# Chapter 4 : Synchronous Machines : Alternators¶

## Example 4.1 page no : 16¶

In :
# Variables
Pole = 4.
Slots = 24.
Phase = 3. 			#number of phases

# Calculations
n = Slots/Pole 			#slots per pole
m = Slots/Pole/Phase 			#slots per pole per phase
beeta = 180/n  			#Slot angle

# results
print "Slot angle : %.f degrees"%beeta

Slot angle : 30 degrees


## Example 4.2 page no : 26¶

In :
import math

# Variables
Slots = 120.
Pole = 8.
Phase = 3.   			#number of phases

# Calculations
n = Slots/Pole         			#Slots per Pole
m = Slots/Pole/Phase   			#Slots per Pole per Phase
beeta = 180/n           			#Slot angle in degree

# Results
print 'Distribution Factor:K_d = %.3f'%(K_d)

Distribution Factor:K_d = 0.957


## Example 4.3 page no : 26¶

In :
import math

# Variables
Slots = 36.
Pole = 4.
Phase = 3. 			#number of phases
n = Slots/Pole    			#Slots per pole
beeta = 180/n     			#Slot angle in degrees

# Calculations
#coil is shorted by 1 slot i.e. by beeta degrees to full pitch dismath.tance
alpha = beeta     			#angle of short pitch
K_c = math.cos(math.radians(alpha/2))  			#Coil span Factor

# Results
print 'Coil Span Factor:K_c = %.4f'%(K_c)

Coil Span Factor:K_c = 0.9848


## Example 4.4 page no : 26¶

In :
import math

# Variables
N_s = 250.  			#Synchronous speed in r.p.m
f = 50.     			#Frequency of generated e.m.f in hertz
Slots = 216.
phi = 30.*10**-3 			#flux per pole in weber

Pole = 120*f/N_s
n = Slots/Pole   			#Slots per Pole
m = n/3          			#Slots per Pole per Phase
beeta = 180/n     			#Slot angle in degree

# Calculations
K_c = 1 			#Coil Span Factor for full pitch coils = 1

Z = Slots*5   			#Z is total no of conductors
Z_ph = Z/3    			#Conductors Per Phase
T_ph = Z_ph/2 			#Turns per phase
E_ph = 4.44*K_c*K_d*f*phi*T_ph 			#induced emf
E_line = E_ph*math.sqrt(3)

# Results
print 'Induced e.m.f across the Terminals is %.2f V'%(E_line)

Induced e.m.f across the Terminals is 1992.90 V


## Example 4.5 page no : 27¶

In :
import math

# Variables
Pole = 16.
N_s = 375. 			#synchronous speed in rpm
Slots = 144.
E_line = 2.657*10**3 			#line value of emf across terminals
f = Pole*N_s/120 			#frequency

# Calculations
K_c = 1  			#assuming full pitch winding Coil span Factor = 1
n = Slots/Pole 			#slots per pole
m = n/3 			#slots per pole per phase

beeta = 180/n
conductors_per_slot = 10
Z = Slots*conductors_per_slot  			#total conductors

Z_ph = Z/3  			#number of conductors per phase
T_ph = Z_ph/2 			#no of turns per phase
E_ph = E_line/math.sqrt(3) 			#phase value of emf across terminals

phi = E_ph/(4.44*K_c*K_d*f*T_ph)     			#E_ph = 4.44*K_c*K_d*f*phi*T_ph

# Results
print 'Frequency of Induced e.m.f is %.0fHz \nFlux per Pole is %.0f mWb'%(f,phi*1000)

Frequency of Induced e.m.f is 50Hz
Flux per Pole is 30 mWb


## Example 4.6 page no : 34¶

In :
import math

# Variables
d = 0.25            			#Diameter in metre
l = 0.3             			#Length in metre
Pole = 4.
A1 = math.pi*d*l/Pole    			#Area of each fundamental pole
f = 50. 			#frequency in hertz
B_m1 = 0.15
B_m3 = 0.03
B_m5 = 0.02 			#Amplitude of 1st 3rd and 5th harmonics
phi_1 = (2/math.pi)*B_m1*A1  			#average value of fundamental flux per pole in weber

# Calculations and Results
#PART A
E_c1 = 1.11*2*f*phi_1  			#R.M.S value of fundamental frequency e.m.f generated in math.single conductor
Coil_span = (13./15)*180  			#math.since winding coil span is 13/15 of pole pitch
alpha = 180-Coil_span

#Pitch factor for 1st 3rd and 5th harmonic

#using E_cx = E_c1 * (B_mx/B_m1)
E_c3 = E_c1 * (B_m3/B_m1)
E_c5 = E_c1 * (B_m5/B_m1)

E_t1 = K_c1 * (2*E_c1)   			#R.M.S Vaue of fundamental frequency EMF generated in 1 turn (in volts)
E_t3 = K_c3 * 2*E_c3
E_t5 = K_c5 * 2*E_c5
E_t = math.sqrt(E_t1**2 +E_t3**2 +E_t5**2)
V = 10*E_t  			#(number of turns per coil )* (Total e.m.f per turn)
print 'Voltage generated per coil is %.1f V'%(V)

# PART B
#E_1ph = 4.44*K_c1*K_d1*phi_1*f*T_ph
T_ph = 200.   			#T_ph = (60 coils  * 10 turns per coil)/3

Total_Conductors = 1200. 			# 60 coils * 10 turns per coil * 2
Conductors_per_Slot = 20. 			#2 conductors per turn * 10  turns per slot
Slots = Total_Conductors/Conductors_per_Slot

n = Slots/Pole
m = n/3
beeta = 180/n   			#Slot angle in degree

E_1ph = 4.44 * K_c1 * K_d1*phi_1 * f * T_ph
# using E_xph =  E_1ph* (B_mx*K_cx*K_dx)/(B_m1*K_c1*K_d1)
E_3ph =  E_1ph* (B_m3*K_c3*K_d3)/(B_m1*K_c1*K_d1)
E_5ph =  E_1ph* (B_m5*K_c5*K_d5)/(B_m1*K_c1*K_d1)
E_ph = math.sqrt( E_1ph**2 + E_3ph**2 + E_5ph**2 )  			#voltage generated per phase
print 'Voltage generated per phase is %.f V'%(E_ph)

#PART c
E_line = math.sqrt(3) * math.sqrt( E_1ph**2 + E_5ph**2  ) 			#terminal voltage
print 'Terminal Voltage is %.1f V '%(E_line)

Voltage generated per coil is 12.4 V
Voltage generated per phase is 235 V
Terminal Voltage is 404.8 V


## Example 4.7 page no : 38¶

In :
import math

# Variables
Ns = 250. 			#Synchronous speed in rpm
f = 50.
Slots = 288.
E_line = 6600.
Pole = 120*f/Ns
n = Slots/Pole  			#slots per pole
m = n/3 			#slots per pole per phase
beeta = 180/n 			#slot angle
conductors_per_slot = 32   			#16 conductors per coil-side  *2 coil-sides per slot

# Calculations
alpha = 2*beeta			# angle of short pitch
K_c = math.cos(math.radians(alpha/2))  			#coil span factor
Z  =  Slots*conductors_per_slot  			#total conductors
Z_ph = Z/3 			#Conductors per phase
T_ph = Z_ph/2 			#turns per phase

E_ph = E_line/math.sqrt(3)
phi = E_ph/(4.44*K_c*K_d*f*T_ph)           			#Because E_ph = 4.44 *K_c *K_d *phi *f *T_ph

# Results
print 'Flux per pole is %.0f mWb '%(phi*1000)

Flux per pole is 12 mWb


## Example 4.8 page no : 40¶

In :
import math

# Variables
Ns = 1500.  			#synchronous speed in rpm
Pole = 4.
Slots = 24.
conductor_per_slot = 8.
phi = 0.05 			#flux per pole in weber
f = Pole*Ns/120 			#frequenccy
n = Slots/Pole 			#slots per pole
m = n  			# as number of phases is 1
beeta = 180/n  			#slot angle

# Calculations

#Full pitch =  n  = 6 slots
#(1/6)th of full pitch  = 1slot
#angle of short pitch  =  1 slot angle
alpha = beeta
K_c = math.cos(math.radians(alpha/2))  			#coil span factor

Z = conductor_per_slot*Slots 			#total conductors
Z_ph = Z 			# as number of phases is 1
T_ph = Z_ph/2 			#turns per phase
E_ph = 4.44*K_c*K_d* phi *f *T_ph  			#induced emf

# Results
print 'Induced e.m.f is %.1f V '%(E_ph)

Induced e.m.f is 662.8 V


## Example 4.9 page no : 41¶

In :
import math

# Variables
Pole = 48.
n = 9. 			#slots per pole
phi = 51.75*10**-3 			#flux per pole in weber
Ns = 125.
f = Ns*Pole/120 			#frequency
K_c = 1. 			#due to full pitch winding
m = n/3 			#slots per pole per phase
beeta = 180/n  			#slot angle

# Calculations
conductor_per_slot = 4*2 			#Each slot has 2 coil sides and each coil side has 4 conductors
Slots = n*Pole
Z = conductor_per_slot*Slots   			#total number of conductors
Z_ph = Z/3 			#conductors per phase
T_ph = Z_ph/2 			#turns per phase
E_ph = 4.44 *K_c *K_d *phi *f *T_ph  			#induced emf

E_line = (math.sqrt(3))*E_ph  			#due to star connection

# Results
print 'Induced e.m.f is %.0f kV '%(E_line/1000)

Induced e.m.f is 11 kV


## Example 4.10 page no : 42¶

In :
import math

# Variables
Slots = 180.
Pole = 12.
Ns = 600. 			#Synchronous speen in rpm
f = Pole*Ns/120 			#frequency
phi = 0.05 			#flux per pole in weber

# Calculations and Results
#Part(i)
#Average EMF in a conductor = 2*f*phi
rms_value_1 = 1.11*2*f*phi 			#rms value of emf in a conductor
print 'i)r.m.s value of e.m.f in a conductor is %.2f V '%(rms_value_1)

#part(ii)
#Average EMF in a turn = 4*f*phi
rms_value_2 = 1.11*4*f*phi			#r.m.s value of e.m.f in a turn
print 'ii)r.m.s value of e.m.f in a turn is %.2f V '%(rms_value_2)

#part(iii)
conductors_per_coilside = 10/2
rms_value_3 = rms_value_2*conductors_per_coilside  			#r.m.s value of e.m.f in a coil
print 'iii)r.m.s value of e.m.f in a coil is %.1f V '%(rms_value_3)

#part(iv)
conductors_per_slot = 10
Z = conductors_per_slot * Slots  			#total number of conductors
Z_ph = Z/3    			#conductors per phase
T_ph = Z_ph/2 			#turns per phase
n = Slots/Pole  			#slots per pole
m = n/3        			#slots per pole per phase
beeta = 180/n   			#slot angle

K_c = 1  			#distribution & coil-span factor
E_ph = rms_value_2*T_ph*K_d*K_c   			#induced emf
print 'iv)per phase induced e.m.f is %.1f V '%(E_ph)

i)r.m.s value of e.m.f in a conductor is 6.66 V
ii)r.m.s value of e.m.f in a turn is 13.32 V
iii)r.m.s value of e.m.f in a coil is 66.6 V
iv)per phase induced e.m.f is 3822.9 V


## Example 4.11 page no : 44¶

In :
import math

# Variables
Pole = 8.
f = 50. 			#frequency
phi = 60.*10**-3 			#flux per pole in weber
Slots = 96.
n = Slots/Pole 			#slots per pole
beeta  =  180/n 			#slot angle
m = n/3 			#slots per pole per phase

# Calculations and Results
coil_pitch = 10*beeta 			#10 slots
alpha = 180-coil_pitch

conductors_per_slot = 4
Z = Slots*conductors_per_slot 			#total conductors
Total_turns = Z/2
T_ph = Total_turns/3  			#turns per phase

#part (i)
E_ph =  4.44 *K_c *K_d *phi *f *T_ph
print '\The phase voltage is %.2f V '%(E_ph)

#part(ii)
E_line = E_ph*math.sqrt(3)
print 'The Line Voltage is %.2f V '%(E_line)

#part(iii)
I_ph = 650
I_l = I_ph  			# Star Connection
kVA_rating = math.sqrt(3)*E_line*I_l
print 'kVA rating is %.1f kVA '%(kVA_rating/1000)

\The phase voltage is 788.57 V
The Line Voltage is 1365.84 V
kVA rating is 1537.7 kVA


## Example 4.12 page no : 45¶

In :
import math

# Variables
Ns = 600. 			#synchronous speed in rpm
Pole = 10.
l = 30./100     			#divided by 100 for centimetre-metre conversion
Pole_pitch = 35./100  			#numerically equal to pi*d/Pole
Phase = 3.
conductors_per_slot = 8.
A1 = Pole_pitch*l 			#Area of each fundamental pole
m = 3.   			#Slot per Pole per Phase
n = Phase*m 			#slots per pole
beeta = 180/n  			#slot angle

B_m1 = 1.
B_m3 = 0.3
B_m5 = 0.2  			#amplitude of 1st 3rd and 5th harmonic
phi_1 = (2/math.pi)*A1*B_m1  			#average value of fundamental flux per pole
f = Ns*Pole/120 			#frequency

# Calculations
Coil_span = (8./9)*180
alpha = 180-Coil_span
#pitch factor for 1st 3rd and 5th harmonic

# using K_dx = math.sin(m*x*beeta*(math.pi/180)/2) /(m*math.sin(x*beeta*(math.pi/180)/2))
#distribution factor for 1st 3rd and 5th harmonic

Slots = n*Pole
Total_conductors = conductors_per_slot * Slots
Total_turns = Total_conductors/2
T_ph = Total_turns/3  			#turns per phase

#EMF of 1st 3rd and 5th harmonic
E_1ph = 4.44 * K_c1 * K_d1*phi_1 * f * T_ph
E_3ph =  E_1ph* (B_m3*K_c3*K_d3)/(B_m1*K_c1*K_d1)
E_5ph =  E_1ph* (B_m5*K_c5*K_d5)/(B_m1*K_c1*K_d1)

# Results
# using E_xph =  E_1ph* (B_mx*K_cx*K_dx)/(B_m1*K_c1*K_d1)
E_ph = math.sqrt( E_1ph**2 + E_3ph**2 + E_5ph**2 )
print 'Phase value of induced e.m.f is %.2f V '%(E_ph)
E_line = math.sqrt(3) * math.sqrt( E_1ph**2 + E_5ph**2  )			#no 3rd harmonic appears in line value
print 'line value of induced e.m.f is %.2f V '%(E_line)

print 'Answer mismatches due to approximation'

Phase value of induced e.m.f is 1711.94 V
line value of induced e.m.f is 2916.65 V


## Example 4.13 page no : 47¶

In :
import math

# Variables
Pole = 16.
phi = 0.03 			#flux per pole
Ns = 375. 			#synchronous speed in rpm

# Calculations and Results
f = Ns*Pole/120 			#frequency
print 'frequency is %.0f Hz '%(f)
Slots = 144
n = Slots/Pole  			#slots per pole
m = n/3 			#slots per pole per phase
beeta = 180/n  			#slot angle
K_c = 1 			#assuming Full-Pitch coil
Conductors_per_slot = 10

Total_conductors = Slots*Conductors_per_slot
Total_turns = Total_conductors/2
T_ph = Total_turns/3  			#turns per phase
E_ph = 4.44* K_c* K_d*phi* f* T_ph
E_line = E_ph*math.sqrt(3)
print 'line voltage is %.2f V '%(E_line)

# note : rounding off error.

frequency is 50 Hz
line voltage is 2657.20 V


## Example 4.14 page no : 48¶

In :
import math

# Variables
Ns = 250. 			#Speed in rpm
f = 50. 			#frequency
I_l = 100.
Slots = 216.
Conductors_per_slot = 5
Pole = 120.*f/Ns
phi = 30.*10**-3			#flux per pole in weber
Z = Slots*Conductors_per_slot  			#Total Conductors
Z_ph = Z/3 			#conductors per phase
T_ph = Z_ph/2 			#turns per phase
n = Slots/Pole 			#slots per pole
m = n/3 			#slots per pole per phase
beeta = 180./n 			#Slot angle

# Calculations

e_av = 2*f*phi     			#Average Value of EMF in each conductor
E_c = 1.11*(2*f*phi)  			#RMS value of EMF in each conductor
E = 2*E_c*K_d 			#RMS value of EMF in each turn
E_ph = T_ph*E 			#RMS value of EMF in each phase
E_line =  E_ph*math.sqrt(3)  			#As Star Connected Alternator

# Results
print 'RMS value of EMF in each phase  =  %.3f V'%(E_ph)
print 'RMS value of EMF line value   =  %.3f V'%(E_line)
kVA_rating = math.sqrt(3)*E_line*I_l
print 'kVA rating is %.3f kVA '%(kVA_rating/1000)

RMS value of EMF in each phase  =  1150.602 V
RMS value of EMF line value   =  1992.902 V
kVA rating is 345.181 kVA


## Example 4.15 page no : 50¶

In :
import math

# Variables
Pole = 10.
Slots = 90.
E_l = 11000.
f = 50.
phi = 0.15 			#flux per pole in weber
n = Slots/Pole 			#slots per pole
m = n/3 			#slots per pole per phase
beeta = 180/n 			#slot angle

# Calculations
K_c = 1 			#coil span factor

E_ph = E_l/math.sqrt(3)
T_ph = ( E_ph/(4.44*K_c*K_d*phi*f)  )
#T_ph should necessarily be an integer
Z_ph = (T_ph)*2

# Results
print 'Required number of armature conductors is %d'%(Z_ph)

# note : rounding off error.

Required number of armature conductors is 397


## Example 4.16 page no : 50¶

In :
import math

# Variables
Pole = 10.
Ns = 600. 			#speen in rpm
conductor_per_slot = 8.
n = 12. 			#slots per pole
Slots = Pole*n
m = n/3 			#slots per pole per phase
beeta = 180/n 			#slot angle
alpha = 2*beeta  			#short by 2 slots

#flux per pole corresponding to 1st 3rd and 5th harmonic
phi_1 = 100*10**-3
phi_3 = (33./100)*phi_1
phi_5 = (20./100)*phi_1

#coil span factor corresponding to 1st 3rd and 5th harmonic

# using K_dx = math.sin(m*x*beeta /2) /(m*math.sin(x*beeta /2))
#distribution factor corresponding to 1st 3rd and 5th harmonic

Z = conductor_per_slot*n*Pole    			#Total Conductors
Zph = Z/3 			#conductors per phase
T_ph = Zph/2 			#turns per phase

f = Ns*Pole/120
E_1ph = 4.44*K_c1*K_d1*phi_1*f*T_ph
E_3ph = 4.44*K_c3*K_d3*phi_3*f*T_ph
E_5ph = 4.44*K_c5*K_d5*phi_5*f*T_ph

E_ph = math.sqrt( E_1ph**2 +  E_3ph**2 + E_5ph**2     )

# Results
print 'Phase value of induced e.m.f is %.0f V '%(E_ph)
E_line = math.sqrt(3)*math.sqrt( E_1ph**2 +  E_5ph**2     )  			#In a line value 3rd harmonic doesnt appear
print 'line value of induced e.m.f is %d V '%(E_line)

# note : rounding off error.

Phase value of induced e.m.f is 3330 V
line value of induced e.m.f is 5691 V


## Example 4.17 page no : 52¶

In :
import math

# Variables
Pole = 6.
Ns = 1000.   			#speed in rpm
d = 28./100  			#Divided by 100 to convert from centimeters to metres
l = 23./100  			#Divided by 100 to convert from centimeters to metres
m = 4.   	    		#slots per pole per phase
B_m1 = 0.87  			#amplitude of 1st harmonic component of flux density
B_m3 = 0.24  			#amplitude of 3rd harmonic component of flux density
Conductors_per_slot = 8
f = Ns*Pole/120 			#frequency
A1 = math.pi*d*l/Pole 			#area of each fundamental pole
phi_1 = (2/math.pi)*A1*B_m1 			#flux per pole in weber
n = m*3 		    	#slots per pole
beeta = 180/n 			#slot angle
alpha = beeta  			#because of 1 slot short

# Calculations
K_c1 = math.cos(math.radians(alpha/2)) 			#coil span factor corresponding to 1st harmonic
K_c3 = math.cos(math.radians(3*alpha/2))			#coil span factor corresponding to 3rd harmonic
# using K_dx = math.sin(m*x*beeta*(math.pi/180)/2) /(m*math.sin(x*beeta*(math.pi/180)/2))

Slots = n*Pole
Z = Slots*Conductors_per_slot 			#total number of conductors
Z_ph = Z/3 			#conductors per phase
T_ph = Z_ph/2 			#turns per phase

E_1ph = 4.44*K_c1*K_d1*phi_1*f*T_ph
E_3ph = E_1ph* (B_m3*K_c3*K_d3)/(B_m1*K_c1*K_d1)      			#using E_xph = E_1ph* (B_mx*K_cx*K_dx)/(B_m1*K_c1*K_d1)
E_ph = math.sqrt( E_1ph**2 +  E_3ph**2     )
print 'r.m.s value of resultant voltage is %.1f V'%(E_ph)
E_line = math.sqrt(3)*E_1ph   			#For line Value 3rd harmonic does not appear
print 'line voltage is %.3f V'%(E_line)

r.m.s value of resultant voltage is 383.7 V
line voltage is 654.560 V


## Example 4.18 page no : 53¶

In :
import math

# Variables
V_L = 125.
V_ph = V_L
VA = 600.*10**3
I_L = VA/(math.sqrt(3)*V_L)     			# Because VA = math.sqrt(3)* V_L * I_L
I_ph = I_L/(math.sqrt(3))

# Calculations and Results
#After Reconnection
V_ph = 125
V_L = V_ph*math.sqrt(3)
print 'New rating in volts is %.3f V'%(V_L)
#Winding Impedances remain the same
I_ph = 1600
I_L = I_ph

print 'New rating in amperes is %.0f A'%(I_L)
kVA = math.sqrt(3)*V_L*I_L*(10**-3)
print 'New rating in kVA is %.0f kVA'%(kVA)

New rating in volts is 216.506 V
New rating in amperes is 1600 A
New rating in kVA is 600 kVA


## Example 4.19 page no : 55¶

In :
import math

# Variables
Pole = 4.
f = 50. 			#frequency
phi = 0.12 			#flux per pole in weber
m = 4.  			# slot per pole per phase
conductor_per_slot = 4.
coilspan = 150.
Ns = 120*f/Pole 			#synchronous speed in rpm
n = m*3  	    		#Slots per pole
beeta = 180/n 			#slot angle

# Calculations

e.m.f generated is 788.57 Vphase, 1365.84 Vline)