sheath-in [o(mr2)K,(mr,)+ Ko(mr,)I(mr) (61.25) sheath-mutual (61.26) 2Tr,,D Lo(mr3)K,(mr,)+ Ko(mr)I(mr2) (61.27) joHo. cos -{+R-4 sheath/pipe-insulation (61.28) where p= resistivity of conductor, D=I(mr3)K,(mT2)-I(mr2)K,(mr3),Y= Eulers constant =1.7811, I modified Bessel function of the first kind of order K= modified Bessel function of the second kind of order jou/p= reciprocal of the complex depth of penetration A submatrix of [Zp] is given in the following form ZniI (61.29) Zpit in Eq(61.29)is the impedance between the jth and kth inner conductors with respect to the pipe inn surface. When j= k, Zpik= Zpipe-in otherwise Zpit is given in Eq (61.31) =pmkm)+y‖ Kn(mq) (61.30) 2πaK,(m q)nu, Kn(mq)-mqKn(mq) In 4+H Ko( mq) K(mq) (61.31) d k cos(nek) 2u nu, K, (mq)-mgkn(mg) n where q is the inside radius of the pipe(Fig. 61.4) The formulation of the potential coefficient matrix of a pipe-type cable is similar to the impedance matrix. 0][P2l 00:P c 2000 by CRC Press LLC© 2000 by CRC Press LLC (61.25) (61.26) (61.27) (61.28) where r = resistivity of conductor, D = I1(mr3)K1(mr2) – I1(mr2)K1(mr3), g = Euler’s constant = 1.7811, Ii = modified Bessel function of the first kind of order i, Ki = modified Bessel function of the second kind of order i, and m = = reciprocal of the complex depth of penetration. A submatrix of [Zp] is given in the following form: (61.29) Zpjk in Eq. (61.29) is the impedance between the jth and kth inner conductors with respect to the pipe inner surface. When j = k, Zpjk = Zpipe-in; otherwise Zpjk is given in Eq. (61.31). (61.30) (61.31) where q is the inside radius of the pipe (Fig. 61.4). The formulation of the potential coefficient matrix of a pipe-type cable is similar to the impedance matrix. (61.32) Z m r D I mr K mr K mr I mr sheath-in = + r 2p 2 0 2 1 3 0 21 3 [ ( ) ( ) ( ) ( )] Z rrD sheath-mutual = r 2p 2 3 Z m r D I mr K mr K mr I mr sheath-out = + r 2p 3 0 3 1 2 0 31 2 [ ( ) ( ) ( ) ( )] Z j qRd qR i i i sheath/pipe-insulation = Ê + - Ë Á ˆ ¯ ˜ wm - p 0 1 2 22 2 2 cosh jw r m/ [ ] Z Z Z Z Z pjk pjk pjk pjk pjk = È Î Í Í ˘ ˚ ˙ ˙ Z m q K mq K mq j d q K mq n K mq mqK mq i n n n rn n pipe-in = + Ê Ë Á ˆ ¯ ˜ ¢ È Î Í Í ˘ ˚ ˙ ˙ = • Â r p wm 2 p m 0 1 2 1 ( ) ( ) ( ) ( )– ( ) Z j q S mq K mq K mq d d q n K mq n K mq mqK mq n pjk jk r j k n jk r n n rn n = + + Ê Ë Á ˆ ¯ ˜ ¢ È Î Í ˘ ˚ ˙ Ï Ì Ô Ô Ó Ô Ô ¸ ˝ Ô Ô ˛ Ô Ô = • Â wm p m q m m 0 0 1 2 1 2 2 1 ln ( ) ( ) cos( ) ( ) ( )– ( ) – [ ] [ ] [] [] [] [ ] [] [] [] [ ] P P P P i i i in = ××× ××× ××× È Î Í Í Í Í ˘ ˚ ˙ ˙ ˙ ˙ 1 2 0 0 0 0 0 0 M MOM