Chapter 7: Synchronous Motor and Brushless dc Motor Drives¶

Example no:7.1,Page no:247¶

In :
from __future__ import division
import cmath

#variable declaration
#ratings of the synchronous motor
Pm1=500*1000   # power rating in W
f=50           # frequency in HZ
Vl=3.3*1000    # line voltage in V
pf=0.8         # power factor lagging
P=4            # number of poles
I=10           # field current in A
Xs=15          # reactance of the windings in ohm
Rs=0           # resistance of the windings in ohm
Wms=50*math.pi # synchronous speed in rad/sec
Pm=Pm1/2       # power at half the rated torque when the losses are neglected

#calculation
V=Vl/math.sqrt(3)  #phase voltage
Is=Pm1/(math.sqrt(3)*Vl*pf)   #rated current

Is=cmath.rect(Is,-rad)        #rated current in vector form
V=cmath.rect(V,0)             #rated phase voltage in rectangular form
E=V-Is*1j*Xs                  #back emf

#(i) when field current has not changed
sin_delta=Pm*Xs/(3*abs(V)*abs(E))
delta=math.asin(sin_delta)                 #angle delta
Is=(V-cmath.rect(abs(E),-delta))/(1j*Xs)   #armature current
Is1=cmath.polar(Is)
x=math.degrees(Is1)           #where x=Is which is the required armature current
power_factor=math.cos(Is1)    #power factor

#(ii) At unity power factor and rated torque
cos_phi=1
Is=Pm1/(3*V)         #since Pm1=3*V*Is
E1=V-Is*1j*Xs
If=abs(E1)/abs(E)*I  #field current

#(iii) At the field current of 12.5 A
If1=12.5      #field current
E2=If1/I*abs(E)
Is=math.sqrt(E2**2-abs(V)**2)/Xs  #since E2=abs(V-Is*1j*Xs)
Pm=3*abs(V)*Is*cos_phi            #power output at the given field current
T=Pm/Wms                          #required torque

#results
print"i)armature current :",round(Is1,2),round(x),"°","A"
print"  power factor",round(power_factor,2),"lagging"
print"\nii)field current at unity power factor at rated torque:",round(If,2),"A"
print"\niii)Required torque is:",round(T,1),"N-m"
print"Note: there is a slight difference in the answer due to the decimal place"
i)armature current : 52.76 -34.0 ° A
power factor 0.83 lagging

ii)field current at unity power factor at rated torque: 14.43 A

iii)Required torque is: 1507.2 N-m
Note: there is a slight difference in the answer due to the decimal place

Example no:7.2,Page no:249¶

In :
import math
from __future__ import division
import cmath

#variable declaration
#ratings of the synchronous motor is same as that of Example-7.1
Pm1=500*1000   # power rating in W
f=50           # frequency in HZ
Vl=3.3*1000    # line voltage in V
pf=0.8         # power factor lagging
P=4            # number of poles
I=10           # field current in A
Xs=15          # reactance of the windings in ohm
Rs=0           # resistance of the windings in ohm
Pm=Pm1/2       # power at half the rated torque when the losses are neglected

#calculation
Wms=50*math.pi     # synchronous speed in rad/sec
V=Vl/math.sqrt(3)  # phase voltage
Is=Pm1/(math.sqrt(3)*Vl*pf)  #rated current

Is=cmath.rect(Is,-rad)       #rated current in vector form
V=cmath.rect(V,0)
E=V-Is*1j*Xs                 #back emf

#(i) at rated current and unity power factor
E1=V-abs(Is)*1j*Xs
delta=cmath.phase(E1)       #phase angle of E1
Pm=3*abs(V)*abs(E1)*math.sin(delta)/Xs   #mechanical power developed
T=Pm/Wms                #braking torque
If=abs(E1)/abs(E)*I     #field current

#(ii) at field current of 15A and 500kW output
If1=15           #field current
Pm=-500*1000    #output power
E2=If1/I*abs(E)
sin_delta=Pm*Xs/(3*abs(V)*abs(E2))
delta=math.asin(sin_delta)               #angle delta
Is=(cmath.rect(E2,abs(delta))-V)/(1j*Xs) #armature current
Is=cmath.polar(Is)
x=(Is)*180/math.pi        #phase angle of Is
power_factor=math.cos(Is) #power factor

#results
print"i)braking torque :",round(T,1),"N-m"
print"  Field current",round(If,2),"A"
print"\nii)armature current :",round(Is,2),round(x,2),"°","A"
print"   power factor",round(power_factor,3),"lagging"
print"\nNote :There is a slight difference in the answers due to the decimal place"
i)braking torque : -3978.9 N-m
Field current 15.68 A

ii)armature current : 87.78 -4.79 ° A
power factor 0.997 lagging

Note :There is a slight difference in the answers due to the decimal place

Example no:7.3,Page no:257¶

In :
import math
from __future__ import division
import cmath
from sympy import Symbol

#variable declaration
#ratings of the synchronous motor
Pm1=6*10**6     # power rating in W
f=50           # frequency in HZ
Vl=11*1000     # line voltage in V
P=6            # number of poles
I=10           # rated field current in A
Xs=9           # reactance of the windings in ohm
Rs=0           # resistance of the windings in ohm
N=120*f/P      # synchronous speed

#calculation
V=Vl/math.sqrt(3)           #phase voltage
Is=Pm1/(math.sqrt(3)*Vl*pf)  #rated current

#(i)to find torque and field current at rated armature current
#   at 750 rpm and 0.8 leading power factor
Is=cmath.rect(Is,rad)       #rated current in vector form
V=cmath.rect(V,0)
E=V-Is*1j*Xs                #back emf

N1=750          #speeed in rpm
f1=N1/N*f       #required frequency
V1=abs(V)*f1/f  #required voltage
Xs1=Xs*f1/f     #required field resistance
E1=V1-Xs1*1j*cmath.rect(abs(Is),math.acos(pf1))   #rated back emf  in complex form
E1_polar=cmath.polar(E1)                          #rated back emf in rectangular form
#at rated field current and 750 rpm
E2=abs(E)*N1/N             #back emf at the given speed N1
If=abs(E1)/E2*f            #field current at the given speed N1
Pm=3*abs(V1)*abs(Is)*pf1   #power input at the given speed N1
Wm1=2*math.pi*N1/60        #angular motor speed in rad/s
T=Pm/Wm1

#(ii) at half the rated motor torque and 1500 rpm and rated field current
Pm=6*10**6       #rated power rating in W
N1=1500          #speeed in rpm
f1=N1/N*f        #required frequency
Xs1=f1/f*Xs      #required field resistance
E1=abs(E)*f1/f   #back emf at rated field current

Wms = Symbol('Wms')           #rated speed in rad/sec
T_rated = Symbol('T_rated')   #rated torque
Wms=Pm/T_rated
Wms_=N1/N*Wms
Pm_= (0.5*T_rated)*Wms_       #required power developed at N1=1500 rpm

sin_delta=Pm_*Xs1/(3*abs(V)*abs(E1))       #since Pm=3*abs(V)*abs(E1)*math.sin(delta)/Xs
delta=math.asin(sin_delta)                 #angle delta
Is=(abs(V)-cmath.rect(E1,-delta))/(1j*Xs1) #armature current
Is1=cmath.polar(Is)                         #aramture current in rectangular form
x1=math.degrees(Is1)
power_factor1=math.cos(Is1)               #power factor

#(iii) at 750 rpm and rated field current from part(i)
N1=750         #speeed in rpm
f1=N1/N*f      #required frequency at N1=750 rpm
V1=abs(V)*f1/f #required voltage at N1=750 rpm
Xs1=Xs*f1/f    #required field resistance
E2=abs(E)*N1/N

Pm=-4.2*10**6 #braking power output
sin_delta=Pm*Xs1/(3*abs(V1)*abs(E2))       #since Pm=3*abs(V)*abs(E1)*math.sin(delta)/Xs
delta=math.asin(sin_delta)                 #angle delta
Is=(cmath.rect(E2,abs(delta))-V1)/(1j*Xs1) #armature current
Is2=cmath.polar(Is)                         #aramture current in rectangular form
x2=math.degrees(Is2)
power_factor2=math.cos(Is2) #power factor

#(iv)from part (ii) at 1500 rpm and from part(iii) the armature current of 349.9 A is taken
Is=Pm1/(math.sqrt(3)*Vl*pf)        #armature current as given from (i)
N1=1500           #speeed in rpm
f1=N1/N*f         #required frequency at N1=1500 rpm
Xs1=f1/f*Xs       #required field resistance
E1=abs(E)*f1/f    #at rated field current
E2=V-1j*Xs1*Is
E2=cmath.polar(E2)

If1=E2/abs(E1)*f   #required field current
Pm=3*abs(V)*E2*math.sin(abs(E2))/Xs1   #power input
T1=Pm/Wm1

#results
print"\ni)Required torque is:",round(T,1),"N-m"
print"  Field current :",round(If,2),"A"
print"\nii)armature current :",round(Is1,1),round(x1,2),"°","A"
print"\niii)armature current :",round(Is2,2),round(x2,2),"°","A"
print"    power factor :",round(power_factor2,3),"lagging"
print"\niv)Field current :",round(If1,2),"A"
print"   Required torque is:",round(T1),"N-m"
print"\nNote :There is a slight difference in the answers due to the decimal place"
i)Required torque is: 50929.6 N-m
Field current : 52.37 A

ii)armature current : 475.5 60.21 ° A

iii)armature current : 334.62 -28.55 ° A
power factor : 0.878 lagging

iv)Field current : 32.07 A
Required torque is: 42441.0 N-m

Note :There is a slight difference in the answers due to the decimal place

Example no:7.4,Page no:265¶

In :
import math
from __future__ import division
import cmath

#variable declaration
#ratings of the synchronous motor
Pm=8*10**6     # power rating in W
f=50           # frequency in HZ
Vl=6600        # line voltage in V
pf=1           # unity power factor
P=6            # number of poles
I=10           # rated field current in A
Xs=2.8         # reactance of the windings in ohm
Rs=0           # resistance of the windings in ohm
Rd=0.1         # Dc link inductor resistance
alpha=140      # constant firing angle in degrees

#calculation
N=120*f/P                   #synchronous speed
V=Vl/math.sqrt(3)           #phase voltage
Is=Pm/(math.sqrt(3)*Vl*pf)  #rated current

Id=math.pi/math.sqrt(6)*Is  #Dc line current
phi=180-alpha  #phase angle between Is and V in degrees

#(i) when motor operates at rated current and 500rpm
N1=500     #motor speed in rpm
f1=N1/N*f  #frequency at N1
V1=f1/f*V  #voltge at N1
Pm1=3*V1*Is*math.cos(math.radians(phi)) #power developed by the motor
#for the 3-phase load commutated inverter
Vds=-Vdl+Id*Rd
cos_alpha_s=Vds/(3*math.sqrt(6)/math.pi*V)
alpha_s1=math.degrees(alpha_s)   #in degrees

#(ii) regenerative braking at 500rpm and at rated motor current
alpha=0  #firing angle
#when firng angle is zero then power factor is unity
pf=1

Pm2=3*V1*Is*pf   #power developed by the motor
Ps=Pm2-Id**2*Rd  #power supplied to the source
Vds=-Vdl+Id*Rd
cos_alpha_s=Vds/(3*math.sqrt(6)/math.pi*V)