# CHAPTER 2.4: OVERHEAD LINE INSULATORS¶

## Example 2.4.1, Page number 183¶

In :
#Variable declaration
V_1 = 9.0       #Potential across top unit(kV)
V_2 = 11.0      #Potential across middle unit(kV)
n = 3.0         #Number of disc insulators

#Calculation
#Case(a)
K = (V_2-V_1)/V_1         #Ratio of capacitance b/w pin & earth to self capacitance
#Case(b)
V_3 = V_2+(V_1+V_2)*K     #Potential across bottom unit(kV)
V = V_1+V_2+V_3           #Voltage between line and earth(kV)
V_l = 3**0.5*V            #Line voltage(kV)
#Case(c)
eff = V/(n*V_3)*100       #String efficiency(%)

#Result
print('Case(a): Ratio of capacitance b/w pin & earth to self-capacitance of each unit, K = %.2f ' %K)
print('Case(b): Line voltage = %.2f kV' %V_l)
print('Case(c): String efficiency = %.f percent' %eff)

Case(a): Ratio of capacitance b/w pin & earth to self-capacitance of each unit, K = 0.22
Case(b): Line voltage = 61.39 kV
Case(c): String efficiency = 76 percent


## Example 2.4.2, Page number 183-184¶

In :
from sympy import Symbol

#Variable declaration
C = Symbol('C')
m = 10.0              #Mutual capacitance of top insulator

#Calculation
X = 1*C+m*C                  #Mutual capacitance
Y = (1.0+2)*C+m*C            #Mutual capacitance
Z = (1.0+2+3)*C+m*C          #Mutual capacitance
U = (1.0+2+3+4)*C+m*C        #Mutual capacitance
V = (1.0+2+3+4+5)*C+m*C      #Mutual capacitance

#Result
print('Mutual capacitance of each unit:')
print(' X = ' +repr(X)+'')
print(' Y = ' +repr(Y)+'')
print(' Z = ' +repr(Z)+'')
print(' U = ' +repr(U)+'')
print(' V = ' +repr(V)+'')

Mutual capacitance of each unit:
X = 11.0*C
Y = 13.0*C
Z = 16.0*C
U = 20.0*C
V = 25.0*C


## Example 2.4.3, Page number 184¶

In :
from sympy import Symbol

#Variable declaration
V = Symbol('V')
n = 3.0             #Number of insulators

#Calculation
V_1 = 155.0/475.0*V          #Potential across top unit
V_2 = 154.0/155.0*V_1        #Potential across middle unit
V_3 = 166.0/155.0*V_1        #Potential across bottom unit
eff = V*100/(n*V_3)          #String efficiency(%)

#Result
print('Voltage across top unit, V_1 = ' +repr(V_1)+'')
print('Voltage across middle unit, V_2 = ' +repr(V_2)+'')
print('Voltage across bottom unit, V_3 = ' +repr(V_3)+'')
print('String efficiency = %.2f percent' %eff)

Voltage across top unit, V_1 = 0.326315789473684*V
Voltage across middle unit, V_2 = 0.32421052631579*V
Voltage across bottom unit, V_3 = 0.349473684210526*V
String efficiency = 95.38 percent


## Example 2.4.4, Page number 184-185¶

In :
#Variable declaration
V_3 = 17.5          #Voltage across line unit(kV)
c = 1.0/8           #Shunt capacitance = 1/8 of insulator capacitance
n = 3.0             #Number of insulators

#Calculation
K = c                       #String constant
V_1 = V_3/(1+3*K+K**2)      #Voltage across top unit(kV)
V_2 = (1+K)*V_1             #Voltage across middle unit(kV)
V = V_1+V_2+V_3             #Voltage between line & earth(kV)
eff = V*100/(n*V_3)         #String efficiency(%)

#Result
print('Line to neutral voltage, V = %.2f kV' %V)
print('String efficiency = %.2f percent' %eff)

Line to neutral voltage, V = 44.24 kV
String efficiency = 84.27 percent


## Example 2.4.5, Page number 185¶

In :
#Variable declaration
n = 8.0               #Number of insulators

#Calculation
A = 1.0/(n-1)         #Line to pin capacitance
B = 2.0/(n-2)         #Line to pin capacitance
C = 3.0/(n-3)         #Line to pin capacitance
D = 4.0/(n-4)         #Line to pin capacitance
E = 5.0/(n-5)         #Line to pin capacitance
F = 6.0/(n-6)         #Line to pin capacitance
G = 7.0/(n-7)         #Line to pin capacitance

#Result
print('Line-to-pin capacitance are:')
print(' A = %.3f*C' %A)
print(' B = %.3f*C' %B)
print(' C = %.3f*C' %C)
print(' D = %.3f*C' %D)
print(' E = %.3f*C' %E)
print(' F = %.3f*C' %F)
print(' G = %.3f*C' %G)

Line-to-pin capacitance are:
A = 0.143*C
B = 0.333*C
C = 0.600*C
D = 1.000*C
E = 1.667*C
F = 3.000*C
G = 7.000*C


## Example 2.4.6, Page number 186¶

In :
#Variable declaration
m = 6.0               #Mutual capacitance
n = 5.0               #Number of insulators

#Calculation
E_4 = (1+(1/m))                                    #Voltage across 4th insulator as percent of E_5(%)
E_3 = (1+(3/m)+(1/m**2))                           #Voltage across 3rd insulator as percent of E_5(%)
E_2 = (1+(6/m)+(5/m**2)+(1/m**3))                  #Voltage across 2nd insulator as percent of E_5(%)
E_1 = (1+(10/m)+(15/m**2)+(7/m**3)+(1/m**4))       #Voltage across 1st insulator as percent of E_5(%)
E_5 = 100/(E_4+E_3+E_2+E_1+1)                      #Voltage across 5th insulator as percent of E_5(%)
E4 = E_4*E_5                                       #Voltage across 4th insulator as percent of E_5(%)
E3 = E_3*E_5                                       #Voltage across 3rd insulator as percent of E_5(%)
E2 = E_2*E_5                                       #Voltage across 2nd insulator as percent of E_5(%)
E1 = E_1*E_5                                       #Voltage across 1st insulator as percent of E_5(%)
eff = 100/(n*E1/100)                               #String efficiency(%)

#Result
print('Voltage distribution as a percentage of voltage of conductor to earth are:')
print(' E_1 = %.2f percent' %E1)
print(' E_2 = %.2f percent' %E2)
print(' E_3 = %.1f percent' %E3)
print(' E_4 = %.1f percent' %E4)
print(' E_5 = %.2f percent' %E_5)
print('String efficiency = %.f percent' %eff)
print('\nNOTE: Changes in obtained answer from that of textbook is due to more precision')

Voltage distribution as a percentage of voltage of conductor to earth are:
E_1 = 34.80 percent
E_2 = 23.94 percent
E_3 = 17.1 percent
E_4 = 13.0 percent
E_5 = 11.17 percent
String efficiency = 57 percent

NOTE: Changes in obtained answer from that of textbook is due to more precision


## Example 2.4.7, Page number 186-187¶

In :
#Variable declaration
n = 3.0              #Number of insulators
C_1 = 0.2            #Capacitance in terms of C
C_2 = 0.1            #Capacitance in terms of C

#Calculation
#Without guard ring
e_2_a = 13.0/13.3                 #Potential across middle unit as top unit
e_1_a = 8.3/6.5*e_2_a             #Potential across bottom unit
E_a = 1+(1/(8.3/6.5))+(1/e_1_a)   #Voltage in terms of e_1
eff_a = E_a/n*100                 #String efficiency(%)
e1_a = 1/E_a                      #Voltage across bottom unit as a % of line voltage
e2_a = 1/(8.3/6.5)*e1_a           #Voltage across middle unit as a % of line voltage
e3_a = 1/e_1_a*e1_a               #Voltage across top unit as a % of line voltage
#With guard ring
e_2_b = 15.4/15.5                 #Potential across middle unit as top unit
e_1_b = 8.3/7.7*e_2_b             #Potential across bottom unit
E_b = 1+(1/(8.3/7.7))+(1/e_1_b)   #Voltage in terms of e_1
eff_b = E_b/n*100                 #String efficiency(%)
e1_b = 1/E_b                      #Voltage across bottom unit as a % of line voltage
e2_b = 1/(8.3/7.7)*e1_b           #Voltage across middle unit as a % of line voltage
e3_b = 1/e_1_b*e1_b               #Voltage across top unit as a % of line voltage

#Result
print('Without guard ring:')
print(' Voltage across bottom unit, e_1 = %.2f*E' %e1_a)
print(' Voltage across bottom unit, e_2 = %.2f*E' %e2_a)
print(' Voltage across bottom unit, e_3 = %.2f*E' %e3_a)
print(' String efficiency = %.1f percent' %eff_a)
print('\nWith guard ring:')
print(' Voltage across bottom unit, e_1 = %.2f*E' %e1_b)
print(' Voltage across bottom unit, e_2 = %.2f*E' %e2_b)
print(' Voltage across bottom unit, e_3 = %.3f*E' %e3_b)
print(' String efficiency = %.2f percent' %eff_b)

Without guard ring:
Voltage across bottom unit, e_1 = 0.39*E
Voltage across bottom unit, e_2 = 0.30*E
Voltage across bottom unit, e_3 = 0.31*E
String efficiency = 86.1 percent

With guard ring:
Voltage across bottom unit, e_1 = 0.35*E
Voltage across bottom unit, e_2 = 0.32*E
Voltage across bottom unit, e_3 = 0.326*E
String efficiency = 95.38 percent


## Example 2.4.8, Page number 187-188¶

In :
#Variable declaration
n = 3.0              #Number of insulators

#Calculation
V_1 = 0.988                 #Voltage across top unit as middle unit
V_3 = 1.362                 #Voltage across bottom unit as middle unit
V_2 = 1/(V_1+1+V_3)         #Voltage across middle unit as % of line voltage to earth
V1 = V_1*V_2*100            #Voltage across top unit as % of line voltage to earth
V2 = V_2*100                #Voltage across middle unit as % of line voltage to earth
V3 = V_3*V_2*100            #Voltage across bottom unit as % of line voltage to earth
eff = 100/(n*V3/100)        #String efficiency(%)

#Result
print('Case(a): Voltage across top unit as a percentage of line voltage to earth, V_1 = %.2f percent' %V1)
print('         Voltage across middle unit as a percentage of line voltage to earth, V_2 = %.2f percent' %V2)
print('         Voltage across bottom unit as a percentage of line voltage to earth, V_3 = %.2f percent' %V3)
print('Case(b): String efficiency = %.2f percent' %eff)

Case(a): Voltage across top unit as a percentage of line voltage to earth, V_1 = 29.49 percent
Voltage across middle unit as a percentage of line voltage to earth, V_2 = 29.85 percent
Voltage across bottom unit as a percentage of line voltage to earth, V_3 = 40.66 percent
Case(b): String efficiency = 81.99 percent


## Example 2.4.9, Page number 188¶

In :
#Variable declaration
n = 3.0              #Number of insulators
V = 20.0             #Voltage across each conductor(kV)
c = 1.0/5            #Capacitance ratio

#Calculation
V_2 = 6.0/5.0            #Voltage across middle unit as top unit
V_1 = V/(1+2*V_2)        #Voltage across top unit(kV)
V_3 = V_2*V_1            #Voltage across bottom unit(kV)
C_x = c*(1+(1/V_2))      #Capacitance required

#Result
print('Case(a): Voltage on the line-end unit, V_3 = %.2f kV' %V_3)
print('Case(b): Value of capacitance required, Cx = %.3f*C' %C_x)

Case(a): Voltage on the line-end unit, V_3 = 7.06 kV
Case(b): Value of capacitance required, Cx = 0.367*C