Chapter 9: Polyphase Induction Motors¶

Example 9.1, Page 513¶

In [1]:
#Variable declaration
f=60.;#in Hertzs
P=4;#no. of poles
N_fL=1755;#in rpm

#Calculations&Results
N_s=120*f/P;
print '(a) synchronous speed of induction motor (in rpm)=%.f'%N_s
s=(N_s-N_fL)/N_s;
print '(b) Slip at full load =%.3f'%s
f_r=s*f;
print '(c) rotor frequency at full load (in Hertzs)=%.1f'%f_r

(a) synchronous speed of induction motor (in rpm)=1800
(b) Slip at full load =0.025
(c) rotor frequency at full load (in Hertzs)=1.5


Example 9.2, Page 521¶

In [2]:
import math

#Variable declaration
V=230;#in volts
f=60.;#in Hertz
P=6;#no. of poles
N_s=120*f/P;#synchronous speed (in rpm)
V_1=V/math.sqrt(3);#per phase voltage (in Volts)
R_2=0.25;#in ohms
R_1=0.5;#in ohms
X_1=0.75;#in ohms
X_2=0.5;#in ohms
X_m=100;#in ohms
R_c=500;#in ohms
s=0.025;#slip

#Calculations
Z_1=complex(R_1,X_1);#in ohms
Z_2=complex((R_2/s),X_2);#in ohms
Z=complex(0.002,0.01)+complex(0.10025,0.0050125);
Z_e=(1/Z);#equivalent impedance (in ohms)
Z_in=Z_1+Z_e;#input impedance (in ohms)
I_1=V_1/Z_in;#in Amperes
theta=math.degrees(math.atan(I_1.imag/I_1.real))
P_in=3*V_1*I_1.real;
P_scl=3*(abs(I_1))**2*R_1;
E_1=V_1-I_1*Z_1;
I_c=E_1/R_c;#core loss current
I_m=complex(0,E_1/X_m)
I_phy=I_c+I_m;#excitation current (in Amperes)
I_2=I_1-I_phy;#rotor current (in Amperes)
P_m=3*abs(I_c)*abs(I_c)*R_c;#core loss (in Watts)
P_ag=P_in-P_scl-P_m;#air gap power (in Watts)
P_rcl=3*abs(I_2)*abs(I_2)*R_2;#rotor copper loss (in Watts)
P_d=P_ag-P_rcl;#power developed (in Watts)
P_o=P_d-150;#output power (in Watts)
Eff=P_o.real/P_in.real

#Result
print 'Efficiency (%%)=%.2f'%(Eff*100)

Efficiency (%)=87.97


Example 9.3, Page 524¶

In [3]:
import math

#Variable declaration
#Refer to data of Exa:9.2
R_1=0.5;#in ohms
R_2=0.25;#in ohms
X_1=0.75;#in ohms
X_2=0.5;#in ohms
R_c=500;#in ohms
s=0.025;#slip

#Calculations
I_c=132.791/500;#Core-loss currrent (in Amperes)
I_m=complex(0,132.791/100);#Magnetization current (in Amperes)
Z_e=R_1+complex((R_2/s),(X_1+X_2));#in ohms
I_2=132.791/Z_e;#rotor current (in Amperes)
I_1=I_2+I_c+I_m;#in Amperes
P_in=3*132.791*I_1.conjugate();#power input (in Watts)
P_scl=3*(abs(I_2))**2*R_1;#stator copper loss (in Watts)
P_rcl=3*(abs(I_2))**2*R_2;#rotor copper loss (in Watts)
P_m=3*(abs(I_c))**2*R_c;# core loss (in Watts)
P_o=P_in.real-P_scl-P_rcl-P_m-150;#power output (in Watts)
Eff=P_o/P_in.real;

#Result
print 'Efficiency (%%)=%.2f'%(Eff*100)

Efficiency (%)=87.96


Example 9.4, Page 527¶

In [4]:
import cmath
import math

#Variable declaration
V=120;#in volts
f=60;#in Hertzs
R_1=0.1;#in ohms
X_1=0.15;#in ohms
R_2=0.2;#in ohms
X_2=0.25;#in ohms

#Calculations&Results
Z_e=R_1+complex(R_2,(X_1+X_2));#Eqv impedance in ohms
Z_e_m = math.sqrt(Z_e.real**2+Z_e.imag**2)
s_p=R_2/(R_2+Z_e_m);
print 'Slip=%.3f'%s_p
P_dm=(3*V**2)/(2*(R_1+R_2+abs(Z_e)));
print 'max power developed (in Kilo-Watts)=%.f'%(P_dm/1000)
N_s=120*f/6;#synchronous speed (in rpm)
w_m=(1-s_p)*w_s;
T_d=P_dm/w_m;
print 'Torque developed (in Newton-meter)=%.2f'%T_d

Slip=0.286
max power developed (in Kilo-Watts)=27
Torque developed (in Newton-meter)=300.80


Example 9.5, Page 529¶

In [6]:
import math

#Variable declaration
#Refer to data of Exa9.4
R_1=0.1;# in ohms
R_2=0.2;# in ohms
X_1=0.15;# in ohms
X_2=0.25;# in ohms
V_1=120;#in Volts

#Calculations&Results
s_b=R_2/math.sqrt(R_1**2+(X_1+X_2)**2);
print '(a) Breakdown slip=%.3f'%s_b
T_dm=(3*V_1**2)/(2*w_s*(R_1+math.sqrt(R_1**2+(X_1+X_2)**2)));
print '(b) Breakdown Torque (in Newton-meter)=%.2f'%T_dm
P_d=T_dm*(1-s_b)*w_s;
print '(c) Power developed by the motor (in Kilo-Watts)=%.2f'%(P_d/1000)

(a) Breakdown slip=0.485
(b) Breakdown Torque (in Newton-meter)=335.52
(c) Power developed by the motor (in Kilo-Watts)=21.71


Example 9.6, Page 531¶

In [7]:
import math

#Variable declaration
f=60.;#in Hertzs
P=8;#no. of poles
R_2=0.02;#in ohms
X_2=0.08;#in ohms

#Calculations&Results
s_b=R_2/X_2;#breakdown slip
print '(a) breakdown slip=%.2f'%s_b
N_s=120*f/P;#synchronous speed (in rpm)
w_s=N_s*2*math.pi/60;
N_m=(1-s_b)*N_s;#motor speed (in rpm)
V_1=120.;#in V
T_dm=(3*V_1**2*s_b)/(2*w_s*R_2)
print 'Breakdown torque (in Newton-meter)=%.3f'%T_dm
T_s=2*1*s_b*T_dm/(1+s_b**2);
print '(b) Starting torque is %.2f'%(T_s/T_dm),"times the max torque";
s_bn=(-(-2.5)-math.sqrt((-2.5)**2-4*1*1))/2;#new breakdown slip
R_2n=s_bn*X_2;
print 'rotor resistance (in ohms)=%.2f'%R_2n

(a) breakdown slip=0.25
Breakdown torque (in Newton-meter)=2864.789
(b) Starting torque is 0.47 times the max torque
rotor resistance (in ohms)=0.04


Example 9.7, Page 535¶

In [8]:
import math

#Variable declaration
f=60;#in Hertzs
P=4;#no. of poles
V_1=230;#in volts
I_2=4.5;#rotor current (in Amperes)
P_d=2*746;#in watts
N_m=1710;#speed of motor in (rpm)

#Calculations
N_s=120*f/P;#Synchronous speed (in rpm)
s=(N_s-N_m)/N_s;#slip
T_d=P_d/w_m;#torque developed (in Newton-meter)
T_dL=T_d*(0.9*230/230)**2;#in Newton-meter
I_2L=I_2*(0.9*230/230);#in Amperes
T_dH=8.33*1.1**2;#in Newton-meter
I_2H=I_2*1.1;#in Amperes

#Result
print "(a) Torque range (in Newton-meter) is:";
print 'minimum value=%.2f'%T_dL
print 'maximum value=%.2f'%T_dH
print "(b) Current range (in Amperes) is:";
print 'minimum value=%.2f'%I_2L
print 'maximum value=%.2f'%I_2H

(a) Torque range (in Newton-meter) is:
minimum value=6.75
maximum value=10.08
(b) Current range (in Amperes) is:
minimum value=4.05
maximum value=4.95


Example 9.8, Page 540¶

In [9]:
import math

#Variable declaration
V_1=208;#in Volts
f=60;#in Hertzs
P=4;#no. of poles
N_m=1710;#in rpm

#Calculations&Results
R_1=2.4/2;#in ohms
print 'R_1 (in ohms)=%.1f'%R_1
W_oc=450./3;#in Watts
P_fw_phy=18./3;#in Watts
P_oc=W_oc-P_fw_phy;#in Watts
V_oc=V_1/math.sqrt(3);#in Volts
I_oc=1.562;#in Amperes
R_c=V_oc**2/P_oc;
print 'R_c=core loss resistance (in ohms)=%.f'%R_c
S_oc=V_oc*I_oc;#in Volt-Ampere
theta_oc=math.acos(W_oc/S_oc);
I_m=I_oc*math.sin(theta_oc);
X_m=V_oc/I_m;
print 'X_m=Magnetization reactance (in ohms)=%.f'%X_m
V_br=27/math.sqrt(3);#in Volts
P_br=59.4/3;#in Watts
I_br=2.77;#In Amperes
R_e=P_br/I_br**2;
R_2=R_e-R_1;
print 'R_2 (in ohms)=%.2f'%R_2
Z_e=V_br/I_br;
X_e=math.sqrt(Z_e**2-R_e**2);
X_1=X_e/2;
X_2=X_1;
print 'X_1 (in ohms)=%.1f'%X_1
print 'X_2 (in ohms)=%.1f'%X_2

R_1 (in ohms)=1.2
R_c=core loss resistance (in ohms)=100
X_m=Magnetization reactance (in ohms)=128
R_2 (in ohms)=1.38
X_1 (in ohms)=2.5
X_2 (in ohms)=2.5


Example 9.10, Page 549¶

In [11]:
import math

#Variable declaration
R=20*10**-6;#in ohms
X=2*10**-3;#in ohms
P=4.;#no. of poles
Q=48.;#no. of bars
S=36.;#no. of slots
#For Stator:
m_1=3.;#no. of phases

#Calculations&Results
n=3*(S/(P*3));#coils per pole per phase
S_p=S/P;#pole span
S_s=180./S_p;#slot span (in electrical degree)
k_p1=math.sin(140/2*math.pi/180);#pitch factor
k_d1=math.sin(3*S_s/2*math.pi/180)/(3*math.sin(S_s/2*math.pi/180));#distribution factor
k_w1=k_p1*k_d1;#winding factor
N_1=10*S/3;#turns per phase
#For Rotor:
k_w2=1;
m_2=Q/P;#no. of phases
N_2=P/2;#turns per phase
a=int((k_w1*N_1/(k_w2*N_2))*math.sqrt(m_1/m_2));
R_2=a**2*R;
print "Rotor Parameters as reffered to stator:";
print 'R_2 (in mili ohms)=%.2f'%(R_2*1000)
X_2=a**2*X;
print 'X_2 (in ohms)=%.3f'%X_2

Rotor Parameters as reffered to stator:
R_2 (in mili ohms)=14.58
X_2 (in ohms)=1.458