# Calculating the high frequency transmission line parameters of power cables

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1 < ' Calculatng the hgh frequency transmsson lne parameters of power cables Authors: Dr. John Dcknson, Laboratory Servces Manager, N 0 RW E B Communcatons Mr. Peter J. Ncholson, Project Assgnment Manager, NORWEB ~ommuncatons ntroducton, n usng electrcty dstrbuton networks for hgh frequency communcatons t would be of sgnfcant value to be able to predct the way that the attenuaton and phase vary as the frequency of the communcaton sgnal s changed. fths were possble then reference to a company's records could quckly reveal the performance of any communcaton system that was to be nstalled. Hardware could be pre-programmed wth a sutable algorthm that would allow certan frequences to be automatcally avoded. The four transmsson lne parameters of capactance, resstance, nductance and conductance are requred n order to start the process of network modellng. n cables desgned prmarly to carry HF sgnals the transmsson lne parameters are all known and kept wthn strct tolerance levels durng manufacture, ths s not necessarly the case for power cables whose prmary functon s the dstrbuton of electrcal energy. Ths paper descrbes the steps requred n order to determne the transmsson lne parameters for three phase dstrbuton cables. UK LV Networks Practcal testng on UK, low voltage electrcty dstrbuton networks has shown that for the frequency range from to 0 MHz the attenuaton can be as lttle as 30 db or as much 90 db for a network length of 250 metres. Ths varaton n attenuaton s caused by reflectons created by mpedance msmatches at the end of each spur and from any ponts on the cable where the cable's electrcal parameters change. A typcal UK low voltage dstrbuton spur comprses a three phase dstrbutor runnng ( geographcally close to the premses to be suppled. ndvdual, sngle or three phase supples are taken from the dstrbutor to locatons where an electrcty supply s requred. n the UK there wll be between 25 and 50 supples from one dstrbutor., t ' Cable Archtecture The feeder cable wll reduce n sze as the dstance from the substaton ncreases. Typcally beng 85 mm2 on leavng the substaton and reducng to 95 mm2 after some dstance. Typcal cable constructon for two types of modem cable s shown n fgure. '!!

2 Sold Alumnum Conductors Fgure, A Sold Alumnum or Stranded Copper Conductor Fgure, B Transmsson lne parameters for the cables shown n fgure can be calculated usng the equatons gven n the followng pages. Ths analyss assumes that the propagaton s va the TEM mode,

3 - -"X\%,-m Capactance [Ref. Fgure 2! There are two values of capactance assocated wth the conductors of both types of 3 phase cables. These are shown schematcally n fgure 2. Sector shaped conductors (Fgure, A) Core to core capactance (C,) can be calculated by treatng the conductors as two parallel plates separated by twce the thckness of the cable nsulaton of length one meter. Core to sheld capactance (C,) can be calculated by takng one thrd of the capactance for two parallel concentrc cylnders of length one meter. Equatons are gven below. Cc =T &a ~ X R X E C, = b 3xln- a ' - By usng fnte element analyss for comparson wth the above equatons an error of between 3% and 5% can be expected. Crcular conductors (Fgure,B) Equatons for the core to core (C,) and screen to core (C,) capactance has been developed emprcally usng fnte element and lnear regresson technques. The followng equatons are only vald for modern 3 phase cable where the conductor centre s of the screen radus from the centre ofthe cable. The fgure of s arrved at by placng 3 maxmum szed crcles wthn a larger crcle.

4 Where r s the conductor radus and R s the screen nternal radus. Accuracy of these equatons s expected to be better than 4%. For older cables where conductor centre s not necessarly at of the screen radus the followng two equatons are needed. Where: "a" s the conductor radus. "b" s the screen nternal radus. "d" s the dstance fom screen centre to conductor centre. ' These equatons are less accurate than those gven prevously wth an expected accuracy of between 6% and 0% based on comparson wth practcal measurement fom actual cables. As these equatons are only applcable to older types of cable the comparsons are not deal because the cable used had been reclamed fom fault stes or fom stes where re-workng of!

5 Resstance [Ref. When alternatng current flows wthn a conductor the self nductance wthn the conductor causes more current to flow on the outsde of the wre than towards the centre. Ths phenomenon s termed the skn effect [Ref 2. As the frequency ncreases ths effect causes an ncrease n the resstance of the cable. At the frequences of nterest ths ncrease n resstance must be accounted for n any calculatons Though the current flow s stll dstrbuted. throughout the cable, when calculatng the resstance t s nhal to assume that All the current flows wthn the "skn depth" of the cable. The skn depth (6) s a hncton of the &equency and can be calculated usng the equaton below. Fgure 3. Slun effect n a round conductor. f the conductor s of sold composton then by usng the above method for calculatng the skn depth and knowng the radus of the conductor the effectve cross sectonal area of current flow can be determned. However, f the conductor s stranded then the area of current flow s agan reduced because of the gaps left at the crcumference of the conductor. See fgure 4. Fgure 4. Magded dagram of outer surface of stranded core showng approxmate area of current flow 3

7 ~ ~ A more accurate value for resstance can now be calculated usng the rato to mod* the cross sectonal area. Ths calculaton assumes the core dameter to be very large when compared to the core conductor dameter. The larger ths rato, the more accurate wll be the approxmaton. A smlar calculaton can be appled to a stranded sheld. On some stranded sheld cables the conductors do not touch each other. n ths case the dstance between conductors qompared to conductor radus must be measured and a decson made as to whether or not to account for a reducton n effectve area as shown. Conductance and nductance The conductance for these types of cables s very hgh and wll not normally affect the results of calculatons. Lf requred then the equatons gven here for capactance can be used to calculate conductance by replacng E wth o. The contrbuton of the nductance to the propagaton of sgnals on these cables s small. n mplementng the transmsson lne equatons the nductance was calculated usng standard equatons for concentrc cylnders and parallel conductors. The accuracy of results could be mproved by applyng the same analyss to the nductance as has been outlned here for the capactance. Concluson Tfe methods descrbed above provde a means of obtanng transmsson lne parameters for three phase electrcty dstrbuton cables. These parameters can n turn be used to determne the attenuaton and phase characterstcs of commucaton sgnals on complex tree and branch type dstrbuton networks. Transmsson lne analyss as appled to three phase underground dstrbuton networks s hghly complex [Ref 3 and not easly summarsed n a short paper however research work wthn NORWEB Communcatons has proved ts vablty. References. J. Dcknson, "Hgh frequency modellng of powerlne dstrbuton networks", Open Unversty, PhD thess, A H. Morton, "Advanced Electrcal Engneerng", Longman Scentfc and Techncal, M. Rddle, S. Ardalan, J. Sue, "Dervaton of voltage and current transfer functons for multconductor transmsson lnes", EEE, SCAS, 989, PP

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