[James_H._Harlow]_Electric_Power_Transformer_Engin(BookSee.org)

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constant for the entire computation. Equation 3.10.1 and Equation 3.10.2 are based on this assumption that the transformer core is linear and that the various elements in the model are not frequency dependent. Work in the last decade has addressed both the nonlinear characteristics of the core and the frequencydependent properties of the materials. Much progress has been made, but their inclusion adds considerably to the model’s complexity and increases the computational difficulty. If the core is nonlinear, the permeability changes as a function of the material properties, past magnetization history, and instantaneous flux magnitude. Therefore, the associated inductance model is time dependent. The basic strategy for solving the transient response of the nonlinear model in Equation 3.10.1 or Equation 3.10.2 is to linearize the transformer’s nonlinear magnetic characteristics at an instant of time based on the flux in the core at that instant. Two other model formulations should be mentioned. De Leon addressed the transient problem using a model based on the idea of duality [26]. The finite element method has found wide acceptance in solving for electrostatic and electromagnetic field distributions. In some instances, it is very useful in solving for the transient distribution in coils and windings of complex shape. 3.10.3.3 Frequency-Domain Solution A set of linear differential equations representing the transient response of the transformer can be solved either in the time domain or in the frequency domain. If the model is linear, the resultant solution will be the same for either method. The frequency-domain solution requires that the components of the input waveform at each frequency be determined; then these individual sinusoidal waves are applied individually to the transformer, and the resultant voltage response throughout the winding determined; and finally the total response in the time domain is determined by summing the component responses at each frequency by applying superposition. An advantage of this method is that it allows the recognition of frequency-dependent losses to be addressed easily. Disadvantages of this method are that it does not allow the modeling of time-dependent switches, nonlinear resistors like ZnO, or the recognition of nonlinear magnetic-core characteristics. 3.10.3.4 Solution in the Time Domain The following briefly discusses the solution of Equation 3.10.1 and Equation 3.10.2. There are numerous methods to solve Equation 3.10.1, but it has been found that when solving the stiff differential equation model of a transformer, a generalization of the Dommel method [28,29] works very well. A lossless lumped-parameter transformer model containing n nodes has approximately n(n + 1)/2 inductors and 3n capacitors. Since the total number of inductors far exceeds the number of capacitors in the network, this methodology reduces storage and computational time by representing each capacitor as an inductor in parallel with a current source. The following system of equations results: where [F(t)] = nodal integral of the voltage vector [I(t)] = nodal injected current vector [H(t)] = past history current and ˆ YF() t I() t H() t (3.10.3) 4 2 Ŷ C (3.10.4) t t G n 2 The lumped-parameter model is composed of capacitances, inductances, and losses computed from the winding geometry, permittivity of the insulation, iron core permeability, and the total number of sections into which the winding is divided. Then the matrix [ Y ˆ] is computed using the integration step size, t. At every time step, the above system of equations is solved for the unknowns in the integral of the voltage vector. The unknown nodal voltages, [E(t)], are calculated by taking the derivative of [F(t)]. t is selected based on the detail of the model and the highest resonant frequency of interest. Normally, t is smaller than one-tenth the period of this frequency. The state-variable formulation shown as Equation 3.10.2 can be solved using differential equation routines available in IMSL or others based on the work of Gear or Adams [30]. The advantage of these routines is that they are specifically written with the solution of stiff systems of equations in mind. The disadvantage of these routines is that they consume considerable time during the solution. 3.10.3.5 Accuracy Versus Complexity Every model of a physical system is an approximation. Even the simplest transformer has a complex winding and core structure and, as such, possesses an infinite number of resonant frequencies. A lumpedparameter model, or for that matter any model, is at best an approximation of the actual device of interest. A lumped-parameter model containing a structure of inductances, capacitances, and resistances will produce a resonant-frequency characteristic that contains the same number of resonant frequencies as nodes in the model. The transient behavior of a linear circuit (the lumped-parameter model) is determined by the location of the poles and zeros of its terminal-impedance characteristic. It follows then that a detailed transformer model must posses two independent characteristics to faithfully reproduce the transient behavior of the actual equipment. First, accurate values of R, L, and C, reflecting the transformer geometry. This fact is well appreciated and documented. Second, the transformer must be modeled with sufficient detail to address the bandwidth of the applied waveshape. In a valid model, the highest frequency of interest would have a period at least ten times larger than the travel time of the largest winding segment in the model. If this second characteristic is overlooked, a model can produce results that appear valid, but may have little physical basis. A final issue is the manner in which the transformer structure is subdivided. If care is not taken, the manner in which the model is constructed will itself introduce significant errors, with a mathematically robust computation but an inaccurate approximation of the physically reality. 3.10.4 Resonant Frequency Characteristic 3.10.4.1 Definitions The steady-state and transient behavior of any circuit, for any applied voltage, is established by the location of the poles and zeros of the impedance function of the lumped-parameter model in the complex plane. The zeros of the terminal-impedance function coincide with the natural frequencies of the model, by definition. McNutt [31] defines terminal resonance as the terminal current maximum and a terminal impedance minimum. In a physical system, there are an infinite number of resonances. In a lumpedparameter model of a system, these are as many resonances as nodes in the model (or the order of the system). Terminal resonance is also referred to as series resonance [32,33]. Terminal antiresonance is defined as a terminal current minimum and a terminal impedance maximum [31]. This is also referred to as parallel resonance [32,33]. McNutt defines internal resonance as an internal voltage maximum and internal antiresonance as an internal voltage minimum. 3.10.4.2 Impedance versus Frequency The terminal resonances for a system can be determined by taking the square root of the eigenvalues of the system matrix, [A], shown in the state-variable representation for the system shown in Equation 3.10.5. [q] · = [A][q] + [B][u] [y] = [C][q] + [D][u] (3.10.5) © 2004 by CRC Press LLC © 2004 by CRC Press LLC
where [A] = state matrix [B] = input matrix [C] = output matrix [D] = direct transmission matrix q = vector of state variables for system ·q = first derivative of [q] u = vector of input variables y = vector of output variables The impedance-vs.-frequency characteristic requires a little more effort. In light of the previous definitions, terminal resonance can be defined as occurring when the reactive component of the terminal impedance is zero. Equivalently, terminal resonance occurs when the imaginary component of the quotient of the terminal voltage divided the injected terminal current is zero. Recalling that, in the Laplace domain, s is equivalent to j with a system containing n nodes with the excited terminal node j, one can rewrite Equation 3.10.1 to obtain: RESISTANCE/REACTANCE (kOhm) 50 40 30 20 RESISTANCE 10 0 10 REACTANCE 20 30 40 50 0 5 10 15 20 25 30 FREQUENCY (kHZ) e s Z s i() 1j() e () s Z j() s 2 2 – – ij() s e s Z s j() jj() – – e s Z s n() nj() (3.10.6) FIGURE 3.10.4 Terminal impedance for a helical winding. The voltage at the primary (node j) is in operational form. Rearranging the terminal impedance is given by: Z ( ) = Z (j ) = Z (s) = e (s) j t jj jj i (s) j (3.10.7) AMPLIFICATION FACTOR 50 40 30 20 10 10 5 0 In these equations, the unknown quantities are the voltage vector and the frequency. It is a simple matter to assume a frequency and solve for the corresponding voltage vector. Solving Equation 3.10.7 over a range of frequencies results in the well-known impedance-vs.-frequency plot. Figure 3.10.4 shows the impedance versus frequency for the example used in Figure 3.10.1 and Figure 3.10.2. 0 100 80 60 40 % WINDING 20 0 30 25 20 15 FREQUENCY (kHZ) 3.10.4.3 Amplification Factor The amplification factor or gain function is defined as: FIGURE 3.10.5 Amplification factor at 5 from 0 to 30 kHz. As shown by Degeneff [27], this results in: N lm, j Voltage between points l and m at frequency Voltage applied at input node j at frequency (3.10.8) 3.10.5 Inductance Model 3.10.5.1 Definition of Inductance Inductance is defined as: N = lm, j [ Z lj(j ) Z mj(jw) ] Z (j ) (3.10.9) It is a simple matter to assume a frequency and solve for the corresponding voltage-vs.-frequency vector. If one is interested in the voltage distribution within a coil at one of the resonant frequencies, this can be found from the eigenvector of the coil at the frequency of interest. If one is interested in the distribution at any other frequency, Equation 3.10.9 can be utilized. This is shown in Figure 3.10.5. jj and if the system is linear: L = d dI L I (3.10.10) (3.10.11) © 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
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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 and 198: Gade, S., Sound Intensity Instrumen
Page 199: nected. This steady-state voltage d
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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