# Chapter 4 - INDUCTANCE AND CAPACITANCE OF TRANSMISSION LINES¶

## Example E1 - Pg 85¶

In [1]:
#calculate Loop inductance,Reactance of transmission line
#Given data :
import math
f=50.##Hz
d=1.*100.##cm
r=1.25/2.##cm
r_dash=r*0.7788##cm
L=0.4*math.log(d/r_dash)##mH
print '%s %.2f' %("Loop inductance per km(mH)",L)#
XL=2.*math.pi*f*L*10.**-3.##ohm/Km
print '%s %.3f' %("Reactance of transmission line(ohm/km)",XL)#

Loop inductance per km(mH) 2.13
Reactance of transmission line(ohm/km) 0.669


## Example E2 - Pg 85¶

In [2]:
#calculate Loop inductance
#Given data :
import math
f=50.##Hz
a=10.##cm**2
l=500./1000.##km
r=math.sqrt(a/math.pi)##cm
d=5.*100.##cm
r_dash=r*0.7788##cm
L=0.4*math.log(d/r_dash)*l##mH
print '%s %.3f' %("Loop inductance per km(mH)",L)#

Loop inductance per km(mH) 1.177


## Example E3 - Pg 85¶

In [3]:
#calculate Loop inductance per km of copper conductor line,Loop inductance per km of steel conductor line
#Given data :
import math
r=1./2.##cm
d=1.*100.##cm
mu=50.##relative permeability
r_dash=r*0.7788##cm
L_cu=.1+0.4*math.log(d/r)##mH
print '%s %.2f' %("Loop inductance per km of copper conductor line(mH)",L_cu)#
L_steel=(mu+4.*math.log(d/r))*10.**-7.*10.**3.##mH
print '%s %.2f' %("Loop inductance per km of steel conductor line(mH)",L_steel*10**3)#

Loop inductance per km of copper conductor line(mH) 2.22
Loop inductance per km of steel conductor line(mH) 7.12


## Example E4 - Pg 86¶

In [4]:
#calculate Geometric mean radius
#Given data :
import math
r=3.##mm
d11=r##mm
d12=2.*r##mm
d34=2.*r##mm
d16=2.*r##mm
d17=2.*r##mm
d14=4.*r##mm
d13=math.sqrt(d14**2.-d34**2.)##mm
d15=d13##mm
Ds1=(0.7788*d11*d12*d13*d14*d15*d16*d17)**(1./7.)##mm
Ds2=Ds1##mm
Ds3=Ds1##mm
Ds4=Ds1##mm
Ds5=Ds1##mm
Ds6=Ds1##mm
Ds7=(2.*r*0.7788*d11*d12*d13*2.*r*2.*r)**(1./7.)##mm
Ds=(Ds1*Ds2*Ds3*Ds4*Ds5*Ds6*Ds7)**(1./7.)##mm
print '%s %.3f' %("Geometric mean radius(mm)",Ds)#
#Answer in the book is wrong

Geometric mean radius(mm) 6.367


## Example E5 - Pg 86¶

In [5]:
#calculate Loop inductance of line
#Given data :
import math
r=1.2##cm
rdash=0.7788*r##cm
d12=0.12*100.##cm
d11dash=(0.2+1.2)*100.##cm
d22dash=(0.2+1.2)*100.##cm
d12dash=(0.2+1.2+0.2)*100.##cm
d21dash=(1.2)*100.##cm
Dm=(d11dash*d12dash*d21dash*d22dash)**(1./4.)##cm
d11=0.93456##cm
d22=0.93456##cm
d12=20.##cm
d21=20.##cm
Ds=(d11*d12*d21*d22)**(1./4.)##cm
L=0.4*math.log(Dm/Ds)##mH/km
print '%s %.3f' %("Loop inductance of line(mH/km)",L)#

Loop inductance of line(mH/km) 1.389


## Example E6 - Pg 87¶

In [6]:
#calculate Loop inductance of line
#Given data :
import math
r=2./2.##cm
rdash=0.7788*r##cm
d12=0.12*100##cm
d11dash=300.##cm
d12dash=math.sqrt(300.**2.+100.**2.)##cm
d21dash=d12dash##cm
d22dash=d11dash##cm
d11=rdash##cm
d22=rdash##cm
d12=100.##cm
d21=100.##cm
Dm=(d11dash*d12dash*d21dash*d22dash)**(1./4.)##cm
Ds=(d11*d12*d21*d22)**(1./4.)##cm
L=0.4*math.log(Dm/Ds)##mH/km
print '%s %.3f' %("Loop inductance of line(mH/km)",L)#

Loop inductance of line(mH/km) 1.421


## Example E7 - Pg 89¶

In [7]:
#calculate Inductance per phase
#Given data :
import math
r=1.24/2##cm
rdash=0.7788*r##cm
d=2.*100.##cm
L=0.2*math.log(d/rdash)##mH
print '%s %.3f' %("Inductance per phase per km(mH)",L)#

Inductance per phase per km(mH) 1.205


## Example E8 - Pg 89¶

In [8]:
#calculate Inductance per phase
#Given data :
import math
r=(20./2.)/10.##cm
d1=4.*100.##cm
d2=5.*100.##cm
d3=6.*100.##cm
rdash=0.7788*r##cm
L=0.2*math.log((d1*d2*d3)**(1./3.)/rdash)##mH
print '%s %.2f' %("Inductance per phase(mH)",L)#

Inductance per phase(mH) 1.29


## Example E9 - Pg 90¶

In [9]:
#calculate Inductance per km of phase1,phase2,phase3
#Given data :
import math
r=4./2.##cm
rdash=0.7788*r##cm
d=300.##cm
d3=6.*100.##cm
LAr=0.2*(math.log(d/rdash)+1./2.*math.log(2.))
LAi=0.866*math.log(2.)##mH
print '%s' %("Inductance per km of phase1(mH)")#
print '%.4f %s %.2f' %(LAr,'-j',LAi)
LB=0.2*math.log(d/rdash)##mH
print '%s %.3f' %("Inductance per km of phase2(mH)",LB)#
LCr=0.2*(math.log(d/rdash)+1./2.*math.log(2.))
LCi=0.866*math.log(2.)##mH
print '%s' %("Inductance per km of phase3(mH)")#
print '%.4f %s %.2f' %(LCr,'+j',LCi)

Inductance per km of phase1(mH)
1.1214 -j 0.60
Inductance per km of phase2(mH) 1.052
Inductance per km of phase3(mH)
1.1214 +j 0.60


## Example E10 - Pg 90¶

In [10]:
#calculate Spacing between adjacent conductors
import math
#Given data :
r=1.2/2.*10.##mm
rdash=0.7788*r##mm
d=3.5*1000.##mm
L=2.*10.**-7.*math.log(d/rdash)##H/m
Lav=1./3.*(L+L+L)##H/m
d=rdash*math.exp(Lav/(2.*10.**-7.)-1./3.*math.log(2.))##mm
print '%s %.4e' %("Spacing between adjacent conductors(m)",d/1000.)#

Spacing between adjacent conductors(m) 2.7780e+00


## Example E11 - Pg 94¶

In [11]:
#calculate Spacing between adjacent conductors
import math
#Given data :
r=20.##mm
rdash=0.7788*r##mm
d=7.*1000.##mm
L=10**-7*math.log(math.sqrt(3.)/2.*d/rdash)##H/m
print '%s %.4f' %("Spacing between adjacent conductors(mH)",L*10.**3./10.**-3.)#

Spacing between adjacent conductors(mH) 0.5964


## Example E12 - Pg 94¶

In [12]:
#calculate Inductance per phase
import math
#Given data :
r=0.9##cm
rdash=0.7788*r*10.**-2.##m
daa_dash=math.sqrt(6.**2.+6.**2.)##m
dbb_dash=7.##m
dcc_dash=daa_dash##m
daa=rdash##m
Dsb=(daa*7.)**(1./2.)##m
Dsc=(daa*daa_dash)**(1./2.)##m
Ds=(Dsa*Dsb*Dsc)**(1./3.)##m
dab=math.sqrt(3.**2.+0.5**2.)##m
dab_dash=math.sqrt(3.**2.+6.5**2.)##m
Dbc=((dab*dab_dash)**2.)**(1./4.)##m
Dca=((6.*6.)**2.)**(1./4.)##m
Dm=(Dab*Dbc*Dca)**(1./3.)##m
L=0.2*math.log(Dm/Ds)##mH/km
print '%s %.4f' %("Inductance per phase(mH/km)",L)#

Inductance per phase(mH/km) 0.6135


## Example E13 - Pg 95¶

In [13]:
#calculate GMD,Deq or Dm,Inductance of 100 km line,Using Alternate method, Inductance of 100 km line
import math
#Given data :
r=5./2.##mm
rdash=2.176*r*10.**-3.##m
daa_dash=math.sqrt(6**2+16**2)##m
dbb_dash=6.##m
dcc_dash=daa_dash##m
dab=8.##m
dab_dash=math.sqrt(6.**2.+8.**2.)##m
dbc=8.##m
dbc_dash=math.sqrt(6.**2.+8.**2.)##m
dca=16.##m
dca_dash=6.##m
Dsa=math.sqrt(rdash*daa_dash)##m
Dsb=math.sqrt(rdash*dbb_dash)##m
Dsc=math.sqrt(rdash*dcc_dash)##m
Ds=(Dsa*Dsb*Dsc)**(1/3)##m
print '%s %.2f' %("GMD(m) : ",Ds)#
Dab=(dab*dab_dash)**(1./2.)##m
Dbc=(dbc*dbc_dash)**(1./2.)##m
Dca=(dca*dca_dash)**(1./2.)##m
Dm=(Dab*Dbc*Dca)**(1./3.)##m
print '%s %.2f' %("Deq or Dm(m) : ",Dm)#
L=0.2*math.log(Dm/Ds)##mH/km
L=L*10.**-3.*100.##H(for 100 km line)
print '%s %.4f' %("Inductance of 100 km line(H)",L)#
#/Alternate method is given below
d1=dab##m
d2=dca_dash##m
L=0.2*math.log(2.**(1./6.))*math.sqrt(d1/rdash)*((d1**2.+d2**2.)/(4.*d1**2.+d2**2.))**(1./6.)##mH
L=L*10.**-3.*100.##H(for 100 km line)
print '%s %.4f' %("Using Alternate method, Inductance of 100 km line(H)",L)#

GMD(m) :  1.00
Deq or Dm(m) :  9.22
Inductance of 100 km line(H) 0.0444
Using Alternate method, Inductance of 100 km line(H) 0.0741


## Example E14 - Pg 97¶

In [14]:
#calculate Inductance per phase
import math
#Given data :
r=5./2.##cm
rdash=0.7788*r*10.**-2.##m
d=6.5##m
s=0.4##m
Ds=math.sqrt(rdash*s)##m
dab=6.5##m
dab_dash=6.9##m
Dbc=Dab##m
dca=13.##m
dca_dash=12.6##m
d_cdash_a=13.4##m
Dm=(Dab*Dbc*Dca)**(1./3.)##m
L=0.2*math.log(Dm/Ds)##mH/km
print '%s %.3f' %("Inductance per phase(mH/km)",L)#

Inductance per phase(mH/km) 0.906


## Example E15 - Pg 98¶

In [15]:
#calculate Inductive reactance of bundled conductor line,Inductive reactance with single conductor
import math
#Given data :
f=50.##Hz
r=3.5/2.##cm
rdash=0.7788*r*10.**-2.##m
d=7.##m
s=40./100.##m
Ds=math.sqrt(rdash*s)##m
dab=7.##m
dab_dash=7.4##m
Dbc=Dab##m
dca=14.##m
dca_dash=13.6##m
d_cdash_a=14.4##m
Dm=(Dab*Dbc*Dca)**(1./3.)##m
L=0.2*math.log(Dm/Ds)##mH/km
XL=2*math.pi*f*L*10.**-3.##ohm/km
print '%s %.2f' %("Inductive reactance of bundled conductor line(ohm/km)",XL)#
#Equivalent single conductor
n=2#
r1=math.sqrt(n*math.pi*r**2./math.pi)##m
r1dash=0.7788*r1*10.**-2.##m
Dm1=(Dab*Dbc*Dca)**(1./3.)##m
L1=0.2*math.log(Dm1/r1dash)##mH/km
XL1=2*math.pi*f*L1*10.**-3.##ohm/km
print '%s %.3f' %("Inductive reactance with single conductor(ohm/km)",XL1)#

Inductive reactance of bundled conductor line(ohm/km) 0.30
Inductive reactance with single conductor(ohm/km) 0.385


## Example E16 - Pg 102¶

In [16]:
#calculate Capacitance of line
import math
#Given data :
r=15./2.##mm
d=1.5*1000.##mm
l=30.##km
epsilon_o=8.854*10.**-12.##permitivity
C=math.pi*epsilon_o/math.log(d/r)*l*1000.##F
print '%s %.4f' %("Capacitance of line(micro F)",C*10**6)#

Capacitance of line(micro F) 0.1575


## Example E17 - Pg 105¶

In [17]:
#calculate Capacitance of line
#Given data :
import math
r=2./2.##cm
d=2.5*100.##cm
l=100.##km
epsilon_o=8.854*10.**-12.##permitivity
C=2*math.pi*epsilon_o/math.log(d/r)*l*1000.##F
print '%s %.4f' %("Capacitance of line(micro F)",C*10.**6.)#

Capacitance of line(micro F) 1.0075


## Example E18 - Pg 105¶

In [18]:
#calculate Capacitance of line
import math
#Given data :
r=2./2./100.##m
d1=3.5##m
d2=5.##m
d3=8.##m
epsilon_o=8.854*10.**-12.##permitivity
CN=2.*math.pi*epsilon_o*1000/math.log((d1*d2*d3)**(1./3.)/r)##F
print '%s %.4f' %("Capacitance of line(micro F)",CN*10.**6.)#

Capacitance of line(micro F) 0.0089


## Example E19 - Pg 105¶

In [19]:
#calculate Capacitance per phase per meter  line,Charging current per phase
import math
#Given data :
f=50.##Hz
VL=220.##KV
r=20./2./1000.##m
d1=3.##m
d2=3.##m
d3=6.##m
epsilon_o=8.854*10.**-12.##permitivity
CN=2.*math.pi*epsilon_o/math.log((d1*d2*d3)**(1./3.)/r)##F
print '%s %.4e' %("Capacitance per phase per meter  line(F)",CN)#
Vph=VL*1000./math.sqrt(3.)##V
Ic=2.*math.pi*f*CN*Vph##A
print '%s %.3f' %("Charging current per phase(mA) : ",Ic*1000.)#

Capacitance per phase per meter  line(F) 9.3737e-12
Charging current per phase(mA) :  0.374


## Example E20 - Pg 106¶

In [20]:
#calculate Capacitance per phase per meter  line,Charging current per phase
import math
#Given data :
f=50.##Hz
VL=110.##kV
r=1.05/2.##cm
d1=3.5##m
d2=3.5##m
d3=7.##m
epsilon_o=8.854*10.**-12.##permitivity
CN=2.*math.pi*epsilon_o/math.log((d1*d2*d3)**(1./3.)*100./r)##F
print '%s %.2e' %("Capacitance per phase per meter  line(F)",CN)#
Vph=VL*1000./math.sqrt(3.)##V
Ic=2.*math.pi*f*CN*Vph##A/m
print '%s %.3f' %("Charging current per phase(A/km) : ",Ic/10**-3)#

Capacitance per phase per meter  line(F) 8.26e-12
Charging current per phase(A/km) :  0.165


## Example E21 - Pg 108¶

In [21]:
#calculate Capacitive reactance too neutral,Charging current
import math
#Given data :
r=2.5/2.*10.**-2.##m
VL=132.##KV
epsilon_o=8.85*10.**-12.##permitivity
f=50.##Hz
dRRdash=math.sqrt(7.**2.+(4.+4.)**2.)##m
dBBdash=dRRdash##m
dYYdash=9.##m
DSR=math.sqrt(r*dRRdash)##m
DSY=math.sqrt(r*dYYdash)##m
DSB=math.sqrt(r*dBBdash)##m
Ds=(DSR*DSB*DSY)**(1./3.)##m
dRY=math.sqrt(4.**2.+(4.5-3.5)**2.)##m
dRYdash=math.sqrt((9.-1.)**2.+4.**2.)##m
dRdashY=math.sqrt((9.-1.)**2.+4.**2.)##m
dRdashYdash=math.sqrt(4.**2.+(4.5-3.5)**2.)##m
DRY=(dRY*dRYdash*dRdashY*dRdashYdash)**(1./4.)##m
DYB=((dRY*dRYdash)**2.)**(1./4.)##m
DBR=((8.*7.)**2.)**(1./4.)##m
Dm=(DRY*DYB*DBR)**(1./3.)##m
C=2*math.pi*epsilon_o/math.log(Dm/Ds)##F/m
C=C/10.**-3.##F/km
X=1./(2.*math.pi*f*C)##ohm
print '%s %.3f' %("Capacitive reactance too neutral(kohm) : ",X/1000.)#
Vph=VL*1000./math.sqrt(3.)##Volt
Ic=2.*math.pi*f*C*Vph##A
print '%s %.4f' %("Charging current(A/km)",Ic)#

Capacitive reactance too neutral(kohm) :  166.599
Charging current(A/km) 0.4574


## Example E22 - Pg 109¶

In [22]:
#calculate Capacitance of 100 km line
import math
#Given data :
d1=8.##m
d2=6.##m
epsilon_o=8.854*10.**-12.##permitivity
r=3.*5./2.*10.**-3.##m
C=4*math.pi*epsilon_o/math.log(2.**(1./3.)*d1/r*((d1**2.+d2**2.)/(4.*d1**2.+d2**2.)**(1./3.)))##F/m
C100=C*100.*1000.*10.**6.##microF
print '%s %.3f' %("Capacitance of 100 km line(micro Farad) : ",C100)#
#answer in the textbook is wrong.

Capacitance of 100 km line(micro Farad) :  1.122


## Example E23 - Pg 110¶

In [23]:
#calculate Capacitance,Charging currentimport math
#Given data :
import math
VL=132.##kV
f=50.##Hz
r=5./2.##cm
rdash=0.7788*r*10.**-2.##m
d=6.5##m
s=0.4##m
epsilon_o=8.854*10.**-12.##permitivity
Ds=math.sqrt(rdash*s)##m
dab=6.5##m
dab_dash=6.9##m
Dbc=Dab##m
dca=13.##m
dca_dash=12.6##m
d_cdash_a=13.4##m
Dm=(Dab*Dbc*Dca)**(1./3.)##m
L=0.2*math.log(Dm/Ds)##mH/km
C=2*math.pi*epsilon_o/math.log(Dm/Ds)##F/m
C=C/10.**-3.##F/km
print '%s %.2e' %("Capacitance per km(F/km) : ",C)#
Vph=VL*1000./math.sqrt(3)##Volt
Ic=2*math.pi*f*C*Vph##A/km
print '%s %.1f' %("Charging current per km(A/km) : ",Ic)#

Capacitance per km(F/km) :  1.23e-08
Charging current per km(A/km) :  0.3


## Example E24 - Pg 112¶

In [24]:
#calculate One conductor ACSR moose conductor line,Inductive reactance per Km per phase(ohm) :,Capacitivetive reactance per Km per phase(Mohm),Three conductor bundled line
#calculate Inductive reactance per km per phase(ohm)
#calculate Capacitivetive reactance per km per phase(Mohm)
import math
#Given data :
VL=132.##kV
f=50.##Hz
r=31.8/2##mm
rdash=0.7788*r##mm
d=10.*1000.##mm
epsilon_o=8.854*10**-12##permitivity
print '%s' %("One conductor ACSR moose conductor line : ")#
LAr=0.2*(math.log(d/rdash)+1./2.*math.log(2.))
LAi=0.866*math.log(2.)##mH/km
LB=0.2*math.log(d/rdash)##mH/km
LCr=0.2*(math.log(d/rdash)+1./2.*math.log(2.))
LCi=0.866*math.log(2.)##mH/km
Lav=(LAr+LAi+LB+LCr+LCi)/3.##mH/km
XL=2*math.pi*f*Lav*10.**-3.##ohm
print '%s %.3f' %("Inductive reactance per Km per phase(ohm) : ",XL)#
d1=10.##m
d2=10.##m
d3=20.##m
CN=2.*math.pi*epsilon_o/math.log((d1*d2*d3)**(1./3.)/(rdash*10.**-3.))/10.**3.##F/km
XC=1./(2.*math.pi*f*CN*10.**6.)##ohm
print '%s %.3f' %("Capacitivetive reactance per Km per phase(Mohm) : ",XC/10**6)#
print '%s' %("Three conductor bundled line : ")#
S=40./100.##m
Ds=(rdash*10.**-3.*S**2.)**(1./3.)##m
Deq=(d1*d2*d3)**(1./3.)##m
Ldash=0.2*math.log(Deq/Ds)##mH/km
XLdash=2*math.pi*f*Ldash*10.**-3.##ohm
print '%s %.3f' %("Inductive reactance per km per phase(ohm) : ",XLdash)#
Ds=(r*10.**-3.*S**2.)**(1./3.)##m
Cdash=2.*math.pi*epsilon_o*10.**3./math.log(Deq/Ds)##microF/km
XC=1./(2.*math.pi*f*Cdash)/10.**6.##Mohm
print '%s %.3f' %("Capacitivetive reactance per km per phase(Mohm) : ",XC)#

One conductor ACSR moose conductor line :
Inductive reactance per Km per phase(ohm) :  0.561
Capacitivetive reactance per Km per phase(Mohm) :  0.396
Three conductor bundled line :
Inductive reactance per km per phase(ohm) :  0.290
Capacitivetive reactance per km per phase(Mohm) :  0.259


## Example E25 - Pg 114¶

In [25]:
#calculate Capacitance per km of line
import math
#Given data :
r=1.5/2.##cm
d=3.*100.##cm
h=6.*100.##cm
epsilon_o=8.854*10.**-12.##permitivity
C=math.pi*epsilon_o/math.log(d/(1.+d**2./4./h**2.)**r)*10.**3.##F
print '%s %.3e' %("Capacitance per km of line(F) : ",C)#

Capacitance per km of line(F) :  4.916e-09


## Example E26 - Pg 114¶

In [26]:
#calculate Part(i) Capacitance per phase per meter length,Part(ii) Capacitance per phase per meter length
import math
#Given data :
r=2./100.##m
d1=4.##m
d2=4.##m
d3=8.##m
epsilon_o=8.854*10.**-12.##permitivity
CN=2.*math.pi*epsilon_o/math.log((d1*d2*d3)**(1./3.)/r)##F
print '%s %.3e' %("Part(i) Capacitance per phase per meter length(F) : ",CN)#
h1=20.##m
h2=20.##m
h3=20.##m
h12=math.sqrt(20.**2.+4.**2.)##m
h23=math.sqrt(20.**2.+4.**2.)##m
h31=math.sqrt(20.**2.+8.**2.)##m
Deq=(d1*d2*d3)**(1./3.)##m
CN=2.*math.pi*epsilon_o/(math.log(Deq/r)-math.log((h12*h23*h31/h1/h2/h3)**(1./3.)) )##F
print '%s %.3e' %("Part(ii) Capacitance per phase per meter length(F) : ",CN)#

Part(i) Capacitance per phase per meter length(F) :  1.006e-11
Part(ii) Capacitance per phase per meter length(F) :  1.013e-11