NGEVTQ
CIPGVKE
TCPUKGPVU
TQITCO
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6JGQT[$QQM
CONTENTS
(Click topic to jump to its location)
CHAPTER
PAGE
1. INTRODUCTION TO THE SOLUTION METHOD USED IN THE EMTP
6
2. LINEAR, UNCOUPLED LUMPED ELEMENTS
14
3. LINEAR, COUPLED LUMPED ELEMENTS
45
4. OVERHEAD TRANSMISSION LINES
64
5. UNDERGROUND CABLES
150
6. TRANSFORMERS
192
7. SIMPLE VOLTAGE AND CURRENT SOURCES
238
8. THREE-PHASE SYNCHRONOUS MACHINE
251
9. UNIVERSAL MACHINE
311
10. SWITCHES
339
11. SURGE ARRESTERS AND PROTECTIVE GAPS
354
12. SOLUTION METHODS IN THE EMTP
360
13. TRANSIENT ANALYSIS OF CONTROL SYSTEMS (TACS)
395
APPENDIX
I – NUMERICAL SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS 413
II – RE-INITIALIZATION AT INSTANTS OF DISCONTINUITIES
431
III – SOLUTION OF LINEAR EQUATIONS, MATRIX REDUCTION AND
INVERSION, SPARCITY
IV – ACTUAL VALUES VERSUS PERUNIT QUANTITIES
V – RECURSIVE CONVOLUTION
434
452
459
VI – TRANSIENT AND SUBTRANSIENT PARAMETERS OF SYNCHRONOUS
MACHINES
460
VII – INTERNAL IMPEDANCE OF STRANDED CONDUCTORS
470
REFERENCES
472
$TCPEJQH5[UVGO'PIKPGGTKPI
$QPPGXKNNG2QYGT#FOKPKUVTCVKQP
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6JGCEVWCNEQORWVCVKQPKPVJG'/62RTQEGGFUCUHQNNQYU/CVTKEGU=)##?CPF=)#$?CTGDWKNVCPF=)##?
KUVTKCPIWNCTK\GFYKVJQTFGTGFGNKOKPCVKQPCPFGZRNQKVCVKQPQHURCTUKV[+PGCEJVKOGUVGRVJGXGEVQTQPVJGTKIJVJCPF
UKFGQH'S D KUCUUGODNGFHTQOMPQYPJKUVQT[VGTOUCPFMPQYPEWTTGPVCPFXQNVCIGUQWTEGU6JGPVJG
U[UVGOQHNKPGCTGSWCVKQPUKUUQNXGFHQT=X# V ?WUKPIVJGKPHQTOCVKQPEQPVCKPGFKPVJGVTKCPIWNCTK\GFEQPFWEVCPEG
OCVTKZ +P VJKU TGRGCV UQNWVKQP RTQEGUU VJG U[OOGVT[ QH VJG OCVTKZ KU GZRNQKVGF KP VJG UGPUG VJCV VJG UCOG
VTKCPIWNCTK\GFOCVTKZWUGFHQTFQYPYCTFQRGTCVKQPUKUCNUQWUGFKPVJGDCEMUWDUVKVWVKQP$GHQTGRTQEGGFKPIVQVJG
PGZVVKOGUVGRVJGJKUVQT[VGTOUJKUVQH'S D D CPF D CTGVJGPWRFCVGFHQTWUGKPHWVWTGVKOGUVGRU
1TKIKPCNN[VJG'/62YCUYTKVVGPHQTECUGUUVCTVKPIHTQO\GTQKPKVKCNEQPFKVKQPU+PUWEJECUGUVJGJKUVQT[
VGTOU JKUV JKUV CPF JKUV KP 'S CTG UKORN[ RTGUGV VQ \GTQ $WV UQQP ECUGU CTQUG YJGTG VJG VTCPUKGPV
UKOWNCVKQPJCFVQDGUVCTVGFHTQORQYGTHTGSWGPE[ QT*\ CEUVGCF[UVCVGKPKVKCNEQPFKVKQPU1TKIKPCNN[UWEJ
CEUVGCF[UVCVGKPKVKCNEQPFKVKQPUYGTGTGCFKPDWVVJKURWVCJGCX[DWTFGPQPVJGRTQITCOWUGTYJQJCFVQWUG
CPQVJGTUVGCF[UVCVGUQNWVKQPRTQITCOVQQDVCKPVJGO0QVQPN[YCUVJGFCVCVTCPUHGTDQVJGTUQOGDWVVJGUGRCTCVG
UVGCF[UVCVGUQNWVKQPRTQITCOOKIJVCNUQEQPVCKPPGVYQTMOQFGNUYJKEJEQWNFFKHHGTOQTGQTNGUUHTQOVJQUGWUGF
KPVJG'/62+VYCUVJGTGHQTGFGEKFGFVQKPEQTRQTCVGCPCEUVGCF[UVCVGUQNWVKQPTQWVKPGFKTGEVN[KPVQVJG'/62
YJKEJYCUYTKVVGPD[,99CNMGT
6JGCEUVGCF[UVCVGUQNWVKQPUJCNNCICKPDGGZRNCKPGFHQTVJGECUGQH(KI7UKPIPQFGGSWCVKQPUCICKP
'S PQYDGEQOGU
+ % + % + % + ' +
YJGTGVJGEWTTGPVU+CTGEQORNGZRJCUQTSWCPVKVKGU*+* GL"PQY(QTVJGNWORGFGNGOGPVUVJGDTCPEJGSWCVKQPUCTG
QDXKQWU(QTVJGTGUKUVCPEG
+ '
8 & 8
4 6JKUQRVKQPKUUVKNNCXCKNCDNGKPVJG'/62DWVKVJCUDGEQOGUQOGYJCVQHCJKUVQTKETGNKECPFJCUUGNFQO
DGGPWUGFCHVGTVJGCFFKVKQPQHCUVGCF[UVCVGUQNWVKQPTQWVKPG(QTUQOGV[RGUQHDTCPEJGUKVOC[PQVGXGPYQTM
=?RE HQTVJGKPFWEVCPEG
8 & 8
LT. + ' LT% 8 & 8
+ '
CPFHQTVJGECRCEKVCPEG
(QTCNKPGYKVJFKUVTKDWVGFRCTCOGVGTU4 . ) % VJGGZCEVUVGCF[UVCVGUQNWVKQPKU
+
+
;UGTKGU %
'
;
UJWPV
&;UGTKGU
&;UGTKGU
8
8
;
;UGTKGU %
UJWPV
KHVJGGSWKXCNGPVBEKTEWKVTGRTGUGPVCVKQPQH(KIKUWUGFYKVJ
(KI'SWKXCNGPVBEKTEWKVHQTCEUVGCF[UVCVG
UQNWVKQPQHVTCPUOKUUKQPNKPG
;UGTKGU '
<UGTKGU
YKVJ <UGTKGU ' ý 4 ) % LT. )
ý )
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) % LT% )
UJWPV
(ý
(ý
UKPJ (ý
(ý
VCPJ
CPFUQOGVKOGUGSWCNN[WUGHWN
;UGTKGU %
;
' EQUJ (ý @ ;UGTKGU
UJWPV
YJGTG(KUVJGRTQRCICVKQPEQPUVCPV
( '
4 ) % LT. ) ) ) % LT% )
(QTVJGNQUUNGUUECUGYKVJ4 CPF) 'S UKORNKHKGUVQ
<UGTKGU ' ý @ LT. ) @
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;
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VCP
;UGTKGU %
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Tý . )% )
Tý ) )
.%
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.%
;
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UJWPV
+HVJGXCNWGQHTýKUUOCNNV[RKECNN[ý#MOCV*\HQTQXGTJGCFNKPGUVJGPVJGTCVKQUUKPJ Z ZCPF
VCPJ Z ZKP'S CUYGNNCUUKP Z ZCPFVCP Z ZKP'S CNNDGEQOG6JKUUKORNKHKGF
BEKTEWKVKUWUWCNN[ECNNGFVJGPQOKPCNBEKTEWKV
<UGTKGU ' ý @ 4 ) % LT. )
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;
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UJWPV
KH Tý KU UOCNN
9KVJVJGGSWKXCNGPVBEKTEWKVQH(KIVJGDTCPEJGSWCVKQPHQTVJGNQUUNGUUNKPGHKPCNN[DGEQOGU
+ ' ;UGTKGU % ;UJWPV 8 & ;UGTKGU 8
0QYYGECPCICKPYTKVGVJGPQFGGSWCVKQPHQTPQFGD[KPUGTVKPI'S CPF KPVQ'S %
% LT% % ;UGTKGU % ;UJWPV 8 & 8 &
8 & LT% 8 & ;UGTKGU 8 ' +
4 LT.
4
LT. (QTCP[V[RGQHPGVYQTMYKVJPPQFGUCU[UVGOQHPUWEJGSWCVKQPUECPDGHQTOGF
=;? =8? ' =+?
YKVJ
=;?U[OOGVTKEPQFCNCFOKVVCPEGOCVTKZYKVJEQORNGZGNGOGPVU
=8?XGEVQTQHPPQFGXQNVCIGU EQORNGZRJCUQTXCNWGU =+?XGEVQTQHPEWTTGPVUQWTEGU EQORNGZRJCUQTXCNWGU #ICKP'S KURCTVKVKQPGFKPVQCUGV#QHPQFGUYKVJWPMPQYPXQNVCIGUCPFCUGV$QHPQFGUYKVJMPQYP
XQNVCIGU6JGWPMPQYPXQNVCIGUCTGVJGPHQWPFD[UQNXKPIVJGU[UVGOQHNKPGCTCNIGDTCKEGSWCVKQPU
=;##? =8#? ' =+#? & =;#$? =8$?
$TKPIKPI VJG VGTO =;#$?=8$? HTQO VJG NGHVJCPF UKFG KP 'S VQ VJG TKIJVJCPF UKFG KP 'S KU VJG
IGPGTCNK\CVKQPQHEQPXGTVKPI6JGXGPKPGSWKXCNGPVEKTEWKVU XQNVCIGXGEVQT=8$?DGJKPFCFOKVVCPEGOCVTKZ=;#$? KPVQ
0QTVQPGSWKXCNGPVEKTEWKVU EWTTGPVXGEVQT=;#$?=8$?KPRCTCNNGNYKVJCFOKVVCPEGOCVTKZ=;#$? 0QTVQPGSWKXCNGPVEKTEWKVU EWTTGPVXGEVQT=;#$?=8$?KPRCTCNNGNYKVJCFOKVVCPEGOCVTKZ=;#$? .+0'#470%172.'&.7/2'&'.'/'065
.KPGCTWPEQWRNGFNWORGFGNGOGPVUCTGTGUKUVCPEGU4UGNHKPFWEVCPEGU.CPFECRCEKVCPEGU%6JG[WUWCNN[
CRRGCTCURCTVUQHGSWKXCNGPVEKTEWKVUYJKEJOC[TGRTGUGPVIGPGTCVQTUVTCPUHQTOGTUUJQTVUGEVKQPUQHVTCPUOKUUKQP
NKPGUQTQVJGTEQORQPGPVUQHCPGNGEVTKERQYGTU[UVGOQTVJG[OC[TGRTGUGPVCEQORQPGPVD[KVUGNH
4GUKUVCPEG4
4GUKUVCPEGGNGOGPVUCTGWUGFVQTGRTGUGPVCOQPIQVJGTVJKPIU
C
ENQUKPICPFQRGPKPITGUKUVQTUKPEKTEWKVDTGCMGTU
D
VQYGTHQQVKPITGUKUVCPEG CUCETWFGCRRTQZKOCVKQP=?QHCEQORNKECVGFHTGSWGPE[FGRGPFGPV
ITQWPFKPIKORGFCPEG E
TGUKUVCPEGITQWPFKPIQHVTCPUHQTOGTCPFIGPGTCVQTPGWVTCNU
F
OGVGTKPITGUKUVCPEGKPRNCEGUYJGTGEWTTGPVUQHDTCPEJXQNVCIGUECPPQVDGQDVCKPGFKPQVJGT
YC[UD[VJG'/62
G
CURCTVUQHGSWKXCNGPVPGVYQTMUGIKPRCTCNNGNYKVJKPFWEVCPEGUVQRTQFWEGRTQRGTHTGSWGPE[
FGRGPFGPVFCORKPI UGG5GEVKQP H
HQTVJGTGRTGUGPVCVKQPQHNQPINKPGUKPNKIJVPKPIUWTIGUVWFKGUKHPQTGHNGEVKQPEQOGUDCEMHTQOVJG
TGOQVGGPFFWTKPIVJGFWTCVKQPVOCZQHVJGUVWF[
'ZCORNG H KUGCUKN[FGTKXGFHTQO'S D KHKVKUCUUWOGFVJCVVJGKPKVKCNEQPFKVKQPUQPVJGNKPGCTG\GTQ+PVJCV
ECUGJKUV VJ HQTVJUKPEGKVVCMGUVKOGJHQTCP[PQP\GTQEQPFKVKQPQEEWTTKPIKPPQFGCHVGTV$VQ
UJQYWRKPPQFG+HPQVJKPIKUEQPPGEVGFVQPQFG QRGPGPFGFNKPG VJGP+YQWNFTGOCKP\GTQHQTVJ
6JG'/62TGEQIPK\GUVJKUUKORNKHKECVKQPKH
J
\GTQKPKVKCNEQPFKVKQPU
VOCZCPF
+HDQVJEQPFKVKQPU CPF CTGOGVVJGPVJG'/62TGRTGUGPVUVJGNKPGUKORN[CUVYQUJWPVTGUKUVCPEGU (KI 6JKU UKORNKHKECVKQP UCXGU PQV QPN[
EQORWVGT VKOG DWV UVQTCIG URCEG CU
YGNNDGECWUGPQJKUVQT[VGTOUJCXG
VQDGUVQTGFHQTVJGNKPG6JKUNQPI
NKPGOQFGNKUOQUVN[WUGFKPNKIJVPKPI
UWTIGUVWFKGUCPF
(KI'SWKXCNGPVEKTEWKVQH
NQPINKPGKHPQTGHNGEVKQPUEQOG
DCEMHTQOQVJGTGPF
+VKURQUUKDNGVQOQFKH[VJKUUKORNKHKECVKQPHQTECUGUUVCTVKPIHTQONKPGCTCEUVGCF[UVCVGEQPFKVKQPUCUYGNNKP
VJCVECUGPQFGUCPFKP(KIYQWNFJCXGCEUVGCF[UVCVGEWTTGPVUQWTEGUEQPPGEVGFVQVJGO
7PHQTVWPCVGN[VJG'/62FQGUPQV[GVEQPVCKPVJKUOQFKHKECVKQP
KPVTCPUKGPVTGEQXGT[XQNVCIGUVWFKGUYJGTGVJGWPHCWNVGFNKPGUNGCXKPIVJGUWDUVCVKQPWPFGTUVWF[CTGRTGHGTCDN[
OQFGNNGFVJKUYC[
6JGGSWCVKQPQHCNWORGFTGUKUVCPEG4DGVYGGPPQFGUMCPFO
KMO V '
X V & XO V
4 M
KUUQNXGFCEEWTCVGN[D[VJG'/62CUNQPICUVJGXCNWGQH4KUPQVWPTGCUQPCDN[UOCNN
'TTQT#PCN[UKU
8GT[NCTIGXCNWGUQH4CTGCEEGRVCDNGCPFFQPQVFGITCFGVJGUQNWVKQPQHVJGEQORNGVGPGVYQTM+PVJG
NKOKVKPIECUG44KVUTGEKRTQECNXCNWG4UKORN[IGVUNQUVKP=)?QH'S VJCVKUKVYKNNPQVJCXGCP[
KPHNWGPEGQPVJGPGVYQTMUQNWVKQPCUKVUJQWNFDG#RTCEVKECNNKOKVHQTXGT[NCTIGTGUKUVCPEGUKUVJGCRRTQZKOCVG
USWCTGTQQVQHVJGNCTIGUVTGCNPWODGTYJKEJVJGEQORWVGTECPJCPFNG GI4KHVJGEQORWVGTCEEGRVU
PWODGTUWRVQ 6JKUKUDGECWUGKPVGTOGFKCVGGZRTGUUKQPUQHVJGHQTO4 :CTGEQORWVGFKPVJGUVGCF[UVCVG
UQNWVKQPKPVJGEQPXGTUKQPHTQOKORGFCPEGUVQCFOKVVCPEGU'ZVTGOGN[NCTIGXCNWGUQH4JCXGDGGPWUGFKPVJGRCUV
VQQDVCKPXQNVCIGFKHHGTGPEGUDGVYGGPPQFGUYKVJUWEJOGVGTKPITGUKUVCPEGDTCPEJGUKPPGYGT'/62XGTUKQPU
XQNVCIGFKHHGTGPEGUECPDGQDVCKPGFFKTGEVN[
8GT[NCTIGTGUKUVCPEGUECPDGWUGFVQTGRNCEGVJGUGTKGU4.GNGOGPVUKPU[OOGVTKEOWNVKRJCUGBEKTEWKVU
KHQPGKUQPN[KPVGTGUVGFKPVJGECRCEKVKXGEQWRNKPICOQPIVJGRJCUGUCUGZRNCKPGFKP(KI6JKUVTKEMTGFWEGU
VJGPWODGTQHPQFGUDWVOQTGKORQTVCPVN[KVCXQKFUCEEWTCE[RTQDNGOUYJKEJOC[QEECUKQPCNN[UJQYWRKHVJGB
EKTEWKVTGRTGUGPVUCXGT[UJQTVNKPGUGEVKQP+PVJGUVGCF[UVCVGUQNWVKQPQH'S VJGXCNWGQHVJGUGTKGU
+VOC[DGYQTVJCFFKPICFKCIPQUVKERTKPVQWVKPVJG'/62KHVJGCFOKVVCPEGUQHVJGUGTKGUCPFUJWPVGNGOGPVU
CTGVQQHCTCRCTVKPQTFGTUQHOCIPKVWFG6JKUYQWNFTGSWKTGCEQORCTKUQPQHT.CPFT%KPVJGUVGCF[UVCVG
UQNWVKQPCPFQH)V.CPF%)VKPVJGVTCPUKGPVUQNWVKQP
(KI%QPXGTUKQPQHPQOKPCNBEKTEWKVHQTUJQTVNKPGKPVQBEKTEWKVYKVJECRCEKVKXGEQWRNKPIQPN[
GNGOGPVKUGPVGTGFCUý=< ?KPVQ=;?YJGTG=< ?KUVJGKORGFCPEGRGTWPKVNGPIVJCPFýVJGNGPIVJQHVJGUJQTV
UGEVKQP(QTCUJQTVNGPIVJýKUUOCNNCPFý=< ?CEEQTFKPIN[TGNCVKXGN[NCTIG#VVJGUCOGVKOGVJGUJWPV
UWUEGRVCPEGUý LT=% ?GPVGTGFKPVQ=;?DGEQOGTGNCVKXGN[UOCNN#UýKUFGETGCUGFVJGECRCEKVKXGEQWRNKPIGHHGEV
YKNNGXGPVWCNN[IGVNQUVKPVJGUQNWVKQP+PCRTCEVKECNECUGQHECRCEKVKXGEQWRNKPIDGVYGGPM8EKTEWKVUCV*\
VJKUCEEWTCE[RTQDNGOUJQYGFWRYKVJVJGUJQTVGUVNKPGUGEVKQPDGKPIMOKVYCUFKUEQXGTGFCEEKFGPVCNN[DGECWUG
VJGUKPINGRTGEKUKQPUQNWVKQP CEEWTCE[CRRTQZFGEKOCNFKIKVU QPCP+$/FKHHGTGFWPGZRGEVGFN[D[HTQO
VJGFQWDNGRTGEKUKQPUQNWVKQP CEEWTCE[
CRRTQZFGEKOCNFKIKVU (QTTGCUQPCDNGUVGRUK\GUQH)VVJGRTQDNGOKUNGUUUGXGTGKPVJGVTCPUKGPVECNEWNCVKQP
CUECPGCUKN[DGUGGPKHLTKP'S KUTGRNCEGFD[)VKP'S +PVJKUGZCORNGYKVJ)VzUVJG
XCNWGQHVJGUGTKGUGNGOGPVYQWNFDGUOCNNGTD[CHCEVQTQHYJKNGVJGXCNWGQHVJGUJWPVGNGOGPVYQWNFDGNCTIGT
D[CHCEVQTQH1TKPQVJGTYQTFUCUKOKNCTCEEWTCE[RTQDNGOYQWNFCRRGCTFWTKPIVJGVTCPUKGPVUKOWNCVKQPKH
VJGNKPGYGTGUJQTVGPGFD[CHCEVQTQH ýOKPUVGCFQHMO 8GT[UOCNNXCNWGUQH4FQETGCVGCEEWTCE[RTQDNGOUHQTVJGUCOGTGCUQPCUFKUEWUUGFKPVJGRTGEGFKPI
RCTCITCRJ 8GT[ UOCNN XCNWGU QH 4 ETGCVG XGT[ NCTIG EQPFWEVCPEG XCNWGU 4 KP VJG OCVTKZ =;? QH UVGCF[UVCVG
UQNWVKQPUCPFKPVJGOCVTKZ=)?QHVTCPUKGPVUQNWVKQPUYJKEJECPUYCORQWVVJGGHHGEVUQHQVJGTGNGOGPVUEQPPGEVGF
VQVJCVTGUKUVCPEG8GT[UOCNNXCNWGUQH4JCXGDGGPWUGFKPVJGRCUVRTKOCTKN[VQUGRCTCVGUYKVEJGUUKPEGGCTNKGT
'/62XGTUKQPUCNNQYGFQPN[QPGUYKVEJVQDGEQPPGEVGFVQCPQFGYKVJWPMPQYPXQNVCIG+PPGYGTXGTUKQPUVJKU
NKOKVCVKQPQPVJGNQECVKQPQHUYKVEJGUPQNQPIGTGZKUVUCPFVJGPGGFHQTWUKPIXGT[UOCNNXCNWGUQH4UJQWNFVJGTGHQTG
PQNQPIGTGZKUVKPVJGUGNCVGTXGTUKQPU
*KPVUCDQWVVJGWUGQHUOCNNTGUKUVCPEGUCTGIKXGPKP=?RRDE
'ZCORNGHQT0GVYQTMYKVJ4GUKUVCPEGU
2TCEVKECNGZCORNGUHQTRWTGN[TGUKUVKXGPGVYQTMUCTGTCVJGTNKOKVGF#UKORNGECUGKUUJQYPKP(KI
#UUWOGCFEXQNVCIGUQWTEGYKVJPGINKIKDNGUQWTEGKORGFCPEGKUEQPPGEVGFVQCNKPGVJTQWIJCEKTEWKVDTGCMGTYKVJ
CENQUKPITGUKUVQTQHXCNWG4ENQUG+HYGCTGKPVGTGUVGFKPYJCVJCRRGPUCHVGTENQUKPIQHEQPVCEV+KPVJGHKTUVUJQTVVKOG
RGTKQFFWTKPIYJKEJTGHNGEVKQPUJCXGPQV[GVEQOGDCEMHTQOVJGTGOQVGGPFVJGPVJKUECUGECPDGUVWFKGFYKVJVJG
EKTEWKVQH(KI D +HYGEJQQUG4ENQUG<YGUGGVJCVVJGXQNVCIGCVVJGUGPFKPIGPFYKNNDGRWYJKEJ
YKNNFQWDNGVQRWCVVJGQRGPTGEGKXKPIGPF6JGTGHQTGPQQXGTXQNVCIGYKNNCRRGCTCUNQPICUEQPVCEV++KUUVKNN
QRGP#TGCNECUGKUQDXKQWUN[OQTGEQORNKECVGFDGECWUG
C PGVYQTM
D GSWKXCNGPVEKTEWKV
(KI'PGTIK\CVKQPQHCXGT[NQPINKPG <UWTIG
KORGFCPEGJVTCXGNVKOG
!EQPVCEV++YKNNENQUG V[RKECNN[VQOUNCVGT CUYGNN
!VJGU[UVGOKUVJTGGRJCUG
!VJGNKPGKUPQVNQUUNGUU
!OWNVKRNGTGHNGEVKQPUYKNNQEEWTCUYGUVWF[CNQPIGTVKOGRGTKQF
!ENQUKPIQHEQPVCEV+FQGUPQVPGEGUUCTKN[QEEWTCVOCZKOWOXQNVCIG CRRTQZKOCVGFCUCFEUQWTEGKP(KI
DWVOC[QEEWTCP[YJGTGQPVJGUKPGYCXG
!VJGUQWTEGKORGFCPEGKUPQV\GTQ
!VJGVJTGGRQNGUKPCVJTGGRJCUGU[UVGOFQPQVENQUGUKOWNVCPGQWUN[CPFDGECWUGQHOCP[QVJGTHCEVQTU
+PCV[RKECNU[UVGOOCZKOWOUYKVEJKPIUWTIGQXGTXQNVCIGUOC[DGVQRWYKVJQWVENQUKPITGUKUVQTU
XGTUWURWKP(KI YJKEJYQWNFV[RKECNN[DGTGFWEGFVQVQRWYKVJENQUKPITGUKUVQTU XGTUWU
RWKP(KI =?
5GNH+PFWEVCPEG.
/CIPGVKECNN[EQWRNGFEKTEWKVUCTGUQRTGXCNGPVKPRQYGTU[UVGOUUVCTVKPIHTQOVJGIGPGTCVQTVJTQWIJVJG
VTCPUHQTOGTVQVJGOCIPGVKECNN[EQWRNGFRJCUGEQPFWEVQTUQHCVJTGGRJCUGNKPGVJCVKPFWEVCPEGUWUWCNN[CRRGCTCU
EQWRNGF KPFWEVCPEGU 6JGTG CTG ECUGU JQYGXGT QH WPEQWRNGF UGNH KPFWEVCPEGU #OQPI QVJGT VJKPIU UGNH
KPFWEVCPEGUCTGWUGFVQTGRTGUGPV
C PQTOCNVJTGG
RJCUGEQPPGEVKQP
D HQWTTGCEVQT
EQORGPUCVKQP
UEJGOG
(KI5JWPVTGCEVQTEQPPGEVKQPU
C
UKPINGRJCUGUJWPVTGCEVQTUCPFPGWVTCNTGCEVQTUKPUJWPVEQORGPUCVKQPUEJGOGU (KI D
RCTVQHFKUEJCTIGEKTEWKVUKPUGTKGUECRCEKVQTUVCVKQPU
E
GSWKROGPVKP*8&%EQPXGTVGTUVCVKQPUUWEJCUUOQQVJKPITGCEVQTUCPQFGTGCEVQTURCTVUQHHKNVGTU
QPVJGCECPFFEUKFG
F
KPFWEVKXGRCTVQHUQWTEGKORGFCPEGUKP6JGXGPKPGSWKXCNGPVEKTEWKVUHQTVJGTGUVQHVJGU[UVGO
YJGPRQUKVKXGCPF\GTQUGSWGPEGRCTCOGVGTUCTGKFGPVKECN (KI G
KPFWEVKXGRCTVQHUKPINGRJCUGPQOKPCNBEKTEWKVUKPVJGUKPINGRJCUGTGRTGUGPVCVKQPQHDCNCPEGF
RQUKVKXGQTPGICVKXGUGSWGPEG QRGTCVKQPQTQH\GTQUGSWGPEGQRGTCVKQP (KI H
RCTVQHGSWKXCNGPVEKTEWKVHQTNQCFU (KI GXGPVJQWIJNQCFOQFGNNKPICVJKIJGTHTGSWGPEKGUKU
CXGT[EQORNKECVGFVQRKE=?CPFNQCFUCTGVJGTGHQTGQTHQTQVJGTTGCUQPUQHVGPKIPQTGF
(KI6JGXGPKPGSWKXCNGPVEKTEWKV
YKVJ<RQU<PGI<\GTQ
(KI6[RKECNRQUKVKXGUGSWGPEGPGVYQTMTGRTGUGPVCVKQP
(KI.QCFOQFGNHQTJCTOQPKEUUVWFKGU=?
I
RCTVQHUWTIGCTTGUVGTOQFGNUVQUKOWNCVGVJGF[PCOKEEJCTCEVGTKUVKEUQHVJGCTTGUVGT=?
J
RCTVUQHGNGEVTQPKEEKTEWKVU
%JQMGEQKNUWUGFHQTRQYGTNKPGECTTKGTEQOOWPKECVKQPUCTGPQTOCNN[KIPQTGFKPUYKVEJKPIUWTIGUVWFKGU
DWVOC[JCXGVQDGOQFGNNGFKPUVWFKGUKPXQNXKPIJKIJGTHTGSWGPEKGU%WTTGPVVTCPUHQTOGTUCTGWUWCNN[KIPQTGF
WPNGUUVJGEWTTGPVVTCPUHQTOGTKVUGNHKURCTVQHVJGKPXGUVKICVKQP GIKPUVWF[KPIVJGFKUVQTVKQPQHVJGUGEQPFCT[
EWTTGPVVJTQWIJUCVWTCVKQPGHHGEVU 6JGGSWCVKQPQHCUGNHKPFWEVCPEG.DGVYGGPPQFGUMCPFOKUUQNXGFCEEWTCVGN[KPVJGCEUVGCF[UVCVG
UQNWVKQPYKVJ'S 6JGQPN[RTGECWVKQPVQQDUGTXGKUVJCVT.MOUJQWNFPQVDGGZVTGOGN[UOCNNHQTVJGUCOG
TGCUQPUCUGZRNCKPGFKP5GEVKQPHQTVJGECUGQHUOCNNTGUKUVCPEGXCNWGU
(QTVJGVTCPUKGPVUKOWNCVKQPVJGGZCEVFKHHGTGPVKCNGSWCVKQP
FK
XM & XO ' . MO
FV
KUTGRNCEGFD[VJGCRRTQZKOCVGEGPVTCNFKHHGTGPEGGSWCVKQP
XM V & XO V % XM V&)V & XO V&)V
K V & KMO V&)V
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< ) U < ) O < ) O
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?
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U
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O
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)
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)
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)
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)
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)
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)
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5KPEGVTCPUHQTOCVKQPVQU[OOGVTKECNEQORQPGPVUKPXQNXGUEQORNGZEQGHHKEKGPVUU[OOGVTKECNEQORQPGPVU
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<
CPF
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6JGDCNCPEGFFKUVTKDWVGFRCTCOGVGTNKPGOQFGNUKPVJG'/62WUGVTCPUHQTOCVKQPUVQ"$EQORQPGPVUFWG
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=XRJCUG? ' =6?=X"$?
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=K"$? ' =6?& =KRJCUG?
=KRJCUG? ' =6? =K"$?
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X
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&
=6? '
&
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'
&
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&
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=6?& ' =6?V
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< ) U%< ) O
+
& F8"FZ '
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)
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)
)
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D
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6JGRQUKVKXGUGSWGPEGKPFWEVCPEGQHQXGTJGCFNKPGUKURTCEVKECNN[EQPUVCPVYJKNGVJGRQUKVKXGUGSWGPEG
TGUKUVCPEGTGOCKPUOQTGQTNGUUEQPUVCPVWPVKNUMKPGHHGEVKPVJGEQPFWEVQTUDGEQOGUPQVKEGCDNGCUUJQYPKP(KI
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TGVWTP
(KI2QUKVKXGCPF\GTQUGSWGPEGTGUKUVCPEGCPFKPFWEVCPEGQHCVJTGGRJCUGNKPG
6JGUJWPVECRCEKVCPEGOCVTKZQHCDCNCPEGFVJTGGRJCUGNKPGDGEQOGUFKCIQPCNKP"$EQORQPGPVUCU
YGNNYKVJ
)
)
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)
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)
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D
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VQRQYGTGPIKPGGTU
%QORCTKUQPYKVJTGUWNVUHTQOJCPFDQQMHQTOWNCU
6JGRQUKVKXGCPF\GTQUGSWGPEGRCTCOGVGTUQDVCKPGFHTQOVJGUWRRQTVKPITQWVKPGU.+0'%1056#065CPF
%#$.'%1056#065OC[FKHHGTHTQOVJQUGQDVCKPGFYKVJJCPFDQQMHQTOWNCU5KPEGUQOG'/62WUGTUOC[OCMG
EQORCTKUQPUKVOC[DGYQTVJYJKNGVQGZRNCKPVJGOCLQTFKHHGTGPEGUHQTCURGEKHKEGZCORNG#UUWOGCV[RKECN
M8NKPGYKVJJQTK\QPVCNRJCUGEQPHKIWTCVKQPYKVJRJCUGUHGGVCRCTVCVCPCXGTCIGJGKIJVCDQXGITQWPFQHHGGV
'CEJRJCUGEQPUKUVUQHCU[OOGVTKECNDWPFNGYKVJUWDEQPFWEVQTUURCEGFKPEJGUCRCTV5WDEQPFWEVQTFKCOGVGT
KPEJGUFETGUKUVCPEGSOKNG)/4KPEJGU6JTQWIJQWVVJKUEQORCTKUQPVJGDWPFNG
EQPFWEVQTUCTGTGRTGUGPVGFCUGSWKXCNGPVEQPFWEVQTUYKVJT
KPEJGUHTQO'S CPF)/4
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Bg
)
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RJCUG
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RJCUG
?OCVTKZ$QVJIKXGRTCEVKECNN[
VJGUCOGCPUYGTYKVJTGUWNVUHTQO'S NQYGTVJCPVJQUGHTQO'S +P=?'S KU
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VJGXCNWGOC[CEVWCNN[DGENQUGTVQOGCUWTGFXCNWGUDGECWUGVJGKPHNWGPEGQHVQYGTUYJKEJKUPGINGEVGFKPCNN
HQTOWNCUV[RKECNN[KPETGCUGUVJGECNEWNCVGF\GTQUGSWGPEGECRCEKVCPEGD[CDQWVVQQPM8NKPGUCDQWV
QPCPFM8NKPGUCPFCDQWVQPM8NKPGU=R?
6JGHQTOWNCUHQT\GTQCPFRQUKVKXGUGSWGPEGKORGFCPEGUKPOQUVJCPFDQQMUCTGDCUGFQPVJGCUUWORVKQP
VJCVRCTCOGVGTCKP'S KUUQUOCNNVJCVQPN[VJGHKTUVVGTOKPVJGUGTKGUQH'S OWUVDGTGVCKPGF(QT
PQTOCNRJCUGURCEKPIUVJKUKURTQDCDN[CTGCUQPCDNGCUUWORVKQPCVRQYGTHTGSWGPE[QT*\6JGPCHVGTCNN
FKCIQPCNCPFQHHFKCIQPCNGNGOGPVUJCXGDGGPCXGTCIGFQWVCOQPIVJGOUGNXGUVJTQWIJVTCPURQUKVKQP
)4 ) U ' )4 ) O .
TB @ &
KP SMO
CPF
): ) U . T @ & = & NP JO M
H
D
?
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): ) O . T @ & = & NP &O M
H
D
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6JKUNGCFUVQVJGGZRTGUUKQP
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)
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KP SMO
)/4GSWKX
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CE
QHEQPFWEVQTJGKIJVYJKEJKURTGUGPVKP< CPF< EQORNGVGN[FKUCRRGCTUJGTGKPVCMKPIVJGFKHHGTGPEG<
U
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< < 'S KUVJGHQTOWNCHQWPFKPOQUVJCPFDQQMU6CDNGEQORCTGUTGUWNVUHTQO'S YKVJ
U
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TGUWNVU HTQO 'S HQT VJG M8 NKPG FGUETKDGF CDQXG YKVJ VJG HQNNQYKPI CFFKVKQPCN CUUWORVKQPU 'CTVJ
TGUKUVKXKV[SOUMKPGHHGEVYKVJKPEQPFWEVQTUKIPQTGFVQNKOKVFKHHGTGPEGUVQKPHNWGPEGQHGCTVJTGVWTP VJCVKU
4 4 CPF)/4
CE
FE
GSWKX
EQPUVCPV 6CDNG#EEWTCVGCPFCRRTQZKOCVGRQUKVKXGUGSWGPEGTGUKUVCPEGCPFKPFWEVCPEG
4
H
*\
#%%74#6'
CPF. HTQO'S RQU
4
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.
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4
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4
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.
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6CDNGUJQYUVJCV.
HTQO'S KUSWKVGCEEWTCVGQXGTCYKFGHTGSWGPE[TCPIGYJGTGCU4
RQU
DGEQOGUNGUU
RQU
CEEWTCVGCUVJGHTGSWGPE[KPETGCUGU GTTQTCV*\DWVYTQPID[QTFGTUQHOCIPKVWFGCVM*\ 6JG
KPETGCUGKP4
KPVJGJKIJGTHTGSWGPE[TCPIGKUECWUGFD[GFF[EWTTGPVUKPVJGGCTVJCUKPFKECVGFKP(KIHQT
RQU
CDKRQNCTFENKPG)TQWPFYKTGUCNUQKPHNWGPEGVJGRQUKVKXGUGSWGPEGKORGFCPEGCUOGPVKQPGFKP5GEVKQP
C $QVJKPHNWGPEGUCTGKIPQTGFKP'S DWVCWVQOCVKECNN[KPENWFGFKPVJGOGVJQFFGUETKDGFJGTG
(KI'FF[EWTTGPVUKPGCTVJ
6JG\GTQUGSWGPEGKORGFCPEGQDVCKPGFHTQO'S KU
)
)
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D
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)/4GSWKX@ FO
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O
EQORCTGUVJGCRRTQZKOCVGTGUWNVUHTQO'S YKVJVJGCEEWTCVGTGUWNVUHTQO'S 6JGKPFWEVCPEG.
\GTQ
KUTGCUQPCDN[CEEWTCVGQXGTCYKFGHTGSWGPE[TCPIG GTTQTCV*\GTTQTCVM*\ DWVVJG
TGUKUVCPEG4
KUNGUUCEEWTCVG GTTQTCV*\GTTQTCVM*\ \GTQ
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4
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*\
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.
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4
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.
O*OKNG
2QUKVKXGCPF<GTQ5GSWGPEG2CTCOGVGTUQH$CNCPEGF/2JCUG.KPGU
6JG'/62ECPJCPFNGDCNCPEGFFKUVTKDWVGFRCTCOGVGTNKPGUPQVQPN[HQTVJGECUGQHCVJTGGRJCUGNKPGDWV
HQTCP[PWODGTQHRJCUGU/(QTVJKUIGPGTCNECUGVJG"$VTCPUHQTOCVKQPQH'S JCUDGGPIGPGTCNK\GF
VQ/RJCUGUYKVJVJGVTCPUHQTOCVKQPOCVTKZ=?
/
/
=6? '
/
&
&
, ,&
, ,&
&
,&
/ /&
/ /&
/
, ,&
&
/&
/ /&
YJGTGCICKP
=6?& ' =6?V
=6?QH'S KUCURGEKCNECUGQH'S HQT/KHYGCUUWOGVJCVVJGRJCUGUCTGPWODGTGFKP'S
CPFKHVJG"$SWCPVKVKGUCTGQTFGTGF$" UKIPTGXGTUCNQP" #RRN[KPIVJKU/RJCUG"$VTCPUHQTOCVKQP VQVJGOCVTKEGUQH/RJCUGDCNCPEGFNKPGURTQFWEGUFKCIQPCNOCVTKEGU
QHVJGHQTO
+PVJG7$%'/62CPFKPQNFGTXGTUKQPUQHVJG$2#'/62-CTTGPDCWGT UVTCPUHQTOCVKQP=?KUWUGF
KPUVGCFYJKEJRTQFWEGUVJGUCOGFKCIQPCNOCVTKEGUDWVFQGUPQVJCXGVJGRTQRGTV[QH'S 6JKURTQRGTV[
KUKORQTVCPVDGECWUGKVOCMGUVJGDCNCPEGFNKPGLWUVCURGEKCNECUGQHVJGWPVTCPURQUGFNKPGFKUEWUUGFKP5GEVKQP
Z’zero
Z’pos
Z’pos
.
.
.
Z’pos
with the first diagonal element being the zero sequence (ground mode) impedance, and the next M-1 diagonal elements
being the positive sequence (aerial mode) impedance,
)
)
)
<\GTQ ' <U % / & <O
)
)
C
)
<RQU ' <U & <O
D
and similarly for the capacitances,
)
)
)
%\GTQ ' %U % / & %O
)
)
C
)
%RQU ' %U & %O
D
To refer to the two distinct diagonal elements as zero and positive sequence may be confusing, because the
concept of sequence values has primarily been used for three-phase lines. "Ground mode" and "aerial mode" may be
more appropriate. Confusion is most likely to arise for double-circuit three-phase lines, where each three-phase line
has its own zero and positive sequence values defined by Eq. (4.50) and (4.51) with symmetrical components used for
each three-phase circuit, while in the context of this section the double-circuit line is treated as a six-phase line with
different zero and positive sequence values defined by Eq. (4.60) and (4.61). The fact that the terms zero and positive
sequence are used for M û 3 as well comes from the generalization of symmetrical components of Section 4.1.4 to M
phases with the transformation matrix [56, p. 155]
U U U/
=5/&RJCUG? '
U U U/
C
U/ U/ U//
with
UKM '
/
GZR 6&L
B
K& M& >
/
D
A special case of interest for symmetric bipolar dc lines11 is M = 2. In this case [T] of Eq. (4.58) and [S] of
Eq. (4.62a) are identical,
=6&RJCUG? '
&
4.1.3.3 Two Identical Three-Phase Lines with Zero Sequence Coupling Only
Just as a transposed single-circuit three-phase line can usually be approximated as a balanced line, so two
identical and parallel three-phase lines can often be approximated as "almost balanced" lines with an impedance matrix
of the form
< )U < )O < )O < )R < )R < )R
< )O < )U < )O < )R < )R < )R
< )O < )O < )U < )R < )R < )R
< )R < )R < )R < )U < )O < )O
< )R < )R < )R < )O < )U < )O
< )R < )R < )R < )O < )O < )U
The transposition scheme of Fig. 4.22 would produce such a matrix form, which implies that the two circuits are only
coupled in zero sequence, but not in positive or negative sequence. Such a complicated transposition scheme is seldom,
if ever, used, but the writer suspects that positive and negative sequence couplings in the more common double-circuit
transposition scheme of Fig. 4.23 is often so weak that the model discussed here may be a useful approximation for the
case of Fig. 4.23 as well.
6QDGEQPUKUVGPVNKPGUYKVJ/CPF/CTGECNNGFUKPINGRJCUGCPFVYQRJCUGNKPGU
TGURGEVKXGN[KPVJKUOCPWCN6JKUFKHHGTUHTQOVJG+'''5VCPFCTFU=R?KPYJKEJEKTEWKVUYKVJQPGRJCUG
EQPFWEVQTCPFQPGPGWVTCNEQPFWEVQT YJKEJEQWNFDGTGRNCEGFD[ITQWPFTGVWTP CUYGNNCUEKTEWKVUYKVJVYQ
RJCUGEQPFWEVQTUCPFQPGPGWVTCNEQPFWEVQT QTITQWPFTGVWTP CTGDQVJECNNGFUKPINGRJCUGEKTEWKVUHQTJKUVQTKECN
TGCUQPU(QT/$VJGFGHKPKVKQPKPVJG+'''5VCPFCTFUKUVJGUCOGCUKPVJKUOCPWCN
(KI&QWDNGEKTEWKVVTCPURQUKVKQPUEJGOGYKVJ\GTQUGSWGPEGEQWRNKPI
QPN[
T h e
matrix of Eq. (4.64) is diagonalized by modifying the transformation matrix of Eq. (4.58) to
=6? '
& &
&
&
& &
&
with [T]-1 = [T]t again, which produces the diagonal matrix
Z’G
Z’IL
Z’L
Z’L
(4.66)
Z’L
Z’L
If each circuit has three-phase sequence parameters Z’zero, Z’pos, and if the three-phase zero sequence coupling between
the two circuits is Z’zero-coupling, then the ground mode G, inter-line mode IL and line mode L values required by the EMTP
are found from
<)
+.
'
<)
\GTQ
&
<)
< ) ) ' < ) \GTQ % < )\GTQ&EQWRNKPI
\GTQ&EQWRNKPI
< ) . ' < ) RQU
with identical equations for the capacitances.
If the two three-phase circuits are not identical, then the transformation matrix of Eq. (4.65) can no longer be
used; instead, [T] depends on the particular tower configuration.
4.1.4 Symmetrical Components
Symmetrical components are not used as such in the EMTP, except that the parameters of balanced lines after
transformation to M-phase , , 0-components are the same as the parameters of symmetrical components, namely zero
and positive sequence values. The supporting routine LINE CONSTANTS does have output options for more detailed
symmetrical component information, however, which may warrant some explanations.
In addition to zero and positive sequence values, LINE CONSTANTS also prints full symmetrical component
matrices. Its diagonal elements are the familiar zero and positive sequence values of the line; they are correct for the
untransposed line as well as for a line which has been balanced through proper transpositions. The off-diagonal
elements are only meaningful for the untransposed case, because they would become zero for the balanced line. For
the untransposed case, these off-diagonal elements are used to define unbalance factors [47, p. 93]. The full symmetrical
component matrices are no longer symmetric, unless the columns for positive and negative sequence are exchanged [27].
This exchange is made in the output of the supporting routine LINE CONSTANTS with rows listed in order "zero, pos,
neg,..." and columns in order "zero, neg, pos,...". With this trick, matrices can be printed in triangular form (elements
in and below the diagonal), as is done with the matrices for individual and equivalent phase conductors.
Symmetrical components for two-phase lines are calculated with the transformation matrix of Eq. (4.63), while
those of three-phase lines are calculated with
CPF =XU[OO? ' =5?& =XRJCUG?
=XRJCUG? ' =5? =XU[OO?
C
identical for currents,
X\GTQ
YJGTG
=5? '
=5?&
'
=XU[OO? ' XRQU
XPGI
C C
C C
C C
C
D
C
and a = ej120E.
The columns in these matrices are normalized12; in that form, [S] is unitary,
=5?& ' =5 Â?V
6JGGNGEVTKEWVKNKV[KPFWUVT[WUWCNN[WUGUWPPQTOCNK\GFVTCPUHQTOCVKQPKPYJKEJVJGHCEVQTHQTVJG=5?OCVTKZ
KUKPUVGCFQH%CPFHQTVJG=5? OCVTKZKPUVGCFQH%6JGU[OOGVTKECNEQORQPGPVKORGFCPEGUCTG
KFGPVKECNKPDQVJECUGUDWVVJGUGSWGPEGEWTTGPVUCPFXQNVCIGUFKHHGTD[CHCEVQTQH%
where "*" indicates conjugate complex and "t" transposition.
For M $ 3, the supporting routine LINE CONSTANTS assumes three-phase lines in parallel. Examples:
M = 6: Two three-phase lines in parallel
M = 9: Three-phase lines in parallel
M = 8: Two three-phase lines in parallel, with equivalent phase conductors no. 7 and 8 ignored in the
transformation to symmetrical components.
The matrices are then transformed to three-phase symmetrical components and not to M-phase symmetrical components
of Eq. (4.62). For example for M = 6 (double-circuit three-phase line),
=< ) U[OO? '
=5?&
=5?&
< ) RJCUG
=5?
=5?
with [S] defined by Eq. (4.68), Eq. (4.70) produces the three-phase symmetrical component values required in Eq.
(4.67).
Balancing of double-circuit three-phase lines through transpositions never completely diagonalizes the
respective symmetrical component matrices. The best that can be achieved is with the seldom-used transposition scheme
of Fig. 4.22, which leads to
=< )
U[OO? '
< )\GTQ&+
<\GTQ&EQWRNKPI
< )RQU&+
< )RQU&+
<\GTQ&EQWRNKPI
< )\GTQ&++
< )RQU&++
< )RQU&++
(4.71)
If both circuits are identical, then Z’zero-I = Z’zero-II = Z’zero , and Z’pos-I = Z’pos-II = Z’pos ; in that case, the transformation matrix
of Eq. (4.65) can be used for diagonalization. The more common transposition scheme of Fig. 4.23 produces positive
and zero sequence coupling between the two
(a)
barrels rolled in
opposite direction
(b)
barrels rolled in
same direction
Fig. 4.23 - Double-circuit transposition scheme
circuits as well, with the nonzero pattern of the matrix in Eq. (4.71) changing to
: *: : * : :* :
: *: : * : :* :
where "X" indicates nonzero terms. Re-assigning the phases in Fig. 4.23(b) to CI, BI, AI, AII, BII, CII from top to
bottom would change the matrix further to cross-couplings between positive sequence of one circuit and negative
sequence of the other circuit, and vice versa,
: *: : * :
:* : : *: :* : : * :
4.1.5 Modal Parameters
From the discussions of Section 4.1.3 it should have become obvious that the solution of M-phase transmission
line equations becomes simpler if the M coupled equations can be transformed to M decoupled equations. These
decoupled equations can then be solved as if they were single-phase equations. For balanced lines, this transformation
is achieved with Eq. (4.58).
Many lines are untransposed, however, or each section of a transposition barrel may no longer be short
compared with the wave length of the highest frequencies occurring in a particular study, in which case each section
must be represented as an untransposed line. Fortunately, the matrices of untransposed lines can be diagonalized as
well, with transformations to "modal" parameters derived from eigenvalue/eigenvector theory. The transformation
matrices for untransposed lines are no longer known a priori, however, and must be calculated for each particular pair
of parameter matrices [Z’phase] and [Y’phase].
To explain the theory, let us start again from the two systems of equations (4.31) and (4.32),
F8RJCUG
&
' =< ) RJCUG? =+RJCUG?
C
' =; ) RJCUG?=8RJCUG?
D
FZ
and
F+RJCUG
&
FZ
with [Y’phase] = j [C’phase] if shunt conductances are ignored, as is customarily done. By differentiating the first equation
with respect to x, and replacing the current derivative with the second equation, a second-order differential equation for
voltages only is obtained,
F 8RJCUG
FZ ' =< ) RJCUG? =; ) RJCUG? =8RJCUG?
C
Similarly, a second-order differential equation for currents only can be obtained,
F +RJCUG
FZ ' =; ) RJCUG? =< ) RJCUG? =+RJCUG?
D
where the matrix products are now in reverse order from that in Eq. (4.73a), and therefore different. Only for balanced
matrices, and for the lossless high-frequency approximations discussed in Section 4.1.5.2, would the matrix products
in Eq. (4.73a) and (4.73b) be identical.
With eigenvalue theory, it becomes possible to transform the two coupled equations (4.73) from phase
quantities to "modal" quantities in such a way that the equations become decoupled, or in terms of matrix algebra, that
the associated matrices become diagonal, e.g., for the voltages,
F 8OQFG
FZ ' =7? =8OQFG?
with [ ] being a diagonal matrix. To get from Eq. (4.73a) to (4.74), the phase voltages must be transformed to mode
voltages, with
=8RJCUG? ' =6X? =8OQFG?
C
=8OQFG? ' =6X?& =8RJCUG?
D
and
Then Eq. (4.73a) becomes
F 8OQFG
FZ ' =6X?& =< ) RJCUG? =; ) RJCUG? =6X? =8OQFG?
C
which, when compared with Eq. (4.74), shows us that
=7? ' =6X?& =< ) RJCUG? =; ) RJCUG? =6X?
D
To find the matrix [Tv] which diagonalizes [Z’phase][Y’phase] is the eigenvalue/eigenvector problem. The diagonal elements
of [ ] are the eigenvalues of the matrix product [Z’phase][Y’phase], and [Tv] is the matrix of eigenvectors or modal matrix
of that matrix product. There are many methods for finding eigenvalues and eigenvectors. The most reliable method
for finding the eigenvalues seems to be the QR-transformation due to Francis [3], while the most efficient method for
the eigenvector calculation seems to be the inverse iteration scheme due to Wilkinson [4, 5]. In the supporting routines
LINE CONSTANTS and CABLE CONSTANTS, the "EISPACK"-subroutines [67] are used, in which the eigenvalues
and eigenvectors of a complex upper Hessenberg matrix are found by the modified LR-method due to Rutishauser. This
method is a predecessor of the QR-method, and where applicable, as in the case of positive definite matrices, is more
efficient than the QR-method [68]. To transform the original complex matrix to upper Hessenberg form, stabilized
elementary similarity transformations are used. For a given eigenvalue
, the corresponding eigenvector [tvk] (= k-th
k
column of [Tv]) is found by solving the system of linear equations
6 =< ) RJCUG? =; ) RJCUG? & 8M=7? > =VXM? ' with [U] = unit or identity matrix. Eq. (4.77) shows that the eigenvectors are not uniquely defined in the sense that they
can be multiplied with any nonzero (complex) constant and still remain proper eigenvectors13, in contrast to the
eigenvalues which are always uniquely defined.
Floating-point overflow may occur in eigenvalue/eigenvector subroutines if the matrix is not properly scaled.
Unless the subroutine does the scaling automatically, [Z’phase][Y’phase] should be scaled before the subroutine call, by
dividing each element by
-( 2g0µ 0), as suggested by Galloway, Shorrows and Wedepohl [39]. This division brings
the matrix product close to unit matrix, because [Z'phase][Y'phase] is a diagonal matrix with elements - 2g0 µ 0 if resistances,
internal reactances and Carson's correction terms are ignored in Eq. (4.7) and (4.8), as explained in Section 4.1.5.2. The
eigenvalues from this scaled matrix must of course be multiplied with - 2g0µ 0 to obtain the eigenvalues of the original
matrix. In [39] it is also suggested to subtract 1.0 from the diagonal elements after the division; the eigenvalues of this
modified matrix would then be the p.u. deviations from the eigenvalues of the lossless high-frequency approximation
of Section 4.1.5.2, and would be much more separated from each other than the unmodified eigenvalues which lie close
together. Using subroutines based on [67] gave identical results with and without this subtraction of 1.0, however.
In general, a different transformation must be used for the currents,
=+RJCUG? ' =6K? =+OQFG?
C
6JKUKUKORQTVCPVKHOCVTKEGU=6 ?QDVCKPGFHTQOFKHHGTGPVRTQITCOUCTGEQORCTGF6JGCODKIWKV[ECPDG
TGOQXGFKPCPWODGTQHYC[UGID[CITGGKPIVJCVVJGGNGOGPVUKPVJGHKTUVTQYUJQWNFCNYC[UDGQTD[
PQTOCNK\KPIVJGEQNWOPUVQC'WENKFGCPXGEVQTNGPIVJQHVJCVKUD[TGSWKTKPIV V V V YKVJV EQPLWICVGEQORNGZQHV+PVJGNCVVGTECUGVJGTGKUUVKNNCODKIWKV[KPVJGUGPUGVJCVGCEJEQNWOPEQWNF
DGOWNVKRNKGFYKVJCTQVCVKQPEQPUVCPVG "CPFUVKNNJCXGXGEVQTNGPIVJ
X
X X
L
X X
and
=+OQFG? ' =6K?& =+RJCUG?
D
because the matrix products in Eq. (4.73a) and (4.73b) have different eigenvectors, though their eigenvalues are
identical. Therefore, Eq. (4.73b) is transformed to
F +OQFG
FZ ' =7? =+OQFG?
with the same diagonal matrix as in Eq. (4.74). While [Ti] is different from [Tv], both are fortunately related to each
other [58],
=6K? ' =6X ?&
V
where "t" indicates transposition. It is therefore sufficient to calculate only one of them.
Modal analysis is a powerful tool for studying power line carrier problems [59-61] and radio noise interference
[62, 63]. Its use in the EMTP is discussed in Section 4.1.5.3. It is interesting to note that the modes in single-circuit
three-phase lines are almost identical with the , , 0-components of Section 4.1.3.1 [58]. Whether the matrix products
in Eq. (4.73) can always be diagonalized was first questioned by Pelissier in 1969 [64]. Brandao Faria and Borges da
Silva have shown in 1985 [65] that cases can indeed be constructed where the matrix product cannot be diagonalized.
It is unlikely that such situations will often occur in practice, because extremely small changes in the parameters (e.g.,
in the 8th significant digit) seem to be enough to make it diagonalizable again. Paul [66] has shown that diagonalization
can be guaranteed under simplifying assumptions, e.g., by neglecting conductor resistances.
The physical meaning of modes can be deduced from the transformation matrices [Tv] and [Ti]. Assume, for
example, that column 2 of [Ti] has entries of (-0.6, 1.0, -0.4). From Eq. (4.78a) we would then know that mode-2 current
flows into phase B in one way, with 60% returning in phase A and 40% returning in phase C.
4.1.5.1 Line Equations in Modal Domain
With the decoupled equations of (4.74) and (4.79) in modal quantities, each mode can be analyzed as if it were
a single-phase line. Comparing the modal equation
F 8OQFG&M
FZ ' 8M 8OQFG&M
with the well-known equation of a single-phase line,
F 8
' (8
FZ
with the propagation constant defined in Eq. (1.15), shows that the modal propagation constant
mode-k
is the square
root of the eigenvalue,
(OQFG&M ' "M % L$M ' (M
with
k
= attenuation constant of mode k (e.g., in Np/km),
k
= phase constant of mode k (e.g., in rad/km).
The phase velocity of mode k is
RJCUG XGNQEKV[ '
T
$M
C
and the wavelength is
YCXG NGPIVJ '
B
$M
D
While the modal propagation constant is always uniquely defined, the modal series impedance and shunt
admittance as well as the modal characteristic impedance are not, because of the ambiguity in the eigenvectors.
Therefore, modal impedances and admittances only make sense if they are specified together with the eigenvectors used
in their calculation. To find them, transform Eq. (4.72a) to modal quantities
&
F8OQFG
FZ
' =6X?& =< ) RJCUG? =6K? =+OQFG?
The triple matrix product in Eq. (4.83) is diagonal, and the modal series impedances are the diagonal elements of this
matrix
=< ) OQFG? ' =6X?& =< ) RJCUG? =6K?
C
=< ) OQFG? ' =6K V? =< ) RJCUG? =6K?
D
or with Eq. (4.80),
Similarly, Eq. (4.72b) can be transformed to modal quantities, and the modal shunt admittances are then the diagonal
elements of the matrix
)
)
=;OQFG? ' =6K?& =;RJCUG? =6X?
C
=; ) OQFG? ' =6X V? =; ) RJCUG? =6X?
D
or with Eq. (4.80),
The proof that both [Z’mode] and [Y’mode] are diagonal is given by Wedepohl [58]. Finally, the modal characteristic
impedance can be found from the scalar equation
< )OQFG&M
<EJCT&OQFG&M '
C
; )OQFG&M
or from the simpler equation
(OQFG&M
<EJCT&OQFG&M '
D
; )OQFG&M
A good way to obtain the modal parameters may be as follows: First, obtain the eigenvalues
and the
k
eigenvector matrix [Tv] of the matrix product [Z’phase][Y’phase ]. Then find [Y’mode ] from Eq. (4.85b), and the modal series
impedance from the scalar equation
< )OQFG&M '
8M
E
; )OQFG&M
The modal characteristic impedance can then be calculated from Eq. (4.86a), or from Eq. (4.86b) if the propagation
constant from Eq. (4.81) is needed as well. If [Ti] is needed, too, it can be found efficiently from Eq. (4.85a)
=6K? ' =; ) RJCUG? =6X? =; ) OQFG?&
E
because the product of the first two matrices is available anyhow when [Y’mode] is found, and the post-multiplication with
[Y’mode]-1 is simply a multiplication of each column with a constant (suggested by Luis Marti). Eq. (4.85c) also
establishes the link to an alternative formula for [Ti] mentioned in [57],
=6K? ' =; ) RJCUG? =6X? =&?
with [D] being an arbitrary diagonal matrix. Setting [D] = [Y’mode]-1 leads us to the desirable condition [Ti] = [Tvt]-1 of
Eq. (4.80). If the unit matrix were used for [D], all modal matrices in Eq. (4.84) and (4.85) would still be diagonal, but
with the strange-looking result that all modal shunt admittances become 1.0 and that the modal series impedances
become k. Eq. (4.80) would, of course, no longer be fulfilled. For a lossless line, the modal series impedance would
then become a negative resistance, and the modal shunt admittance would become a shunt conductance with a value of
1.0 S. As long as the case is solved in the frequency domain, the answers would still be correct, but it would obviously
be wrong to associate such modal parameters with
&
MX
' 4 )K
MZ
CPF
(with R’ negative and G’ = 1.0) in the time domain.
&
MK
' ) )X
MZ
4.1.5.2 Lossless High-Frequency Approximation
In lightning surge studies, many simplifying assumptions are made. For example, the waveshape and amplitude
of the current source representing the lightning stroke is obviously not well known. Similarly, flashover criteria in the
form of volt-time characteristics or integral formulas [8] are only approximate. In view of all these uncertainties, the
use of highly sophisticated line models is not always justified. Experts in the field of lightning surge studies normally
use a simple line model in which all wave speeds are equal to the speed of light, with a surge impedance matrix [Zsurge ]
in phase quantities, where
<KK&UWTIG ' NP JKTK
C
<KM&UWTIG ' NP &KMFKM
D
CNN YCXG URGGFU ' URGGF QH NKIJV
E
with ri being the radius of the conductor, or the radius of the equivalent conductor from Eq. (4.30) in case of a bundle
conductor.14
Typically, each span between towers is represented separately as a line, and only a few spans are normally
modelled (3 for shielded lines, or 18 for unshielded lines in [8]). For such short distances, losses in series resistances
and differences in modal travel times are negligible. The effect of corona is sometimes included, however, by modifying
the simple model of Eq. (4.87) [8].
It is possible to develop a special line model based on Eq. (4.87) for the EMTP, in which all calculations are
done in phase quantities. But as shown here, the simple model of Eq. (4.87) can also be solved with modal parameters
as a special case of the untransposed line. The simple model follows from Eq. (4.72) by making two assumptions for
a "lossless high-frequency approximation":
1.
Conductor resistances and ground return resistances are ignored.
2.
The frequencies contained in the lightning surges are so high that all currents flow on the surface of
the conductors, and on the surface of the earth.
Then the elements of [Z’phase] become
< ) KK ' LT
z
B
NP JKTK < ) KM ' LT
z
B
NP &KMFKM
while [Y’] is obtained by inverting the potential coefficient matrix,
=; )? ' LT =2 )?&
with the elements of [P’] being the same as in Eq. (4.88) if the factor j µ 0/(2 ) is replaced by 1/(2 g0). Then both matrix
)TQWPFYKTGUCTGWUWCNN[TGVCKPGFCURJCUGEQPFWEVQTUKPUWEJUVWFKGU+HVJG[CTGGNKOKPCVGFVJGOGVJQF
QH5GEVKQPOWUVDGWUGFQP=< ?
UWTIG
products in Eq. (4.73) become diagonal matrices with all elements being
8M ' & Tgz
M'/
These values are automatically obtained from the supporting routines LINE CONSTANTS and CABLE
CONSTANTS as the eigenvalues of the matrix products in Eq. (4.73), by simply using the above two assumptions in
the input data (all conductor resistances = 0, GMR/r = 1.0, no Carson correction terms). The calculation of the
eigenvector matrix [Tv] or [Ti] needed for the untransposed line model of Section 4.2 breaks down, however, because
the matrix products in Eq. (4.73) are already diagonal. To obtain [Tv], let us first assume equal, but nonzero conductor
resistances R’. Then the eigenvectors [tvk] are defined by
& Tgz=7? % LT4 )=2 )?& =VXM? ' 8M =VXM?
with the expression in parentheses being the matrix product [Z’phase][Y’phase], and [U] = unit matrix. Eq. (4.91) can be
rewritten as
=2 )?& =VXM? ' 8M&OQFKHKGF =VXM?
with modified eigenvalues
LT 4 ) 8M&OQFKHKGF ' 8M % Tgz
Eq. (4.92) is valid for any value of R’, including zero. It therefore follows that [Tv] is obtained as the eigenvectors of
[P’]-1, or alternatively as the eigenvectors of [P’] since the eigenvectors of a matrix are equal to the eigenvectors of its
inverse. The eigenvalues of [P]-1 are not needed because they are already known from Eq. (4.90), but they could also
be obtained from Eq. (4.93) by setting R’ = 0.
For this simple mode, [Tv] is a real, orthogonal matrix,
=6X? =6X?V ' =7?
=6K? ' =6X?
and therefore,
D.E. Hedman has solved this case of the lossless high-frequency approximation more than 15 years ago [45]. He
recommended that the eigenvectors be calculated from the surge impedance matrix of Eq. (4.87), which is the same as
calculating them from [P’] since both matrices differ only by a constant factor.
One can either modify the line constants supporting routines to find the eigenvectors from [P’] for the lossless
high-frequency approximation, as was done in UBC’s version, or use the same trick employed in Eq. (4.91) in an
unmodified program: Set all conductor resistances equal to some nonzero value R’, set GMR/r = 1, and ask for zero
Carson correction terms. If the eigenvectors are found from [P’], then it is advisable to scale this matrix first by
multiplying all elements with 2 g0.
The lossless high-frequency approximation produces eigenvectors which differ from those of the lossy case
at very high frequencies [61]. This is unimportant for lightning surge studies, but important for power line carrier
problems.
Example: For a distribution line with one ground wire (Fig. 4.24) the lossless high-frequency approximation produces
the following modal surge impedances and transformation matrix,
mode
Zsurge-mode ( )
1
2
3
4
993.44
209.67
360.70
310.62
(KI2QUKVKQPQHRJCUGEQPFWEVQTU#$
%CPFITQWPFYKTG& CXGTCIGJGKIJVCNN
FKOGPUKQPUKPO %QPFWEVQTFKCOGVGT
OO
&
& & & &
Converted to phase quantities, the surge impedance matrix becomes [Tv][Zsurge-mode][Tv]t, or
=<UWTIG&RJCUG? '
U[OOGVTKE
S
The elements from Eq. (4.87) are slightly larger, by a factor of 300,000/299,792, because the supporting routine LINE
CONSTANTS uses 299,792 km/s for the speed of light, versus 300,000 km/s implied in Eq. (4.87).
Representation in EMTP then would be by means of a 4-phase, constant-parameter, lossless line. The following
branch cards are for the first of 4 such cascaded sections:
-11A 2A
-11B 2B
-11C 2C
-11D 2D
993.44 .3E-6
993.44 .3E-6
993.44 .3E-6
993.44 .3E-6
2
2
2
2
4
4
4
4
The modelling of long lines as coupled shunt resistances [R] = [Zsurge-phase] has already been discussed in Section
3.1.3. In the example above, such a shunt resistance matrix could be used to represent the rest of the line after the 4
spans from the substation. Simply using the 4 x 4 matrix would be unrealistic with respect to the ground wire, however,
because it would imply that the ground wire is ungrounded on the rest of the line. More realistic, though not totally
accurate, would be a 3 x 3 matrix obtained from [Z’phase] and [Y’phase] in which the ground wire has been eliminated. This
model implies zero potential everywhere on the ground wire, in contrast to the four spans where the potential will more
or less oscillate around zero because of reflections up and down the towers.
Comparison with More Accurate Models: For EMTP users who are reluctant to use the simple model described in this
section, a few comments are in order. First, let us compare exact values with the approximate values. If we use constant
parameters and choose 400 kHz as a reasonable frequency for lightning surge studies, then we obtain the results of table
4.5 for the test example above, assuming T/D = 0.333 for skin effect correction and internal inductance calculation with
the tubular conductor formula, R’dc = 0.53609 /km, and = 100 m.
Table 4.5 - Exact line parameters at 400 kHz
mode
Zsurge-mode ( )
1
2
3
4
1027.6-j33.9
292.0-j0.5
361.9-j0.5
311.1-j0.5
wave velocity (m/s)
285.35
299.32
299.37
299.32
R’ ( /km)
597.4
7.9
8.2
8.0
The differences are less than 0.5% in surge impedance and wave speed for the aerial modes 2 to 4, and not more than
5% for the ground return mode 1. These are small differences, considering all the other approximations which are made
in lightning surge studies. If series resistances are included by lumping them in 3 places, totally erroneous results may
be obtained if the user forgets to check whether R/4 # Zsurge in the ground return mode. For the very short line length
of 90 m in this example, this condition would still be fulfilled here.
Using constant parameters at a particular frequency is of course an approximation as well, and some users may
therefore prefer frequency-dependent models. For very short line lengths, such as 90 m in the example, most frequencydependent models are probably unreliable, however. It may therefore be more sensible to use the simple model
described here, for which answers are reliable, rather than sophisticated models with possibly unreliable answers.
A somewhat better lossless line model for lightning surge studies than the preceding one has been suggested
by V. Larsen [92]. To obtain this better model, the line parameters are first calculated in the usual way, at a certain
frequency which is typical for lightning surges (e.g., at 400 kHz). The resistances are then set to zero when the matrix
product [Z’phase][Y’phase ] is formed, before the modal parameters are computed. With this approach, [Ti] will always be
real. Table 4.6 shows the modal parameters of this better lossless model. They differ very little from those in Table
4.5.
Table 4.6 - Approximate modal parameters at 400 kHz with R=0
mode
Zsurge-mode ( )
wave velocity (m/s)
1
1026.3
285.50
2
292.0
299.32
3
362.0
299.37
4
311.1
299.32
In particular, the wave velocity of the ground return mode 1 is now much closer to the exact value of Table 4.5. The
transformation matrix which goes with the modal parameters of Table 4.6 is
&
=6K? '
& &
& & &
In this case [Tv] is no longer to [Ti]; Eq. (4.80) must be used instead.
4.1.5.3 Approximate Transformation Matrices for Transient Solutions
The transformation matrices [Tv] and [Ti] are theoretically complex, and frequency-dependent as well. With
a frequency-dependent transformation matrix, modes are only defined at the frequency at which the transformation
matrix was calculated. Then the concept of converting a polyphase line into decoupled single-phase lines (in the modal
domain) cannot be used over the entire frequency range. Since the solution methods for transients are much simpler
if the modal composition is the same for all frequencies, or in other words, if the transformation matrices are constant
with real coefficients, it is worthwhile to check whether such approximate transformation matrices can be used without
producing too much error. Fortunately, this is indeed possible for overhead lines [66, 78].
Guidelines for choosing approximate (real and constant) transformation matrices have not yet been worked out
at the time when these notes are being written. The frequency-dependent line model of J. Marti discussed in Section
4.2.2.6 needs such a real and constant transformation matrix, and wrong answers would be obtained if a complex
transformation matrix were used instead. Since a real and constant transformation matrix is always an approximation,
its use will always produce errors, even if they are small and acceptable. The errors may be small in one particular
frequency region, and larger in other regions, depending on how the approximation is chosen.
One choice for an approximate transformation matrix would be the one used in the lossless approximations
discussed in Section 4.1.5.2. This may be the best choice for lightning surge studies.
For switching surge studies and similar types of studies, the preferred approach at this time seems to be to
calculate [Tv] at a particular frequency (e.g., at 1 kHz), and then to ignore the imaginary part of it. In this approach, [Tv]
should be predominantly real before the imaginary part is discarded. One cannot rely on this when the subroutine
returns the eigenvectors, since an eigenvector multiplied with ej50E or any other constant would still be a proper
eigenvector. Therefore, the columns of [Tv] should be normalized in such a way that its components lie close to the real
axis. One such normalization procedure was discussed by V. Brandwajn [79]. The writer prefers a different approach,
which works as follows:
1.
Ignore shunt conductances, as is customarily done. Then [Y’phase] is purely imaginary. Use Eq. (4.85)
to find the diagonal elements of the modal shunt admittance matrix Y’mode-k-preliminary.
2.
In general, these "preliminary" modal shunt admittances will not be purely imaginary, but j C’mode-kej k
instead. Then normalize [Tv] by multiplying each column with ej
k/2
. With this normalized
transformation matrix, the modal shunt admittances will become j C’mode-k, or purely imaginary as in
the phase domain.
3.
To obtain the approximate (real and constant) transformation matrix, discard the imaginary part of
the normalized matrix from step 2.
4.
Use the approximate matrix [Tv-approx.] from step 3 to find modal series impedances and modal shunt
admittances from Eq. (4.84) and (4.85) over the frequency range of interest. If [Ti] is needed, use
=6K&CRRTQZ? ' =6X&CRRTQZ?&
V
5.
If the line model requires nonzero shunt conductances, add them as modal parameters. Usually, only
conductances from phase to ground are used (with phase-to-phase values being zero); in that case,
the modal conductances are the same as the phase-to-ground conductances if the latter are equal for
all phases.
An interesting method for finding approximate (real and constant) transformation matrices has been suggested
by Paul [66]. By ignoring conductor resistances, and by assuming that the Carson correction terms R’ii + j X’ii in Eq.
(4.7) and R’ik + j X’ik in Eq. (4.8) are all equal (all elements in the matrix of correction terms have one and the same
value), the approximate transformation matrix [Ti-approx.] is obtained as the eigenvectors of the matrix product
=% )
RJCUG?
with all elements of the second matrix being 1. To find [T v-approx.], Eq. (4.96) would have to be used. Wasley and
Selvavinayagamoorthy [93] find the approximate transformation matrices by simply taking the magnitudes of the
complex elements, with an appropriate sign reflecting the values of their arguments. They compared results using these
approximate matrices with the exact results (using complex, frequency-dependent matrices), and report that fairly high
accuracy can be obtained if the approximate matrix is computed at a low frequency, even for the case of double-circuit
lines.
If the M-phase line is assumed to be balanced (Section 4.1.3.2), then the transformation matrix is always real
and constant, and known a priori with Eq. (4.58) and Eq. (4.59). Two identical and balanced three-phase lines with zero
sequence coupling only have the real and constant transformation matrix of Eq. (4.65).
4.2 Line Models in the EMTP
The preceding Section 4.1 concentrated on the line parameters per unit length. These are now used to develop
line models for liens of a specific length.
For steady-state solutions, lines can be modelled with reasonable accuracy as nominal -circuits, or rigorously
as equivalent -circuits. For transient solutions, the methods become more complicated, as one proceeds from the simple
case of a single-phase lossless line with constant parameters to the more realistic case of a lossy polyphase line with
frequency-dependent parameters.
4.2.1 AC Steady-State Solutions
Lines can be represented rigorously in the steady-state solution with exact equivalent -circuits. Less accurate
representations are sometimes used, however, to match the model to the one used in the transient simulation (e.g.,
lumping R in three places, rather than distributing it evenly along the line, or using approximate real transformation
matrices instead of exact complex matrices). For lines of moderate "electrical" length (typically # 100 km at 60 Hz),
nominal -circuits are often accurate enough, and are probably the best models to use for steady-state solutions at power
frequency. If the steady-state solution is followed by a transient simulation, or if steady-state solutions are requested
over a wide frequency range, then the nominal -circuit must either be replaced by a cascade connection of shorter
nominal -circuits, or by an exact equivalent -circuit derived from the distributed parameters.
4.2.1.1 Nominal M-Phase -Circuit
For the nominal M-phase -circuit of Fig. 3.10, the series impedance matrix and the two equal shunt
susceptance matrices are obtained from the per unit length matrices by simply multiplying them with the line length, as
shown in Eq. (4.35) and (4.36). The equations for the coupled lumped elements of this M-phase -circuit have already
been discussed at length in Section 3, and shall not be repeated here.
Nominal -circuits are fairly accurate if the line is electrically short. This is practically always the case if
complicated transposition schemes are studied at power frequency (60 Hz or 50 Hz). Fig. 4.25 shows a typical example,
with three circuits on the same right-of-way. In this case, each of the five transposition sections (1-2, 2-3, 3-4, 4-5, 5-6)
would be represented as a nominal 9-phase -circuit. While a nominal -circuit would already be reasonably accurate
for the total line length of 95 km, nominal -circuits are certainly accurate for each transposition section, since the
longest section is only 35 km long. A comparison between measurements on the de-energized line L3 and computer
results is shown in table 4.7 [80]. The coupling in this case is predominantly capacitive.
(KI6TCPURQUKVKQPUEJGOGHQTVJTGGCFLCEGPVEKTEWKVU
Table 4.7 - Comparison between measurements and EMTP results (voltages on energized line L1 =
372 kV and on L2 = 535 kV)
phase
measurement
EMTP results
Induced voltages on de-energized line L3 if open at
both ends
A
B
C
30 kV
15 kV
10 kV
27.5 kV
13.8 kV
7.8 kV
Grounding currents if de-energized line L3 is
grounded at right end
A
B
C
11 A
5A
1A
10.5 A
3.2 A
1.5 A
Because nominal -circuits are so useful for studying complicated transposition schemes, a "CASCADED PI"
option was added to the BPA EMTP. With this option, the entire cascade connection is converted to one single circuit, which is an exact equivalent for the cascade connection. This is done by adding one "component" at a time, as
shown in Fig. 4.26. The "component" may either be an M-phase -circuit, or other types of network elements such as
shunt reactors or series capacitors. Whenever component k is added, the nodal admittance matrix
(KI5EJGOCVKEKNNWUVTCVKQPQHECUECFKPIQRGTCVKQPHQT-VJEQORQPGPV
for nodes 1, 2, 3 is reduced by eliminating the inner nodes 2, to form the new admittance matrix of the equivalent for
the cascaded components 1, 2, ... K. This option keeps the computational effort in the steady-state solution as low as
possible by not having to use nodal equations for the inner nodes of the cascade connection, at the expense of extra
computational effort for the cascading procedure.
4.2.1.2 Equivalent -Circuit for Single-Phase Lines
Lines defined with distributed parameters at input time are always converted to equivalent -circuits for the
steady-state solution.
For lines with frequency-dependent parameters, the exact equivalent -circuit discussed in Section 1 is used,
with Eq. (1.14) and (1.15). The same exact equivalent -circuit is used for distortionless and lossless line models with
constant parameters.
In many applications, line models with constant parameters are accurate enough. For example, positive
sequence resistances and inductances are fairly constant up to approximately 1 kHz, as shown in Fig. 4.20. But even
with constant parameters, the solution for transients becomes very complicated (except for the unrealistic assumption
of distortionless propagation). Fortunately, experience showed that reasonable accuracy can be obtained if L’ and C’
are distributed and if
4 ' 4 )ý
is lumped in a few places as long as R << Zsurge. In the EMTP, R/2 is lumped in the middle and R/4 at both ends of an
otherwise lossless line, as shown in Fig. 4.27, and as further discussed in Section 4.2.2.5. For this transient
representation, the EMTP uses the same assumption15 in the
6JG'/62UJQWNFRTQDCDN[DGEJCPIGFVQD[RCUUVJKUQRVKQPKHQPN[UVGCF[UVCVGUQNWVKQPUCTGTGSWGUVGF
GKVJGTCVQPGHTGSWGPE[QTQXGTCTCPIGQHHTGSWGPEKGU
(KI.KPGTGRTGUGPVCVKQPYKVJNWORGFTGUKUVCPEGU
steady-state solution, to avoid any discrepancies between ac steady-state initialization and subsequent transient
simulation, even though experiments have shown that the differences are extremely small at power frequency. By using
equivalent -circuits for each lossless, half-length section in Fig. 4.27, and by eliminating the "inner" nodes 1, 2, 3, 4,
an equivalent -circuit (Fig. 1.2) was obtained by R.M. Hasibar with
<UGTKGU ' 4EQUTJ & % ;
'
UJWPV
& & 4
<
4 UKPTJ % L UKPTJ EQUTJ@
4
% <
<
4
4
UKPTJ % L UKPTJ EQUTJ
<
<
<UGTKGU
where
J ' NGPIVJ . )% )
< '
.)
%)
4 ' NGPIVJ @ 4 )
4.2.1.3 Equivalent M-Phase -Circuit
To obtain an equivalent M-phase -circuit, the phase quantities are first transformed to modal quantities with
Eq. (4.84) and (4.85) for untransposed lines, or with Eq. (4.58) and (4.59) for balanced lines. For identical balanced
three-phase lines with zero sequence coupling only, Eq. (4.65) is used. For each mode, an equivalent single-phase circuit is then found in the same way as for single-phase lines; that is, either as an exact equivalent -circuit with Eq.
(1.14) and (1.15), or with Eq. (4.98) and (4.99) for the case of lumping R in three places. These single-phase modal
-circuits each has a series admittance Yseries-mode and two equal shunt admittances 1/2 Yshunt-mode. By assembling these
admittances as diagonal matrices, the admittance matrices of the M-phase -circuit in phase quantities are obtained from
=;UGTKGU? ' =6K? =;UGTKGU&OQFG? =6K?V
and
=;UJWPV? ' =6K? =;UJWPV&OQFG? =6K?V
While it is always possible to obtain the exact equivalent M-phase -circuit at any frequency in this way,
approximations are sometimes used to match the representation for the steady-state solution to the one used in the
transient solution. One such approximation is the lumping of resistances as shown in Fig. 4.27. Another approximation
is the use of real and constant transformation matrices in Eq. (4.100) and (4.101), as discussed in Section 4.1.5.3.
4.2.2 Transient Solutions
Historically, the first line models in the EMTP were cascade connections of -circuits, partly to prove that
computers could match switching surge study results obtained on transient network analyzers (TNA’s) at that time. On
TNA’s, balanced three-phase lines are usually represented with decoupled 4-conductor -circuits, as shown in Fig. 4.28.
This representation can easily be derived from Eq. (4.44) by rewriting it as
&
F8#
FZ
' < ) U & < ) O +# % < ) O +# % +$ % +%
(KI(QWTEQPFWEVQTBEKTEWKVWUGFQP
60# U
for phase A, and similar for phases B and C. The first term in Eq. (4.102) is Z’posIA (or branch A1-A2 in Fig. 4.28),
while the second term is the common voltage drop caused by the earth and ground wire return current IA + IB + IC
(branch N1-N2 in Fig. 4.28). Note, however, that Fig. 4.28 is only valid if the sum of the currents flowing out through
a line returns through the earth and ground wires of that same line. For that reason, the neutral nodes N2, N3, ... must
be kept floating, and only N1 at the sending end is grounded. Voltages with respect to ground at location i are obtained
by measuring between the phase and node Ni. In meshed networks with different R/X-ratios, this assumption is probably
not true. For this reason, and to be able to handle balanced as well as untransposed lines with any number of phases,
M-phase -circuits were modelled directly with M x M matrices, as discussed in Section 4.1.2.4. Voltages to ground
are then simply the node voltages. Comparisons between these M-phase -circuits, and with the four-conductor circuits of Fig. 4.28 confirmed that the results are identical.
The need for travelling wave solutions first arose in connection with rather simple lightning arrester studies,
where lossless single-phase line models seemed to be adequate. Section 1 briefly discusses the solution method used
in the EMTP for such lines. This method was already known in the 1920’s and 1930’s and strongly advocated by
Bergeron [81]; it is therefore often called Bergeron’s method. In the mathematical literature, it is known as the method
of characteristics, supposedly first described by Riemann.
It soon became apparent that travelling wave solutions were much faster and better suited for computers than
cascaded -circuits. To make the travelling wave solutions useful for switching surge studies, two changes were needed
from the simple single-phase lossless line: First, losses had to be included, which could be done with reasonable
accuracy by simply lumping R in three places. Secondly, the method had to be extended to M-phase lines, which was
achieved by transforming phase quantities to modal quantities. Originally, this was limited to balanced lines with builtin transformation matrices, then extended to double-circuit lines, and finally generalized to untransposed lines. Fig. 4.29
compared EMTP results with results obtained on TNA’s, using the built-in transformation matrix for balanced threephase lines and simply lumping R in three places.
Fig. 4.29 - Energization of a three phase line. Computer simulation results (dotted line) superimposed on 8 transient
network analyzer results for receiving end voltage in phase B. Breaker contacts close at 3.05 ms in phase A, 8,05 ms
in phase B, and 5.55 ms in phase C (t=0 when source voltage of phase A goes through zero from negative to positive)
[82]. Reprinted by permission of CIGRE
While travelling wave solutions with constant distributed L’, C’ and constant lumped R produced reasonable
accurate answers in many cases, as shown in Fig. 4.29, there were also cases where the frequency dependence,
especially of the zero sequence impedance, could not be ignored. Choosing constant line parameters at the dominant
resonance frequency sometimes improved the results. Eventually, frequency-dependent line models were developed
by Budner [83], by Meyer and Dommel [84] based on work of Snelson [85], by Semlyen [86], and by Ametani [87].
A careful re-evaluation of frequency-dependence by J. Marti [88] led to a fairly reliable solution method, which seems
to become the preferred option as these notes are being written. J. Marti’s method will therefore be discussed in more
detail.
4.2.2.1 Nominal -Circuits
Nominal -circuits are generally not the best choice for transient solutions, because travelling wave solutions
are faster and usually more accurate. Cascade connections of nominal -circuits may be useful for untransposed lines,
however, because one does not have to make the approximations for the transformation matrix discussed in Section
4.1.5.3. On the other hand, one cannot represent frequency-dependent line parameters and one has to accept the
spurious oscillations caused by the lumpiness. Fig. 4.30 shows these oscillations for the simple case of a single-phase
line being represented with 8 and 32 cascaded nominal -circuits. The exact solution with distributed parameters is
shown for comparison purposes as well. The proper choice of the number of -circuits for one line is discussed in [89],
as well as techniques for damping the spurious oscillations with damping resistances in parallel with the series R-L
branches of the -circuit of Fig. 4.28.
Fig. 4.30 - Voltage at receiving end of a single phase line if a dc voltage of 10 V is connected to the sending end at t=0
(line data: R=0.0376 /mile, L=1.52 mH/mile, C=14.3 nF/mile, length-320 miles; receiving end terminated with shunt
inductance of 100 mH)
The solution methods for nominal -circuits have already been discussed in Section 3.4. With M-phase
nominal -circuits, untransposed lines (or sections of a line) are as simple to represent as balanced lines. In the former
case, one simply uses the matrices of the untransposed line, whereas in the latter case one would use matrices with
averaged equal diagonal and averaged equal off-diagonal elements.
4.2.2.2 Single-Phase Lossless Line with Constant L’ and C’
The solution method for the single-phase case has already been explained in Section 1. The storage scheme
for the history terms is the same as the one discussed in the next Section 4.2.2.3 for M-phase lossless lines, except that
each single-phase line occupies only one section in the table, rather than M section for M modes. Similarly, the
initialization of the history terms for cases starting from linear ac steady-state initial conditions is the same as in Eq.
(4.108).
(KI.KPGCTKPVGTRQNCVKQP
(KI'HHGEVQHNKPGCTKPVGTRQNCVKQPQPUJCTRRGCMU&QVVGFNKPG)V
zUVQOCMGJKPVGIGTOWNVKRNGQH)V5QNKFNKPG)VzU JPQVKPVGIGT
OWNVKRNGQH)V
The solution is exact as long as the travel time is an integer multiple of the step size t. If this is not the case,
then linear interpolation is used in the EMTP, as indicated in Fig. 4.31. Linear interpolation is believed to be a
reasonable approximation for most cases, since the curves are usually smooth rather than discontinuous. If
discontinuities or very sharp peaks do exist, then rounding to the nearest integer multiple of t may be more sensible
than interpolation, however. There is no option for this rounding procedure in the EMTP, but the user can easily
accomplish this through changes in the input data. Fig. 4.32 compares results for the case of Fig. 4.30 with sharp peaks
with and without linear interpolation. The line was actually not lossless in this case, but the losses were represented
in a simple way by subdividing the line into 64 lossless sections and lumping resistances in between and at both ends.
The interpolation errors are more severe if lines are split up into many sections, as was done here. If the line were only
split up into two lossless sections, with R lumped in between and at both ends, then the errors in the peaks would be
less (the first peak would be 18.8, and the second peak would be -15.4).
The accumulation of interpolation errors on a line broken up into many sections, with of each section not
being an integer multiple of t, can easily be explained. Assume that a triangular pulse is switched onto a long, lossless
line, which is long enough so that no reflections come back from the remote end during the time span of the study (Fig.
4.33). Let us look at how this pulse becomes distorted through interpolation as it travels down the line if
(a)
the line is broken up into short sections of travel time 1.5 t each, and
(b)
the line from the sending end to the measuring point is represented as one section ( = k @ 1.5 t, with k = 1,
2, 3,...).
(KI5KPINGRJCUGNQUUNGUUNKPGGPGTIK\GFYKVJVTKCPIWNCT
RWNUG
At any point on the line, the current will be
K '
X
<
and between points 1 and 2 separated by (Fig. 4.33),
X V % J ' X V
This last equation was used in Fig. 4.34, together with linear interpolation, to find the shape of the pulse as it travels
down the line. The pulse loses its amplitude and becomes wider and wider if it is broken up into sections of travel time
1.5 t each. On the other hand, the pulse shape never becomes as badly distorted if the line is represented as one single
section.
What are the practical consequences of this interpolation error? Table 4.8 compares peak overvoltages from
a BPA switching surge study on a 1200 kV three-phase line16, 133 miles long. Each section was split up into two
lossless half-sections, with R lumped in the middle and
Table 4.8 - Interpolation errors in switching surge study with t = 50 µs
Peak overvoltages (MV)
Run
Line model
1
2
3
phase A
phase B
phase C
single section
1.311
1.191
1.496
7 sections
1.276
1.136
1.457
1.342
1.167
1.489
single section with
rounded
at both ends, as explained in Section 4.2.2.4. Run no. 1 shows the results of the normal line representation as one
section. Run no. 2 with subdivision into 7 sections produces differences of 2.6 to 4.7%. In run no. 3 the zero and
positive sequence travel times
0
= 664.93 µs and
1
= 445.74 µs were rounded to 650 and 450 µs, respectively, to make
them integer multiples of t = 50 µs. These changes could be interpreted as a decrease in both L' 0 and C' 0 of 2.25%,
and as an increase in both L'1 and C'1 of 0.96%, with the surge impedances remaining unchanged. Since line parameters
are probably no more accurate than ±5% at best anyhow, these implied changes are quite acceptable. With rounding,
a slightly modified case is then solved without interpolation errors. Whether an option for rounding to the nearest
integer multiple of t should be added to the EMTP is debatable. In general, rounding may imply much larger changes
in L', C' than in this case, and if implemented, warning messages with the magnitude of these implied changes should
be added as well. In Table 4.8, runs no. 3 to 1 differ by no more than 2.3%, and the interpolation error is therefore
acceptable if the line is represented as one section. Breaking the line up into very many sections may produce
unacceptable interpolation errors, however.
If the user is interested in a "voltage profile" along the line, then a better alternative to subdivisions into
sections would be a post-processor "profile program" which would calculate
6JGRTQDNGOQHKPVGTRQNCVKQPGTTQTUKUDCUKECNN[VJGUCOGHQTUKPINGRJCUGCPF/RJCUGNKPGUVJGTGHQTGC
VJTGGRJCUGECUGKURTGUGPVGFJGTGHQTYJKEJFCVCYCUCNTGCF[CXCKNCDNG%JQQUKPICUVGRUK\G)VYJKEJOCMGU
VJGVTCXGNVKOGJCPKPVGIGTOWNVKRNGQH)VKUOQTGFKHHKEWNVHQTVJTGGRJCUGNKPGUJQYGXGTDGECWUGVJGTGCTGVYQ
VTCXGNVKOGUHQTVJGRQUKVKXGCPF\GTQUGSWGPEGOQFGQPDCNCPEGFNKPGU QTVJTGGVTCXGNVKOGUHQTVJGOQFGUQP
WPVTCPURQUGFNKPGU (KI2WNUGCVKPETGOGPVCNFKUVCPEGUFQYPVJGNKPG
voltages and currents at intermediate points along the line from the results at both ends of the line. Such a program is
easy to write for lossless and distortionless lines. Luis Marti developed such a profile algorithm for the more
complicated frequency-dependent line models, which he merged into the time step loop of the EMTP [90]. This was
used to produce moves of travelling waves by displaying the voltage profile at numerous points along the line at time
intervals of t.
Fig. 4.34(a) suggests a digital filtering effect from the interpolation which is similar to that of the trapezoidal
rule described in Section 2.2.1. To explain this effect, Eq. (1.6) must first be transformed from the time domain
X V & KMO V ' XO V&J % KOM V&J
< M
<
into the frequency domain,
+ '
8 & +MO '
< M
8 % +OM
< O
@ G &LTJ
For simplicity, let us assume that voltage and current phasors Vm and Imk at node m are known, and that we want to find
I = Vk/Z - Ikm at node k. Without interpolation errors, Eq. (4.103) provides the answer. If interpolation is used, and if
for the sake of simplicity we assume that the interpolated value lies in the middle of an interval t, then Eq. (4.103)
becomes
)V
+KPVGTRQNCVGF '
&LT J%
&LT J&
8O % +OM @ @ G
%G
<
)V
Therefore, the ratio of the interpolated to the exact value becomes
+KPVGTRQNCVGF
+GZCEV
' EQU T
)V
which is indeed somewhat similar to Fig. 2.10 for the error produced by the trapezoidal rule.
Single-phase lossless line models can obviously only approximate the complicated phenomena on real lines.
Nonetheless, they are useful in a number of applications, for example
(a)
in simple studies where one wants to gain insight into the basic phenomena,
(b)
in lightning surge studies, and
(c)
as a basis for more sophisticated models discussed later.
For lightning surge studies, single-phase lossless line models have been used for a long time. They are
probably accurate enough in many cases because of the following reasons:
(1)
Only the phase being struck by lightning must be analyzed, because the voltages induced in the other
phases will be much lower.
(2)
Assumptions about the lightning stroke are by necessity very crude, and very refined line models are
therefore not warranted.
(3)
The risk of insulation failure in substations is highest for backflashovers at a distance of approx. 2
km or less. Insulation co-ordination studies are therefore usually made for nearby strokes. In that
case, the modal waves of an M-phase line "stay together," because differences in wave velocity and
distortion among the M waves are still small over such short distances. They can then easily be
combined into one resultant wave on the struck phase. There seems to be some uncertainty, however,
about the value of the surge impedance which should be used in such simplified single-phase
representations. It appears that the "self surge impedance" Zii-surge of Eq. (4.87a) should be used. For
nearby strokes it is also permissible to ignore the series resistance. Attenuation caused by corona may
be more important than that caused by conductor losses. At the time of writing these notes, corona
is still difficult to model, and it may therefore be best to ignore losses altogether to be on the safe side.
4.2.2.3 M-Phase Lossless Line with Constant L’ and C’
Additional explanations are needed for extending the method of Section 1 from single-phase lines to M-phase
lines. In principle, the equations are first written down in the modal domain, where the coupled M-phase line appears
as if it consisted of M single-phase lines. Since the solution for single-phase lines is already known, this is
straightforward. For solving the line equations together with the rest of the network, which is always defined in phase
quantities, these modal equations must then be transformed to phase quantities, as schematically indicated in Fig. 4.35.
(KI6TCPUHQTOCVKQPDGVYGGPRJCUGCPFOQFCNFQOCKPQPCVJTGGRJCUGNKPG
For simplicity, let us assume that the line has 3 phases. Then, with the notations from Fig. 4.35, each mode
is described by an equation of the form of Eq. (1.6), or
X V % JKUVC&C V&JC
<C C
V % JKUVD&D V&JD
KC&C V '
KD&D V '
X
<D D
KE&E V '
X V % JKUVE&E V&JE
<E E
where each history term hist was computed and stored earlier. For mode a, this history term would be
JKUVC&C V&JC ' &
X V&JC & KC&C V&JC
<C C
and analogous for modes b and c. These history terms are calculated for both ends of the line as soon as the solution
has been obtained at instant t, and entered into a table for use at a later time step. As indicated in Fig. 4.36, the history
terms of a three-phase line would occupy 3 sections of the history tables for modes a, b, c, and the length of each section
would be
/ t, with
increased
increased
being the travel time of the particular mode increased to the nearest integer multiple
of t17. Since the modal travel times a, b,
c
differ from each other, the 3 sections in this table are generally of different
length. This is also the reason for storing history terms as modal values, because one has to go back different travel
times for each mode in picking up history terms. For the solution at time t, the history terms of Eq. (4.106) are obtained
by using linear interpolation on the top two entries of each mode section.
#UKPINGRJCUGNKPGYQWNFUKORN[QEEWR[QPGUGEVKQPYJGTGCUCUKZRJCUGNKPGYQWNFQEEWR[UKZUGEVKQPUKP
VJKUVCDNG
(KI6CDNGHQTJKUVQT[VGTOUQHVTCPUOKUUKQPNKPGU
After
the
solution in each time step, the entries in the tables of Fig. 4.36 must be shifted upwards by one location, thereby
throwing away the values at the oldest point at t-
. This is then followed by entering the newly calculated history
increased
terms hist(t) at the newest point t. Instead of physically shifting values, the EMTP moves the pointer for the starting
address of each section down by 1 location. When this pointer reaches the end of the table, it then goes back again to
the beginning of the table ("wrap-around table") [91].
The initial values for the history terms must be known for t = 0, - t, -2 t, ... -
. The necessity for
increased
knowing them beyond t = 0 comes from the fact that only terminal conditions are recorded. If the conditions were also
given along the line at travel time increments of t, then the initial values at t = 0 would suffice. For zero initial
conditions, the history table is simply preset to zero. For linear ac steady-state conditions (at one frequency ), the
history terms are first computed as phasors (peak, not rms),
*+56MO ' & 8O & +OM
<
C
where Vm and Imk are the voltage and current phasors at line end m (analogous for HISTmk ). With HIST = *HIST* · ej ,
the instantaneous history terms are then
JKUVMO V ' **+56MO* @ EQU TV % " YKVJ V' &)V &)V (4.108b)
Eq. (4.108) is used for single-phase lines as well as for M-phase lines, except that mode rather than phase quantities
must be used in the latter case.
Eq. (4.106) are interfaced with the rest of the network by transforming them from modal to phase quantities
with Eq. (4.78a),
RJCUG
K&
RJCUG
' ;UWTIG X
RJCUG
% JKUV&
C
with the surge admittance matrix in phase quantities,
&
<C
;UWTIG ' 6K <D
&
6K V
D
&
<E
and the history terms in phase quantities,
JKUVC&C
RJCUG
JKUV&
' 6K JKUVD&D
E
JKUVE&E
For a lossless line with constant L' and C', the transformation matrix [Ti] will always be real, as explained in the last
paragraph of Section 4.1.5.2. It is found as the eigenvector matrix of the product [C'][L'] for each particular tower
configuration, where [L'] and [C'] are the per unit length series inductance and shunt capacitance matrices of the line.
For balanced lines, [Ti] is known a priori from Eq. (4.58), and for identical balanced three-phase lines with zero
sequence coupling only it is known a priori from Eq. (4.65).
The inclusion of Eq. (4.109) into the system of nodal equations (1.8a) for the entire network is quite
straightforward. Assume that for the example of Fig. 4.35, rows and columns for nodes 1A, 1B, 1C follow each other,
as do those for nodes 2A, 2B, 2C (Fig. 4.37). Then the 3 x 3 matrix [Ysurge] enters into two 3 x 3 blocks on the diagonal,
as indicated in Fig. 4.37, while the history terms [hist1-2phase] = [hist1A-2A, hist1B-2B , hist1C-2C ] of Eq. (4.109c) enter into rows
1A, 1B, 1C, on the right-hand side with negative signs. Analogous history terms for terminal 2 enter into rows 2A, 2B,
2C on the right-hand side. While [Ysurge] is entered into [G] only once outside the time-step loop, the history terms must
be added to the right-hand side in each time step.
(KI'PVTKGUHQTCVJTGGRJCUGNKPGKPVQU[UVGOQHGSWCVKQPU
M-phase lossless line models are useful, among other things, for
(a)
simple studies where one wants to investigate basic phenomena,
(b)
in lightning surge studies, where single-phase models are no longer adequate, and
(c)
as a basis for more sophisticated models discussed later.
Lightning surge studies cannot always be done with single-phase models. For simulating backflashovers on
lines with ground wires, for example, the ground wire and at least the struck phase must be modelled ("2-phase line").
Since it is not always known which phase will be struck by the backflashover, it is probably best to model all three
phases in such a situation ("4-phase line"). An example for such a study is discussed in Section 4.1.5.2, with 4-phase
lossless line models representing the distribution line, and single-phase lossless line models representing the towers.
Not included in the data listing are switches (or some other elements) for the simulation of potential flashovers from
the tower top (nodes D) to phases A, B, C.
4.2.2.4 Single and M-Phase Distortionless Lines with Constant Parameters
Distortionless line models are seldom used, because wave propagation on power transmission lines is far from
distortionless. They have been implemented in the EMTP, nonetheless, simply because it takes only a minor
modification to change the lossless line equation into the distortionless line equation.
A single-phase transmission line, or a mode of an M-phase line, is distortionless if
4)
))
'
.)
%)
Losses are incurred in the series resistance R’ as well as in the shunt conductance G’. The real shunt conductance of
an overhead line is very small (close to zero), however. If its value must be artificially increased to make the line
distortionless, with a resulting increase in shunt losses, then it is best to compensate for that by reducing the series
resistance losses. The EMTP does this automatically by regarding the input value R’INPUT as an indicator for the total
losses, and uses only half of it for R’,
)
4)
))
4 +0276
'
'
.)
.)
%)
With this formula, the ac steady-state results are practically identical for the line being modelled as distortionless or with
R lumped in 3 places; the transient response differs mainly in the initial rate of rise. From Eq. (4.111), the attenuation
constant
becomes
" '
4 ) +0276
%)
.)
The factor 1/2 can also be justified by using an approximate expression for the attenuation constant for lines with low
attenuation and low distortion [48, p. 257],
" '
4 ) +0276
) ) +0276
%)
%
.)
.)
%)
which is reasonably accurate if R’ << L’ and G’ << C’. This condition is fulfilled on overhead lines, except at very
low frequencies. Eq. (4.112) is then obtained by dropping the term with G’INPUT and by ignoring the fact that the waves
are not only attenuated but distorted as well.
If a user wants to represent a truly distortionless line where G’ is indeed nonzero, then the factor 1/2 should
of course not be used. The factor 1/2 is built into the EMTP, however, and the user must therefore specify R’INPUT twice
as large as the true series resistance in this case.
With known, an attenuation factor e- ý is calculated (ý = length of line). The lossless line of Eq. (1.6) is then
changed into a distortionless line by simply multiplying the history term of Eq. (1.6b) with this attenuation factor,
JKUVMO V&J ' &
X V&J & KOM V&J @ G &"ý
< O
The surge impedance remains the same, namely %L’/C’.
For M-phase lines, any of the M modes can be specified as distortionless. Mixing is allowed (e.g., mode 1
could be modelled with lumped resistances, and modes 2 and 3 as distortionless).
Better results are usually obtained with the lumped resistance model described next, even though lumping of
resistances in a few places is obviously an approximation, whereas the distortionless line is solved exactly if the travel
time is an integer multiple of t.
4.2.2.5 Single and M-Phase Lines with Lumped Resistances
Experience has shown that a lossy line with series resistance R’ and negligible shunt conductance can be
modelled with reasonable accuracy as one or more sections of lossless lines with lumped resistances in between. The
simplest such approach is one lossless line with two lumped resistances R/2 at both ends. The equation for this model
is easily derived from the cascade connection of R/2 - lossless line - R/2, and leads to a form which is identical with that
of Eq. (1.6),
KMO V '
X V % JKUVMO V&J
<OQFKHKGF M
except that the values for the surge impedance and history terms are slightly modified. With Z, R and calculated from
Eq. (4.99),
<OQFKHKGF ' < %
4
and
JKUVMO V&J ' &
<OQFKHKGF
XO V&J % <&
4
K V&J
OM
This model with R/2 at both ends is not used in the EMTP. Instead, the EMTP goes one step further and lumps
resistances in 3 places, namely R/4 at both ends and R/2 in the middle, as shown in Fig. 4.27. This approach was taken
because the form of the equation still remains the same as in Eq. (4.115), except that
<OQFKHKGF ' < %
4
now. The history term becomes more complicated18, and contains conditions from both ends of the line at t - ,
JKUVMO V&J ' &
<
<OQFKHKGF
XO V&J % <&
4
K V&J
OM
6JGGSWCVKQPCVVJGDQVVQOQHRNGHVEQNWOPKP=?EQPVCKPUCPGTTQT+ CPF+ UJQWNFPQVDG
EQORWVGFHTQO'S D KPUVGCFWUG+ < G VJ JK VJ YKVJVJGPQVCVKQPQH=?YJGTG<KU
<
QH'S (QT+ GZEJCPIGUWDUETKRVUMCPFO
M
M
OQFKHKGF
M
MO
O
O
&
4
<OQFKHKGF
XM V&J % <&
4
K V&J
MO
Users who want to lump resistances in more than 3 places can do so with the built-in three-resistance model,
by simply subdividing the line into shorter segments in the input data. For example, 32 segments would produce lumped
resistances in 65 places. Interestingly enough, the results do not change much if the number of lumped resistances is
increased as long as R << Z. For example, results in Fig. 4.30 for the distributed-parameter case were practically
identical for lumped resistances in 3, 65, or 301 places. Fig. 4.29 shows as well that TNA results are closely matched
with R lumped in 3 places only.
One word of caution is in order, however. The lumped resistance model gives reasonable answers only if R/4
<< Z, and should therefore not be used if the resistance is high. High resistances do appear in lightning surge studies
if the parameters are calculated at a high frequency, e.g., at 400 kHz in Table 4.5, where R’ = 597.4 /km in the zero
sequence mode. Lumping R in 3 places would still be reasonable in the case discussed there where each tower span of
90 m is modelled as one line, since 13.4
is still reasonably small compared with Z = 1028 . If it were used to model
a longer line, say 90 km, then R/4 = 13,400 , which would produce totally erroneous results19. In such a situation it
might be best to ignore R altogether, or to use the frequency-dependent option if higher accuracy is required.
For M-phase lines, any of the M modes can be specified with lumped resistances. Mixing is allowed (e.g.,
mode 1 could be modelled with lumped resistances, and modes 2, ... M as distortionless). The lumped resistances do
not appear explicitly as branches, but are built into Eq. (4.115) (4.116) and (4.117) for each mode. Should a user want
to add them explicitly as branches, e.g., for testing purposes, then they would have to be specified as M x M - matrices
[R] in phase quantities, which could easily be done with the M-phase nominal -circuit input option by setting L =
0 and C = 0. All modes would have to use the lumped resistance model in this set-up, that is, mixing of models would
not be allowed in it.
4.2.2.6 Single and M-Phase Lines with Frequency-Dependent Parameters
The two important parameters for wave propagation are the characteristic impedance
<E '
4 ) % LT. )
) ) % LT% )
and the propagation constant
( '
4 ) % LT. ) ) ) % LT% )
Both parameters are functions of frequency, even for constant distributed parameters R’, L’, G’, C’ (except for lossless
and distortionless lines). The line model with frequency-dependent parameters can handle this case of constant
6JG7$%XGTUKQPQHVJG'/62UVQRUYKVJCPGTTQTOGUUCIGKH4 <+VYQWNFDGCFXKUCDNGVQCFFC
YCTPKPIOGUUCIGCUYGNNCUUQQPCU4IGVUHCKTN[NCTIG GI < distributed parameters20, even though it has primarily been developed for frequency-dependent series impedance
parameters R’( ) and L’( ). This frequency-dependence of the resistance and inductance is most pronounced in the zero
sequence mode, as seen in Fig. 4.20. Frequency-dependent line models are therefore important for types of transients
which contain appreciable zero sequence voltages and currents. One such type is the single line-to-ground fault.
To develop a line model with frequency-dependent parameters which fits nicely into the EMTP, it is best to
use an approach which retains the basic idea behind Bergeron’s method. Let us therefore look at what the expression
v + Zi used by Bergeron looks like now, as one travels down the line. Since the parameters are given as functions of
frequency, this expression must first be derived in the frequency domain. At any frequency, the exact ac steady-state
solution is described by the equivalent -circuit of Eq. (1.13), or in an input-output relationship form more convenient
here,
8M
+MO
EQUJ (ý
' UKPJ (ý
<
<EUKPJ (ý
EQUJ (ý
8O
&+OM
which can be found in any textbook on transmission lines. Assume that we want to travel with the wave from node m
to node k. Then the expression V + ZcI is obtained by subtracting Zc times the second row from the first row in Eq.
(4.120),
8M & <E +MO ' 8O % <E +OM @ G &(ý
C
+MO ' 8M<E & 8O<E % +OM @ G &(ý
D
or rewritten as
with a negative sign on Ikm since its direction is opposite to the travel direction. Eq. (4.121) is very similar to Bergeron’s
method; the expression V + ZcI encountered when leaving node m, after having been multiplied with a propagation
factor e- ý, the same when arriving at node k. This is very similar to Bergeron’s equation for the distortionless line,
except that the factor is e- ý there, and that Eq. (4.121) is in the frequency domain here rather than in the time domain.
Before proceeding further, it may be worthwhile to look at the relationship between the equations in the
frequency and time domain for the simple case of a lossless line. In that case,
<E '
.)
%)
( ' LT . )% ) CPF G &(ý . G &LTJ
Anybody familiar with Fourier transformation methods for transforming an equation from the frequency into the time
domain will recall that a phase sift of e-j in the frequency domain will become a time delay in the time domain.
Furthermore, Zc is now just a constant (independent of frequency), and Eq. (4.121) therefore transforms to
6JKUECUGFKHHGTUHTQOVJGNKPGYKVJNWORGFTGUKUVCPEGUKPCUOWEJCUVJGTGUKUVCPEGDGEQOGUVTWN[FKUVTKDWVGF
PQY
XM V & <E KMO V ' XO V&J % <E KOM V&J
which is indeed Bergeron’s equation (1.6).
For the general lossy case, the propagation factor
# T ' G &(ý ' G &"ý @ G &L$ý
with = + j , contains an attenuation factor e- ý as well as a phase shift e-j ý, which are both functions of frequency.
To explain its physical meaning, let us connect a voltage source Vsource to the sending end m through a source impedance
which is equal to Zc( ), to avoid reflections in m (Fig. 4.38). In that case, V m + Z cI mk = V source. Furthermore, let us
assume that the receiving end k is open. Then from Eq. (4.121),
8M ' 8UQWTEG @ # T
(KI8QNVCIGUQWTEGEQPPGEVGFVQGPFOVJTQWIJOCVEJKPI
KORGFCPEG
that is, the propagation factor is the ratio (receiving end voltage) / (source voltage) of an open-ended line if the line is
fed through a matching impedance Zc( ) to avoid reflections at the sending end21. If Vsource = 1.0 at all frequencies from
dc to infinity, then its time domain transform vsource(t) would be a unit impulse (infinitely high spike which is infinitely
narrow with an area of 1.0), and the integral of vsource(t) would be a unit step. Setting Vsource = 1.0 in Eq. (4.122) shows
that A( ) transformed to the time domain must be the impulse which arrives at the other end k, if the source is a unit
impulse. This response to the unit impulse,
C V ' KPXGTUG (QWTKGT VTCPUHQTO QH 6# T >
will be attenuated (no longer infinitely high), and distorted (no longer infinitely narrow). Fig. 4.39 shows these
responses for a typical 500 kV line of 100 miles length. They were obtained
1PGEQWNFCNUQEQPPGEVCOCVEJKPIKORGFCPEG< T HTQOPQFGMVQITQWPFVQCXQKFTGHNGEVKQPUCVVJG
TGEGKXKPIGPFCUYGNN+PVJCVECUGVJGNGHVJCPFUKFGQH'S DGEQOGU8 TCVJGTVJCP8 0QVGVJCVVJG
TCVKQG (ýUVCTVUHTQOCPFDGEQOGUNGUUVJCPCUVJGNKPGNGPIVJ QTHTGSWGPE[ KUKPETGCUGF6JKUKUKP
EQPVTCUVVQVJGQRGPEKTEWKVTGURQPUG8 8 EQUJ (ý OQTGHCOKNKCTVQRQYGTGPIKPGGTUYJKEJKPETGCUGU
YKVJNGPIVJQTHTGSWGPE[ (GTTCPVKTKUG E
M
M
O
M
(a)
zero sequence mode
(b)
positive sequence
mode with same
scale as (a)
(c)
positive sequence
mode with
expanded scale
Fig. 4.39 - Receiving end response vk(t) = a(t) for the network of Fig. 4.38 if vsource(t) = unit impulse [94]. Reprinted
by permission of J. Marti
from the inverse Fourier transformation of A( ) = exp(- ý) calculated by the LINE CONSTANTS supporting routine
at a sufficient number of points in the frequency domain. The amplitude of the propagation factors A( ) for the case
of Fig. 4.39 is shown in Fig. 4.40.
The unit impulse response of a lossless line would be a unit impulse at t = with an area of 1.0. In Bergeron’s
method, this implies picking up the history term vm/Z + imk at t - with a weight of 1.0. In the more general case here,
history terms must now be picked up at more than one point, and weighted with the "weighting function" a(t). For the
example of Fig. 4.39(a),
C \GTQUGSWGPEGOQFG
D RQUKVKXGUGSWGPEGOQFG
(KI2TQRCICVKQPHCEVQT# T HQTVJGNKPGQH(KI=?4GRTKPVGFD[
RGTOKUUKQPQH,/CTVK
history terms must be picked up starting at
min
min
= 0.6 ms back in time, to approx.
is the travel time of the fastest waves, while
max
max
= 2.0 ms back in time. The value
is the travel time of the slowest waves. Each terms has its own
weight, with the highest weight of approx. 5400 around = 0.7 ms back in time. Mathematically, this weighting of
history at the other end of the line is done with the convolution integral
JKUVRTQRCICVKQP ' &
JOCZ
mJOKP
KO&VQVCN V&W C W FW
which can either be evaluated point by point, or more efficiently with recursive convolution as discussed later. The
expression im-total in Eq. (4.124) is the sum of the line current i mk and of a current which would flow through the
characteristic impedance if the voltage vm were applied to it (expression Imk + Vm/Zc in the frequency domain).
With propagation of the conditions from m to k being taken care of through Eq. (4.124), the only unresolved
issue is the representation of the term Vk/Zc in Eq. (4.121b). For the same 500 kV line used in Fig. 4.39, the magnitude
and angle of the characteristic impedance Zc are shown in Fig. 4.41. If the shunt conductance per unit length G’ were
ignored, as is usually done, Zc would become infinite at
= 0. This complicates the mathematics somewhat, and since
G’ is not completely zero anyhow, it was therefore decided to use a nonzero value, with a default option of 0.03 µs/km.
As originally suggested by E. Groschupf [96] and further developed by J. Marti [94], such a frequency-dependent
impedance can be approximated with a Foster-I R-C
network. Then the line seen from node k becomes a simple R-C network in parallel with a current source histpropagation
(Fig. 4.42(a)). One can then apply the trapezoidal rule of integration to the capacitances, or use any other method of
implicit integration. This transforms each R-C block into a current source in parallel with an equivalent resistance.
Summing these for all R-C blocks produces one voltage source in series with one equivalent resistance, or one current
source in parallel with one equivalent resistance (Fig. 4.42(b)). In the solution of the entire network with Eq. (1.8), the
frequency-dependent line is then simply represented again as a constant resistance Requiv to ground, in parallel with a
current source histRC + histpropagation, which has exactly the same form as the equivalent circuit for the lossless line.
To represent the line in the form of Fig. 4.42 in the EMTP, it is necessary to convert the line parameters into
a weighting function a(t) and into an R-C network which approximates the characteristic impedance. To do this, Zc and
are first calculated with the support routine LINE CONSTANTS, from dc to such a high frequency where both A( )
= exp(- ý) becomes negligibly small and Zc( ) becomes practically constant. J. Marti [94] has shown that it is best to
approximate A( ) and Zc( ) by rational functions directly in the frequency domain. The weighting function a(t) can
then be written down explicitly as a sum of exponentials, without any need for numerical inverse Fourier transformation.
Similarly, the rational function approximation of Zc( ) produces directly the values of R and C in the R-C network in
Fig. 4.42.
C
YKVJ4%PGVYQTM
D
YKVJGSWKXCNGPVTGUKUVCPEGCHVGT
CRRN[KPIKORNKEKVKPVGITCVKQP
(KI(TGSWGPE[FGRGPFGPVNKPGTGRTGUGPVCVKQPUGGPHTQONKPGGPFM
The rational function which approximates A( ) has the form
#CRRTQZ U ' G
&UJOKP
M
U%\ U%\ U%\P
U%R U%R U%RO
with s = j and n < m. The subscript "approx" indicates that Eq. (4.125) is strictly speaking only an approximation to
the given function A( ), even though the approximation is very good. The factor e-j
fact that a(t) in Fig. 4.39 is zero up to t =
min
is included to take care of the
; this avoids fitting exponentials through the portion 0 # t #
min
min
where the
values are zero anyhow (remember that a time shift - in the time domain is a phase shift e-j in the frequency domain).
All poles pi and zeros zi in Eq. (4.125) are negative, real and simple (multiplicity one). With
n < m, the rational
function part of Eq. (4.125) can be expanded into partial fractions,
M
U%\ U%\ U%\P
U%R U%R U%RO
M
'
U%R
%
M
U%R
%
MO
U%RO
The corresponding time-domain form of Eq. (4.15) then becomes
CCRRTQZ V ' MG
' &R V&JOKP
% MG
&R V&JOKP
HQT VJOKP
% MOG
&R O V&JOKP
HQT V$JOKP
This weighting function aapprox(t) is used to calculate the history term histpropagation of Eq. (4.124) in each time step.
Because of its form as a sum of exponentials, the integral can be found with recursive convolution much more efficiently
than with a point-by-point integration. If we look at the contribution of one exponential term kie-pi(t- min),
UK V '
4
mJOKP
K V&W MK G
&RK V & JOKP
FW
then si(t) can be directly obtained from the value si(t -
t) known from the preceding time step, with only 3
multiplications and 3 additions,
UK V ' E @ UK V&)V % E @ K V&JOKP % E @ K V&JOKP&)V
as explained in Appendix V, with c1, c2, c3 being constants which depend on the particular type of interpolation used
for i.
The characteristic impedance Zc( ) is approximated by a rational function of the form [94]
U%\ U%\ U%\P
<E&CRRTQZ U ' M
U%R U%R U%RP
with s = j . All poles and zeros are again real, negative and simple, but the number of poles is equal to the number of
zeros now. This can be expressed as
<E&CRRTQZ U ' M %
M
U%R
%
M
U%R
MP
U%RP
which corresponds to the R-C network of Fig. 4.42, with
4 ' M 4K '
MK
RK
%K '
K'P
MK
Rather than applying the trapezoidal rule to the capacitances in Fig. 4.42, J. Marti chose to use implicit integration with
Eq. (I.3) of Appendix I22, with linear interpolation on i. For each R-c block
K '
XK
4K
% %K
FXK
FV
which has the exact solution
XK V ' G
& "K )V
@ XK V&)V %
V & "K V&W
G
K W FW
%K mV&)V
6JKUOGVJQFKUKFGPVKECNVQVJGTGEWTUKXGEQPXQNWVKQPQH#RRGPFKZ8CRRNKGFVQ'S 9JGVJGT
TGEWTUKXGEQPXQNWVKQPKUDGVVGTVJCPVJGVTCRG\QKFCNTWNGKUUVKNNWPENGCT
with
i
= 1/(RiCi). By using linear interpolation on i, the solution takes the form of
XK V ' 4GSWKX&K @ K V % GK V&)V
with ei(t - t) being known values of the preceding time step (formula omitted for simplicity), or after summing up over
all R-C blocks and R0,
X V ' 4GSWKX @ K V % G V&)V
C
with
P
4GSWKX ' 4 % j 4GSWKX&K
K'
P
CPF G ' j GK
D
K'
which can be rewritten as
KV '
X V % JKUV4%
4GSWKX
The equivalent resistance R equiv enters into matrix [G] of Eq. (1.8), whereas the sum of the history terms hist RC +
histpropagation enters into the right hand side.
The key to the success of this approach is the quality of the rational function approximations for A( ) and
Zc( ). J. Marti uses Bode’s procedure for approximating the magnitudes of the functions. Since the rational functions
have no zeros in the right-hand side of the complex plane, the corresponding phase functions are uniquely determined
from the magnitude functions (the rational functions are minimum phase-shift approximations in this case) [94]. To
illustrate Bode’s procedure, assume that the magnitude of the characteristic impedance in decibels is plotted as a function
of the logarithm of the frequency, as shown in Fig. 4.43 [94]. The basic principle is to approximate the given curve by
straight-line segments which are either horizontal or have a slope which is a multiple of 20 decibels/decade. The points
where the slopes change define the poles and zeros of the rational function. By taking the logarithm on both sides of
Eq. (4.130), and multiplying by 20 to follow the convention of working with decibels, we obtain
NQI*<E&CRRTQZ U * ' NQIM % NQI*U%\*% NQI*U%\P*
& NQI*U%R*@@@& NQI*U%RP*
For s = j , each one of the terms in this expression has a straight-line asymptotic behavior with respect to . For
instance, 20 log *j + z1* becomes 20 log z1 for
<< z1 , which is constant, and 20 log
for
>> zi , which is a straight
line with a slope of 20 db/decade. The approximation to Eq. (4.137) is constructed step by step: Each time a zero corner
(at
= zi) is added, the slope of the asymptotic curve is increased by 20 db, or decreased by 20 db each time a pole
corner (at
= pi) is added. The straight-line segments in Fig. 4.43 are only asymptotic traces; the actual function
becomes a smooth curve without sharp corners. Since the entire curve is traced from dc to the
Fig. 4.43 - Asymptotic approximation of the magnitude of Zc(ω) [94].
Reprinted by permission of J. Marti
highest frequency at which the approximated function becomes practically constant, the entire frequency range is
approximated quite closely, with the number of poles and zeros not determined a priori. J. Marti improves the accuracy
further by shifting the location of the poles and zeros about their first positions. Fig. 4.44 shows the magnitude and
phase errors of the approximation of A( ), and Fig. 4.45 shows the errors for the approximation of Zc( ) for the line
used in Fig. 4.39.
L. Marti has recently shown [95] that very good results can be obtained by using lower-order approximations
with typically 5 poles and zeros rather than the 15 poles and zeros used in Fig. 4.44 and 4.45. Furthermore, he shows
that positive and zero sequence parameters at power frequency (50 or 60 Hz) can be used to infer what the tower
geometry of the line was, and use this geometry in turn to generate frequency-dependent parameters. With this
approach, simple input data (60 Hz parameters) can be used to generate a frequency-dependent line model internally
which is fairly accurate.
(a)
zero sequence mode
(15 zeros and 20 poles)
(b)
positive sequence mode
(13 zeros and 17 poles)
Fig. 4.44 - Errors in approximation of A( ) for line of Fig. 4.39 [94]. Reprinted by permission of J.
Marti
For M-phase lines, any of the M modes can be specified as frequency-dependent, or with lumped resistances,
or as distortionless. Mixing is allowed. A word of caution is in order here, however: At the time of writing these notes,
the frequency-dependent line model works only reliably for balanced lines. For untransposed lines, approximate real
and constant transformation matrices must be used, as explained in Section 4.1.5.3, which seems to produce reasonably
(a)
zero sequence mode
(15 zeros and poles)
(b)
positive sequence mode
(16 zeros and poles)
Fig. 4.45 - Errors in approximation of Zc( ) for line for Fig. 4.39 [94]. Reprinted by permission of
J. Marti
accurate results for single-circuit lines, but not for double-circuit lines. Research by L. Marti into frequency-dependent
transformation matrices in connection with models for underground cables will hopefully improve this unsatisfactory
state of affairs.
Fig. 4.46 - Comparison between voltages at phase b for [94]:
(a) Field test oscillograph
(b) BPA’s frequency-dependence simulation
(c) New model simulation
Reprinted by permission of J. Marti
Field test results for a single-line-to-ground fault from Bonneville Power Administration have been sued by
various authors to demonstrate the accuracy of frequency-dependent line models [84]. Fig. 4.46 compares the field test
results with simulation results from an older method which used two weighting functions a1 and a2 [84], and from the
newer method described here. The peak overvoltage in the field test was 1.60 p.u., compared with 1.77 p.u. in the older
method and 1.71 p.u. in the newer method. Constant 60 Hz parameters would have produced an answer of 2.11 p.u.
70&'4)4170&%#$.'5
6JGTGKUUWEJCNCTIGXCTKGV[QHECDNGFGUKIPUQPVJGOCTMGVVJCVKVKUFKHHKEWNVKHPQVKORQUUKDNGVQFGXGNQR
QPGEQORWVGTRTQITCOYJKEJECPECNEWNCVGVJGRCTCOGVGTU4 . % HQTCP[V[RGQHECDNG
(QTNQYGTXQNVCIGTCVKPIUVJGECDNGUCTGWUWCNN[WPUETGGPGFCPFKPUWNCVGFYKVJRQN[XKP[NEJNQTKFG#P
GZCORNGQHCVJTGGRJCUGM8ECDNGYKVJPGWVTCNEQPFWEVQTCPFCTOQTKUUJQYPKP(KI
(KI#TOQTGFM8ECDNG UVTCPFGFEQPFWEVQTKPUWNCVKQPDGFFKPIHNCV
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0
with elements defined
0
in Section 5.4.1
-Z’m
0
[Z’loop] =
0
(5.38a)
-Z’m
0
0
-Z’m
0 0 -Z’m 0 0 -Z’m 0 0 -Z’m
Z’s
with
Z’m = Z’pipe-mutual from Eq. (5.7c),
(5.38b)
Z’s = Z’pipe-out + Z’insulation + Z’earth.
(5.38c)
The first two terms in Eq. (5.38c) are found from Eq. (5.7b) and (5.6) (Z’insulation = 0 if pipe in contact with earth), and
Z’earth is the earth-return impedance discussed in Section 5.3. Transforming Eq. (5.38a) to phase quantities produces
same matrix as
0
Z’ Z’
....Z’
Z’e
for infinite
0
Z’ Z’
....Z’
Z’e
[Z’phase] =
pipe thickness
.
..............
.
0 0 ...
..............
0
.
(5.39a)
.
Z’ Z’
....Z’
Z’e
Z’e Z’e
...Z’e
Z’s
with Zs’ from Eq. (5.38c)
Ze’ = Zs’ - Zm’
(5.39b)
Z’ = Zs’ - 2Zm’
The last row and column in Eq. (5.39a) represent the pipe quantities, while the first 9 rows and columns refer to core,
sheath (shield tapes), armor (skid wires) of phases a, b, and c.
If the pipe is in contact with the earth, then the shunt admittance matrix is the same as in Section 5.4.1. If it
is insulated, then the potential coefficient matrix of Eq. (5.34) must be expanded with one extra row and column for the
pipe, and the same element
2) '
T
NP RKRG&KPUWNCVKQP
BggT
TRKRG&QWVUKFG
must be added to this expanded matrix,
same as in
0
P’ P’
....P’
Eq. (5.34)
0
P’ P’
....P’
[P’phase] =
.
+
.
0 0 ...
..............
P’ P’
(5.41)
....P’
0
The admittance matrix is then again found by inversion with Eq. (5.37).
5.5 Building of Conductors and Elimination of Grounded Conductors
Conductors are sometimes connected together ("bundled"). For example, the concentric neutral conductors
in the cable of Fig. 5.2 are in contact with each other, and therefore electrically connected. In a pipe-type cable, the
shield tapes and skid wires are in contact with the pipe. In a submarine cable, the sheath is often bonded to the armor
at certain intervals, to avoid voltage differences between the sheath and armor.
In such cases, the connected conductors 1,...m can be replaced by (or bundled into) one equivalent conductor,
by introducing the bundling conditions
I1 + I2 +...Im = Iequiv; V1 = V2 = ... Vm = Vequiv
(5.42)
into the equations for the series impedance and shunt admittance matrices. The bundling procedure for reducing the
equations from m individual to one equivalent conductor is the same as Method 1 of Section 4.1.2.2 for overhead lines,
and is therefore not explained again. It is exact if the conductors are continuously connected with zero connection
resistance (as the neutral conductors in Fig. 5.2), and accurate enough if the connections are made at discrete points with
negligible resistance (as in bonding of the sheath to the armor), as long as the distance between the connection points
is short compared to the wavelength of the highest frequency in the transient simulation.
As in the case of overhead lines with ground wires, some conductors in a cable may be grounded. For example,
the steel pipe of a pipe-type cable can usually be assumed grounded, because its asphalt mastic coating is not an electric
insulation. Also, neutral conductors may be connected to ground at certain intervals, or at both ends. If a conductor
i is grounded, then the condition is simply
Vi = 0
(5.43)
and conductor i can then be eliminated from the system of equations in the same way as described in Section 4.1.2.1.
Again, the elimination is only exact if the conductor is grounded continuously with zero grounding resistance, and
accurate enough if the distance between discrete grounding points is short compared to the wavelength of the highest
frequency.
An example of bundled as well as grounded conductors would be a single-core submarine cable which has its
sheath bonded to the armor. Since the asphalt coating of the armor is not an electric insulation, the armor is in effect
in contact with the sea water, and both sheath and armor are therefore grounded conductors. By eliminating both of
them, the submarine cable can be represented by single-phase equations for the core conductor, with the current return
combined in sea water, armor and sheath. For an overhead line, the equivalent situation would be a single-phase line
with two ground wires.
The case of segmented ground wires in overhead lines discussed in Section 4.1.2.5(b) can exist in cables as
well. For example, if the sheath is grounded at one end, but open and ungrounded at the other end, then the sheath could
be eliminated in the same way as segmented ground wires, provided the cable length is short compared to the wavelength
of the highest frequency. The support routine CABLE CONSTANTS does not have an option for such eliminations.
The user must represent the sheath as an explicit conductor, instead, with one end connected to ground. This offers the
advantage that the induced voltage at the other end can automatically be obtained, if so desired.
5.6 Buried Pipelines
Pipelines buried close to power lines can be subjected to hazardous induction effects, especially during singleline-to-ground faults. To study these effects, one can include the pipeline as an additional conductor into the
transmission line representation (Fig. 5.14(a)).
For steady-state analysis, one can also use the single-phase
representation of Fig. 5.14(b), with an impressed voltage
steady-state analysis, one can also use the single-phase representation of Fig. 5.14(b), with an impressed voltage
C RQN[RJCUGTGRTGUGPVCVKQP
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There is no capacitive coupling between the power line and the pipeline if it is buried in the ground.
As explained later, nominal -circuits can only be used for very short lengths of pipeline (typically # 0.3 km
at 60 Hz). The single-phase representation is therefore preferable for steady-state analysis, because the distributed
parameters of Fig. 5.14(b) are more easily converted into an exact equivalent -circuit than the polyphase parameters
of Fig. 5.14(a). This results in the active equivalent -circuit of Fig. 5.15, with Yseries and Yshunt being the usual
parameters obtained from Eq. (1.14), while Iinduced is an active current [158],
+KPFWEGF '
& F8KPFWEGFFZ
< ) RR
(KI#EVKXGGSWKXCNGPVBEKTEWKV
The correctness of the active -circuit can easily be shown. Starting from the differential equations
&
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FZ
&
F+
' ; ) RR 8
FZ
the introduction of a modified current
Imodified = I + Iinduced
transforms the differential equations into the normal form of the line equations, with the assumption that Iinduced does not
change along the line (dImodified/dx = dI/dx),
F8
' < ) RR +OQFKHKGF
FZ
F+
& OQFKHKGF ' ; ) RR 8
FZ
&
The solution for a line between nodes k and m is given in Eq. (1.13), except that the current is now Imodified, or rewritten,
+MO % +KPFWEGF
+OM & +KPFWEGF
'
;UGTKGU % ;UJWPV
& ;UGTKGU
& ;UGTKGU
8M
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This is exactly the same equations which comes out of the equivalent circuit of Fig. 5.15.
With this single-phase approach, the currents in the power line are assumed to be known, e.g., from the usual
type of short-circuit study. It is also assumed that they are constant over the length of the exposure to the pipeline, and
that the pipeline runs parallel to the power line (mutual impedances constant). If either assumption is not true, then the
power line-pipeline system must be split up into shorter sections as is customarily done in interference studies. The
effect of the pipe on the power line zero sequence impedance is usually ignored, but could be taken into account.
In both representations of Fig. 5.14, the mutual impedances between the pipe and the overhead conductors, as
well as the self impedance of the pipe with earth return, are needed. The mutual impedances are obtained with the
formulas discussed in Section 5.3.4. At 60 Hz, Carson’s formula will give practically identical results as the more
complicated formula of Pollaczek.
The self impedance Z’pp of the pipeline consists of the same three terms shown for the armor in Eq. (5.4). The
first two terms are calculated with Eq. (5.7b) and (5.6), while R’earth is found from the equations discussed in Section
5.3.
For the shunt admittance Y’pp = G’ + j C’, the capacitive part is calculated in the usual way with Eq. (5.13).
In contrast to the underground cable, the shunt conductance G’ of the pipeline can no longer be ignored. The insulation
around pipelines is electrically poor, either originally or because of puncturing during the laying operation. The loss
angle in Eq. (5.14) is so large on pipelines insulated with glass-fiber/bitumen that G’ becomes much larger than C’
at power frequency, and if one part of the shunt admittance is ignored it should be C’ rather than G’. On PVC-insulated
pipelines, G’ may still be smaller than C’, though.
If the shunt resistance of the insulation is relatively small, then the grounding resistance of the pipe should be
connected in series with it2 [170], or
)) '
where
4)
KPUWNCVKQP
% 4 ) ITQWPFKPI
R’insulation = resistance of pipe insulation,
R’grounding = grounding resistance.
A useful formula for the grounding resistance is [170]
4 ) ITQWPFKPI '
with
earth
DGCTVJ
ý
% NP
NP
B
&
ý
ý
J %
J %
%
ý
ý
&
= earth resistivity (e.g., in m),
h = depth of burial of pipe
ý = length of pipe
+HVJGUJGCVJCTOQTQTRKRGQHCPWPFGTITQWPFECDNGQTVJGITQWPFYKTGQHCPQXGTJGCFNKPGKUITQWPFGF
VJGPKVJCUDGGPUVCPFCTFRTCEVKEGVQKIPQTGVJGITQWPFKPITGUKUVCPEG 8 #PCNVGTPCVKXGYQWNFDGVQWUGC
HKPKVGUJWPVCFOKVVCPEG; 4
CUTGEGPVN[UWIIGUVGF=?
ITQWPFKPI
D = outside diameter of pipe.
Grounding grids must generally be analyzed as three-dimensional problems, even if they consist of only one
pipe. The grounding resistance from Eq. (5.47) is therefore no longer an evenly distributed parameter, but depends on
the length. Fortunately, the dependence of G’ on length is very small for typical values of G’insulation [158]. In the region
of measured values for G’ between 0.1 S/km for newly-layed pipelines and 0.3 S/km for older pipelines with glass
fiber/bitumen insulation [170], the dependence of G’ on length is practically negligible, as shown in Fig. 5.16. Treating
G’ as an evenly distributed parameter is therefore a reasonable approximation.
(KI5JWPVEQPFWEVCPEGQHDWTKGFRKRG
Because of G’ >> C’, the wavelength of buried pipelines is significantly shorter than that of underground
cables, as shown in Table 5.3 [170]. Therefore, a nominal -circuit of a circuit which includes a buried pipeline should
not be longer than approximately 0.8 km for
Table 5.3 - Wavelength of pipeline at 50 Hz [170]
G’ (S/km)
wavelength (km)
0.1
41.3
1.0
13.1
10.0
4.13
steady-state analysis, or approximately 0.08 km for switching surge studies [158].
Fig. 5.17 shows a comparison between measured and calculated voltages and currents in a pipeline, induced
by currents in a neighboring power line, with the pipeline representation as discussed here [158].
(KI+PFWEGFXQNVCIGUCPFEWTTGPVUKPCDWTKGFRKRGNKPG
5.7 Partial Conductor and Finite Element Methods
The support routine CABLE CONSTANTS uses analytical formulas which are essentially only applicable to
configurations with axial symmetry. The formulas for the nonconcentric configuration in pipe-type cables (Section 5.5)
are only approximate, and the authors of these formulas themselves suggest improvements along the lines discussed
here.
To find the impedances and capacitances for conductor systems with arbitrary shapes (e.g., for the cable of Fig.
5.1), numerical methods can be used in place of analytical formulas, which are either based on subdivisions into partial
conductors or on finite element methods. There is no support routine yet in the EMTP which uses these numerical
methods. The principle of these methods is therefore only outlined very briefly.
5.7.1 Subdivision into Partial Conductors
With this method, each conductor is subdivided into small "partial" conductors ("subconductors" in [162],
"segments" in [164]), as shown in Fig. 5.18. Various shapes can be used for the partial conductors, with rectangles
being the preferred shape for strip lines in
(KI5WDFKXKUKQPQHVJGOCKPEQPFWEVQTUKPVQRCTVKCN
EQPFWEVQTU
integrated circuits (Fig. 5.19).
(KI5WDFKXKUKQPQHUVTKRNKPGUKPVQRCTVKCNEQPFWEVQTU
QHTGEVCPIWNCTUJCRG=?%QR[TKIJVD[+PVGTPCVKQPCN
$WUKPGUU/CEJKPGU%QTRQTCVKQPTGRTKPVGFD[RGTOKUUKQP
In deriving the equations for the system of partial conductors, uniform current density is assumed within each
partial conductor. Then the voltage drops along a system of n partial conductors at one frequency are described by the
phasor equations
F8FZ
4
F8FZ
&
@
@
. . @ @ .P
4
'
% LT
@
@
F8PFZ
4P
. . @ @ .P
@
@
@ @
@
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+
+
@
@
+P
The diagonal resistance matrix contains the dc resistances, and the full inductance matrix contains the self and mutual
inductances of the partial conductors. The formulas for the matrix elements depend on the shape of the partial
conductor, but they are well known.
To obtain the frequency-dependent impedance of a cable system, the matrices [R] and [L] are first computed.
At each frequency, the complex matrix [Z] = [R] + j [L] is formed, and reduced to the number of actual conductors with
Bundling Method 1 of Section 5.5. For example, if partial conductors 1,...50 belong to the core conductor, and partial
conductors 51,...120 to the sheath, then this bundling procedure will reduce the 120 x 120-matrix to a 2 x 2-matrix,
which produces the frequency-dependent impedances
<EE T
<EU T
<EU T
<UU T
This numerical method works well as long as the conductors are subdivided into sufficiently small partial
conductors. The size of these partial conductors must be of the same order of magnitude as the depth of penetration.
5.7.2 Finite Element Methods
Finite element methods are more powerful than partial conductor methods in one sense, inasmuch as it is not
necessary to assume uniform current density within each element. However, it is very difficult to handle open-boundary
conditions with finite element methods, that is configurations where the magnetic field diminishes gradually as one
moves away from the conductors, with no clearly defined boundary of known magnetic vector potential reasonably close
to the conductors. In situations where a boundary is clearly defined, e.g., in pipe-type cables at high frequency where
the depth of penetration becomes much less than the wall thickness, finite element methods can be quite useful.
With finite element methods, the region inside and outside of the conductors is subdivided into small elements,
usually of triangular shape. Fig. 5.20(a) shows the example of a stranded conductor inside a pipe of radius Rb as the
return path (clearly defined boundary with zero magnetic field A = 0 outside the pipe and zero derivative along the two
edges of the "wedge"). Because of axial symmetry, it is sufficient to analyze the "wedge" shown in Fig. 5.20(a). This
wedge region is then subdivided into triangular elements as shown in Fig. 5.20(b), with longer triangles as one moves
away from the conductor.
The magnetic vector potential A is assumed to vary linearly along the edges and inside of each triangle,
A = ax + by + c,
(5.49)
when a first-order method is used (higher-order methods exist as well). The unknowns are essentially the values of A
in the node points. If they were shown in the z-direction of a three-dimensional picture, then the triangles would appear
in a shape similar to a geodesic dome, with the roof height being the value of A. The equations for finding A are linear
algebraic
C 5VTCPFGFEQPFWEVQTKPUKFGRKRGQHTCFKWU4
D
D 5WDFKXKUKQPQHTGIKQPKPVQVTKCPIWNCTGNGOGPVU
Fig. 5.20 - Analysis of stranded conductor with finite element method [171]. Reprinted by permission of Yin Yanan
equations with a sparse matrix, but the number of node points or the number of equations is usually quite high (146
equations for the example of Fig. 5.20). Once the magnetic vector potential is known in the entire region, the
impedances can be derived from it.
For readers interested in finite element methods for cable impedance calculations, the papers by Konrad, Weiss
and Csendes [165, 166, 167] are a good introduction.
5.8 Modal Parameters
Once the series impedance and shunt admittance matrices per unit length [Z’phase], [Y’phase] are known, the
derivation of modal parameters is exactly the same as described in Section 4.1.5 for overhead lines. They could be used,
for example, to develop exact equivalent -circuits for steady-state solutions as explained in Section 4.2.1.3.
For transient simulations, it is more difficult to use modal parameters, as compared to overhead lines, because
the transformation matrix [Ti] can no longer be assumed to be constant as for a single-circuit overhead line. Fig. 5.21
shows the variation of the elements in the third column of [Ti] for a typical three-phase arrangement of 230 kV singlecore cables with core conductor and sheath in each [155]. Especially around the power frequency of 50 or 60 Hz, the
variations are quite pronounced.
(KI/CIPKVWFGQHVJGGNGOGPVUQHEQNWOPQH=6 ?
K
Above a few kHz, the loop between core conductor and sheath becomes decoupled from the outer loop between sheath
and earth return, because the depth of penetration on the inside of the sheath for loop 1 becomes much smaller than the
sheath thickness. In that case, Ztube-mutual ~ 0. This makes the transformation matrix constant above a few kHz, as evident
from Fig. 5.21. For a single-phase single-core cable with sheath and armor, the three modes are identical with the 3
loops described in Eq. (5.1) at high frequency where Z’12 ~ 0 and Z’23 ~ 0. The transformation matrix between loop and
phase quantities of Eq. (5.9),
=6 ?&
K
' CPF =6K? ' & & 5.9 Cable Models in the EMTP
Co-author: L. Marti
As of now (Summer 1986), there are no specific cable models in the BPA EMTP. The only way to simulate
cables is to fit cable data into the models available for overhead lines. It has long been recognized, of course, that this
is only possible in a limited number of cases. A method specifically developed for cables, as discussed in Section
5.9.2.3, will hopefully be implemented in late 1986 or early 1987. It has already been tested extensively in the UBC
EMTP.
5.9.1 Ac Steady-State Solutions
In principle, there is no difficulty in representing cables as nominal or equivalent -circuits in the same way
as overhead lines (Section 4.2.1). If nominal -circuits are used, it should be realized that the wavelength of
underground cables is shorter than on overhead lines. If a nominal -circuit should not be longer than 100 km at 60 Hz
for overhead lines, the limit is more typically 30 km for underground cables. If a pipeline is modelled, the limit can be
as low as 1 km, as discussed in Section 5.6.
Underground cables are often very short compared to the length of connected overhead lines. In such cases,
the (complicated) series impedances have very little effect on the results because the system sees the cable essentially
as a shunt capacitance. The cable can then be modelled as a simple lumped capacitance.
5.9.2 Transient Solutions
The accurate representation of cables with frequency-dependent impedances and frequency-dependent
transformation matrices is discussed in Section 5.9.2.3. Situations where simpler models should be accurate enough
are discussed first.
5.9.2.1 Short Cables
If a rectangular wave pulse travels on an overhead line and hits a relatively short underground cable, then the
cable termination is essentially seen as a lumped capacitance. The voltage then builds up exponentially with a time
constant of T = Zoverhead•Ccable, shown in Fig. 5.22(a). If the cable is modelled somewhat more accurately as a lossless
distributed-parameter line, then the voltage build-up has the staircase shape of Fig. 5.22(b), with the average of the
sending and receiving end curve being more or less the same as the continuous curve in Fig. 5.22(a). As long as the
travel time [] of the cable is short compared to the time constant T, reasonably accurate results can be obtained if the
cable is represented as a lumped capacitance.
(a)
Cable represented as lumped
(b)
Cable represented as
capacitance
lossless transmission line
------ sending end of cable
...... receiving end of cable
Fig. 5.22 - Voltage build-up in a cable connected to an overhead line
Nominal -circuit representations have often been suggested as approximate cable models. They obviously
represent the capacitance effect correctly, but the pronounced frequency-dependence in the series impedance is ignored.
Nominal -circuits give reasonable answers probably only in those cases in which the simpler lumped capacitance
representation is already accurate enough.
5.9.2.2 Single-Phase Cables
There are situations where single-phase representations are possible. An example is a single-phase submarine
cable in which the sheath and armor are bonded together, with the armor being in contact with the sea water. In such
a case, the sheath and armor can be eliminated from Eq. (5.10), which results in the reduced single-phase equation
&
F8E
FZ
' < ) EQTG +E
with Z’core being the impedance of the core conductor with combined current return through sheath, armor and sea water.
Coupling to the cables of the other two phases can be ignored because the three cables are layed relatively far apart, to
reduce the risk of anchors damaging more than one phase in the same mishap.
When the equations have been reduced to single-phase equations, then it is straightforward to use the
frequency-dependent overhead line model described in Section 4.2.2.6.
Sometimes it is not necessary to take the frequency-dependence in the series impedances into account. For
example, single phase SF6-busses have been modelled quite successfully for fast transients with two decoupled lossless
single-phase lines, one for the inside coaxial loop and a second one for the outside loop between the enclosure and the
earth-return. The coupling between the two loops through the enclosure is negligible at high frequencies because the
depth of penetration is much less than the enclosure wall thickness. The only coupling occurs through reflections at
the terminations. Agreement between simulation results from such simple models and field tests has been excellent
[169].
5.9.2.3 Polyphase Cables [155]
The simple overhead line models with constant parameters discussed in Section 4.2.2 are of limited use for
underground cables for two reasons:
(a)
The transformation matrix [Ti] is frequency-dependent up to a few kHz, though a constant [Ti] would be
acceptable for transients which contain only high frequencies (e.g., lightning surge studies).
(b)
The modal parameters (e.g., wave velocity and attenuation) are more frequency-dependent than on overhead
lines, as shown in Fig. 5.23 for three single-core cables with core and sheath [150].
C #VVGPWCVKQP
D 8GNQEKV[
Fig. 5.23 - Modal parameters as a function of frequency [150]. Reprinted by permission of IEE and
the authors
To derive an accurate model for an n-conductor cable system between nodes k and m, we can start from the
phasor equation (4.121) for the overhead line, if we replace that scalar equation, which was written for one phase or
mode, by a matrix equation for the n conductors,
=;E? =8M? & =+MO? ' =#? 6=;E? =8O? % =+OM?>
with [Yc] = [Zc]-1 = characteristic admittance matrix in phase quantities,
[A] = e-j ký = propagation factor matrix.
Eq. (5.51) is transformed to modal quantities, with
=+? ' =6K? =+OQFG?
C
=8? ' =6K V?& =8OQFG?
D
and
which yields
=+MO&OQFG? ' =;E&OQFG? =8M&OQFG? & =#OQFG? 6 =;E&OQFG? =8O&OQFG? % =+OM&OQFG? >
(5.53)
with both [Yc-mode] and [Amode] being diagonal matrices,
=;E&OQFG? ' =6K?& =;E? =6K V?&
C
=#OQFG? ' =6K?& =#? =6K?
D
The diagonal element of [Amode] is obtained from the i-th eigenvalue
#OQFG&K ' G
&ý 8
i
of the product jY’phasek jZ’phasek,
E
K
and [Ti] is the matrix of eigenvectors of the same product [Y’phase ] [Z’phase ]. Eq. (5.53) consists of n decoupled (scalar)
equations, with one equation for each mode.
Transforming these scalar equations into the time domain is the same procedure as described in Section 4.2.2.6
for the overhead line. For mode i, the second term in Eq. (5.53) is found with the same convolution integral as in Eq.
(4.124),
JKUVRTQRCICVKQP ' &
JOCZ
mJOKP
KO&VQVCN V&W C W FW HQT GCEJ OQFG
with the current im-total being the sum of the line current imk and of a current which would flow through the characteristic
impedance of mode i if the voltage vm of mode i were connected across it. Only known history terms appear in Eq.
(5.55), and histpropagation can therefore be found by n recursive convolutions for the n modes, in the same say as in Section
4.2.2.6. The modal propagation factors are very similar in shape to those of an overhead line, as shown for Amode-3 ( )
in Fig. 5.24.
(KI/CIPKVWFGQHRTQRCICVKQPHCEVQTHQTOQFGQHCEQPFWEVQT
U[UVGO VJTGGUKPINGEQTGECDNGUYKVJEQTGCPFUJGCVJKPGCEJ
With
propagation of the conditions from m to k being taken care of through Eq. (5.55), the only unresolved issue in the modal
domain equations is the representation of the term YcVk in Eq. (5.53). Again, the frequency dependence of Yc is similar
to that of an overhead line, as shown in Fig. 5.25, and can be represented with the same type of Foster-I R-C network
shown in Fig. 4.42(a), and reproduced here as Fig. 5.26. By applying the trapezoidal rule of integration to the
capacitances, or by using recursive convolution as discussed in Appendix V, the R-C
(KI/CIPKVWFGQHEJCTCEVGTKUVKECFOKVVCPEGHQTOQFG UCOG
EQPFWEVQTU[UVGOCUKP(KI
C
4%PGVYQTM
D
'SWKXCNGPVTGUKUVCPEGCHVGTCRRN[KPIKORNKEKVKPVGITCVKQPVQ%
(KI4GRTGUGPVCVKQPQHQPGOQFGUGGPHTQOUKFGM
network is converted into an equivalent conductance Gequiv in parallel with a known current source histRC + histpropagation.
After the network solution at each time step, the current flowing through the characteristic impedance represented by
the R-C network must be calculated for both ends of the cable from Gequivv + histRC, because this term is needed after
the elapse of travel time to form the expression im-total needed in Eq. (5.55).
From Fig. 5.26(b), it can be seen that each mode is now represented by the scalar, algebraic equation
ikm(t) = Gequivvk(t) + (histRC + histpropagation)
(5.56)
with an analogous equation for imk(t) at the other end. If the transformation matrix were constant and real, then Eq.
(5.56) could very easily be transformed back to phase quantities,
[ikm-phase(t)] = [Ti][Gequiv][Ti]t[vk-phase] + [Ti][hist]
as explained in Eq. (4.109) for the overhead line. As shown in Fig. 5.21, the transformation matrix [Ti] of cables is very
much frequency-dependent, and the transformation back to phase quantities now requires convolutions based on Eq.
(5.52),
=KRJCUG V ? '
=XOQFG V ? '
V
=V V&W ? =KOQFG W ? FW
m&4 K
V
m&4
=VK V&W ?V =XRJCUG W ? FW
C
D
where [ti] is a matrix obtained from the inverse Fourier transform of the frequency-dependent matrix [Ti]. Similar to
the curve fitting used for the modal characteristic impedances, each element of [Ti] is again approximated by rational
functions of the form
O
MK
K'
LT % RK
6z< T ' M % j
with k0, ki and pi being real constants which, when transformed into the time domain, becomes
O
Vz< V ' M F V % j MK GZR &RKV W V
K'
With the simple sum of exponentials in Eq. (5.59), recursive convolution can be applied again (Appendix V). Then,
the convolution integrals in Eq. (5.57) can be split up into a term containing the yet unknown voltages and currents at
time t, and the known history terms which can be updated recursively,
=KRJCUG V ? ' =V? =KOQFG V ? % =JKUVEWTTGPV?
C
=XOQFG V ? ' =V?V =XRJCUG V ? % =JKUVXQNVCIG?
D
with [t0] being a real, constant n x n-matrix. With Eq. (5.60), the transformation of the modal equations (5.56) to phase
quantities is now fairly simple,
=KMO&RJCUG V ? ' =)RJCUG? =XM&RJCUG V ? % =JKUVRJCUG?
C
=)RJCUG? ' =V? =)RJCUG? =V?V
D
with
and the history term
[histphase] = [histcurrent] + [t0]{[Gequiv][histvoltage]
+ [histRC] + [histpropagation]}
(5.61c)
Since the form of Eq. (5.61a) is identical to that of Eq. (4.109) for the overhead line with constant [Ti], adding the model
to the EMTP is the same as described there. The extra effort lies essentially in the evaluation of the two extra history
vectors [histcurrent] and [histvoltage]. After the network solution at each time step, Eq. (5.60) is used to obtain the modal
quantities from the phase quantities.
The principle of the frequency-dependent cable model is fairly simple, but its correct implementation depends
on many intricacies, which are described in [155]. In particular, it is important to normalize the eigenvectors in such
a way that the elements of [Ti] as well as the modal surge admittances Yc-mode-i both become minimum phase shift
functions. This is achieved by making one element of each eigenvector a real and constant number through the entire
frequency range. Furthermore, standard eigenvalue/eigenvector subroutines do not produce smooth curves of [Ti] and
[Yc-mode] as a function of frequency, because the order in which the eigenvalues are calculated often changes as one
moves from one frequency point to the next. This problem was solved by using an extension of the Jacobi method for
complex symmetric matrices. Symmetry is obtained by reformulating the eigenproblem
=; ) RJCUG? =< ) RJCUG? =Z? ' 8=Z?
in the form
=*? =T? ' 8=T?
C
=*? ' =.?V =< ) RJCUG? =.?
D
=Z? ' =.? =T?
E
where
and
with [L] being the lower triangular matrix obtained from the Choleski decomposition of [Y’phase] [157]. The Choleski
decomposition is a modification of the Gauss elimination method, as explained in Appendix III. One can also replace
[L] in Eq. (5.62) with the square root of [Y’phase] obtained from
=; ) RJCUG? ' =:? =7? =:?&
where [
1/2
] is the diagonal matrix of the square roots of the eigenvalues, and [X[ is the eigenvector matrix of [Y’phase].
Both approaches are very efficient if G’ is ignored, or if tan is constant for all dielectrics in the cable system, because
[L] or [Y’phase]1/2 must then only be computed once for all frequencies.
For parallel single core cables layed in the ground (not in air), [Y’] is diagonal if loop equations are used. In
that case it is more efficient to find the eigenvalues and eigenvectors for [Y’loop][Z’loop], where both [L] and [Y’loop]1/2
become the same diagonal matrix with %Y’loop-i as its elements. The conversion back to phase quantities is trivial with
Eq. (5.50).
The reason why the Jacobi procedure produces smooth eigenvectors is that the Jacobi algorithm requires an
initial guess for the solution of the eigenvectors. This initial guess is readily available from the solution of the
eigenproblem of the preceding frequency step; consequently, the order of the eigenvectors from one calculation to the
next is not lost.
Figure 5.27(a) shows the magnitude of the elements of row 3 of the eigenvector matrix [Ti] for the same 6conductor system as in Fig. 5.24, when standard eigenvalue/eigenvector routines are used. Fig. 5.27(b), on the other
hand, shows the same elements of [Ti] calculated with the modified Jacobi algorithm.
As an application for this cable model, consider the case of three 230 kV single-core cables (with core and
sheath), buried side by side in horizontal configuration, with a length of 10 km. A unit-step voltage is applied to the
core of phase A, and the cores of phases B and C as well as all three sheaths are left ungrounded at both ends. The unitstep function was approximated as a periodic rectangular pulse of 10 ms duration and a period of 20 ms with a Fourier
series containing 500 harmonics,
X V ' C % j 6 CK EQU TKV % DK UKP TKV >
K'
The wave front of this approximation is shown in Fig. 5.28. Choosing a Fourier series
(KI(QWTKGTUGTKGUCRRTQZKOCVKQPQHWPKVUVGR
approximation for the voltage source offered the advantage that exact answers could be found as well, by using ac
steady-state solutions with exact equivalent -circuits (Section 4.2.1.3) at each of the 500 frequencies, and by
superimposing them. Fig. 5.29 and 5.30 show the EMTP simulation results in the region of the third pulse,
superimposed on the exact answers. The two
C 5VCPFCTFGKIGPXCNWGGKIGPXGEVQTUWDTQWVKPGU
D /QFKHKGF,CEQDKCNIQTKVJO
Fig.
5.27
Magnitude of the elements of row 3 of [Ti] (same 6-conductor system as in Fig. 5.24)
curves are indistinguishable in this third pulse region where the phenomena have already become more or less periodic.
This shows that the EMTP cable model is capable of producing highly accurate answers. The insert on the right-hand
side of Fig. 5.29 shows the response to the first pulse, where the EMTP simulation results differ slightly from the exact
answers, not because of inaccuracies in the model but because the EMTP starts from zero initial conditions while the
exact answer assumes periodic behavior even for t < 0.
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EQPPGEVKQPYKNNCNUQETGCVGVJGEQTTGEVRJCUGUJKHVCWVQOCVKECNN[ UKFGNCIIKPIDGJKPFUKFGD[EHQTDCNCPEGF
RQUKVKXGUGSWGPEGQRGTCVKQPKPVJKURCTVKEWNCTECUG (KI6YQEQWRNGFEQKNU
+NN%QPFKVKQPKPIQH+PFWEVCPEG/CVTKZ
6JGHQWTGNGOGPVUKPVJG=:?OCVTKZQH'S EQPVCKPDCUKECNN[VJGKPHQTOCVKQPHQTVJGGZEKVKPIEWTTGPV
OCIPGVK\KPITGCEVCPEG: RW YKVJVJGUJQTVEKTEWKVTGCEVCPEGDGKPITGRTGUGPVGFKPFKTGEVN[VJTQWIJVJG
O
UOCNNFKHHGTGPEGUDGVYGGP: CPF: CPFDGVYGGP: CPF: +HCNNHQWTXCNWGUYGTGTQWPFGFVQQPGFKIKVDGJKPF
VJGFGEKOCNRQKPV : : : .RW VJGPVJGUJQTVEKTEWKVTGCEVCPEGYQWNFDGEQORNGVGN[NQUV:
UJQTV
+POQUVUVWFKGUKVKUVJGUJQTVEKTEWKVTGCEVCPEGTCVJGTVJCPVJGOCIPGVK\KPITGCEVCPEGJQYGXGTYJKEJKPHNWGPEGU
VJG TGUWNVU +V KU VJGTGHQTG KORQTVCPV VJCV =:? DG ECNEWNCVGF CPF RWV KPVQ VJG FCVC HKNG YKVJ XGT[ JKIJ CEEWTCE[
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:
:
UJQTV
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:
UGGP HTQO UKFG KUUVKNNTGCUQPCDN[CEEWTCVG+VKUJKIJN[TGEQOOGPFGFVQECNEWNCVG:
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QH< < < YQWNFJCXGVQDGCEEWTCVGVQYKVJKPvVQCEJKGXGCPCEEWTCE[QHvHQT:
6JKU
UJQTV
CEEWTCE[RTQDNGOKUQPGQHVJGTGCUQPUYJ[< < < ECPPQVDGOGCUWTGFFKTGEVN[KPVGUVUKHVJKUFCVCKUVQEQPVCKP
VJGUJQTVEKTEWKVVGUVKPHQTOCVKQPDGUKFGUVJGGZEKVCVKQPVGUVKPHQTOCVKQP/CVJGOCVKECNN[=:?KUCNOQUVUKPIWNCTCPF
VJGTGHQTGKNNEQPFKVKQPGFVJGOQTGUQVJGUOCNNGTVJGGZEKVKPIEWTTGPVKU'ZRGTKGPEGJCUUJQYPVJCVVJGKPXGTUKQP
QH=:?KPUKFGVJG'/62FQGUPQVECWUGCP[RTQDNGOUCUNQPICUXGT[JKIJCEEWTCE[KUWUGFKPVJGKPRWVFCVC
2TQDNGOUOC[CRRGCTQPNQYRTGEKUKQPEQORWVGTUJQYGXGT6JGCWVJQTVJGTGHQTGRTGHGTUKPXGTUGKPFWEVCPEGOCVTKZ
TGRTGUGPVCVKQPUCUFKUEWUUGFKP5GEVKQP
6JTGG9KPFKPI6TCPUHQTOGTU
6JGKORGFCPEGOCVTKZQHUKPINGRJCUGVJTGGYKPFKPIVTCPUHQTOGTUECPDGQDVCKPGFKPCUKOKNCTYC[YKVJVJG
YGNNMPQYPUVCTEKTEWKVWUGFKP(KI+PVJCVEKTEWKVVJGOCIPGVK\KPITGCEVCPEGKUWUWCNN[EQPPGEVGFVQVJGUVCT
RQKPVDWVUKPEGKVUWPUCVWTCVGFXCNWGKUOWEJNCTIGTVJCPVJGUJQTVEKTEWKVTGCEVCPEGUKVEQWNFDGEQPPGEVGFVQGKVJGT
VJGRTKOCT[UGEQPFCT[QTVGTVKCT[UKFGCUYGNN#UUWOKPIVJCVVJGGZEKVKPIEWTTGPVHQTVJGGZCORNGQH(KIKU
OGCUWTGFHTQOVJGRTKOCT[UKFGYKVJGZEKVCVKQPNQUUGUKIPQTGFVJGOCIPGVK\KPITGCEVCPEGKPVJGUVCTRQKPVYQWNF
VJGPDGRW6JGP
=:? ' RW
6JGRCTVKEWNCTEQPPGEVKQPYQWNFCICKPDGGUVCDNKUJGFVJTQWIJVJGPQFGPCOGUCVDQVJGPFUQHVJGDTCPEJGU(QT
GZCORNGVJGVJTGGDTCPEJGUEQWNFDGEQPPGEVGFHTQOPQFG*#VQITQWPF.#VQ.$CPF6#VQ6$6QEQPXGTV
'S VQCEVWCNXCNWGUFKXKFGCNNGNGOGPVUD[VJGRQYGTTCVKPI5
CPFOWNVKRN[VJGHKTUVTQYCPFEQNWOPYKVJ
TCVKPI
XQNVCIGTCVKPI8 VJGUGEQPFTQYCPFEQNWOPYKVJ8 CPFVJGVJKTFTQYCPFEQNWOPYKVJ8 6JG=4?CPF=:?OCVTKEGUECPGKVJGTDGFGTKXGFD[JCPFQTVJG[ECPDGQDVCKPGFHTQOVJGUWRRQTVTQWVKPGU
:(14/'4$%64#0QT64'.')KPVJG$2#XGTUKQPQHVJG'/626JGNCVVGTVYQUWRRQTVTQWVKPGUYGTG
FGXGNQRGFHQTVJTGGRJCUGWPKVUDWVECPDGWUGFHQTUKPINGRJCUGWPKVUCUYGNN
+PXGTUG+PFWEVCPEG/CVTKZ4GRTGUGPVCVKQPQH5KPING2JCUG6YQCPF6JTGG9KPFKPI6TCPUHQTOGTU
+HVJGGZEKVKPIEWTTGPVKUKIPQTGFVJGPVJGQPN[YC[VQTGRTGUGPVVTCPUHQTOGTUKUYKVJOCVTKEGU=4?CPF=.? YJKEJCTGJCPFNGFD[VJG'/62CUFGUETKDGFKP5GEVKQP6JGCWVJQTRTGHGTUVJKUTGRTGUGPVCVKQPQXGTCNNQVJGTU
DGECWUGVJGOCVTKEGU=4?CPF=.? CTGPQVKNNEQPFKVKQPGFCPFDGECWUGCP[XCNWGQHGZEKVKPIEWTTGPVKPENWFKPI\GTQ
ECPDGWUGF6JGDWKNVKPUVCTEKTEWKVKPVJG$2#XGTUKQPQHVJG'/62WUGUVJKUTGRTGUGPVCVKQPKPVGTPCNN[CUYGNN
(QTVJTGGRJCUGVTCPUHQTOGTUVJGEQPXGTUKQPQHVJGVGUVFCVCVQ=4?CPF=.? OCVTKEGUKUDGUVFQPGYKVJVJG
UWRRQTVTQWVKPG$%64#0(QTUKPINGRJCUGWPKVUCPFHQTVJTGGRJCUGVTCPUHQTOGTUYJGTG< .< VJGEQPXGTUKQP
\GTQ
RQU
KUHCKTN[UKORNGCPFECPGCUKN[DGFQPGD[JCPFCUGZRNCKPGFPGZV
6YQ9KPFKPI6TCPUHQTOGTU
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=T.? OCVTKEGUKPRWECPVJGPDGYTKVVGPFQYPD[KPURGEVKQPHTQOVJGGSWKXCNGPVEKTEWKVQH(KI CHVGTVJG
OCIPGVK\KPIKPFWEVCPEGJCUDGGPTGOQXGF =4RW? '
4 RW
4 RW
CPF =T .RW?& '
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&
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&
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6JGKPXGTUGDTCPEJTGCEVCPEGOCVTKZ=T. ? KUVJGYGNNMPQYPPQFGCFOKVVCPEGOCVTKZQHCUGTKGUDTCPEJYKVJRW
RW
TGCEVCPEG: (QTVJGGZCORNGQH(KIYKVJGZEKVKPIEWTTGPVKIPQTGFVJGRWOCVTKEGUYQWNFDG
RW
=4RW? '
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&
& 6JGOCVTKEGUKP'S CTGEQPXGTVGFVQCEVWCNXCNWGUYKVJ
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4 RW8
5TCVKPI
4 RW8
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5
TCVKPI
8
5TCVKPI
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&
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8 8
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KP S
KP 5
C
D
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CRRCTGPVRQYGTTCVKPI
8 8 XQNVCIGTCVKPIU
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UKORN[UGV8 8 KP'S 6QQDVCKP=.? VJGOCVTKZKP'S KUUKORN[OWNVKRNKGFYKVJT
#U CNTGCF[ OGPVKQPGF KP 5GEVKQP VJG VYQ EQWRNGF DTCPEJGU FGUETKDGF D[ 'S ECP CNUQ DG
TGRTGUGPVGFCUUKZWPEQWRNGFDTCPEJGU+IPQTKPIVJGTGUKUVCPEGUHQTVJGUCMGQHVJKUCTIWOGPVCPFUGVVKPI
5TCVKPI
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L :RW 8
8
V ' 8
RTQFWEGUVJGUVGCF[UVCVGDTCPEJGSWCVKQPU CPFVJGCNVGTPCVGTGRTGUGPVCVKQPUYKVJWPEQWRNGFDTCPEJGUQH(KI
6JTGG9KPFKPI6TCPUHQTOGTU
5GRCTCVKPI4CPF:KUOQTGEQORNKECVGFPQY6JGTGHQTG4UJCNNDGKIPQTGFKPVJGHQNNQYKPIGZRNCPCVKQPU
4GUKUVCPEGUECPDGKPENWFGFJQYGXGTKHVJGUWRRQTVTQWVKPGU$%64#0QT64'.')CTGWUGF UGG5GEVKQP
CPF 6JGUVCTVKPIRQKPVKUVJGYGNNMPQYPUVCTEKTEWKVQH(KI+VUTGCEVCPEGUCTGHQWPFHTQOVJGRW
UJQTVEKTEWKVTGCEVCPEGU:
:
*.RW
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(KI5VCTEKTEWKVHQTVJTGGYKPFKPI
VTCPUHQTOGTYKVJRWXCNWGUDCUGFQPXQNVCIG
TCVKPIUQTYKVJCEVWCNXCNWGUTGHGTTGFVQQPG
UKFG
:
DCUGFQPVJGXQNVCIGTCVKPIUCPFQPGEQOOQPRQYGTDCUG5
.6RW
DCUG
5KPEGVJGRQYGTVTCPUHGTTCVKPIU5 DGVYGGP
*.
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*6
+HYGEJQQUG5
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KPUCOGWPKVUCURQYGTTCVKPIU5 5 5
DCUG
*.
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:*.RW
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:.6RW
5.6
%
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5*6
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:*6RW
5*6
%
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%
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VJGP
5.6
5*6
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:*6RW
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5.6
&
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5*.
(QTVJGGZCORNGWUGFKP5GEVKQPYKVJ: RW: RW: RWDCUGFQP
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:*RW ' & :.RW ' :6RW ' 0GZVVJGYGNNMPQYPUVCTFGNVCVTCPUHQTOCVKQPKUWUGFVQEQPXGTVVJGUVCTEKTEWKVQH(KIKPVQVJGFGNVC
EKTEWKVQH(KI
(KI&GNVCEKTEWKV
YJKEJIKXGUWUVJGUWUEGRVCPEGU
$*.RW '
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YKVJ
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% :.RW :6RW % :*RW :6RW
D
(QTVJGPWOGTKECNGZCORNG
$*.RW ' $*6RW ' $.6RW ' & 0QVGVJCVVJGUWUEGRVCPEGUKP'S C CTGPQVVJGTGEKRTQECNUQHVJGUJQTVEKTEWKVTGCEVCPEGU:WUGFKP'S 6JGRWOCVTKZ=T. ? DCUGFQP5
RW
DCUG
KUGCUKN[QDVCKPGFHTQO(KIYKVJVJGTWNGUHQTPQFCNCFOKVVCPEG
OCVTKEGUCU
5WUEGRVCPEG$KUWUGFJGTGHQTVJGTGEKRTQECNQHTGCEVCPEG:6JKUKUPQVUVTKEVN[EQTTGEVDGECWUG
UWUEGRVCPEGKUVJGKOCIKPCT[RCTVQHCPCFOKVVCPEG YJKEJKORNKGU$: $*.RW % $*6RW
& $*.RW
& $*6RW
& $*.RW
$*.RW % $.6RW
& $.6RW
& $*6RW
& $.6RW
$*6RW % $.6RW
=T.RW ?& '
QTHQTVJGPWOGTKECNGZCORNG
& & =T.RW
?&
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6JGOCVTKZ=T. ? KPCEVWCNXCNWGUKUHQWPFCU
RW
UV TQY CPF EQNWOP QH OWNVKRNKGF YKVJ 8*
T. & '
PF TQY CPF EQNWOP QH OWNVKRNKGF YKVJ 8.
KP 5
TF TQY CPF EQNWOP QH OWNVKRNKGF YKVJ 86
6JKUOCVTKZYKNNEQPVCKPVJGEQTTGEVVWTPUTCVKQU+HCNNSWCPVKVKGUCTGVQDGTGHGTTGFVQQPGUKFGUC[UKFG*VJGP
UKORN[UGV8 8 8 KP'S 5KPEGVJGRWXCNWGUCTGDCUGFQP/8#VJGXQNVCIGUKP'S OWUV
.
6
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DGKPM8
/CVTKZ4GRTGUGPVCVKQPQH5KPING2JCUG0%QKN6TCPUHQTOGTU
6JGPGYGTUWRRQTVTQWVKPGU$%64#0CPF64'.')CTGPQVNKOKVGFVQVJGRCTVKEWNCTECUGQHVYQQTVJTGG
EQKNUDWVYQTMHQTCP[PWODGTQHEQKNU+HGCEJYKPFKPIKUTGRTGUGPVGFCUQPN[QPGEQKN VJGPVTCPUHQTOGTUYKVJ
OQTGVJCPVJTGGEQKNUYKNNUGNFQODGGPEQWPVGTGFDWVKHGCEJYKPFKPIKUTGRTGUGPVGFCUCPCUUGODN[QHEQKNUVJGP
VTCPUHQTOGTOQFGNUHQTOQTGVJCPVJTGGEQKNUCTGFGHKPKVGN[PGGFGF$TGCMKPIQPGYKPFKPIWRKPVQCPCUUGODN[QH
EQKNUOC[YGNNDGTGSWKTGFHQT[GVVQDGFGXGNQRGFJKIJHTGSWGPE[OQFGNUYKVJUVTC[ECRCEKVCPEGU
6QGZRNCKPVJGEQPEGRVQPN[UKPINGRJCUG0EQKNVTCPUHQTOGTUCTGEQPUKFGTGFKPVJKUUGEVKQP6JGGZVGPUKQP
VQVJTGGRJCUGWPKVUKUFGUETKDGFKP5GEVKQP(QTUWEJCP0EQKNVTCPUHQTOGTVJGUVGCF[UVCVGGSWCVKQPUYKVJC
DTCPEJKORGFCPEGOCVTKZ=<?CTG
#EQKNKUCPCUUGODNCIGQHUWEEGUUKXGEQPXQNWVKQPUQHCEQPFWEVQTYJGTGCUCYKPFKPIKUCPCUUGODN[QH
EQKNU=?5KPEGCYKPFKPIOC[GKVJGTDGTGRTGUGPVGFCUQPGQTCUOQTGEQKNUVJGOQTGIGPGTCNVGTOEQKNKU
WUGFJGTG
8
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+
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+
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80
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GPGTIK\GFCPFCNNQVJGTEQKNUCTGQRGPEKTEWKVGFVJGPVJGOGCUWTGFXCNWGUHQT+ CPF8 8 RTQFWEGEQNWOPMQH
M
0
VJG=<?OCVTKZ
<KM ' 8K +M
7PHQTVWPCVGN[VJGUJQTVEKTEWKVKORGFCPEGUYJKEJFGUETKDGVJGOQTGKORQTVCPVVTCPUHGTEJCTCEVGTKUVKEUQH
VJGVTCPUHQTOGTIGVNQUVKPUWEJGZEKVCVKQPOGCUWTGOGPVUCUOGPVKQPGFKP5GEVKQP+VKUVJGTGHQTGOWEJDGVVGT
VQWUGVJGDTCPEJCFOKVVCPEGOCVTKZHQTOWNCVKQP
=+? ' =;? =8?
YJKEJKUVJGKPXGTUGTGNCVKQPUJKRQH'S 'XGPVJQWIJ=<?DGEQOGUKPHKPKVGHQT\GTQGZEKVKPIEWTTGPVQTKNN
EQPFKVKQPGF HQT XGT[ UOCNN GZEKVKPI EWTTGPVU =;? FQGU GZKUV CPF KU KP HCEV VJG YGNNMPQYP TGRTGUGPVCVKQP QH
VTCPUHQTOGTUWUGFKPRQYGTHNQYUVWFKGU(WTVJGTOQTGCNNGNGOGPVUQH=;?ECPDGQDVCKPGFFKTGEVN[HTQOVJGUVCPFCTF
UJQTVEKTEWKVVGUVFCVCYKVJQWVJCXKPIVQWUGCP[GSWKXCNGPVEKTEWKVU6JKUKUGURGEKCNN[KORQTVCPVHQT0 DGECWUG
VJGUVCTEKTEWKVUCVWTCDNGVTCPUHQTOGTEQORQPGPVKPVJG$2#'/62 KUKPEQTTGEVHQTOQTGVJCPVJTGGEQKNU
(QTCPKPVGTOGFKCVGUVGRKPQDVCKPKPI=;?VJGVTCPUHGTEJCTCEVGTKUVKEUDGVYGGPEQKNUCTGPGGFGF.GVVJGUG
VTCPUHGTEJCTCEVGTKUVKEUDGGZRTGUUGFCUXQNVCIGFTQRUDGVYGGPEQKNKCPFVJGNCUVEQKN0
TGFWEGF
<
TGFWEGF
<
8&80
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8&80
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+
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+
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<0&0&
80&&80
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+0&
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EQOOQPDCUGRQYGT5
DCUG
CPFQPVJGVTCPUHQTOGTXQNVCIGTCVKPIUQHVJG0EQKNU OWUVDG\GTQQT
(TQOJGTGQPKVKUDGUVVQYQTMYKVJRWSWCPVKVKGUQTYKVJSWCPVKVKGUTGHGTTGFVQQPGUKFGVQCXQKFECTT[KPI
VJGVWTPUTCVKQUVJTQWIJCNNVJGFGTKXCVKQPU
0
j +M RW ' M'
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HKTUVUJQYPD[5JKRNG[=?(QTCUJQTVEKTEWKVVGUVDGVYGGPKCPF0QPN[+ KP'S KUPQP\GTQCPF8
KRW
0
6JGPVJGKVJTQYDGEQOGU
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TGFWEGF
8K RW ' <K K RW
+K RW
6JGKORGFCPEGKPVJKUGSWCVKQPKUVJGUJQTVEKTEWKVKORGFCPEGDGVYGGPEQKNUKCPF0D[FGHKPKVKQP
TGFWEGF
UJQTV
<K K RW
' <K0 RW
DCUGFQPQPGEQOOQPDCUGRQYGT5 6JGQHHFKCIQPCNGNGOGPV<
DCUG
KUHQWPFD[TGNCVKPITQYUKCPFMQH'S
TGFWEGF
KMRW
VQVJGUJQTVEKTEWKVVGUVDGVYGGPKCPFM(QTVJKUVGUV+
+ CPF8
MRW
KRW
MRW
YKVJCNNQVJGTEWTTGPVUDGKPI
\GTQ6JGPTQYUKCPFMDGEQOG
TGFWEGF
8K RW & 80 RW ' <K K RW
TGFWEGF
& 80 RW ' <M K RW
TGFWEGF
& <K M RW
TGFWEGF
& <M M RW
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TGFWEGF
% <M M RW
C
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QTCHVGTUWDVTCEVKPI'S D HTQO C YKVJ<
8K RW ' <K K RW
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D
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TGFWEGF
<K M RW
UJQTV
UJQTV
UJQTV
<K 0 RW % <M 0 RW & <K M RW
'
DCUGFQPQPGEQOOQPDCUGRQYGT5
E
QT
UJQTV
KMRW
6JKUEQORNGVGUVJGECNEWNCVKQPQHVJGOCVTKZGNGOGPVUQH'S HTQO
DCUG
VJGUJQTVEKTEWKVVGUVFCVCYJKEJKUPQTOCNN[UWRRNKGFD[VJGOCPWHCEVWTGT
'S ECPPQVDGGZRCPFGFVQKPENWFGCNNEQKNUUKPEGCNNOCVTKZGNGOGPVUYQWNFDGEQOGKPHKPKVGYKVJVJG
GZEKVKPIEWTTGPVDGKPIKIPQTGF6QIGVVQVJGCFOKVVCPEGOCVTKZHQTOWNCVKQP 'S KUHKTUVKPXGTVGF
TGFWEGF
=;RW
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TGFWEGF
? ' =<RW
+PVJKUKPXGTUGTGNCVKQPUJKRVJGXQNVCIG8
EQNNGEVGFKPVQC0VJEQNWOPHQT8
QHVJGNCUVEQKNCNTGCF[GZKUVUCPFCNNVGTOUCUUQEKCVGFYKVJKVECPDG
0RW
6JG0VJTQYKUETGCVGFD[VCMKPIVJGPGICVKXGUWOQHTQYU0DCUGF
0RW
QP'S 6JKUTGUWNVUKPVJGHWNNOCVTKZTGRTGUGPVCVKQP
+ RW
; RW ; RW ;0 RW
8 RW
+ RW
; RW ; RW ;0 RW
8 RW
'
C
;0 RW ;0 RW ;00 RW
+0 RW
80 RW
YKVJ
TGFWEGF
;K M RW ' ;K M RW
HTQO 'S 0&
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D
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F
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6QEQPXGTVHTQORWVQCEVWCNXCNWGUCNNGNGOGPVUKP'S CTGOWNVKRNKGFD[VJGQPGEQOOQPDCUGRQYGT5
DCUG
CPFGCEJTQYCPFEQNWOPKKUOWNVKRNKGFYKVJ8 K
(QTVTCPUKGPVUVWFKGUVJGTGUKUVCPEGCPFKPFWEVCPEGRCTVUOWUVDGUGRCTCVGFKPCYC[UKOKNCTVQVJCVQH
5GEVKQP6JKUKUDGUVCEEQORNKUJGFD[DWKNFKPI=<
?QPN[HTQOVJGTGCEVCPEGRCTVQHVJGUJQTVEKTEWKVVGUVFCVC
TGFWEGF
YJKEJKU
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4
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M RW
GKVJGT RW NQCF NQUUGU KP UJQTVEKTEWKV VGUV DGVYGGP K CPF M QT UWO QH RW YKPFKPI
TGUKUVCPEGU
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=?l+'''
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FGVGTOKPGU VJG YCXGUJCRG QH VJG XQNVCIG +H VJG FKUVTKDWVKQP KU UKPWUQKFCN CU CUUWOGF KP VJG KFGCN OCEJKPG
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URCEGJCTOQPKEUCPFKVKUVJKUGHHGEVYJKEJRTQFWEGUVJGJCTOQPKEUKPU[PEJTQPQWUOCEJKPGU
6JGTGCTGOCP[FKHHGTGPVYC[UQHTGRTGUGPVKPIUCVWTCVKQPGHHGEVU=?CPFKVKUPQVEQORNGVGN[ENGCTCVVJKU
VKOG YJKEJ QPG EQOGU ENQUGUV VQ HKGNF VGUV TGUWNVU /QTG TGUGCTEJ QP VJKU VQRKE KU PGGFGF #V VJKU VKOG VJG
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6JGHNWZNKPMCIGQHGCEJYKPFKPIKPVJGFQTSCZKUECPDGTGRTGUGPVGFCUVJGUWOQHCNGCMCIGHNWZ YJKEJ
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6JGPQPNKPGCTEJCTCEVGTKUVKEQH(KIOCMGUKVKORQUUKDNGVQWUGVJGKPKVKCNK\CVKQPRTQEGFWTGQH5GEVKQP
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RQUKVKXGUGSWGPEGEQORQPGPVUECPDGECNEWNCVGFCPFVJGPWUGFVQEQTTGEVVJGKPVGTPCNXCTKCDNGUHQTUCVWTCVKQPGHHGEVU
5KPEGVJKUEQTTGEVKQPFQGUPQVEJCPIGVJGVGTOKPCNXQNVCIGUCPFEWTTGPVUVJGEQORNGVGPGVYQTMUQNWVKQPFQGUPQVJCXG
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6JGGSWKXCNGPVNKPGCTOCEJKPGRTQFWEGUEQTTGEVKPKVKCNEQPFKVKQPUHQTVJGFKHHGTGPVOQFGNWUGFKPVJGVTCPUKGPV
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HTCOGECPPQVDGWUGFDGECWUGVJCVYQWNFOCMGVJGKPFWEVCPEGUVKOGFGRGPFGPVTCVJGTVJCPEQPUVCPV
6JGHNWZGU8FU8SU8QCQPVJGU[PEJTQPQWUN[TQVCVKPICZGUCTGPQYRTGFKEVGFNKPGCTN[
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8FU V&)V
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9KVJCNNSWCPVKVKGUQPVJGTKIJVJCPFUKFGMPQYP GKVJGTHTQOVJGRTGEGFKPIVKOGUVGRQTHTQORTGFKEVKQP VJGVGTOKPCN
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X V
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XFU V
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XSU V
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KPVJGEQORNGVGGNGEVTKEPGVYQTMKUQPN[WUGFQPVJGCTOCVWTGUKFGYJGTGCUEQORGPUCVKQPDCUGFKPVGTHCEGUCTGUVKNN
OCKPVCKPGFHQTVJGHKGNFUVTWEVWTGYKPFKPIUCPFHQTVJGOGEJCPKECNU[UVGO9KVJVJKUKPOKPFVJGUQNWVKQPRTQEGUU
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9KVJVJGWPKXGTUCNOCEJKPGTGRTGUGPVGFCUXQNVCIGUQWTEGUDGJKPF4C KORNGOGPVGFCUEWTTGPVUQWTEGUKP
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GSWKXCNGPVQRGPEKTEWKVXQNVCIGUQH'S KHVJGTGCTGCP[GZVGTPCNPGVYQTMUEQPPGEVGFVQVJGHKGNF
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CPFUKPINGRJCUGEQORGPUCVKQPQPVJGTGUV 'ZVTCEVCUYGNNVJGQRGPEKTEWKVXQNVCIGUQHVJGPGVYQTMYJKEJ
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8F
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8S
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QH5GEVKQP+VCNUQTGSWKTGUNGUUEQORWVCVKQPVKOG+VUPWOGTKECNUVCDKNKV[ECPDGRCTVN[CVVTKDWVGFVQVJGDCEMYCTF
'WNGTOGVJQFKP'S #UUJQYPKP#RRGPFKZ+VJGDCEMYCTF'WNGTOGVJQFKUKFGPVKECNVQVJGVTCRG\QKFCN
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5CVWTCVKQP
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OCEJKPGOQFGNQH1PVCTKQ*[FTQYJKEJKPENWFGUUCVWTCVKQPGHHGEVUKPVJGNGCMCIGHNWZGUCUYGNN
6JGUCVWTCVKQPEWTXGQHVJGWPKXGTUCNOCEJKPGKUCRRTQZKOCVGFCUVYQRKGEGYKUGNKPGCTUGIOGPVUHQTVJGF
CZKUVJGSCZKUQTHQTDQVJ (KI $[WUKPIVJGUVCTEKTEWKVQH(KI
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8O F ' /F KO F
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TQWPFYKVJPQRTQPQWPEGFUCNKGPE[CUKPOQUVKPFWEVKQPOCEJKPGUCPFKPTQWPFTQVQTU[PEJTQPQWUOCEJKPGUVJGP
VJKUFGEQWRNGFCRRTQCEJNGCFUVQWPCEEGRVCDNGTGUWNVU6JGTGHQTGCVQVCNUCVWTCVKQPQRVKQPKUCXCKNCDNGYJKEJWUGU
CUQNWVKQPOGVJQFXGT[UKOKNCTVQVJCVFKUEWUUGFKP5GEVKQP
59+6%*'5
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QRGPRQUKVKQPCPF\GTQXQNVCIG 4 KPVJGENQUGFRQUKVKQP+HVJGUYKVEJDGVYGGPPQFGUMCPFOKUQRGPVJGP
DQVJPQFGUCTGTGRTGUGPVGFKPVJGU[UVGOQHPQFCNGSWCVKQPUYJGTGCUHQTVJGENQUGFUYKVEJDQVJMCPFODGEQOG
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VQCFFCECRCEKVCPEGHTQOMVQOHQTVJGTGRTGUGPVCVKQPQHVJGUVTC[ECRCEKVCPEGQTVJG4%ITCFKPIPGVYQTMQHCP
CEVWCNEKTEWKVDTGCMGT6JGEJCTCEVGTKUVKEUQHVJGCTEKPVJGEKTEWKVDTGCMGTCTGPQV[GVOQFGNNGFDWVYQTMKUKP
RTQITGUUVQKPENWFGVJGOKPHWVWTGXGTUKQPU
5YKVEJGUCTGPQVPGGFGFHQTVJGEQPPGEVKQPQHXQNVCIGCPFEWTTGPVUQWTEGUKHVJG[CTGEQPPGEVGFVQVJG
PGVYQTMCVCNNVKOGU6JGUQWTEGRCTCOGVGTU656#46CPF65612ECPDGWUGFKPRNCEGQHUYKVEJGUVQJCXGEWTTGPV
UQWTEGUVGORQTCTKN[EQPPGEVGFHQT656#46#V#65612CUGZRNCKPGFKP5GEVKQP(QTXQNVCIGUQWTEGUVJKUFGHKPKVKQP
YQWNFOGCPVJCVVJGXQNVCIGKU\GTQHQTV656#46CPFHQTV
65612YJKEJKORNKGUCUJQTVEKTEWKVTCVJGTVJCPC
FKUEQPPGEVKQP6JGTGHQTGUYKVEJGUCTGPGGFGFVQFKUEQPPGEVXQNVCIGUQWTEGU
5YKVEJGUCTGCNUQWUGFVQETGCVGRKGEGYKUGNKPGCTGNGOGPVUCUFKUEWUUGFKP5GEVKQP
$CUKE5YKVEJ6[RGU
6JGTGCTGHKXGDCUKEUYKVEJV[RGUKPVJG'/62YJKEJCTGCNNOQFGNNGFCUKFGCNUYKVEJGU6JG[FKHHGTQPN[
KPVJGETKVGTKCDGKPIWUGFVQFGVGTOKPGYJGPVJG[UJQWNFQRGPQTENQUG
6KOG%QPVTQNNGF5YKVEJ
6JKUV[RGKUKPVGPFGFHQTOQFGNNKPIEKTEWKVDTGCMGTUFKUEQPPGEVQTUCPFUKOKNCTUYKVEJKPIFGXKEGUCUYGNN
CUUJQTVEKTEWKVU6JGUYKVEJKUQTKIKPCNN[QRGPCPFENQUGUCV6%.15'+VQRGPUCICKPCHVGT612'0 KHVOCZ GKVJGT
CUUQQPCUVJGCDUQNWVGXCNWGQHVJGUYKVEJEWTTGPVHCNNUDGNQYCWUGTFGHKPGFEWTTGPVOCTIKPQTCUUQQPCUVJG
EWTTGPVIQGUVJTQWIJ\GTQ FGVGEVGFD[CUKIPEJCPIG CUKPFKECVGFKP(KI(QTVJGUKOWNCVKQPQHEKTEWKV
DTGCMGTUVJGNCVVGTETKVGTKQPHQTQRGPKPIUJQWNFPQTOCNN[DGWUGF6JGVKOGDGVYGGPENQUKPICPFQRGPKPIECPDG
FGNC[GFD[CWUGTFGHKPGFVKOGFGNC[
C EWTTGPVIQKPIVJTQWIJ\GTQ
D EWTTGPVNGUUVJCPOCTIKP
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6JGENQUKPIVCMGURNCEGCVVJGVKOGUVGRPGCTGUVVQ6%.15'KPVJG7$%XGTUKQP (KI C CPFCVVJGVKOG
UVGRYJGTGV$6%.15'HQTVJGHKTUVVKOGKPVJG$2#XGTUKQP (KI D C 7$%XGTUKQP
D $2#XGTUKQP
(KI%NQUKPIQHVKOGEQPVTQNNGFUYKVEJ
6%.15'UKIPCNUVQVJG'/62VJCVVJGUYKVEJUJQWNFDGENQUGFHTQOVJGXGT[DGIKPPKPI+HVJGUKOWNCVKQP
UVCTVUHTQOCWVQOCVKECNN[ECNEWNCVGFCEUVGCF[UVCVGEQPFKVKQPUVJGPVJGUYKVEJYKNNDGTGEQIPK\GFCUENQUGFKPVJG
UVGCF[UVCVGRJCUQTUQNWVKQP
6JG$2#'/62JCUCPCFFKVKQPCNVKOGEQPVTQNNGFUYKVEJV[RG 6#%5EQPVTQNNGFUYKVEJV[RG KPYJKEJ
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6JKUUYKVEJKUWUGFVQUKOWNCVGRTQVGEVKXGICRUICRUKPUWTIGCTTGUVGTUHNCUJQXGTUCETQUUKPUWNCVQTUGVE
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CUVJGCDUQNWVGXCNWGQHVJGXQNVCIGCETQUUVJGUYKVEJGZEGGFUCWUGTFGHKPGFDTGCMFQYPQTHNCUJQXGTXQNVCIG(QT
VJKU EJGEMKPI RTQEGFWTG VJG XQNVCIG XCNWGU CTG CXGTCIGF QXGT VJG NCUV VYQ VKOG UVGRU VQ HKNVGT QWV PWOGTKECN
QUEKNNCVKQPU1RGPKPIQEEWTUCVVJGHKTUVEWTTGPV\GTQRTQXKFGFCWUGTFGHKPGFFGNC[VKOGJCUCNTGCF[GNCRUGF6JKU
ENQUGQRGPE[ENGTGRGCVUKVUGNHYJGPGXGTVJGXQNVCIGGZEGGFUVJGDTGCMFQYPQTHNCUJQXGTXQNVCIGCICKPCUKPFKECVGF
KP(KI
(KI4GRGVKVKQPQHENQUGQRGPQRGTCVKQPHQTICRUYKVEJ
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EQPUVCPVDWVFGRGPFUQPVJGUVGGRPGUUQHVJGKPEQOKPIYCXG6JKUFGRGPFGPEGKUWUWCNN[UJQYPKPVJGHQTOQHC
XQNVCIGVKOGEJCTCEVGTKUVKE (KI YJKEJECPDGOGCUWTGFKPVJGNCDQTCVQT[HQTUVCPFCTFKORWNUGYCXGUJCRGU
7PHQTVWPCVGN[VJGYCXGUJCRGUQHRQYGTU[UVGOVTCPUKGPVUCTGWUWCNN[XGT[KTTGIWNCTCPFXQNVCIGVKOGEJCTCEVGTKUVKEU
ECPUGNFQODGWUGFVJGTGHQTG#PCN[VKECNOGVJQFUDCUGFQPVJGKPVGITCVKQPQHCHWPEVKQP
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V
mV
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EQWNFGCUKN[DGKORNGOGPVGF+P'S XQCPFMCTGEQPUVCPVUCPFDTGCMFQYPQEEWTUCVKPUVCPVVYJGTGVJG
KPVGITCNXCNWG(DGEQOGUGSWCNVQCWUGTFGHKPGFXCNWG=?(QTMVJKUKUVJGGSWCNCTGCETKVGTKQPQH&-KPF
=?0GKVJGTVJGXQNVCIGVKOGEJCTCEVGTKUVKEPQT'S JCUDGGPKORNGOGPVGFUQHCT
6JG $2# '/62 JCU CP CFFKVKQPCN ICR UYKVEJ V[RG 6#%5EQPVTQNNGF UYKVEJ V[RG KP YJKEJ VJG
DTGCMFQYPQTHNCUJQXGTKUEQPVTQNNGFD[CHKTKPIUKIPCNTGEGKXGFHTQOVJG6#%5RCTVQHVJG'/62 5GEVKQP 9KVJVJCVHGCVWTGXQNVCIGVKOGEJCTCEVGTKUVKEUQTETKVGTKCKPVJGHQTOQH'S ECPDGUKOWNCVGFKP6#%5D[
UMKNNGFWUGTU
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6JKUUYKVEJKUWUGFVQUKOWNCVGFKQFGUYJGTGEWTTGPVECPHNQYKPQPN[QPGFKTGEVKQPHTQOCPQFGOVQ
ECVJQFGM (KI 6JGFKQFGUYKVEJENQUGUYJGPGXGTXO$XM XQNVCIGXCNWGUCXGTCIGFQXGTVYQUWEEGUUKXGVKOG
UVGRUVQHKNVGTQWVPWOGTKECNQUEKNNCVKQPU CPFQRGPUCHVGTVJGGNCRUGQHCWUGTFGHKPGFVKOGFGNC[CUUQQPCUVJG
EWTTGPVKOMDGEQOGUPGICVKXGQTCUUQQPCUKVUOCIPKVWFGDGEQOGUNGUUVJCPCWUGTFGHKPGFOCTIKP
(KI&KQFGUYKVEJ
+PVJGCEUVGCF[UVCVGUQNWVKQPVJGFKQFGUYKVEJECPDGURGEKHKGFCUGKVJGTQRGPQTENQUGF
6J[TKUVQT5YKVEJ 6#%5%QPVTQNNGF
6JKUUYKVEJKUVJGDWKNFKPIDNQEMHQT*8&%EQPXGTVGTUVCVKQPU+VDGJCXGUUKOKNCTN[VQVJGFKQFGUYKVEJ
GZEGRVVJCVVJGENQUKPICEVKQPWPFGTVJGEQPFKVKQPQHXO$XMQPN[VCMGURNCEGKHCHKTKPIUKIPCNJCUDGGPTGEGKXGFHTQO
VJG6#%5RCTVQHVJG'/62 5GEVKQP /GCUWTKPI5YKVEJ
#OGCUWTKPIUYKVEJKUCNYC[UENQUGFKPVJGVTCPUKGPVUKOWNCVKQPCUYGNNCUKPVJGCEUVGCF[UVCVGUQNWVKQP
+VKUWUGFVQQDVCKPEWTTGPVQTRQYGTCPFGPGTI[KPRNCEGUYJGTGVJGUGSWCPVKVKGUCTGPQVQVJGTYKUGCXCKNCDNG
6JGPGGFHQTVJGOGCUWTKPIUYKVEJCTQUGDGECWUGVJG'/62FQGUPQVECNEWNCVGEWTTGPVUHQTEGTVCKPV[RGU
QHDTCPEJGUKPVJGWRFCVKPIRTQEGFWTGKPUKFGVJGVKOGUVGRNQQR6JGUGDTCPEJGUCTGGUUGPVKCNN[VJGRQN[RJCUGEQWRNGF
DTCPEJGUYKVJNWORGFQTFKUVTKDWVGFRCTCOGVGTU6JGWRFCVKPIRTQEGFWTGUEQWNFDGEJCPIGFHCKTN[GCUKN[VQQDVCKP
VJGEWTTGPVUCUCPCNVGTPCVKXGVQVJGOGCUWTKPIUYKVEJ
5VCVKUVKECN&KUVTKDWVKQPQH5YKVEJKPI1XGTXQNVCIGU
5KPEGEKTEWKVDTGCMGTUECPPGXGTENQUGKPVQCVTCPUOKUUKQPNKPGGZCEVN[UKOWNVCPGQWUN[HTQODQVJGPFUVJGTG
KUCNYC[UCUJQTVRGTKQFFWTKPIYJKEJVJGNKPGKUQPN[ENQUGFQTTGENQUGFHTQOQPGGPFYKVJVJGQVJGTGPFUVKNNQRGP
6TCXGNNKPIYCXGUCTGVJGPTGHNGEVGFCVVJGQRGPGPFYKVJVJGYGNNMPQYPFQWDNKPIGHHGEVCPFVTCPUKGPVQXGTXQNVCIGU
QH RW CV VJGTGEGKXKPIGPFCTGVJGTGHQTGVQDGGZRGEVGF+PTGCNKV[VJGQXGTXQNVCIGUECPDGJKIJGTHQTVJG
HQNNQYKPITGCUQPU
C
VJGNKPGKUVJTGGRJCUGYKVJVJTGGFKHHGTGPVOQFGRTQRCICVKQPXGNQEKVKGU
D
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ETGCVGTCVJGTEQORNKECVGFTGHNGEVKQPU
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VJGNKPGECRCEKVCPEGOC[UVKNNDGEJCTIGFWRHTQOCRTGEGFKPIQRGPKPIQRGTCVKQP VTCRRGFEJCTIG
KPTGENQUKPIQRGTCVKQPU F
VJGOCIPKVWFGQHVJGQXGTXQNVCIGFGRGPFUQPVJGKPUVCPVQHENQUKPI RQKPVQPYCXGUJCRG G
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RQUUKDNGUYKVEJKPIUWTIGQXGTXQNVCIGDGECWUGVJCVRCTVKEWNCTGXGPVJCUCNQYRTQDCDKNKV[QHGXGTQEEWTTKPICPF
DGECWUG VJG NKPG KPUWNCVKQP EQWNF PQV DG FGUKIPGF GEQPQOKECNN[ HQT VJCV UKPING JKIJ XCNWG (WTVJGTOQTG KV KU
KORQUUKDNG QT XGT[ FKHHKEWNV VQ MPQY YJKEJ EQODKPCVKQP QH RCTCOGVGTU YQWNF RTQFWEG VJG JKIJGUV RQUUKDNG
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RQUUKDN[YKVJXCTKCVKQPQHQVJGTRCTCOGVGTUVQQDVCKPCUVCVKUVKECNFKUVTKDWVKQPQHUYKVEJKPIUWTIGQXGTXQNVCIGU6JKU
KUWUWCNN[UJQYPKPVJGHQTOQHCEWOWNCVKXGHTGSWGPE[FKUVTKDWVKQP (KI (KI%WOWNCVKXGHTGSWGPE[FKUVTKDWVKQPQHTGEGKXKPIGPF
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QHVJGUYKVEJKPIQRGTCVKQPU+PUWNCVKQPFGUKIPHQTYKVJUVCPFKPICEGTVCKPQXGTXQNVCIGQHVGPTGHGTUVQCRTQDCDKNKV[
6JGYKVJUVCPFXQNVCIGQHKPUWNCVQTUFQGUPQVQPN[FGRGPFQPVJGRGCMXCNWGDWVQPVJGYCXGUJCRGCUYGNN(QT
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2 I EQPFWEVCPEGFGRGPFGPVJGCVFKUUKRCVKQP
6JG RCTCOGVGTUJ V CPF2 I CTGFGRGPFGPVQPVJGEJCTCEVGTKUVKEUQHVJGRCTVKEWNCTEKTEWKVDTGCMGT#FGVCKNGF
KPXGUVKICVKQPKPVQVJGWUGHWNPGUUQHXCTKQWUCTEGSWCVKQPUKURTGUGPVN[DGKPIFQPGD[%+)4'9QTMKPI)TQWR
2TCEVKECN#RRNKECVKQPQH#TE2J[UKEUKP%KTEWKV$TGCMGTU +HJKIJHTGSWGPE[QUEKNNCVKQPUFGXGNQRKPVJGCTEEWTTGPVRTKQTVQKPVGTTWRVKQPCUVJG[UQOGVKOGUFQKP
UYKVEJKPIQHHUOCNNKPFWEVKXGEWTTGPVUQTKPQVJGTEWTTGPVEJQRRKPIUKVWCVKQPUVJGPTGKIPKVKQPOC[QEEWTYKVJKP
E[ENGCHVGTEWTTGPVKPVGTTWRVKQP VJGVGTOTGUVTKMGKUWUGFVQFGUETKDGTGUWORVKQPQHEWTTGPVEQPFWEVKQPKHKVQEEWTU
E[ENGQTNQPIGTCHVGTEWTTGPVKPVGTTWRVKQPYJKEJOQUVNKMGN[QEEWTUKPVJGKPVGTTWRVKQPQHECRCEKVKXGEWTTGPVU (QT FGEKFKPI YJGVJGT TGKIPKVKQP QEEWTU VJG CTE GSWCVKQP QH 'S ECPPQV DG WUGF +PUVGCF VJG VTCPUKGPV
TGEQXGT[XQNVCIGKUEQORCTGFCICKPVJGFKGNGEVTKEUVTGPIVJYJKEJKPETGCUGUCUCXQNVCIGKUEQORCTGFCICKPUVVJG
FKGNGEVTKEUVTGPIVJYJKEJKPETGCUGUCUCHWPEVKQPQHVKOGCPFKHKVGZEGGFUKVVJGPTGKIPKVKQPQEEWTU(QTVJG
DTGCMFQYPKVUGNH6QGRNGT UGSWCVKQPECPDGWUGFYJKEJKUQHVJGHQTO=?
I '
V
K W FW
M U m
YJGTG
MEQPUVCPV
UICRURCEKPI
KEWTTGPVKPICR UVCTVKPIHTQOCPGZVTGOGN[UOCNNXCNWG XXQNVCIGCETQUUICR
574)'#44'56'45#0&2416'%6+8')#25
6QRTQVGEVIGPGTCVQTUVTCPUHQTOGTUECDNGU5(DCUUGUCPFQVJGTFGXKEGUCICKPUVNGXGNUQHQXGTXQNVCIGU
YJKEJEQWNFRGTOCPGPVN[FGUVTQ[VJGKTPQPUGNHTGUVQTKPIKPUWNCVKQPUWTIGCTTGUVGTUCTGKPUVCNNGFCUENQUGCURQUUKDNG
VQVJGRTQVGEVGFFGXKEG5JQTVEQPPGEVKQPUCTGKORQTVCPVVQCXQKFVJGFQWDNKPIGHHGEVQHVTCXGNNKPIYCXGUQPQRGP
GPFGF NKPGU GXGP KH VJG[ CTG UJQTV DWUUGU 5WTIG CTTGUVGTU JCXG PQTOCNN[ PQV DGGP WUGF HQT VJG RTQVGEVKQP QH
VTCPUOKUUKQPNKPGUDGECWUGQPGECPGCUKN[TGEQXGTHTQOKPUWNCVQTHNCUJQXGTUYKVJHCUVQRGPKPICPFTGENQUKPIQHEKTEWKV
DTGCMGTU UGNHTGUVQTKPI KPUWNCVKQP 5QOG WVKNKVKGU CTG UVWF[KPI VJG RQUUKDKNKV[ QH WUKPI UWTIG CTTGUVGTU QP
VTCPUOKUUKQPNKPGUVQQVQNKOKVUYKVEJKPIUWTIGQXGTXQNVCIGU
2TQVGEVKXGICRUCTGUGNFQOWUGFPQYCFC[UGZEGRVKPVJGRTQVGEVKQPQHUGTKGUECRCEKVQTUVCVKQPU
2TQVGEVKXG)CRU
2TQVGEVKXGICRUCTGETWFGRTQVGEVKQPFGXKEGU6JG[EQPUKUVQHCKTICRUDGVYGGPGNGEVTQFGUQHXCTKQWUUJCRGU
'ZCORNGUCTGJQTPUQTTKPIUQPKPUWNCVQTUCPFDWUJKPIUQTTQFICRUQPQTPGCTVTCPUHQTOGTU6JG[FQRTQVGEV
CICKPUVQXGTXQNVCIGUD[EQNNCRUKPIVJGXQNVCIGVQRTCEVKECNN[\GTQCHVGTURCTMQXGTDWVVJG[GUUGPVKCNN[RTQFWEGCUJQTV
EKTEWKVYJKEJOWUVVJGPDGKPVGTTWRVGFD[EKTEWKVDTGCMGTU#NUQVJGKTXQNVCIGVKOGEJCTCEVGTKUVKE (KI TKUGU
UVGGRN[HQTHCUVHTQPVUYJKEJOCMGUVJGRTQVGEVKQPCICKPUVHCUVTKUKPIKORWNUGUSWGUVKQPCDNG
2TQVGEVKXGURCTMICRUCTGUVKNNWUGFVQRTQVGEVUGTKGUECRCEKVQTU6JGTGVJGURCTMQXGTFQGUPQVKPETGCUGVJG
VTCPUOKUUKQPNKPGEWTTGPVDWVCEVWCNN[TGFWEGUKVDGECWUGVJGNKPGKORGFCPEGKPETGCUGUYJGPVJGUGTKGUECRCEKVQTKU
D[RCUUGF 5KPEG VJGURCTMICRKUWPCDNGVQKPVGTTWRVVJGEWTTGPVCD[RCUUEKTEWKVDTGCMGTOWUVDGENQUGFVQ
GZVKPIWKUJVJGCTEKPVJGURCTMICR (KI 6JKUD[RCUUDTGCMGTOWUVDGQRGPGFCICKPKHVJGUGTKGUECRCEKVQT
KUVQDGTGKPUGTVGF+PVJGHWVWTGRTQVGEVKXGURCTMICRUOC[DGTGRNCEGFD[OGVCNQZKFGUWTIGCTTGUVGTU
(KI5GTKGUECRCEKVQTRTQVGEVKQPUEJGOG
2TQVGEVKXGICRUCTGUKOWNCVGFKPVJG'/62YKVJVJGICRUYKVEJFKUEWUUGFKP5GEVKQP
5WTIG#TTGUVGTU
6JGTGCTGVYQDCUKEV[RGUQHUWTIGCTTGUVGTUPCOGN[UKNKEQPECTDKFGUWTIGCTTGUVGTUCPFOGVCNQZKFGUWTIG
CTTGUVGTU7PVKNCDQWV[GCTUCIQQPN[UKNKEQPECTDKFGCTTGUVGTUYGTGWUGFDWVVJGOGVCNQZKFGCTTGUVGTKUSWKEMN[
TGRNCEKPIVJGQNFGTV[RGVQVJGGZVGPVVJCVUQOGOCPWHCEVWTGTURTQFWEGQPN[OGVCNQZKFGCTTGUVGTUPQY
5KNKEQP%CTDKFG5WTIG#TTGUVGT
5KNKEQPECTDKFGCTTGUVGTUEQPUKUVQHCUKNKEQPECTDKFGTGUKUVQTYKVJCPQPNKPGCTXKEJCTCEVGTKUVKEKPUGTKGUYKVJ
C URCTM ICR (KI 6JGURCTMICREQPPGEVUVJGCTTGUVGTVQVJGU[UVGOYJGPVJGQXGTXQNVCIGGZEGGFUVJG
URCTMQXGTXQNVCIGCPFVJGTGUKUVQTNKOKVUVJGHQNNQYEWTTGPVCPFGPCDNGUVJGCTTGUVGTVQTGUGCN KPVGTTWRVVJGEWTTGPV
KPVJGICR 6QHCEKNKVCVGTGUGCNKPIUQECNNGFCEVKXGURCTMICRUJCXGDGGPFGUKIPGFKPYJKEJCPCTEXQNVCIGDWKNFU
WRCHVGTUQOGVKOG#TGUKUVQTDNQEMKPUGTKGUYKVJVJGICRKUPQVXGT[JKIJ V[RKECNN[EO CPFVQRTQFWEGVJG
FGUKTGFURCTMQXGTXQNVCIGCPFPQPNKPGCTTGUKUVCPEGHQTCRCTVKEWNCTXQNVCIGNGXGNOCP[UWEJDNQEMUCTGUVCEMGF
VQIGVJGTKPCUGTKGUEQPPGEVKQP6QCEJKGXGTGCUQPCDN[WPKHQTO
(KI0QPNKPGCTEJCTCEVGTKUVKEQHCM8UKNKEQPECTDKFGUWTIGCTTGUVGT
XQNVCIG FKUVTKDWVKQP CNQPI VJG UVCEM RCTCNNGN 4% ITCFKPI PGVYQTMU CTG WUGF YJKEJ CTG PQTOCNN[ KIPQTGF KP
UKOWNCVKQPU
5KNKEQPECTDKFGCTTGUVGTUCTGOQFGNNGFKPVJG'/62CUCPQPNKPGCTTGUKUVCPEGKPUGTKGUYKVJCICRYJKEJJCU
CEQPUVCPVURCTMQXGTXQNVCIG+PTGCNKV[VJGURCTMQXGTXQNVCIGFGRGPFUQPVJGUVGGRPGUUQHVJGKPEQOKPIYCXGCU
UJQYPKP(KI=?5KPEGUWTIGUKPCU[UVGOJCXGXGT[KTTGIWNCTUJCRGUTCVJGTVJCPVJGNKPGCTTKUGWUGFKP
VJGOGCUWTGOGPVUQH(KIVJGUVGGRPGUUFGRGPFGPEGQHVJGURCTMQXGTXQNVCIGKUPQVGCU[VQKORNGOGPVCU
CNTGCF[ FKUEWUUGF KP 5GEVKQP 6JG PQPNKPGCT TGUKUVCPEG KP UGTKGU YKVJ VJG ICR KU GKVJGT UQNXGF YKVJ VJG
EQORGPUCVKQPOGVJQF 5GEVKQP QTYKVJVJGRKGEGYKUGNKPGCTTGRTGUGPVCVKQP 5GEVKQP +PUKNKEQPECTDKFGUWTIGCTTGUVGTUYKVJEWTTGPVNKOKVKPIICRUCXQNVCIGDWKNFUWRCETQUUVJGICRCHVGT
VQzUYJKEJKUDGUVOQFGNNGFCUCPKPUGTVGFTCORV[RGXQNVCIGUQWTEG=?CUUJQYPKP(KI6JKUTCOR
XQNVCIGUQWTEGKUPQVRCTVQHVJG'/62CTTGUVGTOQFGNPQYDWVKVECPGCUKN[DGCFFGFCUCPGZVTCXQNVCIGUQWTEG
CHVGTQPGVTKCNTWPVQFGVGTOKPGYJGPURCTMQXGTQEEWTU6JKUICRXQNVCIGKUQPN[KORQTVCPVKPUYKVEJKPIUWTIG
UVWFKGU+PNKIJVPKPIUWTIGUVWFKGUKVECPDGKIPQTGFDGECWUGQHVJGVKOGFGNC[QHVQzU7UGHWN+'''
IWKFGNKPGUHQTOQFGNNKPIUKNKEQPECTDKFGCTTGUVGTUCTGHQWPFKP=?
C
D
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OGFKWOXQNVCIG
JKIJXQNVCIGNKIJVPKPIUWTIGRTQVGEVKQP
JKIJXQNVCIGNKIJVPKPICPFUYKVEJKPIUWTIGRTQVGEVKQP
(KI#TTGUVGTURCTMQXGTXQNVCIGVKOGEJCTCEVGTKUVKEHQTYCXGHTQPVUYKVJNKPGCTTKUG=?
4GRTKPVGFD[RGTOKUUKQPQH2NGPWO2WDNKUJKPI%QTRCPF$TQYP$QXGTK1GTNKMQP
+VKUFQWDVHWNYJGVJGTXGT[UQRJKUVKECVGFOQFGNUYKVJF[PCOKEEJCTCEVGTKUVKEUUWEJCUVJGV[RGOQFGTP
UV[NG5K%UWTIGCTTGUVGTDCUGFQP=?KPVJG$2#'/62CTGWUGHWNDGECWUGKVYQWNFDGCNOQUVKORQUUKDNGVQ
QDVCKPVJGTGSWKTGFFCVC$TCWPGT=?JCUFGXGNQRGFCOQFGNYKVJF[PCOKEEJCTCEVGTKUVKEUYKVJURGEKCNTGHGTGPEG
VQ)+5KPUWNCVKQPEQQTFKPCVKQPYJKEJCRRGCTUVQTGSWKTGNGUUFCVCVJCPVJGV[RGCTTGUVGT
(KI#TTGUVGTICREJCTCEVGTKUVKE
/GVCN1ZKFG5WTIG#TTGUVGT
/GVCNQZKFGQT\KPEQZKFGUWTIGCTTGUVGTUCTGJKIJN[PQPNKPGCTTGUKUVQTUYKVJCPCNOQUVKPHKPKVGUNQRGKPVJG
PQTOCNXQNVCIGTGIKQPCPFCPCNOQUVJQTK\QPVCNUNQRGKPVJGQXGTXQNVCIGRTQVGEVKQPTGIKQPCUUJQYPKP(KI
6JG[YGTGQTKIKPCNN[ICRNGUUDWVUQOGOCPWHCEVWTGTU
(KI8QNVCIGEWTTGPVEJCTCEVGTKUVKEQHCM8ICRNGUUOGVCNQZKFGUWTIG
CTTGUVGT=?l+'''
JCXGTGKPVTQFWEGFICRUKPVQVJGFGUKIP+VUPQPNKPGCTTGUKUVCPEGKUTGRTGUGPVGFD[CRQYGTHWPEVKQPQHVJGHQTO
K ' R
X
XTGH
S
YJGTGRXTGHCPFSCTGEQPUVCPVU V[RKECNXCNWGUHQTSVQ 5KPEGKVKUFKHHKEWNVVQFGUETKDGVJGGPVKTGTGIKQP
YKVJQPGRQYGTHWPEVKQPVJGXQNVCIGTGIKQPJCUDGGPFKXKFGFKPVQUGIOGPVUKPVJG$2#'/62YKVJGCEJUGIOGPV
FGHKPGF D[ KVU QYP RQYGT HWPEVKQP +P VJG 7$% '/62 QPN[ QPG HWPEVKQP KU CNNQYGF UQ HCT (QT XQNVCIGU
UWDUVCPVKCNN[DGNQYXTGHVJGEWTTGPVKUGZVTGOGN[UOCNN GIKR R HQTXXTGH CPFCNKPGCT
TGRTGUGPVCVKQPKUVJGTGHQTGWUGFKPVJKUNQYXQNVCIGTGIKQP+PVJGOGCPKPIHWNQXGTXQNVCIGRTQVGEVKQPTGIKQPVYQ
UGIOGPVUYKVJRQYGTHWPEVKQPU CTGWUWCNN[UWHHKEKGPV
6JGUVCVKEEJCTCEVGTKUVKEQH'S ECPDGGZVGPFGFVQKPENWFGF[PCOKEEJCTCEVGTKUVKEUUKOKNCTVQJ[UVGTGUKU
GHHGEVUVJTQWIJVJGCFFKVKQPQHCUGTKGUKPFWEVCPEG.YJQUGXCNWGECPDGGUVKOCVGFQPEGVJGCTTGUVGTEWTTGPVKU
CRRTQZKOCVGN[MPQYPHTQOCVTKCNTWP=?#OGVCNQZKFGUWTIGCTTGUVGTOQFGNHQTHCUVHTQPVEWTTGPVUWTIGUYKVJ
VKOGVQETGUVKPVJGTCPIGQHVQzUYCURTQRQUGFCPFEQORCTGFCICKPUVNCDQTCVQT[VGUVUD[&WTDCM=?6JG
DCUKEKFGCKUVQFKXKFGVJGUKPINGPQPNKPGCTTGUKUVCPEGKPVQORCTCNNGNPQPNKPGCTTGUKUVCPEGUYJKEJCTGUGRCTCVGFD[
NQYRCUUHKNVGTUCUKNNWUVTCVGFKP(KIHQTVYQRCTCNNGNPQPNKPGCTKVKGUYJKEJKUWUWCNN[UWHHKEKGPVKPRTCEVKEG6JG
4. EKTEWKVKUVJGNQYRCUUHKNVGTYJKEJUGRCTCVGUVJGVYQPQPNKPGCTTGUKUVCPEGUFGHKPGFD[K Q X Q CPFK X 6JG
KPFWEVCPEG.QTGRTGUGPVUVJGUOCNNDWVHKPKVGKPFWEVCPEGCUUQEKCVGFYKVJVJGOCIPGVKEHKGNFUKPVJGKOOGFKCVGXKEKPKV[
QHVJGUWTIGCTTGUVGTYJKNG4QKUWUGFQPN[VQFCORPWOGTKECNQUEKNNCVKQPU UGG5GEVKQP %KUVJGUVTC[
ECRCEKVCPEGQHVJGUWTIGCTTGUVGT6JGOQFGNQH(KIECPGCUKN[DGETGCVGFHTQOGZKUVKPI'/62GNGOGPVU+H
VJTGGUWEJOQFGNUYGTGEQPPGEVGFVQRJCUGUCDEVJGPVJGUKZPQPNKPGCTTGUKUVCPEGUYQWNFJCXGVQDGUQNXGFYKVJ
VJGEQORGPUCVKQPOGVJQFYKVJCUKZRJCUG6JGXGPKPGSWKXCNGPVEKTEWKV
(KI6YQUGEVKQPUWTIGCTTGUVGTOQFGNHQTHCUV
HTQPVUWTIGU=?
#UQOGYJCVFKHHGTGPVOQFGN (KI JCUDGGPRTQRQUGFD[-PGEJV=?+VEQPUKUVUQHCPQPNKPGCT
TGUKUVCPEG4 X COQTGQTNGUUEQPUVCPVECRCEKVCPEG%CPFCNKPGCTDWVHTGSWGPE[FGRGPFGPVKORGFCPEG< T 0Q+'''IWKFGNKPGUJCXG[GVDGGPRWDNKUJGFHQTVJGOQFGNNKPIQHOGVCNQZKFGUWTIGCTTGUVGTU6JGGPGTI[
CDUQTDGFKPVJGOKUCPKORQTVCPVFGUKIPHCEVQTCPFUJQWNFVJGTGHQTGDGEQORWVGFKPYJCVGXGTV[RGQHOQFGNKUWUGF
5KPEGGPGTI[CDUQTRVKQPOC[EJCPIGCUVJGU[UVGOKUGZRCPFGFKVKUKORQTVCPVVQEJGEMYJGVJGTTCVKPIUYJKEJYGTG
CRRTQRTKCVGKPKVKCNN[OC[RQUUKDN[DGGZEGGFGFKPHWVWTG[GCTU'PGTI[CDUQTRVKQPECRCDKNKV[KURTQDCDN[OQTGQHC
NKOKVCVKQPHQTUYKVEJKPIUWTIGUVJCPHQTNKIJVPKPIUWTIGU6JGUJCTREJCPIGHTQOVJGCNOQUVXGTVKECNVQVJGCNOQUV
JQTK\QPVCNUNQRGYJKEJNKOKVUQXGTXQNVCIGUCNOQUVKFGCNN[CVVJGCTTGUVGTNQECVKQPEQWNFRTQFWEGQUEKNNCVKQPUYKVJ
QXGTUJQQVCVNQECVKQPUUQOGFKUVCPEGHTQOVJGCTTGUVGTGURGEKCNN[KPUWDUVCVKQPUYKVJNQPIDWUTWPU6JKUOC[DG
CPQVJGTHCEVQTYQTVJYCVEJKPIHQT
/GVCNQZKFGUWTIGCTTGUVGTUCTGIGPGTCNN[UQNXGFYKVJVJGEQORGPUCVKQPOGVJQFKPVJG'/62YKVJKVGTCVKQPU
WUKPI0GYVQP UOGVJQFCUGZRNCKPGFKP5GEVKQP6JGRKGEGYKUGNKPGCTTGRTGUGPVCVKQPKUNGUUWUGHWNDGECWUG
VJGJKIJN[PQPNKPGCTEJCTCEVGTKUVKEQH'S KUPQVGCUKN[FGUETKDGFD[RKGEGYKUGNKPGCTUGIOGPVU
(KI#NVGTPCVKXGUWTIGCTTGUVGTOQFGN
+HVJGUWTIGCTTGUVGTKUGSWKRRGFYKVJCUJWPVURCTMICRCUKNNWUVTCVGFKP(KIVJGPKVKUUVKNNTGRTGUGPVGF
CUCPQPNKPGCTTGUKUVCPEGKPVJGUQNWVKQPRTQEGUUGZEGRVVJCVVJGHWPEVKQPHQT
(KI/GVCNQZKFGUWTIG
CTTGUVGTYKVJUJWPVURCTMICR
VJCVTGUKUVCPEGYKNNEJCPIGCDTWRVN[HTQO4 K 4 V DGHQTGURCTMQXGTVQ4 K CHVGTURCTMQXGT+HVJGUWTIGCTTGUVGT
KUGSWKRRGFYKVJCUGTKGUURCTMICRVJGPCXGT[JKIJTGUKUVCPEGKUCFFGFVQ4 K 4 K VQTGRTGUGPVVJGUGTKGUICR
DGHQTGURCTMQXGT
51.76+10/'6*1&5+06*''/62
6JGDCUKEVJGQT[DGJKPFVJGUQNWVKQPOGVJQFUHQTVJGVTCPUKGPVUKOWNCVKQPCPFHQTVJGCEUVGCF[UVCVGRJCUQT
UQNWVKQPJCUCNTGCF[DGGPGZRNCKPGFKP5GEVKQP'ZVGPUKQPUQHVJGDCUKEVJGQT[HQTOQTGEQORNKECVGFPGVYQTM
GNGOGPVUJCXGOQUVN[DGGPFKUEWUUGFKPVJGUGEVKQPUFGCNKPIYKVJVJGUGGNGOGPVU9JCVTGOCKPUVQDGGZRNCKPGFJGTG
CTGVJGXCTKQWUQRVKQPUHQTJCPFNKPIPQPNKPGCTKVKGUVJGNQCFHNQYQRVKQPCPFOGVJQFUHQTKPKVKCNK\KPIXCTKCDNGUYKVJ
OQTGVJCPQPGHTGSWGPE[EQORQPGPV
+PENWUKQPQH0QPNKPGCT'NGOGPVU
6JG OQUV EQOOQP V[RGU QH PQPNKPGCT GNGOGPVU CTG PQPNKPGCT KPFWEVCPEGU HQT VJG TGRTGUGPVCVKQP QH
VTCPUHQTOGT CPF UJWPV TGCEVQT UCVWTCVKQP CPF PQPNKPGCT TGUKUVCPEGU HQT VJG TGRTGUGPVCVKQP QH UWTIG CTTGUVGTU
0QPNKPGCTGHHGEVUKPU[PEJTQPQWUCPFWPKXGTUCNOCEJKPGUCTGJCPFNGFKPVJGOCEJKPGGSWCVKQPUFKTGEVN[CPFCTG
VJGTGHQTGPQVFGUETKDGFJGTG
7UWCNN[VJGPGVYQTMEQPVCKPUQPN[CHGYPQPNKPGCTGNGOGPVU+VKUVJGTGHQTGUGPUKDNGVQOQFKH[VJGYGNN
RTQXGPNKPGCTOGVJQFUOQTGQTNGUUVQCEEQOOQFCVGPQPNKPGCTGNGOGPVUTCVJGTVJCPVQWUGNGUUGHHKEKGPVPQPNKPGCT
UQNWVKQPOGVJQFUHQTVJGGPVKTGPGVYQTM6JKUJCUDGGPVJGRJKNQUQRJ[YJKEJJCUDGGPHQNNQYGFKPVJG'/626JTGG
OQFKHKECVKQPUEJGOGUJCXGDGGPWUGFQXGTVJG[GCTUPCOGN[
EWTTGPVUQWTEGTGRTGUGPVCVKQPUYKVJVKOGNCI)V PQNQPIGTWUGF EQORGPUCVKQPOGVJQFUCPF
RKGEGYKUGNKPGCTTGRTGUGPVCVKQPU
%WTTGPV5QWTEG4GRTGUGPVCVKQPYKVJ6KOG.CI)V
#UUWOGVJCVVJGPGVYQTMEQPVCKPUCPQPNKPGCTKPFWEVCPEGYKVJCIKXGPHNWZEWTTGPVEJCTCEVGTKUVKE8 K CPF
VJCVVJGPGVYQTMKULWUVDGKPIUQNXGFCVKPUVCPVV#NNSWCPVKVKGUCTGVJGTGHQTGMPQYPCVV)VKPENWFKPIHNWZ8 V)V YJKEJKUHQWPFD[KPVGITCVKPIVJGXQNVCIGCETQUUVJGPQPNKPGCTKPFWEVCPEGWRVQV)V2TQXKFGF)VKUUWHHKEKGPVN[
UOCNNQPGEQWNFWUG8 V)V VQHKPFCEWTTGPVK V)V HTQOVJGPQPNKPGCTEJCTCEVGTKUVKECPFKPLGEVVJKUCUCEWTTGPV
UQWTEGDGVYGGPVJGVYQPQFGUVQYJKEJVJGPQPNKPGCTKV[KUEQPPGEVGFHQTVJGUQNWVKQPCVKPUVCPVV+PRTKPEKRNGCP[
PWODGTQHPQPNKPGCTKVKGUEQWNFDGJCPFNGFVJKUYC[
(KIUJQYUCEWTTGPVNKOKVKPIFGXKEGYJGTGVJKUUKORNGOGVJQFYCUWUGFHQTVJGVYQ
C
EKTEWKVFKCITCO DTCPEJ
HGGFKPIPGVYQTMDTCPEJGU
DGVYGGPCPFEWTTGPV
NKOKVKPIFGXKEGUJWPV4KP
NQCFUYKVEJKPUJQTV
EKTEWKV
D
EJCTCEVGTKUVKEUQH
UCVWTCDNGTGCEVQTU
DQVJKFGPVKECN
(KI%WTTGPVNKOKVKPIFGXKEG
PQPNKPGCTKPFWEVCPEGU6JGUKOWNCVKQPTGUWNVUCTGRNQVVGFKP(KI6JGPWOGTKECNQUEKNNCVKQPUCTQWPFV
E[ENGUUGGOVQDGECWUGFD[VJGVKOGNCI)VUKPEGVJG[FKUCRRGCTYKVJOQTGUQRJKUVKECVGFVGEJPKSWGUKP(KI
5KPEGVJKUOGVJQFKUXGT[GCU[VQKORNGOGPVKVOC[DGWUGHWNKPURGEKCNECUGURTQXKFGFVJCVVJGUVGRUK\G
)VKUUWHHKEKGPVN[UOCNN+VKUPQVCDWKNVKPQRVKQPKPCP[QHVJGCXCKNCDNG'/62XGTUKQPUJQYGXGT
(KI5KOWNCVKQPTGUWNVUYKVJEWTTGPVUQWTEGTGRTGUGPVCVKQP
%QORGPUCVKQP/GVJQF
+PGCTNKGTXGTUKQPUQHVJG'/62VJGEQORGPUCVKQPOGVJQFYQTMGFQPN[HQTCUKPINGPQPNKPGCTKV[KPVJG
PGVYQTMQTKPECUGQHOQTGPQPNKPGCTKVKGUKHVJG[YGTGCNNUGRCTCVGFHTQOGCEJQVJGTVJTQWIJFKUVTKDWVGFRCTCOGVGT
NKPGU+VCRRGCTUVJCVVJGV[RGPQPNKPGCTKPFWEVCPEGKPVJG$2#'/62UVKNNJCUVJKUTGUVTKEVKQPKORQUGFQPKVDWV
HQTOQUVQVJGTV[RGUOQTGPQPNKPGCTKVKGUYKVJQWVVTCXGNVKOGUGRCTCVKQPCTGCNNQYGFPQY
6JGGZVGPUKQPQHVJGEQORGPUCVKQPOGVJQFVQOQTGVJCPQPGPQPNKPGCTKV[YCUHKTUVKORNGOGPVGFHQTOGVCN
QZKFGUWTIGCTTGUVGTUCPFNCVGTWUGFHQTQVJGTPQPNKPGCTGNGOGPVUCUYGNN
+PEQORGPUCVKQPDCUGFOGVJQFUVJGPQPNKPGCTGNGOGPVUCTGGUUGPVKCNN[UKOWNCVGFCUEWTTGPVKPLGEVKQPUYJKEJ
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VTCPUHQTOGTYJKEJOCMGUVJGCFOKVVCPEGOCVTKZQPVJGIGPGTCVQTUKFGUKPIWNCT QTGZVTGOGN[KNNEQPFKVKQPGF 6JG
UWOQHVJGVJTGGEWTTGPVU+# +$ +%DGEQOGUUNKIJVN[PQP\GTQ DGECWUGQHTQWPFQHHGTTQTU KPVJGKVGTCVKQP
RTQEGUUCPFVJKUGZVTGOGN[UOCNN\GTQUGSWGPEGEWTTGPVYKNNDGKPLGEVGFKPVQCPKPHKPKVG\GTQUGSWGPEGKORGFCPEG
QPVJGFGNVCUKFGYJKEJRTQFWEGUCNCTIG\GTQUGSWGPEGXQNVCIG6JKUECWUGUVJGOGVJQFVQFKXGTIG
(KI$2#VGUVECUGHQTNQCFHNQYQRVKQP
6Q TGUQNXG VJKURTQDNGOQPGECPEQPPGEVUJWPVKORGFCPEGUVQPQFGU#$%VQOCMGVJGOCVTKZ
PQPUKPIWNCT5KPEGPQFGKUCU[PEJTQPQWUOCEJKPGCPFUKPEGUWEJCOCEJKPGUJQWNFRTQRGTN[DGTGRTGUGPVGFCU
EWTTGPVUQWTEGUKPRCTCNNGNYKVJVJGPGICVKXGUGSWGPEGKORGFCPEGUVQJCPFNGWPDCNCPEGFECUGU 5GEVKQP CPCVWTCN
EJQKEGYQWNFDG
))
<UJWPV ' L
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:F % :S
9KVJVJKUUJWPVKORGFCPEGCPFYKVJCOQFKHKECVKQPQH'S VQCEEQWPVHQTVJGEWTTGPVKPVJKUKORGFCPEG
+M J '
4M&URGEKHKGF & L 3M&URGEKHKGF
8 J&
%
<UJWPV
8M( J&
VJGEWTTGPVUQWTEGKVGTCVKQPUEQPXGTIGKPKVGTCVKQPUYJKEJKUHCUVGTVJCP4CUOWUUGP UOGVJQFDWVOWEJUNQYGT
VJCPGZRGEVGFHTQOVJGGZRGTKGPEGYKVJVJGPQFGVGUVU[UVGOQH(KI
+VKUMPQYPHTQOUVCDKNKV[UVWFKGUVJCVUJWPVKORGFCPEGUURGGFWREQPXGTIGPEGKHVJG[CTGFGVGTOKPGFKPUWEJ
CYC[VJCVVJG[YQWNFRTQFWEGVJGURGEKHKGFRQYGTCVVJGTCVGFXQNVCIGQT
<UJWPV ' &
*8TCVGF*
2URGEKHKGF & L 3URGEKHKGF
YJGTG23KUPGICVKXGHQTNQCFUCPFRQUKVKXGHQTIGPGTCVKQP9KVJVJKUUJWPVKORGFCPEGVJGEWTTGPVUQWTEGKVGTCVKQP
OGVJQFFQGUKPFGGFEQPXGTIGSWKEMN[KPKVGTCVKQPU(KIUJQYU
C 4GCNRCTVQH8#N
D +OCIKPCT[RCTVQH8#N
(KI8QNVCIGUCUCHWPEVKQPQHKVGTCVKQPUVGRHQTVGUVECUGQH(KI
VJGTGCNCPFKOCIKPCT[RCTVQHXQNVCIG8#HQT4CUOWUUGP UOGVJQFCPFHQTVJGEWTTGPVUQWTEGKVGTCVKQPOGVJQFYKVJ
<UJWPVHTQO'S CPFHTQO'S (TQOCEQPXGTIGPEGUVCPFRQKPV<UJWPVHTQO'S KUQDXKQWUN[
DGUVDWVKHQPGYCPVUVQTGRTGUGPVVJGU[PEJTQPQWUOCEJKPGRTQRGTN[KPWPDCNCPEGFECUGU<UJWPVHTQO'S UJQWNFDGWUGF
#;CPUVCTVGFKORNGOGPVKPIVJKUOGVJQFKP1PVCTKQ*[FTQKP#HGYKUUWGUTGOCKPVQDGTGUQNXGF
1PGKUVJGVTGCVOGPVQH2*8*PQFGUYJGTGCEVKXGRQYGTCPFXQNVCIGOCIPKVWFGCTGURGEKHKGF+HQPGKUYKNNKPIVQ
RTGECNEWNCVGVJGKPVGTPCNKORGFCPEGQHVJGPGVYQTMCUUGGPHTQOGCEJ2 *8*PQFGVJGPQPGECPEQPUVTWEVCP
CRRTQZKOCVG6JGXGPKPGSWKXCNGPVEKTEWKVCHVGTVJGUQNWVKQPCVGCEJKVGTCVKQPUVGR9KVJ8MPQYPCPF+MPQYPDGKPIVJG
MPQYPXCNWGUCVC2*8*PQFGCPFYKVJ<6JGXJCXKPIDGGPRTGECNEWNCVGFQPEGCPFHQTCNNVJGQRGPEKTEWKVXQNVCIG
86JGXQHVJG6JGXGPKPGSWKXCNGPVEKTEWKV (KI KUUKORN[
86JGX ' 8MPQYP & <6JGX +MPQYP
D 4GNCVKQPUJKRDGVYGGP 8 CPF3
C %KTEWKV
(KI6JGXGPKPGSWKXCNGPVEKTEWKV
+HYGCUUWOGVJCVVJKU6JGXGPKPGSWKXCNGPVEKTEWKVKUPQYEQTTGEVN[FGHKPGFD[86JGXCPF<6JGXCPFVJCV8CPF+CTG
CNNQYGFVQEJCPIGVJGPYGQDVCKPCTGNCVKQPUJKRDGVYGGP3CPF*8*
3 ' $ *86JGX* *8* EQU" & *8* % ) *86JGX* *8* UKP"
YJGTG"KUVJGCPINGDGVYGGP86JGXCPF8CPF) L$<6JGX5KPEGVJGCPING"KUOQTGTGNCVGFVQCEVKXGRQYGT
HNQYUVJCPTGCEVKXGRQYGTHNQYUKVKUTGCUQPCDNGVQCUUWOGVJCV"FQGUPQVEJCPIGKHVJG3*8*TGNCVKQPUJKRKU
CNVGTGF9KVJ")$CPF*86JGX*MPQYPCPFYKVJVJGURGEKHKGFPQFGXQNVCIGOCIPKVWFGDGKPIWUGFHQT*8*'S
ECPDGUQNXGFHQT36JKUXCNWGQH3KUVJGPWUGFKPECNEWNCVKPIVJGEWTTGPVHQTVJGPGZVKVGTCVKQPUVGRHTQO
'S (QTVJGPQFGVGUVU[UVGOQH(KIVJKUOGVJQFEQPXGTIGFKPKVGTCVKQPUYJGPPQFGUCPF
YGTGVTGCVGFCU2*8*PQFGUYKVJPQFGUCPFTGOCKPKPI23PQFGU
6JGVTGCVOGPVQH23PQFGUCPF2*8*PQFGUKPVJTGGRJCUGWPDCNCPEGFECUGUKUUVKNNWPFGTFGXGNQROGPV
6QQDVCKPTGCNKUVKECPUYGTUVJGWUGTECPPQVURGEKH[RQYGTQTXQNVCIGOCIPKVWFGXCNWGUOQTGQTNGUUCTDKVTCTKN[CV
GCEJQPGQHVJGVJTGGRJCUGU+PUVGCFQPGOWUVMPQYJQYVJGNQCFQTIGPGTCVQTTGCEVUVQWPDCNCPEGFEQPFKVKQPU
#UGZRNCKPGFKP5GEVKQPU[PEJTQPQWUOCEJKPGUOWUVDGOQFGNNGFCUU[OOGVTKECNXQNVCIGUQWTEGUDGJKPF QT
U[OOGVTKECNEWTTGPVUQWTEGUKPRCTCNNGNYKVJ CZKORGFCPEGOCVTKZECNEWNCVGFHTQO<RQU<PGICPF<\GTQ6Q
QDVCKP TGUWNVU YJKEJ CTG RJ[UKECNN[ RQUUKDNG VJG UCOG TGRTGUGPVCVKQP YQWNF JCXG VQ DG WUGF KP VJG NQCF HNQY
KVGTCVKQPU5KOKNCTOQFGNUXCNKFHQTWPDCNCPEGFEQPFKVKQPUYQWNFJCXGVQDGFGXGNQRGFHQTVJGWPKXGTUCNOCEJKPG
CPFHQTQVJGTFGXKEGUYJKEJCRRGCTKPVJG'/62CUNQCFUQTIGPGTCVQTU
5VGCF[5VCVG5QNWVKQPUYKVJQWV*CTOQPKEU
6JGNKPGCTCEUVGCF[UVCVGRJCUQTUQNWVKQPCVQPGHTGSWGPE[JCUCNTGCF[DGGPFGUETKDGFKP5GEVKQPCPFVJG
OQFGNUQHVJGXCTKQWUGNGOGPVUYJKEJOWUVDGWUGFKPVJCVUQNWVKQPJCXGDGGPFKUEWUUGFKPVJGTGURGEVKXGUGEVKQPU
6JG TQWVKPGHQTVJGUVGCF[UVCVGUQNWVKQPYCUCFFGFD[,99CNMGTQTKIKPCNN[VQQDVCKPCEUVGCF[UVCVGKPKVKCN
EQPFKVKQPUCWVQOCVKECNN[.CVGTKVDGECOGCWUGHWNVQQNQPKVUQYPGIHQTUVWF[KPIEQORNKECVGFEQWRNKPIGHHGEVU
DGVYGGPEKTEWKVUQPVJGUCOGTKIJVQHYC[ GZCORNGKP(KI 6JG'/62VJGTGHQTGJCUCPQRVKQPVQVGTOKPCVG
VJGTWPCHVGTVJGUVGCF[UVCVGUQNWVKQPCPFVQVCDWNCVGVJGRJCUQTXCNWGUKPCUOWEJFGVCKNCUVJGWUGTOC[YKUJ
6JGXGPKP'SWKXCNGPV%KTEWKVU
+HCNCTIGPGVYQTMKUVQDGUQNXGFTGRGCVGFN[YKVJQPN[CHGYRCTCOGVGTUXCTKGFGCEJVKOGVJGPKVOC[DG
DGUVVQIGPGTCVGCP/RJCUG6JGXGPKPGSWKXCNGPVEKTEWKVHQTVJGNCTIGPGVYQTMHKTUVCUKNNWUVTCVGFKP(KI6JG
RCTCOGVGTU QHVJG6JGXGPKPGSWKXCNGPVEKTEWKVECPGCUKN[DGQDVCKPGFYKVJVJG'/62HTQO/ UVGCF[UVCVG
UQNWVKQPUQHVJGNCTIGPGVYQTMCUHQNNQYU
(KI2CTCOGVGTXCTKCVKQPKPDTCPEJGU/
4GOQXGVJGDTCPEJGU/ YJKEJOC[DGEQWRNGFCOQPIVJGOUGNXGU HTQOVJGNCTIGPGVYQTM1DVCKPC
UVGCF[UVCVGUQNWVKQPCPFTGEQTFVJGQRGPEKTEWKVDTCPEJXQNVCIGUCETQUUVJGPQFGRCKTUCCDDCVVJG
NQECVKQPUYJGTGVJG/DTCPEJGUYGTGTGOQXGF+HVJGDTCPEJGUCTGCNNEQPPGEVGFHTQOPQFGVQITQWPFVJGP
VJGUGDTCPEJXQNVCIGUCTGUKORN[PQFGXQNVCIGU6JKUHKTUVUVGCF[UVCVGUQNWVKQPRTQFWEGUVJGQRGPEKTEWKV
XQNVCIGXGEVQTQHVJG/RJCUG6JGXGPKPGSWKXCNGPVEKTEWKV
8C&C
8D&D
=86JGX ? '
8/&/
(KPFVJGKORGFCPEGOCVTKZ
<CC <CD <C/
=<6JGX ? ' <DC <DD <D/
</C </D <//
QHVJG6JGXGPKPGSWKXCNGPVEKTEWKVEQNWOPD[EQNWOPYKVJ/UVGCF[UVCVGUQNWVKQPU(KTUVUJQTVEKTEWKVCNN
XQNVCIGUQWTEGUKPVJGNCTIGPGVYQTM GCUKGUVYC[VQFQVJKUKUVQUGVVJGKTCORNKVWFGUVQ\GTQUKORN[TGOQXKPI
VJGOHTQOVJGFCVCHKNGYQWNFETGCVGQRGPEKTEWKVU CPFECPEGNCNNEWTTGPVUQWTEGU UGVCORNKVWFGUVQ\GTQQT
TGOQXGVJGOHTQOFCVCHKNG 6QQDVCKPEQNWOPMQH=<6JGX?EQPPGEVQPGEWTTGPVUQWTEGQH# 4/5 VQ
PQFGMCPFCUGEQPFEWTTGPVUQWTEGQH# 4/5 VQPQFGMCPFCUMHQTCUVGCF[UVCVGUQNWVKQP6JGP
VJGGNGOGPVUQHVJGMVJEQNWOPQH=<6JGX?CTGUKORN[VJG4/5DTCPEJXQNVCIGU
<CM
8C&C
<DM
8D&D
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</M
8/&/
9KVJ=86JGX?CPF=<6JGX?MPQYPVJGNCTIG WPEJCPIGF RCTVQHVJGPGVYQTMKUFGUETKDGFD[VJG/RJCUG
6JGXGPKPGSWKXCNGPVEKTEWKVQH(KIYKVJKVUDTCPEJGSWCVKQP
=8? ' =86JGX ? & =<6JGX ? =+?
(KI/DTCPEJ6JGXGPKPGSWKXCNGPVEKTEWKV
YJGTG=8?CPF=+?CTGDTCPEJXQNVCIGUCPFEWTTGPVU+HVJGUGDTCPEJGUCTGRCUUKXGYKVJCDTCPEJKORGFCPEGOCVTKZ
=<DTCPEJ?YJQUGXCNWGUCTGVQDGXCTKGFTGRGCVGFN[VJGP
=8? ' =<DTCPEJ? =+?
YJKEJECPDGUQNXGFYKVJ'S HQTVJGEWTTGPVU
=+? ' 6 =<6JGX ? % =<DTCPEJ ? >& =86JGX ?
'/62WUGTUOC[YCPVVQYTKVGVJGKTQYPRTQITCOVQUQNXG'S TCVJGTVJCPWUGVJG'/62HQTKV
5WEJC6JGXGPKPGSWKXCNGPVEKTEWKVYCUWUGFD[$2#VQUVWF[TGUQPCPEGRTQDNGOUQPCUJWPVEQORGPUCVGF
VTCPUOKUUKQPNKPGYJKEJKUUYKVEJGFQHHCVDQVJGPFUDWVYJKEJKUUVKNNECRCEKVKXGN[EQWRNGFVQRCTCNNGNEKTEWKVUQPVJG
UCOGTKIJVQHYC[$GECWUGQHEQORNKECVGFVTCPURQUKVKQPUEJGOGUVJGEQORNGVGPGVYQTMKUHCKTN[NCTIGYJGTGCU
VJG6JGXGPKPGSWKXCNGPVEKTEWKVQHVJGPGVYQTMUGGPHTQOVJGVJTGGUJWPVTGCEVQTEQPPGEVKQPRQKPVU#$%JCUQPN[
CZ=<6JGX?OCVTKZ(KIUJQYUVJGTGUWNVUQHVJKUUVWF[KPYJKEJVJGKPFWEVCPEGQHVJGUJWPVTGCEVQTUYCU
XCTKGFKPUOCNNUVGRUHTQOVQ*
(KI4GUQPCPEGKPUJWPVTGCEVQTU
6JGXGPKPGSWKXCNGPVEKTEWKVUCTGKPRTKPEKRNGQPN[XCNKFCVVJGHTGSWGPE[CVYJKEJVJG[CTGECNEWNCVGF+PVJG
RTGEGFKPIGZCORNGKVYQWNFDGMPQYPJQYGXGTVJCVCPQRGPGPFGFNKPGKUUGGPD[VJGUJWPVTGCEVQTCUCECRCEKVCPEG
WRVQUQOGHTGSWGPE[YC[CDQXG*\CPFVJCVVJGEQWRNKPIVQGPGTIK\GFNKPGUKUECRCEKVKXGCUYGNN6JG6JGXGPKP
KORGFCPEGKUVJGTGHQTG<6JGX LT% QTKPVJGOCVTKZECUG
=%?& ' L T =<6JGX ?
6JGECRCEKVCPEGOCVTKZTGRTGUGPVCVKQPYQWNFVJGPDGWUGHWNHQTVTCPUKGPVUVWFKGU WRVQCEGTVCKPHTGSWGPE[ CUYGNN
(TGSWGPE[5ECP
6JGHKTUVCFFKVKQPVQVJGUVGCF[UVCVGUQNWVKQPTQWVKPGYCUCNQQRVQXCT[VJGHTGSWGPE[CWVQOCVKECNN[HTQO
HOKPVQHOCZGKVJGTKPNKPGCTUVGRUQH)HQTQPCNQICTKVJOKEUECNG#VGCEJHTGSWGPE[VJGUQNWVKQPKUQDVCKPGFKPVJG
UCOGYC[CUDGHQTG6JKUQRVKQPJCUDGEQOGMPQYPCUHTGSWGPE[UECP+PUVGCFQHIGVVKPIXQNVCIGUCPFEWTTGPVU
CUCHWPEVKQPQHVKOGVJGKTOCIPKVWFGUCPFCPINGUCTGQDVCKPGFCUCHWPEVKQPQHHTGSWGPE[6JKUQRVKQPKUXGT[WUGHWN
HQT HKPFKPI VJG HTGSWGPE[FGRGPFGPV KORGFCPEG QH C PGVYQTM UGGP HTQO C RCTVKEWNCT NQECVKQP 6Q QDVCKP VJG
KORGFCPEGCNNXQNVCIGUQWTEGUCTGUJQTVEKTEWKVGFCPFCNNEWTTGPVUQWTEGUCTGTGOQXGF#EWTTGPVUQWTEGQH#KU
VJGPCFFGFCETQUUVJGVYQPQFGUDGVYGGPYJKEJVJGKORGFCPEGKUVQDGQDVCKPGF6JGDTCPEJXQNVCIGYKNNDGGSWCN
VQVJGKORGFCPEG(KIUJQYUCPGZCORNGYJGTGVJGKORGFCPEGDGVYGGPVYQRJCUGUYCUEQORWVGFYKVJVJG
HTGSWGPE[UECPQRVKQPQHVJG'/62CUYGNNCUKPFKTGEVN[OGCUWTGFYKVJCRJCUGVQRJCUGHCWNV VKOGTGURQPUG
(QWTKGTVTCPUHQTOGFVQHTGSWGPE[TGURQPUG CURCTVQHCUVWF[VQKPXGUVKICVGRQVGPVKCNUWDU[PEJTQPQWUTGUQPCPEG
RTQDNGOU
(KI%QORCTKUQPDGVYGGPKORGFCPEGECNEWNCVGFYKVJ
HTGSWGPE[UECPCPFOGCUWTGFKORGFCPEG=?l
+'''
&KHHGTGPV(TGSWGPEKGUKP&KUEQPPGEVGF2CTVU
6JG$2#'/62KUECRCDNGQHHKPFKPIVJGUVGCF[UVCVGUQNWVKQPKPPGVYQTMUYKVJUQWTEGUJCXKPIFKHHGTGPV
HTGSWGPEKGURTQXKFGFVJCVVJGPGVYQTMKUFKUEQPPGEVGFKPVQUWDPGVYQTMUYKVJGCEJUWDPGVYQTMQPN[EQPVCKPKPI
UQWTEGUYKVJVJGUCOGHTGSWGPE[6JGPGGFHQTVJKUECRCDKNKV[CTQUGRTKOCTKN[KPEQPPGEVKQPYKVJWPKXGTUCNOCEJKPG
KPKVKCNK\CVKQP 5GEVKQP (QTGZCORNGVJGCTOCVWTGYKPFKPIUQHCU[PEJTQPQWUOCEJKPGCPFVJGEQPPGEVGFRQYGT
U[UVGOOWUVDGUQNXGFHQTCEEQPFKVKQPUYJGTGCUVJGHKGNFEKTEWKVTGSWKTGUCFEUQNWVKQP
6JGUCOGECRCDKNKV[ECPDGWUGFVQJCPFNGVTCRRGFEJCTIGQPCPKUQNCVGFNKPGCPF*8&%NKPMU+PVJGNCVVGT
ECUGVJGEQPXGTVGTUCTGGKVJGTTGRTGUGPVGFCUKORGFCPEGUQTEWTTGPVUQWTEGUQPVJGCEUKFGCPFCUXQNVCIGUQWTEGU
QPVJGFEUKFG6JKUKIPQTGUVJGEWTTGPVJCTOQPKEUQPVJGCEUKFGCPFVJGXQNVCIGJCTOQPKEUQPVJGFEUKFGDWVKV
FQGURTQFWEGTGCUQPCDNGKPKVKCNEQPFKVKQPUHQTVJGVTCPUKGPVUKOWNCVKQP(KIUJQYUUKOWNCVKQPTGUWNVUHQTCFE
VTCPUOKUUKQPNKPGYKVJUKZRWNUGEQPXGTVGTUYJKEJYGTGEQPPGEVGFVJTQWIJEQPXGTVGTVTCPUHQTOGTUVQCEPGVYQTMU
(KI4GEVKHKGTCPFKPXGTVGTXQNVCIGYKVJUKOWNCVKQP
UVCTVKPIHTQOCRRTQZKOCVGKPKVKCNEQPFKVKQPU
TGRTGUGPVGFCUVJTGGRJCUG6JGXGPKPGSWKXCNGPVEKTEWKVU#VNGCUVKPVJKUECUGVJGHKPCNUVGCF[UVCVGYCUTGCEJGF
CNOQUVKOOGFKCVGN[
6JGVJGQT[DGJKPFVJKUUKPINGUQNWVKQPYKVJOWNVKRNGHTGSWGPEKGUKUXGT[UKORNG#UUWOGVJCVVJGTGCTGVYQ
UWDPGVYQTMUCPFYKVJHTGSWGPEKGUHCPFH5KPEGVJG[OWUVDGFKUEQPPGEVGF'S JCUVJGHQTO
=;?
=8?
=;? =8?
'
=+?
=+?
CPFCNNVJCVKUTGSWKTGFKUVJCVHTGSWGPE[HDGWUGFKPHQTOKPI=;?CPFHKPHQTOKPI=;?
(QTFEUQNWVKQPUCPKPFWEVCPEGDTCPEJDGEQOGUCUJQTVEKTEWKVCPFVJGVYQPQFGUVJGTGHQTGEQNNCRUGKPVQ
QPGPQFG6QUQNXGFEEQPFKVKQPUGZCEVN[YQWNFVJGTGHQTGTGSWKTGRTQITCOOQFKHKECVKQPUYJKEJJCXGDGGPTGICTFGF
CUCNQYRTKQTKV[KVGOWPVKNPQY+PUVGCFFEUQWTEGUCTGTGRTGUGPVGFCUCEUQWTEGUQHVJGHQTO8EQUTVQT+EQUTV
YKVJTDGKPIXGT[NQY V[RKECNN[H*\ +PFWEVCPEGUCTGVJGPXGT[NQYKORGFCPEGUTCVJGTVJCPUJQTVEKTEWKVU
5VGCF[5VCVG5QNWVKQPYKVJ*CTOQPKEU
5VGCF[UVCVGJCTOQPKEUKPJKIJXQNVCIGVTCPUOKUUKQPU[UVGOUCTGRTKOCTKN[RTQFWEGFD[VTCPUHQTOGT CPF
RQUUKDN[ UJWPV TGCEVQT UCVWTCVKQP D[ *8&% EQPXGTVGT UVCVKQPU CPF D[ NCTIG TGEVKHKGT NQCFU GI CNWOKPWO
TGFWEVKQPRNCPVU +PTGEVKHKGTUCPFKPXGTVGTUVJGOCIPKVWFGQHJCTOQPKEUKUTGCUQPCDN[YGNNMPQYPCPFVJGUG
JCTOQPKEU ECP VJGTGHQTG DG TGRTGUGPVGF CU IKXGP EWTTGPV QT XQNVCIG UQWTEGU KP JCTOQPKE NQCF HNQY RTQITCOU
URGEKHKECNN[FGUKIPGFHQTJCTOQPKEUUVWFKGU+PEQPVTCUVJCTOQPKEUIGPGTCVGFD[VTCPUHQTOGTUCVWTCVKQPFGRGPF
ETKVKECNN[ QP VJG RGCM OCIPKVWFG CPF YCXGHQTO QH VJG XQNVCIG CV VJG VTCPUHQTOGT VGTOKPCNU YJKEJ KP VWTP CTG
KPHNWGPEGFD[VJGJCTOQPKEEWTTGPVUCPFVJGHTGSWGPE[FGRGPFGPVPGVYQTMKORGFCPEGU
6TCPUKGPVUKOWNCVKQPUYKVJVJG'/62YKNNEQPVCKPJCTOQPKEUGHHGEVUGKVJGTHTQOVTCPUHQTOGTUCVWTCVKQPQH
HTQO EQPXGTVGTU +H VJG UKOWNCVKQP KU ECTTKGF QWV NQPI GPQWIJ VQ NGV VJG VTCPUKGPVU UGVVNG FQYP VQ UVGCF[UVCVG
EQPFKVKQPUVJGPVJGYCXGHQTOUYKNNEQPVCKPVJGJCTOQPKEUYKVJTGCUQPCDNGCEEWTCE[WRVQCEGTVCKPQTFGTFGRGPFKPI
QPVJGUVGRUK\G)V#(QWTKGTCPCN[UKURTQITCOKUCXCKNCDNGCUCUWRRQTVTQWVKPGKPVJG'/62VQCPCN[\GUWEJ
YCXGHQTOU6JKUCRRTQCEJKUFKUEWUUGFHKTUV6JGTGCTGECUGUJQYGXGTYJGTGKVYQWNFDGFGUKTCDNGVQJCXGVJG
JCTOQPKEUCNTGCF[KPENWFGFKPVJGUVGCF[UVCVGKPKVKCNEQPFKVKQPUDGECWUGUVGCF[UVCVGJCTOQPKEUFQUQOGVKOGUJCXG
CPGHHGEVQPVJGVTCPUKGPVU6JKUKUFKUEWUUGFPGZVKP5GEVKQP
*CTOQPKEUHTQO.KPGCTCE5VGCF[5VCVG5QNWVKQPHQNNQYGFD[6TCPUKGPV5KOWNCVKQP
#UKORNGOGVJQFHQTQDVCKPKPIUCVWTCVKQPIGPGTCVGFJCTOQPKEUKUVQRGTHQTOCVTCPUKGPVUKOWNCVKQPYKVJVJG
'/62YJKEJUVCTVUHTQOCRRTQZKOCVGNKPGCTCEUVGCF[UVCVGEQPFKVKQPU(QTVJGKPKVKCNCEUVGCF[UVCVGUQNWVKQPVJG
OCIPGVK\KPIKPFWEVCPEGUQHVTCPUHQTOGTUCTGTGRTGUGPVGFD[VJGKTWPUCVWTCVGFXCNWGU+PVJGVTCPUKGPVUKOWNCVKQPVJG
QPN[FKUVWTDCPEGUYKNNVJGPDGVJGFGXKCVKQPUDGVYGGPVJGNKPGCTCPFPQPNKPGCTOCIPGVK\KPIKPFWEVCPEGTGRTGUGPVCVKQPU
6JGVTCPUKGPVUECWUGFD[VJGUGFGXKCVKQPUYKNNQHVGPUGVVNGFQYPVQVJGFKUVQTVGFUVGCF[UVCVGYKVJKPCHGYE[ENGU
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+++
RTQPQWPEGFYKVJVJGURCTUKV[VGEJPKSWGUFKUEWUUGFKP5GEVKQP+++
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OCVTKZ
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6JKUPQVCVKQPCNGNGICPEGKUFGEGKXKPIJQYGXGTDGECWUGKVKIPQTGUVJGEQORWVCVKQPCNDWTFGPQHQDVCKPKPIVJGKPXGTUG
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WRRGTVTKCPIWNCTRCTVYKVJQRVKOCNQTFGTKPIDCUGQP5EJGOG6JGHKNNKPGNGOGPVUCTGKPFKECVGFD[VJGU[ODQN1
YJGTGCU:KPFKECVGUVJGQTKIKPCNGNGOGPVU$GECWUGQHHKNNKPVJGPWODGTQHQHHFKCIQPCNGNGOGPVUKPVJGWRRGT
VTKCPIWNCTOCVTKZITQYUHTQOVQDWVVJKUKUUVKNNXGT[URCTUGEQORCTGFYKVJGNGOGPVUKPCHWNNOCVTKZ
+++
(KI+++0QP\GTQRCVVGTPQHU[OOGVTKEOCVTKZDGHQTGVTKCPIWNCTK\CVKQPUJQYPDGNQY
FKCIQPCNCPFCHVGTVTKCPIWNCTK\CVKQPUJQYPCDQXGFKCIQPCNHQTCPGVYQTMYKVJPQFGU
+++4QYD[TQY'NKOKPCVKQPYKVJ5VCVKE5VQTCIG
9JKNGVJGTGCTGOCP[XCTKCVKQPUQHVJGDCUKE)CWUUGNKOKPCVKQPCPFCUUQEKCVGFURCTUGUVQTCIGUEJGOGUVJG
DGUVEJQKEGHQTRQYGTU[UVGOCPCN[UKUUGGOUVQDGTQYD[TQYGNKOKPCVKQPYKVJUVCVKEUVQTCIG6JKUKUVJGUEJGOG
WUGFKPVJG'/62+VUVYQDCUKEEQPEGRVUCTG
C
6JGPQP\GTQRCVVGTPQHVJGVTKCPIWNCTK\GF=;?QT=)?OCVTKZPGGFPQVDGMPQYPKPCFXCPEG GXGPVJQWIJ
+++
KVEQWNFDGQDVCKPGFCUCD[RTQFWEVQHVJGTGQTFGTKPIUWDTQWVKPG DWVVJGPQFGUOWUVDGTGPWODGTGFPGCT
QRVKOCNN[HQTOKPKOWOHKNNKP
D
#UGCEJTQYQHVJGWRRGTVTKCPIWNCTOCVTKZKUDWKNVKVKUUVQTGFCYC[QPEGCPFHQTCNNCPFPGXGTEJCPIGF
CICKP UVCVKEUVQTCIG 5KPEGVJGPQFCNOCVTKEGUCTGU[OOGVTKEKPVJGUVGCF[UVCVGCUYGNNCUKPVJGVTCPUKGPV
UQNWVKQPRCTVVJGNQYGTVTKCPIWNCTOCVTKZKUPQVPGGFGF
%QPEGRV D TWNGUQWVVJGVGZVDQQMCRRTQCEJVQGNKOKPCVKQPUJQYPKP5GEVKQP+++KPYJKEJ\GTQUCTGRTQFWEGF
EQNWOPD[EQNWOPKPVJGNQYGTVTKCPIWNCTOCVTKZDGECWUGVJGTGUWNVKPITGFWEGFOCVTKEGUEJCPIGYKVJHKNNKPGNGOGPVU
HTQOQPGGNKOKPCVKQPUVGRVQVJGPGZV
4QYD[TQYGNKOKPCVKQPQPVJGOCVTKZGNGOGPVUYKVJUVCVKEUVQTCIGYQTMUDCUKECNN[CUHQNNQYU (KI+++ (KI+++4QYD[TQYGNKOKPCVKQP
5GVGNKOKPCVKQPUVGRM
+PETGCUGMD[
5VQRRTQEGUUKHM
$WKNFTQYMQH=;?HTQODTCPEJVCDNGUKPCQPGFKOGPUKQPCNYQTMKPITQYCTTC[ QTVTCPUHGTFCVCKPVQ
0
YQTMKPITQYKH=;?KUCNTGCF[CXCKNCDNG 7UGGKVJGTCHWNNYQTMKPITQYUEJGOGQTCRCEMGFYQTMKPITQY
UEJGOGCUFKUEWUUGFKP5GEVKQP+++
+PVJGYQTMKPITQYGNKOKPCVG; KHPQP\GTQ D[CFFKPIVJGCRRTQRTKCVGOWNVKRNGQHTQYQHVJGCNTGCF[
M
GZKUVKPIRCTVQHVJGWRRGTVTKCPIWNCTOCVTKZ6JGPGNKOKPCVG; KHPQP\GTQ KPCPCPCNQIQWUYC[VJGP;
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GVEWRVQ;
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0QVGVJCVTQYUQHVJGGZKUVKPIRCTVQHVJGWRRGTVTKCPIWNCTOCVTKZCTGQPN[TGECNNGFHTQO
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UVQTCIGDWVPQVOQFKHKGF UVCVKEUVQTCIG 5VQTGVJGFKCIQPCNGNGOGPVKPCVCDNGQHNGPIVJ0 QTKVUTGEKRTQECNQPEQORWVGTUYJGTGFKXKUKQPVCMGUOQTG
VKOGVJCPOWNVKRNKECVKQP CPFCFFVJGPQP\GTQGNGOGPVU;OVQVJGTKIJVQHKV O
M KPEQORCEVHQTOVQ
VJGGZKUVKPIRCTVQHVJGWRRGTVTKCPIWNCTOCVTKZGIYKVJVJGTQYRQKPVGTEQNWOPKPFGZUEJGOGQH5GEVKQP
+++5KPEGVJGOCVTKEGUKPVJG'/62CTGU[OOGVTKEQPN[VJGWRRGTVTKCPIWNCTOCVTKZJCUVQDGUVQTGF
4GVWTPVQUVGR
+HVJGTGKUQPN[QPGUQNWVKQPCUKPVJGUVGCF[UVCVGKPKVKCNK\CVKQPVJGPVJGTKIJVJCPFUKFG=+?KURTQEGUUGF
VQIGVJGTYKVJ=;?CUKHKVYGTGCPGZVTC 0 VJEQNWOP
(QTVJGTGRGCVUQNWVKQPUKPVJGVTCPUKGPVUQNWVKQPRCTVVJGFQYPYCTFQRGTCVKQPUCTGOCFGYKVJVJGTQYUQH
VJGWRRGTVTKCPIWNCTOCVTKZUKPEGVJGGNGOGPVUQHTQYMQHVJGWRRGTVTKCPIWNCTOCVTKZFKXKFGFD[VJGPGICVKXG
FKCIQPCN GNGOGPV CTG VJG OWNVKRNKECVKQP HCEVQTU WUWCNN[ UVQTGF KP EQNWOP M QH VJG NQYGT VTKCPIWNCT OCVTKZ +H
+++
;4'% - KUTGEKRTQECNQHVJGFKCIQPCNGNGOGPVKH4 + KUTKIJVJCPFUKFGKH;7 , CTGVJGQHHFKCIQPCNGNGOGPVUQH
TQYMQHVJGWRRGTVTKCPIWNCTOCVTKZCPFKH/7 , CTGVJGEQNWOPKPFKEGUOQHVJGUGGNGOGPVUVJGPGNKOKPCVKQPUVGR
MQHCTGRGCVUQNWVKQPYQWNFDGOQTGQTNGUUUKOKNCTVQVJGHQNNQYKPI(QTVTCPUVCVGOGPVU
#4 - ;4'% &1, DGIKPPKPIQHTQY GPFQHTQY
//7 ,
4 / 4 / ;7 , #
+++9QTMKPI4QY
(QTVJGYQTMKPITQYFKUEWUUGFKPUVGRQHVJGRTGEGFKPIUGEVKQPCPFKPFKECVGFKP(KI+++CHWNNTQYECP
DGWUGFYKVJCQPGFKOGPUKQPCNCTTC[QHFKOGPUKQP0KPYJKEJ\GTQGNGOGPVUCTGCEVWCNN[TGRTGUGPVGFD[\GTQ
XCNWGU+PVJGGNKOKPCVKQPUQHUVGRCPFKPUVQTKPIPQP\GTQGNGOGPVUKPUVGRQHVJGRTGEGFKPIUGEVKQPGCEJGNGOGPV
OWUVDGEJGEMGFYJGVJGTKVKUPQP\GTQ6JKUEQUVUGZVTCEQORWVGTVKOGYJKEJKUVJGRTKEGQPGJCUVQRC[HQTVJG
UKORNKEKV[ QH VJG HWNN YQTMKPI TQY UEJGOG YJGTG HKNNKPU HCNN PCVWTCNN[ KPVQ VJGKT RTQRGT NQECVKQP FWTKPI VJG
GNKOKPCVKQPRTQEGUUQHUVGR RTGEGFKPIUGEVKQP #HWNNYQTMKPITQYUEJGOGECPDGWUGFKPUKVWCVKQPUYJGTGVJG
GZVTCVKOGQHEJGEMKPIHQT\GTQUKUPQVXGT[KORQTVCPVKPVJGVQVCNEQORWVGTVKOG6JKUKUOQTGQTNGUUVJGECUGKP
VJGVTCPUKGPVUQNWVKQPRCTVQHVJG'/62YJGTGVJG=)?OCVTKZKUQPN[VTKCPIWNCTK\GFQEECUKQPCNN[PCOGN[CVVJG
DGIKPPKPIQHVJGVKOGUVGRNQQRCPFYJGPGXGTUYKVEJGUEJCPIGVJGKTRQUKVKQP6JGTGHQTGVJG7$%'/62CPFQNFGT
XGTUKQPUQHVJG$2#'/62WUGCHWNNYQTMKPITQYUEJGOG0QCFFKVKQPCNUVQTCIGKUPGGFGFHQTVJCVTQYDGECWUG
VJG QPGFKOGPUKQPCN CTTC[ PGGFGF KP VJG VKOG UVGR NQQR HQT VJG TKIJVJCPF UKFG KU CXCKNCDNG CV VJG VKOG QH
VTKCPIWNCTK\CVKQP
(QTWVOQUVURGGFRCEMGFYQTMKPITQYUEJGOGUUJQWNFDGWUGFGURGEKCNN[KHVJGOCVTKZKUTGVTKCPIWNCTK\GF
HTGSWGPVN[6JKUUKVWCVKQPCTQUGYKVJVJGUKOWNCVKQPQH*8&%U[UVGOUYJGTGVJGUYKVEJGUTGRTGUGPVKPIVJGXCNXGU
QRGPCPFENQUGCHVGTGXGT[UVGRUQTUQ0GYGTXGTUKQPUQHVJG$2#'/62VJGTGHQTGWUGCRCEMGFYQTMKPITQY
UEJGOGYJKEJKUGUUGPVKCNN[VJGUCOGCUVJGQPGFGUETKDGFKP=?KPVCDNG+:+PURKVGQHVJGPGEGUUKV[QHCFFKVKQPCN
KPFGZKPIVCDNGUGPQWIJUVQTCIGURCEGCPFEQORWVGTVKOGKUUCXGFVQLWUVKH[VJGCFFKVKQPCNEQORNKECVKQPUKVGPVCKNU
+++4QY2QKPVGT%QNWOP+PFGZ5VQTCIG5EJGOG
$GHQTG FKUEWUUKPI VJG EQORCEV UVQTCIG QH VJG WRRGT VTKCPIWNCT OCVTKZ KV UJQWNF DG TGOGODGTGF VJCV C
UGRCTCVGCTTC[KUWUGFHQTVJGYQTMKPITQYGI
4'#.419 0 HQTCHWNNYQTMKPITQYCPFTGCN PQVEQORNGZ OCVTKZGNGOGPVU
4'#.419 / HQTCRCEMGFYQTMKPITQYYKVJ/0CEEQTFKPIVQ=?
+06')'40':6 / -1. / #TQYRQKPVGTEQNWOPKPFGZUEJGOGKPVJGHQTOFKUEWUUGFPGZVQTKPCUKOKNCTHQTOUGGOUVQQHHGTVJGDGUV
EJQKEGKPVGTOUQHGCUGQHCEEGUUCPFGEQPQO[QHURCEG0QVGVJCVUWEJCUEJGOGECPPQVQPN[DGWUGFHQTUVQTKPI
VJGVTKCPIWNCTOCVTKZKPEQORCEVHQTODWVCNUQHQTUVQTKPIVJGQTKIKPCN=;?OCVTKZKPEQORCEVHQTO
6JG FKCIQPCN GNGOGPVU QT VJGKT TGEKRTQECNU QP EQORWVGTU YJGTG FKXKUKQP VCMGU PQVKEGCDN[ NQPIGT VJCP
+++
OWNVKRNKECVKQP CTGUVQTGFKPCQPGFKOGPUKQPCNCTTC[QHFKOGPUKQP0UC[KP;&+#) (KI+++ 6JGPQP\GTQQHH
FKCIQPCN GNGOGPVU QH VJG WRRGT VTKCPIWNCT OCVTKZ CTG UVQTGF TQYD[TQY KP CPQVJGT QPGFKOGPUKQPCN CTTC[ QH
FKOGPUKQPNCTIGTVJCP0 V[RKECNN[ 0 UC[KP;7YKVJVJGUVCTVKPICFFTGUUQHGCEJTQYCXCKNCDNGHTQOCTQY
RQKPVGTVCDNGQHNGPIVJ0 UC[KP-56#466JGCFFTGUUQHVJGNCUVGPVT[KPCTQYKUUKORN[VJGUVCTVKPICFFTGUU
QHVJGHQNNQYKPITQYOKPWUYJKEJGZRNCKPUYJ[CP 0 VJGPVT[KUPGGFGFKP-56#461DXKQWUN[VJGEQNWOP
PWODGTUIGVNQUVYJGPGNGOGPVUCTGRCEMGFKPVQ;76JGTGHQTGCPGZVTCEQNWOPKPFGZVCDNGUC[/7KUTGSWKTGF
CUKPFKECVGFKP(KI+++6JGQXGTJGCFDWTFGPQHVJKUGZVTCVCDNG
VCDNGQHNGPIVJVCDNGQHNGPIVJVCDNG;7HQTGNGOGPVUQHWRRGT
0HQTFKCIQPCN0 HQTTQYVTKCPIWNCTOCVTKZCPFVCDNG/7
GNGOGPVU QTRQKPVGTHQTCUUQEKCVGFEQNWOPKPFKEGU
VJGKTTGEKRTQECNU
;&+#)-56#46;7/7
TGCN KPVGIGT TGCN KPVGIGT
00
0 . 0
(KI+++4QYRQKPVGTEQNWOPKPFGZUVQTCIGUEJGOG
QHVJKUGZVTCVCDNGYJKEJDGEQOGUNGUUHQTEQORNGZGNGOGPVUKPVJGUVGCF[UVCVGUQNWVKQPDGECWUGQPN[;7YQWNFJCXG
VQDGTGRNCEGFD[VYQVCDNGUDWV/7YQWNFUVKNNDGCUKPINGVCDNGKUVTKXKCNHQTNCTIGOCVTKEGUYJGPVJGVQVCNUVQTCIG
TGSWKTGOGPVUCTGEQORCTGFYKVJVJGCNVGTPCVKXGQHUVQTKPICHWNNOCVTKZKPCVYQFKOGPUKQPCNCTTC['ZRGTKGPEGJCU
UJQYPVJCVVJGPWODGTQHYQTFUHQTEQORCEVUVQTCIGKURTQRQTVKQPCNVQ0 CUKUVJGEQORWVGTVKOG YJGTGCUVJG
PWODGTQHYQTFUTGSWKTGFHQTUVQTKPICHWNNOCVTKZKURTQRQTVKQPCNVQ0 6CDNG+++ +H0KP6CDNG+++
6CDNG+++5VQTCIGTGSWKTGOGPVHQTWRRGTVTKCPIWNCTOCVTKZCPFHQTXGEVQTUPGGFGFHQTTGRGCVUQNWVKQPUKPECUGQH
U[OOGVTKEOCVTKEGU=?
TGCNGNGOGPVU
EQORCEVUVQTCIG
HWNNUVQTCIG
EQORNGZGNGOGPVU
0
0 0 0
0
0 0 0
VJGPYQTFUYQWNFDGPGGFGFHQTCTGCNOCVTKZCPFYQTFUHQTCEQORNGZOCVTKZEQORCTGFYKVJ
YQTFUCPFYQTFUTGURGEVKXGN[HQTUVQTKPICHWNNOCVTKZ6JGUCXKPIUCTGCPFTGURGEVKXGN[
+++
+++5RGEKCN6GEJPKSWGUHQT5[OOGVTKE/CVTKEGU
6JGOCVTKEGU=;?HQTVJGUVGCF[UVCVGUQNWVKQPCUYGNNCU=)?HQTVJGVTCPUKGPVUQNWVKQPCTGDQVJU[OOGVTKE
5[OOGVT[ECPDGGZRNQKVGFKPVYQYC[U
C
6JGNQYGTVTKCPIWNCTOCVTKZPGGFPQVDGUVQTGFHQTTGRGCVUQNWVKQPUCPF
D
QPN[GNGOGPVUVQVJGTKIJVQHVJGFKCIQPCNOWUVDGRTQEGUUGFKPUVGRQH5GEVKQP+++YJKEJEWVUVJG
QRGTCVKQPEQWPVKPVJGGNKOKPCVKQPRTQEGUUHQTVJGOCVTKZCRRTQZKOCVGN[KPJCNH0QUCXKPIUECPDGICKPGF
KPTGRGCVUQNWVKQPUJQYGXGT
2QKPV C JCUDGGPFKUEWUUGFKP5GEVKQP+++KVTGUWNVUHTQOVJGHCEVVJCVCEQNWOPQHVJGNQYGTVTKCPIWNCT
OCVTKZKUGSWCNVQVJGTGURGEVKXGTQYQHVJGWRRGTVTKCPIWNCTOCVTKZFKXKFGFD[VJGPGICVKXGFKCIQPCNGNGOGPV6JKU
RQKPVKUGZRNQKVGFKPVJG'/62
2QKPV D KUVTWGDGECWUGVJGOWNVKRNKECVKQPHCEVQTUPGGFGFHQTVJGGNKOKPCVKQPQH; ; KPUVGRQH
M
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5GEVKQP+++CTGCNTGCF[CXCKNCDNGKPVJGFQVVGFEQNWOPQH(KI+++6JKUKUPQVGZRNQKVGFKPVJG'/62UKPEGKV
KUFKHHKEWNVVQCEEGUUVJKUFQVVGFEQNWOPFKTGEVN[YKVJQWVUQOGCFFKVKQPCNKPFGZKPIVCDNGU#UKORNGYC[QWVQHVJKU
RTQDNGOYQWNFDGVQUVQTGGNGOGPVUQHVJGFQVVGFEQNWOPCUVJG[CTGETGCVGFKPCUGRCTCVGVCDNGHQTVJGTQYUQHVJG
NQYGTVTKCPIWNCTOCVTKZDWVVJCVYQWNFFGHGCVVJGCFXCPVCIGQHRQKPV C CDQXG
6JGTGKUUQOGCFXCPVCIGKPFKXKFKPIVJGTQYUQHVJGWRRGTVTKCPIWNCTOCVTKZD[VJGFKCIQPCNGNGOGPVKPVJG
NQQR YJGTG VJG[ CTG UVQTGF KP UVGR QH 5GEVKQP +++ 6JKU YC[ 0 OWNVKRNKECVKQPU CTG UCXGF KP VJG
DCEMUWDUVKVWVKQPQHVJGTGRGCVUQNWVKQPCVVJGGZRGPUGQHQPGGZVTCOWNVKRNKECVKQPHQTGCEJQHHFKCIQPCNGNGOGPVKP
VJGVTKCPIWNCTOCVTKZ9JGPVJG'/62YCUHKTUVYTKVVGPKVYCUCUUWOGFVJCVVJGOCVTKZYKNNQPN[DGVTKCPIWNCTK\GF
QEECUKQPCNN[ DGHQTGGPVGTKPIVJGVKOGUVGRNQQRCPFYJGPGXGTUYKVEJGUEJCPIGVJGKTRQUKVKQP 6JGTGHQTGFKXKUKQP
D[VJGFKCIQPCNGNGOGPVUYCUEJQUGPVQMGGRVJGQRGTCVKQPEQWPVKPVJGTGRGCVUQNWVKQPUKPUKFGVJGVKOGUVGRNQQRCU
NQYCURQUUKDNG+PUKOWNCVKPI*8&%U[UVGOUVJGUCXKPIUKPVJGVKOGUVGRNQQROC[DGEQOGNGUUVJCPVJGGZVTC
QRGTCVKQPUPGGFGFHQTOWNVKRN[KPIVJGQHHFKCIQPCNGNGOGPVUYKVJVJGTGEKRTQECNQHVJGFKCIQPCNGNGOGPV
9KVJFKXKUKQPD[VJGFKCIQPCNGNGOGPVUVJGOCVTKZRTQEGUUKUQPN[OQFKHKGFKPUVGR QH5GEVKQP+++
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MO
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//7 ,
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4GVWTPVQUVGR
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YKVJVJGHQNNQYKPINQQR FKCIQPCNGNGOGPVGZENWFGF #4 &1::, DGIKPPKPIQHTQY GPFQHTQY
//7 ,
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6JGWUGQHRGTWPKVSWCPVKVKGUJCUDGGPEWUVQOCT[HQTUQOCP[[GCTUKPVJGGNGEVTKERQYGTKPFWUVT[VJCVKV
KUPQVCNYC[UTGEQIPK\GFVJCVCEVWCNXCNWGUECPDGWUGFLWUVCUGCUKN[CPFVJCVVJGRGTWPKVU[UVGOOC[JCXGQWVNKXGF
KVUWUGHWNPGUU6JKUYTKVGTUGGUPQCFXCPVCIGUKPYQTMKPIYKVJRGTWPKVSWCPVKVKGUCPFHGGNUOWEJOQTGEQOHQTVCDNG
YKVJCEVWCNXCNWGU
6JGYKFGURTGCFWUGHRGTWPKVSWCPVKVKGURTQDCDN[UVCTVGFYKVJPGVYQTMCPCN[\GTUKPVJG U(QTRQYGT
HNQYCPFUJQTVEKTEWKVUVWFKGUQPPGVYQTMCPCN[\GTURGTWPKVSWCPVKVKGUQHHGTGFVYQCFXCPVCIGUPCOGN[UECNKPIQH
KORGFCPEGUVQXCNWGUCXCKNCDNGQPVJGCPCN[\GTCPFVJGRQUUKDKNKV[QHTGRTGUGPVKPIVTCPUHQTOGTUCUUKORNGUGTKGU
KORGFCPEGUCUNQPICUVJGKTVWTPUTCVKQYCUKFGPVKECNVQVJGTCVKQQHVJGDCUGXQNVCIGUQPVJGVYQUKFGU5QOGYJCV
UKOKNCTCTIWOGPVUHQTRGTWPKVSWCPVKVKGUEQWNFDGOCFGKPVJGGCTN[FC[UQHFKIKVCNEQORWVGTUYKVJHKZGFRQKPV
CTKVJOGVKEYJGTGVJGQTFGTQHOCIPKVWFGQHKPVGTOGFKCVGCPFHKPCNTGUWNVUJCFVQDGCDQWVVJGUCOG1POQFGTP
EQORWVGTUYKVJHNQCVKPIRQKPVCTKVJOGVKEVJGTGKUPQTGCUQPJQYGXGTYJ[QPGUJQWNFP VYQTMFKTGEVN[YKVJCEVWCN
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R?+VJCUDGGPEWUVQOCT[VQWUGQPGEQOOQPDCUGRQYGT5 CRRCTGPVRQYGT HQTVJGGPVKTGU[UVGO V[RKECNN[
DCUG
/8# CPFCFKHHGTGPVDCUGXQNVCIGHQTGCEJXQNVCIGNGXGN GI8
M8CPF8
DCUG
M8KPC
DCUG
M8U[UVGO CUVJGDCUGXCNWGU6JGPVJGRGTWPKVSWCPVKVKGUKPCUKPINGRJCUGPGVYQTMCTG
8
+RW ' +CEVWCN @ DCUG
5DCUG
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8DCUG
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5 YQWNFDGVJGUKPINGRJCUGDCUGRQYGTQHQPGYKPFKPI GI/8# CPF8CUGYQWNFDGVJGDCUGXQNVCIG
DCUG
CETQUUGCEJYKPFKPIPCOGN[VJGRJCUGVQITQWPFDCUGXQNVCIG GI/M8CPF/M8 +PFGNVCEQPPGEVGF
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CETQUUGCEJYKPFKPIYQWNFPQYDGVJGRJCUGVQRJCUGDCUGXQNVCIG GIM8CPFM8 DCUG
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+''' 5VCPFCTF &KEVKQPCT[ QH 'NGEVTKECN CPF 'NGEVTQPKEU 6GTOU 5GEQPF 'FKVKQP RWDNKUJGF D[ +'''
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4