# Chapter 10: Single-Phase Motors¶

## Example 10.1, Page 571¶

In [1]:
#Variable declaration
P=4;#no. of poles
f=60.;#frequency in Hertzs
R2=12.5;#rotor resistance (in ohms)

#Calculations&Results
N_s=120*f/P;#synchronous speed of motor(in rpm)
N_m=1710;#speed of motor in clockwise direction (in rpm)
s=(N_s-N_m)/N_s;
print '(a) slip in forward direction=%.2f'%s
s_b=2-s;
print '(b) slip in backward direction=%.2f'%s_b
#effective rotor resistance
R_f=0.5*R2/s;#(in forward branch)
print 'effective rotor resistance in forward branch (in ohms)=%.f'%R_f
R_b=0.5*R2/s_b;#(in backward direction)
print 'effective rotor resistance in backward branch (in ohms)=%.3f'%R_b
(a) slip in forward direction=0.05
(b) slip in backward direction=1.95
effective rotor resistance in forward branch (in ohms)=125
effective rotor resistance in backward branch (in ohms)=3.205

## Example 10.2, Page 576¶

In [2]:
import math

#Variable declaration
V=120.;#in volts
f=60.;#frequency in Hertzs
P=4.;#no. of poles
R1=2.5;#in ohms
X1=complex(0,1.25)
R2=3.75;
X2=complex(0,1.25)
X_m=complex(0,65)
N_m=1710;#speed of motor (in rpm)
P_c=25;#core lossv(in Watts)
P_fw=2;#friction and windage loss (in Watts)

#Calculations&Results
N_s=120*f/P;#synchronous speed of motor
s=(N_s-N_m)/N_s;#slip
Z_f=(X_m*((R2/s)+X2)*0.5)/((R2/s)+(X2+X_m));#forward impedance
Z_b=(X_m*((R2/(2-s))+X2)*0.5)/((R2/(2-s))+(X2+X_m));#backward impedance
Z_in=R1+X1+Z_f+Z_b;
I_1=V/Z_in;
P_in=V*I_1.conjugate()
I_2f=X_m*I_1/((R2/s)+(X1+X_m));#forward current
I_2b=X_m*I_1/((R2/(2-s))+(X1+X_m));#backward current
P_agf=0.5*(R2/s)*(abs(I_2f))**2;#air gap power in forward path
P_agb=0.5*(R2/(2-s))*(abs(I_2b))**2;#air gap power in backward path
P_ag=P_agf-P_agb;#net air gap power
P_d=(1-s)*P_ag;#gross power developed
P_o=P_d-P_c-P_fw;#net power output
w_m=2*(math.pi)*N_m/60;
T_s=P_o/w_m;
print 'shaft torque (in Newton-meter)=%.3f'%T_s
Eff=P_o/P_in.real;
print 'Efficiency of motor (%%)=%.2f'%(Eff*100)
shaft torque (in Newton-meter)=1.295
Efficiency of motor (%)=65.86

## Example 10.3, Page 590¶

In [3]:
import math

#Variable declaration
V1=230.;#in volts
f=50.;#frequency in Hz
P=6.;#no. of poles
R1=34.14;#in ohms
X1=complex(0,35.9)
R_a=149.78;
X2=complex(0,29.32)
X_m=complex(0,248.59)
R2=23.25;
a=1.73;
P_c=19.88;#core loss
P_fw=1.9;#friction and windage loss
N_m=940.;#speed of motor in rpm
N_s=120.0*f/P;#synchronous speed of motor

#Calculations&Results
s=(N_s-N_m)/N_s;#slip
X_c=complex(0,1/(2*math.pi*f*C));#reactance of capacitance
Z_f=(X_m*((R2/s)+X2)*0.5)/((R2/s)+(X2+X_m));#forward impedance
Z_b=(X_m*((R2/(2-s))+X2)*0.5)/((R2/(2-s))+(X2+X_m));#backward impedance
Z_11=R1+X1+Z_f+Z_b;#in ohms
Z_12=-1*complex(0,a*(Z_f-Z_b));#in ohms
Z_21=-Z_12;#in ohms
Z_22=a*a*(Z_f+Z_b+X1)+R_a-X_c;#in ohms
I_1=V1*(Z_22-Z_12)/(Z_11*Z_22-Z_12*Z_21);#current in main winding
I_2=V1*(Z_11-Z_21)/(Z_11*Z_22-Z_12*Z_21);#current in auxilary winding
I_L=I_1+I_2;
print '(a) magnitude of line current (in Amperes)=%.3f'%(math.sqrt(I_L.real**2+I_L.imag**2))
print '  phase of line current (in Degree)=%.2f'%math.degrees(math.atan(I_L.imag/I_L.real))
P_in=V1*I_L.conjugate();
print '(b) power input (in Watts)=%.3f'%P_in.real
P_agf=complex((I_1*Z_f),(-I_2*a*Z_f))*I_1.conjugate()+complex((I_2*a*a*Z_f),(I_1*a*Z_f))*I_2.conjugate();#air gap power developed by forward field
P_agb=complex((I_1*Z_b),(I_2*a*Z_b))*I_1.conjugate()+complex((I_2*a*a*Z_b),(-I_1*a*Z_b))*I_2.conjugate();#air gap power developed by backward field
P_ag=P_agf.real-P_agb.real
P_d=(1-s)*P_ag;#power developed
P_o=P_d-P_c-P_fw;#output power
print '(c) Efficiency of motor (%%)=%.1f'%(P_o*100/P_in.real)
T_s=P_o/w_m;
print '(d) shaft torque (in Newton-meter)=%.3f'%T_s
V_c=I_2*X_c;
print '(e) magnitude of voltage across capacitor (in Volts)=%.3f'%(math.sqrt(V_c.real**2+V_c.imag**2))
print 'phase of voltage across capacitor (in Degree)=%.2f'%(math.degrees(math.atan(V_c.imag/V_c.real)))
#for starting torque
s=1;
s_b=1;
w_s=2*math.pi*N_s/60;
Z_f=(X_m*((R2/s)+X2)*0.5)/((R2/s)+(X2+X_m));#forward impedance
Z_b=(X_m*((R2/(2-s))+X2)*0.5)/((R2/(2-s))+(X2+X_m));#backward impedance
Z_11=R1+X1+Z_f+Z_b;#in ohms
Z_12=complex(0,(-a*(Z_f-Z_b)));#in ohms
Z_21=-Z_12;#in ohms
Z_22=a*a*(Z_f+Z_b+X1)+R_a-X_c;#in ohms
I_1s = V1/Z_11;#current in main winding
I_2s=V1/Z_22
#I_2s=V1*(Z_11-Z_21)/(Z_11*Z_22-Z_12*Z_21);#current in auxilary winding
I_Ls=I_1s+I_2s;
P_in=V1*I_Ls.conjugate();
P_agf=complex((I_1s*Z_f),(-I_2s*a*Z_f))*I_1s.conjugate()+complex((I_2s*a*a*Z_f),(I_1s*a*Z_f))*I_2s.conjugate();#air gap power developed by forward field
P_agb=complex((I_1s*Z_b),(I_2s*a*Z_b))*I_1s.conjugate()+complex((I_2s*a*a*Z_b),(-I_1s*a*Z_b))*I_2s.conjugate();#air gap power developed by backward field
P_ag=P_agf-P_agb;
T_s=P_ag.real/w_s;
print '(f) starting torque (in Newton-meter)=%.2f'%T_s
(a) magnitude of line current (in Amperes)=1.207
phase of line current (in Degree)=-23.48
(b) power input (in Watts)=254.563
(c) Efficiency of motor (%)=57.5
(d) shaft torque (in Newton-meter)=1.488
(e) magnitude of voltage across capacitor (in Volts)=444.666
phase of voltage across capacitor (in Degree)=-68.29
(f) starting torque (in Newton-meter)=0.52

## Example 10.4, Page 597¶

In [4]:
import math

#Variable declaration
R_m=2.5;#main winding resistance
R_a=100.;#auxilary winding resistance
#blocked-rotor test
V_bm=25.;#voltage (in Volts)
I_bm=3.72;#current (in Amperes)
P_bm=86.23;#power (in Watts)
#with auxilary winding open no load test
V_nL=115;#voltage (in Volts)
I_nL=3.2;#current (in Amperes)
P_nL=55.17;#power (in Watts)
#with main winding open blocked rotor test
V_ba=121;#voltage (in Volts)
I_ba=1.2;#current (in Amperes)
P_ba=145.35;#power (in Watts)

#Calculations&Results
Z_bm=V_bm/I_bm;
R_bm=P_bm/I_bm**2;
X_bm=math.sqrt(Z_bm**2-R_bm**2);
X1=0.5*X_bm;
X2=X1;
R2=R_bm-R_m;
print 'X1 (in ohms)=%.2f'%X1
print 'X2 (in ohms)=%.2f'%X2
print 'R2 (in ohms)=%.2f'%R2
Z_nL=V_nL/I_nL;
R_nL=P_nL/I_nL**2;
X_nL=math.sqrt(Z_nL**2-R_nL**2);
X_m=2*X_nL-0.75*X_bm;
P_r=P_nL-I_nL**2*(R_m+0.25*R2);
print 'P_r (in Watts)=%.f'%(int(P_r))
print 'X_m (in ohms)=%.2f'%X_m
Z_ba=V_ba/I_ba;
R_ba=P_ba/I_ba**2;
R_2a=R_ba-R_a;
alpha=math.sqrt(R_2a/R2);
print 'alpha=%.1f'%alpha
X1 (in ohms)=1.26
X2 (in ohms)=1.26
R2 (in ohms)=3.73
P_r (in Watts)=20
X_m (in ohms)=69.17
alpha=0.5

## Example 10.5, Page 606¶

In [5]:
import math

#Variable declaration
V_s=120;#in Volts
P_rot=80;#rotational loss (in Watts)
N_m=8000;#speed of motor (in rpm)
pf=0.912;#lagging
theta=-math.degrees(math.acos(pf))

#Calculations&Results
I_a=17.58*complex(math.cos(theta*math.pi/180),math.sin(theta*math.pi/180));#in Amperes
Z_s=complex(0.65,1.2);#series field winding impedance (in ohms)
Z_a=complex(1.36,1.6);#armature winding impedance (in ohms)
E_a=V_s-I_a*(Z_s+Z_a);#induced emf (in Volts)
print '(a) induced emf in the armature (in Volts)=%.1f'%(math.sqrt(E_a.real**2+E_a.imag**2))
print 'phase of induced emf in the armature (in Degree)=%.2f'%(math.degrees(math.atan(E_a.imag/E_a.real)))
P_d=E_a*I_a.conjugate();
P_o=P_d.real-P_rot;
print '(b) power output (in Watts)=%.2f'%P_o
w_m=2*math.pi*N_m/60;#rated speed of motor (in rad/sec)
T_s=P_o/w_m;
print '(c) shaft torque (in Newton-meter)=%.2f'%T_s
P_in=V_s*abs(I_a)*pf;
Eff=P_o*100/P_in;
print '(d) Efficiency (%%)=%.1f'%Eff
(a) induced emf in the armature (in Volts)=74.1
phase of induced emf in the armature (in Degree)=-24.22
(b) power output (in Watts)=1222.75
(c) shaft torque (in Newton-meter)=1.46
(d) Efficiency (%)=63.6