Gross C.A. "Power Transformer The electrical Engineering Handbook Ed. Richard C. Dorf Boca raton crc Press llc. 2000
Gross, C.A. “Power Transformers” The Electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton: CRC Press LLC, 2000
64 Power Transformers 64.1 Transformer Construction The Transformer Core. Core and Shell Types. Transformer windings. Taps 64.2 Power Transformer Modeling The Three-Winding Ideal Transformer Equivalent Circuit. A Practical Three-Winding Transformer Equivalent Circuit. The Two-Winding Transformer 64.3 Transformer Performance 64.4 Transformers in Three-Pha Phase Shift in Y-A Connections Determining Per-Phase Equivalent Circuit Val Charles a. gross Transformers: An Example 64.5 Autotransformers 64.1 Transformer Construction The Transformer Core The core of the power TRANSFORMER is usually made of laminated cold-rolled magnetic steel that is grain oriented such that the rolling direction is the same as that of the flux lines. This type of core construction tends to reduce the eddy current and hysteresis losses. The eddy current loss P is proportional to the square of the product of the maximum flux density Bx(T), the frequency f(Hz), and thickness t(m)of the individual steel Pe=K,(Bmtf)2 (W) (64.1) K is dependent upon the core dimensions, the specific resistance of a lamination sheet, and the mass of the core. also Ph=Kh In Eq (64.2), P, is the hysteresis power loss, n is the Steinmetz constant(1.5< n<2.5)and kh is a constant dependent upon the nature of core material and varies from 3 X 10-m to 20X10-'m, where m= core mass in rams. The core loss therefore is P c 1999 by CRC Press LLC
© 1999 by CRC Press LLC 64 Power Transformers 64.1 Transformer Construction The Transformer Core • Core and Shell Types • Transformer Windings • Taps 64.2 Power Transformer Modeling The Three-Winding Ideal Transformer Equivalent Circuit • A Practical Three-Winding Transformer Equivalent Circuit • The Two-Winding Transformer 64.3 Transformer Performance 64.4 Transformers in Three-Phase Connections Phase Shift in Y–D Connections • The Three-Phase Transformer • Determining Per-Phase Equivalent Circuit Values for Power Transformers: An Example 64.5 Autotransformers 64.1 Transformer Construction The Transformer Core The core of the power TRANSFORMER is usually made of laminated cold-rolled magnetic steel that is grain oriented such that the rolling direction is the same as that of the flux lines. This type of core construction tends to reduce the eddy current and hysteresis losses. The eddy current loss Pe is proportional to the square of the product of the maximum flux density BM (T), the frequency f (Hz), and thickness t (m) of the individual steel lamination. Pe = Ke(BMtf )2 (W) (64.1) Ke is dependent upon the core dimensions, the specific resistance of a lamination sheet, and the mass of the core. Also, Ph = Kh f BM n (W) (64.2) In Eq. (64.2), Ph is the hysteresis power loss, n is the Steinmetz constant (1.5 < n < 2.5) and Kh is a constant dependent upon the nature of core material and varies from 3210–3m to 20210–3m, where m = core mass in kilograms. The core loss therefore is Pe = Pe + Ph (64.3) Charles A. Gross Auburn University
国国围群能 FIGURE 64.1 230kVY: 17.IkVA 1153-MVA 3o power transformer.(Photo courtesy of General Electric Company. Core and Shell Types Transformers are constructed in either a shell or a core structure. The shell-type transformer is one where the windings are completely surrounded by transformer steel in the plane of the coil Core-type transformers are those that are not shell type. a power transformer is shown in Fig. 64.1 Multiwinding transformers, as well as polyphase transformers, can be made in either shell- or core-type designs core shell Transformer windings The windings of the power transformer may be either copper or aluminum. These conductors are usually made of conductors having a circular cross section; however, larger cross-sectional area conductors may require a rectangular cross section for efficient use of winding space. temperature is the sum of the ambient and the temperature rise. The temperature rise in a transforme.e The life of a transformer insulation system depends, to a large extent, upon its temperature. The intrinsic to that transformer at a fixed load. The ambient temperature is controlled by the environment the transformer is subjected to. The better the cooling system that is provided for the transformer, the higher the kVA rating for the same ambient. For example, the kVA rating for a transformer can be increased with forced air(fan)cooling. Forced oil and water cooling systems are also used. Also, the duration of operating time at high temperature directly affects insulation life c 1999 by CRC Press LLC
© 1999 by CRC Press LLC Core and Shell Types Transformers are constructed in either a shell or a core structure. The shell-type transformer is one where the windings are completely surrounded by transformer steel in the plane of the coil. Core- type transformers are those that are not shell type. A power transformer is shown in Fig. 64.1. Multiwinding transformers, as well as polyphase transformers, can be made in either shell- or core-type designs. Transformer Windings The windings of the power transformer may be either copper or aluminum. These conductors are usually made of conductors having a circular cross section; however, larger cross-sectional area conductors may require a rectangular cross section for efficient use of winding space. The life of a transformer insulation system depends, to a large extent, upon its temperature. The total temperature is the sum of the ambient and the temperature rise. The temperature rise in a transformer is intrinsic to that transformer at a fixed load. The ambient temperature is controlled by the environment the transformer is subjected to. The better the cooling system that is provided for the transformer, the higher the “kVA” rating for the same ambient. For example, the kVA rating for a transformer can be increased with forced air (fan) cooling. Forced oil and water cooling systems are also used. Also, the duration of operating time at high temperature directly affects insulation life. FIGURE 64.1 230kVY:17.1kVD 1153-MVA 3f power transformer. (Photo courtesy of General Electric Company.)
Other factors that affect transformer insulation life are vibration or mechanical stress, repetitive expansion and contraction, exposure to moisture and other contaminants, and electrical and mechanical stress due to overvoltage and short-circuit currents Paper insulation is laid between adjacent winding layers. The thickness of this insulation is dependent on the expected electric field stress. In large transformers oil ducts are provided using paper insulation to allow a path for cooling oil to flow between coil elements. The short-circuit current in a transformer creates enormous forces on the turns of the windings. The short circuit currents in a large transformer are typically 8 to 10 times larger than rated and in a small transformer are 20 to 25 times rated. The forces on the windings due to the short-circuit current vary as the square of the current, so whereas the forces at rated current may be only a few newtons, under short-circuit conditions these forces can be tens of thousands of newtons. These mechanical and thermal stresses on the windings must be taken into consideration during the design of the transformer. The urrent-carrying components must be clamped firmly to limit ment. The solid insulation material should be precompressed formed to avoid its collapse due to the thermal expansion of the ings. Power transformer windings typically have taps, as shown. The effect on transformer models is to change the turns ratio 64.2PowerTransformerModeling is proportional to the product of voltage and current, for a specified power level, low current levels can exist only at high voltage, and vice versa The Three-Winding Ideal Transformer Equivalent Circuit Consider the three coils wrapped on a common core as shown in Fig. 64. 2(a). For an infinite core permeability (p) and windings made of material of infinite conductivity(o): VI= N N (64.4) dt where o is the core flux. This produces N For sinusoidal steady state performance: V. V N V3=Ⅵ where V, etc are The circuit symbol is shown in Fig. 64.2(b). Ampere's law requires that 手,a= enclosed=0 (64.7) c 1999 by CRC Press LLC
© 1999 by CRC Press LLC Other factors that affect transformer insulation life are vibration or mechanical stress, repetitive expansion and contraction, exposure to moisture and other contaminants, and electrical and mechanical stress due to overvoltage and short-circuit currents. Paper insulation is laid between adjacent winding layers. The thickness of this insulation is dependent on the expected electric field stress. In large transformers oil ducts are provided using paper insulation to allow a path for cooling oil to flow between coil elements. The short-circuit current in a transformer creates enormous forces on the turns of the windings. The shortcircuit currents in a large transformer are typically 8 to 10 times larger than rated and in a small transformer are 20 to 25 times rated. The forces on the windings due to the short-circuit current vary as the square of the current, so whereas the forces at rated current may be only a few newtons, under short-circuit conditions these forces can be tens of thousands of newtons. These mechanical and thermal stresses on the windings must be taken into consideration during the design of the transformer. The current-carrying components must be clamped firmly to limit movement. The solid insulation material should be precompressed and formed to avoid its collapse due to the thermal expansion of the windings. Taps Power transformer windings typically have taps, as shown. The effect on transformer models is to change the turns ratio. 64.2 Power Transformer Modeling The electric power transformer is a major power system component which provides the capability of reliably and efficiently changing (transforming) ac voltage and current at high power levels. Because electrical power is proportional to the product of voltage and current, for a specified power level, low current levels can exist only at high voltage, and vice versa. The Three-Winding Ideal Transformer Equivalent Circuit Consider the three coils wrapped on a common core as shown in Fig. 64.2(a). For an infinite core permeability (m) and windings made of material of infinite conductivity (s): (64.4) where f is the core flux. This produces: (64.5) For sinusoidal steady state performance: (64.6) where V, etc. are complex phasors. The circuit symbol is shown in Fig. 64.2(b). Ampere’s law requires that (64.7) v N d dt v N d dt v N d dt 1 1 = 2 = 2 3 = 3 f f f v v N N v v N N v v N N 1 2 1 2 2 3 2 3 3 1 3 1 = = = V N N V V N N V V N N 1 V 1 2 2 2 2 3 3 3 3 1 = = = 1 ˆ ˆ H × = dl i = Ú enclosed 0
2 N, 3 FIGURE 64.2 Ideal three-winding transformer(a)Ideal three-winding transformer;(b)schematic symbol;(c)per-unit rival 0=N1i1+N2i2+N3l3 Transform Eq (64.8)into phasor notation N11+N22+N33=0 quations(64.6)and(64.)are basic to understanding transformer operation. Consider Eq. (64.6). Also note that-Vp-V2, and-V3 must be in phase, with dotted terminals defined positive. Now consider the total input complex power -S (64.10) Hence, ideal transformers can absorb neither real nor reactive power It is customary to scale system quantities(V, S, 2) into dimensionless quantities called per-unit values. The basic per-unit scaling equation is Per-unit values actual value The base value always carries the same units as the actual value, forcing the per-unit value to be dimensionless Base values normally selected arbitrarily are Vbase and Sase. It follows that
© 1999 by CRC Press LLC 0 = N1i1 + N2i2 + N3i3 (64.8) Transform Eq. (64.8) into phasor notation: (64.9) Equations (64.6) and (64.9) are basic to understanding transformer operation. Consider Eq. (64.6). Also note that –V1, –V2, and –V3 must be in phase, with dotted terminals defined positive. Now consider the total input complex power –S. (64.10) Hence, ideal transformers can absorb neither real nor reactive power. It is customary to scale system quantities (V, I, S, Z) into dimensionless quantities called per-unit values. The basic per-unit scaling equation is The base value always carries the same units as the actual value, forcing the per-unit value to be dimensionless. Base values normally selected arbitrarily are Vbase and Sbase. It follows that: FIGURE 64.2 Ideal three-winding transformer. (a) Ideal three-winding transformer; (b) schematic symbol; (c) per-unit equivalent circuit. NI N I NI 11 2 2 3 3 ++= 0 S VI V I VI =++ = 11 2 2 3 3 *** 0 Per-unit value = actual value base value I S V Z V I V S base base base base base base base base = = = 2