Chapter 11: Speed and Torque Control¶
Example 11.1, Page number: 561¶
In [4]:
from __future__ import division
#Variable declaration:
Vdc=240 #DC supply (V)
D=0.75 #Duty cycle
Rf=187 #field resistance(ohm)
Lf=4.2 #field winding inductance(H)
T=1 #switching period(msec)
#Calculations:
If=D*(Vdc/Rf)
tau=Lf/Rf #time constant(msec)
del_if=(2*Vdc/Rf)*(T/tau)*D*(1-D)
#Results:
print "Avg field current:",round(If,2),"A"
print "Magnitude of currnet ripple:",round(del_if,1),"mA"
Example 11.2, Page number: 563¶
In [1]:
from __future__ import division
from sympy import *
from math import *
#Variable declaration:
n1=1800 #r/min
n2=3600 #r/min
Va=240 #terminal voltage(V)
Ifo=0.34 #No-load field current(A)
Ra=0.05 #Armature resistance(ohm)
Rsh=187 #Shunt field resistance(ohm)
#Calculations:
wm=symbols('wm')
wm1=float(2*pi*n1/60)
wm2=float(2*pi*n2/60)
def Pload(wm):
return (22.4*(120*pi)**-3)*(wm)**3
T1=Pload(wm1)*1000/wm1
T2=Pload(wm2)*1000/wm2
Kf=Va/(Ifo*wm2)
def If(T,wm):
return (Va/(2*Kf*wm))*(1+sqrt(1-(4*wm*T*Ra)/Va**2))
Rf1tot=round(Va/float(If(T1,wm1)))
Rf2tot=round(Va/float(If(T2,wm2)))
Rrh1=Rf1tot-Rsh
Rrh2=Rf2tot-Rsh
#Results:
print "----------------------------------------------------------------"
print "r/min Tload[N.m] If[A] R(f)tot[ohm] Rrheostat[ohm]"
print "----------------------------------------------------------------"
print n1,"\t ",round(float(T1),1),"\t\t ",round(float(If(T1,wm1)),3),"\t",Rf1tot,"\t ",Rrh1
print n2,"\t ",round(float(T2),1),"\t\t ",round(float(If(T2,wm2)),3),"\t",Rf2tot,"\t ",Rrh2
print "----------------------------------------------------------------"
Example 11.3, Page number: 567¶
In [7]:
from __future__ import division
%matplotlib inline
from math import *
from pylab import *
#Variable declaration:
Rf=109 #Field resistance(ohm)
Vf=300 #Rated field voltage(V)
Ra=0.084 #Armature resistance(ohm)
Kf=0.694 #Geometric constant(A.rad/sec)
#Calculations:
If=Vf/Rf #Resulting field current(A)
w_rated=2500*(pi/30) #Rated speed(rad/sec)
P_rated=100*746 #Watts
T_rated=P_rated/w_rated #Nm
Va=[0]*102
NoLoadRPM=[0]*102
FullLoadRPM=[0]*102
for n in range(1,102,1):
Va[n-1]=250*(1+(n-1)/100)
T=0 #Zero torque
w=(Va[n-1]-T*Ra/(Kf*If))/(Kf*If)
NoLoadRPM[n-1]=w*30/pi
T=T_rated
w=(Va[n-1]-T*Ra/(Kf*If))/(Kf*If)
FullLoadRPM[n-1]=w*30/pi
print"The plot is as shown:"
plot(Va,NoLoadRPM)
plot(Va[20] ,NoLoadRPM[20] ,'r+')
plot (Va[50] , NoLoadRPM[50] , 'r+')
plot (Va[80] ,NoLoadRPM[80] , 'r+')
plot (Va, FullLoadRPM,'.')
plot (Va[20] ,FullLoadRPM[20] ,'o')
plot (Va[50] , FullLoadRPM[50] , ' o' )
plot (Va[80] , FullLoadRPM[80] ,'o' )
title('Speed vs Armature voltage')
xlabel('Armature voltage [V] ')
ylabel('Speed [r/min] ')
annotate('+ = Zero torque',xy=(270,2300))
annotate('o = Full load torque',xy=(270,2100))
ylim(1000,2500)
xlim(250,500)
show()
Example 11.4, Page number: 571¶
In [3]:
from __future__ import division
from sympy import *
from math import *
#Variable declaration:
Rf=109 #Field resistance(ohm)
Vf=300 #Rated field voltage(V)
n1=2000 #rpm
T_rated=285 #Rated torque(Nm)
n2=1975 #Dropped rpm
Kf=0.694 #Geometric constant(A.rad/sec
Ra=0.084 #Armature resistance(ohm)
#Calculations:
If=Vf/Rf #Resulting field current(A)
wm1=2*pi*n1/60
w_ref=wm1
Vao=Kf*If*wm1
Ia=T_rated/(Kf*If)
wm2=2*pi*n2/60
Ea=Kf*If*wm2
Va=Ea+Ia*Ra
G=symbols('G')
x=solve(Vao-round(Va)+G*(w_ref-wm2),G)
#Results:
print "Armature voltage,Vao:",round(Va,0),"V"
print "Multiplicative constant,G:",float(round(x[0],2)),"A.sec/rad"
Example 11.5, Page number: 573¶
In [4]:
from __future__ import division
from math import *
from sympy import *
#Variable Declaration:
Km=0.22 #torque constant(V/(rad/sec))
Ra=1.03 #ohm
Pl=100 #Power load(W)
Va1=40 #Armature voltage(V)
Va2=50 # " " "
#Calculations:
wm1=(Va1/(2*Km))*(1+sqrt(1-(4*Pl*Ra/Va1**2)))
wm2=(Va2/(2*Km))*(1+sqrt(1-(4*Pl*Ra/Va2**2)))
#Results:
print "for Va=40 V, wm=",round(wm1,1),"rad/sec"
print "for Va=50 V, wm=",round(wm2,1),"rad/sec"
Example 11.6, Page number: 575¶
In [5]:
from __future__ import division
from math import *
#Variable declaration:
Rf=109 #Field resistance(ohm)
Vf=300 #Rated field voltage(V)
Ra=0.084 #Armature resistance(ohm)
Kf=0.694 #Geometric constant(A.rad/sec)
Tfl=285 #Full load torque(Nm)
nf=2500 #Speed at full load(r/min)
#wm=2500 #rated r/min
#for part (1):
n1=2000 #r/min
n2=2500 #r/min
#Calculations:
#part (a):
If=Vf/Rf
w1=n1*2*pi/60
w2=n2*2*pi/60
Ea1=Kf*If*w1 #Avg Amature voltage(V)
Ea2=Kf*If*w2
Ia1=n1*Tfl/(nf*Kf*If)
Ia2=n2*Tfl/(nf*Kf*If)
Va1 = Ea1 + Ia1*Ra
Va2 = Ea2 + Ia2*Ra
Tl1=(n1/nf)*Tfl
Tl2=(n2/nf)*Tfl
#part (b):
# The dynamic equation governing the speed of the motor is
# J*(dwm/dt)=Tmech-Tload
# wm=(pi/30)*n & wr=(pi/30)*nf
# Tload= (Tfl/wf)*wm
# Tmech = Kf*If*Ia=Kf*If*(Va-Ea)/Ra #Under armature-voltage control
# Thus the governing differential equation is
# d(wm)/dt + 48.4*wm - 24.7*Va = 0
# wm = wf + (wi-wf)*exp(-t/tau) #tau=1/48.4=20.7 msec
# n = 2500- 50*exp( -t/tau )
# The armature current will decrease exponentially with the
# same 20.7 msec time constant from an initial value of
# (Vf - Vi)/Ra = 1190 A to its final value of 149 A.
# Ia = 149 + 1041*exp(-t/tau)
#part (c):
# J*d(wm)/dt = Tmech-Tload = Tf-(Tf/wm)*wm
# or d(wm)/dt + 1.18*wm - 310 = 0
#In this case, the speed will rise exponentially to wm=wf=262 rad/sec as
# wm = 262-53*exp(-t/tau) #tau=1/1.18=845 msec
#Results:
print "part(a):\n"
print "-------------------------------------------------"
print "r/min\tw[rad/s]\tVa(V)\tIa(A)\tTload[Nm]"
print "-------------------------------------------------"
print n1,"\t",round(w1),"\t\t",round(Va1),"\t",round(Ia1),"\t",Tl1,"Nm"
print n2,"\t",round(w2),"\t\t",round(Va2),"\t",round(Ia2),"\t",Tl2,"Nm"
print "-------------------------------------------------"
print "\npart (b):"
print " The resultant motor speed, n = 2500 - 50*exp(-t/tau) where tau=20.7 msec"
print "\npart (c):"
print " The resultant motor speed, wm = 262 - 53*exp(-t/tau) where tau=845 msec"
Example 11.7, Page number: 581¶
In [2]:
from __future__ import division
import math
#Variable Calculations:
f1=60 #Initial frequency(Hz)
f2=50 #Changed frequency(Hz)
Xs=0.836 #Saturated synch reactance(ohm)
Va=1+0j #Armature voltage(V p.u)
Ia=1+0j #Armature current(A p.u)
If_rated=2.84 #Rated field current(A)
p=6 #No. of poles
#Calculations:
#for part (a):
ns1=120*f1/p
ns2=120*f2/p
Eaf=Va-1j*Xs*Ia*exp(1j*0) #field voltage(V)
Ifo=abs(Eaf)*If_rated #motor field current(A)
#for part(b):
#Eaf= (wm/wmo)*(If/Ifo)*Eafo
If=Ifo
#Results:
print "part(a):"
print "(i) The motor speed:",ns1,"r/min"
print "(ii) The motor field current:",round(Ifo,2),"A"
print "part(b):"
print "(i) The changed speed:",ns2,"A"
print "(ii) The mototr field current:",round(If,2),"A"
Example 11.8, Page number: 588¶
In [1]:
from __future__ import division
from math import *
from sympy import *
#Variable declaration:
iF=2.84 #rated field current(A)
Vbase=220 #base voltage(V)
Eaf=220/sqrt(3) #Rms voltage line-to-neutral(V)
f=60 #Hz
p=6 #poles
P_rated=45*10**3 #rated power(W)
Xs_pu=0.836 #per unit synchronous reactance(ohm)
#Calculations:
we=2*pi*f
Laf=sqrt(2)*Eaf/(we*iF) #Armature field reactance(H)
T_rated=P_rated/(we*2/p)
#setting rated values to reference values.
Tref=T_rated
iFref=iF
iQ=round((2/3)*(2/p)*Tref/(Laf*iFref),2)
iD=0
#since theta_me=wc*t, iD=0,we have,
t=symbols('t')
wc=120*pi
def ia(t):
return iD*cos(wc*t)-iQ*sin(wc*t)
def ib(t):
return iD*cos(wc*t-2*pi/3)-iQ*sin(wc*t-2*pi/3)
def ic(t):
return iD*cos(wc*t+2*pi/3)-iQ*sin(wc*t+2*pi/3)
Ibase=P_rated/(sqrt(3)*Eaf)
Imax=round(ia((pi/(2*wc))))
Ia=1j*abs(round(Imax/sqrt(2)))
Eaf=1j*we*Laf*iF/sqrt(2)
Zbase=Vbase**2/P_rated
Xs=Xs_pu*Zbase
Va=1j*Xs*Ia+Eaf #line-to-neutral voltage
Vt=abs(sqrt(3)*Va)/Vbase #p.u terminal voltage(line-to-line)(V)
#Results:
print "part(a):"
print "\tia(t)=",ia(t),"A"
print "\tib(t)=",ib(t),"A"
print "\tic(t)=",ic(t),"A"
print "part(b):"
print "\tTerminal voltage:",round(float(Vt),2),"per unit"
Example 11.9, Page number: 591¶
In [10]:
from __future__ import division
from math import *
#Variable declaration:
iF=2.84 #rated field current(A)
Vrated=220 #rated terminal voltage,l-l(V)
Ibase=118 #base current(A)
Eaf=220/sqrt(3) #Rms voltage, line-to-neutral(V)
f=60 #Hz
p=6 #poles
P_rated=45*10**3 #rated power(W)
Xs=0.899 #Synchronous reactance(ohm)
Xs_pu=0.836 #per unit synchronous reactance(ohm)
Tref=358 #Reference torque(Nm) (from Ex11.8)
#Calculations:
Va=Vrated/sqrt(3) #base voltage, line to neutral(V)
we=2*pi*f
wm=(2/p)*we
Laf=sqrt(2)*Eaf/(we*iF) #Armature field reactance(H)
Ia=Tref*wm/(3*Va)
Ls=Xs/we #Synchronous inductance(mH)
delta=-atan(we*Ls*Ia/Va)
iQ_ref=sqrt(2)*Ia*cos(delta)
iD_ref=sqrt(2)*Ia*sin(delta)
iF_ref=(2./3)*(2/p)*Tref/(Laf*iQ_ref)
#since motor is running at rated voltage, base voltage and rated voltage
# are assumed to be same.
Va_pu=Va/Va
Ia_pu=Ia/Ibase
#Results:
print "The reqd motor field current:",round(iF_ref,2),"A"
print "Per unit voltage:",Va_pu,"p.u"
print "Per unit current:",round(Ia_pu),"p.u"
Example 11.10, Page number: 593¶
In [11]:
from __future__ import division
from math import *
#Variable declaration:
ns=4000 #rated speed(rpm)
Va=220 #rated voltage(V)
Ls=1.75*10**-3 #synchronous inductance(H)
Prated=25000 #Watts
n=3200 #rated OC speed(rpm)
p=2 #No. of poles
#Calculations:
#for part(a):
Eaf=Va/sqrt(3)
wm=ns*pi/30 #rad/sec
Trated=Prated/wm
we=(p/2)*n*pi/30
lambdaPM=sqrt(2)*Eaf/we #flux linked wth permanent magnet(Wb)
Tref=Trated*0.65 #since motor is operated at 65% of Trated
iQref=(2./3)*(2/p)*(Tref/lambdaPM)
#for part(b:)
lambdaD=lambdaPM #since iD=0
lambdaQ=Ls*iQref
lambdaa=sqrt((lambdaD**2+lambdaQ**2)/2) #rms line-to-neutral armature flux(Wb)
lambdaa_base=Eaf/wm
lambda_pu=lambdaa/lambdaa_base
#for part(c)
lambdaD=sqrt(2*(lambdaa_base)**2-lambdaQ**2)
iDref=(lambdaD-lambdaPM)/Ls
Ia=sqrt((iDref**2+iQref**2)/2) #rms armature current(A)
Ibase=Prated/(sqrt(3)*Va)
I_pu=Ia/Ibase
#Results:
print "(a) Required quadrature-axis current:",round(iQref,1),"A"
print "(b) Resultant armature flux linkage",round(lambda_pu,2),"p.u"
print "(c) iD:",round(iDref,1),"A"
print " Rms value of armature current:",round(Ia),"A"
print " Per unit value of armature current:",round(I_pu,2),"A"
Example 11.11, Page number: 600¶
In [12]:
from __future__ import division
from math import *
#Variable declaration:
V10=230/sqrt(3)
Nph=3
p=4
fe0=60
R1=0.095 #Armature resistance(ohm)
R2=0.2 #Rotor resistance(ohm)
X10=0.680 #Armature leakage reactance(ohm)
X20=0.672 #Rotor leakage reactance(ohm)
Xm0=18.7 #Inductice reactance(ohm)
#Calculations:
#taking two frequency values:
fe1=40
fe2=60
for m in range(1,3,1):
if m==1:
fe=fe1
else:
fe=fe2
X1=X10*(fe/fe0)
X2=X20*(fe/fe0)
Xm=Xm0*(fe/fe0)
V1=V10*(fe/fe0)
ws=4*pi*fe/p
ns=120*fe/p
V1eq=abs(V1*1j*Xm/(R1+1j*(X1+Xm)))
Z1eq=1j*Xm*(R1+1j*X1)/(R1+1j*(X1+Xm))
R1eq=Z1eq.real
X1eq=Z1eq.imag
#Search over the slip until the Pload = Pmech
s=0 #slip initialised to 0
error=1
while error >=0:
s=s+0.00001
rpm=ns*(1-s)
wm=ws*(1-s)
Tmech=(1/ws)*Nph*V1eq**2*(R2/s)
Tmech = Tmech/((R1+R2/s)**2 + (X1+X2)**2)
Pmech=Tmech*wm
Pload=10.5*10**3*(rpm/1800)**3
error=Pload-Pmech
print "\nFor fe =",fe,"Hz :"
print "\tTerminal voltage=",round(V1*sqrt(3)),"V l-l"
print "\trpm =",round(rpm)
print "\tslip =",round(100*s,1),"%"
print "\tPload =",round(Pload/1000,2),"kW"
Example 11.12, Page number: 608¶
In [13]:
from __future__ import division
from math import *
#Variable declaration:
V10=230/sqrt(3)
Nph=3
p=4
fe0=60
R1=0.095 #Armature resistance(ohm)
R2=0.2 #Rotor resistance(ohm)
X10=0.680 #Armature leakage reactance(ohm)
X20=0.672 #Rotor leakage reactance(ohm)
Xm0=18.7 #Inductice reactance(ohm)
n=1680 #rpm
Pmech=9.7*10**3 #Electromagnetic power(W)
#Calculations:
we0=2*pi*fe0
Lm=Xm0/we0
LS=Lm+X10/we0
LR=Lm+X20/we0
Ra=R1
RaR=R2
lambda_rated=sqrt(2)*V10/we0
lambdaDR=lambda_rated
#for specified operating condition
wm=n*(pi/30)
Tmech=Pmech/wm
iQ=(2/3)*(2/p)*(LR/Lm)*(Tmech/lambdaDR)
iD=lambdaDR/Lm
Ia=sqrt((iD**2+iQ**2)/2)
wme=(p/2)*wm
we=wme+(RaR/LR)*(iQ/iD)
fe=we/(2*pi)
Va=sqrt(((Ra*iD-we*(LS-Lm**2/LR)*iQ)**2 + (Ra*iQ+we*LS*iD)**2)/2)
#Results:
print "Rms amplitude of the armature current:",round(Ia,1),"A"
print "The electrical frequency:",round(fe,1),"Hz"
print "Rms terminal voltage:",round(sqrt(3)*Va,1),"V line-line"
Example 11.13, Page number: 610¶
In [6]:
from __future__ import division
%matplotlib inline
from math import *
from pylab import *
#Variable declaration:
P_rated = 12*10**3 #Watts
V_rated = 230 #Rated line-line voltage(v)
Va_rated = 230/sqrt(3) #Rated line to neutral(V)
fe_rated = 60 #Hz
we_rated = 2*pi*fe_rated #rad/sec
lambda_rated = sqrt(2)*Va_rated/we_rated #Wb
I_rated = P_rated/(sqrt(3)*V_rated) #A
Ipeak_base = sqrt(2)*I_rated #A
p = 4 #poles
V10=V_rated/sqrt(3)
R1=0.095 #Armature resistance(ohm)
R2=0.2 #Rotor resistance(ohm)
X10=0.680 #Armature leakage reactance(ohm)
X20=0.672 #Rotor leakage reactance(ohm)
Xm0=18.7 #Inductice reactance(ohm)
#Calculations:
Lm = Xm0/we_rated;
LS = Lm + X10/we_rated;
LR = Lm + X20/we_rated;
Ra = R1
RaR = R2
#operating point:
n = 1680 #rpm
lambdaDR=lambda_rated
wm = n*pi/30
wme = (p/2)*wm
Pmech = 9.7*10**3
Tmech = Pmech/wm
lambda_DRpu=[0]*42
iDpu=[0]*42
Iapu=[0]*42
fe=[0]*42
Vapu=[0]*42
for n in range(1,43,1):
lambdaDR = (0.8+(n-1)*0.4/40)*lambda_rated
lambda_DRpu[n-1]=lambdaDR/lambda_rated
iQ=(2/3)*(2/p)*(LR/Lm)*(Tmech/lambdaDR)
iD=(lambdaDR/Lm)
iDpu[n-1]=iD/Ipeak_base
iQR=-(Lm/LR)**iQ
Ia=sqrt((iD**2+iQ**2)/2)
Iapu[n-1]=Ia/I_rated
we=wme-(RaR/LR)*(iQ/iD)
fe[n-1]=we/(2*pi)
Va_rms=sqrt(((Ra*iD-we*(LS-Lm**2/LR)*iQ)**2 +(Ra*iQ+ we*LS*iD)**2)/2)
Vapu[n-1]=Va_rms/Va_rated
#Results:
print "The required plot is as shown:"
plot(iDpu,Iapu)
plot(iDpu,Vapu,':')
xlabel('i_D [per unit] ')
ylabel('per unit')
annotate('Ia',xy=(0.21,1.05))
annotate('Va',xy=(0.21,0.85))
show()
