H

around the closed path of flux

(1.13)

If the relative permeability of the magnetic path is assumed as infinite, the integral

of magnetic field intensity around the closed path is zero Hence, in the r.m.s.notations,

which is the same result as in equation 1.11

Thus, for an ideal transformer (zero winding resistance, no leakage flux, linearB-H curve with an infinite permeability, no core losses), it can be summarized as,

(1.15)and

Schematic representation of the transformer in figure 1.2 is shown in figure 1.3.The polarities of voltages depend upon the directions in which the primary andsecondary windings are wound It is common practice to put a dot at the end of thewindings such that the dotted ends of the windings are positive at the same time,meaning that the voltage drops from the dotted to unmarked terminals are inphase Also, currents flowing from the dotted to unmarked terminals in thewindings produce an mmf acting in the same direction in the magnetic circuit

If the secondary winding in figure 1.2 is loaded with an impedance Z2,

Substituting from equation 1.15 for V2 and I2,

(1.18)Hence, the impedance as referred to the primary winding 1 is

The ideal transformer transforms direct voltage, i.e., DC voltages on primaryand secondary sides are related by turns ratio This is not a surprising resultbecause for the ideal transformer, we have assumed infinite core materialpermeability with linear (non-saturating) characteristics permitting core flux to

rise without limit under a DC voltage application When a DC voltage (Vd1) is

applied to the primary winding with the secondary winding open-circuited,

(1.21)Thus, is constant (flux permitted to rise with time without any limit) and

is equal to (Vd1/N1) Voltage at the secondary of the ideal transformer is

Figure 1.3 Schematic representation of transformer

(1.22)However, for a practical transformer, during the steady-state condition, current

has a value of Vd1/R1, and the magnetic circuit is driven into saturation reducing

eventually the value of induced voltages E1 and E2 to zero (in saturation there ishardly any change in the flux even though the current may still be increasing till

the steady state condition is reached) The current value, Vd1/R1, is quite high,resulting in damage to the transformer

1.3.2 Practical transformer

Analysis presented for the ideal transformer is merely to explain the fundamentals

of transformer action; such a transformer never exists and the equivalent circuit of

a real transformer shown in figure 1.4 is now developed

Whenever a magnetic material undergoes a cyclic magnetization, two types oflosses, eddy and hysteresis losses, occur in it These losses are always present intransformers as the flux in their ferromagnetic core is of alternating nature Adetailed explanation of these losses is given in Chapter 2

The hysteresis loss and eddy loss are minimized by use of a better grade of core

material and thinner laminations, respectively The total no-load current, I0,

consists of magnetizing component (Im ) responsible for producing the mutual flux

and core loss component (Ic ) accounting for active power drawn from the

source to supply eddy and hysteresis losses The core loss component is in phasewith the induced voltage and leads the magnetizing component by 90° With thesecondary winding open-circuited, the transformer behaves as a highly inductivecircuit due to magnetic core, and hence the no-load current lags the applied

voltage by an angle slightly less than 90° (Im is usually much greater than Ic) In the

Figure 1.4 Practical transformer

equivalent circuit shown in figure 1.5, the magnetizing component is represented

by the inductive reactance Xm , whereas the loss component is accounted by the

resistance Rc.

Let R1 and R2 be the resistances of windings 1 and 2, respectively In a practicaltransformer, some part of the flux linking primary winding does not link thesecondary This flux component is proportional to the primary current and is

responsible for a voltage drop which is accounted by an inductive reactance XL1

(leakage reactance) put in series with the primary winding of the ideal

transformer Similarly, the leakage reactance XL2 is added in series with the

secondary winding to account for the voltage drop due to flux linking only thesecondary winding One can omit the ideal transformer from the equivalentcircuit, if all the quantities are either referred to the primary or secondary side ofthe transformer For example, in equivalent circuit of figure 1.5 (b), all quantitiesare referred to the primary side, where

(1.23)(1.24)This equivalent circuit is a passive lumped-T representation, valid generally forsinusoidal steady-state analysis at power frequencies For higher frequencies,capacitive effects must be considered, as discussed in Chapter 7 For any transientanalysis, all the reactances in the equivalent circuit should be replaced by thecorresponding equivalent inductances

Figure 1.5 Equivalent circuit

While drawing a vector diagram, it must be remembered that all the quantities

in it must be of same frequency Actually, the magnetization (B-H) curve of corematerial is of non-linear nature, and it introduces higher order harmonics in themagnetizing current for a sinusoidal applied voltage of fundamental frequency Inthe vector diagram, however, a linear B-H curve is assumed neglecting harmonics.The aspects related to the core magnetization and losses are dealt in Chapter 2 For

figure 1.5(a), the following equations can be written:

Vector diagrams for primary and secondary voltages/currents are shown in figure

1.6 The output terminal voltage V2 is taken as a reference vector along x-axis Theload power factor angle is denoted by θ2 The induced voltages are in phase andlead the mutual flux (r.m.s value of ) by 90° in line with equations 1.1 and

1.6 The magnetizing component (Im ) of no-load current (I0) is in phase with

whereas the loss component Ic leads by 90° and is in phase with the induced

voltage E1 The core loss is given as