CC BY
10
Electrical Insulation and Cable Engineering
UDC 621.3.022: 621.315.2(3) https://doi.org/10.20998/2074-272X.202L6.05
M.I. Baranov, S.G. Buriakovskyi, V.V. Kniaziev
Destruction of polymer insulation and threshold amplitudes of current pulses of different temporal shapes for electric wires and cables in the low- and high-current circuits of pulse power engineering, electrical engineering and electronic devices
Goal. Development of engineering method for settlement of threshold amplitudes Impk of single-pulse current ip(t) of different temporal shapes for electric wires and cables with polyethylene (PET), polyvinylchloride (PVC) and rubber (R) half-length insulation, used in modern pulsed power engineering, electrical engineering and electronics in their low- and high-current circuits. Methodology. Basis of the theoretical and applied electrical engineering, electrical power engineering, electrophysics bases of technique of high-voltage and large pulsed currents, bases of low- and high-current electronics, measuring technique, electromagnetic compatibility and standardization. Results. Development of engineering method is executed on close calculation determination of threshold amplitudes Impk of single-pulse axial-flow current ip(t) of different temporal shapes for electric wires and cables with copper (aluminum) current-carrying parts and PET, PVC and R half-length insulation, used in the ow- and high-current circuits of pulsed electrical power engineering, electrical engineering and electronics. Electrothermal resistibility of half-length insulation of the examined cable and wire products (CWP), proper maximum to the possible temperatures of heating of current-carrying and insulating parts of the probed wires and cables and shutting out the offensive of the phenomenon destruction in the indicated insulation of CWP, was fixed based on this method. Calculation analytical correlations are obtained for finding in probed CWP of threshold numeral values of Impk amplitudes ofpulses of current ip(t), time-varying both on aperiodic dependence of type i/ip with duration of their front if and duration of their pulses Tp and by law of exponential attenuation sinewave. It is shown that at Imp>Impk destruction of their half-length insulation, resulting in the decline of service life of CWP, will come from the thermal overheat of current-carrying parts of the examined electric wires and cables. The examples of practical application of the offered method are resulted upon settlement for a radiofrequency coaxial cable RC 50-4-11 with middle sizes is easily soiled with continuous PET insulation of threshold amplitudes of Impk of standard aperiodic pulses of current ip(t) from nano-, micro- and millisecond temporal ranges of shape of t/tp=5 ns/200 ns, t/tp=10 fs/350 fs and t/tp=7 ms/160 ms. It is shown that with the proper growth of parameter Tp>>Tf for flow on a continuous copper tendon and split copper shell of radiofrequency coaxial cable RC 50-4-11 with middle sizes is easily soiled indicated homopolar pulses of current ip(t) substantial diminishing of their threshold amplitudes of Impk (with 531,2 кА for the nanosecond pulse of current of type 5 ns/200 ns to 1.84 кА for the millisecond impulse of current of type of 7 ms/160 ms takes place). Originality. An engineering method is first developed for close settlement of threshold numeral values of Impk amplitudes of single-pulse axial-flow current ip(t) of arbitrary peak-temporal parameters for electric wires and cables with copper (aluminum) current-carrying parts and PET, PVC and R half-length insulation. Practical value. Application in electrical engineering practice of the offered engineering method for determination of threshold amplitudes Impk of the indicated pulses of axial-flow current ip(t) for the probed electric wires and cables will allow considerably to increase service life of examined CWP. References 20, tables 2.
Key words: electrical wires and cables with polymer insulation, electrothermal resistance of cable and wire products, destruction of insulation, threshold amplitudes of current pulses for wires and cables.
Запропонована тженерна методика за розрахунковим визначенням порогових амплтуд Impk iMnynbcie струму ip(t) pi3Hoi часовоi форми для електричних проводiв i кабелiв з полiетиленовою, полiвiнiлхлоридною i гумовою iзоляцieю, широко вживаних в галyзi жпульсно1 енергетики, високовольтно1 сильнострумово1 техшки, вимiрювальноi техтки i електротки, а також в системах iмпyльсного електроживлення, контролю, управлтня роботою i дiагностики стану функцюнування електротехтчних пристро1в рiзного загальногромадянського i втськового призначення. В якостi вихiдного критерiального положення при виборi порогових амплтуд Impk iмпyльсiв струму ip(t) довыьних амплтудно-часових параметрiв для вказаних проводiв i кабелiв була вибрана термiчна сттюсть 1х поясно'1 iзоляцii, яка вiдповiдаe гранично допустимим короткочасним температурам нагрiвy мiдних (алюмШевих) i iзоляцiйних частин до^джуваних кабелiв (проводiв) i що не допускае настання явища деструкцп в iзоляцii дано1 кабельно-провiдниковоi продукцп. Приведет приклади практичного використання запропонованоi методики за розрахунковим визначенням порогових амплтуд Impk стандартних аперiодичних iмпyльсiв струму ip(t) часовоi форми 5 нс/200 нс, 10 мкс/350 мкс i 7 мс/160 мс для радючастотного коакЫального середньогабаритного кабелю марки РК 50-4-11 зi суцыьною полiетиленовою iзоляцiею. Бiбл. 20, табл. 2. Ключовi слова: електричш проводи i кабелi з полiмерною iзoлящeю, електротермiчна стшккть кабельно-проввдниково1 продукщ1, деструкщя iзoлящl, noporoBi ампщтуди iмпульсiв струму для проводiв i кабелiв.
Предложена инженерная методика по расчетному определению пороговых амплитуд Impk импульсов тока ip(t) различной временной формы для электрических проводов и кабелей с полиэтиленовой, поливинилхлоридной и резиновой изоляцией, широко применяемых в области импульсной энергетики, высоковольтной сильноточной техники, измерительной техники и электроники, а также в системах импульсного электропитания, контроля, управления работой и диагностики состояния функционирования электротехнических устройств различного общегражданского и военного назначения. В качестве исходного критериального положения при выборе пороговых амплитуд Impk импульсов тока ip(t) произвольных амплитудно-временных параметров для указанных проводов и кабелей была выбрана термическая стойкость их поясной изоляции, соответствующая предельно допустимым кратковременным температурам нагрева медных (алюминиевых) и изоляционных частей исследуемых кабелей (проводов) и не допускающая наступления явления деструкции в изоляции
© M.I. Baranov, S.G. Buriakovskyi, V.V. Kniaziev
рассматриваемой кабельно-проводниковой продукции. Приведены примеры практического использования предлагаемой методики по расчетному определению пороговых амплитуд 1трк стандартных апериодических импульсов тока гр(Г) временной формы 5 нс/200 нс, 10 мкс/350 мкс и 7 мс/160 мс для радиочастотного коаксиального среднегабаритного кабеля марки РК 50-4-11 со сплошной полиэтиленовой изоляцией. Библ. 20, табл. 2.
Ключевые слова: электрические провода и кабели с полимерной изоляцией, электротермическая стойкость кабельно-проводниковой продукции, деструкция изоляции, пороговые амплитуды импульсов тока для проводов и кабелей.
State-of-the-art and urgency of the problem. In
the field of traditional electric power engineering, there is an electrical engineering approach to the engineering selection of the cross-sections SC of electrical wires and cables with polymer insulation, used for both long-term and short-term modes of their operation [1]. This approach is based on the thermal stability of such a cable and wire product (CWP) under conditions of direct exposure to it of short-circuit (SC) current with specified amplitude-temporal parameters (ATPs). In this case, the thermal resistance of electric cables (wires) of any design is limited by the maximum permissible short-term temperature 8S of heating of their metal and insulating parts in the mode of one-, two and three-phase SC in the electrical network [1]. According to electrical data from [1], the specified temperature 8lS should not exceed for those used in power circuits with alternating current of power frequency 50 Hz of non-insulated (bare) copper and aluminum buses (wires) in the SC mode of the highest level of 250 °C and 200 °C, and for cables (insulated wires) with copper (aluminum) cores, polyvinylchloride (PVC), rubber (R) and polyethylene (PET) insulation - respectively, the temperature level of 150 °C and 120 °C. In addition, it is known that in the field of industrial power engineering, the long-term permissible heating temperature 8n of conductive (insulating) parts of various electrical wires and cables is limited by the conditions of reliable operation of electrical contacts and contact connections of their circuits, as well as by the operating conditions of their insulation [1]. Here, the maximum long-term permissible heating temperature 8n for the main types of bare wires (buses) and cables (wires) with PVC, R and PET insulation, which are under current load in industrial electric power circuits, should not numerically exceed the level of 70 °C and 65 °C, respectively [1]. Taking into account the above electrical data and the fulfillment of those initial conditions that the cable (wire) before the AC current acts on it was fully electrically loaded and had temperature 8U, and upon the onset of the SC mode, it heated up to temperature 8lS, in [1] to select the minimum permissible cross-section Slmin of the electric wire (cable), the following calculated analytical relationship is recommended:
Sl min = Bk2 I Ck , (1)
tt
where Bk = j ¿2 (t)dt is the Joule (action) integral of the SC
0
current ik(t) with its duration tk of flow in the CWP, A2-s;
Ck is the constant coefficient, A-s12/m2.
We point out that the engineering method for
calculating the Joule integral Bk in (1) and the numerical
values of the coefficient Ck corresponding to the indicated operating conditions of wires and cables in power electric circuits for the CWP are given in [1]. The values of the cross-sections Slmin of the CWP found by (1) will correspond to the operating mode in the electric power circuits of wires and cables when the heating temperature of their current-carrying and insulating parts does not exceed the maximum permissible short-term temperature 8lS and when the thermal resistance of the indicated CWP is ensured.
ATPs of current pulses ip(t) of nano-, micro- and millisecond time ranges generated and used in the field of high-voltage pulse technology (HPT) [2] and in other areas of modern pulsed low- and high-current power engineering, electrical engineering and electronics (for example, in equipment of pulse electrical technologies and accelerator technology [3]) to achieve various scientific and technological goals, usually do not correspond to ATPs of AC SC current in industrial power circuits. In this regard, the practical application of relationship (1) for the calculation determination of the cross-sections SC of wires (cables) in low- and high-current electrical circuits of the indicated pulse technology is fundamentally impossible. In addition, the indicated electrical engineering approach to the choice in the industrial electric power industry of the cross-sections SC of electric wires (cables) does not allow determining the threshold amplitudes Impk of pulse currents ip(t) with various ATPs, above which destructive processes in its polymer insulation will begin to manifest in the considered CWP. As is known, the phenomenon of destruction of such insulation, accompanied by irreversible violations of its electrical insulating properties, can be caused by an external (internal) damaging or destabilizing effect on wires and cables of various physical factors (for example, an alternating (pulsed) current flowing through their current-carrying parts, ionizing and electromagnetic radiation) [1, 4, 5]. Within the framework of the applied problem we are solving, only one destabilizing factor is considered, due to the pulse current ip(t) flowing through the electric wires and cables with various ATPs. At certain (threshold) values Impk of the amplitude Imp of current pulses ip(t) of one or another temporal shape flowing through the current-carrying parts (inner core and outer shell) of the CWP, due to the intense Joule heating of these metal parts of the wires (cables), processes of thermal destruction in their belt insulation can occur [1, 6, 7]. It should be noted that at values of the amplitudes Imp of pulse currents ip(t) of hundreds of kiloamperes, destructive processes in the polymer insulation of the studied CWP can also arise from the action of large electrodynamic forces on cables (wires) [2]. In practice, it is important to know such
threshold values Impk of the amplitudes Imp of current pulses ip(t) of various temporal shapes, leading to a violation of the electrical insulating properties of wires (cables) used in the field of HPT, measuring technology, electronics, in systems of power supply, control, operation control and diagnostics of the state of functioning of electrical devices, and a decrease in their service life.
At present, when developing and creating in the world new polymeric insulating materials with various nano- and microstructural structures of electrical engineering and other (including medical and biological) purposes, the issues of behaviour of polymer insulation under conditions of the action of destabilizing (damaging) physical factors are given increased attention [8-11].
The goal of the paper is the development of an engineering methodology for calculating the threshold amplitudes Impk of single current pulses ip(t) of various temporal shapes for electrical wires and cables with PET, PVC and R belt insulation used in modern pulsed power engineering, electrical engineering and electronics in their low- and high-current circuits.
Problem definition. Consider insulated wires and cables with copper (aluminum) inner cores (i=1) and outer shells (i=2), with PET, PVC and R belt insulation used in electrical circuits of HPT and other low- and high-current electrical engineering (electronics) [1, 12]. We assume that single current pulses ip(t) flow in their longitudinal direction along the round solid or split copper (aluminum) cores and shells of the indicated wires and cables of electrical circuits of pulsed electrical engineering (electronics) devices, the ATPs of which can correspond to nano-, micro- and millisecond time ranges with amplitudes Imp, varying in a wide range from 1 A to 1000 kA [2, 3]. We assume that wires and cables of finite length l0 are placed in the surrounding air with temperature equal to 80 = 20 °C [13]. Let us use the condition of the adiabatic nature of electrothermal processes flowing at the time of action of the pulsed axial current ip(t) no more than 1000 ms in the materials of the cores (shells) of the studied CWP, in which the effect of heat transfer from the surfaces of their current-carrying parts, having current temperature 8C>80, and their thermal conductivity of their electrically conductive materials and insulation for Joule heating of the metal parts of the cores (shells) of wires (cables) are neglected. We believe that the thermal resistance of wires (cables) of circuits of HPT and other above-mentioned electrical engineering (electronics) devices when exposed to pulse current ip(t) is limited by their maximum permissible short-term heating temperature 8CiS, which depends on the degree of decrease in the mechanical strength of the core (shell) material and thermal conditions of operation of the insulation of the CWP in the mode of its short-term heating by current pulses of nano-, micro- or millisecond duration, flowing through their current-carrying parts. As in [14], we assume that the temperature value 8CiS corresponds to the known from [1] maximum permissible short-term temperature 8lS of heating of wires and cables by SC currents of power frequency. Then, in accordance with the
data from [1], in the electrical low- and high-current circuits of the considered electrical engineering for their insulated wires (cables) with copper and aluminum conductors (shells) and PVC (R) insulation, the value 8CiS will be numerically approximately 8CiS~ 150 °C , and for their CWP with the indicated cores (shells) and PET insulation - 8CiS ~ 120 °C . It is required to calculate in an approximate form the threshold amplitudes Impk of single current pulses ip(t) of various ATPs from nano-, micro-and millisecond time ranges flowing through electrical wires and cables with copper (aluminum) cores (shells) and PET, PVC and R belt insulation.
Electrical engineering approach to the selection of the permissible minimum cross-sections SCil of wires and cables with pulse current of various ATPs. For the permissible minimum cross-sections SCil of the current-carrying conductors (shells) of the investigated electrical wires (cables) with pulse current ip(t) of arbitrary ATPs, from the equation of their heat balance in the adiabatic mode of the CWP operation in low- and high-current circuits, the following approximate calculated relationship can be obtained [14 ]:
SCil - (JCiA )1/2 1 Cl ■
(2)
where Jca =
p
J ip (t)dt - the action integral of a single
0
current pulse ip(t) with its duration tp and given ATPs, A2-s; Ci = (JClS - JCll)1/2 is the constant coefficient, A-s1/2/m2; Jcis, Jaa are the current integrals for current-carrying conductors (shells) of electrical wires (cables) of low-current and power circuits of various electrical devices, the permissible short-term and long-term permissible heating temperatures of the CWP material which correspond to the values adopted above: 8lS = 8CiS and 8ll, A2-s/m4.
To find the numerical values of the current integrals JCiS and JCn included in (2), the following analytical expressions can be used [14]:
JciS - Y0iPm ln[% Aw Vis -3>) +1]; (3) Jcii - Y0tP- ln[c0iA0i (°u -^0) +1], (4)
where y0i, c0i, fi0i are, respectively, the specific electrical conductivity, specific volumetric heat capacity and thermal coefficient of specific electrical conductivity of the material of the core (shell) of the wire (cable) of the electric circuit before the impact on the considered CWP of the pulsed current ip(t) with arbitrary ATPs.
Table 1 shows the numerical values of the electrophysical parameters y0i, c0i h fi0i, included in the calculation relationships (3), (4), at room temperature of the air surrounding the electrical wires and cables under consideration, equal to 80 = 20 °C [13].
Table 2 shows calculated according to (2) - (4) taking into account the quantitative data of Table 1 numerical values of the coefficient Ci for insulated wires and cables with copper (aluminum) cores (shells) with PVC, R and PET insulation for two cases possible in real
practice of their exploitation: the case of their preliminary current load (JCU 4 0) and the case of their complete de-energizing (Jan = 0).
Table 1
Basic electrophysical parameters of the material of current-carrying cores (shells) of insulated wires and cables in low- and high-current circuits of modern electrical engineering (electronics) at O0 = 20 °C [13]
Material of the Numerical value of the parameter
core (shell) of Yoi, C0i, Po,
the wire (cable) 107-(Q-m)-1 106-J/(m3-°C) 10-9-m3/J
Copper 5,81 3,92 1,31
Alunimum 3,61 2,70 2,14
Table 2
Numerical values of the coefficient Cl for insulated wires and cables with copper (aluminum) cores (shells) in low- and high-current circuits of modern electrical engineering (electronics) with nano-, micro- and millisecond current pulses ip(t) of various ATPs
As for the calculated definition in (2) of the action integral JCA of a single current pulse ip(t) with one or another ATPs, we first consider the case of a change in this type of electric current in time t according to the aperiodic law of the following form [2,15]:
ip (t) = kpiImp [exp(-ait) - exp(-«2t)], (5)
where a1 ~ 0.76/xp, a2 ~ 2.37/Tf are the shape coefficients of the aperiodic current pulse with the given ATPs, flowing in low- and high-current circuits of HPT, pulsed electrical engineering and electronics; kp1 = [(a1/a2)m -- (a1/a2)"]-1 is the normalizing factor; m = a1/(a2-a1); n = a2/(a2-a1); t, tp are, respectively, the rise time at the level of (0.1-0.9) Imp and the duration of the current pulse at the level of 0.5 Imp; Imp is the amplitude of the current pulse ip(t) flowing through the wire (cable).
In this electrophysical case, the expression for the action integral JCiA flowing in low- and high-current circuits of the considered technique of the current pulse ip(t) takes at tp=3rp according to (2), (5) the following approximate analytical form:
JCiA * khI2mp[0,658Tp - 0,633V]. (6)
Next, consider the electrophysical case when changes in time t of the pulse current ip(t) acting on the electric wires (cables) of the indicated circuits of electrical engineering (electronics) occur according to the law of a damped sinusoid [2, 13]:
ip (t) = kp21 mp1 exp(-#) sin(®t), (7)
where d=Ap/Tp is the current damping coefficient; m=2n/Tp is the circular frequency of current oscillations; Tp is the
period of current oscillations; Ap=ln(Imp1/Imp3) is the logarithmic decrement of pulse current ip(t) oscillations with the first Imp1 and the third Imp3 amplitudes in electrical circuits; kp2=[exp(-Ap/2n-arcctgAp/2n)sin(arcctgAp/2n)]-1 is the normalizing factor for the damped sinusoidal current ip(t) flowing in the wire (cable).
For the temporal shape (7) of the change in the wire (cable) of the current pulse ip(t) at tp=3Tp in (2), the calculation expression for the action integral JCiA of the current pulse ip(t) flowing in the investigated low- and high-current circuits of modern electrical engineering takes the following approximate analytical form:
Jca * k2p2I2mp1[Tp(4Ap)-1 - ApTp(4A2p +16^2)-1]. (8)
Knowing the numerical values of Imp, Imp1, t, tp, Ap and Tp from regulatory documents or experimental data, taking into account the calculated estimate of the values of the normalizing coefficients kp1 and kp2 for the indicated temporal shapes of changes in the pulse current ip(t) according to (2) - (8) we can in approximate form calculate (with an error of no more than 5 %), the permissible minimum cross-sections Saa of cores (shells) of electrical wires and cables used in circuits of HPT, power electrical engineering and electronics.
With regard to the applied problem being solved, the threshold values Impk of the amplitude Imp of the current pulse ip(t) of the given temporal shape will correspond to the permissible short-term heating temperature 8CiS = 0lS with this pulse current ip(t) of the electric wire and cable with the selected insulation. Therefore, from (2), taking into account (6), (8) and the data of Table 2, when the relation Sa = Saa is fulfilled, the calculated threshold values Impk of the amplitudes Imp and Imp1 of the axial current pulses ip(t) for the time shapes indicated according to (5), (7) flowing along the investigated insulated wires and cables in low- and high-current circuits of electrical engineering and electronics can also be determined in the considered approximation.
Calculation estimation of threshold amplitudes Impk of current pulses ip(t) of various ATPs for electrical wires and cables. In accordance with the above expressions (2), (5), (6) for the threshold value Impk of the amplitude Imp of the aperiodic (unipolar) pulse of the axial current ip(t), acting on the current-carrying and insulating parts of the investigated electrical wires (cables), under the condition SCl=SCu, the following approximate calculated analytical relationship can be obtained:
Impk * SaClk-pl[0,658Tp -0,633^]-1/2 . (9)
From (9) it can be seen that for given temporal parameters of the front ty and duration tp of the current pulse ip(t), known structural characteristics of wires and cables (values of their cross-sections SCi) and the selected operating mode of the CWP with the studied polymer insulation and the specified materials of its cores and shells (the known value of the coefficient Cl according to the data in Table 2), finding the desired value of the amplitude Impk will be reduced to determining,
Type of insulation in a wire (cable) of a low- and high-current circuit of electrical engineering and electronics Material of the core (shell) of the wire (cable) Numerical value of C, 108 A-s1/2/m2
Jcu=0 Jaft-0
PVC, R Copper 1,506 1,160
Aluminum 0,972 0,745
PET Copper 1,355 0,957
Aluminum 0,877 0,616
according to (5), the numerical value of the normalizing coefficient kp1>1.
From (2), (7), (8) under the condition SCi = SCii for the threshold value Impk of the first amplitude Imp\ of the damped sinusoidal pulse current ip(t) in the insulated wire and cable, the following approximate calculated analytical expression follows:
Impk - SaClk~p2[Tp(4Ap)_1 - ApTp(4Ap +16^rT1'Z. (10)
Similarly to (9), using (10) to find the calculated value Impk for a particular wire (cable) with known characteristics SCi and Cl (see Table 2) at given temporal parameters Ap h Tp for the discharge pulse current ip(t) flowing through the CWP will actually be reduced to the calculation according to (7) of the numerical value of the normalizing coefficient kp2>1.
According to (9), (10), the threshold values Impk of the amplitudes Imp of aperiodic and damped sinusoidal current pulses ip(t) are directly proportional to the cross-sections SCi of the metal cores (shells) of the electrical wires and cables under study. In addition, the sought values of Impk are actually inversely proportional to the temporal parameters xp12 h Tp12 for current pulses p(t)flowing through the CWP.
Note that the calculated relations (9), (10) for determining the threshold values Impk of the amplitudes Imp of the axial current pulses ip(t), varying in time t according to (5), (7) according to the aperiodic dependence and the law of the exponentially decaying sinusoid, cover a wide nomenclature of temporary shapes and ATPs of used in electrophysical practice single current pulses ip(t) flowing through current-carrying parts of wires and cables with PVC, R and PET belt insulation in modern pulsed power engineering, electrical engineering and electronics.
In low- and high-current circuits of the considered power engineering, electrical engineering and electronics with the temporal shapes of current pulses ip(t) flowing through their CWP used according to (5), (7) at Imp>Impk, thermal overheating of the current-carrying parts of wires and cables will lead to destruction of their insulation, which reduces the working life of the CWP used in them.
Calculation estimation of the heating temperature 0Ci of electrical wires and cables by current pulses ip(t) of various ATPs. For the purpose of computational verification of the formulas (9), (10) for choosing the threshold amplitudes Impk of the pulse current in the considered CWP, let us estimate the temperature 8Ci of the Joule heating of the current-carrying parts of cables (wires) through which single current pulses ip(t) with specified ATPs flow. For this, we use the well-known nonlinear dependence of the electrical conductivity y0i of the material of the core (shell) of the wire (cable) on its current temperature 8Ci [13]:
Y0i -720, [1 + % A 0Ci -M"1, (11) where y20i is the specific electrical conductivity y0i of the conductive material of the current-carrying parts of the CWP at the temperature of the surrounding air 8c,=80=20 °C
1
.2\-1i-1/2
(for copper and aluminum these values of y20i are indicated in Table 1).
For used in CWP of pulsed power engineering, electrical engineering and electronics basic metals, (11) describes the temperature changes in their parameter y0i with an error of no more than ±5 % [13].
Taking into account (11) and data from [13], the solution of the inhomogeneous differential equation of thermal conductivity applied to the metal parts of the investigated cables (wires) of the adopted length l0 with pulse current ip(t) of various ATPs for the current temperature 8Ci of their Joule heating by the specified current under the initial condition [8Ci|(t=0)-80] = 0 can be written in the following approximate form:
% -00 + (C0A0i )-1[exp(Jc,a72-01,A0, /sC, ) -1]. (12)
It follows from (12) that the current temperature 8Ci of heating by pulsed current ip(t) of various ATPs of current-carrying cores (shells) of the considered CWP is inversely proportional to the specific volumetric heat capacity c0i (heat capacity per unit volume of metal) of their conductive materials, which for most metals in the solid phase varies depending on their temperature within ±10 % of its average numerical value [13]. We point out that for the thermophysical parameter c0i, an equality of the form [13] is fulfilled: ci—cpipi, where cpi is the heat capacity per unit mass of the homogeneous conductive material of the CWP (J/kg-°C) with its density p (kg/m3). Therefore, in the investigated electrophysical case, we can say that the overall parameters of the CWP (except for the cross-section SCi of its cores and shells) at given ATPs of current pulses i ip(t) flowing through its metal parts do not affect the heating of cables (wires). This thermal process is attended, according to (12), by mainly the specific thermophysical parameters of the CWP (y0i, c0i and fi0l) and the ATPs of the pulsed current ip(t) [13]. For short (with extremely small values of tp and Tp) current pulses ip(t), the heating zone of the CWP will be localized in very thin layers of its metal cores and shells. Taking into account the accepted assumptions and (12), we can conclude that in the considered adiabatic approximation, the length l0 of the cable (wire) and, accordingly, the total mass of the metal parts of the CWP at t<3zp or t<3Tp does not have a noticeable effect on the pulse heating temperature 8Ci of these parts of the CWP. This influence on the temperature level 8Ci will increase after the passage of the considered current pulses ip(t) through the current-carrying parts of the CWP, when, due to the thermal conductivity of their metal, the temperature will begin to equalize along the thickness of these parts.
In the case when the relation SCi = SCil is fulfilled for the current-carrying metal parts of the cables and wires under consideration, taking into account (2), expression (12) is simplified and takes the following form:
0
Ci
> 00 + (% A )-1 [exp^-AC-) -1],
(13)
where Cl is the constant coefficient, the numerical values of which for the considered polymer insulation of cables (wires) and the specified operating modes of their electrical circuits are given in Table 2.
Relation (13) can be just used in the calculated verification of the obtained expressions (9), (10) to find the numerical values of the threshold amplitudes Impk of the pulse current ip(t) in the considered CWP. According to the conditions we have adopted, at SCi = SCil, the calculated according to (13) temperature 8Ci of heating of the current-carrying parts of the studied CWP should not exceed the accepted normalized permissible short-term temperature 8CiS for it.
Examples of calculating the threshold amplitudes Impk of current pulses ip(t) of nano-, micro- and millisecond temporal ranges. As the investigated CWP, we choose a short radiofrequency coaxial medium-sized cable with solid PET insulation, brand RC 50-4-11 [12], having, at l0<10 m, a round solid copper core with diameter of 1.37 mm (SC1~ 1.474 mm2) and tinned braided copper shell (braid with twisting density of at least 95 %) with inner diameter of 4.6 mm and wall thickness of 0.15 mm (SC2~2,059 mm2). We assume that this cable is placed in an air atmosphere at room temperature 80=20 °C with the fulfillment of the condition for the current integral JCll=0 in its electrical circuit (without preliminary current load of the cable at 8l=80). From the given design data, it can be seen that the copper core of the selected RF cable brand RC 50-4-11 with cross-section SC1~ 1.474 mm2 in comparison with its reverse external current conductor (copper braid with cross-section SC2~2.059 mm2) will be less resistant to the electrothermal action of a current pulse ip(t) longitudinally flowing through them in opposite directions with specified ATPs. Let a single current pulse ip(t) flowing through the current-carrying parts of the adopted coaxial cable has an aperiodic temporal shape. Therefore, the specified core of the cable of the RC 50-4-11 brand can be an internal local hotbed of overheating of the current-carrying parts of this cable. In this regard, the continuous belt PET insulation adjacent to the copper core of the RC 50-4-11 radiofrequency cable may experience the effect of increased levels of the temperature field caused by the Joule heating of this copper core by the adopted current pulse ip(t) flowing through it. It is the copper core and the adjacent cylindrical zone of PET insulation of the RF cable of the adopted in applied calculations values Impk that will be the weak «links» in a possible chain of destructive processes in the cable under consideration. Taking into account the above, it can be concluded that the calculation estimation of the threshold values Impk of the amplitude Imp of the used current pulse ip(t) of a given time shape for a radiofrequency cable of the brand RC 50-4-11 should be tied to the electrothermal state of a single-wire round copper core (SC1~1.474 mm2) of this cable experiencing the thermal effect of an aperiodic current pulse ip(t).
1. First, we use a standard nanosecond current pulse of a temporal shape t/tp=5 ns/200 ns, which was used in a number of countries when simulating in high-current discharge electric circuits of HPT with multi-wire air systems of field formation and, accordingly, in their working air volumes with those tested for electromagnetic compatibility (durability) technical objects of various
dimensions of a powerful electromagnetic pulse of a high-altitude nuclear explosion [4, 16]. From (5) we find that for this calculation case, the shape factors a1 and a2 of the nanosecond current pulse ip(t) take the following numerical values: a1~3.8-106 s-1; a2~4.7-108 s-1. In this case, for a given temporal shape of a unipolar current pulse ip(t), the normalizing factor kp1 according to (5) turns out to be approximately equal to kp1~1.049. Then from (9) at kp1~1.049, Cl=1.355-108 A-s1/2/m2 (see the corresponding data in Table 2) and SCi=SC1=1.474 mm2 (cross-section of the copper cable core) for the threshold numerical value Impk the amplitude Imp of the considered aperiodic current pulse ip(t) of the temporal shape t/tp=5 ns/200 ns in relation to the RF coaxial cable of the brand RC 50-4-11, we find that 7^531.2 kA.
2. Next, consider the standard microsecond aperiodic current pulse ip(t) of the time shape Ty/Tp=10 ^s/350 ^s, which is now used in accordance with the requirements of the current International Standard IEC 62305-1-2010 [17] when testing power electrical equipment for resistance to the direct action of powerful short lightning electric discharges on it [18]. From (9) at
kp1 ~ 1.054 (a1
2.17-103 s-1; a2
2.37-105 s-1),
Ci = 1.355-10 A-s /m and section of a solid copper core SCi=SC1=1.474 mm2 of the RC 50-4-11 RF cable under study for the threshold numerical value Impk of the amplitude Imp of the considered aperiodic pulse of the axial current ip(t) of the time shape tJtp = 10 |is/350 ^s in the adopted cable we find that Impk ~ 12.66 kA.
3. At the end of the examples of applied calculations of threshold amplitudes Impk for the CWP, we use the standard millisecond aperiodic current pulse ip(t) of the temporal shape t/tp=7 ms/160 ms, which is now used in accordance with the requirements of the current US regulatory document SAE ARP 5412: 2013 [19] during full-scale electromagnetic tests of the main units and systems of aviation equipment for lightning resistance to direct exposure to them by the long-term component of the artificial lightning current [20]. For this current pulse ip(t)in accordance with (9) at kp1~1.078 (a1~4.75 s-1; a2~3.38-102 s-1), Ci = 1.355-108 A-s1/2/m2 and a given cross-section of a copper core SCi = SC1 = 1.474 mm2 of a radiofrequency coaxial cable of the RC 50-4-11 brand, it follows that the threshold numerical value Impk of the amplitude Imp of the specified axial current pulse ip(t) of the temporal shape t/tp=7 ms/160 ms for it will be equal to about Impk^ 1.84 kA.
One of the indicators of the reliability of the electrical engineering approach used by us and the approximate calculated relationship (9) obtained on its basis, used in the above examples of determining the threshold values Impk of the amplitudes Imp of unipolar current pulses ip(t) of nano-, micro- and millisecond duration for a radiofrequency coaxial cable brand RC 50-4-11, is that the performance according to (13) in relation to these practical cases of finding the numerical values Impk of the estimated calculation of the heating temperature 8Ci (at 80 = 20 °C and JCU = 0) of the round solid copper core of the specified cable leads to a result
equal to 8Ci ~ 119.9 °C. It can be seen that the calculated temperature level 8Ci of Joule heating of the CWP in the cases under study does not exceed the permissible short-term temperature 8CiS ~ 120 °C, which is typical for electrical cables with PET insulation.
Conclusions.
1. An engineering technique has been developed for the approximate calculation of the threshold amplitudes Impk of single pulses of axial current ip(t) of various temporal shapes for electrical wires and cables with copper (aluminum) current-carrying parts and PET, PVC and R belt insulation used in low- and high-current pulse circuits of power engineering, electrical engineering and electronics. This technique is based on the electrothermal resistance of the polymer insulation of the considered CWP, which corresponds to the permissible short-term heating temperatures of current-carrying and insulating parts of its wires and cables and does not allow the occurrence of the phenomenon of thermal destruction in the belt insulation of the CWP.
2. Calculation analytical relationships (9), (10) are obtained for finding the threshold numerical values Impk of the amplitudes Imp of the current pulses ip(t), which vary in time t according to the aperiodic dependence and according to the law of the exponentially decaying sinusoid, in the studied CWP. It is shown that at Imp>Impk, due to thermal overheating of the current-carrying parts of the wires and cables under consideration, destruction of their belt insulation will occur, leading to a decrease in the service life of the CWP.
3. Examples of practical application of the proposed engineering methodology for the calculation definition for a radiofrequency coaxial medium-sized cable of the RC 50-4-11 brand with solid PET belt insulation of threshold amplitudes Impk of standard aperiodic current pulses ip(t) from nano-, micro- and millisecond temporal ranges of the shape t/tp=5 ns/200 ns, r/rp=10 |is/350 ^s, and Tf/rp=7 ms/160 ms are presented. It was found that with a corresponding increase in the parameter Tp>>Tf for the indicated unipolar single pulses of current ip(t) flowing through a round solid copper conductor and a hollow split tinned copper braid of this cable, there is a significant decrease in their threshold amplitudes Impk (from 531.2 kA for a nanosecond current pulse of 5 ns/200 ns to 1.84 kA for millisecond current pulse of 7 ms/160 ms).
Acknowledgment. The work was supported by the Ministry of Education and Science of Ukraine (Project DB No. 0121U109546).
Conflict of interest. The authors of the paper declare no conflict of interest.
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M.I. Baranov1, Doctor of Technical Science, Professor, S.G. Buriakovskyi Doctor of Technical Science, Professor, V.V. Kniaziev1, Candidate of Technical Science, Leader Research Scientist,
1 Research and Design Institute «Molniya»
of National Technical University
«Kharkiv Polytechnic Institute»,
47, Shevchenko Str., Kharkiv, 61013, Ukraine,
e-mail: baranovmi@kpi.kharkov.ua (Corresponding Author),
sergbyr@i.ua, knyaz2@i.ua
Received 12.10.2021 Accepted 15.11.2021 Published 03.12.2021
How to cite this article:
Baranov M.I., Buriakovskyi S.G., Kniaziev V.V. Destruction of polymer insulation and threshold amplitudes of current pulses of different temporal shapes for electric wires and cables in the low- and high-current circuits of pulse power engineering, electrical engineering and electronic devices. Electrical Engineering & Electromechanics, 2021, no. 6, pp. 31-38. doi: https://doi.org/10.20998/2074-272X.202L6.05.
