# Chapter 05 : Current And Voltage Relations On A Transmission Line¶

## Example 5.1, Page No 101¶

In [1]:
import math
#initialisation of variables
D_12 = 23.8
D_23 = 23.8
D_31 = 47.6 #in ft
l = 230.0  #in mi
f = 60  #in Hz
P = 125e6  #in W
V = 215e3  #in V

#Calculations
D_eq = (D_12 * D_23 * D_31)**(1.0/3)

#From Table A.1 and A.2 for 30ft Rook
#z = R + i(Xa + Xd)
z = complex(0.1603,(0.415+0.4127))

#From Table A.1 and A.3 for 30ft Rook
y = complex(1e-6 / ( 0.0950 + 0.1008))

#Calculations
yl = 0.0455806 + complex(0.4750793)
Z_c = 404.43706 - complex(38.802997)
V_r = V / math.sqrt(3)
I_r = P / (math.sqrt(3)*V)
print(" Sending end voltage= {0:.2f}+{1:.2f}i".format(yl.real, yl.imag))

math.cosh_yl = math.cosh(yl.real) * math.cos(yl.imag) + complex(math.sinh(yl.real)) * math.sin(yl.imag)
math.sinh_yl = math.sinh(yl.real) * math.cos(yl.imag) + complex(math.cosh(yl.real)) * math.sin(yl.imag)

V_s = V_r * math.cosh_yl + I_r * Z_c * math.sinh_yl
I_s = I_r * math.cosh_yl + V_r * math.sinh_yl / Z_c
print( "Sending end voltage = {0:.2f}".format(abs(V_s.real)))
print("Angle = %.2f V " %(math.degrees(math.atan2(V_s.imag,V_s.real))))
print( "Sending end voltage = {0:.2f}".format(abs(I_s.real)))
print("Angle = %.2f V " %(math.degrees(math.atan2(I_s.imag,I_s.real))))

Line_voltage = math.sqrt(3) * abs(V_s) / 1000.0
Line_current = abs(I_s)
Power_factor = math.cos(math.atan2(V_s.imag,V_s.real) - math.atan2(I_s.imag,I_s.real))
Power = math.sqrt(3) * Line_voltage * Line_current * Power_factor

#Results
print(" Sending end line voltage = %.1f kV " %Line_voltage)
print(" Sending end line current = %.1f A " %Line_current)
print(" Sending end power = %.0f kW " %Power)

voltage_regulation = (((abs(V_s)/abs(math.cosh_yl)) - V_r)/V_r)*100
print(" Voltage Regulation = %.1f percent " %voltage_regulation)
B = 0.0020656
y1 = 2 * math.pi / B
Velocity = f * y1
print(" Wavelength = %.0f mi " %y1)
print(" Velocity = %.0f mi/s " %Velocity)

 Sending end voltage= 0.52+0.00i
Sending end voltage = 208167.01
Angle = 0.00 V
Sending end voltage = 567.06
Angle = 0.00 V
Sending end line voltage = 360.6 kV
Sending end line current = 567.1 A
Sending end power = 354129 kW
Voltage Regulation = 47.3 percent
Wavelength = 3042 mi
Velocity = 182509 mi/s


## Example 5.2, Page No 103¶

In [2]:
import math
#initialisation of variables
l = 230.0  #in mi
f = 60.0  #in Hz
P = 125e6  #in W
V = 215e3  #in V

#From Table A.1 and A.2 for 30ft Rook
#z = R + i(Xa + Xd)
z = 0.1603 + complex(0.415+0.4127)

#From Table A.1 and A.3 for 30ft Rook
y = complex(1e-6 / ( 0.0950 + 0.1008))

#Calculations
yl = 0.0455806 + complex(0.4750793)
Z_c = 404.43706 - complex(38.802997)
V_r = V / math.sqrt(3)
I_r = P / (math.sqrt(3)*V)

math.cosh_yl = math.cosh(yl.real) * math.cos(yl.imag) + complex(math.sinh(yl.real)) * math.sin(yl.imag)
math.sinh_yl = math.sinh(yl.real) * math.cos(yl.imag) + complex(math.cosh(yl.real)) * math.sin(yl.imag)

#Per Unit calculations
Base_impedance = V**2 / P
Base_current = P / (math.sqrt(3)*V)
Z_c_pu = Z_c / Base_impedance
V_r_pu = (V / math.sqrt(3)) / (V / math.sqrt(3))
I_r_pu = (P / (math.sqrt(3)*V)) / Base_current

V_s_pu = V_r_pu * math.cosh_yl + I_r_pu * Z_c_pu * math.sinh_yl
I_s_pu = I_r_pu * math.cosh_yl + V_r_pu * math.sinh_yl / Z_c_pu

Line_voltage = abs(V_s_pu)*V / 1000
Line_current = abs(I_s_pu)*Base_current

#Results
print(" Sending end line voltage = %.1f V " %Line_voltage)
print(" Sending end line current = %.1f A " %Line_current)

 Sending end line voltage = 360.6 V
Sending end line current = 567.1 A


## Example 5.3, Page No 106¶

In [3]:
import math
#initialisation of variables
l = 230  #in mi
f = 60  #in Hz
P = 125e6  #in W
V = 215e3  #in V

#From Table A.1 and A.2 for 30ft Rook
#z = R + complex(Xa + Xd)
z = 0.1603 + complex(0.415+0.4127)

#From Table A.1 and A.3 for 30ft Rook
y = complex(1e-6 / ( 0.0950 + 0.1008))

#Calculations
yl = 0.0455806 + complex(0.4750793)
Z_c = 404.43706 - complex(38.802997)
cosh_yl = math.cosh(yl.real) * math.cos(yl.imag) + complex(math.sinh(yl.real)) * math.sin(yl.imag)
sinh_yl = math.sinh(yl.real) * math.cos(yl.imag) + complex(math.cosh(yl.real)) * math.sin(yl.imag)

#Equivalent pi circuit
Z1 = Z_c * sinh_yl
Y1_2 = (cosh_yl - 1)/(Z_c * sinh_yl)

#Results
print('Equivalent PI circuit')
print( "Total series impedance of the line = {0:.2f}".format(abs(Z1.real)))
print("Angle = %.2f ohm in series arm " %(math.degrees(math.atan2(Z1.imag,Z1.real))))
print( "Total Shunt admittance of the line = {0:.2f}".format(abs(Y1_2.real)))
print("Angle = %.2f ohm in each shunt arm " %(math.degrees(math.atan2(Y1_2.imag,Y1_2.real))))

#Nominal pi Circuit
Z = l * z
Y_2 = y * l/2

print('Nominal PI circuit')
print( "Total series impedance of the line = {0:.2f}".format(abs(Z.real)))
print("Angle = %.2f ohm in series arm " %(math.degrees(math.atan2(Z.imag,Z.real))))
print( "Total Shunt admittance of the line = {0:.2f}".format(abs(Y_2.real)))
print("Angle = %.2f ohm in each shunt arm " %(math.degrees(math.atan2(Y_2.imag,Y_2.real))))

zp = ((abs(Z)-abs(Z1))/abs(Z1))*100
yp = ((abs(Y_2)-abs(Y1_2))/abs(Y1_2))*100

print(" Line impedace of the series arm of the nominal pi exceeds that of equivalent pi by %.1f percent " %zp)
print(" Conductance of the shunt arms of the nominal pi is %.0f percent less than that of equivalent pi " %abs(yp))

Equivalent PI circuit
Total series impedance of the line = 199.09
Angle = 0.00 ohm in series arm
Total Shunt admittance of the line = 0.00
Angle = 0.00 ohm in each shunt arm
Nominal PI circuit
Total series impedance of the line = 227.24
Angle = 0.00 ohm in series arm
Total Shunt admittance of the line = 0.00
Angle = 0.00 ohm in each shunt arm
Line impedace of the series arm of the nominal pi exceeds that of equivalent pi by 14.1 percent
Conductance of the shunt arms of the nominal pi is 16 percent less than that of equivalent pi


## Example 5.4 Page No 111¶

In [4]:
import math
#initialisation of variables
l = 230.0  #in mi
f = 60.0   #in Hz
P = 125e6  #in W
V = 215e3  #in V

#From Table A.1 and A.2 for 30ft Rook
#z = R + i(Xa + Xd)
z = 0.1603 + complex(0.415+0.4127)

#From Table A.1 and A.3 for 30ft Rook
y =complex(1e-6 / ( 0.0950 + 0.1008))

#Calculations
yl = 0.0455806 + complex(0.4750793)
Z_c = 404.43706 - complex(38.802997)

cosh_yl = math.cosh(yl.real) * math.cos(yl.imag) + complex(math.sinh(yl.real)) * math.sin(yl.imag)
sinh_yl = math.sinh(yl.real)*math.cos(yl.imag) + complex(math.cosh(yl.real)) * math.sin(yl.imag)

#Equivalent pi circuit
Z1 = Z_c * sinh_yl
Y1_2 = (math.cosh_yl - 1)/(Z_c * sinh_yl)
A = cosh_yl
D = cosh_yl
B = Z1
C = sinh_yl / Z_c

print('For an uncompensated line')
print( "A = D = {0:.2f}".format(abs(A.real)))
print("Angle = %.2f v " %(math.degrees(math.atan2(A.imag,A.real))*180/math.pi))
print( "B = {0:.2f}".format(abs(B.real)))
print("Angle = %.2f ohm " %(math.degrees(math.atan2(B.imag,B.real))*180/math.pi))
print( "C = {0:.2f}".format(abs(C.real)))
print("Angle = %.2f ohm " %(math.degrees(math.atan2(C.imag,C.real))*180/math.pi))

#For a series compensation factor of 70%
cf = 0.7
B1 = Z1 - complex(cf) * l * (0.415 + 0.4127) #X_a = 0.415 ohm/mi,X_d = 0.4127 in
A1 = B1 * Y1_2 + 1
C1 = 2 * Y1_2 + B1 * (Y1_2)**2

#Results
print('For a series compensation factor  of 70%')
print( "A = D = {0:.2f}".format(abs(A1.real)))
print("Angle = %.2f " %(math.degrees(math.atan2(A1.imag,A1.real))))
print( "B = {0:.2f}".format(abs(B1.real)))
print("Angle = %.2f ohm " %(math.degrees(math.atan2(B1.imag,B1.real))))
print( "C = {0:.2f}".format(abs(C1.real)))
print("Angle = %.2f ohm " %(math.degrees(math.atan2(C1.imag,C1.real))))

For an uncompensated line
A = D = 1.14
Angle = 0.00 v
B = 199.09
Angle = 0.00 ohm
C = 0.00
Angle = 0.00 ohm
For a series compensation factor  of 70%
A = D = 1.05
Angle = 0.00
B = 65.83
Angle = 0.00 ohm
C = 0.00
Angle = 0.00 ohm


## Example 5.5, Page No 112¶

In [5]:
import math
#initialisation of variables
l = 230.0  #in mi
f = 60.0   #in Hz
P = 125e6  #in W
V = 215e3  #in V

#From Table A.1 and A.2 for 30ft Rook
#z = R + i(Xa + Xd)
z = 0.1603 + complex(0.415+0.4127)

#From Table A.1 and A.3 for 30ft Rook
y = complex(1e-6 / ( 0.0950 + 0.1008))

#Calculations
yl = 0.0455806 + complex(0.4750793)
Z_c = 404.43706 - complex(38.802997)
V_r = V / math.sqrt(3)
I_r = P / (math.sqrt(3)*V)

cosh_yl = math.cosh(yl.real) * math.cos(yl.imag) + complex(math.sinh(yl.real)) * math.sin(yl.imag)
sinh_yl = math.sinh(yl.real) * math.cos(yl.imag) + complex(math.cosh(yl.real)) * math.sin(yl.imag)

V_s = V_r * cosh_yl + I_r * Z_c * sinh_yl
I_s = I_r * cosh_yl + V_r * sinh_yl / Z_c

#Equivalent pi circuit
Z1 = Z_c * sinh_yl
Y1_2 = (cosh_yl - 1)/(Z_c * sinh_yl)

#Total capacitive Susceptance
B_c = complex(y) * l

#For 70% Compensation
cf = 0.7
B_L = - B_c * 0.7

#From appendix
A = 1
D = 1
B = 0
C = -complex(B_L)

#From Table A.6 for combining two networks in series
A_eq = cosh_yl + Z1 * C
voltage_regulation = ((abs(V_s)/abs(A_eq))-V_r)*100.0/V_r
print(" Voltage regulation in percent ")
print(voltage_regulation)

 Voltage regulation in percent
28.7687144089