# Chapter 8: Synchronous Motors¶

## Example 8.1, Page 477¶

In [1]:
import math

#Variable declaration
R_s=complex(0,5);#synchronous reactance of motor
P_o=10*746;#power output (in Watts)
P_rot=230.;#rotational loss (in Watts)
P_d=P_o+P_rot;#power developed (in Watts)
V=230.;#in volts
V_a=V/math.sqrt(3);#rms value of per phase voltage
P_fw=70.;#feild winding loss
theta=math.acos(pf);

#Calculations
I_ao=P_d/(pf*V*math.sqrt(3));
P_in=P_d+P_fw;
Eff=(P_o/P_in)*100;
I_a=I_ao*complex(math.cos(theta),math.sin(theta));
E_a=V_a-(I_a*R_s);

#Results
print 'efficiency (%%)=%.1f'%Eff
print 'magnitude of generated voltage (in Volts)=%.1f'%(abs(E_a))
print 'Phase angle of generated voltage (in Degree)=%.1f'%(math.degrees(math.atan(E_a.imag/E_a.real)))

efficiency (%)=96.1
magnitude of generated voltage (in Volts)=248.8
Phase angle of generated voltage (in Degree)=-22.8


## Example 8.2, Page 480¶

In [2]:
import math

#Variable declaration
V=480.;#in volts
V_a=V/math.sqrt(3);#per phase applied voltage
I_a=50.;#in Amperes
R_a=0.5;#armature winding resistance
X_d=complex(0,3.5);#d-axis reactance
X_q=complex(0,2.5);#q-axis reactance

#Calculations&Results
E_ao=V_a-(I_a*R_a)-(I_a*X_q);
delta=math.atan(E_ao.imag/E_ao.real);
I_d=I_a*math.sin(abs(delta))*complex(math.cos(90+delta),math.sin(90+delta));#d-axis current
E_a=E_ao-(I_d*(X_d-X_q));
E_L=E_a*math.sqrt(3);
print 'rms value of excitation voltage (in Volts)=%.2f'%(abs(E_L))
P_d=3*E_ao*I_a.conjugate()
print 'power developed by motor (in Kilo-Watts)=%.2f'%(P_d.real/1000)

rms value of excitation voltage (in Volts)=522.10
power developed by motor (in Kilo-Watts)=37.82


## Example 8.3, Page 483¶

In [3]:
import math

#Variable declaration
V=440.;#in volts
V_a=V/math.sqrt(3);#per phase voltage

#Calculations&Results
X_s=complex(0,36./3);#per phase reactance
E_ao=560/math.sqrt(3);#per-phase excitation voltage
P_d=9000;#power developed (in Watts)
delta=math.degrees(math.asin(-P_d*12/(3*V_a*E_ao)))
E_a=E_ao*complex(math.cos(delta),math.sin(delta));
I_a=(V_a-E_a)/X_s;
alpha=math.atan(I_a.imag/I_a.real);
print '(a) Power factor=%.2f'%(math.cos(alpha))
print '(b) power angle (in Degree)=%.f'%delta
E_L=(math.sqrt(3))*E_a*complex(math.cos(30),math.sin(30));
print '(c) line to line excitation voltage (in Volts)=%.f'%(abs(E_L))
print 'phase angle of line to line excitation voltage (in Degree) %.1f'%(math.degrees(math.atan(E_L.imag/E_L.real)))
T_d=P_d/w_m;
print '(d) Torque developed (in Newton-meter)=%.2f'%T_d

(a) Power factor=0.98
(b) power angle (in Degree)=-26
(c) line to line excitation voltage (in Volts)=560
phase angle of line to line excitation voltage (in Degree) 49.4
(d) Torque developed (in Newton-meter)=47.75


## Example 8.4, Page 486¶

In [4]:
import math

#Variable declaration
pf=0.8;#lagging
theta=-math.degrees(math.acos(pf))
V_a=120;#in V
X_d=2.7;#d-axis reactance (in ohms/phase)
X_q=1.7;#q-axis reactances (in ohms/phase)

#Calculations&Results
I_a=40*complex(math.cos(-36.87*math.pi/180),math.sin(-36.87*math.pi/180));#in Amperes
E_a_dash=V_a-complex(0,(I_a*X_q));#in Volts
delta=math.degrees(math.atan(E_a_dash.imag/E_a_dash.real));#in degree
alpha=abs(theta-delta);#in degree
I_d=abs(I_a)*math.sin(alpha*math.pi/180)*complex(math.cos((-34.48-90)*math.pi/180),math.sin((-34.48-90)*math.pi/180));
E_a=E_a_dash-complex(0,(I_d*(X_d-X_q)))
E_a_m=math.sqrt(E_a.real**2+E_a.imag**2)
print '(a) per-phase excitation voltage(in Volts)=%.3f'%(abs(E_a_m))
print 'phase angle of excitation voltage (in degree)=%.2f'%(math.degrees(math.atan(E_a.imag/E_a.real)))
P_df=(3*V_a*abs(E_a)*math.sin(34.48*math.pi/180))/X_d;
print '(b) power developed due to field excitation(in Watts)=%.2f'%P_df
P_ds=((X_d-X_q)*math.sin(2*34.48*math.pi/180)*3*V_a**2)/(2*X_d*X_q);
print '(c) power developed due to saliency of motor (in Watts)=%.2f'%P_ds
P_d=P_df+P_ds;
print '(d) total power developed (in Watts)=%.f'%P_d
P_r=0.05*P_d;#rotational loss (in Watts)
P_in=3*V_a*I_a.conjugate();#power input (in Watts)
P_o=P_in.real-P_r;#power output (in Watts)
Eff=(P_o/P_in.real)*100;
print '(e) Efficiency (in %%)=%.f'%Eff
#refer to eqn 8.24
A=(3*120*abs(E_a))/X_d;
B=3*(X_d-X_q)*120**2/(2*X_d*X_q);
P_dm=A*math.sin(63.4*math.pi/180)+B*math.sin(2*63.4*math.pi/180);
print '(f) maximum power developed (in Watts)=%.f'%P_dm

(a) per-phase excitation voltage(in Volts)=94.418
phase angle of excitation voltage (in degree)=-34.48
(b) power developed due to field excitation(in Watts)=7126.90
(c) power developed due to saliency of motor (in Watts)=4392.14
(d) total power developed (in Watts)=11519
(e) Efficiency (in %)=95
(f) maximum power developed (in Watts)=15025


## Example 8.5, Page 491¶

In [5]:
import math

#Variable declaration
Zs = complex(0.5,3)  #armature winding resistance,ohms
Vs = 69.28*complex(math.cos(0*math.pi/180),math.sin(0*math.pi/180))  #V
Ia1 = 10*complex(math.cos(36.87*math.pi/180),math.sin(36.87*math.pi/180)) #A
Ea1 = 87.54*complex(math.cos(-17.96*math.pi/180),math.sin(-17.96*math.pi/180))

#Calculations&Results
Sm = math.degrees(math.atan(Zs.imag/Zs.real))  #degrees
Ea2 = 87.54*complex(math.cos(-Sm*math.pi/180),math.sin(-Sm*math.pi/180))
Ia2 = (Vs-Ea2)/Zs  #A
Ia2_mag = math.sqrt(Ia2.real**2+Ia2.imag**2)
Ia2_phase = math.degrees(math.atan(Ia2.imag/Ia2.real))
print "The line current is %.2f with phase = %.2f A"%(Ia2_mag,Ia2_phase)

Pdma = Ea2*Ia2_mag*complex(math.cos(-Ia2_phase*math.pi/180),math.sin(-Ia2_phase*math.pi/180))
print "The pre-phase maximum power developed is %.f W/phase"%Pdma.real

Pdm = 3*Pdma
print "Total power developed = %.f W"%Pdm.real

The line current is 33.64 with phase = -22.98 A
The pre-phase maximum power developed is 1580 W/phase
Total power developed = 4740 W


## Example 8.6, Page 496¶

In [6]:
import math

#Variable declaration
V=208;#in Volts
V_a=V/math.sqrt(3);#in volts
P=7200;#in Watts
X_a=4;#synchronous reactance
pf=0.8;#lagging
theta=-math.degrees(math.acos(pf))

#Calculations&Results
I_a=(P/(3*V_a*pf))*complex(math.cos(theta*math.pi/180),math.sin(theta*math.pi/180));#Armature current (in Amperes)
E_a=V_a-(I_a*complex(0,X_a));#in Volts
E_a_m=math.sqrt(E_a.real**2+E_a.imag**2)
E_an=1.5*abs(E_a_m);#new excitation voltage (in Volts)
delta_n=-math.degrees(math.asin(P*X_a/(3*E_an*V_a)));#new torque angle
I_an=(V_a-E_an*complex(math.cos(delta_n*math.pi/180),math.sin(delta_n*math.pi/180)))/complex(0,4);
print '(a) New armature current (in Ampere)=%.3f'%(abs(I_an))
print 'Phase angle of new armature current (in Degree)=%.2f'%(math.degrees(math.atan(I_an.imag/I_an.real)))
pf_n=math.cos(math.atan(I_an.imag/I_an.real));
print '(b) New Power factor=%.3f'%pf_n

(a) New armature current (in Ampere)=20.059
Phase angle of new armature current (in Degree)=4.93
(b) New Power factor=0.996


## Example 8.7, Page 499¶

In [7]:
import math

#Calculations&Results
theta_L=math.degrees(math.acos(0.6));#lag (in degree)
S_L=100*complex(math.cos(53.13*math.pi/180),math.sin(53.13*math.pi/180));#in KVA
#for synchronous motor:
x = complex(math.cos(theta_m*math.pi/180),math.sin(theta_m*math.pi/180))
S_m=(10./0.5)*x.conjugate();#in Watts
S_t=S_L+S_m;#overall power (in Watts)
pf=math.cos(math.atan(S_t.imag/S_t.real));
print 'overall power factor=%.2f'%pf
#for power factor=0.9
theta_t=25.84;
S_tn=(S_t.real/0.9)*complex(math.cos(theta_t*math.pi/180),math.sin(theta_t*math.pi/180));#in KVA
S_mn=S_tn-S_L;#in KVA
pf_n=math.cos(math.atan(S_mn.imag/S_mn.real));
print 'power factor of motor to improve overall power factor to 0.9=%.2f'%pf_n

overall power factor=0.74
power factor of motor to improve overall power factor to 0.9=0.21