Chapter 7 : Synchronous Motors

Example 7.1 Page no : 27

In [1]:
import math 
import numpy as np

# Variables
V_l = 400.
R_a = 0.2
X_s = 2.  			#armature resistance and synchronous reactance
I_L = 25.
I_aph = I_L
V_ph = V_l/math.sqrt(3)
Z_s = complex(R_a,X_s) 			#synchronous impedance
angle = math.atan(Z_s.imag/Z_s.real)
angle = math.degrees(angle)
E_Rph=I_aph*abs(Z_s)
theta = (math.pi/180.)*angle 

# Calculations and Results
#case 1
phi = math.acos(0.8) 			#lagging
E_bph =  math.sqrt( (E_Rph)**2 + (V_ph)**2  -2*E_Rph*V_ph*math.cos(theta-phi)  )
print 'i)Back EMF induced with 0.8 lagging pf is %.3f V'%(E_bph)

#case 2
phi = math.acos(0.9) 			#leading
E_bph =  math.sqrt( (E_Rph)**2 + (V_ph)**2  -2*E_Rph*V_ph*math.cos(theta+phi)  )
print 'ii)Back EMF induced with 0.8 lagging pf is %.3f V'%(E_bph)

#case 3
phi = math.acos(1)
E_bph =  math.sqrt( (E_Rph)**2 + (V_ph)**2  -2*E_Rph*V_ph*math.cos(theta)  )
print 'iii)Back EMF induced with 0.8 lagging pf is %.3f V'%(E_bph)

i)Back EMF induced with 0.8 lagging pf is 200.386 V
ii)Back EMF induced with 0.8 lagging pf is 252.678 V
iii)Back EMF induced with 0.8 lagging pf is 231.406 V

Example 7.2 Page no : 28

In [2]:
import math 
from numpy import cos
# Variables
V_l = 500.
R_a = 0.4
X_s = 4. 			#armature resistance and synchronous reactance
Z_s = complex(R_a,X_s)			#synchronous impedance
angle = math.degrees(math.atan(Z_s.imag/Z_s.real))
theta = (math.pi/180)*angle			#phasemag returns angle in degrees,not radians
V_ph = V_l/math.sqrt(3)
I_l = 50.
I_aph = I_l
E_Rph = I_aph*abs(Z_s)

# Calculations and Results		
#case 1
E_bline = 600
E_bph = E_bline/math.sqrt(3)
phi = math.acos( (-E_bph**2 + E_Rph**2 + V_ph**2 )/(2*E_Rph*V_ph) ) -theta    			#leading
#because E_bph =  math.sqrt( (E_Rph)**2 + (V_ph)**2  -2*E_Rph*V_ph*math.cos(theta+phi)  )
print 'i)power factor is %.4f leading'%(cos(phi))

#case 2
E_bline = 380
E_bph = E_bline/math.sqrt(3)
phi =  theta-math.acos( (-E_bph**2 + E_Rph**2 + V_ph**2 )/(2*E_Rph*V_ph) )     			#leading
#because E_bph =  math.sqrt( (E_Rph)**2 + (V_ph)**2  -2*E_Rph*V_ph*math.cos(theta-phi)
print 'ii)power factor is %.4f lagging'%(cos(phi))

i)power factor is 0.9977 leading
ii)power factor is 0.8197 lagging

Example 7.3 Page no : 29

In [3]:
import math 

# Variables
V_L = 6600.
P_out = 500.*10**3
eta = 83./100 			#efficiency
R_a = 0.3
X_s = 3.2 			#armature resistance and synchronous reactance
Z_s = complex(R_a,X_s)  			#synchronous impedance
angle = math.degrees(math.atan(Z_s.imag/Z_s.real))
theta = (math.pi/180)* angle 			#phasemag returns the angle in degrees not radians
phi = math.acos(0.8) 			#leading
V_ph = V_L/math.sqrt(3)
P_in = P_out/eta

# Calculations and Results
I_L =    P_in/ (math.sqrt(3) * V_L *  math.cos(phi) )

# because P_in = math.sqrt(3) * V_L * I_L * math.cos(phi)
I_aph = I_L
E_Rph = I_aph*abs(Z_s)
E_bph =  math.sqrt( (E_Rph)**2 + (V_ph)**2  -2*E_Rph*V_ph*math.cos(theta+phi)  )
print 'i) Generated EmF on full loaad is %.2f V'%(E_bph)

delta =  math.degrees(math.asin( (E_Rph/E_bph))*math.sin(theta+phi)  )
#This is obtained after applying sune rule to triangle OAB from thre phasor diagram
print 'ii) load angle is %.2f degrees'%(delta)

i) Generated EmF on full loaad is 3925.33 V
ii) load angle is 2.64 degrees

Example 7.4 Page no : 32

In [4]:
import math 
from numpy import roots

# Variables
V_L = 500.
V_ph = V_L/math.sqrt(3)
phi = math.acos(0.9) 				#lagging
output_power = 17.*10**3
R_a = 0.8 							#armaature reactance
mechanical_losses = 1300. 			#mechanical losses is W
P_m = output_power+mechanical_losses   			#gross mechanical power developed

#  P_m =  input_power - stator losses
#  input_power =  3* V_ph * I_aph * math.cos(phi)
#  Stator losses =  3*I_aph**2*R_a
# solving above equations we get 2.4 I_a**2 - 779/.4225*I_a + 18300  =  0
I_a_eqn = [2.4, -779.4225, 18300]
I_a_roots = roots(I_a_eqn)
I_a = I_a_roots[1]  			#neglecting higher value
I_aph = I_a
print 'Current drawn by the motor is %.3f A'%(I_a)

input_power =  3* V_ph * I_aph * math.cos(phi)
eta = 100*output_power/input_power
print 'Full load efficiency is %.2f percent'%(eta)

Current drawn by the motor is 25.478 A
Full load efficiency is 85.61 percent

Example 7.5 Page no : 44

In [5]:
import math 

# Variables
#subscript 1 is for industrial load and 2 for synchronous motor
P_1 = 800. 			# Active power in KW
phi_1 = math.acos(0.6) 			#lagging
Q_1 = P_1*math.tan(phi_1)   			#reactive power by load 1

# Calculations and Results
output_power = 200.
eta = 91./100 			#efficiency of synchronous motor
input_power =  output_power/eta
P_2 = input_power			# active power drawn by synchronous motor
P_T = P_1 + P_2  			#combined total load of industry and synchronous motor
phi_T = math.acos(0.92 )			#lagging
Q_T = P_T* math.tan(phi_T)  			#from power triangle
Q_2 =  Q_T - Q_1  			#it turns out to be negative indicating its leading nature
S_2 = math.sqrt( P_2**2 + Q_2**2    )
print 'Desired kVA rating of Synchronous motor  is %.3f kVA'%(S_2)

phi_2 =  math.atan (Q_2/P_2)
print 'Power factor of synchronous motor is %.4f LEADING'%(math.cos(phi_2))

Desired kVA rating of Synchronous motor  is 669.353 kVA
Power factor of synchronous motor is 0.3283 LEADING

Example 7.6 Page no : 47

In [6]:
import math 

# Variables
V_L = 400.
output_power = 37.3*1000 			#Watts on full load
Z_s = complex(0.2,1.6) 			#synchronous impedance
angle = math.degrees(math.atan(Z_s.imag/Z_s.real))
theta = (math.pi/180)*angle 			#phase mag returns the angle in degrees and not raidians
phi = math.acos(0.9) 			#leading
V_ph = V_L/math.sqrt(3)
eta = 88. 			#efficiency in percentage

# Calculations
input_power = 100*output_power/eta
I_L = input_power/(math.sqrt(3)*V_L*math.cos(phi))
I_aph = I_L
E_Rph = I_aph*abs(Z_s)

E_bph =  math.sqrt( (E_Rph)**2 + (V_ph)**2  -2*E_Rph*V_ph*math.cos(theta+phi)  )
E_line = math.sqrt(3)*E_bph

# Results
print 'Induced EMF is %.2f V and its line value is %.2f V'%(E_bph,E_line)

Induced EMF is 285.65 V and its line value is 494.75 V

Example 7.7 Page no : 48

In [7]:
import math 

# Variables
V_L = 400.
input_power = 20.*1000 
R_a = 0.
X_s = 4. 			#armature reactance and synchronous reactance
Z_s = complex(R_a,X_s)   			#synchronous impedance
angle = math.degrees(math.atan(Z_s.imag))

theta = (math.pi/180)*angle  			#phase mag returns the angle in degrees and not raidians
V_ph = V_L/math.sqrt(3)
E_bline = 550. 			#star connection
E_bph = E_bline/math.sqrt(3)

# Calculations
I_a_cos_phi = input_power/(math.sqrt(3)*V_L)  			#product of I_a and math.cos(phi)
I_a_sin_phi =  (   math.sqrt(E_bph**2- (abs(Z_s)*I_a_cos_phi)**2   )  -V_ph )/abs(Z_s)			#from triangle DAB
phi = math.atan(I_a_sin_phi/I_a_cos_phi)
I_a = I_a_cos_phi/math.cos(phi)        

# Results
print 'Motor power fctor is %.3f Leading'%(cos(phi))
print 'Current drawn by the motor is %.2f A'%(I_a)

Motor power fctor is 0.872 Leading
Current drawn by the motor is 33.11 A

Example 7.8 Page no : 50

In [8]:
import math 

# Variables
V_L = 3300.
V_ph = V_L/math.sqrt(3)
R_a = 2.
X_s = 18. 			#armature reactance and synchronous reactance
Z_s = complex(R_a,X_s)			#synchronous impedance
angle = math.degrees(math.atan(Z_s.imag/Z_s.real))

theta = (math.pi/180)*angle 			#phasemag returns angle in degrees not radians
E_bline = 3800.
E_bph = E_bline/math.sqrt(3)

#part(i)
P_m_max  =  (E_bph*V_ph/abs(Z_s))- (E_bph**2/abs(Z_s))*math.cos(theta)
print 'i)Max total mechanical power developed that motor can develop is %.2f W per phase'%(P_m_max)

#part(ii)
#from phasor diagram applying math.comath.sine rule to triangle OAB
E_Rph = math.sqrt(  E_bph**2 + V_ph**2 -2*E_bph*V_ph*math.cos(theta) ) 
I_aph =  E_Rph/abs(Z_s)
print 'ii)Current at max power developed is %.1f A'%(I_aph)

copper_loss = 3* I_aph**2 * R_a
P_in_max_total = 3 * P_m_max 			#input power at max power developed
total_P_in =  P_in_max_total + copper_loss 			#total input power 
pf = total_P_in/(math.sqrt(3)*I_aph*V_L)
print 'Power factor at max power developed is %.3f leading'%(pf)


# 'Answer in part1 mismatched because of improper approximation in book'

i)Max total mechanical power developed that motor can develop is 201452.30 W per phase
ii)Current at max power developed is 151.4 A
Power factor at max power developed is 0.857 leading

Example 7.9 Page no : 51

In [10]:
import math 

# Variables
V_L = 500.
R_a = 0.03
X_s = 0.3 			#armature reactance and synchronous reactance
Z_s = complex(R_a,X_s)			#synchronous impedance
angle = math.degrees(math.atan(Z_s.imag/Z_s.real))

theta = (math.pi/180)*angle 			#phasemag returns angle in degrees not radians
phi = math.acos(0.8)
eta = 93/100.
output_power = 100.*746
input_power = output_power/eta
I_L = input_power/(math.sqrt(3)*V_L*math.cos(phi))
I_aph = I_L
E_Rph = I_aph*abs(Z_s)

#from the phasor diagram
E_bph  =    math.sqrt( E_Rph**2 + (V_L/math.sqrt(3))**2 - 2*E_Rph*(V_L/math.sqrt(3))*math.cos(phi+theta)  ) 

cu_losses = 3*(I_aph)**2*R_a 			#total copper losses
P_m =  input_power - cu_losses  			#total mechanical power developed

print 'EMF developed per phase is %.4f V \nTotal mechanical power developed is %.1f watts'%(E_bph,P_m)

EMF developed per phase is 308.1880 V 
Total mechanical power developed is 79008.6 watts

Example 7.10 Page no : 53

In [11]:
import math 

# Variables
V_L = 415.
V_ph = V_L 			#due to delta connection
E_bline = 520.
R_a = 0.5
X_s = 4. 			#armature reactance and synchronous reactance
Z_s = complex(R_a,X_s) 			#synchronous impedance
angle = math.degrees(math.atan(Z_s.imag/Z_s.real))

theta = (math.pi/180.)*angle 			#phasemag returns angle in degrees not radians

delta = theta 			#for maximum power
P_m_max  =  (E_bline*V_ph/abs(Z_s))- (E_bline**2/abs(Z_s))*math.cos(theta)
P_m_max_total =  3* P_m_max
fi_loss = 1000.  			#frictional and iron losses
P_out_total  =  P_m_max_total-fi_loss  

HP_output =  P_out_total/746 			#converting watts to horse power
print 'HP output for maximum power output is %.2f HP'%(HP_output)

#from the phasor diagram
E_Rph = math.sqrt(  E_bline**2 + V_ph**2 -2*E_bline*V_ph*math.cos(delta) ) 
I_aph =  E_Rph/abs(Z_s)
I_L = I_aph*math.sqrt(3)
print 'Line current is %f A'%(I_L)
cu_loss_total = 3*(I_aph)**2*R_a  			#total copper losses
input_power = P_m_max_total+ cu_loss_total
pf = input_power/(math.sqrt(3)*I_L*V_L)  			#leading
print 'Power factor for maximum power output is %.2f leading '%(pf)

eta = 100*P_out_total /input_power
print 'Efficiency for maximum power output is %.2f percent'%(eta)

# 'Answer might mismatch because of improper approximation done in book'

HP output for maximum power output is 180.48 HP
Line current is 268.015479 A
Power factor for maximum power output is 0.89 leading 
Efficiency for maximum power output is 78.48 percent

Example 7.11 Page no : 54

In [13]:
import math 

# Variables
P = 8.
f = 50. 			#Pole and frequency
N_s = 120.*f/P 			#synchronous speed
V_L = 6.6*10**3 
V_ph = V_L/math.sqrt(3)
Z_s = complex(0.66,6.6) 			#synchronous impedance
angle = math.degrees(math.atan(Z_s.imag/Z_s.real))

theta = (math.pi/180)*angle 			#phasemag returns angle in degree  not radians
E_bph = 4500.
input_power = 2500.*10**3
I_a_cosphi = input_power/(math.sqrt(3)*V_L)   			#Its product of I_a and math.cos(phi);I_a = I_l for star conneted load

# Calculations
#applying math.comath.sine rule to triangle ABC from phasor diagram and solve 
#math.tan(phi)**2 + 5.2252 math.tan(phi)-2.2432 = 0
p = [1, 5.2252, -2.2432]
tan_phi = roots(p)
phi = math.atan(tan_phi[1])
pf = math.cos(phi)
I_a =  I_a_cosphi/ math.cos(phi)

#apply math.sine rule to triangle ABC
delta =    math.asin(I_a*abs(Z_s)*math.sin(theta+phi)/E_bph)
P_m = 3*E_bph*I_a*math.cos(delta+phi)
T_g  =  P_m/(2*math.pi*N_s/60)

# Results
print 'i)Torque developed is %f N-m'%(T_g)
print 'ii)Input current is %.4f A'%(I_a)
print 'iii)Power factor is %.4f leading'%(pf)
print 'iv)Power angle is %.2f degrees '%((180/math.pi)*delta)

# note : rounding off error.

i)Torque developed is 30437.047497 N-m
ii)Input current is 235.4472 A
iii)Power factor is 0.9288 leading
iv)Power angle is 19.48 degrees

Example 7.12 Page no : 57

In [14]:
import math 
from numpy import roots

# Variables
input_power = 15.*10**3
V_L = 400.
V_ph = V_L/math.sqrt(3)
E_b = 480.
E_bph = E_b/math.sqrt(3)
Z_s = complex(1,5) 			#synchronous impedance
angle = math.degrees(math.atan(Z_s.imag/Z_s.real))

theta = (math.pi/180)*angle 			#phasemag returns angle in degree  not radians

# Calculations
I_a_cosphi = input_power/(math.sqrt(3)*V_L)  			#product of I_a & math.cos(phi)
#Applying math.comath.sine rule to triangle OAB and solving
#math.tan(phi)**2+ 4.101*math.tan(phi)-1.7499 = 0
p = [1,4.101,-1.7449]
tan_phi = roots(p)
phi = math.atan(tan_phi[1])  			#ignoring negative vaule
I_a =  I_a_cosphi/ math.cos(phi)

#applying math.sine rule to Triangle OAB
delta = math.asin( I_a*abs(Z_s)* math.sin(theta+phi)/E_bph )

# Results
print 'Load angle is %.1f degrees'%(delta*180/math.pi)
print 'Armature current is %.4f A'%(I_a)
print 'Power factor is %.3f leading'%(cos(phi))

Load angle is 24.9 degrees
Armature current is 23.2283 A
Power factor is 0.932 leading

Example 7.13 Page no : 59

In [19]:
import math 
from numpy import roots

# Variables
V_L = 400.
V_ph = V_L/math.sqrt(3)
E_b = 460.
E_bph = E_b/math.sqrt(3)
input_power = 3.75*10**3
Z_s = complex(1,8) 			#synchronous impedance
angle = math.degrees(math.atan(Z_s.imag/Z_s.real))

theta = math.degrees(angle) 			#phasemag returns angle in degree ,not radians
I_L_cos_phi  =  input_power/(math.sqrt(3)*V_L)

# Calculations and Results
#Applying math.comath.sine rule to triangle OAB and solving further
#math.tan(phi)**2 + 458.366*math.tan(phi) -450.65  = 0 
p = [1,458.366-450.65]
tan_phi = roots(p)

phi = math.atan(tan_phi[0])  			#ignoring negative value
print 'Required power factor is %.4f leading'%(cos(phi))
I_L = I_L_cos_phi /math.cos(phi)
print 'Required current is %.4f A'%(I_L)

# roots() python gives some different answer. Kindly check.

Required power factor is 0.1285 leading
Required current is 42.1134 A

Example 7.14 Page no : 60

In [20]:
import math 

# Variables
#subscript 1 indicates induction motor 1
P_1 = 350.
phi_1 = math.acos(0.7071) 			#lagging
Q_1 = P_1*math.tan(phi_1)			#from power triangle

#subscript 2 indicates induction motor 2
P_2 = 190.

# Calculations
#subscript T indicates total
P_T = P_1+P_2
phi_T = math.acos(0.9) 			#lagging
Q_T = P_T*math.tan(phi_T)

Q_2 = Q_T-Q_1
kva_rating = math.sqrt(P_2**2+ Q_2**2)

# Results
print 'kVA rating of synchronous motor is %.2f kVA'%(kva_rating)

kVA rating of synchronous motor is 209.59 kVA

Example 7.15 Page no : 61

In [21]:
import math 

# Variables
V_L = 400.
V_ph = V_L/math.sqrt(3)
Pole = 6.
f = 50.
R_a = 0.2
X_s = 3. 			#armature reactance and synchronous reactance
Z_s = complex(R_a,X_s)			#synchronous impedance
theta = math.atan(Z_s.imag/Z_s.real)       # calculates angle in radians

N_s = 120*f/Pole  			#synchronous speed

# Calculations
#subscript 1` refers to load 1
I_a1 = 20.
phi_1 = math.acos(1)
E_R1 =  I_a1* abs(Z_s)
E_bph  =   math.sqrt( E_R1**2 + V_ph**2 - 2*E_R1*V_ph*math.cos(phi_1+theta)  )

#subscript 2` refers to load 2
I_a2 = 60.
E_R2 =  I_a2* abs(Z_s)
phi_2 =  math.acos ((E_R2**2 + V_ph**2 -E_bph**2 )/(2*E_R2*V_ph))   -theta   			#new power factor

input_power = math.sqrt(3)*V_L*I_a2*math.cos(phi_2)
cu_loss = 3*I_a2**2*R_a
P_m = input_power-cu_loss
T_g  = P_m /(2*math.pi*N_s/60)   			#gross mechanical power developed

# Results
print 'Gross torque developed is %.4f N-m and new power factor is %.4f lagging'%(T_g,cos(phi_2))

Gross torque developed is 357.1971 N-m and new power factor is 0.9518 lagging

Example 7.16 Page no : 63

In [22]:
import math 

# Variables
V_L = 3300.
V_ph = V_L/math.sqrt(3)
E_bph = V_ph
Z_s = complex(0.5,5) 			#synchronous impedance
angle = math.degrees(math.atan(Z_s.imag/Z_s.real))

theta = (math.pi/180)*angle 			#phasemag returns angle in degrees, not radians
P = 8.
f = 50. 			#pole and frequency
delta_mech = 3. 			#mechanical angle in degrees by which rotor is behind
delta_elec = (P/2)*delta_mech  			#delta mech converted to electrical degrees
E_Rph = math.sqrt(  E_bph**2 + V_ph**2 -2*E_bph*V_ph*math.cos(math.radians(delta_elec) )) 
I_aph =  E_Rph/abs(Z_s)

#from the phasor diagram 
phi = theta- math.asin( math.sin(math.radians(delta_elec))*E_bph/E_Rph )
pf = math.cos(phi)
print 'power factor of the motor is %.5f lagging'%(pf)

power factor of the motor is 0.99999 lagging

Example 7.17 Page no : 64

In [23]:
import math 

# Variables
V_L = 400.
V_ph = V_L/math.sqrt(3)
E_bph = V_ph
P = 4.
f = 50.			#Pole and frequency
delta_mech = 4*(math.pi/180) 			#mechanical angle in degrees by which rotor is behind
delta_elec =  delta_mech *(P/2) 			#delta_mech convertd to electrical degrees
Z_s = complex(0,2) 			#synchronous impedance

# Calculations
#referring to phasor diagram
BC =   E_bph*math.sin(delta_elec)
AB =   E_bph
OA =   V_ph

AC =   math.sqrt(AB**2-BC**2)
OC =   OA-AC
phi = math.atan(OC/BC)
OB = math.sqrt(OC**2 + BC**2)
I_a = OB/abs(Z_s)

# Results
print 'Armature current drawn by the motor is %.4f A'%(I_a)

Armature current drawn by the motor is 16.1096 A

Example 7.18 Page no : 65

In [24]:
import math 

# Variables
V_L = 400.
V_ph = V_L/math.sqrt(3)
input_power = 5472.
Z_s = complex(0,10)  			#synchronous impedance
I_L_cosphi = input_power/(math.sqrt(3)*V_L) 			#product of I_L & math.cos(phi)
BC = 10*I_L_cosphi
AB = V_ph
OA = V_ph

# Calculations
#from Triangle ABC in phasor diagram
AC  =  math.sqrt(AB**2- BC**2)
OC  =  OA - AC

			#from Triangle OCB 
OB = math.sqrt( OC**2+ BC**2 )
E_Rph  =  OB
I_L = E_Rph/abs(Z_s)

phi = math.atan(OC/BC)
pf = math.cos(phi)
delta = math.atan(BC/AC)  			#load angle

# Results
print 'Power factor is %.4f lagging'%(pf)
print 'Load angle is %.0f degrees'%(delta*180/math.pi)
print 'Armature current is %.3f A'%(I_L)

Power factor is 0.9848 lagging
Load angle is 20 degrees
Armature current is 8.020 A

Example 7.19 Page no : 67

In [25]:
import math 

# Variables
V_L = 6600.
V_ph = V_L/math.sqrt(3)
Z_s = complex(2,20) 			#synchronous impedance
angle = math.degrees(math.atan(Z_s.imag/Z_s.real))

theta = (math.pi/180) * angle 			#phasemag returns angle in degrees not radians
P_1 = 1000*10**3
P_2 = 1500*10**3
phi_1 = math.acos(0.8)  			#leading

# Calculations
I_L1 = P_1/(math.sqrt(3)*V_L*math.cos(phi_1))
I_a1ph = I_L1
E_R1ph = I_a1ph*abs(Z_s)
E_bph =   math.sqrt(    V_ph**2 + E_R1ph** -2*V_ph*E_R1ph*math.cos(theta+phi_1)  )
I_a2_cosphi_2 = P_2/(math.sqrt(3)*V_L)

#Refer to the phasor diagram and solving for I_y
#404I_y**2 -152399.968 I_y -4543000 = 0
p = [404, -152399.968, -4543000]
ans = roots(p)
I_y = abs(ans[1]) 			#becuase root 1 is too high and root is -ve

I_a2 = complex(I_a2_cosphi_2,I_y)
phi_2 = math.degrees(math.atan(I_a2.imag/I_a2.real))

print 'Required power factor is %.3f leading'%(math.cos(math.radians(phi_2)))

Required power factor is 0.978 leading

Example 7.20 Page no : 69

In [26]:
import math 

V_L = 2300.
V_ph = V_L/math.sqrt(3)
I_L = 200.
I_a = I_L
Z_s = complex(0.2,2.2) 			#synchronous impedance
theta = math.atan(Z_s.imag/Z_s.real)
phi = math.acos(0.5)

# Calculations
E_Rph = I_a*abs(Z_s)
E_bph  =    math.sqrt( E_Rph**2 + V_ph**2 - 2*E_Rph*V_ph*math.cos(phi+theta)  ) 

# Results
print 'Generated EMF per phase is %.3f V'%(E_bph)

Generated EMF per phase is 1708.045 V

Example 7.21 Page no : 69

In [28]:
import math 

# Variables
V_L = 3300.
V_ph = V_L/math.sqrt(3)
E_bline = 3800.
E_bph = E_bline/math.sqrt(3)

R_a = 2.
X_s = 18. 			#armature resistance and synchronous reactance
Z_s = complex(R_a,X_s) 			#synchronous impedance
theta = math.atan(Z_s.imag/Z_s.real)

#part(i)
P_m_max  =  (E_bph*V_ph/abs(Z_s))- (E_bph**2/abs(Z_s))*math.cos(theta)  			#maximum total mechanical power
print 'i)Maximum total mechanical power that the motor can develop is %.2f W per phase'%(P_m_max )
#part(ii)
delta = theta 			#for max P_m
E_Rph = math.sqrt(  E_bph**2 + V_ph**2 -2*E_bph*V_ph*math.cos(delta) ) 
I_aph =  E_Rph/abs(Z_s)
print 'ii)Current at maximum power developed is %.1f A'%(I_aph)
cu_loss_total  =  3*I_aph**2*R_a  			#total copper loss
P_m_max_total = 3*P_m_max 			#total maximum total mechanical power
P_in_total  =  P_m_max_total+ cu_loss_total  			#total input power

pf = P_in_total/(math.sqrt(3)*V_L*I_aph)
print '    Power factor at maximum power developed is %.3f leading'%(pf)

# note : rounding off error.

i)Maximum total mechanical power that the motor can develop is 201452.30 W per phase
ii)Current at maximum power developed is 151.4 A
    Power factor at maximum power developed is 0.857 leading

Example 7.22 Page no : 71

In [29]:
import math 

#subscript 1 refers to load 1
I_1 = 18.
phi_1 = math.acos(0.8)
V_L = 440.
S_1 = math.sqrt(3)*I_1*V_L /1000  			#kVA for load 1
P_1 = S_1*math.cos(phi_1)
Q_1 = S_1*math.sin(phi_1)

# Calculations
P_out = 6.
eta_motor = 88./100
P_2 = P_out/eta_motor

P_T = P_1+P_2
phi_T = math.acos(1)  			#total power factor angle
Q_T = P_T*math.tan(phi_T)

Q_2 =   Q_T - Q_1   			#kVAR supplied by motor
#this will have  a negative sign just indicating its leading nature 
phi_2 = math.atan(abs(Q_2)/P_2)
pf = math.cos(phi_2)  			#leading
S_2 = P_2/math.cos(phi_2)  			#kVA input to the motor

# Results
print 'kVA input to the motor is %.3f kVA '%(S_2)
print 'Power factor when driving a 6kW mechanical load is %.4f leading'%(pf)

kVA input to the motor is 10.688 kVA 
Power factor when driving a 6kW mechanical load is 0.6379 leading

Example 7.23 Page no : 72

In [31]:
import math 

# Variables
output_power = 8.*10**3
V_L = 400.
V_ph = V_L/math.sqrt(3)
R_a = 0.
X_s = 8.			#armature resistance and syncronous reactance
Z_s = complex(R_a,X_s) 			#synchronous impedance
theta = math.atan(Z_s.imag)  			# returns angle in radians 
eta = 88./100
input_power = output_power/eta

# Calculations and Results
#minimum current occurs at max power factors
phi = math.acos(1)
I_a_min = input_power/(math.sqrt(3)*V_L*math.cos(phi)) 			#required minimum current 
print 'Minimum current is %.3f A'%(I_a_min)
E_R =  I_a_min * abs(Z_s)
E_bph  =    math.sqrt( E_R**2 + V_ph**2 - 2*E_R*V_ph*math.cos(phi+theta)  ) 
print 'Induced EMF at full-load is %.3f V'%(E_bph)

# note : rounding off error.

Minimum current is 13.122 A
Induced EMF at full-load is 241.534 V

Example 7.24 Page no : 73

In [33]:
import math 

# Variables
R_a = 0.8
X_s = 5.
Z_s = complex(R_a,X_s) 			#armature resistance and syncronous reactance
theta = math.atan(Z_s.imag/Z_s.real)  			# returns angle in radians 
alpha = (math.pi/2) - theta
V_t = 3300/math.sqrt(3)
P_e_in = 800./(3) 			#per phase
phi = math.acos(0.8)  			#leading
Q_e_in = -P_e_in*math.tan(phi)

# Calculations
# using the following equation
# P_e_in =  V_t**2*R_a/(abs(Z_s))**2   +   V_t*E_b*math.sin(delta-alpha)/abs(Z_S)
# Q_e_in =  V_t**2*X_s/(abs(Z_s))**2   -   V_t*E_b*math.cos(delta-alpha)/abs(Z_S)
E_b_sin_delta_minus_9  =  407.2
E_b_cos_delta_minus_9  = 2413.6
#solving further
delta  =  math.atan(E_b_sin_delta_minus_9/E_b_cos_delta_minus_9 ) + 9
E_b = E_b_sin_delta_minus_9/math.sin(math.radians(delta-9))

P_e_in_new  =  1200*10**3/3
# using the following equation again
# P_e_in =  V_t**2*R_a/(abs(Z_s))**2   +   V_t*E_b*math.sin(delta-alpha)/abs(Z_S)
# Q_e_in =  V_t**2*X_s/(abs(Z_s))**2   -   V_t*E_b*math.cos(delta-alpha)/abs(Z_S)

alpha  = delta - math.asin(math.radians(P_e_in_new - V_t**2*R_a/(abs(Z_s))**2 ) / (V_t*E_b/abs(Z_s)))
Q_e_in_new =  V_t**2*X_s/(abs(Z_s))**2   -   V_t*E_b*math.cos(math.radians(delta - alpha))/abs(Z_s)

pf = math.cos ( math.atan(abs(Q_e_in_new/P_e_in_new)))

# Results
print 'New power factor is %.2f leading '%(pf)

New power factor is 0.01 leading

Example 7.25 Page no : 74

In [34]:
import math 

# Variables
V_L = 6.6*10**3
V_ph = V_L/math.sqrt(3)
P_in = 900.*10**3
R_a = 0.
X_s = 20. 			#armature resistance and synchronous reactance 
Z_s = complex(R_a,X_s) 			#synchronous impedance
theta = math.atan(Z_s.imag)  			# returns angle in radians 
E_b_L = 8.6*10**3
E_bph = E_b_L/math.sqrt(3) 

# Calculations
#refer to phasor diagram
OA = V_ph
AB = E_bph 			#OB =  E_Rph

I_a_cosphi = P_in/(math.sqrt(3)*V_L)  			#I_a*math.cos(phi)
BC = I_a_cosphi*abs(Z_s) 			#BC is a vector in phasor diagram

OC = math.sqrt(AB**2 -BC**2  )- OA  			#from phasor diagram
I_a_sinphi = OC/abs(Z_s)  			#product of I_a and math.sin(phi)
phi =  math.atan (I_a_sinphi/I_a_cosphi)
I_a = I_a_cosphi/math.cos(phi) 			#product of I_a and math.cos(phi)

# Results
print 'Motor current is %.3f A'%(I_a)
print 'Power factor of motor is %f leading'%(cos(phi))
print 'Note:There is slight mismatch in answer due to the approximation made during I_a* sinphi calculation'

Motor current is 90.643 A
Power factor of motor is 0.868564 leading
Note:There is slight mismatch in answer due to the approximation made during I_a* sinphi calculation

Example 7.26 Page no : 75

In [35]:
import math 


# Variables
#subscipt 1 refers to factory load
P_1 = 1800.
phi_1 = math.acos(0.6) 			#lagging
Q_1 = P_1*math.tan(phi_1)

#Subscript 2 refers to synchronous condenser
P_2 = 0.

# Calculations
#Subscript T refers to combination of condenser and factory load
P_T = P_1+P_2
phi_T = math.acos(0.95) 			#lagging
Q_T = P_T*math.tan(phi_T)

kva_rating = math.sqrt(P_T**2+ Q_T**2)

Q_2 = Q_T - Q_1

# Results
print 'i)kVA rating of synchronous condender is %.3f kVA    Minus sign indicates leading nature'%(Q_2)
print 'ii)kVA rating of total factory is %.4f kVA'%(kva_rating)

i)kVA rating of synchronous condender is -1808.369 kVA    Minus sign indicates leading nature
ii)kVA rating of total factory is 1894.7368 kVA

Example 7.27 Page no : 77

In [36]:
import math 

# Variables
I_1 = 35.
phi_1 = math.acos(0.8)
V_L = 440.
S_1 = math.sqrt(3)*I_1*V_L /1000 			#in kVA


# Calculations
P_1 = S_1*math.cos(phi_1)
Q_1 = S_1*math.sin(phi_1)

P_out = 12. 			#motor load
eta_motor = 85./100
P_2 = P_out/eta_motor

P_T = P_1 + P_2
phi_T = math.acos(1)
Q_T = P_T * math.tan(phi_T)


Q_2 = Q_T - Q_1  			#kVA supplied by motor
#negative sign of Q_2 indicates its leading nature
phi_2 =  math.atan(abs(Q_2)/P_2)
S_2 = P_2/math.cos(phi_2)

# Results
print 'Power factor when motor supplies 12kW load is %.4f leading'%(math.cos(phi_2))
print 'kVA input to the motor is %.3f kVA'%(S_2)

Power factor when motor supplies 12kW load is 0.6615 leading
kVA input to the motor is 21.341 kVA

Example 7.28 Page no : 78

In [37]:
import math 

# Variables
V_L = 400
V_ph = V_L/math.sqrt(3)
Z_s = complex(0.5,4)  			#synchronous impedance
theta = math.atan(Z_s.imag/Z_s.real)  			# returns angle in radians 

I_aph = 60.
phi = math.acos(0.866) 			#leading
power_losses = 2*10**3

# Calculations
E_bph  =    math.sqrt( (I_aph*abs(Z_s))**2 + (V_ph)**2 - 2*(I_aph*abs(Z_s))*(V_ph)*math.cos(phi+theta)  ) 
delta = theta 			#for P_m_max
P_m_max  =  (E_bph*V_ph/abs(Z_s))- (E_bph**2/abs(Z_s))*math.cos(delta)
P_m_max_total =  3 * P_m_max
P_out_max =   P_m_max_total- power_losses

# Results
print 'Maximum power output is %.4f kW'%(P_out_max*10**-3)

Maximum power output is 51.3899 kW

Example 7.29 Page no : 79

In [38]:
import math 

# Variables
V_L = 6.6*10**3
V_ph = V_L/math.sqrt(3)
I_L = 50.
I_aph = I_L
Z_s = complex(1.5,8) 			#synchronous impedance
theta = math.atan(Z_s.imag/Z_s.real)  			# returns angle in radians 
E_Rph = I_aph*abs(Z_s)

# Calculations and Results
#part(i)
phi = math.acos(0.8)
P_in =  math.sqrt(3)*V_L*I_L*math.cos(phi)   			#for both lag and lead supplied power will be the same
print 'i)Power supplied to the motor is %.3f kW'%(P_in*10**-3)
#part(ii)
E_bph_lag  =    math.sqrt( E_Rph**2 + V_ph**2 - 2*E_Rph*V_ph*math.cos(theta-phi)  )  			#for lagging power factor
#Note that E_bph_lag > V_ph
print 'ii)Induced EMF for 0.8 power factor lag is %.3f V'%(E_bph_lag)
E_bph_lead  =    math.sqrt( E_Rph**2 + V_ph**2 - 2*E_Rph*V_ph*math.cos(theta+phi)  )  			#for leading power factor
#Note that E_bph_lead < V_ph
print '    Induced EMF for 0.8 power factor lead is %.3f V'%(E_bph_lead)

i)Power supplied to the motor is 457.261 kW
ii)Induced EMF for 0.8 power factor lag is 3521.267 V
    Induced EMF for 0.8 power factor lead is 4007.170 V

Example 7.30 Page no : 80

In [40]:
import math 

# Variables
V_L = 400.
V_ph = V_L/math.sqrt(3)
P_out = 7.5*735.5
eta = 85./100 			#efficiency
R_a = 0.
X_s = 10. 			#armature resistance and synchronous reactance
Z_s = complex(R_a,X_s)			#synchronous impedance
theta = math.atan(Z_s.imag)  			# returns angle in radians 

# Calculations and Results
P_in = P_out/eta
phi = math.acos(1) 			#for mimimum current power factor is maximum
I_L = P_in/(math.sqrt(3)*V_L*math.cos(phi))  
I_aph = I_L
print 'Minimum current is %.3f A at full load condition '%(I_L)

E_Rph =  I_aph*abs(Z_s)
E_bph  =    math.sqrt( E_Rph**2 + V_ph**2 - 2*E_Rph*V_ph*math.cos(phi+theta)  ) 
print 'and corresponding EMF is %.4f V'%(E_bph)

# note : rounding off error.

Minimum current is 9.367 A at full load condition 
and corresponding EMF is 240.4216 V

Example 7.31 Page no : 80

In [42]:
import math 

# Variables
V_L = 3.3*10**3
V_ph = V_L/math.sqrt(3)
V_t = V_ph
Pole = 24.
f = 50.  			#Pole and frequency
P = 1000.*10**3
R_a = 0
X_s = 3.24 			#armature resistance and synchronous reactance
Z_s = complex(R_a,X_s) 			#synchronous impedance
theta = math.atan(Z_s.imag)  			# returns angle in radians 
phi = math.acos(1)
I_aph = P/(math.sqrt(3)*V_L*math.cos(phi))

# Calculations
E_Rph = I_aph*abs(Z_s)
E_bph  =   math.sqrt( E_Rph**2 + V_ph**2 - 2*E_Rph*V_ph*math.cos(phi+theta)  ) 

P_m_max = 3*(E_bph*V_ph/abs(Z_s)) 			#maximum power that can be delivered
N_s = 120*f/Pole 			#synchronous speed
T_max = P_m_max /(2*math.pi*N_s/60) 			#maximum torque that can be developed

# Results
print 'Maximum power and torque the motor can deliver is %.3f kW and %.2f *10**3 Nm respectively'%(P_m_max*10**-3,T_max/1000)

# rounding off error.

Maximum power and torque the motor can deliver is 3211.633 kW and 122.68 *10**3 Nm respectively