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OUTPUT EQUATION OF SINGLE PHASE TRANSFORMER

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## OUTPUT EQUATION OF SINGLE PHASE TRANSFORMER

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**OUTPUT EQUATION OF SINGLE PHASE TRANSFORMER**· The equation which relates the rated kVA output of a transformer to the area of core and window is called output equation. · In transformers the output kVA depends on flux density and ampere-turns. · The flux density is related to core area and the ampere-turns is related to window area. · The simplified cross-section of core type and shell type single phase transformers are shown in figures (4-1) and (4-2). · The low voltage winding is placed nearer to the core in order to reduce the insulation requirement. · The space inside the core is called window and it is the space available for accommodating the primary and secondary winding. · The window area is shared between the winding and their insulations.**· The induced emf in a transformer,**· Emf per turn, · The window in single phase transformer contains one primary and one secondary winding.**· The window space factor Kw is the ratio of conductor area**in window to total area of window. · Conductor area in window, · The current density is same in both the windings. Therefore Current density, · Area of cross - section of primary conductor, · Area of cross - section of secondary conductor,**· If we neglect magnetizing mmf then primary ampere turns**is equal to secondary ampere turns. Therefore ampere turns, · Total copper area in window, Ac = Copper area of primary winding + Copper area of secondary winding = (Number of primary turns x area of cross-section of primary conductor) + (Number of secondary turns x area of cross - section of secondary conductor)**· On equating the above equations, we get,**· Therefore Ampere turns, · The kVA rating of single phase transformer is given by · On substituting for E and AT from equations we get, The above equation is the output equation of single phase transformer.**OUTPUT EQUATION OF THREE PHASE TRANSFORMER**· The simplified cross-section of core type three phase transformer is shown in figure. · The cross-section has three limbs and two windows. · Each limb carries the low voltage and high voltage winding of a phase. · The induced emf per phase, · Emf per turn, · In case of three phase transformer, each window has two primary and two secondary windings.**· The window space factor K is the ratio of conductor area**in window to total area of window, · Therefore Conductor area in the window, · The current density is same in both the windings. where, I = Primary current per phase = Secondary current per phase · Area of cross - section of primary conductor, · Area of cross - section of secondary conductor,**· If we neglect magnetizing mmf then primary ampere turns**per phase is equal to secondary ampere turns per phase. · Total copper area in window, Ac = (2 x Number of primary turns x area of cross-section of primary conductor) + ( 2 x Number of secondary turns x area of cross - section of secondary conductor) · On equating we get,**· The kVA rating of three phase transformer is given by,**· On substituting for E and AT from equations we get, · The above equation is the output equation of three phase transformer.**DESIGN OF CORES**· For core type transformer the cross-section may be rectangular, square or stepped. · When circular coils are required for distribution and power transformers, the square and stepped cores are used. · For shell type transformer the cross-section may be rectangular. · When rectangular cores are used the coils are also rectangular in shape. · The rectangular core is suitable for small and low voltage transfor · In core type transformer with rectangular cores, the ratio of depth to width of the core is 1.4 to 2. · In shell type transformers with rectangular cores the width of the central limb is 2 to 3 times the depth of the core. · The figure shows the cross-section of transformer cores.**· In square cores the diameter of the circumscribing circle**is larger than the diameter of stepped cores of same area of cross-section. · Thus when stepped cores are used the length of mean turn of winding is reduced with consequent reduction in both cost of copper and copper loss. · However with larger number of steps a large number of different sizes of laminations have to be used. · This results in higher labor charges for shearing and assembling different types of laminations. · Let d = diameter of circumscribing circle · Also, d = diagonal of the square core and a = side of square · Diameter of circumscribing circle,**· Therefore Side of square,**· Gross core area, Agj = area of square = a2 · Let stacking factor, Sf = 0.9 · Net core area, Ai = Stacking factor x Gross core area = 0.9 x 0.5 d2 = 0.45 d2 · Area of circumscribing circle,**· Another useful ratio for the design of transformer core**is core area factor. · It is the ratio of net core area and square of the circumscribing circle TWO STEPPED CORE FOR CRUCIFORM CORE · In stepped cores the dimensions of the steps should be chosen, such as to occupy maximum area within a circle. The dimensions of the two step to give maximum area for the core in the given area of circle are determined as follows. · Let, a = Length of the rectangle b = Breadth of the rectangle d = Diameter of the circumscribing circle**Also, d= Diagonal of the rectangle**ϴ= Angle between the diagonal and length of the rectangle. · The cross-section of two stepped core is shown in figure.**MULTI-STEPPED CORES**· We can prove that the area of circumscribing circle is more effectively utilized by increasing the number of steps. · The most economical dimensions of various steps for a multi-stepped core can be calculated as shown for cruciform (or two stepped) core. The results are tabulated in table.**CHOICE OF FLUX DENSITY IN THE CORE**• · The flux density decides the area of cross-section of core and core loss. • · Higher values of flux density results in smaller core area, lesser cost, • reduction in length of mean turn of winding, higher iron loss and large • magnetizing current. • · The choice of flux density depends on the service condition (i.e., distribution • or transmission) and the material used for laminations of the core. • · The laminations made with cold rolled silicon steel can work with higher flux • densities than the laminations made with hot rolled silicon steel. • · Usually the distribution transformers will have low flux density to achieve • lesser iron loss. • · When hot routed silicon steel is used for laminations the following values can • be used for maximum flux density (Bm) • Bm = 1.1 to, 1.4 Wb/m2 - For distribution transformers • Bm= 1.2 to 1.5 Wb/m2 - For power transformers • · When cold rolled silicon steel is used for laminations, the following values • can be used for maximum flux density (Bm) • Bm = 1.55 Wb/m - For transformers with voltage rating upto 132 kV • Bm= 1.6 Wb/m - For transformers with voltage rating 132 kV to 275 kY • Bm= 1.7 Wb/m - For transformers with voltage rating 275 kV to 400 Kv**OVERALL DIMENSIONS OF THE TRRNSFORMER**· The main dimensions of the transformer are Height of window (Hw) and Width of window (Ww). · The other important dimensions of the transformer are width of largest stamping (a), diameter of circumscribing circle (d), and distance between core centres (D), height of yoke (Hy), depth of yoke (Dy), overall height of transformer frame (H) and overall width of transformer frame (W). · These dimensions for various types of transformers are shown in figures.**· The above figure shows a vertical and horizontal**cross-section of the core and winding assembly of a core type single phase transformer. · The following figure shows a vertical and horizontal cross-section of the core and winding assembly of a core type three phase transformer. · The next figure shows a vertical and horizontal cross-section of a shell type single phase**· The figure shows a vertical and horizontal cross-section**of a shell type single phase transformer.**TRANSFORMER CONSTRUCTION**· A transformer consists of two windings coupled through a magnetic medium. · The two windings work at different voltage level. · The two windings of the transformer are called High voltage winding and Low voltage winding. · Both the windings are wound on a common core. · One of the winding is connected to ac supply and it is called primary. · The other winding is connected to load and it is called secondary. · The transformer is used to transfer electrical energy from high voltage winding to low voltage winding or vice-versa through magnetic field. · The construction of transformers varies greatly, depending on their applications, winding voltage and current ratings and operating frequencies. • · The two major types of construction of transformers (used in transmission and • distribution of electrical energy) are core type and shell type. • · Depending on the application, these transformers can be classified as • distribution transformers and power transformers. • · The transformer is extremely important as a component in many different • types of electric circuits, from small-signal electronic circuits to high voltage • power transmission systems. • · The most important function performed by transformers are, • Changing voltage and current level in an electric system. • Matching source and load impedances for maximum power • transfer in electronic and control circuitry. • Electrical isolation.**CORE TYPE**TRANSFORMER · In core type transformer, the magnetic core is built of laminations to form a rectangular frame and the windings are arranged concentrically with each other around the legs or limbs. · The top and bottom horizontal portion of the core are called yoke. · The yokes connect the two limbs and have a cross sectional area equal to or greater than that of limbs. · Each limb carries one half of primary and secondary. · The two windings are closely coupled together to reduce the leakage reactance. · The low voltage winding is wound near the core and high voltage winding is wound over low voltage winding away from core in order to reduce the amount of insulating materials required.**SHELL TYPE TRANSFORMER**· In shell type transformers the windings are put around the central limb and the flux path is completed through two side limbs. · The central limb carries total mutual flux while the side limbs forming a part of a parallel magnetic circuit carry half the total flux. · The cross sectional area of the central limb is twice that of each side limbs.**Choice of core length**• The factors to be considered for the choice of core length are • Cost • Ventilation • Voltage between adjacent commutator segments • Specific magnetic loading • When the length of the core is large, the ratio of inactive copper to active • copper will be small. Hence the machine may cost less. • When the core length is very large then ventilation of the core will be • difficult. The centre portion of the core tends to attain a high temperature rise and • so the core must be ventilated (or cooled) by special methods. • An expression for maximum value of core length can be derived as shown • below. • For better commutation the voltage between adjacent commutator • segments at load is to be limited to 30V. To achieve this, the induced • emfin a conductor should not exceed 7.5/TcNc • where, Tc = Turns per coil • Nc= Number coils between adjacent segments • Nc= 1, for simplex lap winding • Nc= p/2, for simplex wave winding • p = Number of poles**CHOICE OF SPECIFIC MAGNETIC**LOADING • The choice of average gap density or specific magnetic loading depends on • the following • Flux density in teeth • Frequency of flux reversal • Size of machine • Large values of flux density in teeth results in increased field mmf. • Higher values of field mmf increase the iron loss, copper loss and cost of • copper. • The Bav is chosen such that the flux density at the root of the teeth does not • exceed 2.2 Wb/m2. • If the frequency of flux reversals is high then iron losses in armature core and • teeth would be high. Therefore we should not use a high value of flux density in • the air gap of machines which have a high frequency. • It is possible to use increased values of flux density as the size of the machine • increases. • As the diameter D of the machine increases, the width of the tooth also • increases, permitting an increased value of gap flux density without causing • saturation in the machine. • The value of Bg varies between 0.55 to 1.15 Wb/m2 and the corresponding • values Of Bav are 0.4 to 0.8 Wb/m2**CHOICE OF SPECIFIC**ELECTRIC LOADING • The choice of specific electric loading depends on the following • Temperature rise 2. Size of machine 3. Speed of machine • Armature reaction 5. Voltage 6. Commutation • A higher value of ac results in a high temperature rise of windings. • The temperature rise depends on the type of enclosure and cooling techniques • employed in the machine. • If the speed of machine is high, the ventilation of the machine is better and • therefore, greater losses can be dissipated. Thus a higher value of ac can be used • for machine having high speed. • In high voltage machines, large space is required for insulation and therefore • there is less space for conductors. This means that in high voltage machines, the • space left for conductors is less and therefore we should use a small value of ac. • In large size machines it is easier to find space for accommodating • conductors. Hence specific electric loading can be increased with increase in • linear dimensions. • With high values of ac, armature reaction will be severe. To counter this, the • field mmf is increased and so the cost of the machine goes high. • High values of ac worsen the commutation condition in machines. From the • point of view of commutation a small value of ac is desirable. The value of ac • usually lies between 15000 to 50000 amp.cond/m.**TRANSFORMER CORE**DESIGN • The core of a transformer consists of core and winding. • The armature core is cylindrical in shape with slots on the outer periphery of • the armature. • The core is formed with circular laminations of thickness 0.5 mm. • The winding is placed on the slots in the armature core. • The design of armature core involves the design of main dimensions D & L, • number of slots, slot dimensions and depth of core. • Number of transformer slots • The factors to be considered for selection of number of transformers slots are • Slot width (or pitch) • Cooling of armature conductors • Flux pulsations • Commutation • Cost**A large number of slots results in smaller slot pitch and so**the width of tooth • is also small. This may lead to difficulty in construction. • But large number of slots will lead to less number of conductors per slot and • so the cooling of armature conductors is better. • If the air-gap reluctance per pair of pole is constant then the flux pulsations • and oscillations can be avoided. • It can be proved that the air-gap reluctance is constant if the slots per pole is • an integer plus 1/2. • For sparkless commutation the flux pulsations and oscillations under the • interpolemust be avoided. This can be achieved with large number of slots per • pole. • In fact, the number of slots in the region between the tips of two adjacent • poles should be at least 3. • The slots per pole should be greater than or equal to 9, for better commutation. • When large number of slots are used the cost of lamination and the cost of • insulation will be high**Depth of**transformer core • The depth of core cannot be independently designed, because it depends on • the diameter of armature (D), inner diameter of armature (Di) and the depth of slot • (ds). The figure shows the cross-section of armature. • From figure, • D = Di + 2dc + 2ds**Depth of core, d = 1/2(D-Di-2ds)**• After estimating D, Di and ds the available depth of core dc can be calculated. • With this value of dc, the flux density in the core can be estimated and if it • does not exceed 1.5 Wb/m2, then the available depth of core is sufficient. • Otherwise we have to increase the diameter of the armature D to give • ufficientdepth for core. The usual value of flux density in the core is 1.0 to 1.5 • Wb/m2 • Finally, the depth of the core is given by, • dc= ½( φ/iBLc) • where, φ= Flux per pole • Li = Net iron length of the armature • B = Flux density in the core**Types of**transformer winding • DC machines employ two general types of double layer windings. They are • Simplex lap winding • Simplex wave winding • These two types of windings primarily differ from each other in the following • two factors. • The number of circuits between the positive and negative brushes, • i.e., number of parallel paths. • The manner in which the coil ends are connected to the commutatorsegments. • In simplex lap winding the number of parallel paths is equal to number of • poles, whereas in simplex wave winding the number of parallel paths is two. • In simplex lap winding the finish of a coil is connected to start of next coil. In • simplex wave winding the finish of a coil is connected to start of a coil which is • lying one pitch away from the finish. • The simplex lap or wave windings are suitable for most of the dc machines • used for various applications. But occasionally the number of parallel paths has to • be increased to a value more than that provided by simplex windings. In such • cases the multiplex windings are employed. • .**Definition of various terms used in transformer**winding • Conductor: The active length of copper or aluminium wire in the slot is • called conductor. • Turn: Two conductors connected for additive emf is called a turn. The two • conductors of a turn are placed approximately a pole pitch apart. • Coil: A coil consists of a number of turns and it is the principal element of • armature winding. The coil with single turn is called single turn coil and the • coil with several turns is called multi turn coil. • Coil side: The active portions of the conductors in a coil are called coil sides. • A coil will have two sides and they are upper coil side and lower coil side. • Usually the top coil side is represented by solid line and bottom coil side by • dotted line. The top coil side is placed in the upper portion of a slot and the • bottom coil side is placed at the lower portion of another slot. The distance • between the two coil sides is kept approximately as one pole pitch. • Overhang: The end portion of the coil connecting the two coil sides is called • overhang.**Coil span: The distance between the two coil sides of a coil**is called coil • span. It is expressed in terms of number or slots or in electrical degrees. • Full pitch coil: When the coil span is equal to pole pitch, the coils are called • full pitched coils. • Short pitched or chorded coil: When the coil span is less than the pole pitch, • the coils are called short pitched or short chorded coils. • Single layer winding: When the coil sides are arranged in a single layer in a • slot, the winding is called single layer winding. • Double layer winding : When the coil sides are arranged in two layers in a • slot, the winding is called double layer winding. • Back Pitch (Yb): The distance between top and bottom coil sides of a coil • measured around the back of the armature (away from the commutator) is • called the back pitch. The back pitch is measured in terms of coil sides. Since • Ybis difference between odd and even number, it is always an odd number. • The back pitch of a coil determines the size of the coil and is nearly equal to • coil sides per pole or pole pitch.**Front Pitch (Yf): The distance between two coil sides**connected to the same • commutatorsegment is called the front pitch (Yf). The front pitch determines • the type of the winding only and it does not affect the size of the coils. • Winding Pitch (Y): The distance between the starts of two consecutive coils • measured in terms of coil sides is called winding pitch (Y). The winding pitch • is always an even integer. • Y = Yb - Yf for lap winding • Y = Yb + Yf for wave winding • Commutator Pitch (Yc): The distance between the two commutator segments • to which the two ends (start and finish) of a coil are connected is called the • commutatorpitch (Yc) and it is measured in terms of commutator segment. • Number of armature coils: The number of turns per coil and the number of • coils are so chosen that the voltage between adjacent commutator segments is • limited to a value where there is no possibility of a flashover. Normally, the • maximum voltage between adjacent segments at load should not exceed 30V.**SIMPLEX**LAP WINDING • In simplex lap winding the finish of a coil is connected to start of next coil. • This winding scheme results in a number of parallel paths which is equal to • number of poles. • The simplex lap winding is a closed winding. In a closed winding if we trace • the winding starting from one point, we will reach the same point after traveling • through all the turns. • But the electrical circuit closes through external load in case of generator and • through external supply in case of motor. The simplex winding has one closed • electrical circuit. (i.e., all the parallel paths electrically closes through external • load or supply). • The two types of simplex lap winding used are • progressive lap winding and • Retrogressive lap winding. • In the progressive lap winding the joining to the commutator progress around • the commutator in the same direction as the coils progress around the armature, as • shown in figure.**In the retrogressive lap winding the joining to the**commutator segment • progresses around the commutator in the opposite direction to the progress of • coils around the armature, as shown in figure.**Steps for designing of lap winding for**transformers • Step 1: Find the range of slots from the range of slot pitch. Armature slot • pitch, Ysa = 25 to 35 mm. Slots, itD/y where D is diameter of armature • Step 2: In the above range of slots, list the values of slots which are multiples • of pole pairs. • Step 3: In order to reduce flux pulsations, the slots per pole should be an • integer ± 1/2. The integer should be in the range of 8 to 16. List all the • multiples of integer ± 1/2 from the list obtained in step 2. • Step 4: Choose the suitable slot from the list obtained in step 3. • Step 5: Estimate the total number of armature conductors, Z using the • equation of induced ernf. Find the conductors per slot and choose it to the • nearest even number. • Step 6: Find the minimum number of coils. • Step 7: Assume, u = 2, 4, 6, 8 etc., where u = coil sides per slot. • Step 8: For each value of u, calculate the number of coils. Choose the number • of coils such that, it is greater than minimum number of coils. Also the value • of u should be a divisor of conductors per slot. • Step 9: Once the number of coils and slots are finalized, Estimate the new • value of total number of conductors and number of turns per coil. • Total armature conductors, Z = Slots x Conductor per slot. • Number of turns per coil = Z/2C. • If a suitable value of C is not obtained to satisfy the above condition, then • make another choice of slots from the list obtained in step 3.**COOLING OF TRANSFORMERS**• · The losses developed in the transformer cores and windings are converted into • thermal energy and cause heating of corresponding transformer parts. • · The heat dissipation in transformer occurs by Conduction, Convection and • Radiation. • · The paths of heat flow in transformer are the following • From internal most heated spots of a given part (of core or • winding) to their outer surface in contact with the oil. • From the outer surface of a transformer part to the oil that cools it. • From the oil to the walls of a cooler, eg. Wall of tank. • From the walls o the cooler to the cooling medium air or w • In the path 1 mentioned above heat is transferred by conduction. In the path 2 • and 3 mentioned above heat is transferred by convection of the oil. In path 4 • the heat is dissipated by both convection and radiation.**· The various methods of cooling transformers are**• Air natural • Forced circulation of oil • Air blast • Oil forced-air natural • Oil natural • Oil forced-air forced • Oil natural-air forced • Oil forced-water forced • Oil natural-water forced • · The choice of cooling method depends upon the size, type of application and • type of conditions obtaining at the site where the transformer in installed. · Air natural is used for transformers up to 1.5 MVA. Since cooling by air is not so effective and proves insufficient for transformers of medium sizes, oil is used as a coolant. · Oil is used for almost all transformers except for the transformers used for special applications. · Both plain walled and corrugated walled tanks are used in oil cooled transformer. · In oil natural-air forced method the oil circulating under natural head transfers heat to tank walls. The air is blown through the hollow space to cool the transformer.**· In oil natural-water forced method, copper cooling coils**are mounted above the transformer core but below the surface of oil. Water is circulated through the cooling coils to cool the transformer. · In oil forced-air natural method of cooling, oil is circulated through the transformer with the help of a pump and cooled in a heat exchanger by natural circulation of air. · In oil forced-air forced method, oil is cooled in external heat exchanger using air blast produced by fans. · In oil forced-water forced method, heated oil is cooled in a water heat exchanger. In this method pressure of oil is kept higher than that of water to avoid leakage of oil. · Natural cooling is suitable up to 10 MVA. The forced oil and air circulation are employed for transformers of capacities 3Q MVA and upwards.