The electrical engineering handbook
Gross, C.A. “Power Transformers” The Electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton: CRC Press LLC, 2000 © 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 P e is proportional to the square of the product of the maximum flux density B M (T), the frequency f (Hz), and thickness t (m) of the individual steel lamination. P e = K e ( B M tf ) 2 (W) (64.1) K e is dependent upon the core dimensions, the specific resistance of a lamination sheet, and the mass of the core. Also, P h = K h fB M n (W) (64.2) In Eq. (64.2), P h is the hysteresis power loss, n is the Steinmetz constant (1.5 < n < 2.5) and K h is a constant dependent upon the nature of core material and varies from 3 ן 10 –3 m to 20 ן 10 –3 m , where m = core mass in kilograms. The core loss therefore is P e = P e + P h (64.3) Charles A. Gross Auburn University © 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:17.1kV D 1153-MVA 3 f power transformer. (Photo courtesy of General Electric Company.) © 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 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 current-carrying components must be clamped firmly to limit move- ment. The solid insulation material should be precompressed and formed to avoid its collapse due to the thermal expansion of the wind- ings. 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) vN d dt vN d dt vN d dt 11 22 33 === fff 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 VV N N VV N N V 1 1 2 22 2 3 33 3 1 1 === ˆ ˆ Hdli×= = ò enclosed 0 © 1999 by CRC Press LLC 0 = N 1 i 1 + N 2 i 2 + N 3 i 3 (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 – V 1 , – V 2 , and – V 3 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 V base and S base . 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++= SVIVIVI=++= 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 © 1999 by CRC Press LLC When per-unit scaling is applied to transformers V base is usually taken as V rated as in each winding. S base is common to all windings; for the two- winding case S base is S rated , since S rated is common to both windings. Per-unit scaling simplifies transformer circuit models. Select two primary base values, V 1base and S 1base . Base values for windings 2 and 3 are: (64.11) and (64.12) By definition: (64.13) It follows that (64.14) Thus, Eqs. (64.3) and (64.6) scaled into per-unit become: (64.15) (64.16) The basic per-unit equivalent circuit is shown in Fig. 64.2(c). The extension to the n -winding case is clear. A Practical Three-Winding Transformer Equivalent Circuit The circuit of Fig. 64.2(c) is reasonable for some power system applications, since the core and windings of actual transformers are constructed of materials of high m and s , respectively, though of course not infinite. However, for other studies, discrepancies between the performance of actual and ideal transformers are too great to be overlooked. The circuit of Fig. 64.2(c) may be modified into that of Fig. 64.3 to account for the most important discrepancies. Note: R 1 , R 2 , R 3 Since the winding conductors cannot be made of material of infinite conductivity, the windings must have some resistance. X 1 , X 2 , X 3 Since the core permeability is not infinite, not all of the flux created by a given winding current will be confined to the core. The part that escapes the core and seeks out parallel paths in surrounding structures and air is referred to as leakage flux. R c , X m Also, since the core permeability is not infinite, the magnetic field intensity inside the core is not zero. Therefore, some current flow is necessary to provide this small H. The path provided in the circuit for this “magnetizing” current is through X m . The core has internal power losses, referred to as core loss, due to hystereses and eddy current phenomena. The effect is accounted for in the resistance R c . Sometimes R c and X m are neglected. V N N VV N N V 2 2 1 13 3 1 1 base base base base == SSSS 123 base base base base === I S V I S V I S V 1 1 2 2 3 3 base base base base base base base base base === I N N II N N I 2 1 2 13 1 3 1 base base base base == VVV 123 pu pu pu == III 123 0 pu pu pu ++= © 1999 by CRC Press LLC The circuit of Fig. 64.3 is a refinement on that of Fig. 64.2(c). The values R 1 , R 2 , R 3 , X 1 , X 2 , X 3 are all small (less than 0.05 per-unit) and R c , X m , large (greater than 10 per-unit). The circuit of Fig. 64.3 requires that all values be in per-unit. Circuit data are available from the manufacturer or obtained from conventional tests. It must be noted that although the circuit of Fig. 64.3 is commonly used, it is not rigorously correct because it does not properly account for the mutual couplings between windings. The terms primary and secondary refer to source and load sides, respectively (i.e., energy flows from primary to secondary). However, in many applications energy can flow either way, in which case the distinction is meaningless. Also, the presence of a third winding (tertiary) confuses the issue. The terms step up and step down refer to what the transformer does to the voltage from source to load. ANSI standards require that for a two-winding transformer the high-voltage and low-voltage terminals be marked as H1-H2 and X1-X2, respec- tively, with H1 and X1 markings having the same significance as dots for polarity markings. [Refer to ANSI C57 for comprehensive information.] Additive and subtractive transformer polarity refer to the physical posi- tioning of high-voltage, low-voltage dotted terminals as shown in Fig. 64.4. If the dotted terminals are adjacent, then the transformer is said to be subtractive, because if these adjacent terminals (H1-X1) are connected together, the voltage between H2 and X2 is the difference between primary and secondary. Similarly, if adjacent terminals X1 and H2 are connected, the voltage (H1-X2) is the sum of primary and secondary values. The Two-Winding Transformer The device can be simplified to two windings. Common two-winding transformer circuit models are shown in Fig. 64.5. (64.17) FIGURE 64.3A practical equivalent circuit. FIGURE 64.4Transformer polarity terminology: (a) subtractive; (b) additive. ZZZ e =+ 12 © 1999 by CRC Press LLC (64.18) Circuits (a) and (b) are appropriate when –Z m is large enough that magnetizing current and core loss is negligible. 64.3 Transformer Performance There is a need to assess the quality of a particular transformer design. The most important measure for performance is the concept of efficiency, defined as follows: (64.19) where P out is output power in watts (kW, MW) and P in is input power in watts (kW, MW). The situation is clearest for the two-winding case where the output is clearly defined (i.e., the secondary winding), as is the input (i.e., the primary). Unless otherwise specified, the output is understood to be rated power at rated voltage at a user-specified power factor. Note that SL = P in – P out = sum of losses The transformer is frequently modeled with the circuit shown in Fig. 64.6. Transformer losses are made up of the following components: Electrical losses: I ¢ 1 2 R eq = I 1 2 R 1 + I 2 2 R 2 (64.20a) Primary winding loss = I 1 2 R 1 (64.20b) Secondary winding loss = I 2 2 R 2 (64.20c) FIGURE 64.5Two-winding transformer-equivalent circuits. All values in per-unit. (a) Ideal case; (b) no load current negligible; (c) precise model. Z RjX RjX m cm cm = + () h= P P out in © 1999 by CRC Press LLC Magnetic (core) loss: P c = P e + P h = V 1 2 /R c (64.21) Core eddy current loss = P e Core hysterisis loss = P h Hence: SL = I ¢ 1 2 R eq + V 1 2 /R c (64.22) A second concern is fluctuation of secondary voltage with load. A measure of this situation is called voltage regulation, which is defined as follows: (64.23) where V 2FL = rated secondary voltage, with the transformer supplying rated load at a user-specified power factor, and V 2NL = secondary voltage with the load removed (set to zero), holding the primary voltage at the full load value. A complete performance analysis of a 100 kVA 2400/240 V single-phase transformer is shown in Table 64.1. 64.4 Transformers in Three-Phase Connections Transformers are frequently used in three-phase connections. For three identical three-winding transformers, nine windings must be accounted for. The three sets of windings may be individually connected in wye or delta in any combination. The symmetrical component transformation can be used to produce the sequence equiv- alent circuits shown in Fig. 64.7 which are essentially the circuits of Fig. 64.3 with R c and X m neglected. The positive and negative sequence circuits are valid for both wye and delta connections. However, Y–D connections will produce a phase shift which is not accounted for in these circuits. FIGURE 64.6Transformer circuit model. FIGURE 64.7Sequence equivalent transformer circuits. Voltage Regulation (VR)= -VV V NL FL FL 22 2 © 1999 by CRC Press LLC TABLE 64.1 Analysis of a Single-Phase 2400:240V 100-kVA Transformer Voltage and Power Ratings HV (Line-V) LV (Line-V) S (Total-kVA) 2400 240 100 Test Data Short Circuit (HV) Values Open Circuit (LV) Values Voltage = 211.01 240.0 volts Current = 41.67 22.120 amperes Power = 1400.0 787.5 watts Equivalent Circuit Values (in ohms) Values referred to HV Side LV Side Per-Unit Series Resistance = 0.8064 0.008064 0.01400 Series Reactance = 4.9997 0.049997 0.08680 Shunt Magnetizing Reactance = 1097.10 10.9714 19.05 Shunt Core Loss Resistance = 7314.30 73.1429 126.98 Power Factor Efficiency Voltage Power Factor Efficiency Voltage (—) (%) Regulation (%) (—) (%) Regulation (%) 0.0000 lead 0.00 –8.67 0.9000 lag 97.54 5.29 0.1000 lead 82.92 –8.47 0.8000 lag 97.21 6.50 0.2000 lead 90.65 –8.17 0.7000 lag 96.81 7.30 0.3000 lead 93.55 –7.78 0.6000 lag 96.28 7.86 0.4000 lead 95.06 –7.27 0.5000 lag 95.56 8.26 0.5000 lead 95.99 –6.65 0.4000 lag 94.50 8.54 0.6000 lead 96.62 –5.89 0.3000 lag 92.79 8.71 0.7000 lead 97.07 –4.96 0.2000 lag 89.56 8.79 0.8000 lead 97.41 –3.77 0.1000 lag 81.09 8.78 0.9000 lead 97.66 –2.16 0.0000 lag 0.00 8.69 1.0000 — 97.83 1.77 Rated load performance at power factor = 0.866 lagging. Secondary Quantities; LOW Voltage Side Primary Quantities; HIGH Voltage Side SI Units Per-Unit SI Units Per-Unit Voltage 240 volts 1.0000 Voltage 2539 volts 1.0577 Current 416.7 amperes 1.0000 Current 43.3 amperes 1.0386 Apparent power 100.0 kVA 1.0000 Apparent power 109.9 kVA 1.0985 Real power 86.6 kW 0.8660 Real power 88.9 kW 0.8888 Reactive power 50.0 kvar 0.5000 Reactive power 64.6 kvar 0.6456 Power factor 0.8660 lag 0.8660 Power factor 0.8091 lag 0.8091 Efficiency = 97.43%; voltage regulation = 5.77%. . 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. 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