[James_H._Harlow]_Electric_Power_Transformer_Engin(BookSee.org)

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- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - R pi - - - - i R ii j k - - - - - - - - - - L ii - - - - C pi R gij C gij n n-1 . . . . . . . .i j k 4 3 2 1 % VOLTAGE 150 100 50 50 0 100 0 ↑ GROUND 50 100 TRANSIENT VOLTAGE RESPONSE FOR 100 SECTION DISK WINDING MODELED AS 50 SECTION PAIRS ← STANDARD FULL WAVE 150 200 0 20 80 60 40 % WINDING 100 TIME (µsec) Mutual Inductances should be considered, i.e. M ji is between segments j and i. FIGURE 3.10.2 Voltage versus time for helical winding. FIGURE 3.10.1 Sample of section used to model example coil. 100 INITIAL AND PSEUDOFINAL VOLTAGE DISTRIBUTIONS 3.10.2 Surges in Windings 3.10.2.1 Response of a Simple Coil <strong>Transformer</strong> windings are complex structures of wire and insulation. This is the result of many contradictory requirements levied during the design process. In an effort to introduce the basic concepts of transient response, a very simple disk coil was modeled and the internal transient response computed. The coil consisted of 100 identical continuous disk sections of 24 turns each. The inside radius of the coil is 318.88 mm; the space between each disk coil is 5.59 mm; and the coil was assumed to be in air with no iron core. Each turn was made of copper 7.75 mm in height, 4.81 mm in the radial direction, with 0.52 mm of insulation between the turns. For this example, the coil was subjected to a full wave with a 1.0 per-unit voltage. Figure 3.10.1 provides a sketch of the coil and the node numbers associated with the calculation. For this example, the coil has been subdivided into 50 equal subdivisions, with each subdivision a section pair. Figure 3.10.2 contains the response of the winding as a function of time for the first 200 sec. It should be clear that the response is complex and a function of both the applied excitation voltage and the characteristics of the coil itself. 3.10.2.2 Initial Voltage Distribution If the voltage distribution along the helical coil shown in Figure 3.10.2 is examined at times very close to time zero, it is observed that the voltage distribution is highly nonuniform. For the first few tenths of a second, the distribution is dominated exclusively by the capacitive structure of the coil. This distribution is often referred to as the initial (or short time) distribution, and it is generally highly nonuniform. This initial distribution is shown in Figure 3.10.3. For example, examining the voltage gradient over the first 10% of the winding, one sees that the voltage is 82% rather that the anticipated 10%, or one sees a rather large enhancement or gradient in some portions of the winding. The initial distribution shown in Figure 3.10.3 is based on the assumption that the coil knows how it is connected, i.e., it requires some current to flow in the winding, and this requires some few tens of a nanosecond. The initial distribution can be determined by evaluating the voltage distribution for the % VOLTAGE 90 80 70 60 50 40 30 20 10 PSEUDOFINAL 0 0 10 20 30 40 50 60 70 80 90 100 % WINDING FIGURE 3.10.3 Initial and pseudo-final voltage distribution. capacitive network of the winding and ignoring both the inductive and resistive components of the transformer. This discussion is applicable for times greater than approximately 0.25 sec. This is the start of the transient response for the winding. For times smaller than 0.25 sec, the distribution is still dictated by capacitance, but the transformer’s capacitive network is unaware that it is connected. This is addressed in the chapter on very fast transients and in the work of Narang et al. [11]. 3.10.2.3 Steady-State Voltage Distribution The steady-state voltage distribution depends primarily on the inductance and losses of the windings structure. This distribution, referred to by Abetti [15] as the pseudo-final, is dominated primarily by the self-inductance and mutual inductance of the windings and the manner in which the winding is con- INITIAL © 2004 by CRC Press LLC © 2004 by CRC Press LLC
nected. This steady-state voltage distribution is very nearly (but not identical) to the turns ratio. For the simple winding shown in Figure 3.10.1, the distribution is known by inspection and shown in Figure 3.10.3, but for more complex windings it can be determined by finding the voltage distribution of the inductive network and ignoring the effect of winding capacitances. 3.10.2.4 Transient-Voltage Distribution Figure 3.10.2 shows the transient response for this simple coil. The transient response is the voltage the coil experiences when the voltage within the coil is in transition between the initial voltage distribution and the steady-state distribution. It is the very same idea as pulling a rubber-band away from its stretched (but steady-state position) and then letting it go and monitoring its movement in space and time. What is of considerable importance is the magnitude and duration of these transient voltages and the ability of the transformer’s insulation structure to consistently survive these voltages. The coil shown in Figure 3.10.1 is a very simple structure. The challenge facing transformer design engineers since the early 1900s has been to determine what this voltage distribution would be for a complex winding structure of a commercial transformer design. 3.10.3 Determining Transient Response 3.10.3.1 History Considerable research has been devoted to determining the transformer’s internal transient-voltage distribution. These attempts started at the beginning of this century and have continued at a steady pace for almost 100 years [12–14]. The earliest attempts at using a lumped-parameter network to model the transient response was in 1915. Until the early 1960s, these efforts were of limited success due to computational limitations encountered when solving large numbers of coupled stiff differential equations. During this period the problem was attacked in the time domain using either a standing-wave approach or the traveling-wave method. The problem was also explored in the frequency domain and the resultant individual results combined to form the needed response in the time domain. Abetti introduced the idea of a scale model, or analog, for each new design. Then, in the 1960s, with the introduction of the highspeed digital computer, major improvements in computational algorithms, detail, accuracy, and speed were obtained. In 1956 Waldvogel and Rouxel [16] used an analog computer to calculate the internal voltages in the transformer by solving a system of linear differential equations with constant coefficients resulting from a uniform, lossless, and linear lumped-parameter model of the winding, where mutual and self-inductances were calculated assuming an air core. One year later, McWhirter et al. [17], using a digital and analog computer, developed a method of determining impulse voltage stresses within the transformer windings; their model was applicable, to some extent, for nonuniform windings. But it was not until 1958 that Dent et al. [18] recognized the limitations of the analog models and developed a digital computer model in which any degree of nonuniformity in the windings could be introduced and any form of applied input voltage applied. During the mid 1970s, efforts at General <strong>Electric</strong> [19–21] focused on building a program to compute transients for core-form winding of completely general design. By the end of the 1970s, an adequate linear lossless model of the transformer was available to the industry. However, adequate representation of the effects of the nonlinear core and losses was not available. Additionally, the transformer models used in insulation design studies had little relationship with lumped-parameter transformer models used for system studies. Wilcox improved the transformer model by including core losses in a linear-frequency-domain model where mutual and self-impedances between winding sections were calculated considering a grain-oriented conductive core with permeability r [22]. In his work, Wilcox modeled the skin effect, the losses associated with the magnetizing impedance, and a loss mechanism associated with the effect of the flux radial component on the transformer core during short-circuit conditions. Wilcox applied his modified modal theory to model the internal voltage response on practical transformers. Vakilian [23] modified White’s inductance model [19] to include the saturable characteristics of the core and established a system of ordinary differential equations for the linearized lumped-parameter induction resistive and capacitive (LRC) network. These researchers then used Gear’s method to solve for the internal voltage response in time. During the same year, Gutierrez and Degeneff [24,25] presented a transformer-model reduction technique as an effort to reduce the computational time required by linear and nonlinear detailed transformer models and to make these models small enough to fit into the electromagnetic transients program (EMTP). In 1994 de Leon and Semlyen [26] presented a nonlinear three-phase-transformer model including core and winding losses. This was the first attempt to combine frequency dependency and nonlinearity in a detailed transformer model. The authors used the principle of duality to extend their model to three-phase transformers. 3.10.3.2 Lumped-Parameter Model Transient response is a result of the flow of energy between the distributed electrostatic and electromagnetic characteristics of the device. For all practical transformer winding structures, this interaction is quite complex and can only be realistically investigated by constructing a detailed lumped-parameter model of the winding structure and then carrying out a numerical solution for the transient-voltage response. The most common approach is to subdivide the winding into a number of segments (or groups of turns). The method of subdividing the winding can be complex and, if not addressed carefully, can affect the accuracy of the resultant model. The resultant lumped-parameter model is composed of inductances, capacitances, and losses. Starting with these inductances, capacitances, and resistive elements, equations reflecting the transformer’s transient response can be written in numerous forms. Two of the most common are the basic admittance formulation of the differential equation and the statevariable formulation. The admittance formulation [27] is given by: Is () 1 G s C n E() s (3.10.1) s The general state-variable formulation is given by Vakilian [23] describing the transformer’s lumpedparameter network at time t: L 0 0 die / dt C 0 0 den / dt 0 0 U dfe / dt r T r t T 0 ie G Is t en T 0 (3.10.2) where the variables in Equation 3.10.1 and Equation 3.10.2 are: [i e ] = vector of currents in the winding [e n ] = model’s nodal voltages vector [f e ] = windings’ flux-linkages vector [r] = diagonal matrix of windings series resistance [T] = windings connection matrix [T] t = transpose of [T] [C] = nodal capacitance matrix [U] = unity matrix [I(s)] = Laplace transform of current sources [E(s)] = Laplace transform of nodal voltages [ n ] = inverse nodal inductance matrix = [T][L] –1 [T] t [L] = matrix of self and mutual inductances [G] = conductance matrix for resistors connected between nodes [I s ] = vector of current sources In a linear representation of an iron-core transformer, the permeability of the core is assumed constant regardless of the magnitude of the core flux. This assumption allows the inductance model to remain © 2004 by CRC Press LLC © 2004 by CRC Press LLC
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ELECTRIC POWER TRANSFORMER ENGINEER
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I wish to recognize the interest of
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Contents 1 Theory and Principles Ch
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1.3 Equivalent Circuit of an Iron-C
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designer starts to make a design fo
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• Transient voltages generated du
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the transformer’s bushings and, m
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% regulation = [(V NL - V FL )/V FL
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In core-form transformers, the wind
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FIGURE 2.1.14 Layer windings (singl
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the transformer design, which may b
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consumer’s service circuit is a d
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FIGURE 2.2.3 Single-phase transform
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is installed so that the tank never
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above which persons might be burned
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FIGURE 2.2.16 Two-bushing subway. (
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carbon steel or the very expensive
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FIGURE 2.2.31 Radial-style dead fro
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FIGURE 2.2.36 Complete transformer
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voltage in each winding simultaneou
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mounted transformers can have arres
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V S + V S I V L Z=R+jX a) V L V S V
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Z = jX (2.3.19) 0.7 Then the curren
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X1 L* X1 S =X1 L X2 L* X3 L* a) S L
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150 V(%) 100 50 0 -50 40 80 T(s) 12
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2.3.8.1.2 Series Connection • The
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A six-pulse single-way transformer
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The degree of coupling also influen
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where I h = current for the hth har
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In the case of liquid-filled transf
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FIGURE 2.4.21 A 36-pulse system mad
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Vacuum-pressure encaps ulated — T
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FIGURE 2.6.1 Typical wiring and sin
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a) H1 b) Ip X2 FLUX H2 LEAKAGE FLUX
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and for CTs is TCF = RCF - (PA/2600
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RCF x Bx Bt RCF t - RCF 0 co
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2000 1000 5% Vex BANDWITH FROM NOMI
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This may be more useful when operat
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also some uncertainty in repeatabil
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FIGURE 2.6.25 Standard two-element
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2.7 Step-Voltage Regulators Craig A
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Source Bypass Switch Source Bypass
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2.7.4 Theory A step-voltage regulat
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FIGURE 2.7.22 Equalizer winding inc
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TABLE 2.7.4 Increase in Ampere Load
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References IEEE, Standard Requireme
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300% FIGURE 2.8.4 Schematic of a co
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TABLE 2.8.1 Typical Sizing Workshee
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Input with Interruption Ferro Outpu
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FIGURE 2.8.20A Performance of a rid
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• Input frequency or frequency ra
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References 1. Sola/Hevi-Duty Corp.,
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FIGURE 2.9.5 Typical phase-reactor
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FIGURE 2.9.10 Two generator arrange
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• Overloading of lines • Increa
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FIGURE 2.9.23 34-kV, 25-MVAR (per p
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V 90 ( X X ) s T (2.9.10a) where,
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Therefore, Line reactive losses = (
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ate of rise of transient-recovery v
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the LLCR is provided in IEEE Std. C
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aluminum shielding, i.e., an oil-im
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Leo J. Savio ADAPT Corporation Ted
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3.1.2.2.2 Heat Dissipation A second
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FIGURE 3.2.1 Solid-type bushing. ma
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1.6 cm. This means that most of the
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where HS = steady-state bushing ho
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the conductivity of the water-solub
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of 178 ohm-m, and a test duration o
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2. Easley, J.K. and Stockum, F.R.,
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FIGURE 3.3.5 Reactance-type LTC, in
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FIGURE 3.3.10 LTC with delta connec
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3.3.5 Selection of Load Tap Changer
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FIGURE 3.3.19 Effect of the leakage
Page 147 and 148: References Goosen, P.V. , Transform
Page 149 and 150: If i 1 is the full-load current rat
Page 151 and 152: TABLE 3.4.1 Loading Recommendations
Page 153 and 154: 3.5.4 Three-Phase Transformer Conne
Page 155 and 156: FIGURE 3.5.4 Interconnected star-gr
Page 157 and 158: 3.6.1.1 Standards ANSI standards fo
Page 159 and 160: ANSI standards. LTC control setting
Page 161 and 162: FIGURE 3.6.7 Discharging the capaci
Page 163 and 164: FIGURE 3.6.10 Standard switching-im
Page 165 and 166: voltage of the excited winding, rea
Page 167 and 168: FIGURE 3.6.18 Current, voltage, and
Page 169 and 170: TABLE 3.6.3 Transformer Short-Circu
Page 171 and 172: FIGURE 3.7.3 Control for voltage re
Page 173 and 174: FIGURE 3.7.6A Feeder with power-fac
Page 175 and 176: 3. Circulating current (current bal
Page 177 and 178: FIGURE 3.7.10 Control block diagram
Page 179 and 180: Primary Current (Amps) 60 40 20 0 -
Page 181 and 182: current transformer having two prim
Page 183 and 184: 87R1 87BL1 (A) Independent Harmonic
Page 185 and 186: Winding 1 Secondary Currents Data A
Page 187 and 188: Y Y 20 18 3 rd Harmonic CTR1=40 CTR
Page 189 and 190: 230 kV 3 180 MVA 138 kV 5 CTR1 = 2
Page 191 and 192: 29. Concordia, C. and Rothe, F.S.,
Page 193 and 194: FIGURE 3.9.1 Measurement of pressur
Page 195 and 196: H 2m 1. Vertical Forced Air 2. Prin
Page 197: Gade, S., Sound Intensity Instrumen
Page 201 and 202: where [A] = state matrix [B] = inpu
Page 203 and 204: where C = capacitance between the t
Page 205 and 206: L ijo = N i N j ijo (3.10.31) 1.4
Page 207 and 208: % VOLTAGE % VOLTAGE 100 BIL 90 50 3
Page 209 and 210: 29. Degeneff, R.C., Reducing Storag
Page 211 and 212: All connections should be cleaned a
Page 213 and 214: 6. Costs to set up and use a mobile
Page 215 and 216: Records of relay operation must be
Page 217 and 218: • Has heating occurred as the res
Page 219 and 220: a reference value) of the currents
Page 221 and 222: The next data-processing step is to
Page 223 and 224: temperature. As the transformer coo
Page 225 and 226: 3.13.3.2 Instrument Transformers Th
Page 227 and 228: FIGURE 3.13.5 Analysis of bushing s
Page 229 and 230: temperature. Under abnormal conditi
Page 231 and 232: 3.14.1.2 Accredited Standards Commi
Page 233 and 234: FIGURE 3.14.4 IEC technical-committ
Page 235 and 236: TABLE 3.14.2 Relevant Documents for
Page 241: TABLE 3.14.13 Relevant Documents fo
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[James_H._Harlow]_Electric_Power_Transformer_Engin(BookSee.org)

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