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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 257 28 "Appendix B
2--Dielectric Arcs" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 258 0 "" }
{TEXT 259 26 "Aleksandar Donev, 12/17/00" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }{TEXT 256 0 "" }{TEXT -1 99 "This worksheet develops th
e conjugate functions and related derivatives for the cost functions f
or " }{TEXT 260 15 "dielectric arcs" }{TEXT -1 99 ". The procedure is \+
identical to the one in Appendix B1, so many of the comments have been
 ommitted." }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "restart:" "6#%(
restartG" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "with(plots):" "6#
-%%withG6#%&plotsG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "The propose
d form for the voltage-current characteristic is now given as the inve
rse " }{XPPEDIT 18 0 "I(V)" "6#-%\"IG6#%\"VG" }{TEXT -1 6 ", not " }
{XPPEDIT 18 0 "V(I)" "6#-%\"VG6#%\"IG" }{TEXT -1 1 ":" }}}{EXCHG 
{PARA 0 "> " 0 "" {XPPEDIT 263 1 "J:=v->v*(1+tanh((v-u)/xi))/R:" "6#>%
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&,&F'F.%\"uG!\"\"F.%#xiGF7F.F.%\"RGF7F,F,F," }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 21 "Specific numbers are:" }}}{EXCHG {PARA 0 "> " 0 "" 
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$$\"1nm\"HYt7v\"Feu$\"1Wgm^Q40qFeu7$$\"1+++q(G**y\"Feu$\"1k)***z]rfrFe
u7$$\"1nm;9@BM=Feu$\"1Wmmc%GpL(Feu7$$\"1LLL`v&Q(=Feu$\"1HLL8-V&\\(Feu7
$$\"1++DOl5;>Feu$\"1+++XhUkwFeu7$$\"1++v.Uac>Feu$\"1+++:o<EyFeu7$$\"\"
#F($\"\")F(-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$Q\"v6\"Q%i(
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dy know how to invert this from the previous Appendix B1:" }}}{EXCHG 
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2*3`i\"Fip7$$\"+v>:nmFip$\"+%*zym;Fip7$$\"+0a#o$oFip$\"+^j?4<Fip7$$\"+
`Q40qFip$\"+jMF^<Fip7$$\"+\"3:(frFip$\"+q(G**y\"Fip7$$\"+e%GpL(Fip$\"+
:@BM=Fip7$$\"+:-V&\\(Fip$\"+av&Q(=Fip7$$\"+ZhUkwFip$\"+Pl5;>Fip7$$\"+<
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0 "" {TEXT -1 131 "Finding the power function now is a but trickier. A
 straightforward attempt at integraing the LambertW function fails in \+
this case:" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "assume(xi>0,u>0
,R>0):" "6#-%'assumeG6%2\"\"!%#xiG2F'%\"uG2F'%\"RG" }}}{EXCHG {PARA 0 
"> " 0 "" {XPPEDIT 19 1 "f:=unapply(simplify(Int(V(J),J=0..i)),i);" "6
#>%\"fG-%(unapplyG6$-%)simplifyG6#-%$IntG6$-%\"VG6#%\"JG/F1;\"\"!%\"iG
F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%#i|irG6\"6$%)operatorG
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#$\"+bhT`%*!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "But we can use \+
the equivalent formula that does not use the inverse, " }{XPPEDIT 18 
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129 ".  The result is analytical, but the expression is rather complic
ated and not shown (hand simplification is needed in this case):" }}}
{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 261 1 "f := unapply(simplify(int(U*d
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{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "From here on it's the same as in A
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