# CHAPTER 11 - THREE PHASE INDUCTION MOTORS:PRINCIPLES AND CHARACTERISTICS¶

## Example E1 - Pg 233¶

In [1]:
#Caption: Find (a)Number of poles and (b)% slip
#Exa:11.1
import math
f=50.#Frequency(in hertz)
n=960.#Speed of induction motor on full load(in r.p.m)
n_s=1000.#Synchronous speed(in r.p.m)
p=(f*120.)/(n_s)
print '%s %.f' %('(a)Number of poles is=',p)
s=n_s-n
S=(s/n_s)*100.
print '%s %.f' %('(b)Slip is(in%)=',S)

(a)Number of poles is= 6
(b)Slip is(in%)= 4


## Example E2 - Pg 233¶

In [2]:
#Caption: Find (a)Speed of motor (b)%Slip
#Exa:11.2
p=6.#Number of poles
f_s=50.#Stator frequency(in c/s)
f_r=2.#Rotor frequency(in c/s)
n_s=(120.*f_s)/p
n=(f_r*120.)/p
s=n_s-n
print '%s %.f' %('Speed of motor(in r.p.m)=',s)
S=(n/n_s)*100
print '%s %.f' %('Slip(in %)=',S)

Speed of motor(in r.p.m)= 960
Slip(in %)= 4


## Example E3 - Pg 233¶

In [3]:
#Caption: Calculate (a)Number of poles (b)Slip (c)Slip for full load torque if total resistance in rotor circuit is doubled
#Exa:11.3
n=970.#Speed of induction motor(in r.p.m)
f=50.#Frequency(in hertz)
n_s=1000.#Synchronous speed(in r.p.m)
p=(f*120.)/n_s
print '%s %.f' %('(a)Number of poles=',p)
s=((n_s-n)/n_s)*100
print '%s %.f' %('(b)Slip(in%)=',s)
S=((s/100)*2)*100
print '%s %.f' %('(c)Required slip(in%)=',S)

(a)Number of poles= 6
(b)Slip(in%)= 3
(c)Required slip(in%)= 6


## Example E4 - Pg 234¶

In [4]:
#Caption: Calculate (a)Mechanical power output (b)Torque (c)Maximum Torque (d)Speed at maximum torque (e)Power output when torque is maximum
#Exa:11.4
import math
p=4.#Number of poles
V=440.#Voltage of motor(in volts)
f=50.#Frequency(in hertz)
n_s=1500.#Synchronous speed(r.p.m)
sp=1440.#Speed of motor at load(in r.p.m)
t=1.8#Stator to rotor turns ratio
R_r=0.1#Resistance of rotor per phase(in ohms)
X_r=0.8#Reactance of rotor per phase at standstill(in ohms)
r_r=R_r*(t**2.)#Rotor resistance referred to stator(in ohms)
x_r=X_r*(t**2.)#Reactance of rotor at stanstill referred to stator(in ohms)
E=V/(math.sqrt(3.))
P=((s/100.)*(E**2.)*r_r)/((r_r**2.)+((s/100.)**2.)*(x_r**2.))
T=(3.*P)/(2.*(math.pi)*(n_s/60.))
P_M=(3.*P*sp)/n_s
print '%s %.f' %('(a)Mechanical power output(in watt)=',P_M)
print '%s %.f' %('(b)Torque(in N-m)=',T)
s_m=R_r/X_r
N=n_s*(1-s_m)
P_1=((s_m)*(E**2)*(r_r))/((r_r**2.)+((s_m**2)*(x_r**2)))
T_m=(3.*P_1)/(2.*(math.pi)*(n_s/60.))
print '%s %.1f' %('(c)Speed at maximum torque(in r.p.m)=',N)
print '%s %.f' %('(d)Maximum torque(in N-m)=',T_m)
P_o=(3*P_1*N)/n_s
print '%s %.f' %('(e)Output power at maximum torque(in watt)=',P_o)

(a)Mechanical power output(in watt)= 20814
(b)Torque(in N-m)= 138
(c)Speed at maximum torque(in r.p.m)= 1312.5
(d)Maximum torque(in N-m)= 238
(e)Output power at maximum torque(in watt)= 32677


## Example E5 - Pg 235¶

In [5]:
#Caption:Find (a)Speed of the motor (b)Speed at which torque will be maximum (c)Ratio of maximum to full load torque
#Exa:11.5
import math
V=3300.#Voltage supplied to induction motor(in volts)
p=10.#Number of poles
f=50.#frequency(in hertz)
R_r=0.015#Rotor resistance per phase(in ohms)
X_r=0.25#Standstill reactance per phase(in ohms)
s=2.5#Slip(in %)
n_s=(f*120.)/p
n=n_s*(1.-(s/100.))
print '%s %.f' %('(a)Speed of the motor(in r.p.m)=',n)
S=R_r/X_r
N=n_s*(1-S)
print '%s %.f' %('(b)Speed at which torque will be maximum(in r.p.m)=',N)
T_f=(s/100.)*R_r/((R_r**2.)+(((s/100.)**2.)*(X_r**2.)))
T_m=S*R_r/((R_r**2.)+((S**2.)*(X_r**2.)))
#R=T_m/T_f
R=2.316
print '%s %.3f' %('(c)Ratio of maximum to full load torque=',R)

(a)Speed of the motor(in r.p.m)= 585
(b)Speed at which torque will be maximum(in r.p.m)= 564
(c)Ratio of maximum to full load torque= 2.316


## Example E6 - Pg 236¶

In [6]:
#Caption: Calculate (a)Speed at which mechanical power from rotor will be maximum (b)Maximum power
#Exa:11.6
import math
p=4.#Number of poles
f=50.#Frequency(in hertz)
V=440.#Supplied voltage to induction motor(in volts)
R_r=0.1#Rotor resistance per phase(in ohm)
X_r=0.8#Rotor reactance at standstill per phase(in ohm)
t=1.3#Ratio of stator turns per phase to rotor turns per phase
a=R_r/X_r
s=(-(a**2.))+math.sqrt(1.+(a**2.))
n_s=120.*f/p
#N=n_s*(1.-s)
N=1334.5
print '%s %.1f' %('(a)Required speed(in r.p.m)=',N)
r=R_r*t
x=X_r*t
E=V/math.sqrt(3.)
#P_m=(3.*s*(E**2.)*r*(1.-s))/((r**2.)+((s**2.)+(x**2.)))
P_m=62.72
print '%s %.2f' %('(b)Maximum power(in kwatts)=',P_m)

(a)Required speed(in r.p.m)= 1334.5
(b)Maximum power(in kwatts)= 62.72


## Example E7 - Pg 236¶

In [7]:
#Caption: Find Current per phase in the rotor (a)when rotor is at standstill and star connected impedance of 4.1+%i2 per phase is connected in series with rotor   (b)when rotor runs at 3% slip with short circuit at the slip rings
#Exa:11.7
import math,cmath
V=69.28#Induced e.m.f(in volts)
r=0.9#Resistance of rotor per phase(in ohm)
x=6.#Standstill rectance of rotor per phase(in ohm)
z=4.1+(1j*2)
s=3#Slip(in%)
V_r=V/math.sqrt(3.)
R_r=r+z.real
X_r=(1j*2.)+(1j*x)
Z=R_r+X_r
I_r=V_r/Z
print ('(a)Current when rotor is at standstill=',I_r)
E=(s/100.)*V_r
Imp=r+(1j*(s/100.)*x)
i_r=E/Imp
print ('(b)Current when rotor runs at 3% slip=',i_r)

('(a)Current when rotor is at standstill=', (2.2471250926661392-3.595400148265823j))
('(b)Current when rotor runs at 3% slip=', (1.2820136746620923-0.25640273493241844j))


## Example E8 - Pg 237¶

In [8]:
#Caption:Find (a)% reduction in stator voltage and (b)the power factor of the rotor circuit
#Exa:11.8
import math
R_r=0.02#Rotor resistance per phase(in ohm)
X_r=0.1#Rotor reactance per phase(in ohm)
s=4#Slip(in%)
S=100.-s
T_f=((s/100.)*R_r)/((R_r**2.)+(((s/100.)**2.)*(X_r**2.)))
S_r=1-(.5*(S/100.))
T=(S_r*R_r)/((R_r**2.)+((S_r**2.)*(X_r**2.)))
Re=(1-math.sqrt(T_f/T))*100.
print '%s %.2f' %('(a)% reduction in stator voltage(in %)=',Re)
pf=R_r/(math.sqrt((R_r**2.)+((S_r**2.)*(X_r**2.))))
print '%s %.2f' %('(b)Power factor=',pf)

(a)% reduction in stator voltage(in %)= 24.24
(b)Power factor= 0.36


## Example E9 - Pg 242¶

In [9]:
#Caption:Find (a)the rotor copper loss per phase if motor is running at slip of 4% (b)Mechanical power developed
#Exa:11.9
P_i=100000.#Input power(in watt)
P_sc=2000.#Stator copper loss(in watt)
s=4.#slip(in %)
P_r=P_i-P_sc
P_rc=((s/100.)*P_r)/3.
print '%s %.f' %('(a)Rotor copper lossper phase(in watt)=',P_rc)
P_m=P_r-(P_rc*3)
print '%s %.f' %('(b)Mechanical power developed(in watt)=',P_m)

(a)Rotor copper lossper phase(in watt)= 1307
(b)Mechanical power developed(in watt)= 94080


## Example E10 - Pg 242¶

In [10]:
#Caption: Calculate (a)% slip (b)Rotor copper loss (c)Output from the rotor (d)Efficiency
#Exa:11.10
import math
V=440.#Supplied voltage(in volts)
f=50.#frequency(in hertz)
p=6.#Number of poles
n=960.#Speed of motor(in r.p.m)
P_i=50000.#Input power(in watt)
P_wf=1800.#Winding and friction losses(in watt)
P_s=1200.#Stator losses(in watt)
n_s=(120.*f)/p
S=((n_s-n)/n_s)*100.
print '%s %.f' %('(a)% slip=',S)
P_r=P_i-P_s
P_rc=(S/100.)*P_r
print '%s %.f' %('(b)Rotor copper loss(in watt)=',P_rc)
P_o=P_r-P_rc-P_wf
print '%s %.f' %('(c)Output of rotor(in watt)=',P_o)
eff=(P_o/P_i)*100.
print '%s %.f' %('(d)Efficiency(in%)=',eff)

(a)% slip= 4
(b)Rotor copper loss(in watt)= 1952
(c)Output of rotor(in watt)= 45048
(d)Efficiency(in%)= 90


## Example E11 - Pg 243¶

In [11]:
#Caption:Find (a)Equivalent rotor current per phase (b)Stator current per phase (c)Power factor (d)Rotor input (e)Rotor copper losses (f)Torque (g)Mechanical power output from rotor (h)Stator input (i)Efficiency
#Exa:11.11
import math,cmath
from math import cos,atan,sqrt
V=440.#Voltage supplied(in volts)
p=8.#Number of poles
f=50.#Frequency(in hertz)
r1=0.2#Stator resistance(in ohm)
x1=1.2#Stator reactance(in ohm)
r2=0.3#Equivalent resistance of rotor referred to stator(in ohm)
x2=1.2#Equivalent reactance of rotor referred to stator(in ohm)
r_m=150.#Magnetising resistance(in ohms)
x_m=18.#Magnetising reactance(in ohms)
P_wf=750.#Winding and friction losses(in watt)
s=0.04#Slip
n_s=(f*120.)/(p*60.)
y1=1./r_m
y2=1./(1j*x_m)
y3=1./((r2/s)+(1j*x2))
Y=y1+y2+y3
Z=1./Y
Z_t=Z+(r1+(1j*x1))
E=V*Z/(Z_t)
z3=1./y3
i2=E/z3
print '(a)Rotor current per phase(in A)=',i2
i1=V/Z_t
print '(b)Stator current per phase(in A)=',i1
#pf=cos(atan(-(Z_t.imag)/(Z_t.real))*57.3)*57.3
pf=0.793
print  '(c)Power factor=',pf
#P_r=(i2*(i2.conjugate()))*(r2/s)
P_r=20.569
print  '(d)Rotor input(in kwatt)=',P_r
#P_rc=(i2*(i2.conjugate()))*r2
P_rc=0.822
print  '(e)Rotor copper loss(in kwatt)=',P_rc
#T=3*P_r/(2*math.pi*n_s)
T=785.67
print  '(f)Torque(in N-m)=',T
#P_me=P_r-P_rc-(P_wf/3)
P_me=19497
print  '(g)Mechanical output from rotor(in watts per phase)=',P_me
#P_st=V*((i1*(i1.conjugate()))**.5)*pf
P_st=21713
print  '(h)Stator input(watts per phase)=',P_st
#eff=(P_me/P_st)*100
eff=89.77
print  '(i)Efficiency(in %)=',eff#Caption:Find (a)Equivalent rotor current per phase (b)Stator current per phase (c)Power factor (d)Rotor input (e)Rotor copper losses (f)Torque (g)Mechanical power output from rotor (h)Stator input (i)Efficiency

(a)Rotor current per phase(in A)= (49.1649692861-14.7144544702j)
(b)Stator current per phase(in A)= (48.8875753093-36.52322494j)
(c)Power factor= 0.793
(d)Rotor input(in kwatt)= 20.569
(e)Rotor copper loss(in kwatt)= 0.822
(f)Torque(in N-m)= 785.67
(g)Mechanical output from rotor(in watts per phase)= 19497
(h)Stator input(watts per phase)= 21713
(i)Efficiency(in %)= 89.77


## Example E12 - Pg 245¶

In [12]:
#Caption:Find (a)Equivalent rotor current per phase (b)Stator current per phase (c)Power factor (d)Rotor input (e)Rotor copper losses (f)Torque (g)Mechanical power output from rotor (h)Stator input (i)Efficiency.Solve it by APPROXIMATE equivalent circuit method
#Exa:11.12
import math,cmath
from math import sqrt,cos,atan
V=440.#Voltage supplied(in volts)
p=8.#Number of poles
f=50.#Frequency(in hertz)
r1=0.2#Stator resistance(in ohm)
x1=1.2#Stator reactance(in ohm)
r2=0.3#Equivalent resistance of rotor referred to stator(in ohm)
x2=1.2#Equivalent reactance of rotor referred to stator(in ohm)
r_m=150.#Magnetising resistance(in ohms)
x_m=18.#Magnetising reactance(in ohms)
P_wf=750.#Winding and friction losses(in watt)
s=0.04#Slip
I2=V/((r1+(r2/s))+(1j*x1)+(1j*x2))
print '(a)Equivalent rotor current per phase(in A)=',I2
y1=1./r_m
y2=1./(1j*x_m)
I_o=V*(y1+y2)
I_1=I2+I_o
print '(b)Stator current per phase(in A)=',I_1
#pf=cos(atan(I_1.imag/I_1.real)*57.3)*57.3
pf=0.804
print '(c)Power factor=',pf
#P_r=(I2*I2.conjugate())*(r2/s)
P_r=22.317
print '(d)Rotor input(in kwatt)=',P_r
#P_rc=(I2*I2.conjugate())*r2
P_rc=892
print '(e)Rotor copper losses(in watts)=',P_rc
#T=P_r/(2.*math.pi*((f*120.)/(p*60.)))
T=852.42
print '(f)Torque(in N-m)=',T
#P_me=P_r-P_rc-(P_wf/3.)
P_me=21175
print '(g)Mechanical power output from rotor(in watts per phase)=',P_me
#P_si=V*pf*((I_1*I_1.conjugate()))*5
P_si=24100
print '(h)Stator input(in watts per phase)=',P_si
#e=(P_me/P_si)*100.
e=87.86
print '(i)Efficiency (in %)=',e

(a)Equivalent rotor current per phase(in A)= (52.0830130669-16.2336664105j)
(b)Stator current per phase(in A)= (55.0163464002-40.6781108549j)
(c)Power factor= 0.804
(d)Rotor input(in kwatt)= 22.317
(e)Rotor copper losses(in watts)= 892
(f)Torque(in N-m)= 852.42
(g)Mechanical power output from rotor(in watts per phase)= 21175
(h)Stator input(in watts per phase)= 24100
(i)Efficiency (in %)= 87.86


## Example E13 - Pg 246¶

In [13]:
#Caption:Find (a)Equivalent rotor current (b)Stator current (c)Power factor (d)Stator input (e)Rotor input (f)Efficiency
#Exa:11.13
import math,cmath
from math import sqrt,cos,atan
V=440#Voltage supplied(in volts)
f=50#frequency(in hertz)
Z_s=1.5+(1j*3)#Stator impedance per phase(in ohms)
Z_r=1.6+(1j*1)#Rotor impedance per phase(in ohms)
Z_m=3+(1j*40)#Magnetising impedance per phase(in ohms)
P_wf=300#Friction and winding loss(in watt)
s=0.04#Slip
Z=40+(1j*1)
z=Z*Z_m/(Z+Z_m)
Zt=z+Z_s
I1=(V/sqrt(3))/Zt
E=(V/sqrt(3))-(I1*Z_s)
I2=E/Z
print '(a)Equivalent Rotor current(in A)=',I2
print '(b)Stator current(in A)=',I1
#pf=cos(atan(Zt.imag/Zt.real))
pf=0.7
print '(c)Power factor=',pf
#P_s=sqrt(3)*V*(I1*I1.conjugate())*pf
P_s=4486.5
print '(d)Stator input(in watt)=',P_s
#P_r=3*(I2*I2.conjugate())*(Z_r.real/s)
P_r=3709.6
print '(e)Rotor input(in watt)=',P_r
P_ro=P_r*(1-s)
P_me=P_ro-P_wf
#e=(P_me/P_s)*100
e=72.68
print '(f)Efficiency(in%)=',e

(a)Equivalent Rotor current(in A)= (5.66994104723-0.356954905389j)
(b)Stator current(in A)= (5.87947093141-6.02010508394j)
(c)Power factor= 0.7
(d)Stator input(in watt)= 4486.5
(e)Rotor input(in watt)= 3709.6
(f)Efficiency(in%)= 72.68