Chapter 17 : AC Machines

Pg: 774 Ex: 17.1

In [1]:
from __future__ import division
from cmath import polar
from math import sin,cos,pi,sqrt
P_rot=900#      #rotational losses
V_L=440*complex(cos(0),sin(0))#
R_s=1.2#
X_s=1J*2#
X_m=1J*50#
R_r_1=0.6#
R_r_2=19.4#
X_r=1J*0.8#
n_m=1746#      #machine operating speed in rpm
W_m=n_m*2*pi/60#      #speed in radians per second
n_s=1800#      #synchronous speed for a four-pole monitor
s=(n_s-n_m)/n_s#      #slip
Z_s=R_s+X_s+(X_m*(R_r_1+R_r_2+X_r))/(X_m+R_r_1+R_r_2+X_r)#      #impedance seen by the source
Z_s_max=polar(Z_s)[0]
phi=polar(Z_s)[1]
Z_s_phi=(phi.real)#      #removing negligible imaginary part(if any is there)
PF=cos(Z_s_phi)#      #power factor
V_s=V_L#      #phase voltage
I_s=V_s/Z_s#      #phase current
I_s_max=polar(I_s)[0]
I_s_phi=polar(I_s)[1]
I_L=I_s_max*sqrt(3)#      #line current
P_in=3*I_s*V_s*PF#      #input power
V_x=I_s*(X_m*(R_r_1+R_r_2+X_r))/(X_m+R_r_1+R_r_2+X_r)#
I_r=V_x/(X_r+R_r_1+R_r_2)#
I_r_max=polar(I_s)[0]
I_r_phi=polar(I_r)[1]#
P_s=3*R_s*I_s_max**2#      #copper loss in stator
P_r=3*R_r_1*I_r_max**2#      #copper loss in rotor
P_dev=3*(1-s)*R_r_1*I_r_max**2/s#      #developed power
#we may verify that P_in=P_dev+P_s+P_r to within rounding error
P_in=P_dev+P_s+P_r#      #input power
P_o=P_dev-P_rot#      #output power
T_o=P_o/W_m#      #output torque
eff=P_o*100/P_in#      #efficiency
print " All the values in the textbook are approximated hence the values in this code differ from those of Textbook"
print 'Power factor : %0.2f'%PF
print 'line current = %0.2f amperes'%I_L
print 'output power = %0.2f watts'%P_o
print 'output torque = %0.2f Nm'%T_o
print 'efficiency percentage : %0.2f'%eff
 All the values in the textbook are approximated hence the values in this code differ from those of Textbook
Power factor : 0.89
line current = 37.58 amperes
output power = 26494.77 watts
output torque = 144.91 Nm
efficiency percentage : 88.50

Pg: 775 Ex: 17.2

In [2]:
from __future__ import division
from cmath import polar
from math import sin,cos,pi,sqrt
s=1#      #slip for starting
V_L=440*complex(cos(0),sin(0))#
f=60#
R_s=1.2#
X_s=1J*2#
X_m=1J*50#
R_r_1=0.6#
R_r_2=19.4#
X_r=1J*0.8#
Z_eq=X_m*(R_r_1+X_r)/(X_m+R_r_1+X_r)#      #equivalent impedance to the right in the figure in textbook
Z_s=R_s+X_s+Z_eq#
I_s=V_s/Z_s#      #starting phase current
I_s_max=polar(I_s)[0]
phi=polar(I_s)[1]
I_L=sqrt(3)*I_s_max#      #starting line current
#I_L here is almost six times larger than in previous example. It is a typical characteristic of induction motors.
P_ag=3*(Z_eq.real)*I_s_max**2#      #power crossing air gap
W_s=2*pi*(60)#
T_dev=P_ag/(W_s/2)#
print " All the values in the textbook are approximated hence the values in this code differ from those of Textbook"
print 'Starting line current = %0.2f A'%I_L
print 'Torque = %0.2f Nm'%T_dev
 All the values in the textbook are approximated hence the values in this code differ from those of Textbook
Starting line current = 229.99 A
Torque = 163.08 Nm

Pg: 776 Ex: 17.3

In [3]:
from __future__ import division
from math import pi,sqrt
V_L=220#
V_s=V_L/sqrt(3)#      #phase voltage
I_s=31.87#
P_s=400#      #total stator copper losses
P_r=150#      #total rotoe copper losses
P_rot=500#      #rotational losses
PF=0.75#      #power factor
P_in=3*V_s*I_s*PF#      #input power
P_ag=P_in-P_s#      #air-gap power
P_dev=P_in-P_s-P_r#      #developed power
P_o=P_dev-P_rot#      #output power
eff=P_o*100/P_in#      #efficiency
print " All the values in the textbook are approximated hence the values in this code differ from those of Textbook"
print 'Power crossing the air gap = %0.2f watts'%P_ag
print 'developed power = %0.2f watts'%P_dev
print 'output power = %0.2f watts'%P_o
print 'effciency percentage : %0.2f'%eff      #this value is given wrong in the textbook
 All the values in the textbook are approximated hence the values in this code differ from those of Textbook
Power crossing the air gap = 8708.08 watts
developed power = 8558.08 watts
output power = 8058.08 watts
effciency percentage : 88.47

Pg: 777 Ex: 17.4

In [4]:
from __future__ import division
from math import pi,sqrt,acos,sin,cos,atan,asin

P_dev_1=50*746#      #developed power
V_L=480#      #line voltage
PF=0.9#      #power factor
f=60#      #frequency
P=8#      #number of poles
X_s=1.4#      #synchronous reactance
#CASE a
n_s=120*f/P#      #speed of machine in rpm
W_s=n_s*2*pi/60#      #speed in radians per second
T_dev=P_dev_1/W_s#      #developed torque
print " All the values in the textbook are approximated hence the values in this code differ from those of Textbook"
print 'CASE a:'
print 'speed = %0.2f rpm'%n_s
print 'developed torque = %0.2f Nm'%T_dev
#CASE b
V_a=V_L#      #phase voltage
I_a_max=P_dev_1/(3*V_a*PF)#      #phase current
phi=acos(PF)#
I_a=I_a_max*complex(cos(phi),sin(phi))#
E_r=V_a-1J*X_s*I_a#      #voltage induced by rotor
E_r_max=sqrt(((E_r.real)**2)+((E_r.imag)**2))#
E_r_phi=atan((E_r.imag)/(E_r.real))#
TA=-E_r_phi#      #torque angle
print 'CASE b:'
print 'Phase current:'
print 'peak value = %0.2f amperes'%I_a_max
print 'phase angle = %0.2f degrees'%(phi*180/pi)
print 'Voltage induced by rotor:'
print 'peak value = %0.2f volts'%E_r_max
print 'phase angle = %0.2f degrees'%(E_r_phi*180/pi)
print 'torque angle = %0.2f degrees'%(TA*180/pi)
#CASE c
#excitation constant means the values of I_f, B_r and E_r are constant
P_dev_2=100*746#
sin_t=P_dev_2*sin(TA)/P_dev_1#      #developed power is proportional to sin_t
t=asin(sin_t)#
E_r=E_r_max*complex(cos(-t),sin(-t))#      #E_r is constant in magnitude
I_a=(V_a-E_r)/(1J*X_s)#      #new phase current
I_a_max=sqrt(((I_a.real)**2)+((I_a.imag)**2))#
I_a_phi=atan((I_a.imag)/(I_a.real))#
PF=cos(I_a_phi)#
print 'CASE c:'
print 'Phase current:'
print 'peak value = %0.2f amperes'%I_a_max
print 'phase angle = %0.2f degrees'%(I_a_phi*180/pi)
print 'Voltage induced by rotor:'
print 'peak value = %0.2f volts'%E_r_max
print 'phase angle = %0.2f degrees'%(-t*180/pi)
print 'torque angle = %0.2f degrees'%(t*180/pi)
print 'power factor : %0.2f'%(PF)
 All the values in the textbook are approximated hence the values in this code differ from those of Textbook
CASE a:
speed = 900.00 rpm
developed torque = 395.77 Nm
CASE b:
Phase current:
peak value = 28.78 amperes
phase angle = 25.84 degrees
Voltage induced by rotor:
peak value = 498.88 volts
phase angle = -4.17 degrees
torque angle = 4.17 degrees
CASE c:
Phase current:
peak value = 52.71 amperes
phase angle = 10.61 degrees
Voltage induced by rotor:
peak value = 498.88 volts
phase angle = -8.36 degrees
torque angle = 8.36 degrees
power factor : 0.98

Pg: 778 Ex: 17.5

In [5]:
from __future__ import division
from math import pi,sqrt,acos,sin,cos,atan,asin
from cmath import polar
V_a=480#      #phase voltage
f=60#      #frequency
P_dev=200*746#      #developed power
PF=0.85#      #power factor
I_f_1=10#      #field current
X_s=1.4#      #synchronous resistance
phi=acos(PF)#
I_a_1_max=P_dev/(3*V_a*PF)#      #phase current
I_a_1_phi=-phi#
I_a_1=I_a_1_max*complex(cos(-phi),sin(-phi))#
E_r_1=V_a-1J*X_s*I_a_1#      #rotor induced voltage
E_r_1_max=polar(E_r_1)[0]
E_r_1_phi=polar(E_r_1)[1]
#to achieve 100 percent power factor, increase I_a until it is in phase with V_a
I_a_2=P_dev/(3*V_a*cos(0))#
E_r_2=V_a-1J*X_s*I_a_2#
E_r_2_max=polar(E_r_2)[0]
E_r_2_phi=polar(E_r_2)[1]
I_f_2=I_f_1*E_r_2_max/E_r_1_max#      #magnitude of E_r proportional to field current
print " All the values in the textbook are approximated hence the values in this code differ from those of Textbook"
print 'The new field current to achieve 100%% power factor = %0.2f amperes'%I_f_2
 All the values in the textbook are approximated hence the values in this code differ from those of Textbook
The new field current to achieve 100% power factor = 12.05 amperes