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1、Section 1EMC OF ELECTRICAL POWER EQUIPMENTMUTUAL INDUCTANCE BETWEEN WIRES IN HELICALLYTWISTED POWER CABLESBernd W. Jaekel, GermanySiemens AG, Automation and Drives, Germany, e-mail: bernd.jaekelAbstract. The arrangement of the conductors in a multi-core power cable leads to a situation where various
2、 conductor loops are built up. One or several loops are formed by the phase and neutral conductors with the operational current flowing in these conductors. A further loop is built up by the protective earth conductor which is connected to the equipotential bonding system at several locations. The a
3、rea of this loop is essentially arranged outside of the power cable. The inductive coupling from the phase conductor loops into that loop causes common mode voltages in the protective earth system with consequent common mode currents. It can be demonstrated that this effect even takes place in the c
4、ase of balanced phase currents in the cable. Numerical simulations and parameter studies were carried out in order to describe this effect quantitatively and to investigate the influence of different cable parameters onto the resulting common mode voltages. 53IntroductionPower cables represent compo
5、nents of an entire power supply network which can be carried out in different types. If an earthed system is required, i.e. a system which is connected to the local reference earth, mainly two types of supply networks can be distinguished: TN-C and TN-S. From an EMC point of view a TN-S power networ
6、k should definitely be preferred 1. In this type of network the neutral and protective earth (PE) conductors are strictly separated except at one net point where both conductors are connected, normally at the transformer or the switchgear. This type of installation prevents that any operational curr
7、ents flow outside of the phase and neutral conductors. No cable net currents should exist and therefore the equipotential bonding system is generally assumed to be free of any operational currents. But when looking in more detail at this type of network and at the physical structure of power cables
8、some physical mechanisms can be identified which nevertheless lead to the generation of common mode voltages and common mode currents even in the case of balanced loaded TN-S power net systems.Low Voltage Power CablesMulti-core low voltage power cables consist of the phase conductors and depending o
9、n the grounding arrangement of the power supply network of a neutral conductor and/or a PE conductor. An example for the structure of a power cable is shown in Fig. 1 for a cable of type NYY. Each of the conductors as well as the entire conductor arrangement are covered by an insulation for which a
10、material is chosen depending on the specific requirements and fields of applications 2. PE-conductorphase-conductorsThe n individual insulated conductors are twisted together and each conductor can be represented by a helical line. An appropriate cylindrical coordinate system for describing the spat
11、ial arrangement of a conductor is shown in Fig. 2 together with the relevant parameters such as a as the radius of the helical line with respect to the centre line of the cable and the pitch distance p as the twist length of the cable, i.e. the length of the cable per rotation of the conductors. For
12、 simplicity reasons only one conductor is shown. The further n-1 conductors can be represented as similar lines and they are rotated by an angle F = 3600/n with respect to that one shown in Fig. 2. Fig. 1: Multi-core power cable of type NYYCommon Mode Voltages in Power CablesThe magnetic flux densit
13、y B caused by the currents in the individual conductors can be calculated by means of the Biot-Savart law, as long as the situation at the power frequency range is considered:(1)I represents the phasor of the exciting alternating current, r with its cylindrical coordinates r, F, z denotes the observ
14、ation point and r with its cylindrical coordinates r, F, z means a variable point on the line current. Though this expression can be easily solved in the case of straight wires, the situation is relatively complex in the case of power cables where the various conductors are twisted and each conducto
15、r can be represented by a helical solenoid (see Fig. 2). In power cables where a protective earth conductor (PE-conductor) is twisted along with the phase conductors an inductive coupling exists between the phase conductor loops and a loop built up by the PE-conductor and its connecting structures t
16、o the equipotential bonding system. This effect can be explained by means of a schematic sketch of a 4-conductor cable as shown in Fig. 3. For reasons of clarity the twisting of the conductors is not shown in the figure.Fig. 2: A twisted conductor (helical line) in a cylindrical coordinate system 3T
17、he conductors L1, L2 for example form a spatial loop in which the phase current IL1-L2 flows. Corresponding loops are built up by the arrangement of conductors L1-L3 and L2-L3 with the loop currents IL1-L3 and IL2-L3, respectively. A further loop results from the PE conductor which is connect to the
18、 equipotential bonding system by conductive structures. This loop is shown as PE-Loop in Fig. 3.Fig. 3: Mechanism for generation of common mode voltagesThe induced voltage UPE in the PE-Loop can be derived by means of the mutual inductances between the various phase conductor loops and the PE-Loop o
19、r accordingly by the mutual inductance between the current carrying phase conductors and the PE-Loop:(2)with the mutual inductances MLi-PE (i = 1,2,3) to be derived by(3)BLi represents the magnetic flux density caused by the current Ii in conductor Li (i = 1, 2, 3) and S the area of the PE-Loop 4.Fi
20、g. 3 shows the situation for a four-conductor cable. From the cross-section of the entire cable configuration it can be seen that there is no total symmetry when looking at the three mutual inductances between the phase conductors and the protective earth loop. Hence a net mutual inductance results
21、leading to a net induced common mode voltage and a common mode current in the case of a closed loop, respectively.The Biot-Savart integral (1) needed in order to determine BLi, however, cannot be calculated analytically for a current in a helical conductor arrangement. The magnetic vector potential
22、has to be used and a series expansion of the reciprocal distance between the observation point and a variable point on the conductor has to be introduced. Using some well-known trigonometric theorems together with Bessel functions the following equations for the different components of the magnetic
23、flux density vector can be derived for observation points outside (r a) the helical arrangement 5:(4)(5)(6)with is thecoordinate of the point where the helix intersects the plane : modified Bessel functions of first and second kind of order n (): their derivatives)In 5 the corresponding investigatio
24、ns are expanded to the situation of a twisted three-phase arrangement. Furthermore an approximation is given there to estimate the field strength versus distance to the twisted phase conductor arrangement.There are different possibilities to determine the amplitude of the induced voltage UPE: by ana
25、lytically performed integration techniques by numerical simulations or by measurements.The integration of the magnetic flux density across the area of the PE loop results in the total magnetic flux and subsequently in the induced voltage. This approach, however, represents a very complex task, becau
26、se on the one hand the expressions (4), (5) and (6) with the Bessel functions have to be integrated and on the other side these equations which are valid for observation points outside the cable have to be considered as well as corresponding equations which describe the situation inside the power ca
27、ble 5. No analytical results or approximate procedures were found in the technical literature which offer solutions.Numerical SimulationsWithin the frame of numerical simulations the twisted conductors as well as the generated PE-loop have to be modelled spatially. A schematic representation for the
28、 physical model of a twisted power cable is given in Fig. 4. For the simulations the geometrical data of a cable of type 4x25 mm2 are used where a twist length (pitch) of 0.4 m was considered. The PE-conductor of the cable is connected to conductive structures (of equipotential bonding system) and a
29、 closed loop results. The dimensions of the loop are length L and width W.LW(a) twisted cable with PE-loop(b) detail of (a) showing individual twisted conductors and connection of PE-loop conductors outside the cableFig. 4: Spatial model of a power cableThe simulations were performed by means of the
30、 computer program CONCEPT which bases on the Method of Moments 6. This method and program allows modelling of all the conductive structures of the arrangement under consideration. The phase conductors are excited by means of a three phase voltage source and all the voltages and currents can be calcu
31、lated which are induced in any conductive element of the model.The amplitude of the induced common mode voltage depends on the actual cable parameters and the cable installation condition. In a first step the dependency on the phase current amplitudes and the phase current frequencies in the range f
32、rom several Hz to several hundreds of Hz, i.e. in the electrically low frequency range were investigated. In both cases a linear correlation between induced voltage and current and frequency, respectively, could be found. This clearly indicates that the physical mechanism which causes the induced vo
33、ltage is an inductive coupling between the current carrying phase conductors and the PE-loop. Furthermore the voltage induced into the PE-loop was calculated for varying lengths of the loops, i.e. for varying longitudinal dimensions of the resulting loop built up by the PE-conductor of the power cab
34、le and the equipotential bonding system. The results are given in Fig. 5.Fig. 5: Induced voltage versus length L of the PE-loopThe induced voltage linearly increases with increasing longitudinal length L of the loop. This behaviour can be expected due to the resulting conductor configuration where t
35、he arrangement of the PE conductor is constant with respect to the phase conductors. Hence the PE-loop is exposed to constant magnetic fields along the cable length resulting in a constant induced voltage per length unit. This behaviour was found also by means of experimental investigations 7. From
36、the slope of the line a coefficient can be derived which expresses the induced voltage per cable/loop length and which is about 0.44 V/A/m/Hz.In the case of a balanced three phase system with phase current I and taking into account the phase relationship, equation (2) can be simplified in order to d
37、escribe the relation between the induced voltage UPE, the frequency f of the currents and their amplitudes:(7)with MNET as the net mutual inductance derived from the superposition of the individual mutual inductances. According to this relation a mutual inductance MNET = 70 nH per meter length resul
38、ts when the induced voltage UPE is considered as derived above.The mutual inductance is expected to depend on several cable parameters. Corresponding investigations concerning the amplitudes of the induced voltage UPE were performed for various cable parameters and cable configurations 8. From those
39、 results can be derived that there is nearly no dependency on the twist length (pitch) of the power cable conductors at least for practical twist lengths of more than about 40 cm. Furthermore there is only a small impact of the helical radius of the power cable conductors. This fact was found also b
40、y means of measurements where the results of a 4x95mm2 power cable are nearly the same as in the present case of a 4x25 mm2 power cable 7.Since the inductive coupling has to consider the PE-loop dimensions a significant impact of the loop width W might be expected. For the same power cable as descri
41、bed above (4x25mm2, 40 cm pitch length) the width of the PE-loop was varied in the range between 5 cm and 100 cm. This variation reflects potential influences due to variation of the cable laying above ground or above equivalent equipotential bonding structures. The corresponding results concerning
42、the voltage induced in the PE-loop are shown in Fig. 6. The induced voltage and hence also the mutual inductance MNET does not show any significant change with varying PE loop width.Fig. 6: Induced voltage versus width W of the loop for two exemplary loop lengths of 1.2m and 2mThis result can be exp
43、lained by the fact that the most dominant part of the voltage is induced by those magnetic fields in the very close vicinity of the phase conductors. The magnetic flux density which results from all the phase conductors decreases rapidly and no relevant contributions exist for distances of more than
44、 some helical conductor radii because the resultant magnetic flux density decreases nearly exponentially with increasing distance from the cable 9. Hence for practical estimations the influence of the loop width W can be neglected.Furthermore the situation was investigated when five conductors (5x25
45、mm2) are used instead of four. This variation reflects the usage of five-conductor cables used in TN-S networks where the N and PE conductor separation is established. Simulation results are described in 8 and a mutual inductance MNET = 95 nH per meter length can be derived from those. This higher v
46、alue and hence higher values for induced voltages UPE can be explained qualitatively by the higher asymmetry of the PE conductor (see Fig. 7) in a five conductor arrangement where only three phase conductors carry a current.ConclusionIn power cables with twisted PE conductors a common mode voltage i
47、s induced by the currents in the phase conductors. This is valid even in the case of balanced currents and is due to the fact that the PE conductor has a certain asymmetry with respect to the phase conductors resulting in a net magnetic flux through the loop built up by the PE conductor and structur
48、es of the equipotential bonding system.Fig. 7: Arrangement of conductors in a five-conductor power cableThe amplitude of the induced voltage depends strongly on the loop length but only slightly on the loop width and on cable parameters such as twist length or conductor cross-section. Hence a mutual
49、 inductance per unit length can be derived to express the induced voltage. It is in the range of about 70 100 nH/m and can be used to estimate induced common mode currents.It shall be mentioned that this phenomenon takes place for cables with twisted PE conductors only. It does not exist in the case of cables with concentric PE conductors w
