# CHAPTER 6: UNDERGROUND CABLES¶

## Example 6.1, Page number 191-192¶

In :
C_m = 0.28      #Capacitance b/w ant 2 cores(micro-F/km)
f = 50.0        #Frequency(Hz)
V_L = 11.0      #Line voltage(kV)

import math
C = 2*C_m                           #Capacitance b/w any conductor & shield(micro-F/km)
w = 2*math.pi*f                     #Angular frequency
I_c = V_L*10**3*w*C*10**-6/3**0.5   #Charging current/phase/km(A)
Total = 3**0.5*I_c*V_L              #Total charging kVAR/km

print('Charging current/phase/km , I_c = %.3f A' %I_c)
print('Total charging kVAR/km = %.2f ' %Total)

Charging current/phase/km , I_c = 1.117 A
Total charging kVAR/km = 21.29


## Example 6.2, Page number 193-194¶

In :
E_c = 100.0     #Safe working stress(kV/cm) rms
V = 130.0       #Operating voltage(kV) rms
d = 1.5         #Diameter of conductor(cm)

import math
ln_D = 2*V/(E_c*d)+math.log(d)
D = math.exp(ln_D)
thick_1 = (D-d)/2               #Insulation thickness(cm)

d_2 = 2*V/E_c
D_2 = 2.718*d_2                 #Sheath diameter(cm)
thick_2 = (D_2-d_2)/2           #Insulation thickness(cm)

print('(i)  Internal sheath radius = %.2f cm' %thick_1)
print('(ii) Internal sheath radius = %.2f cm' %thick_2)

(i)  Internal sheath radius = 3.49 cm
(ii) Internal sheath radius = 2.23 cm


## Example 6.3, Page number 194-196¶

In :
d = 3.0         #Diameter of conductor(cm)
D = 8.5         #Sheath diameter(cm)
e_r1 = 5.0      #Permittivity of inner dielectric
e_r2 = 3.0      #Permittivity of outer dielectric
E_c = 30.0      #Safe working stress(kV/cm) rms

import math
E_i = E_c
D_1 = e_r1/e_r2*d
thick_1 = (D_1-d)/2     #Thickness of first layer(cm)
thick_2 = (D-D_1)/2     #Thickness of second layer(cm)

V_1 = E_c*d*math.log(D_1/d)/2       #Voltage across first layer(kV)
V_2 = E_i*D_1*math.log(D/D_1)/2     #Voltage across second layer(kV)
V = V_1 + V_2                       #Permissible conductor voltage(kV)

V_3 = E_c*d*math.log(D/d)/2         #Permissible conductor voltage(kV) for homogeneous permittivity of 5

print('Case(i) :')
print('Thickness of first layer = %.2f cm' %thick_1)
print('Thickness of second layer = %.2f cm' %thick_2)
print('\nCase(ii) :')
print('Permissible conductor voltage = %.2f kV' %V)
print('\nCase(iii) :')
print('Permissible conductor voltage if a homogeneous insulation of permittivity 5 is used , V = %.2f kV' %V_3)
print('\nNOTE : ERROR : Relative permittivity of outer dielectric is 3 & not 9 as given in textbook')

Case(i) :
Thickness of first layer = 1.00 cm
Thickness of second layer = 1.75 cm

Case(ii) :
Permissible conductor voltage = 62.78 kV

Case(iii) :
Permissible conductor voltage if a homogeneous insulation of permittivity 5 is used , V = 46.87 kV

NOTE : ERROR : Relative permittivity of outer dielectric is 3 & not 9 as given in textbook


## Example 6.4, Page number 196-197¶

In :
E = 40.0        #Safe working stress(kV/cm) rms
d = 1.5         #Conductor diameter(cm)
D = 6.7         #Sheath diameter(cm)
t = 0.1         #Thickness of lead tube(cm)

import math
r = d/2                     #Conductor radius(cm)
R = D/2                     #Sheath radius(cm)
r_i = r+((R-r)/2)-t/2       #Internal radius of intersheath(cm)
r_e = r_i + t               #External radius of intersheath(cm)
V_1 = E*r*math.log(r_i/r)   #Voltage across conductor & intersheath(kV)
V_2 = E*r_e*math.log(R/r_e) #Voltage across intersheath & earthed sheath(kV)
V = V_1 + V_2               #Safe working voltage with intersheath(kV)
V_no = E*r*math.log(R/r)    #Safe working voltage without intersheath(kV)

print('Safe working voltage with intersheath , V = %.2f kV' %V)
print('Safe working voltage without intersheath , V = %.2f kV' %V_no)

Safe working voltage with intersheath , V = 68.65 kV
Safe working voltage without intersheath , V = 44.90 kV