Chapter01 - Poly phase ac machines

Example 1.1 Page 24

In [99]:
from __future__ import division
# Given data
Zinner=0.01+0.5*1J #Impedence at standstill of inner cage in ohm
Zouter=0.05+0.1*1J #Impedence at standstill of outer cage in ohm

#Calculations
#Part (a) : at starting
R1=Zinner.real #in ohm
R2=Zouter.real #in ohm
X1=Zinner.imag #in ohm
X2=Zouter.imag #in ohm
#Formula : Ts=3/ws*V_dash**2*R2/(R2**2+X2**2)
TsoBYTsi=(R2/(R2**2+X2**2))/(R1/(R1**2+X1**2))
print "Part(a) Ratio of Torque : " ,round(TsoBYTsi)
#Part(b) : slip =5%
S=5/100 #slip
#Formula : T=3/ws*V_dash**2*(R2/S)**2/((R2/S)+X2**2)
ToBYTi=((R2/S)/((R2/S)**2+X2**2))/((R1/S)/((R1/S)**2+X1**2))
print "Part(b) Ratio of Torque : " ,round(ToBYTi,3)

Part(a) Ratio of Torque :  100.0
Part(b) Ratio of Torque :  1.436

Example 1.1 Page: 41

In [100]:
from __future__ import division
# Given data
P=4 #No. of poles
f=50 #in Hz
N=1410 #in rpm

#Calculations
Ns=120*f/P #in r
print "Synchronous speed = %0.f rpm" %Ns
S=(Ns-N)/Ns #Full load slip
S=S*100 #in %
print "Full load slip = %0.f %%  "%S 

Synchronous speed = 1500 rpm
Full load slip = 6 %

Example 1.2 Page 42

In [101]:
# Given data
P=2 #No. of poles
f=50 #in Hz
S=2 #in %#Calculations
S=S/100 #unitless
Ns=120*f/P #in rpm
N=Ns*(1-S)
print "Speed of motor = %0.f rpm " %N

Speed of motor = 2940 rpm

Example 1.3 Page 42

In [102]:
# Given data
P=4 #No. of poles
f=50 #in Hz
N=1470 #in rpm

#Calculations
Ns=120*f/P #in rpm
S=(Ns-N)/Ns #Slip
fr=S*f #induced emf frequency in Hz
print "Induced emf frequency = %0.2f Hz" %fr

Induced emf frequency = 1.00 Hz

Example 1.4 Page 42

In [103]:
import math as mh
# Given data
P=4 #No. of poles
f=50 #in Hz
K=1/2 #rotor to stator turns
N=1455 #in rpm
E1_line=415 #in volt#Calculations
Ns=120*f/P #in rpm
S=(Ns-N)/Ns #Slip
fr=S*f #induced emf frequency in Hz
print "(i) Frequency of rotor emf in running condition = %0.2f Hz" %fr
N2BYN1=K #rotor to stator turns
N1BYN2=1/K #stator to rotor turns
E1ph=E1_line/mh.sqrt(3) #
#Formula : E2ph/E1ph=K
E2ph=E1ph*K #in volt
print "(ii) Rotor induced emf at standstill = %0.2f V"%E2ph
E2r=S*E2ph #in volt
print "(iii) Rotor induced emf at running condition = %0.3f V" %E2r

(i) Frequency of rotor emf in running condition = 1.50 Hz
(ii) Rotor induced emf at standstill = 119.80 V
(iii) Rotor induced emf at running condition = 3.594 V

Example 1.5 Page 43

In [104]:
import math as mh
# Given data
P=4 #No. of poles
f=50 #in Hz
R2=0.2 #in ohm
X2=1 #in ohm
N=1440 #in rpm
E2_line=120 #in volt#Calculations
E2ph=E2_line/mh.sqrt(3) #
cosfi_2=R2/mh.sqrt(R2**2+X2**2) #lagging power factor
I2=E2ph/mh.sqrt(R2**2+X2**2) #in Ampere/phase
print "(i) Rotor power factor = %0.3f lagging" %cosfi_2
print "(i) Rotor Current = %0.2f Ampere per phase" %I2
Ns=120*f/P #in rpm
S=(Ns-N)/Ns #Slip
cosfi_2r=R2/mh.sqrt(R2**2+(S*X2)**2) #lagging power factor
I2r=S*E2ph/mh.sqrt(R2**2+(S*X2)**2) #in Ampere
print "(ii) Rotor power factor = %0.4f lagging" %cosfi_2r
print "(ii) Rotor Current = %0.3f Ampere" %I2r

(i) Rotor power factor = 0.196 lagging
(i) Rotor Current = 67.94 Ampere per phase
(ii) Rotor power factor = 0.9806 lagging
(ii) Rotor Current = 13.587 Ampere

Example 1.6 Page 44

In [105]:
import math as mh
# Given data
P=4 #No. of poles
f=50 #in Hz
R2=0.1 #in ohm
X2=1 #in ohm
N=1440 #in rpm
E1_line=400 #in volt
Kdash=2 #stator turns by rotor turns

#Calculations
K=1/Kdash #rotor turns by stator turns
Ns=120*f/P #in rpm
E1ph=E1_line/mh.sqrt(3) #
#Formula : E2ph/E1ph=K
E2ph=E1ph*K #in volt
S=(Ns-N)/Ns #Slip
ns=Ns/60 #synchronous speed in rps
T=3/(2*mh.pi*ns)*(S*E2ph**2*R2)/(R2**2+(S*X2)**2) #in N-m
print "Torque devloped on full load = %0.2f N-m "%T 

Torque devloped on full load = 87.81 N-m

Example 1.7 Page 45

In [106]:
from math import sqrt, pi
# Given data
P=4 #No. of poles
f=50 #in Hz
Kdash=4 #stator turns by rotor turn
R2=0.01 #in ohm
X2=0.1 #in ohm
E1_line=400 #in volt

#Calculations
K=1/Kdash #rotor turns by stator turns
Ns=120*f/P #in rpm
E1ph=E1_line/sqrt(3) #
#Formula : E2ph/E1ph=K
E2ph=E1ph*K #in volt
#(i) at start S=1
ns=Ns/60 #in rps
K=3/2/pi/ns 
Tst=K*E2ph**2*R2/(R2**2+X2**2) #in N-m
print "(i) Starting Torque = %0.2f N-m" %Tst
#part (ii) 
Sm=R2/X2 #slip for max torque
print "(ii) Slip at which max torque devloped = %0.2f %% "%(Sm*100) 
#Part (iii)
N=Ns*(1-Sm) #in rpm
print "(iii) Speed at which max torque occur = %0.2f rpm" %N
#Part (iv)
Tm=K*E2ph**2/2/X2 #in N-m
print "(iv) Maximum  torque = %0.2f N-m" %Tm
#Part (v)
Sf=4 #in %
Sf=Sf/100 #slip
Tfl=K*Sf*E2ph**2*R2/(R2**2+(Sf*X2)**2) #in N-m
print "(v) Full load Torque devloped = %0.2f N-m  "%Tfl 

(i) Starting Torque = 63.03 N-m
(ii) Slip at which max torque devloped = 10.00 % 
(iii) Speed at which max torque occur = 1350.00 rpm
(iv) Maximum  torque = 318.31 N-m
(v) Full load Torque devloped = 219.52 N-m

Example 1.8 Page 46

In [107]:
# Given data
P=24 #No. of poles
f=50 #in Hz
R2=0.016 #in ohm
X2=0.265 #in ohm
N=247 #in rpm

#Calculations
Ns=120*f/P #in rpm
Sf=(Ns-N)/Ns #full load slip
Sm=R2/X2 #max slip
Tfl_BY_Tm=2*Sm*Sf/(Sm**2+Sf**2) #unitless
print "Ratio of full load torque to max torque : " ,round(Tfl_BY_Tm,4)
Tst_BY_Tm=2*Sm/(1+Sm**2) #unitless
print "Ratio of starting torque to max torque : " ,round(Tst_BY_Tm,4)

Ratio of full load torque to max torque :  0.3824
Ratio of starting torque to max torque :  0.1203

Example 1.9 Page 47

In [108]:
# Given data
R2=0.04 #in ohm
X2=0.2 #in ohm

#Calculations
R2dash=X2 #in ohm (for Tm=Tst)
#formula : R2dash=R2+rex
Rex=R2dash-R2 #in ohm/phase
print "(i) External resistance required = %0.2f ohm/phase"%Rex 
from sympy import symbols, solve
k, E2, R2dash = symbols('k, E2, R2dash')
# For Tst = 1/2*Tm
Tm = k*E2**2/2/X2
Tst = k*E2**2*R2/(R2**2+X2**2)
#R2dash = R2+Rex
##Putting R2 dash in place of R2
Tst = k*E2**2*R2dash/(R2dash**2+X2**2)
#Equating Tst = 1/2*Tm gives
expr = R2dash**2-4*X2*R2dash+X2**2
R2dash = solve(expr, R2dash)
R2dash = R2dash[0] #ohm  as R2 can't >X2
Rex = R2dash-R2
print "(ii) External resistance required = %0.4f ohm/phase" %Rex

(i) External resistance required = 0.16 ohm/phase
(ii) External resistance required = 0.0136 ohm/phase

Example 1.10 Page 48

In [109]:
# Given data
f=50 #in Hz
P=8 #no. of poles
Tsh=190 #in N-m
fr=1.5 #in Hz
MechLoss=700 #in watts

#Calculations
S=fr/f #Slip
Ns=120*f/P #in rpm
N=Ns*(1-S) #in rpm
Pout=Tsh*2*pi*N/60 #in watts
Pm=Pout+MechLoss #in watts
#formula-:  P2:Pc:Pm=1:S:1-S
Pc=Pm*S/(1-S) #in watts
print "Rotor Copper loss = %0.3f W" %Pc

Rotor Copper loss = 469.326 W

Example 1.11 Page 49

In [110]:
# Given data
f=50 #in Hz
P=4 #no. of poles
Pin=50 #in kW
N=1440 #in rpm
StatorLoss=1000 #in watts
FrictionalLoss=650 #in watts

#Calculations
Ns=120*f/P #in rpm
S=(Ns-N)/Ns #Slip

N=Ns*(1-S) #in rpm
P2=Pin-StatorLoss/1000 #in KW
#formula-:  P2:Pc:Pm=1:S:1-S
Pc=S*P2 #in KW
Pm=P2-Pc #in KW
Pout=Pm-FrictionalLoss/1000 #in KW
Eff=Pout/Pin*100 #in %
print "Full load efficiency = %0.2f %%" %Eff

Full load efficiency = 92.78 %

Example 1.12 Page 50

In [111]:
# Given data
f=50 #in Hz
phase=3 #no. of phase
P=4 #no. of poles
Tsh=300 #in N-m
Tlost=50 #in N-m
fr=120 #in cycles/min
fr=fr/60 #in Hz
I2r=60 #in Ampere/phase

#Calculations
S=fr/f #slip
print "(i) Slip = %0.2f %%" %(S*100)
Ns=120*f/P #in rpm
N=Ns*(1-S) #in rpm
Pout=Tsh*2*pi*N/60 #watts
print "(ii) Net output Power = %0.3f kW" %(Pout/1000)
FricLoss=Tlost*2*pi*N/60 #in watts
Pm=Pout+FricLoss #in watts
#formula-:  P2:Pc:Pm=1:S:1-S
Pc=S*Pm/(1-S) #copper loss in Watts
PcPERphase=Pc/phase #Copper loss per phase in watts
print "(iii) Rotor copper loss per phase = %0.4f W"%PcPERphase 
P2=Pc/S #in watts
Eff=Pm/P2*100 #in %
print "(iv) Rotor efficiency = %0.2f %% "%Eff 
#Formula : CuLossPerPhase=I2r**2*R2 #in watts
R2=PcPERphase/I2r**2 #in ohm/phase
print "(v) Rotor resistance per phase = %0.4f W/phase "%R2 

(i) Slip = 4.00 %
(ii) Net output Power = 45.239 kW
(iii) Rotor copper loss per phase = 733.0383 W
(iv) Rotor efficiency = 96.00 % 
(v) Rotor resistance per phase = 0.2036 W/phase

Example 1.13 Page 51

In [112]:
# Given data
Pout=25 #in KW
f=50 #in Hz
phase=3 #no. of phase
P=4 #no. of poles
N=1410 #in rpm
MechLoss=850 #in watts
StatLossBYCuLoss=1.17 
I2r=65 #in Ampere

#Calculations
Ns=120*f/P #in rpm
S=(Ns-N)/Ns #slip
Pm=Pout*1000+MechLoss #in watts
print "Gross mechanical power devloped = %0.2f W" %Pm
#formula-:  P2:Pc:Pm=1:S:1-S
Pc=S*Pm/(1-S) #copper loss in Watts
print "Rotor Copper Losses = %0.2f W"%Pc
R2=Pc/phase/I2r**2 #in ohm/phase
print "Rotor resistance per phase = %0.2f ohm" %R2
StatorLoss=1.7*Pc #in watts
P2=Pc/S #in Watts
Pin=P2+StatorLoss #in watts
Eff=Pout*1000/Pin*100 #in %
print "Full laod Efficiency = %0.2f %%" %Eff

Gross mechanical power devloped = 25850.00 W
Rotor Copper Losses = 1650.00 W
Rotor resistance per phase = 0.13 ohm
Full laod Efficiency = 82.49 %

Example 1.14 Page 52

In [113]:
# Given data
Pout=24 #in KW
P=8 #no. of poles
N=720 #in rpm
VL=415 #in volt
IL=57 #in Ampere
f=50 #in Hz
phase=3 #no. of phase
cosfi=0.707 #power factor
MechLoss=1000 #in watts
Rs=0.1 #in ohm/phase

#Calculations
Ns=120*f/P #in rpm
S=(Ns-N)/Ns #slip
Pm=Pout*1000+MechLoss #in watts
#formula-:  P2:Pc:Pm=1:S:1-S
#Pc=S*Pm/(1-S) #copper loss in Watts
Tsh=Pout*10**3/(2*pi*N/60) #in N-m
print "Shaft Torque = %0.3f N-m" %Tsh
T=Pm/((2*pi*N/60)) #in N-m
print "Gross torque devloped = %0.3f N-m"%T
Pc=S*Pm/(1-S) #copper loss in Watts
print "Rotor Cu losses = %0.2f W" %Pc
P2=Pc/S #in watts
Pin=sqrt(3)*VL*IL*cosfi #in watts
Is=IL #stator current per phase in Ampere
StatorCuLoss=3*Is**2*Rs #in watts
print "Stator Copper losses = %0.2f W" %StatorCuLoss
StatorLosses=Pin-P2 #in watts
StatorIronLoss=StatorLosses-StatorCuLoss #in watts
print "Stator Iron losses = %0.2f W" %StatorIronLoss
Eff=Pout*10**3/Pin*100 #in %
print "Efficiency = %0.2f %%" %Eff

Shaft Torque = 318.310 N-m
Gross torque devloped = 331.573 N-m
Rotor Cu losses = 1041.67 W
Stator Copper losses = 974.70 W
Stator Iron losses = 1950.60 W
Efficiency = 82.85 %

Example 1.15 Page 54

In [114]:
from __future__ import division
import cmath
from numpy import array, sqrt, pi
from math import degrees, cos
# Given data
Poles=12 #no. of poles
V1=420 #in volt
f=50 #in Hz
r1=2.95 #in watts
x1=6.82 #in watts
r2dash=2.08 #in watts
x2dash=4.11 #in ohm/phase
ImLine=6.7 #in Ampere
TcoreLoss=269 #in watts
S=3 #slip in %

#Calculations
S=S/100 #slip
Im=ImLine/sqrt(3) #in Ampere
Im_bar=Im*cmath.exp(1J*(-pi/2))*(r1+1J*x1) #in Ampere


from sympy import symbols, solve
E1 = symbols('E1')
V1 = E1+Im_bar
#V1=(E1+real(Im_bar))+imag(Im_bar) 
#Equating magnitude of both sides gives a polynomial for E1
V1=420 # Volt
poly = (E1+Im_bar.real)**2+Im_bar.imag**2-V1**2
E1 = solve(poly, E1)
E1 = E1[1] # V # #discarding -ve value
Xo=E1/Im #in ohm

#Zeq=Xo*exp(1J*(pi/2))*(r2dash/S)/(1J*Xo+1J*x2dash+r2dash/S) 
Zeq = complex(0,Xo)*complex(r2dash/S,x2dash)/(complex(0,Xo)+complex(r2dash/S,x2dash))
Zin=r1+1J*x1+Zeq #in ohm
I1=V1/Zin #in Ampere
print "Input current : Magnitude =",round(abs(I1),3)," & phase =",round(degrees(cmath.phase(I1)),2) ,"degree"
# Note : In the textbook +ve sign of phase is shown by mistake.
cosfi=cos(round(degrees(cmath.phase(I1)),2)*pi/180) #lagging power factor
print "Power factor =",round(cosfi,4),"lagging" 
I2r_dash=I1*complex(0,Xo)/complex(r2dash,(Xo+x2dash)) #in Ampere
print "Rotor current : Magnitude =",round(abs(I2r_dash),3)," & phase =",round(degrees(cmath.phase(I2r_dash)),2) ,"degree"
Ns=120*f/Poles #in rpm
T=9.55*3*I2r_dash.real**2*r2dash/S/Ns #in N-m
print "Torque developed =",round(T,3),"N-m" 
Zth=complex(r1,x1)*complex(0,Xo)/(complex(r1,x1)+complex(0,Xo)) #in Ohm
Rth=Zth.real #in ohm
Xth=Zth.imag #in ohm
Vth=V1*complex(0,Xo)/complex(r1,(Xth+Xo)) #in Volt
Ws=(2*pi*Ns/60) #in rad/sec
Tm=(3/Ws)*0.5*(Vth.real)**2/(Rth+sqrt(Rth**2+(Xth+x2dash)**2)) #in N-m
print "Maximum torque devloped = %0.2f N-m "%(Tm) 
Sm=r2dash/sqrt(Rth**2+(Xth+x2dash)**2) #slip
Nm=Ns*(1-Sm) #
print "Speed at maximum torque = %0.1f rpm" %Nm 
#Answer for rotor equivalent Current and Torque developed is wrong in the book.

Input current : Magnitude = 6.732  & phase = -40.04 degree
Power factor = 0.7656 lagging
Rotor current : Magnitude = 6.469  & phase = -38.91 degree
Torque developed = 100.678 N-m
Maximum torque devloped = 331.11 N-m 
Speed at maximum torque = 404.4 rpm

Example 1.16 Page 57

In [115]:
from math import acos, atan
from cmath import exp
# Given data
V=440 #in volt
P=8 #no. of poles
Pout=40 #in KW
f=50 #in Hz
phase=3 #no. of phase
R1=0.1 #in ohm
X1=0.4 #in ohm
R2dash=0.15 #Equivalent rotor resistance in ohm
X2dash=0.44 #Equivalent rotor reactance in ohm
I0=20*exp(1J*-acos(0.09)) #in Ampere
N=727.5 #in rpm
MechLoss=1000 #in watts
CoreLoss=1250 #in watts

#Calculations
Ns=120*f/P #in rpm
S=(Ns-N)/Ns #slip
RLdash=R2dash*(1-S)/S
V1=V/sqrt(3) #in volt
R1e=R1+R2dash #in ohm
X1e=X1+X2dash #in ohm
I2rdash=V1/(R1e+RLdash+1J*X1e) #in Ampere
I1bar=I0+I2rdash #in Ampere
InputCurrent=abs(I1bar) #in Ampere
InputPF=cos(atan((I1bar.imag)/(I1bar.real))) #
print "(i) Input Current = %0.3f Ampere & PF = %0.4f lagging" %(InputCurrent,InputPF)
T=3*abs(I2rdash)**2*R2dash/S/(2*pi*Ns/60) #in N-m
print "(ii) Torque Developed = %0.4f N-m"%T
P2=3*abs(I2rdash)**2*R2dash/S #in Watts
#Formula : P2:Pc:Pm=1:S:1-S
Pm=(1-S)*P2 #in Watts
TotPout=Pm-MechLoss #in watts
print "(iii) Output power = %0.2f Watts" %TotPout
TotCuLoss=3*abs(I2rdash)**2*R1e #in watts
Eff=TotPout/(TotPout+TotCuLoss+CoreLoss+MechLoss)*100 #in %
print "(iv) Efficiency = %0.2f %%" %Eff

(i) Input Current = 57.518 Ampere & PF = 0.8744 lagging
(ii) Torque Developed = 461.3394 N-m
(iii) Output power = 34146.51 Watts
(iv) Efficiency = 89.37 %

Example 1.17 Page 59

In [116]:
# Given data
Z1=0.07+1J*0.4 #in ohm
Z2dash=0.08+1J*0.2 #in ohm
V1_line=200 #in volt

#Calculations
R1=Z1.real #in ohm
X1=Z1.imag #in ohm
R2dash=Z2dash.real #in ohm
X2dash=Z2dash.imag #in ohm
R1e=R1+R2dash #in ohm
X1e=X1+X2dash #in ohm
Z1e=R1e+1J*X1e #in ohm
Z1e_mag=abs(Z1e) #magnitude of Z1e in ohm
V1PerPhase=V1_line/sqrt(3) #in volt
Pout_max=3*V1PerPhase**2/2/(R1e+Z1e) #
S=R2dash/(R2dash+Z1e_mag) #
print "Slip = %0.2f %% "%(S*100)

Slip = 11.45 %

Example 1.18 Page 59

In [117]:
# Given data
P=4 #in poles
f=50 #in Hz
Pout=30 #in HP
VL=400 #in volt
Eta=0.8 #Efficiency
cosfi=0.75 #lagging power factor

#Calculations
Pout=Pout*735.5 #in Watts
Pin=Pout/Eta #in Watts
#Formula : Pin=sqrt(3)*VL*IL*cosfi
IL=Pin/sqrt(3)/VL/cosfi #in Ampere
print "Current by the mains = %0.2f A" %IL

Current by the mains = 53.08 A

Example 1.19 Page 60

In [118]:
# Given data
P=4 #in poles
Pout=37 #in HP
f=50 #in Hz
N=1425 #in rpm
MechLoss=3 #in HP
StatorLoss=2500 #in watts
VL=500 #in volt
cosfi=0.9 #power factor

#Calculations
Ns=120*f/P #in rpm
S=(Ns-N)/Ns #slip
print "(i) Slip is :  " ,S
Pout=Pout*735.5 #in Watts
MechLoss=MechLoss*735.5 #in Watts
Pin=Pout+MechLoss #in Watts
#Formula : P2:Pc:Pin=1:5:1-S
Pc=(S/(1-S))*Pin #in watts
print "(ii) Rotor Cu Loss = %0.2f W" %Pc
P2=Pc/S #in Watts
Pin=P2+StatorLoss #in watts
print "(iii) Total power input = %0.2f W" %Pin
Eta=Pout/Pin*100 #in %
print "(iv) Efficiency = %0.2f %% "%Eta 
fr=S*f #in Hz
fr=fr*60 #in cycles/min
print "(v) No. of cycles per minute : " ,fr
#Part (ii) & (iii) answer is wrong in the book.

(i) Slip is :   0.05
(ii) Rotor Cu Loss = 1548.42 W
(iii) Total power input = 33468.42 W
(iv) Efficiency = 81.31 % 
(v) No. of cycles per minute :  150.0

Example 1.20 Page 61

In [119]:
# Given data
E2line=60 #in Volt
R2=0.6 #in ohm
X2=4 #in ohm
Rx=5 #in ohm
Xx=2 #in ohm
S=4 #in %

#Calculations
E2ph=E2line/sqrt(3) #in volt
ZT=R2+1J*X2+Rx+1J*Xx #
I2=E2ph/abs(ZT) #in Ampere
print "(i) Rotor Current per phase = %0.4f A" %I2
S=S/100 #slip
Z2r=R2+1J*S*X2 #in ohm
I2r=S*E2ph/abs(Z2r) #in Ampere
print "(ii) Rotor Current per phase = %0.4f Ampere "%I2r 

(i) Rotor Current per phase = 4.2207 A
(ii) Rotor Current per phase = 2.2314 Ampere

Example 1.21 Page 62

In [120]:
# Given data
P=4 #no. of poles
f=50 #in Hz
P2=3000 #in watts

#Calculations
Ns=120*f/P #in rpm
T=P2/(2*pi*Ns/60) #in N-m
print "Torque Devloped = %0.3f N-m "%T 
T=T*(2*pi*Ns/60) #in syn. Watt
print "Torque Devloped = %0.f syn. Watt "%T 

Torque Devloped = 19.099 N-m 
Torque Devloped = 3000 syn. Watt

Example 1.22 Page 62

In [121]:
# Given data
E1Line=1000 #in volt
R2=0.01 #in ohm
X2=0.2 #in ohm
I2st=200 #in Ampere
ratio=3.6 #ratio of stator to rotor turns

#Calculations
K=1/ratio #ratio of rotor to stator turns
E1ph=E1Line/sqrt(3) #in Volt
E2ph=K*E1ph #in volt
#Let R2dash=R2+Rx
#Formula : I2st=E2ph/sqrt(R2dash**2+X2**2) 
R2dash=sqrt((E2ph/I2st)**2-X2**2)
Rx=R2dash-R2 #in ohm
print "External resistance required per phase = %0.4f ohm "%Rx 

External resistance required per phase = 0.7665 ohm

Example 1.23 Page 63

In [122]:
# Given data
VL=400 #in volt
E1Line=VL #in volt
P=4 #no. of poles
S=5 #in %
f=50 #in Hz
R2=0.15 #in ohm
X2=1 #in ohm
ratio=2 #ratio of stator to rotor turns

#Calculations
S=S/100 #slip
E1ph=E1Line/sqrt(3) #in Volt
K=1/ratio #ratio of rotor to stator turns
E2ph=K*E1ph #in volt
Ns=120*f/P #in rpm
ns=Ns/60 #in rps
T=(3/2/pi/ns)*S*E2ph**2*R2/(R2**2+(S*X2)**2) #in N-m
print "(i) Total Torque devloped = %0.2f N-m "%T 
Tm=(3/2/pi/ns)*E2ph**2/2/X2 #in N-m
print "(ii) Maximum Torque = %0.2f N-m"%Tm 
Sm=R2/X2 #maximum slip
N=Ns*(1-Sm) #in rpm
print "(iii) Speed at maximum torque = %0.2f rpm "%N 

(i) Total Torque devloped = 76.39 N-m 
(ii) Maximum Torque = 127.32 N-m
(iii) Speed at maximum torque = 1275.00 rpm

Example 1.24 Page 64

In [123]:
from math import cos, atan
# Given data
VL=400 #in volt
f=50 #in Hz
P=6 #no. of poles
Z1=0.3+1J*0.4 #in ohm
Z2dash=0.2+1j*0.4 #in ohm
X0=20 #Magnetic reactance in ohm
R0=100 #resistance for core loss in ohm
S=4 #in %
StatorLoss=2 #in KW
MechLoss=2 #in KW
#Calculations
R1=Z1.real #in ohm
R2dash=Z2dash.real #in ohm
X1=Z1.imag #in ohm
X2dash=Z2dash.imag #in ohm
S=S/100 #slip
V1=VL/sqrt(3) #in volt
Ns=120*f/P #in rpm
Ri=R2dash*(1-S)/S #in ohm
R1e=R1+R2dash #in ohm
X1e=X1+X2dash #in ohm
I2rdash=V1/(R1e+Ri+1J*X1e) #in Ampere
Ic=V1/R0 #in Ampere
Im=V1/(1J*X0) #in Ampere
I0=(Ic+Im) #in Ampere
CoreLoss=Ic**2*R0 #Core loss per phase in Watts
I1=I0+I2rdash #in Ampere
Istator=abs(I1) #in Ampere
cosfi=cos(atan(I1.imag/I1.real)*pi/180) #lagging power factor

Pc=3*abs(I2rdash)**2*R2dash #in Watts
#Here P2:P0:Pm=1:S:1-S
Pm=Pc*(1-S)/S #in watts
Pout=Pm-MechLoss*1000 #in watts
StatorCuLoss=3*abs(I1)**2*R1 #in watts
TotLoss=CoreLoss*3+StatorCuLoss+Pc+MechLoss*1000 #in watts
Eff=Pout/(Pout+TotLoss)*100 #in %
N=Ns*(1-S) #in rpm
print "(a) Motor Speed = %0.2f rpm "%N
print "(b) Stator current = %0.2f Ampere "%Istator 
print "(c) Power factor lagging : " ,round(cosfi,5)
print "(d) Motor Output = %0.2f W" %Pout
print "(e) Efficiency = %0.2f %% "%Eff 
#Answer of Pout & PF is wrong in the book.

(a) Motor Speed = 960.00 rpm 
(b) Stator current = 48.38 Ampere 
(c) Power factor lagging :  0.99998
(d) Motor Output = 24731.64 W
(e) Efficiency = 78.38 %

Example 1.25 Page 66

In [124]:
# Given data
V=440 #in volt
f=50 #in Hz
P=4 #no. of poles
X1=5.2 #in ohm
R2dash=1.2 #in ohm
X2dash=4.5 #in ohm

#Calculations
#Rth=R1 #in ohm
#Xth=X1 #in ohm
#Formula : R2dash/Sm=sqrt(X1**2+X2dash**2)
Sm=R2dash/(X1+X2dash) #Maximum Slip
I1=V/sqrt(3)/sqrt((R2dash/Sm)**2+(X1+X2dash)**2) #in Ampere
I2dash=I1 #in Ampere(Neglecting I0)
Ns=120*f/P #in rpm
Tmax=3*I2dash**2*R2dash/Sm/2/pi/Ns*60 #in N-m
print "Maximum Torque = %0.2f N-m "%Tmax 
print "Maximum Slip = %0.2f %% "%(Sm*100) 

Maximum Torque = 63.53 N-m 
Maximum Slip = 12.37 %

Example 1.26 Page 67

In [125]:
from __future__ import division
from sympy import symbols, solve
R2dash = symbols('R2dash')
# Given data
f=50 #in Hz
P=4 #no. of poles
X2=0.1 #in ohm
R2=0.02 #in ohm
#Tst=2/3*Tmax
TstByTm=2/3 #ratio
#Calculations
#Tm proportional to E2**2/(2*X2)
#formula : TstByTm=(E2**2*R2dash/(R2dash**2+X2**2))/(E2**2/(2*X2))
expr = (R2dash**2)*TstByTm-2*X2*R2dash+TstByTm*X2**2
R2dash=solve(expr, R2dash) #
R2dash=R2dash[0] #discarding higher value bcoc R2dash < X2
Rex=R2dash-R2 #in ohm
print "Extra resistance required = %0.3f ohm "%Rex 

Extra resistance required = 0.018 ohm

Example 1.27 Page 68

In [126]:
# Given data
VL=3300 #in volt
f=50 #in Hz
P=10 #no. of poles
X2=0.25 #in ohm
R2=0.015 #in ohm
Sfl=2.5 #Slip in %

#Calculations
Ns=120*f/P #in rpm
N=Ns*(1-Sfl/100) #in rpm
print "(1.) The speed of motor, N = %0.2f rpm "% N 
Sm=R2/X2 #Max Slip
Nm=Ns*(1-Sm) #Max speed in rpm
print "(2.) Speed of motor, Ns = %0.2f rpm"%Nm 
TmByTfl=Sm*R2/(R2**2+(Sm*X2)**2)*(R2**2+(Sfl/100*X2)**2)/(Sfl/100)/R2 #ratio
print "(3.) Ratio of max torque to full load torque : " ,round(TmByTfl,4)
#Answer of 1st part is wrong in the book.

(1.) The speed of motor, N = 585.00 rpm 
(2.) Speed of motor, Ns = 564.00 rpm
(3.) Ratio of max torque to full load torque :  1.4083

Example 1.28 Page 69

In [127]:
from math import sin, acos, degrees
from cmath import phase
# Given data

V0=400 #in volt
f=50 #in Hz
P=10 #no. of poles
R1=1.75 #in ohm
X1=5.5 #in ohm
R2dash=2.25 #in ohm
X2dash=6.6 #in ohm
I0=3.8 #in Ampere
W0=310 #in watts
S=4 #in %

#Calculations
S=S/100 #slip in ratio
#Formula : W0=sqrt(3)*V0*I0*cos_fi0
cos_fi0=W0/sqrt(3)/V0/I0 #power factor
sin_fi0=sin(acos(cos_fi0)*pi/180) 
Ic=I0*cos_fi0 #in Ampere
Im=I0*sin_fi0 #in Ampere
Vph=V0/sqrt(3) #in Volt
R0=Vph/Ic #in ohm
X0=Vph/Im #in ohm
Ns=120*f/P #in rpm
RLdash=R2dash*(1-S)/S #in ohm
R1e=R1+R2dash #in ohm
X1e=X1+X2dash #in ohm
I2rdash=Vph/(R1e+RLdash+1J*X1e) #in Ampere
print "Rotor Current : Magnitude =",round(abs(I2rdash),3),"A & Phase =",round(degrees(phase(I2rdash)),2),"degree"
I0_bar=Ic-1J*Im #in Ampere
I1_bar=I0_bar+I2rdash #Supply current in Ampere
print "Supply Current : Magnitude =",round(abs(I1_bar),3),"A & Phase =",round(degrees(phase(I1_bar)),2),"degree"
cosfi=cos(pi/180*atan((I1_bar.imag)/(I1_bar.real))) #Lagging power factor
print "Power factor = %0.4f lagging  "%cosfi
Pc=3*abs(I2rdash)**2*R2dash #in watts
#Formula : P2:Pc:Pm=1:S:1-S
Pm=Pc*(1-S)/S ##in watts
print "Mechanical power devloped = %0.2f N-m "%Pm 
N=Ns*(1-S) #in rpm
w=2*pi*N/60 #in rad/sec
T=Pm/w #in N-m
print "Gross load torque = %0.3f N-m  "%T 
# Supply current and power factor ans is wrong in te textbook.

Rotor Current : Magnitude = 3.898 A & Phase = -11.78 degree
Supply Current : Magnitude = 4.356 A & Phase = -11.82 degree
Power factor = 1.0000 lagging  
Mechanical power devloped = 2461.25 N-m 
Gross load torque = 40.804 N-m

Example 1.29 Page 70

In [128]:
# Given data
PA=12 #no. of poles
Ns=500 #in rpm
N=1440 #in rpm

#Calculations
#Formula : Ns=120*f/PA
f=Ns/120*PA #in Hz
PM=4 #assumed for motor
Ns=120*f/PM #in rpm(For motor)
S=(Ns-N)/Ns*100 #slip in %
print "Slip = %0.2f %% "%S 

Slip = 4.00 %

Example 1.30 Page 71

In [129]:
# Given data
R2=0.04 #in ohm
X2=0.2 #in ohm
TstByTm=50 #in %

#Calculations
Sm=1 #slip for max Torque
R2dash=Sm*X2 #in ohm
Rx=R2dash-R2 #in ohm
print "(i) External resistance required for max Torque = %0.2f ohm"%Rx 
TstByTm=TstByTm/100 #in ratio
#Formula : Tst proportional to E2**2*R2dash/(R2dash**2+X2**2)
#Formula : Tm Proportional to E2**2/2/X2
from sympy import symbols, solve
R2dash = symbols('R2dash')
expr = R2dash**2+X2**2-2*X2*R2dash/TstByTm#Polynomial for R2dash
R2dash=solve(expr, R2dash) #
R2dash=R2dash[0] #discarding higher value bcoc R2dash < X2
Rx=R2dash-R2 #in ohm
print "(ii) Extra resistance required for 50%% max Torque at start = %0.4f ohm"%Rx 

(i) External resistance required for max Torque = 0.16 ohm
(ii) Extra resistance required for 50% max Torque at start = 0.0136 ohm

Example 1.31 Page 72

In [130]:
# Given data
f=50 #in Hz
P=8 #no. of poles
Sf=40 #in %
R2=0.001 #in ohm/phase
X2=0.005 #in ohm/phase

#Calculations
Sf=Sf/100 #slip
#Formula : T proportional to S*R2/(R2**2+(S*X2)**2)
Sm=R2/X2 #slip for max Torque
TmByTfl=Sm*R2/(R2**2+(Sm*X2)**2)*(R2**2+(Sf*X2)**2)/Sf/R2 #in ratio
print "Ratio of max torque to full load torque :  " ,TmByTfl
Ns=120*f/P #in rpm
N=Ns*(1-Sm) #in rpm
print "Speed for maximum torque = %0.2f rpm "%N 

Ratio of max torque to full load torque :   1.25
Speed for maximum torque = 600.00 rpm

Example 1.32 Page 73

In [131]:
# Given data
f=50 #in Hz
P=4 #no. of poles
VL=400 #in volt
E2=100 #in volt
R2=50 #in milli ohm
X2=0.5 #in ohm

#Calculations
R2=R2*10**-3 #in ohm
Sm=R2/X2 #Maximum Slip
ns=(120*f/P)/60 #in rpS
Tmax=3/2/pi/ns*Sm*E2**2*R2/(R2**2+(Sm*X2)**2) #in N-m
print "Maximum Torque = %0.2f N-m  "%Tmax 
print "Slip at which Tmax occur : " ,Sm

Maximum Torque = 190.99 N-m  
Slip at which Tmax occur :  0.1

Example 1.33 Page 73

In [132]:
 
# Given data
#Tst=100% of Tfl #in %
#Tm=100% of Tfl #in %
TstByTfl=100/100 #ratio
TmByTfl=200/100 #ratio

#Calculations
#Formula : T proportional to S*E2**2*R2/(R2**2+(S*X2)**2)
#Formula : TstByTm=2*Sm/(Sm**2+1)
TstByTm=TstByTfl/TmByTfl #Calculating TstByTm


from sympy import symbols, solve, N
Sm, Sfl = symbols('sm, Sfl')
expr = Sm**2-4*Sm+1#Polynomial for R2dash
Sm=solve(expr, Sm) #
Sm=Sm[0] #Discarding value > 1
print "Slip at which max Torque occurs = %0.2f %%"%(Sm*100) 
#Formula : 1/TstByTm=(Sm**2+Sfl**2)/(2*Sm*Sfl)
Expr = Sm**2+Sfl**2-2*(2*Sm*Sfl)
Sfl=solve(Expr,Sfl) 
Sfl=Sfl[0] #Discarding value >= 1
print "Full load slip = ",N(Sfl*100,4) ,"%"

Slip at which max Torque occurs = 26.79 %
Full load slip =  7.180 %

Example 1.34 Page 74

In [133]:
# Given data
P=8 #no. of Poles
f=50 #in Hz
Tm=150 #in N-m
N=650 #in rpm
R2=0.6 #in ohm
S=4 #in %

#Calculations
S=S/100 #Slip
Ns=120*f/P #in rpm
Sm=(Ns-N)/Ns #Maximum Slip
#Formula : T proportional to S*E2**2*R2/(R2**2+(S*X2)**2)
X2=R2/Sm #in ohm
T=Tm*S*(R2**2+(Sm*X2)**2)/Sm/(R2**2+(S*X2)**2) #In N-m
print "Torque at 4%% slip = %0.4f N-m  "%T 

Torque at 4% slip = 82.5688 N-m

Example 1.35 Page 75

In [134]:
from math import atan, cos
# Given data
V=400 #in volts
P=4 #no. of Poles
f=50 #in Hz
r1=0.15 #in ohm
x1=0.44 #in ohm
r2dash=0.12 #in ohm
x2dash=0.44 #in ohm
xm=30 #in ohm
S=4 #in %

#Calculations
S=S/100 #Slip
RLdash=r2dash*(1-S)/S #in ohm
V1=V/sqrt(3) #in volt
I2rdash=V1/(r1+r2dash+RLdash+1J*(x1+x2dash)) #in Ampere
I0=V1/(1J*xm) #in Ampere
I1=I0+I2rdash #in Ampere
print "Stator Current in Ampere : " 
print "Magnitude is %0.3f A"%(abs(I1))," & angle = %0.2f degree"%degrees((atan((I1.imag)/(I1.real))) )
cosfi=cos(atan((I1.imag)/(I1.real))) #lagging power factor
print "Power factor = %0.4f lagging "%cosfi 

Stator Current in Ampere : 
Magnitude is 73.059 A  & angle = -21.43 degree
Power factor = 0.9308 lagging

Example 1.36 Page 76

In [135]:
# Given data
VL=440 #in volts
P=4 #no. of Poles
f=50 #in Hz
#Zleak=0.3+%i*5.5+0.25/S #in ohm/phase
K=2.5 #Stator to rotor voltage ratio
T=150 #in N-m
N=1250 #in rpm
R1 = 0.1 # ohm
X1e = 1.83 # ohm


#Calculations
Ns=120*f/P #in rpm
S=(Ns-N)/Ns #slip
Zleakage=1/3*(0.3+1J*5.5+0.25/S) #in ohm/phase
V1=VL/sqrt(3) #in volt

from sympy import symbols, solve
R2x = symbols('R2x')
#expr = R2x**2*T*S*2*pi*ns/S**2+R2x*T*S*(2*pi*ns*0.2/S+T*S*2*pi*ns*0.01+T*S*2*pi*ns*1.83**2-3*(V1**2)*R2x) #equating
#R2x**2*T*S*2*pi*ns/S**2+R2x*T*S*2*pi*ns*0.2/S+T*S*2*pi*ns*0.01+T*S*2*pi*ns*1.83**2-3*(V1**2)*R2x=0 #equating
Idash_2r_sqr = V1**2/((R1*R2x/S)**2+X1e**2)
expr = 35.856*R2x**2-48*R2x+3.3589
R2x = solve(expr, R2x)
R2x = R2x[1] # discarding lower value
ns=Ns/60 #in rps
#P=[T*S*2*pi*ns/S**2 T*S*2*pi*ns*0.2/S-3*(V1**2) T*S*2*pi*ns*0.01+T*S*2*pi*ns*1.83**2] #polynomial for value of R2xdash
Rx_stator=R2x-0.083 #in ohm
print "External resistance referred to stator = %0.4f ohm "%Rx_stator 
Rx_rotor=Rx_stator/K**2 #in ohm/phase
print "External resistance rotor side = %0.3f ohm/phase "%Rx_rotor 

External resistance referred to stator = 1.1816 ohm 
External resistance rotor side = 0.189 ohm/phase

Example 1.37 Page 77

In [136]:
 
# Given data
Nnl=1485 #in rpm
Nfl=1350 #in rpm
f=50 #in Hz

#Calculations
Ns=1500 #nearest syn speed to Nfl in rpm(Assumed)
#Formula : Ns=120*f/P
P=120*f/Ns #no. of poles
print "Part (i)" 
print "No. of poles : " ,P
Snl=(Ns-Nnl)/Ns #slip
print "Part (ii)" 
print "No load Slip = %0.2f %%"%(Snl*100) 
Sfl=(Ns-Nfl)/Ns #slip
print "No load Slip = %0.2f %% "%(Sfl*100) 
fr_nl=f*Snl #in Hz
fr_fl=f*Sfl #in Hz
print "Part (iii)" 
print "No load frequency = %0.2f Hz  "%fr_nl 
print "Full load frequency = %0.2f Hz "%fr_fl 
#Part (iv)
print "On No Load : " 
N1=120*fr_nl/P #speed of rotor field with respect to rotor conductor in rpm
print "Speed of rotor field with respect to rotor conductor = %0.2f rpm  "%N1 
Rf_wrtS=1500 #in rpm
Rf_wrtSF=0 #in rpm
print "Rotor field with respect to stator = %0.2f rpm"%Rf_wrtS 
print "Rotor field with respect to stator field = %0.2f rpm "%Rf_wrtSF 
print "On Full Load : " 
N2=120*fr_fl/P #speed of rotor field with respect to rotor conductor in rpm
print "Speed of rotor field with respect to rotor conductor = %0.2f rpm "%N2 
Rf_wrtS=1500 #in rpm
Rf_wrtSF=0 #in rpm
print "Rotor field with respect to stator = %0.2f rpm "% Rf_wrtS 
print "Rotor field with respect to stator field = %0.2f rpm  "%Rf_wrtSF 
#Answer of no load slip is wrong in the book.

Part (i)
No. of poles :  4.0
Part (ii)
No load Slip = 1.00 %
No load Slip = 10.00 % 
Part (iii)
No load frequency = 0.50 Hz  
Full load frequency = 5.00 Hz 
On No Load : 
Speed of rotor field with respect to rotor conductor = 15.00 rpm  
Rotor field with respect to stator = 1500.00 rpm
Rotor field with respect to stator field = 0.00 rpm 
On Full Load : 
Speed of rotor field with respect to rotor conductor = 150.00 rpm 
Rotor field with respect to stator = 1500.00 rpm 
Rotor field with respect to stator field = 0.00 rpm

Example 1.38 Page 78

In [137]:
# Given data
VL=3.3 #in KV
P=20 #no of poles
f=50 #in Hz
R2=0.025 #in ohm/phase
X2=0.28 #in ohm/phase
N=294 #Full load speed in rpm

#Calculations
Sm=R2/X2 #Max Slip
print "Slip at max torque = %0.2f %%" %(Sm*100)
Ns=120*f/P #in rpm
Sfl=(Ns-N)/Ns #Full load slip
#Formula : T proportional to S*E2**2*R2/(R2**2+(S*X2)**2)
TmByTfl=Sm/(R2**2+(Sm*X2)**2)*((R2**2+(Sfl*X2)**2))/Sfl #ratio
print "Ratio of max to full load torqiue : " ,round(TmByTfl,3)

Slip at max torque = 8.93 %
Ratio of max to full load torqiue :  2.344

Example 1.39 Page 79

In [138]:
 
# Given data
Pin=50 #in KW
StatorLoss=800 #in watts
f=50 #in Hz
fr=90 #cycles/min

#Calculations
fr=fr/60 #in Hz
S=fr/f #slip
P2=Pin*1000-StatorLoss #watts
#Formula : P2:Pc:Pm=1:S:1-S
Pm=P2*(1-S) #in watts
print "Total Mechanical power devloped = %.f W" %Pm

Total Mechanical power devloped = 47724 W

Example 1.40 Page 79

In [139]:
# Given data
P=4 #no. of poles
VL=200 #in volt
f=50 #in Hz
R2=0.1 #in ohm/phase
X2=0.9 #in ohm/phase
S=4 #in %
K=0.67 #rotor to stator turns  

#Calculations
S=S/100 #slip
E1ph=VL/sqrt(3) #in volt
E2ph=K*E1ph #in volt
Ns=120*f/P #in rpm
ns=Ns/60 #in rps
T=3/2/pi/ns*S*E2ph**2*R2/(R2**2+(S*X2)**2) #in N-m
print "Total torque at 4%% slip = %0.4f N-m  "%T
Tm=3/2/pi/ns*E2ph**2/2/X2 #in N-m
print "Maximum torque developed = %0.4f N-m  "%Tm
Sm=R2/X2 #Max Slip
Nm=Ns*(1-Sm) #in rpm
print "Speed at max Torque = %0.3f rpm "%Nm 
Pmax=Tm*2*pi*Nm/60 #in watts
print "Maximum mechanical power = %0.4f W" %Pmax

Total torque at 4% slip = 40.4786 N-m  
Maximum torque developed = 63.5064 N-m  
Speed at max Torque = 1333.333 rpm 
Maximum mechanical power = 8867.1605 W

Example 1.41 Page 80

In [140]:
# Given data
P=6 #no. of poles
f=50 #in Hz
Tsh=150 #in N-m
fr=1.5 #in Hz
Tlost=10 #in N-m

#Calculations
S=fr/f #slip
Ns=120*f/P #in rpm
N=Ns*(1-S) #in rpm
RotationalLoss=Tlost*2*pi*N/60 #in watts
Pout=Tsh*2*pi*N/60 #in watts
Pm=Pout+RotationalLoss #in watts
#Formula : P2:Pc:Pm=1:S:1-S
Pc=Pm*S/(1-S) #in watts
print "Rotor Copper Loss = %0.4f W" %Pc
P2=Pc/S #in watts
print "Input to the rotor = %0.2f W" %P2
StatorLoss=700 #in watts(assumed)
Pin=P2+StatorLoss #in watts]
Eff=Pout/Pin*100 #in %
print "Efficiency = %0.2f %%" %Eff

Rotor Copper Loss = 502.6548 W
Input to the rotor = 16755.16 W
Efficiency = 87.29 %

Example 1.42 Page 81

In [141]:
# Given data
Sfl=5 #in %
IscByIfl=6 #ratio

#Calculations
Sfl=Sfl/100 #slip
TstByTfl=1 #as Tfl=Tst
#Let X= tapping on transformer
X=sqrt(TstByTfl/(IscByIfl**2)/Sfl) #Tapping on transformer
print "Tapping on auto transformer : " ,round(X,4)
IstByIfl=X**2*IscByIfl #supply starting current to full load current 
print "The supply starting current is",round(IstByIfl,2),"times of full load current." 

Tapping on auto transformer :  0.7454
The supply starting current is 3.33 times of full load current.

Example 1.43 Page 81

In [142]:
# Given data
TmByTfl=2.5 #ratio
R2=0.4 #in ohm/phase
X2=4 #in ohm/phase

#Calculations
#Formula : Tm=K*E2**2/2/X2 and Tst=K*E2**2*R2/(R2**2+X2**2)
#E2=E2/sqrt(3) #for star delta starter
TstByTfl=(TmByTfl*2*X2)*R2/(R2**2+X2**2)/3 #calculated from above equations
print "ratio of starting torque to full load torque is : " ,round(TstByTfl,3)

ratio of starting torque to full load torque is :  0.165

Example 1.44 Page 82

In [143]:
# Given data
Zouter=0.05+1J*0.10 #in ohm
Zinner=0.01+1J*0.60 #in ohm
S=3 #in %

#Calculations
R2o=(Zouter.real) #in ohm
R2i=(Zinner.real) #in ohm
X2o=(Zouter.imag) #in ohm
X2i=(Zinner.imag) #in ohm
S=S/100 #slip
#Formula : T=3/2/pi/ns*(S*E2**2*R2/(R2**2+(S*X2)**2))
S=1 #at starting
TouterByTinner=R2o/R2i*(R2i**2+X2i**2)/(R2o**2+X2o**2) #
print "(i) Ratio of torque due to two cages at starting : " ,round(TouterByTinner)
S=3/100 #Slip at running
TouterByTinner=R2o/R2i*(R2i**2+(S*X2i)**2)/(R2o**2+(S*X2o)**2) #
print "(ii) Ratio of torque due to two cages when running at 3% slip : ",round(TouterByTinner,5)

(i) Ratio of torque due to two cages at starting :  144.0
(ii) Ratio of torque due to two cages when running at 3% slip :  0.84496

Example 1.45 Page 83

In [144]:
# Given data
Zi=0.6+1J*7 #in ohm
Zo=3.5+1J*1.5 #in ohm
Sfl=6 #in %

#Calculations
#At starting S=1
Ro=(Zo.real) #in ohm
Ri=(Zi.real) #in ohm
Xo=(Zo.imag) #in ohm
Xi=(Zi.imag) #in ohm
Zeq1=Zi*Zo/(Zi+Zo) #equivalent impedence in ohm
Req1=(Zeq1.real) #in ohm
#I2=V/Zeq
#Tst=I2**2*R2 #in N-m

#During full load
S=Sfl/100 #slip 
Zi=Ri/S+1J*Xi #in ohm
Zo=Ro/S+1J*Xo #in ohm
Zeq2=Zi*Zo/(Zi+Zo) #equivalent impedence in ohm
Req2=(Zeq2.real) #in ohm
#I2=V/Zeq
#Tfl=I2**2*R2 #in N-m
TstByTfl=(1/abs(Zeq1)**2)/(1/abs(Zeq2)**2)*Req1/Req2 #ratio
print "Starting torque is",round(TstByTfl*100,0),"% of full load torque." 
#Answer in the book is not accurate.

Starting torque is 301.0 % of full load torque.

Example 1.46 Page 85

In [145]:
# Given data
P=4 #no. of poles
f=50 #in Hz
R2=0.2 #in ohm per phase
X2=1 #in ohm per phase
Sf=4 #full load slip in %
N2=1260 #reduced speed in rpm

#Calculations
Sf=Sf/100 #full load slip
Ns=120*f/P #in rpm
S2=(Ns-N2)/Ns #new value of slip
#Let new resistance is R2dash
#Formula : T proportional to S*E2**2*R2/(R2**2+(S*X2)**2)
#T1=T2 as load is same
from sympy import symbols, solve
R2dash = symbols('R2dash')
expr = (R2dash**2)*Sf*(E2**2)*R2-R2dash*(R2**2+(Sf*X2)**2)*(S2*E2**2)+Sf*(E2**2)*R2*(S2*X2)**2
#P=[Sf*R2 -[R2**2+(Sf*X2)**2]*(S2) Sf*R2*(S2*X2)**2] #polynomial for R2dash
R2dash=solve(expr,R2dash) 
R2dash=R2dash[1] #discarding smaller value as R2dash cant be < R2
Rex=R2dash-R2
print "Extra resistance required = %0.2f ohm per phase "%Rex 

Extra resistance required = 0.60 ohm per phase

Example 1.47 Page 85

In [146]:
 
# Given data
P=4 #no. of poles
V2=415 #in volt
f=50 #in Hz
E2ByE1=1.75 #stator to rotor turn ratio
Z2=0.1+1J*0.9 #in ohm
I2=60 #in Ampere at startX2=1 #in ohm per phase
Sf=4 #full load slip in %
N2=1260 #reduced speed in rpm

#Calculations
R2=(Z2.real) #in ohm
X2=(Z2.imag) #in ohm
E1ph=V2/sqrt(3) #in volt
E2ph=E1ph/E2ByE1 #in Volt
#Formula : I2=E2ph/sqrt(R2dash**2+X2**2)
R2dash=sqrt((E2ph/I2)**2-X2**2) #in ohm
Rex=R2dash-R2 #in ohm per phase
print "Extra resiatance required = %0.3f ohm"%Rex 

Extra resiatance required = 1.997 ohm

Example 1.48 Page 86

In [147]:
# Given data
P=16 #no. of poles
PM=4 #no. of poles of modulating function
n=1 #assumed
r=4 #assumed
f=50 #in Hz

#Calculations
check=n/r==1/3*(1-PM/P) 
if check :
    print "Equation is satisfied with -ve sign." 
    P2=P+PM 

check=n/r==1/3*(1+PM/P) 
if check :
    print "Equation is satisfied with +ve sign."
    P2=P-PM 

Ns1=120*f/P #in rpm
Ns2=120*f/P2 #in rpm
print "Two speeds = %0.2f rpm & %0.2f rpm"%(Ns1,Ns2)

Equation is satisfied with -ve sign.
Two speeds = 375.00 rpm & 300.00 rpm

Example 1.49 Page 87

In [148]:
 
# Given data
PA=4 #no. of poles
PB=6 #no. of poles
f=50 #in Hz

#Calculations
Ns=120*f/PA #in rpm, A running alone
print "(1.) If A running alone, Speed = %0.2f rpm" %Ns
Ns=120*f/PB #in rpm, B running alone
print "(2.) If B running alone, Speed = %0.2f rpm" %Ns
Ns=120*f/(PA+PB) #in rpm, Cumulative cascade
print "(3.) For Cumulative cascade, Speed = %0.2f rpm"%Ns
Ns=120*f/(PA-PB) #in rpm, Differential cascade
print "(4.) Differential cascade, Speed = %0.2f rpm"%Ns

(1.) If A running alone, Speed = 1500.00 rpm
(2.) If B running alone, Speed = 1000.00 rpm
(3.) For Cumulative cascade, Speed = 600.00 rpm
(4.) Differential cascade, Speed = -3000.00 rpm

Example 1.50 Page 87

In [149]:
 
# Given data
PA=4 #no. of poles
PB=6 #no. of poles
f=50 #in Hz
fr2=1 #in Hz

#Calculations
Nsc=120*f/(PA+PB) #synchronous speed of set in rpm
S=1 #Slip
N=Nsc-(S/f)*Nsc #combined speed of set in rpm
print "Combibned speed of set = %0.2f rpm"%N 
NSA=120*f/PA #in rpm
SA=(NSA-N)/NSA #slip 
print "Slip of machine A = %0.2f %% "%(SA*100) 
fr1=SA*f #in Hz
NSB=120*fr1/PB #in rpm
SB=(NSB-N)/NSB #slip
print "Slip of machine B = %0.1f %%"%(SB*100) 

Combibned speed of set = 588.00 rpm
Slip of machine A = 60.80 % 
Slip of machine B = 3.3 %

Example 1.51 Page 88

In [150]:
# Given data
P=4 #no. of poles
f=50 #in Hz
R2=0.25 #in ohm per phase
X2=2 #in ohm per phase
N1=1455 #ion rpm
N2=N1*83/100 #in rpm
 
#Calculations
Ns=120*f/P #synchronous speed in rpm
S1=(Ns-N1)/Ns #Slip
S2=(Ns-N2)/Ns #Slip at reduced speed
#Formula : T proportional to S*E2**2*R2/(R2**2+(S*X2)**2)
T1ByT2=1 #as T1=T2 & For T2: R2dash Rex+R2
from sympy import symbols, solve
R2dash = symbols('R2dash')
expr = R2dash**2-1.7067*R2dash+0.1519
R2dash=solve(expr, R2dash) #in ohjm per phase
R2dash=R2dash[1] #neglecting lower value
Rex=R2dash-R2 #in ohm per phase
print "External resistance required = %0.4f ohm per phase "%Rex 

External resistance required = 1.3625 ohm per phase

Example 1.52 Page 89

In [151]:
# Given data
P=6 #no. of poles
f=50 #in Hz
Sf=3 #in %
R2=0.2 #in ohm per phase
 
#Calculations
Sf=Sf/100 #Slip 
Ns=120*f/P #in rpm
N1=Ns*(1-Sf) #in rpm
N2=N1*90/100 #in rpm
S2=(Ns-N2)/Ns #new slip
#Formula : T=K*S*E2**2*R2/R2**2 #S*X2 is neglected
#Sf/R2=S2/(R2+r)   if Tfl=T20
r=(S2*R2)/Sf-R2 #Extra resistance required in ohm
print "Extra resistance necessery in series = %0.2f ohm "%r 

Extra resistance necessery in series = 0.65 ohm

Example 1.53 Page 90

In [152]:
# Given data
IscByIfl=5 #ratio
Sf=5 #in %
K=50 #tapping in %
 
#Calculations
Sf=Sf/100 #Slip 
#(i) Start delta 
TstByTfl=1/3*IscByIfl**2*Sf #ratio
print "(i) Starting torque is",round(TstByTfl*100,2),"% of full load torque." 
#(ii) Auto Transformer having 50% tapping
K=K/100 #tapping
TstByTfl=K**2*IscByIfl**2*Sf #ratio
print "(ii) Starting torque is",(TstByTfl*100),"% of full load torque." 

(i) Starting torque is 41.67 % of full load torque.
(ii) Starting torque is 31.25 % of full load torque.

Example 1.54 Page 90

In [153]:
# Given data
P=6 #no. of poles
f=50 #in Hz
Ifl=60 #in Ampere
N=940 #speed in rpm
Tfl=150 #in N-m
Isc=300 #in Ampere
 
#Calculations
Ns=120*f/P #in rpm
Sf=(Ns-N)/Ns #Slip full load
#Formula : Tst/Tfl=(Isc/Ifl)**2*Sf
Tst=(Isc/Ifl)**2*Sf*Tfl #in N-m
print "Starting Torque = %0.2f N-m  "%Tst 
#For Start delta 
Tst=1/3*(Isc/Ifl)**2*Sf*Tfl #in N-m
print "Starting Torque for star delta starter = %0.2f N-m "%Tst 
Isc=sqrt(3*Tst/Tfl/Sf)*Ifl #in Ampere
print "Starting current for star delta starter = %0.2f Ampere "%Isc 

Starting Torque = 225.00 N-m  
Starting Torque for star delta starter = 75.00 N-m 
Starting current for star delta starter = 300.00 Ampere

Example 1.55 Page 91

In [154]:
 
# Given data
TmByTfl=2.2 #ratio
R2=0.5 #in ohm per phase
X2=5 #in ohm per phase
K=70 #tapping in %
 
#Calculations
#Formula :Tst proportional to E2**2*R2/(R2**2+X2**2) 
#Formula :Tm proportional to E2**2/(2*X2) 
#Formula :Tfl proportional to 1/4.4*E2**2/X2
TstByTfl=R2/(R2**2+X2**2)*TmByTfl*2*X2 #ratio for direct on line
print "Ratio of starting torque to full load torque for direct on line starter : " ,round(TstByTfl,4)
TstByTfl=(1/sqrt(3))**2*R2/(R2**2+X2**2)*TmByTfl*2*X2 #ratio for star delta starting
print "Ratio of starting torque to full load torque for star delta starter : " ,round(TstByTfl,4)
TstByTfl=(K/100)**2*R2/(R2**2+X2**2)*TmByTfl*2*X2 #ratio for auto transformer starting
print "Ratio of starting torque to full load torque for auto transformer starter : " ,round(TstByTfl,4)

Ratio of starting torque to full load torque for direct on line starter :  0.4356
Ratio of starting torque to full load torque for star delta starter :  0.1452
Ratio of starting torque to full load torque for auto transformer starter :  0.2135

Example 1.56 Page 92

In [155]:
# Given data
TmByTfl=3 #ratio
Sm=0.1 #slip at max Torque
 
#Calculations
TstByTfl_dol=2*Sm/(1+Sm**2)*TmByTfl #ratio for D.O.L starter
print "Ratio of starting torque to full load torque for D.O.L starter : " ,round(TstByTfl_dol,3)

TstByTfl=1/3*TstByTfl_dol #ratio for star delta starting
print "Ratio of starting torque to full load torque for star delta starter : " ,round(TstByTfl,3)
#Anser of first part is not given in the book.

Ratio of starting torque to full load torque for D.O.L starter :  0.594
Ratio of starting torque to full load torque for star delta starter :  0.198

Example 1.57 Page 92

In [156]:
# Given data
VL=400 #in volt
Ist=1200 #in Ampere
Eff=0.85 #Efficiency
cosfi=0.8 #power factor
IstByIrated=5 #ratio
 
#Calculations
I2_rated=Ist/IstByIrated #in Ampere
KWrating=sqrt(3)*VL*I2_rated*cosfi*Eff #in KW
#To have star delta styarter tapping Xo=1/sqrt(3)
#Ist=X0**2*IstByIrated*IL
X0=1/sqrt(3) #tapping
IL=Ist/X0**2/IstByIrated #in Ampere
KWmax=sqrt(3)*VL*IL*cosfi*Eff/1000 #in KW
print "Maximum KW rating with star delta starter : " ,round(KWmax,3)

Maximum KW rating with star delta starter :  339.205

Example 1.58 Page 93

In [157]:
 
# Given data
IscByIfl=3*180/100 #ratio
TstByTfl=0.35 #ratio
X=80/100 #tapping
 
#Calculations
#Formula : TstByTfl=1/3*(IscByIfl**2)*Sfl
Sfl=TstByTfl/IscByIfl**2*3 #slip at full load
IstByIsc=X**2 #ratio
IstByIfl=IstByIsc*IscByIfl #ratio
print "Starting current is",round(IstByIfl,1),"imes of full load current." 
TstByTfl=X**2*IscByIfl**2*Sfl #ratio
print "Starting torque is",round(TstByTfl*100,2),"% of full load torque." 

Starting current is 3.5 imes of full load current.
Starting torque is 67.2 % of full load torque.

Example 1.59 Page 93

In [158]:
from numpy import real
# Given data
Zouter=0.05+1J*0.11 #in ohm
Zinner=0.015+1J*0.5 #in ohm
 
#Calculations
R2odash=(Zouter.real) #in ohm
X2odash=(Zouter.imag) #in ohm
R2idash=real(Zinner.real) #in ohm
X2idash=(Zinner.imag) #in ohm
TouterByTinner=R2odash/(R2odash**2+X2odash**2)*(R2idash**2+X2idash**2)/R2idash #ratio
print "Ratio of Torque due to two windinga : " ,round(TouterByTinner,2)

Ratio of Torque due to two windinga :  57.13

Example 1.60 Page 194

In [159]:
# Given data
PA=4 #no. of poles
PB=4 #no. of poles
f=50 #in Hz
V=440 #in volt

#calculations
#Independently with A
Ns=120*f/PA #in rpm
print "Independently with A, Synchrpnous speed Ns = %0.2f rpm " %Ns
#Independently with B
Ns=120*f/PB #in rpm
print "Independently with B, Synchrpnous speed Ns = %0.2f rpm"%Ns
#Running as cumulative cascaded
Ns=120*f/(PA+PB) #in rpm
print "Running as cumulative cascaded, Synchrpnous speed Ns = %0.2f rpm"%Ns
#Running as differentially cascaded
print "Running as differentially cascaded, Synchrpnous speed Ns is undefined." 

Independently with A, Synchrpnous speed Ns = 1500.00 rpm 
Independently with B, Synchrpnous speed Ns = 1500.00 rpm
Running as cumulative cascaded, Synchrpnous speed Ns = 750.00 rpm
Running as differentially cascaded, Synchrpnous speed Ns is undefined.

Example 1.61 Page 94

In [160]:
# Given data
IscByIfl=3*180/100 #ratio
TstByTfl=35/100 #ratio
X=75 #tapping in %

#calculations
X=X/100 #tapping

#Star delta starting
#Formula : TstByTfl=1/3*IscByIfl*Sfl
Sfl=TstByTfl*3/IscByIfl**2 #slip at full load

#Auto transformer  starting
IstByIsc=X**2 #ratio
IstByIfl=X**2*IscByIfl #ratio
print "Starting current is",(IstByIfl*100),"% of fulll load current." 
TstByTfl=X**2*IscByIfl**2*Sfl #ratio
print "Starting torque is",(TstByTfl*100),"% of fulll load torque." 
#Answer of starting current in terms of full load curremt is not given in the book.

Starting current is 303.75 % of fulll load current.
Starting torque is 59.0625 % of fulll load torque.

Example 1.62 page 95

In [161]:
# Given data
VL=400 #in volt
f=50 #in Hz
I=100 #i Ampere

#calculations
#D.O.L starter
IL=I*sqrt(3) #in Ampere
print "(i) The line current for direct on line starting = %0.3f Ampere "%IL 
#In star delta starter
Vph=VL/sqrt(3) #in Volt
Iph=I/sqrt(3) #in Ampere
print "(ii) Starting phase current for star delta starting = %0.3f A" %Iph
print "(ii) Starting line current for star delta starting = %0.3f A" %Iph
#Auto transformer starter
K=70/100 #tapping of auto transformer
Vph=VL/sqrt(3) #in Volt
Vline=K*VL #in volt
Ist_phase=Vline*I/VL #in Ampere
print "(iii) Starting phase current of motor = %0.2f A" %Ist_phase
Ist_line=Ist_phase*sqrt(3) #in Ampere
print "(iii) Starting line current of motor = %0.2f A" %Ist_line
IsupplyLine=K*Ist_line #in Ampere
print "(iii) Supply line current of motor = %0.2f A" %IsupplyLine

(i) The line current for direct on line starting = 173.205 Ampere 
(ii) Starting phase current for star delta starting = 57.735 A
(ii) Starting line current for star delta starting = 57.735 A
(iii) Starting phase current of motor = 70.00 A
(iii) Starting line current of motor = 121.24 A
(iii) Supply line current of motor = 84.87 A

Example 1.63 Page 97

In [162]:
# Given data
P=12 #no. of poles
Ns=500 #in rpm
Nr=1440 #in rpm

#calculations
f1=P*Ns/120 #in Hz
Nsm=1500 #in rpm (Assumed closed synchronous speed)
S=(Nsm-Nr)/Nsm #slip
print "Slip of the motor = %0.2f %%"%(S*100) 
Pm=120*f1/Nsm #no. of poles of the motor
print "No. of poles of the motor : " ,Pm

Slip of the motor = 4.00 %
No. of poles of the motor :  4.0

Example 1.64 Page 97

In [163]:
# Given data
P=4 #no. of poles
f1=50 #in Hz
f2=1.5 #in Hz

#calculations
S=f2/f1 #slip
print "Slip = %0.2f %% "%(S*100) 
Ns=120*f1/P #in rpm
N=(1-S)*Ns #in rpm
print "Running speed of motor = %0.2f rpm"%N 

Slip = 3.00 % 
Running speed of motor = 1455.00 rpm

Example 1.65 Page 97

In [164]:
# Given data
P=6 #no. of poles
f1=50 #in Hz
S0=1 #in %
Sfl=3 #in %

#calculations
S0=S0/100 #slip
Sfl=Sfl/100 #slip
Ns=120*f1/P #in rpm
print "(a) Synchronous speed = %0.2f rpm "%Ns 
N0=(1-S0)*Ns #in rpm
print "(b) No Load speed = %0.2f rpm "%N0 
Nfl=(1-Sfl)*Ns #in rpm
print "(c) Full load speed = %0.2f rpm "%Ns 
f2_st=f1*S0 #in Hz
print "(d) Frequeny of rotor current at standstill = %0.2f Hz "%f2_st 
f2_fl=f1*Sfl #in Hz
print "(e) Frequeny of rotor current at full load = %0.2f Hz"%f2_fl 
#Answer of part (c) & part(d) is wrong. Calcultion mistake & slip is not divided by 100.

(a) Synchronous speed = 1000.00 rpm 
(b) No Load speed = 990.00 rpm 
(c) Full load speed = 1000.00 rpm 
(d) Frequeny of rotor current at standstill = 0.50 Hz 
(e) Frequeny of rotor current at full load = 1.50 Hz

Example 1.66 Page 98

In [165]:
# Given data
P=4 #no. of poles
f1=50 #in Hz
S=4 #in %
R2=1 #in ohm/phase
X2=4 #in ohm/phase

#calculations
Ns=120*f1/P #in rpm
S=S/100 #slip
#part (a)
N=(1-S)*Ns #in rpm
print "(a) Speed of the motor = %0.2f rpm "%N 
#part (b)
f2=S*f1 #in Hz
print "(b) Frequency of rotor emf = %0.2f Hz "%f2 
#part (i)
Z2=R2+1J*X2 #in ohm
cosfi=cos(atan((Z2.imag)/(Z2.real))) #power factor
print "(i) Power factor at standstill = %0.4f lag "%cosfi 
#part (ii)
N=1400 #speed in rpm (given for this part)
S1=(Ns-N)/Ns #slip
Z2s1=R2+1J*S1*X2 #in ohm
cosfi=cos(atan((Z2s1.imag)/(Z2s1.real))) #power factor at 1400 rpm speed
print "(ii) Power factor at 1400 rpm = %0.4f lag "%cosfi 

(a) Speed of the motor = 1440.00 rpm 
(b) Frequency of rotor emf = 2.00 Hz 
(i) Power factor at standstill = 0.2425 lag 
(ii) Power factor at 1400 rpm = 0.9662 lag

Example 1.67 Page 99

In [166]:
from numpy import real, imag
# Given data
E=60 #in volt
Zrotor=0.8+1J*6 #rotor impedence in ohm/phase
Zstator=4+1J*3 #stator impedence in ohm/phase
S=5 #in %

#calculations
E2=E/sqrt(3) #emf induced/phase in volt
Ztotal=Zstator+Zrotor #in ohm/phase
#Part (a)
I2=E2/Ztotal #in Ampere
print "Part(a) Magnitude is",round(abs(I2),3),"& angle = %0.2f degree"%degrees(atan((I2.imag)/(I2.real)))
#Part (b)
S=S/100 #slip
R2=real(Zrotor) #in ohm/phase
X2=imag(Zrotor) #in ohm/phase
I2s=S*E2/(R2+S*1J*X2) #in ampere
print "Part(b) Magnitude is",round(abs(I2s),2),"& angle = %0.2f degree "%degrees(atan((I2s.imag)/(I2s.real)))
#Answer of part (b) is wrong in the book.

Part(a) Magnitude is 3.396 & angle = -61.93 degree
Part(b) Magnitude is 2.03 & angle = -20.56 degree

Example 1.68 Page 99

In [167]:
# Given data
Pis=60 #in KW
phase=3 #no. of phase
S=3 #in %
StatorLaser=1 #in KW

#calculations
S=S/100 #slip 
StatorOutput=Pis-StatorLaser #in KW
RotorInput=StatorOutput #in KW
RotorCuLoss=S*RotorInput #in KW
RotorCuLoss_phase=S*RotorInput/phase #in KW/phase
print "Rotor Copper loss per phase = %0.2f kW "%RotorCuLoss_phase 
MechPower=RotorInput-RotorCuLoss #in KW
print "Total mechanical power devloped = %0.2f kW"%MechPower

Rotor Copper loss per phase = 0.59 kW 
Total mechanical power devloped = 57.23 kW

Example 1.69 Page 100

In [168]:
# Given data
P=6 #no. of poles
f1=50 #in Hz
f2=1.5 #in Hz
Zo=150 #useful Torque in N-m
FrictionLoss=10 #in N-m
Psc=700 #stator loss in watt

#calculations
Ns=120*f1/P #in rpm
S=f2/f1 #slip
Nr=(1-S)*Ns #in rpm
wr=2*pi*Nr/60 #in rad/sec
Po=Zo*wr #in watts
Pmd=(Zo+FrictionLoss)*wr #in watts
#Part (a)
Prc=S/(1-S)*Pmd #in watts
print "(a) Rotor Copper Loss = %0.4f kW "%(Prc/1000)
#Part (b)
Pi=Pmd+Prc+Psc #in watts
print "(b) Input to the motor = %0.4f kW"%(Pi/1000)
#Part (c)
Eff=Po/Pi #Effiiency
print "(d) Efficiency = %0.2f %%"%(Eff*100)

(a) Rotor Copper Loss = 0.5027 kW 
(b) Input to the motor = 17.4552 kW
(d) Efficiency = 87.29 %

Example 1.70 Page 100

In [169]:
# Given data
V=440 #in Volt
f=50 #in Hz
phase=3 #no. of phase
P=6 #no. of poles
Pin=80 #rotor input in KW
f1=50 #in Hz
f2=100 #in rotation/min
I=65 #rotor current in Ampere

#calculations
f2=f2/60 #in Hz
S=f2/f1 #slip
print "Slip = %0.3f p.u" %S
Ns=120*f/P #in rpm
Nr=Ns*(1-S) #in rpm
print "Rotor speed = %0.2f rpm "%Nr 
RotorCuLoss=S*Pin*1000 #in Watts
Pmd=Pin*1000-RotorCuLoss #Mechanical powre developed /in watts
Pmd=Pmd/746 #in HP
print "Mechanical power developed = %0.2f HP "%Pmd 
RotorCuLoss_phase=RotorCuLoss/phase #in watts/phase
print "Rotor Coopper Loss per phase = %0.f W" %RotorCuLoss_phase
R2=RotorCuLoss_phase/I**2 #in ohm
print "Rotor resistance per phase = %0.4f ohm"%R2 

Slip = 0.033 p.u
Rotor speed = 966.67 rpm 
Mechanical power developed = 103.66 HP 
Rotor Coopper Loss per phase = 889 W
Rotor resistance per phase = 0.2104 ohm

Example 1.71 Page 101

In [170]:
# Given data
f1=50 #in Hz
phase=3 #no. of phase
P=6 #no. of poles
Nr=960 #in rpm
GearCuLoss=250 #in watt
Power=25 #in HP
MechLoss=1000 #in watts
I2=35 #in Ampere

#calculations
Ns=f1*120/P #in rpm
S=(Ns-Nr)/Ns #slip
#Formula : RotorCuLoss=S/(1-S)*MechDevPower
#3*I2**2*R2+GearCuLoss=S/(1-S)*(Power*746+MechLoss)
R2=(S/(1-S)*(Power*746+MechLoss)-GearCuLoss)/3/I2**2 #in ohm
print "Resistance per phase = %0.5f ohm "%R2 

Resistance per phase = 0.15476 ohm

Example 1.72 Page 101

In [171]:
# Given data
V=500 #in Volt
f1=50 #in Hz
phase=3 #no. of phase
P=6 #no. of poles
Nr=995 #in rpm
Pm=20 #mech power in KW
StatorLoss=1500 #in watts
pf=0.87 #power facator

#calculations
Ns=f1*120/P #in rpm
S=(Ns-Nr)/Ns #slip
print "(a) Slip is : " ,S,'pu'
Prc=S/(1-S)*Pm*1000 #in watts
print "(b) Rotor I**2*R Loss = %0.2f Watts"%Prc 
RotorInput=Prc/S #in watts
TotalInput=RotorInput+StatorLoss #in watts
print "(c) Total input = %0.2f kW "%(TotalInput/1000) 
LineCurrent=TotalInput/sqrt(3)/V/pf #in Ampere
print "(d) Line current = %0.1f A" %LineCurrent
fr=S*f1 #in Hz
print "(e) Rotor frequency = %0.2f HZ "%fr 

(a) Slip is :  0.005 pu
(b) Rotor I**2*R Loss = 100.50 Watts
(c) Total input = 21.60 kW 
(d) Line current = 28.7 A
(e) Rotor frequency = 0.25 HZ

Example 1.73 Page 102

In [172]:
# Given data
StatorLoss=2 #in KW
StatorInput=90 #stator input in KW
S=4 #in %

#calculations
S=S/100 #slip
StatorOutput=StatorInput-StatorLoss #in KW
Pri=StatorOutput #rotor input in KW
Pcr=S*Pri #in KW
print "Rotor Copper Loss = %0.2f kW "%Pcr 
Pm=Pri-Pcr #in KW
print "Rotor mechanical power developed = %0.2f kW"%Pm 

Rotor Copper Loss = 3.52 kW 
Rotor mechanical power developed = 84.48 kW

Example 1.74 Page 102

In [173]:
# Given data
emf=60 #in volt
R2=0.6 #in ohm
X2=4 #in ohm
Rrh=5 #in ohm
Xrh=2 #in ohm
S=4 #in %

#calculations
S=S/100 #slip
E2=emf/sqrt(3) #in volt
Rt=R2+Rrh #in ohm
Xt=X2+Xrh #in ohm
I2=E2/sqrt(Rt**2+Xt**2) #in Ampere
print "(a) Current per phase in rotor = %0.2f A" %I2
E2s=S*E2 #in volt
Z2s=sqrt(R2**2+(S*X2)**2) #in ohm
I2s=E2s/Z2s #in Ampere
print "(b) Current per phase in rotor = %0.3f A" %I2s

(a) Current per phase in rotor = 4.22 A
(b) Current per phase in rotor = 2.231 A

Example 1.75 Page 103

In [174]:
# Given data
R2=0.05 #in ohm
X2=0.1 #in ohm

#calculations
R2dash=X2 #for max Torque
r=R2dash-R2 #in ohm
print "External resistance per phase required = %0.2f ohm "%r 

External resistance per phase required = 0.05 ohm

Example 1.76 Page 103

In [175]:
# Given data
P=6 #no. of poles
f=50 #in Hz
Sfl=4 #in %
Z2=0.01+1J*0.05 #in ohm

#calculations
S=Sfl/100 #slip
R2=real(Z2) #in ohm
X2=imag(Z2) #in ohm
Sm=R2/X2 #slip at max speed
Ns=120*f/P #in rpm
Nm=(1-Sm)*Ns #in rpm
TmaxByTfl=(S**2+Sm**2)/2/S/Sm #ratio
print "Maximum Torque is",(TmaxByTfl),"times of full load torque." 

Maximum Torque is 2.6 times of full load torque.

Example 1.77 Page 103

In [176]:
# Given data
P=6 #no. of poles
f=50 #in Hz
Tmax=30 #in N-m
Nm=960 #in rpm
S=5 #in %
R2=0.6 #in ohm

#calculations
S=S/100 #slip
Ns=120*f/P #in rpm
Sm=(Ns-Nm)/Ns #slip at max speed
X2=R2/Sm #in ohm
Tau_s=2*S*Sm/(S**2+Sm**2)*Tmax #in N-m
print "Torque exerted by the motor = %0.2f N-m "%Tau_s 

Torque exerted by the motor = 29.27 N-m

Example 1.78 Page 104

In [177]:
# Given data
P=4 #no. of poles
f=50 #in Hz
Tmax=110 #in N-m
Nm=1360 #in rpm
R2=0.25 #in ohm
TstByTmax=1/2 #ratio

#calculations
Ns=120*f/P #in rpm
Sm=(Ns-Nm)/Ns #slip at max speed
X2=R2/Sm #in ohm
#Formula : Tmax=K*E2**2/2/X2 and Tst=K*E2**2*(R2+r)/((R2+r)**2+X2**2)
from sympy import symbols, solve
RT = symbols('RT')
expr = TstByTmax*RT**2+TstByTmax*X2**2-RT*2*X2

RT=solve(expr, RT)[0] #in ohm
r=RT-R2 #in ohm
r=r #leaving higher value as Tmax goes with S>1 for this value
print "Resistance required in series = %0.3f ohm  "%r

Resistance required in series = 0.468 ohm

Example 1.79 Page 105

In [178]:
# Given data
P=16 #no. of poles
f=50 #in Hz
Z2=0.02+1J*0.15 #in ohm
Nr=360 #in rpm

#calculations
Ns=120*f/P #in rpm
Sfl=(Ns-Nr)/Ns #slip at full load
R2=real(Z2) #in ohm
X2=imag(Z2) #in ohm
Sm=R2/X2 #slip at max torque
Nm=(1-Sm)*Ns #in rpm
print "(a) Speed at which max Torque occurs = %0.2f rpm "%Nm 
TmaxByTfl=(Sfl**2+Sm**2)/2/Sfl/Sm #ratio
print "(b) Ratio of maximum to full load torque : " ,round(TmaxByTfl,4)
R2dash=X2 #for max Torque
r=R2dash-R2 #in ohm
print "(c) External resistance per phase required = %0.2f ohm "%r 

(a) Speed at which max Torque occurs = 325.00 rpm 
(b) Ratio of maximum to full load torque :  1.8167
(c) External resistance per phase required = 0.13 ohm

Example 1.80 Page 106

In [179]:
# Given data
P=6 #no. of poles
f=50 #in Hz
N=940 #in rpm
Output=7 #in KW
Nm=800 #in rpm
TotalLaser=840 #in watts

#calculations
Ns=120*f/P #in rpm 
S=(Ns-N)/Ns #slip
Sm=(Ns-Nm)/Ns #slip at max Torque
Pmd=Output*1000+TotalLaser #in watts
#Formula : Pmd=2*pi*N*Td/60
Tdfl=Pmd/2/pi/N*60 #in N-m
Tst=Tdfl*(S**2+Sm**2)/S/(1+Sm**2) #in N-m
print "Starting tiorque = %0.2f N-m "%Tst

Starting tiorque = 55.65 N-m

Example 1.81 Page 107

In [180]:
# Given data
P=4 #no. of poles
f=50 #in Hz
VL=200 #in volt
R2=0.1 #in ohm
X2=0.9 #in ohm
Te2ByTe1=0.67 #ratio of rotor to stator turns
S=4 #in %

#calculations
S=S/100 #slip
Ns=120*f/P #in rpm
E1=VL/sqrt(3) #in volt
E2=E1*Te2ByTe1 #in volt
Td=3*S*E2**2*R2/2/pi/(Ns/60)/(R2**2+(S*X2)**2) #in N-m
print "(a) Total torque at 4%% slip = %0.2f N-m"%Td 
Tmax=3*E2**2/2/pi/(Ns/60)/(2*X2) #in N-m
print "(b) Total torque at 4%% slip = %0.2f N-m "%Tmax 
Sm=R2/X2 #slip at max torque
Nm=(1-Sm)*Ns #speed at Tmax in rpm
print "(c) Speed at maximum torque = %0.2f rpm "%Nm 
Pmd_max=2*pi*Nm/60*Tmax #in N-m
print "(d) Maximum mechanical power = %0.2f N-m "%Pmd_max 

(a) Total torque at 4% slip = 40.48 N-m
(b) Total torque at 4% slip = 63.51 N-m 
(c) Speed at maximum torque = 1333.33 rpm 
(d) Maximum mechanical power = 8867.16 N-m

Example 1.82 Page 107

In [181]:
from math import sqrt
# Given data
TstByTfl=1 #ratio
TmaxByTfl=2 #ratio

#calculations
TstByTmax=TstByTfl/TmaxByTfl #ratio
#Formula : TstByTmax=2*Sm/(1+Sm**2)

from sympy import symbols, solve 
Sm, S = symbols('Sm, S')
expr = TstByTmax*Sm**2-2*Sm+TstByTmax
Sm=solve(expr, Sm) #slip at max torque
Sm=Sm[0] #neglecting the higher value
print "(a) Slip at which max torque occurs : %0.3f" %Sm
#Formula : TflByTmax=2*S*Sm/(S**2+Sm**2) 
expr = S**2-TmaxByTfl*2*S*Sm+Sm**2
S=solve(expr,S) #slip at max torque
S=S[0] #neglecting the higher value
print "(b) Full load slip : %0.4f" %S
#I2stByI2fl**2=(Sm**2+S**2)/S**2/(1+Sm**2)
I2stByI2fl=sqrt((Sm**2+S**2)/S**2/(1+Sm**2)) #ratio
print "(c) Rotor current at starting at full load current : %0.4f"%I2stByI2fl 

(a) Slip at which max torque occurs : 0.268
(b) Full load slip : 0.0718
(c) Rotor current at starting at full load current : 3.7321

Example 1.83 Page 109

In [182]:
# Given data
Zst=25 #in N-m

#calculations
#K=2*Zst*R2
KbyR2=2*Zst #calculation
#(a) Tst=K*2*R2/((2*R2)**2+R2**2)
Tst=KbyR2*2/(2**2+1) #in N-m
print "(a) Starting torque = %0.2f N-m "%Tst 
#(b) Tst=K/2*R2/((R2/2)**2+R2**2)
Tst=KbyR2/2/((1/2)**2+1) #in N-m
print "(b) Starting torque = %0.2f N-m "%Tst 

(a) Starting torque = 20.00 N-m 
(b) Starting torque = 20.00 N-m

Example 1.84 Page 109

In [183]:
# Given data
P=4 #no. of poles
f=50 #in Hz
R2=0.4 #in ohm
X2=4 #in ohm

#calculations
Ns=120*f/P #in rpm
Sm=R2/X2 #slip at max Torque
Nm=Ns*(1-Sm) #in rpm
print "Speed at Max Torque = %0.2f rpm "%Nm 
TmaxByTst=(1+Sm**2)/2/Sm #ratio
print "Ratio of max Torque to starting Torque : " ,TmaxByTst
#After adding additional resistance
from sympy import symbols, solve
r = symbols('r')
TstByTm=1/2 #given ratio
expr = TstByTm-2*X2*(R2+r)/((R2+r)**2+X2**2) #ratio
r=solve(expr, r) #in ohm
r=r[0] #leaving higher value
print "Required resistance value = %0.2f ohm "%r 
#Answer of resistance is wrong in the book.

Speed at Max Torque = 1350.00 rpm 
Ratio of max Torque to starting Torque :  5.05
Required resistance value = 0.67 ohm

Example 1.85 Page 110

In [184]:
 
# Given data
VL=440 #in volt
f=50 #in Hz
X2byR2=3 #ratio
TmByTfl=4 #ratio

#Calculations
Sm=1/X2byR2 #Maximum slip
#Formula : TmByTfl=(Sm**2+S**2)/(2*S*Sm)
from sympy import symbols, solve, N
S = symbols('S')
expr = 9*S**2-24*S+1
S=solve(expr, S)
S=N(S[0],3)
print "(i) Full load slip : " ,S
TstByTfl=(Sm**2+S**2)/(S*(1+Sm**2)) #ratio
print "(ii) Ratio of starting torque to full load torque : " ,round(TstByTfl,4)
V1=VL/sqrt(3) #in volt
#Tst=Tfl : K*V11**2R2/(R2**2+X2**2)=R*V1*S*R2/(R2**2+(S*X2)**2)
V11=sqrt(S*V1**2*(1+X2byR2**2)/(1+(S*X2byR2)**2)) #in volt
Linevoltage=V11*sqrt(3) #in volt
print "(c) Line Voltage  = %0.2f Volt" %Linevoltage
#Note : Answer of line voltage is wrong in the book due to calculation mistake.

(i) Full load slip :  0.0423
(ii) Ratio of starting torque to full load torque :  2.4002
(c) Line Voltage  = 284.01 Volt