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High Electric and Magnetic Field Engineering. Cable Engineering
UDC 621.3.022: 621.315.3: 537.311.8 doi: 10.20998/2074-272X.2019.5.06
M.I. Baranov
A CHOICE OF CRITICAL SECTIONS OF ELECTRIC WIRES AND CABLES IN POWER CIRCUITS OF ELECTRICAL EQUIPMENT OF POWER INDUSTRY
Purpose. Implementation of close calculation determination of critical sections of Sa and critical amplitudes of density of alternating current SCi of frequency 50 Hz in wires and cables of power circuits of electrical equipment of power industry, characterized flowing in it at malfunctions of operation current ik(t) of short circuit (SC) with set amplitude-temporal parameters (ATP). Methodology. Scientific and technical bases of power engineering, electrophysics bases of technique of high-voltage and large pulsed currents, theoretical bases of the electrical engineering. Results. The results of the developed electrical engineering approach are resulted in a calculation choice on the condition of electric explosion (EE) of current-carrying parts of cable and conductor products (CCP) of critical sections of SCi for the copper (aluminum) cores of the uninsulated wires, and also for the insulated wires and cables with a polyvinilkhloride (PVC), rubber (R) and polyethylene (PET) insulation with copper (aluminum) cores (shells) on which in the power circuits of electrical equipment of the general industrial equipment in malfunction the current of SC ik(t) flows with set ATP. On the basis of determination of sizes of the real critical sections SCi for the indicated wires and cables the calculation numeral estimation of critical amplitudes of density SCi of SC current ik(t) is executed with set ATP in current-carrying parts of investigated CCP ofpower circuits of the examined electrical equipment. It is determined that in the power circuits of electric equipment of the general industrial installations (for permanent time of slump of Ta=50 ms of aperiodic to the constituent of current of SC) critical amplitudes of density SCi of SC current ik(t) at time of his disconnecting tk=100 ms in copper (aluminum) cores for the uninsulated wires and insulated wires (cables) with copper (aluminum) cores (shells), PVH, R and PET it is numeral made an insulation according to approximately 1,57 (1,18) kA/mm2. At time of disconnecting tk=160 ms of SC current ik(t) in the power circuits of the examined electrical equipment (Ta=50 ms) critical amplitudes of density SCi of SC current ik(t) for the copper (aluminum) cores (shells) of indicated CCP become accordingly numeral equal approximately 1,33 (0,99) kA/mm2. Originality. First by a calculation way taking into account information for ATP of SC current ik(t) and quantitative values of time of his disconnecting tk in the power circuits of electrical equipment of the general industrial installations the numerical values of critical sections SCi and critical amplitudes of density SCi of AC SC current ik(t) are certain for the uninsulated wires, and also insulated wires and cables with copper (aluminum) cores (shells), PVH, R and PET insulation. Practical value. Obtained results for the critical sections SCi and amplitudes of density SCi of AC SC current ik(t) of frequency 50 Hz (at the period of oscillations of Tp=20 ms for of periodic constituent of emergency current of SC) can be utilized in power industry in practice at a choice thermally by a bar to the action of large SC currents ik(t) of CCP with copper (aluminum) bars (shells), intended for reliable operation in the power circuits of electrical equipment of industrial power industry. References 8, tables 5.
Key words: power industrial, electrical equipment, electric wires and cables of circuits of electrical equipment, calculation choice of critical sections of wires and cables in the circuits of electrical equipment.
Hadaui результати розробленого електротехтчиого тдходу до розрахункового вибору по yMoei електричного вибуху (ЕВ) струмопровiдних частин кабельно-пров1дниково1 продукци критичних nерерiзiв SCi неiзольовaних дротiв, а також iзольовaних дротiв i кабелв з потвштхлоридною (ПВХ), гумовою (Г) i полетиленовою (ПЕТ) Ьолящею з мiдними (алюмШевими) жилами (оболонками), по яких в силових колах електрообладнання загальнопромислового призначення в аваршному режимi протiкae струм ik(t) короткого замикання (КЗ) b заданими ампмтудно-часовими параметрами (АЧП). На niдстaвi даного тдходу здшснений реальний вибiр критичних перерiзiв SCi для вказаних дротiв (кабетв) силових кт електрообладнання, що розглядаеться Виконана розрахункова чисельна ощнка критичних амплтуд щтьностей SCi струму ik(t) КЗ ii заданими АЧП в дротах i кабелях силових кт вказаного електрообладнання. Отримаш дат сприятимуть забезпеченню термiчноí <тйко<т електричних неiзольовaних дротiв, а також дротiв i кабелв з ПВХ, Г i ПЕТ iзоляцiею, як широко застосовуються в силових колах електрообладнання загальнопромислового призначення. Бiбл. 8, табл. 5.
Ключовi слова: промислова електроенергетика, електрообладнання, електричш дроти i кабелi юл електрообладнання, розрахунковий вибiр критичних перерiзiв дроив i кабелiв в колах електрообладнання.
Приведены результаты разработанного электротехнического подхода к расчетному выбору по условию электрического взрыва (ЭВ) токонесущих частей кабельно-проводниковой продукции критических сечений SCi неизолированных проводов, а также изолированных проводов и кабелей с поливинилхлоридной (ПВХ), резиновой (Р) и полиэтиленовой (ПЭТ) изоляцией с медными (алюминиевыми) жилами (оболочками), по которым в силовых цепях электрооборудования общепромышленного назначения в аварийном режиме протекает ток ik(t) короткого замыкания (КЗ) с заданными амплитудно-временными параметрами (АВП). На основании данного подхода осуществлен реальный выбор критических сечений SCi для указанных проводов (кабелей) силовых цепей рассматриваемого электрооборудования. Выполнена расчетная численная оценка критических амплитуд плотностей SCi тока ik(t) КЗ с заданными АВП в проводах и кабелях силовых цепей указанного электрооборудования. Полученные данные будут способствовать обеспечению термической стойкости электрических неизолированных проводов, а также проводов и кабелей с ПВХ, Р и ПЭТ изоляцией, широко применяемых в силовых цепях электрооборудования общепромышленного назначения. Библ. 8, табл. 5.
Ключевые слова: промышленная электроэнергетика, электрооборудование, электрические провода и кабели цепей электрооборудования, расчетный выбор критических сечений проводов и кабелей в цепях электрооборудования.
Introduction. In [1] the results of a calculation and experimental determination of critical sections Sa and critical amplitudes of current densities 8Ci for non- and insulated electric wires (cables) containing metal cores
(i=1) and shells (i=2) and used in discharge circuits of high-voltage high-current pulse technology are presented. The basis for this choice of Sa cross-sectional values of
© M.I. Baranov
conductive wires (shells) and current density amplitudes dCi for cable-conductor products (CCP) with pulsed axial current varying in the nano-, micro-, and millisecond time ranges was known in electrophysics the electric explosion (EE) condition [2, 3] of metal cores (shells) of the indicated wires and cables, leading to the sublimation of their current-carrying parts and the CCP failure. In the field of power industry in emergency operation of the power circuits of its electrical equipment, accompanied by the flow of short-circuit (SC) current ik(t) through the current-carrying parts of the CCP with frequency f = 50 Hz and amplitude of up to (40-125) kA [4], at unreasonable use in the power circuits of the electric equipment of the CCP, cases of the occurrence of the EE phenomenon of copper (aluminum) cores and shells (reverse conductors) of their wires and cables are also possible. Insufficient attention was paid by electrical engineers and power engineers to such emergency conditions in power circuits of electrical equipment of general industrial use with similar dire consequences for their CCP. In this regard, the calculation determination of the critical cross-sections SCi of electric wires (cables) used in power circuits of electrical equipment of general industrial use is an urgent applied problem in the field of power engineering.
The goal of the paper is to perform an approximate calculation determination of critical sections SCi and critical amplitudes of densities dCi of AC current with frequency f = 50 Hz in the wires and cables of power circuits of electrical equipment of power industry, characterized by flowing in them in emergency modes of SC current ik(t) with specified amplitude-temporal parameters (ATPs).
1. Problem definition. Let us consider widely used in power circuits of electrical equipment of general industrial use uninsulated copper and aluminum wires, as well as insulated wires and cables with copper (aluminum) inner cores and outer shells-current conductors with the initial specific conductivity y0i of their non-magnetic material, having polyvinyl chloride (PVC), rubber (R) and polyethylene (PET) belt insulation [4, 5]. We assume that along round continuous or split copper (aluminum) cores (shells) of the indicated wires and cables of power electric circuits of the considered electrical equipment in emergency mode in their longitudinal direction current ik(t) of the three-phase SC with specified ATPs flows. We indicate that this particular type of SC current is the calculated emergency current for the electrical equipment under study [4]. We believe that the wires and cables under consideration are located in an ambient air environment, the temperature of which corresponds to room temperature and is equal to 60=20 °C [2, 4]. The above value of the specific conductivity y0i of the core (shell) material of the CCP corresponds to this temperature. We believe that the wires (cables) under consideration, before exposure to their current-carrying parts of the SC current ik(t) with specified ATPs, can be both de-energized and under nominal current load. In this regard, the initial temperature 80i of the material of the current-carrying parts of the CCP, depending on the current mode of operation of the power circuits of the electrical
equipment, can correspond to 60=20 °C or the value of the long-term allowable heating temperature 6u of their material. It is known that for non- and insulated wires and cables with PVC, R and PET insulation, the temperature 8n does not numerically exceed the levels of 70 and 65 °C regulated by current requirements, respectively [4, 6]. We use the assumption that the axial SC current ik(t) is almost uniformly distributed over the cross-section St of the core (shell) of the wire (cable) of the electrical equipment considered. The justification for this assumption is that the minimum penetration depth A,- of the magnetic field (skin layer thickness) from the SC current k(t) in the quasi-stationary approximation into the considered nonmagnetic conductive materials, determined from the calculated relation of the form A^[1/(nf«0y0i)]1/2 [2], where y0i is the electrical conductance of the core (shell) material of the CCP at 00=20 °C, and /i0=4n-10-1 H/m is the magnetic constant, numerically for copper is approximately 9.3 mm, and for aluminum is 11.8 mm. It can be seen that the presented numerical values of Ai are comparable with the real radii (thicknesses) of the current-carrying conductors (shells) of wires and cables used in electrical equipment circuits of general industrial use. We use the condition of the adiabatic nature of the occurring at times of action of the SC current ik(t) in the power circuits of the specified electrical equipment of no more than 1000 ms in the materials of the conductors (shells) of the investigated CCP of the electrothermal processes under which heat transfer from surfaces of current-carrying parts having the current temperature 0ci>00i, and the thermal conductivity of the layers of its conductive materials of the core (shell) and insulation on the Joule heating of the current-carrying parts of the checkpoint are neglected. It is required by calculation to determine in approximate form the critical sections Sa of current-carrying parts for uninsulated copper (aluminum) wires, as well as for insulated wires and cables with copper (aluminum) cores (shells), PVC, R and PET insulation used in power circuits of electrical equipment of general industrial applications and experiencing in emergency operation mode the direct effect of the axial SC current i ik(t) with specified ATPs. In addition, based on the calculation of the values of the critical sections Sci, it is necessary to determine the values of the critical amplitudes of the densities dCi of AC current with frequency f = 50 Hz in the current-carrying parts of wires and cables of power circuits of electrical equipment of power industry, through which SC current ik(t) can flow.
The electrical engineering approach to the calculation of critical sections SCi and current densities da in wires and cables of circuits of electrical equipment for general industrial purposes. To find critical cross-sections SCi of conductive cores (shells) of considered non-insulated and insulated with PVC, R and PET insulation electrical wires and cables in power circuits of electrical equipment with axial SC current ik(t) of specified ATPs, from the equation of their heat balance in the adiabatic Joule mode heating of the current-carrying parts of the CCP the following calculation relation follows [3]:
Sa = Jca)1/2/Da, (1)
tk
where JCiA= j i2(t)dt - the Joule (action) integral for the
0
SC current ik(t) with duration tk of its flow through the cores (shells) of the CCP, A2-s; DCi = (Ja-Ju)m, A-s1/2/m2; Jti is the critical value of the current integral for the material of cores (shells) of wires and cables of electrical equipment circuits [2], A2-s/m4; JH is the value of the current integral for the material of conductors (shells) of wires and cables of electrical equipment circuits, the long-term permissible heating temperature of which with the rated current corresponds to the known value 0,, [4], A2-s/m4.
In (1), the value of the current integral Jli is calculated from the following analytical expression [3]:
Ji = Y0iP— ln[i (0 -00) +1], (2) where c0i, are accordingly, the specific heat attributed to the unit volume of the material of the core (shell) of the wire (cable) and the thermal coefficient of the specific conductivity of this CCP material before the SC current ik(t) flows through it at 00=20 °C.
It can be seen from (2) that at 0,i=00 (the power-off mode of the CCP), the desired current integral is zeroed (J,,=0).
In Table 1 at 00=20 °C, numerical values are given for such basic characteristics of copper and aluminum conductors (shells) of wires (cables) of the studied power circuits as Y0i, C0„ fa and Ja [2].
Table 1
Thermophysical characteristics of copper (aluminum) cores (shells) of considered wires and cables of power circuits of electrical equipment before action on them of SC current ik(t) (at 00=20 °C) [2]
Material of the core (shell) of the wire (cable) Numerical value of the characteristic
y0i, 107 (Q-m)-1 cot, 106J/ (m3-°C) fiot,10~9' m3/J Ja, 1017 A2-s-m-4
Copper 5.81 3.92 1.31 1.95
Aluminum 3.61 2.70 2.14 1.09
Table 2, taking into account the use of data of (1), (2) and Table 1, shows the numerical values of the coefficient DCi necessary for calculating, according to (1), the critical cross-section SCi of the conductive core (sheath) of the considered wire (cable) in the power circuit of electrical equipment used in power industry.
Table 2
Numerical values of the coefficient DCi for non- and insulated wires (cables) with copper (aluminum) cores (shells) in general-
purpose electrical equipment circuits
Type of insulation in the wire (cable) of the electrical installation Material of the core (shell) of the wire (cable) Numerical value of DCi, 108 A-s1/2/m2
Jiè0 Ji=0
Without insulation Copper 4.299 4.415
Aluminum 3.236 3.301
PVC, R Copper 4.299 4.415
Aluminum 3.236 3.301
PET Copper 4.299 4.415
Aluminum 3.236 3.301
From the data of Table 2 it can be seen that the current mode of operation of the CCP under consideration (wires and cables of electrical circuits are fully loaded with rated current (J,#0) or at J,,=0 are completely de-energized) slightly affects the numerical values of the calculated coefficient DCi (up to 3 %). Its quantitative values are determined mainly by the type of conductive material of the core (shell) of the considered CCP. For copper and aluminum, the difference in the numerical values of DCi is approximately 25 %.
It follows from (1) that, for the numerical values of the coefficient DCi found (see Table 2), the determination of the critical cross-sections SCi of the copper (aluminum) cores (shells) of the investigated CCP is reduced to the quantitative determination of the action integral JCiA of the SC current ik(t) flowing during time tk over the current-carrying parts of the selected wires and cables.
2.1Approximate calculation at SC integral of the action integral JaA of the emergency current. As in [7, 8], we assume that the SC current ik(t) in the circuits with the considered CCP is described by the following temporal dependence:
ik (t) = Imk [exp(-t/Ta) - cos(2n /Tp)], (3)
where Imk is the amplitude of the steady SC current ik(t) in the power circuit of the electrical equipment; Ta, Tp are, respectively, the decay time constant of the aperiodic component and the period of oscillations of the periodic component of the emergency SC current ik(t) in the circuit with the CCP.
Then, based on (1) and (3), the calculated expression for the integral of the action of the SC current ik(t) in power circuits with the CCP takes the following analytical form [7]:
JCA = IIk f°,5tk + 0,25n"Tp sin(2ntk /Tp ) x
X cos(2^tk /Tp) - 2Ta2Tp2(Tp + 4^2Ta2)-1 [exp(-tk /Ta) x (4)
x [l-nT- sin(2ntk /Tp) - T- cos(2^tk /Tp) + Ta_1] ] +
+ 0,5Ta [1 - exp(-2tk / Ta)]}.
In Table 3, for the case Ta=50 ms (Tp=20 ms), the numerical values of the action integral JCiA calculated by (4) for the SC current ik(t) are given for the steady SC current amplitudes Imk and its durations (shutdown times) tk in power circuits of electrical equipment for general industrial use according to the requirements of [4, 8]. Knowing the numerical values of the coefficient DCi (see Table 2) and the action integral JCA of the SC current ik(t) (see Table 3), from (1) the numerical values of the critical sections SCi of the current-carrying parts of the considered CCP in power electrical installations circuits for general industrial use can be relatively easy determined. Taking into account the assumptions made, from the relation of the form da~Imk/SCi, the critical amplitudes of the AC current densities dCi in the cores (shells) materials of the studied wires (cables) for the SC emergency fault mode can be quantified.
2.2. The results of the calculated choice of critical sections Sa and current densities 8a in wires and cables of electrical equipment circuits for general industrial purposes. Table 4 shows the results of approximate calculation according to (1), taking into
Table 3
Numerical values of the action integral JCA for the SC current
i£f) flowing in power circuits of general-purpose electrical equipment (Tp=20 ms; Ta=50 ms) calculated by the relation (4)
The value of the amplitude Imk of the steady SC current i^f) in the power circuit of an industrial electrical installation, kA Values of the action integral JCA for the SC current i/(t) by (4), 107-A2-s
fk=100 ms fk=160 ms
30 6.75 9.45
50 18.75 26.25
70 36.75 51.45
100 75.00 105.00
125 117.18 164.06
Type of insulation in the wire (cable) of the electrical installation Material of the core (shell) of the wire (cable) Values of section SCi, mm2
Amplitude Imk of the steady SC current, kA
30 50 70 100
Without insulation Copper 19.11 31.85 44.59 63.70
Aluminum 25.38 42.31 59.24 84.63
PVC, R Copper 19.11 31.85 44.59 63.70
Aluminum 25.38 42.31 59.24 84.63
PET Copper 19.11 31.85 44.59 63.70
Aluminum 25.38 42.31 59.24 84.63
Table 5
Numerical values of critical cross-sections SCi for bare wires and insulated wires (cables) with copper and aluminum cores (shells) in power circuits of general industrial electrical equipment with amplitude Imk of SC current ik(t) varying from 30 to 100 kA (for tk=160 ms and Ta=50 ms)
account the data in Table 2, 3 of critical cross-sections SCi of copper (aluminum) cores (shells) of uninsulated (bare) wires and insulated wires (cables) of power circuits of general industrial electrical equipment (Tp=20 ms) for the case when J/#0, tk=100 ms and Ta=50 ms, and the amplitude Imk of the steady SC current ik(t) varies discretely in the range (30-100) kA. From the data of Table 4 it follows that the critical amplitudes of the densities dC^ImkISCi of the SC current ik(t) at the time of its flow (switch off) in the power circuits of electrical installations equal to tk=100 ms for uninsulated wires and insulated wires (cables) with PVC, R and PET insulation with copper (aluminum) cores (shells) in the circuits of electrical equipment for general industrial use (Tp=20 ms; Ta=50 ms) are numerically approximately 1.57 kAImm2 and 1.18 kAImm2, respectively. It is important to note that these values of the critical amplitudes of the densities dCi of the SC current ik(t) in the materials of the current-carrying parts of the wires (cables) of the power circuits of the electrical equipment do not depend on the level of the amplitude Imk of the steady SC emergency fault current of the power frequency of 50 Hz.
Table 5 presents the results of the calculated determination according to (1), taking into account the data in Table 2, 3, of critical cross-sections Sa of copper (aluminum) cores (shells) of bare wires and insulated wires (cables) of power circuits of electrical equipment of general industrial use (Tp=20 ms) for the case when J#0, tk=160 ms and Ta=50 ms, and the amplitude Imk of the steady SC current ik(t) varies discretely in the range (30-100) kA.
Table 4
Numerical values of critical cross-sections SCi for bare wires and insulated wires (cables) with copper and aluminum cores (shells) in power circuits of general industrial electrical equipment with amplitude Imk of SC current ik(t) varying from 30 to 100 kA (for tk=100 ms and Ta=50 ms)
Type of insulation in the wire (cable) of the electrical installation Material of the core (shell) of the wire (cable) Values of section SCi, mm2
Amplitude Imk of the steady SC current, kA
30 50 70 100
Without insulation Copper 22.61 37.68 52.76 75.37
Aluminum 30.04 50.06 70.09 100.13
PVC, R Copper 22.61 37.68 52.76 75.37
Aluminum 30.04 50.06 70.09 100.13
PET Copper 22.61 37.68 52.76 75.37
Aluminum 30.04 50.06 70.09 100.13
From the data of Table 5 it follows that the critical amplitudes of the densities SC^ImkISCi of the SC current ik(t) at time tk=160 ms of its flow (switch off) in the power circuits of the electrical equipment under consideration for uninsulated wires and insulated wires (cables) with PVC, R and PET insulation with copper (aluminum) cores (shells) in the circuits of electrical equipment for general industrial use (Tp=20 ms; Ta=50 ms) numerically are approximately 1.33 kAImm2 and 0.99 kAImm2, respectively. Here, the numerical values of the critical amplitudes of the densities 8Ci of the SC current ik(t) in the copper (aluminum) cores (shells) of the CCP of the industrial electric power equipment under consideration, indicated for the calculation case (tk=160 ms; Ta=50 ms), as well as for the previous case (tk=100 ms; Ta=50 ms), do not depend on the quantitative values of the amplitude Imk of the steady SC current. In addition, from the data of Table 4, 5 it follows that the quantitative values of the critical cross-sections SCi and critical amplitudes of densities 8Ci of AC SC current ik(t) for the current-carrying parts of the investigated CCP, which is widely used in power circuits of electrical equipment of general industrial use, do not depend on any type of insulation (air or solid) used in the considered electric wires and cables of electric power devices.
Conclusions.
1. The proposed electrical engineering approach allows, according to the condition of the EE in the atmospheric air of the current-carrying parts of the CCP, to approximate the critical cross-sections SCi and the amplitudes of the densities 8Ci of the AC axial current for uninsulated wires with copper (aluminum) cores, as well as for insulated wires and cables with copper (aluminum) cores (shells), PVC, R and PET insulation, through which AC SC current ik(t) with given ATPs flows during emergency operation of power electrical equipment for general industrial use.
2. The used approximate calculation relations (1)-(4) made it possible for two real cases (tk=100 ms; tk=160 ms) at Ta=50 ms to establish for a discrete change in the amplitude Imk of the steady SC current ik(t) in the range (30-100) kA specific capabilities of the proposed electrical engineering approach for the selection of critical sections Sa and density amplitudes 8Ci of AC current in
the indicated wires and cables of power circuits of electrical equipment of general industrial use, in the current-carrying parts of which in emergency operating modes of the considered power electrical equipment large axial SC currents ik(t) flow.
3. It was established by calculation that the critical amplitudes of the densities dCpIm/JSCi of the axial SC current ik(t) in copper (aluminum) cores of uninsulated wires and insulated wires (cables) with copper (aluminum) cores (shells), PVC, R and PET insulation at Ta=50 ms for the case of switch off time tk=100 ms of the SC current ik(t) in the power circuits of electrical equipment numerically equal approximately 1.57 (1.18) kA/mm2, and for a case of tk=160 ms - 1.33 (0.99) kA/mm2.
4. The results obtained for critical cross-sections SCi and density amplitudes dCi of AC axial current with frequency f = 50 Hz (Tp=20 ms) can be used in electric power practice to choice thermally resistant to large SC axial currents ik(t) the CCP with copper (aluminum) cores (shells), designed to operate in power circuits of electrical equipment of power industry.
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Received 04.02.2019
M.I. Baranov, Doctor of Technical Science, Professor, Scientific-&-Research Planning-&-Design Institute «Molniya», National Technical University «Kharkiv Polytechnic Institute», 47, Shevchenko Str., Kharkiv, 61013, Ukraine, phone +38 057 7076841, e-mail: baranovmi@kpi.kharkov.ua
How to cite this article:
Baranov M.I. A choice of critical sections of electric wires and cables in power circuits of electrical equipment of power industry. Electrical engineering & electromechanics, 2019, no.5, pp. 35-39. doi: 10.20998/2074-272X.2019.5.06.
