# CHAPTER 9: SYNCHRONOUS GENERATOR¶

## Example 9.1, Page number 295¶

In [1]:
#Variable declaration
N = 300.0       #Speed of water turbine(rpm)
f = 50.0        #Frequency of induced voltage(Hz)

#Calculation
P = 120*f/N     #Number of poles

#Result
print('Number of poles of the generator , P = %.f poles' %P)
print('Alternatively , %.f pairs of north(N) and south(S) poles' %(P/2))

Number of poles of the generator , P = 20 poles
Alternatively , 10 pairs of north(N) and south(S) poles


## Example 9.2, Page number 299-300¶

In [1]:
import math

#Variable declaration
P = 8.0        #Number of poles
m = 3.0        #Number of phase
S = 144.0      #Number of slots

#Calculation
T_p = S/P               #Pole pitch(slots)
slots_1 = 180/T_p       #Pole pitch per slot(degree)
gamma = 2*slots_1       #Short pitch angle(degree)
y = gamma*math.pi/180   #Short pitch angle(radian)
k_p = math.cos(y/2)     #Pitch factor

#Result
print('Pitch factor , k_p = %.2f ' %k_p)

Pitch factor , k_p = 0.98


## Example 9.3, Page number 300-301¶

In [1]:
import math

#Variable declaration
P = 4.0     #Number of poles
m = 3.0     #Number of phase
S = 40.0    #Number of slots
s_1 = 1.0   #Coil span
s_2 = 9.0   #Coil span

#Calculation
T_p = S/P               #Pole pitch(slots)
T_c = s_2-s_1           #Coil pitch for coil spans 1 to 9(slots)
slots = 180/T_p         #Pole pitch per slot(degree)
y = T_p-T_c             #Short pitch angle(slots)
gamma = y*slots         #Short pitch angle(degree)
y = gamma*math.pi/180   #Short pitch angle(radian)
k_p = math.cos(y/2)     #Pitch factor

#Result
print('Pitch factor , k_p = %.2f ' %k_p)

Pitch factor , k_p = 0.95


## Example 9.4, Page number 302¶

In [1]:
import math

#Variable declaration
P = 4.0     #Number of poles
S = 48.0    #Number of slots
m = 3.0     #Number of phase

#Calculation
T_p = S/P                                        #Pole pitch(slots)
slot = 180/T_p                                   #Pole pitch per slot(degree)
a = slot*math.pi/180                             #Pole pitch per slot(radian)
n = S/(P*m)                                      #Number of slots or coils per pole per phase
k_d = math.sin(n*a/2)/(n*math.sin(a/2))          #Distribution factor

#Result
print('Distribution factor , k_d = %.2f ' %k_d)

Distribution factor , k_d = 0.96


## Example 9.5, Page number 304-305¶

In [1]:
import math

#Variable declaration
P = 12.0         #Number of poles
S = 180.0        #Number of slots
phi_m = 0.05     #Flux per pole(Wb)
N = 600.0        #Speed of machine(rpm)
m = 3.0          #Number of phase

#Calculation
T_p = S/P                                   #Pole pitch(slots)
slot = 180/T_p                              #Slots per pole(degree)
n = S/(P*m)                                 #Number of slots or coils per pole per phase
a = slot*math.pi/180                        #Pole pitch per slot(radian)
k_d = math.sin(n*a/2)/(n*math.sin(a/2))     #Distribution factor
k_p = 1.0                                   #Pitch factor
Z = (180/m)*slot                            #Number of conductors per phase
T = Z/2                                     #Number of turns per phase
f = P*N/120                                 #Frequency(Hz)
E = 4.44*k_p*k_d*f*phi_m*T                  #Induced voltage(V)
E_L = 3**0.5*E                              #Line voltage(V)

#Result
print('Line voltage , E_L = %.1f V' %E_L)
print('\nNOTE : Changes in obtained answer from that of textbook answer is due to precision i.e more number of decimal places')

Line voltage , E_L = 7945.7 V

NOTE : Changes in obtained answer from that of textbook answer is due to precision i.e more number of decimal places


## Example 9.6, Page number 305-307¶

In [1]:
import math

#Variable declaration
P = 4.0         #Number of poles
m = 3.0         #Number of phase
f = 50.0        #Frequency(Hz)
phi_m = 0.05    #Flux per pole(Wb)
n = 6.0         #Number of slots per pole per phase
cond = 5.0      #Conductors per layer
no_layer = 2.0  #Number of layer winding

#Calculation
T_p = n*m                                     #Slots per pole
slot = 180/T_p                                #Slots per pole(degree)
a = slot*math.pi/180                          #Pole pitch per slot(radian)
T_c = (5.0/6)*T_p                             #Coil pitch is 5/6 of full pitch
gamma = T_p-T_c                               #Short pitch angle(slots)
y_angle = gamma*slot                          #Short pitch(angle)
k_p = math.cos(y/2)                           #Pitch factor
k_d = math.sin(n*a/2)/(n*math.sin(a/2))       #Distribution factor
T = 1.0/2*n*P*cond*no_layer                   #Number of turns in any phase
E = 4.44*k_p*k_d*f*phi_m*T                    #Voltage per phase(V)

#Result
print('Voltage per phase , E = %.2f V' %E)
print('\nNOTE : Changes in obtained answer from that of textbook answer is due to precision i.e more number of decimal places')

Voltage per phase , E = 1230.19 V

NOTE : Changes in obtained answer from that of textbook answer is due to precision i.e more number of decimal places


## Example 9.7, Page number 307-308¶

In [1]:
import math

#Variable declaration
P = 10.0           #Number of poles
m = 3.0            #Number of phase
f = 50.0           #Frequency(Hz)
n = 3.0            #Number of slots per pole per phase
phi_m1 = 0.05      #Fundamental component of flux(Wb)
phi_m3 = 0.006     #Third harmonic component of flux(Wb)
T_c = 150.0        #Coil span(degree)
cond = 5.0         #Conductors per layer
no_layer = 2.0     #Number of layer winding

#Calculation
T_p = n*m                                     #Slots per pole
slot = 180/T_p                                #Slots per pole(degree)
a = slot*math.pi/180                          #Pole pitch per slot(radian)
gamma = 180-T_c                               #Short pitch angle(degree)
y = gamma*math.pi/180                         #Short pitch angle(radian)
T = 1.0/2*P*n*cond*no_layer                   #Number of turns
k_p1 = math.cos(y/2)                          #Fundamental pitch factor
k_d1 = math.sin(n*a/2)/(n*math.sin(a/2))      #Fundamental distribution factor
E_1 = 4.44*k_p1*k_d1*f*phi_m1*T               #Fundamental emf per phase(V)
k_p3 = math.cos(3*y/2)                        #Third harmonic pitch factor
k_d3 = math.sin(3*n*a/2)/(n*math.sin(3*a/2))  #Third harmonic distribution factor
E_3 = 4.44*k_p3*k_d3*f*phi_m3*T               #Voltage(V)
E = (E_1**2+E_3**2)**0.5                      #Induced voltage per phase(V)

#Result
print('rms value of induced voltage per phase , E = %.1f V' %E)
print('\nNOTE : Changes in obtained answer from that of textbook answer is due to precision i.e more number of decimal places')

rms value of induced voltage per phase , E = 1546.5 V

NOTE : Changes in obtained answer from that of textbook answer is due to precision i.e more number of decimal places


## Example 9.8, Page number 312¶

In [1]:
import math

#Variable declaration
kVA = 50.0        #Ratings(kVA)
V_t = 220.0       #Voltage(V)
R_a = 0.011       #Effective resistance(ohm)
X_s = 0.09        #Synchronous reactance(ohm)
pf = 0.85         #Lagging power factor

#Calculation
phi = math.acos(pf)                                                #Power factor angle(radians)
I_a = kVA*10**3/V_t                                                #Armature current(A)
E_f = ((V_t*pf+I_a*R_a)**2+(V_t*math.sin(phi)+I_a*X_s)**2)**0.5    #Induced voltage per phase(V)
VR = ((E_f-V_t)/V_t)*100                                           #Voltage regulation(percent)

#Result
print('No-load induced voltage per phase , E_f = %.1f V' %E_f)
print('Voltage regulation , VR = %.1f percent' %VR)
print('\nNOTE : Changes in obtained answer from that of textbook answer is due to precision i.e more number of decimal places')

No-load induced voltage per phase , E_f = 233.5 V
Voltage regulation , VR = 6.1 percent

NOTE : Changes in obtained answer from that of textbook answer is due to precision i.e more number of decimal places


## Example 9.9, Page number 314¶

In [1]:
import math

#Variable declaration
kVA = 200.0          #Rating(kVA)
V_t = 33.0*10**3     #Voltage(V)
R_a = 0.54           #Armature resistance(ohm)
V_L = 415.0          #Voltage between lines for SC test(V)
I_sh = 25.0          #Short circuit current(A)
pf = 0.9             #Lagging power factor

#Calculation
#For case(i)
V_P = V_L/3**0.5                                                   #Phase voltage during short circuit test(V)
Z_s = V_P/I_sh                                                     #Synchronous impedance(ohm)
#For case(ii)
X_s = (Z_s**2-R_a**2)**0.5                                         #Synchronous reactance(ohm)
#For case(iii)
I_a = kVA*1000/(3**0.5*V_t)                                        #Full load current(A)
V_ta = V_t/3**0.5                                                  #Voltage per phase of alternator(V)
phi = math.acos(pf)                                                #Power factor angle(radians)
E_f = ((V_ta*pf+I_a*R_a)**2+(V_ta*math.sin(phi)+I_a*X_s)**2)**0.5  #No-load voltage per phase(V)
VR = ((E_f-V_ta)/V_ta)*100                                         #Voltage regulation

#Result
print('(i)   Synchronous impedance , Z_s = %.1f ohm' %Z_s)
print('(ii)  Synchronous reactance , X_s = %.2f ohm' %X_s)
print('(iii) Voltage regulation , VR = %.2f percent' %VR)
print('\nNOTE : ERROR : In textbook calculation , R_a is taken instead of X_s in calculation of E_f')

(i)   Synchronous impedance , Z_s = 9.6 ohm
(ii)  Synchronous reactance , X_s = 9.57 ohm
(iii) Voltage regulation , VR = 0.09 percent

NOTE : ERROR : In textbook calculation , R_a is taken instead of X_s in calculation of E_f


## Example 9.10, Page number 317¶

In [1]:
import math

#Variable declaration
MVA = 30.0      #Rating(MVA)
V = 20.0        #Supply voltage(kV)
N = 1800.0      #Speed(rpm)
V_t = 15.0      #Voltage per phase(kV)
E_f = 10.0      #Per phase terminal voltage(kV)
delta = 40.0    #Power angle(degree)
X_s = 6.0       #Per phase synchronous reactance(ohm)

#Calculation
#For case(i)
P = 3*V_t*E_f*math.sin(d)/X_s   #3-phase power delivered to the load(MW)
#For case(ii)
P_max = 3*V_t*E_f/X_s           #Three phase maximum power(MW)

#Result
print('(i)  Three-phase real power delivered to the load , P = %.2f MW' %P)
print('(ii) Three-phase maximum power , P_max = %.f MW' %P_max)

(i)  Three-phase real power delivered to the load , P = 48.21 MW
(ii) Three-phase maximum power , P_max = 75 MW


## Example 9.11, Page number 321-322¶

In [1]:
import math

#Variable declaration
kVA = 25.0        #Rating(kVA)
V = 440.0         #Suppy voltage(V)
f = 50.0          #Supply frequency(Hz)
pf = 0.8          #Lagging power factor
R_a = 0.3         #Resistance of machine per phase(ohm)
X_d = 5.0         #Reactance of machine per phase(ohm)
X_q = 3.0         #Reactance of machine per phase(ohm)

#Calculation
#For case(i)
phi = math.acos(pf)                                #Power factor angle(radians)
phi_deg = phi*180/math.pi                          #Power factor angle(degree)
V_t = V/3**0.5                                     #Terminal voltage per phase(V)
I_a = kVA*10**3/(3**0.5*V)                         #Armature current(A)
tan_d = (I_a*X_q*pf/(V_t+I_a*X_q*math.sin(phi)))
d_angle = d*180/math.pi                            #Torque angle(degree)
#For case(ii)
I_d = I_a*math.sin(d+phi)                          #Direct axis component of the current(A)
E_f = V_t*math.cos(d)+I_d*X_d                      #Induced voltage per phase(V)
#For case(iii)
VR = ((E_f-V_t)/V_t)*100                           #Voltage regulation(percent)

#Result
print('(i)   Torque angle , δ = %.2f° ' %d_angle)
print('(ii)  Induced voltage per phase , E_f = %.2f V' %E_f)
print('(iii) Voltage regulation , VR = %.2f percent' %VR)
print('\nNOTE : Changes in obtained answer from that of textbook answer is due to precision i.e more number of decimal places')

(i)   Torque angle , δ = 14.12°
(ii)  Induced voltage per phase , E_f = 373.80 V
(iii) Voltage regulation , VR = 47.15 percent

NOTE : Changes in obtained answer from that of textbook answer is due to precision i.e more number of decimal places


## Example 9.12, Page number 324-325¶

In [1]:
import math
import cmath

#Variable declaration
E_1 = 220.0                             #Induced voltage per phase by alternator 1(V)
E_2 = 220*cmath.exp(1j*5*math.pi/180)   #Induced voltage per phase by alternator 2(V)
Z_1 = complex(0,3)                      #Impedance of alternator 1(ohm)
Z_2 = complex(0,4)                      #Impedance of alternator 2(ohm)

#Calculation
#For case(i)
#For case(ii)
V_t = I*Z                                            #Terminal voltage(V)
#For case(iii)
I_a1 = ((E_1-E_2)*Z+E_1*Z_2)/(Z_1*Z_2+Z*(Z_1+Z_2))   #Armature current(A)
P_1 = abs(V_t*I_a1)*math.cos(cmath.phase(V_t)-cmath.phase(I_a1))*10**-3  #Power per phase delivered by the first alternator(W)

#Result
print('(i)   Load current , I = %.1f∠%.f° A' %(abs(I),cmath.phase(I)*180/math.pi))
print('(ii)  Terminal voltage , V_t = %.f∠%.f° V' %(abs(V_t),cmath.phase(V_t)*180/math.pi))
print('(iii) Power per phase delivered by the first alternator , P_1 = %.1f kW' %P_1)
print('\nNOTE : ERROR : In textbook case(iii) power calculation current I is taken instead of I_a1')

(i)   Load current , I = 41.6∠-17° A
(ii)  Terminal voltage , V_t = 208∠-17° V
(iii) Power per phase delivered by the first alternator , P_1 = 4.4 kW

NOTE : ERROR : In textbook case(iii) power calculation current I is taken instead of I_a1