Calculation of Cable Parameters for Different Cable Shapes

8/10/2019 Calculation of Cable Parameters for Different Cable Shapes

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Calculation of Cable Parameters for Different Cable Shapes

H. Lorenzen 1, J. Timmerberg 2, S. Mylvaganam 3 1UAS OOW, 26389 Wilhelmshaven, Germany, Department of Electrical Engineering2UAS OOW, 26389 Wilhelmshaven, Germany, Department of Engineering and Management3Telemark University College, 3901 Porsgrunn, Norway, Department Technology / Engineering

[email protected]

Abstract: Efforts involving simulation of transmission linenetworks necessitate the accurate values of the

parameters of the lines. In this paper, as an at-tempt in estimating such parameters, the parame-ters of high voltage asymmetric power line arecalculated. In the process of estimation, the three

phase equivalent circuit model is used. The resis-tance and inductance of such lines are dependenton frequency. The parameters are estimated boththeoretically and using COMSOL MultiphysicSoftware. The results are compared and dis-cussed with focus on field theoretical considera-tions.

Keywords: Partial wire technique; low voltagequad core cable; power quality calculations;equivalent pi/T-network.

1. IntroductionIncreasing demand based on trends towards

embedded power-generation forces scatteredharmonic excitation for the electrical transmis-sion system. Because of new methods for analys-ing Harmonic Resonance-Phenomenon there is ademand on detailed, all phase representation ofall relevant system components like generators,transformers and lines, i.e. generation, transmis-sion, supply and end consumption.

The manufacturers of transmission lines requirealso such detailed knowledge of cable parame-ters in conjunction with quality assurance and inmeeting the needs of the customers. In addition,comes the economic consideration with respectto maximum utilisation of the copper cross sec-tion. In addition, the increasing demand for fullcustom design of cables meeting the customerspecifications both mechanically (such as maxi-mum tensile strength , Youngs Modulus) andelectrically (such as full knowledge of the im-

pedance, i.e. resistance, inductance and capaci-tance, preferably as functions of frequency).

The emphasis on the knowledge of frequencydependency of the electrical parameters, makesthe COMSOL Multiphysic program an attractivetool for accomplishing the task of estimating

these parameters with ease. We present an appli-cation of COMSOL Multiphysic Program in theestimation of these parameters for a segmentcable.

Transmission line models are normally based onthe transmission line theory very often based onthe ladder network with discrete components as

shown in Figure 1.

Figure 1: Ladder network of a transmission line in-volving discrete components

To obtain symmetry, either lateral capacitanceand conductance or longitudinal resistance andinductance are divided into equal halves whichlead to the common equivalent - or T-networks.For both models the cable parameters R, L, C,(G) are needed.

The dashes " ' " are used to imply parameter

values per unit length, for example ' / R R l= .For coaxial cables used in communication tech-nology or screened cables for medium voltage

Figure 2: Low voltage segment cable

Excerpt from the Proceedings of the COMSOL Conference 2008 Hannover


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power distribution, these parameters can easily be determined, if the inductive coupling is negli-gible. But many cables have a design for whichthe analytical solution cannot be applied. Toshow the complexity involved, the structure of acommonly used low-voltage power distributioncable is shown in Figure 2.

The manufacturers of such cables usually givethe nominal inductance B L and load current L I .However, the need for detailed simulation de-mands the knowledeg of all the parameters givenin Figure 1. The usage of Comsol MultiphysicProgram in the estimation of these parameters isdiscussed in the following sections.

The calculation of the cable parameters are car-ried out in the Multiphysic Program modulesElectrostatic or AC-Electromagnetic de-

pending on the type of parameters needed. To begin with, the cable geometry for the cableshown in Figure 1, is designed with the comfort-able internal GUI. This geometry is then used inthe above mentioned modules to calculate R', C'and L'.

2. Cable capacitanceThe discrete capacitance values can be calcu-

lated independent of frequency. To calculate thesix partial capacities C of the segment cableshown in Figure 1, the voltage is applied to twoconsecutive conductors, repeating these connec-tions until we have exhausted the whole cablecluster. The electric field in the isolator is calcu-lated first. Then the energy of the electric field isdetermined performing the needed integration inthe postprocessing of data in Multiphysic Pro-gram. By means of a linear equation systemevolving out of the system shown in Figure 1,the desired capacitance values can be deter-mined.

The cross-section of the cable shown in Figure 3,having the segments numbered from 0 to 3 isused in the simulations. As mentioned before,the consecutive segments are coupled to a volt-age source of 1 kV. The conductor with +attached to the segment number is connected tothe positive terminal of the 1kV-source. The fol-lowing independent cases are studied: (1+),(2+), (1+2+), (1+2+3+), (2+3+), (1+2+3+). Theunnamed cable segment in each case, is obvi-

ously earthed for the case study involving themodule Electrostatic .

Figure 3: Cross-section of the cablewith the numbered cable sections. Selec-tion of cable segments for applying volt-

ages discussed in text.

For all the cases mentioned above, the energyeW of the electrostatic field is calculated. Then

the well known equation (1) between electro-static energy eW , voltage U and capacitancevalue C is used to estimate the cable capacitance.

212e

W CU = (1)

The conducting earth is assumed to be at23.5ar r = .

2.1 Energy of the field with applied voltage onsegment 1In the case of segment 1 being supplied with1kV, i.e. (1+), all the other conductors and thescreening conductor are earthed. For the electro-static case, the simplified equivalent circuit forthe case of segment 1 connected to 1 kV, is givenin Figure 4.

Figure 4: Voltage on conductor segment 1 with allthe other conductors and screen earthed


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In the case of (1+), the capacitances' ' '

10 31 12, ,C C C

are in parallel connection, givingthe following equation (2) for the energy

( )' ' ' 2(1 ) 10 31 1212eW C C C U + = + + (2)

Figure 5: Voltage distribution for the case with 1kVon conductor segment 1, and all the other parts

earthed. Red implying high end and blue the low endof the voltage scale.

It is of interest to observe the voltage distributionof the case under study, as shown in Figure 5. Inthe Postprocessing stage of the Comsol Mul-tiphysic, the energy intensity is evaluated to be

-4(1 ) 1.558726 10 Ws/meW + = .

2.2 Electrostatic energy in the case of voltageapplied to conductor segments 1 and 2

In the next simulation, the case of (1+2+), i.e. theapplication of 1kV on conductor segments, isstudied. In the equivalent circuit, the capacitan-ces ' ' ' '10 20 31 23, , ,C C C C are now coupled asknown in Figure 6.

2

0

3

1

C 23

C 12C 31

C 30C 10 C 200V

1000V

Figure 6: Voltage on conductor segments 1 and 2,with screen and conductor 3 earthed

As discussed before, the electrostatic energyintensity in the case of conductor segments 1 and2 being energized is given by

( )' ' ' ' 2(1 2 ) 10 20 31 23-4

12

=1.589495 10 Ws/m

eW C C C C U + + = + + + (3)

and the corresponding voltage distribution isshown in Figure 7.

Figure 7: Voltage distribution with 1kV on conductor

segments 1 and 2 with the other parts earthed. . Redimplying high end and blue the low end of the voltagescale

2.3 Complete set of capacitancesAfter performing the simulations involving thesix different cases, the various capacitance val-ues are connected to each other and the suppliedvoltage thorough a linear system of equations, asgiven in equation (4) .

'(1 )10

'(1 2 )20

'30

2'12'23'31

1 0 0 1 0 1

1 1 0 0 1 1

: : : : : : :1: : : : : : :

: : : : : : :

: : : : : : :

e

e

W C

W C

C

U C

C

C

+

+ +

=

(4)

For the cable geometry shown in Figure 1, thecapacitance values are hence given by,


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'10'20'30'

12'23'31

146

17.4146

147

147

7.24

C

C C nF

kmC

C

C

=

These values of the cable capacitance values andtheir validity is discussed in section 4.

3. Equivalent resistance and inductanceThe inductance L' and the resistance R' in

high voltage cables can be calculated using themodule AC-electromagnetic, which helps toaccount for skin-effect, as the cable consists oflarge diameter conductors and carries currentswith high harmonics. The current density ishence not homogeneous. The cable parameters,depending on the frequency, are graphically rep-resented usually. The case of thin screen sur-rounding the cable segments is not addressed inthe present simulations.

The equivalent circuit is now reduced to a sys-tem containing capacitances and inductances, asshown in Figure 8.

Figure 8: Equivalent circuit showing discrete valuesof llongitudinal resistances and inductances

The equivalent circuit shown in Figure 1 is de-scribed by the system equations, given by (5):

3

0

( )k k k k kl lll k

U R j L I M I =

= + + (5)

with 0 ... 3k = , k U the applied voltage on con-ductor k , resistance and inductance of conductork being Rk and Lk , and the mutual inductances

between conductor k and l being kl lk M M = ,with k,l = 0,1,..3.

Then the sequential supply of currents to thedifferent conductor segments is carried out toestimate the voltages thus generated in the circuitgiven in Figure 8. The values of current is givenusing the menu Options/Constants as _ I soll .With the additional equation, available under themenu Physics/GlobalEquations , the appliedvoltages _ 2U err and _ 3U err gives rise to thecurrents:

_ 2 _ 3ltr A

I ist S dA I ist = = (6)

Figure 9: Global Equations as used in the simulationof the circuit shown in Figure 8 for the cable configu-

ration shown in Figure 5.

3.1 Currents in 2 und 3The cable segments 2 and 3 are supplied with

currents, so that 2 _ I I soll=

. The equivalentcircuit is shown in Figure 10.

Figure 10: Equivalent circuit for the case involvingcurrent supply to conductors 1 and 2.

The current density field distribution for the caseof segments 2 and 3 being supplied with currentsof frequency 200 kHz with the cable geometryshown in Figure 3 is now shown in Figure 11.The current density over the cross section in thedirection of the x-axis (red line) of the cable sec-tion 2 is also shown in the plot given Figure 11.As expected, the skin-effect is clearly seen asone move towards the edges of the conductorsegment.


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Figure 11: Magnetic field distribution and currentdistribution for the case of conductor 2 and 3 fed with

currents

The results are obtained using the menu Phys-ics/SubomainSettings and the menu Postproc-essing/GlobalVariables. The voltages _ 2U err und _ 3U err are hence obtained. The imped-ance of the conductor segment 2 is then calcu-lated using

( )2 2*2 2* 2 2 232

_ 2 _ 2

U U err Z R j L M

I I ist

= = + =

(7)

3.2 All inductances

For the system under study, we have 3 unknownresistances

( )1 2 3T R R R R=

And 10 unknown inductances,

( )1 2 3 12 13 13 30... ...T L L L L M M M M =

Using 10 different current loadings, the imped-ances can be coupled to a linear system of equa-tions for a given frequency.

4. Verification using analytical field theo-

retical methodsWhen the focus is on the verification of the

results, independent theoretical estimates areuseful. Simulation tools such as COMSOL Mul-tiphysics are easy to handle and can help us toestimate and visualize various physical charac-teristics/phenomena, and deliver the results veryfast, even for very for complex geometries.However results produced using such versatiletools should be verified using independent meth-ods. The results of simulation using COMSOL

Multiphysics are analytically verified with dif-ferent field theoretical considerations. The fre-quency dependent capacitance values are esti-mated using coupled capacitance models. Thefrequency dependent inductance values involv-ing skin effect of the cable is estimated using

partial wire method.

4.1 The capacitance ValuesIn an idealised version of the geometry, the cablesegments could be represented by parallel platecapacitors, as the parallel surfaces of the seg-ments facilitate such a simplification of the ge-ometry. With the original dimensions of Figure3, the twin capacitor system shown in Figure 12evolves.

The distance between the plates is then5.7d mm= with an area of 21.2 1 A mm x m= .

The capacitance values for Comsol Multiphysics

can be estimated using 212

W CU = . Fringe

fields and fields outside the plates in the simpli-fied model, lead to a difference of 26% com-

pared to capacitance values obtained with simu-lations. To compensate these effects, the distance

between cable segments 2, 3, 4 is reduced to0.2Kontr d mm= . The estimate is then made using

the parallel plate capacitor model.

The energy intensity for the actual modelWFemLab and parallel capacitor model WKontare found to be in the ratio

Figure 12: Idealised geometry with distance betweensegments 0.2 mm


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-3

-3

2.150 10 Ws/m

2.205 10 Ws/m

Kontr

FemLab

W

W

= ,

implying a good set of tallying values.

4.2 The Inductance ValuesThe verification of the frequency dependent in-ductance values follows using partial wire me-thod [3], [4]. The bulk conductor is considered asa cluster of closely lying conductors as shown inFigure 13. Each conductor in such a set of clus-tered conductors has its own resistance and in-ductance and in addition mutual inductances toeach and every other conductor in the cluster.The verification is now limited to conductor seg-

ments 2 and 3 only.

Figure 13: Bulk of conductor viewed as a cluster ofsnugly lying set of single conductors in line with par-

tial wire techniquesThe equivalent electrical circuit for the systemshown in Figure 13 is described by the systemequations below, with the standard notationsadopted before, but modified to the partial wiresystem under discussion now..

1

( )m

k k k k kl lll k

U R j L I M I =

= + + (8)with 1 ...k m= and kl lk M M = , written in ma-trix form as

U Z I = bzw. 1 I Z U = (9)

The impedance matrix is then given by

11 11 12 13

21 22 22 23

31 32 33 33

:

:

:

: : : :

R j L M M

M R j L M Z

M M R j L

+ + = +

(10)

The inductance and mutual inductance for paral-lel thin conductors can now be estimated usingwell known formulae, e.g. [1], [2]. The necessary

calculations involving the matrices definedabove can be done using MATLAB. The partialwire system for 2 93 x conductors is given in Fig-ure 14.

Figure 14: Distribution of the cluater of condutorsfor segments 2 and 3 of cable shown in Figure 3.

The current distribution at 50 f Hz= is given inFigure 15. For the sake of comparison, the con-ductivity of segment 0 and 1 is set to zero.

Figure 15: Current distribution when only conductorsegments 2 and 3 are fed

Figure 16 shows the resistance and inductancevalues of the segments based on partial wire wiremethod and the values obtained with ComsolMultiphysics for the actual cable geometry(shown by the blue circles).

The difference in the resistance values can beattributed to non-conducting areas (filling factorof the circles with respect to the bulk conductor).This can be improved by improving the model.


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Figure 16: Inductance and resistance values estimated(continuous lines) using partial wire method compared

to estimates using Comsol Multiphysics (circles)

5. ConclusionsUsing Comsol Multiphysics program and the

modules Electrostatic and AC Electromag-netic the impedance values of a three segment

power cable are estimated for different frequen-cies. The estimated values are then comparedwith values obtained using partial wire methodsfor inductances and simplified ideal models forcapacitances. The results obtained for the se-

lected cable show that these values tally well.The results show that the need for electrical char-acterisation of the cables produced by cablemanufacturers can be addressed to some extentsatisfactorily using the techniques described inthe paper, although some modifications and ad-ditional work are required to improve perform-ance.

6. References[1] John David Jackson; Klassische Elektro-dynamik, de Gruyter, 2002

[2] Karl Kpfmller; Theoretische Elektro-technik : Eine Einfhrung, 17. Auflage, 2006

[3] Jung-Hsiang Wang; Modeling of Frequency-Dependent Impedance of the Third Rail Used inTraction Power Systems, IEEE Trans. on PowerDelivery, Vol. 15, No. 2, April 2000

[4] Alexander W. Barr; Calculation of Frequen-cy-Dependent Impedance for Conductors of Rec-tangular Cross Section, AMP Journal of Techno-logy Vol. 1 November, 1991

f in Hz

0/ R R

0/ L L

Calculation of Cable Parameters for Different Cable Shapes

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