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EMTDC
functions with increasing accuracy as the thickness of the annulus
becomes small compared with the radius of the annulus. The
advantage to using these formulae over the Bessel function based
impedances is mainly stability and speed of solution. The error
introduced by the approximation equations is negligible in the time
domain.
The equations for the individual impedances z1 to z6 in Figure 8-7
are given as follows. Both the Bessel function and equivalent
approximate formula are presented for comparison:
Internal Impedance of Core Outer Surface (Solid Cylinder):
Approximate:
m​ρ1​ ​
0.3179 · ​ρ1​ ​
_____
_________
 ​ 
​Z1​ ​= ​ 
 ​ 
 
· coth(0.733 · ​mr​1​) + ​
2π · ​r1​ ​
π · r​
​​ 12​​ ​
Bessel:
(mr1)
mρ1 I_____
0
_____
​ 
Z1 = jω · ​ 
  
​· ​
2π · r1 I1(mr1)
(8-8)
Internal Impedance of Core Outer Surface (Annulus):
Approximate:
mρ1
ρ1
_____
____________
Z1 = ​ 
 ​ 
· coth[m(r1 - r0)] + ​ 
 
  ​
2π · r1
2π · r1 · (r0 + r1)
Bessel:
µ
(mr1) · K1(mr0) + K0(mr1) · I1(mr0)
µ
0 1 ___
0
1 1
____
___________________________
Z1 = jω · ​  ​ 
 · mr
   
    ​
​    ​ · ​
2π
I1(mr1) · K1(mr0) - K1(mr1) · I1(mr0)
1
(8-9)
Core Outer Insulator Impedance:
(  )
µ
µ
r__2
0 2
____
Z2 = jω · ​  ​ 
 · In​ ​r ​   ​
2π
1
(8-10)
Internal Impedance of Sheath Inner Surface:
Approximate:
mρ2
ρ2
_____
____________
Z3 = ​ 
  
​· coth[m(r3 - r2)] + ​ 
 
  ​
2π · r2
2π · r2 · (r3 + r2)
Bessel:
µ
µ
I0(mr2) · K1(mr3) + K0(mr2) · I1(mr3)
0 3 ___
1 ______________________
____
Z3 = jω · ​  ​ 
 · ​    ​ · ​
   
    ​
2π mr2 I1(mr3) · K1(mr2) - K1(mr3) · I1(mr2)
(8-11)
EMTDC User’s Guide
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