Chapter 11 - INSULATED CABLES

Example E1 - Pg 273

In [1]:
#calculate Insulation resistance of cable
#Given data :
import math
rho=5.*10.**14.*10.**-2##ohm-m
l=5.*1000.##m
r1=1.25##m
r2=r1+1.##m
R_ins=rho/(2.*math.pi*l)*math.log(r2/r1)##ohm
print '%s %.2f' %("Insulation resistance of cable(Mohm) :",R_ins/10.**6.)#

Insulation resistance of cable(Mohm) : 93.55

Example E2 - Pg 274

In [2]:
#calculate Insulation resistance of cable
#Given data :
import math
rho=5.*10.**14.*10.**-2##ohm-m
l=5.*1000.##m
r1=2.5##m
r2=r1+1.##m
R_ins=rho/(2.*math.pi*l)*math.log(r2/r1)##ohm
print '%s %.2f' %("Insulation resistance of cable(Mohm) :",R_ins/10.**6.)#

Insulation resistance of cable(Mohm) : 53.55

Example E3 - Pg 274

In [3]:
#calculate Resistivity
#Given data :
import math
l=3000.##cm
d1=1.5##cm
r1=d1/2.##cm
d2=5.##cm
r2=d2/2.##cm
R_INS=1800.##Mohm
rho=R_INS*10**6*(2*math.pi*l)/math.log(r2/r1)##ohm-m
print '%s %.2e' %("Resistivity (ohm-m) :",rho)#

Resistivity (ohm-m) : 2.82e+13

Example E4 - Pg 276

In [4]:
#calculate Maximum electrostatic stress,Minimum electrostatic stress,Capacitance per km length,Charging Current per phase per km length
#Given data :
import math
V1=11000.##Volt
f=50.##Hz
a=0.645##cm**2
d=math.sqrt(4.*a/math.pi)##cm
d=d/100.##m
D=2.18/100.##m
epsilon_r=3.5##relative permitivity
V=V1*math.sqrt(2.)/math.sqrt(3.)##V(assuming 3 phase system)
gmax=2.*V/d/math.log(D/d)##V/m
gmax=gmax/10.**5.##KV/cm
print '%s %.2f' %("Maximum electrostatic stress(kV/cm)",gmax)#
gmin=2.*V/D/math.log(D/d)##V/m
gmin=gmin/10.**5.##kV/cm
print '%s %.3f' %("Minimum electrostatic stress(kV/cm)",gmin)#
C=0.024*epsilon_r/math.log10(D/d)##micro F
print '%s %.2e' %("Capacitance per km length(F)",C*10.**-6)##
Vp=V1/math.sqrt(3.)##V
Ic=2.*math.pi*f*C*10.**-6*Vp##A
print '%s %.2f' %("Charging Current per phase per km length(A)",Ic)#

Maximum electrostatic stress(kV/cm) 22.58
Minimum electrostatic stress(kV/cm) 9.387
Capacitance per km length(F) 2.20e-07
Charging Current per phase per km length(A) 0.44

Example E5 - Pg 277

In [5]:
#calculate Maximum electrostatic stress,Total charging
import math
#Given data :
VL=33.*1000.##Volt
f=50.##Hz
l=3.4##km
d=2.5##cm
radial_thick=0.6##cm
epsilon_r=3.1##relative permitivity
V=VL*math.sqrt(2.)/math.sqrt(3.)##V(assuming 3 phase system)
D=d+2.*radial_thick##cm
D=D/100.##cm
d=d/100.##m
gmax=2.*V/d/math.log(D/d)##V/m
print '%s %.2e' %("Maximum electrostatic stress(V/m)",gmax)#
C=0.024*epsilon_r*l/math.log10(D/d)##micro F
Vp=VL/math.sqrt(3.)##V
Ic=2.*math.pi*f*C*10.**-6*Vp##A
kVA=math.sqrt(3.)*VL*Ic*10.**-3##kVAR
print '%s %.2f' %("Total charging kVA(kVAR)",kVA)#

Maximum electrostatic stress(V/m) 5.50e+06
Total charging kVA(kVAR) 508.29

Example E6 - Pg 278

In [23]:
#calculate Diameter of conductor,Internal diameter of sheath
import math
#Given data :
VL=10.*1000.##Volt
Emax=23.##kV/cm
gmax=Emax*10.**5.##V/m
d=2.*VL/gmax##m
print '%s %.1f' %("Diameter of conductor(mm)",d*10.**3.)#
D=math.e*d##m
print '%s %.2f' %("Internal diameter of sheath(mm)",D*10.**3.)#

Diameter of conductor(mm) 8.7
Internal diameter of sheath(mm) 23.64

Example E7 - Pg 278

In [24]:
#calculate Diameter of conductor,Internal diameter of sheath
import math
#Given data :
VL=132.*1000.##Volt
gmax=60.##kV/cm(peak)
gmax=gmax/math.sqrt(2.)*10.**5.##V/m(rms)
V=VL/math.sqrt(3.)##Volt
d=2.*V/gmax##m
print '%s %.1f' %("Diameter of conductor(mm)",d*10.**3.)#
D=math.e*d##m
print '%s %.2f' %("Internal diameter of sheath(mm)",D*10.**3.)#

Diameter of conductor(mm) 35.9
Internal diameter of sheath(mm) 97.66

Example E8 - Pg 280

In [8]:
#calculate RMS value of max safe working voltage
import math
#Given data :
r=0.5##cm
R=3.5##cm
r1=1.##cm
g1max=34.##kV/cm(peak)
epsilon_r=5.##relative permitivity
g2max=g1max*r/r1/epsilon_r##kV/cm(peak)
Vpeak=r*g1max*math.log(r1/r)+r1*g2max*math.log(R/r1)##kV
Vrms=Vpeak/math.sqrt(2.)##kV
print '%s %.2f' %("RMS value of max safe working voltage(kV)",Vrms)#

RMS value of max safe working voltage(kV) 11.34

Example E9 - Pg 281

In [25]:
##calculate Radial thickness of inner dielectric,Radial thickness of outer dielectric,Maximum working voltage
import math
#Given data :
g1max=60.##kV/cm
g2max=50.##kV/cm
epsilon_r1=4.##relative permitivity
epsilon_r2=2.5##relative permitivity
D=5.##cm(sheat inside diameter)
d=1.##cm
#g1max/g2max=epsilon_r2*d1/(epsilon_r1*d)
d1=g1max/g2max/epsilon_r2*(epsilon_r1*d)##cm
t_inner=(d1-d)/2.##cm
print '%s %.1f' %("Radial thickness of inner dielectric(mm)",t_inner*10.)#
t_outer=(D-d1)/2.##cm
print '%s %.1f' %("Radial thickness of outer dielectric(mm)",t_outer*10.)#
Vpeak=g1max/2.*d*math.log(d1/d)+g2max/2*d1*math.log(D/d1)##kV
Vrms=Vpeak/math.sqrt(2.)##kV
print '%s %.2f' %("Maximum working voltage(rms in kV)",Vrms)#

Radial thickness of inner dielectric(mm) 4.6
Radial thickness of outer dielectric(mm) 15.4
Maximum working voltage(rms in kV) 46.32

Example E10 - Pg 281

In [26]:
##calculate Working voltage(rms) for the cable
#Given data :
import math
r=1.##cm
R=2.5##cm
d=2.*r##cm
D=2.*R##cm
epsilon_r1=5.##relative permitivity
epsilon_r2=4.##relative permitivity
epsilon_r3=3.##relative permitivity
gmax=40.##KV/cm
#epsilon_r1*d=epsilon_r2*d1=epsilon_r3*d2
d1=(epsilon_r1/epsilon_r2)*d##cm
d2=(epsilon_r1/epsilon_r3)*d##cm
Vpeak=gmax/2.*(d*math.log(d1/d)+d1*math.log(d2/d1)+d2*math.log(D/d2))##kV
Vrms=Vpeak/math.sqrt(2.)##kV
print '%s %.1f' %("Working voltage(rms) for the cable (kV)",Vrms)#

Working voltage(rms) for the cable (kV) 35.6

Example E11 - Pg 282

In [27]:
#calculate Potential gradient at the surface of conductor,Maximum stress in the outer dielectric,Stress at the surface of outer dielectric
import math
#Given data :
Vs=66.##kV
d=1.##cm
d1=1.+2.*1.##cm
D=3.+2.*1.##cm
epsilon_r1=3.##relative permitivity
epsilon_r2=2.5##relative permitivity
g2maxBYg1max=d*epsilon_r1/(d1*epsilon_r2)#
Vmax=Vs*math.sqrt(2.)/math.sqrt(3.)##kV
#Vmax=g1max*d/2*log(d1/d)+g2max*d1/2*log(D/d1)##kV
g1max=Vmax/(d/2.*math.log(d1/d)+g2maxBYg1max*d1/2.*math.log(D/d1))##kV/cm
print '%s %.f' %("Potential gradient at the surface of conductor(kV/cm)",g1max)#
g2max=g1max*g2maxBYg1max##kV/cm
print '%s %.1f' %("Maximum stress in the outer dielectric(kV/cm)",g2max)#
Stress=g2max*d1/D##kV/cm
print '%s %.2f' %("Stress at the surface of outer dielectric(kV/cm)",Stress)#

Potential gradient at the surface of conductor(kV/cm) 63
Maximum stress in the outer dielectric(kV/cm) 25.2
Stress at the surface of outer dielectric(kV/cm) 15.11

Example E12 - Pg 282

In [28]:
#calculate Potential gradient at the surface of conductor,Maximum stress in the outer dielectric
import math
#Given data :
Vs=66.##kV
d=2.##cm
d1=2.+2.*1.##cm
D=4.+2.*1.##cm
epsilon_r1=5.##relative permitivity
epsilon_r2=3.##relative permitivity
g2maxBYg1max=d*epsilon_r1/(d1*epsilon_r2)#
Vmax=Vs*math.sqrt(2.)/math.sqrt(3.)##kV
#Vmax=g1max*d/2*log(d1/d)+g2max*d1/2*log(D/d1)##kV
g1max=Vmax/(d/2.*math.log(d1/d)+g2maxBYg1max*d1/2.*math.log(D/d1))##kV/cm
print '%s %.2f' %("Potential gradient at the surface of conductor(kV/cm)",g1max)#
g2max=g1max*g2maxBYg1max##kV/cm
print '%s %.1f' %("Maximum stress in the outer dielectric(kV/cm)",g2max)#

Potential gradient at the surface of conductor(kV/cm) 39.37
Maximum stress in the outer dielectric(kV/cm) 32.8

Example E13 - Pg 283

In [29]:
#calculate Inner diameter of lead sheath
import math
#Given data :
Vs=66.##kV
r=0.5##cm
g1max=50.##kV/cm
g2max=40.##kV/cm
g3max=30.##kV/cm
epsilon_r1=4.##relative permitivity
epsilon_r2=4.##relative permitivity
epsilon_r3=2.5##relative permitivity
#Q=2*%pi*epsilon0*epsilon_r1*r*g1max=2*%pi*epsilon0*epsilon_r2*r*g2max=2*%pi*epsilon0*epsilon_r3*r*g3max
r1=epsilon_r1*r*g1max/(epsilon_r2*g2max)##cm
r2=epsilon_r2*r1*g2max/(epsilon_r3*g3max)##cm
Vmax=Vs*math.sqrt(2.)##kV
#Vmax=g1max*r*log(r1/r)+g2max*r1*log(r2/r1)+g3max*r2*log(R/r2)##kV
R=math.exp((Vmax-g1max*r*math.log(r1/r)-g2max*r1*math.log(r2/r1))/g3max/r2)*r2##cm
D=2.*R##cm
print '%s %.1f' %("Inner diameter of lead sheath(cm)",D)#

Inner diameter of lead sheath(cm) 14.9

Example E14 - Pg 283

In [14]:
#calculate Voltage between sheath & intersheath
import math 
#Given data :
Vrms=66.##kV
Vmax=Vrms*math.sqrt(2.)##kV
gmax=60.##kV/cm
d=2.*Vmax/math.e/gmax##cm
d1=math.e*d##cm
V1=Vrms/math.e##kV
dV=Vrms-V1##kV(Voltage between sheath & intersheath)
print '%s %.2f' %("Voltage between sheath & intersheath(kV)",dV)#

Voltage between sheath & intersheath(kV) 41.72

Example E15 - Pg 284

In [15]:
#calculate Maximum stress without intersheath,Peak voltage on 1st intersheath,Peak voltage on 2nd intersheath
import math
#Given data :
Vs=66.##kV
Vmax=Vs*math.sqrt(2.)/math.sqrt(3.)##kV
D=6.##cm
d=2.5##cm
d1=math.e*d##cm
gmax=2.*Vmax/d/math.log(D/d)##kV/cm
print '%s %.2f' %("Maximum stress without intersheath(kV/cm)",gmax)#
#d1/d=d2/d1=D/d2=alfa(say)
alfa=(D/d)**(1./3.)#
d1=alfa*d##cm
d2=alfa*d1##cm
gmax=Vmax/(d/2*math.log(d1/d)+d1/2.*math.log(d2/d1)+d2/2.*math.log(D/d2))##kV/cm
V1max=gmax*d/2.*math.log(d1/d)##kV
V2max=gmax*d1/2.*math.log(d2/d1)##kV
Vpeak1=Vmax-V1max##kV
print '%s %.4f' %("Peak voltage on 1st intersheath(kV)",Vpeak1)#
Vpeak2=Vpeak1-V2max##kV
print '%s %.4f' %("Peak voltage on 2nd intersheath(kV)",Vpeak2)#

Maximum stress without intersheath(kV/cm) 49.24
Peak voltage on 1st intersheath(kV) 40.8452
Peak voltage on 2nd intersheath(kV) 23.3815

Example E16 - Pg 286

In [16]:
#calculate Capacitance of core to neutral,Capacitance between any two core,Charging current per phase
import math
#Given data :
Vs=11.##kV
f=50.##Hz
l=2.5*1000.##m
C_all3=1.8##micro F
Cdash=1.5##micro F(2*Cc+Cs)
Cs=C_all3/3.##micro F
Cc=(Cdash-Cs)/2.##micro F
C_N=3.*Cc+Cs##micro F
print '%s %.2f' %("Capacitance of core to neutral(micro F)",C_N)#
C_2=C_N/2.##micro F
print '%s %.3f' %("Capacitance between any two core(micro F)",C_2)#
Vp=Vs*1000./math.sqrt(3.)##Volt
Ic=2.*math.pi*f*Vp*C_N*10.**-6##A
print '%s %.2f' %("Charging current per phase(A)",Ic)#

Capacitance of core to neutral(micro F) 1.95
Capacitance between any two core(micro F) 0.975
Charging current per phase(A) 3.89

Example E17 - Pg 287

In [30]:
#calculate kVA taken by the cable
import math
#Given data :
l=10.##km
Vs=10.##kV
f=50.##Hz
C=0.3##micro F/km(between any two core)
C2=l*C##micro F(between any two core)
C_N=2.*C2##micro F
Vp=Vs*1000./math.sqrt(3.)##Volt
Ic=2.*math.pi*f*Vp*C_N*10.**-6##A
kVA=3.*Vp*Ic/1000.##kVAR
print '%s %.1f' %("kVA taken by the cable(kVAR)",kVA)#

kVA taken by the cable(kVAR) 188.5

Example E18 - Pg 287

In [31]:
#calculate Capacitance between any two cores,Capacitance between any two shorted conductors and third conductor
import math
#Given data :
Cs3=1.##micro F/km(between shorted conductor)
Cs=Cs3/3.##micro F
Cdash=0.6##micro F(Cdash=2*Cc+Cs : between two shorted conductor)
Cc=(Cdash-Cs)/2.##micro F
C2=1./2.*(3*Cc+Cs)##micro F
print '%s %.4f' %("Capacitance between any two cores(micro F)",C2)#
C2dash=2.*Cc+2./3.*Cs##micro F
print '%s %.3f' %("Capacitance between any two shorted conductors and third conductor(micro F)",C2dash)#

Capacitance between any two cores(micro F) 0.3667
Capacitance between any two shorted conductors and third conductor(micro F) 0.489

Example E19 - Pg 287

In [19]:
#calculate Total charging kVAR,Maximum stress in the cable
import math
#Given data :
Vs=33.##kV
f=50.##Hz
l=3.4##km
d=2.5##cm
D=d+2.*0.6##cm
epsilon_r=3.1##relative permitivity
C=0.024*epsilon_r/math.log10(D/d)*l*1000.*1000.*10.**-6## F/phase
Vp=Vs*1000./math.sqrt(3.)##Volt
Ic=2.*math.pi*f*C*10.**-6*Vp##A
kVAR=3.*Vp*Ic*10.**-3##kVAR
print '%s %.2f' %("Total charging kVAR : ",kVAR)#
Emax=Vp/(d/2.*math.log(D/d))*10.**-3##kV/cm
print '%s %.2f' %("Maximum stress in the cable(kV/cm) ",Emax)#

Total charging kVAR :  508.29
Maximum stress in the cable(kV/cm)  38.88

Example E20 - Pg 291

In [21]:
#calculate Capacitance of the cable,Charging current,Dielectric loss,Equivalent insulation resistance
import math
#Given data :
Vs=11.##kV
f=50.##Hz
D=2.##cm
d=0.5##cm
epsilon_r=3.5##relative permitivity
pf=0.05##power factor
C=0.024*epsilon_r/math.log10(D/d)*10.**-6## F/km
print '%s %.4f' %("Capacitance of the cable(micro F)",C*10.**6.)#
Vp=Vs*1000./math.sqrt(3.)##Volt
Ic=2.*math.pi*f*C*Vp##A
print '%s %.3f' %("Charging current(A)",Ic)#
fi=math.acos(pf) *180./math.pi##degree
dela= 90.-fi##degree(Dielectric loss angle)
loss_dielectric=2*math.pi*f*C*Vp**2*math.tan(dela*math.pi/180.)##W
print '%s %.1f' %("Dielectric loss(W)",loss_dielectric)#
R_INS=Vp**2./loss_dielectric##ohm
print '%s %.3f' %("Equivalent insulation resistance(Mohm)",R_INS/10.**6.)#

Capacitance of the cable(micro F) 0.1395
Charging current(A) 0.278
Dielectric loss(W) 88.5
Equivalent insulation resistance(Mohm) 0.456

Example E21 - Pg 292

In [22]:
#calculate Loss angle,No load current drawn by cable
import math 
#Given data :
Vs=11.##kV
f=50.##Hz
C_N_by_2=2.5##micro F(between 2 core 1 core shorted)
C_N=C_N_by_2*2.##micro F
Vp=Vs*1000./math.sqrt(3.)##Volt
Ic=2.*math.pi*f*Vp*C_N*10.**-6##A
R_INS2=810.##kohm
R_INS=R_INS2/2.##kohm
dela=math.atan(1./(R_INS*10.**3.*2.*math.pi*f*C_N*10.**-6)) *180/math.pi##degree
print '%s %.2f' %("Loss angle(degree)",dela)#
Ie=Vp/R_INS/1000.##A
I=math.sqrt(Ic**2.+Ie**2.)##A
print '%s %.3f' %("No load current drawn by cable(A)",I)#

Loss angle(degree) 0.09
No load current drawn by cable(A) 9.976