Chapter 38: Magnetic materials

Example 1, page no. 694

In [1]:
from __future__ import division
import math
import cmath
#initializing  the  variables:
A  =  12.5;#  in  cm2
x  =  500;#  horizontal  axis  1  cm  =  500  A/m
y  =  0.2;#  vertical  axis  1  cm  =  0.2  T
f  =  50;#  in  Hz

 #calculation:  
 #hysteresis  loss  per  cycle
HL  =  A*x*y
 #At  50  Hz  frequency,  hysteresis  loss
HLf  =  HL*f


#Results
print  "\n\n  Result  \n\n"
print  "\n(a)hysteresis  loss  per  cycle  is  =  ",HL,"  J/m3"
print  "\n(b)At  50  Hz  frequency,  hysteresis  loss  ",HLf,"  W/m3"


  Result  



(a)hysteresis  loss  per  cycle  is  =   1250.0   J/m3

(b)At  50  Hz  frequency,  hysteresis  loss   62500.0   W/m3

Example 2, page no. 695

In [2]:
from __future__ import division
import math
import cmath
#initializing  the  variables:
n  =  1.6;#  the  Steinmetz  index
f1  =  50;#  in  Hz
f2  =  25;#  in  Hz
Bm1  =  1.5;#  in  Tesla
Bm2  =  1.1;#  in  Tesla
Ph1  =  62500;#  in  W/m3
v  =  1;

#calculation:  
 #hysteresis  loss  Ph  =  kh*v*f*(Bm)**n
kh  =  Ph1/(v*f1*(Bm1)**n)
 #When  f  =  25  Hz  and  Bm  =  1.1  T,
Ph2  =  kh*v*f2*(Bm2)**n


#Results
print  "\n\n  Result  \n\n"
print  "\n  hysteresis  loss  When  f  =  25  Hz  and  Bm  =  1.1  T,  is  =  ",round(Ph2,2),"  W/m3"


  Result  



  hysteresis  loss  When  f  =  25  Hz  and  Bm  =  1.1  T,  is  =   19025.33   W/m3

Example 3, page no. 695

In [3]:
from __future__ import division
import math
import cmath
#initializing  the  variables:
csa  =  0.002;#  in  m2
l  =  1;#  in  m
a  =  400/0.01;#  10  mm  =  400  A/m  
b  =  0.1/0.01;#  10  mm  =  0.1  T  
A  =  0.01;#  in  m2
f  =  80;#  in  Hz

#calculation:  
 #hysteresis  loss  per  cycle
HL  =  A*a*b
 #At  a  frequency  of  80  Hz,
 #hysteresis  loss
HLf  =  HL*f
 #Volume  of  ring
v  =  csa*l
 #hysteresis  loss
Ph  =  HLf*v


#Results
print  "\n\n  Result  \n\n"
print  "\n  the  hysteresis  loss  at  a  frequency  of  80  Hz  is  ",Ph,"  W"


  Result  



  the  hysteresis  loss  at  a  frequency  of  80  Hz  is   640.0   W

Example 4, page no. 696

In [1]:
from __future__ import division
import math
import cmath
#initializing  the  variables:
Phi1  =  0.01;#  in  Wb
Phi2  =  0.008;#  in  Wb
csa  =  0.008;#  in  m2
v  =  0.005;#  in  m3
f  =  50;#  in  Hz
n  =  1.7;#  the  Steinmetz  constant
Ph1  =  100;#  in  Watt

 #calculation:  
 #maximum  flux  density
Bm1  =  Phi1/csa
 #hysteresis  loss  Ph1  =  kh*v*f*(Bm1)**n
kh  =  Ph1/(v*f*(Bm1)**n)
 #When  the  maximum  core  flux  is  8  mWb,
Bm2  =  Phi2/csa
 #hysteresis  loss,  Ph2
Ph2  =  kh*v*f*(Bm2)**n


#Results
print  "\n\n  Result  \n\n"
print  "value  of hysteresis  loss  when  maximum  core  flux  is  8  mWb  and  the  frequency  is  50  Hz  is  ",round(Ph2,2),"  W"


  Result  


value  of hysteresis  loss  when  maximum  core  flux  is  8  mWb  and  the  frequency  is  50  Hz  is   68.43   W

Example 5, page no. 699

In [5]:
from __future__ import division
import math
import cmath
#initializing  the  variables:
Pe1  =  10;#  in  W/m3
f1  =  50;#  in  Hz
f2  =  30;#  in  Hz

 #calculation:  
 #When  the  eddy  current  loss  is  10  W/m3,  frequency  f  is  50  Hz.
 #constant  k
k  =  Pe1/(f1**2)
 #When  the  frequency  is  30  Hz,  eddy  current  loss,
Pe2  =  k*(f2**2)


#Results
print  "\n\n  Result  \n\n"
print  "\neddy  current  loss  per  cubic  metre  is  ",Pe2,"  W/m3"


  Result  



eddy  current  loss  per  cubic  metre  is   3.6   W/m3

Example 6, page no. 699

In [6]:
from __future__ import division
import math
import cmath
#initializing  the  variables:
Pe  =  100;#  in  W/m3
f1  =  50;#  in  Hz
t1  =  0.0005;#  in  m
x  =  1/3;
f2  =  250;#  in  Hz
Bm1 = 1;
 #calculation:  
 #Pe  =  ke*(Bm1*f1*t1)**2
 #Hence,  at  50  Hz  frequency
ke  =  Pe/(Bm1*f1*t1)**2
 #At  250  Hz  frequency
Bm2  =  x*Bm1
t2  =  ((Pe/ke)**0.5)/(Bm2*f2)


#Results
print  "\n\n  Result  \n\n"
print  "\nlamination  thickness  is  ",t2,"m"


  Result  



lamination  thickness  is   0.0003 m

Example 7, page no. 700

In [2]:
from __future__ import division
import math
import cmath
#initializing  the  variables:
Ph1  =  40;#  in  W
Pe1  =  20;#  in  W
f1  =  50;#  in  Hz
x  =  1/2;
f2  =  60;#  in  Hz
t1  =  1;
 #calculation:  
 #hysteresis  loss  Ph  =  kh*v*f*(Bm)**n  =  k1*f
 #Thus  when  the  hysteresis  is  40  W  and  the  frequency  50  Hz,
k1  =  Ph1/f1
 #If  the  frequency  is  increased  to  60  Hz,
Ph2  =  k1*f2
 #eddy  current  loss,  Pe  =  ke*(Bm1*f1*t1)**2  =  k2*f**2
 #since  the  flux  density  and  lamination  thickness  are  constant.
 #When  the  eddy  current  loss  is  20  W  the  frequency  is  50  Hz.  Thus
k2  =  Pe1/(f1**2)
 #If  the  frequency  is  increased  to  60  Hz,
Pe2  =  k2*(f2**2)
 #hysteresis  loss  Ph  =  kh*v*f*(Bm)**n,  is  independent  of  the  thickness  of  the  laminations.  
    #Thus,  if  the  thickness  of  the  laminations  is  halved,  the  hysteresis  loss  remains  at  
Phb2  =  Ph1
 #eddy  current  loss,  Pe  =  ke*(Bm1*f1*t1)**2  =  k2*t**3
k3  =  Pe1/(t1**3)
t2  =  0.5*t1
Peb2  =  k3*t2**3
 #total  core  loss  when  the  thickness  of  the  laminations  is  halved  is  given  by
TL  =  Phb2  +  Peb2


#Results
print  "\n\n  Result  \n\n"
print  "\n(a)If  the  frequency  is  increased  to  60  Hz,hysteresis  loss  is  ",Ph2,"  W  and  eddy  current  loss  ",  Pe2,"  W"
print  "\n(b)the  total  core  loss  when  the  thickness  of  the  laminations  is  halved  ",TL,"  W"


  Result  



(a)If  the  frequency  is  increased  to  60  Hz,hysteresis  loss  is   48.0   W  and  eddy  current  loss   28.8   W

(b)the  total  core  loss  when  the  thickness  of  the  laminations  is  halved   42.5   W

Example 8, page no. 701

In [8]:
from __future__ import division
import math
import cmath
#initializing  the  variables:
V1  =  500;#  in  Volts
V2  =  1000;#  in  Volts
Ph1  =  400;#  in  W
Pe1  =  150;#  in  W
f1  =  50;#  in  Hz
n  =  1.6;#  Steinmetz  index
f2  =  100;#  in  Hz

 #calculation:  
 #hysteresis  loss  Ph  =  k1*f*(E/f)**n
 #At  500  V  and  50  Hz
k1  =  Ph1/(f1*(V1/f1)**1.6)
 #At  1000  V  and  100  Hz,
Ph2  =  k1*f2*(V2/f2)**1.6
 #eddy  current  loss,  Pe  =  k2*E**2
 #At  500  V,
k2  =  Pe1/(V1**2)
 #At  1000  V,
Pe2  =  k2*(V2**2)
 #the  new  total  core  loss
TL  =  Ph2  +  Pe2


#Results
print  "\n\n  Result  \n\n"
print  "\n  the  new  total  core  loss  ",TL,"  W"


  Result  



  the  new  total  core  loss   1400.0   W

Example 9, page no. 702

In [1]:
from __future__ import division
import math
import cmath
#from pylab import *
%matplotlib inline
#initializing  the  variables:
f1  =  50;#  in  Hz
f2  =  60;#  in  Hz

#calculation:
k1  =  0.5
k2  =  0.032
f = [30, 50, 70,90]
Pcf = [1.5, 2.1, 2.7, 3.4]
fig  = plt.figure()
ax = fig.add_subplot(1, 1, 1)
ax.plot(f,Pcf,'-')
xlabel('f')
ylabel('Pc/f')
show()
HL1 = k1*f1
ECL1 = k2*f1**2

HL2 = k1*f2
ECL2 = k2*f2**2

#Results
print  "\n\n  Result  \n\n"
print  "\n  (a) at Frequency = 50 Hz, hysteresis loss is", HL1," W and eddy current loss is", ECL1," W"
print  "\n  (b) at Frequency = 60 Hz, hysteresis loss is", HL2," W and eddy current loss is", ECL2," W"

Welcome to pylab, a matplotlib-based Python environment [backend: module://IPython.zmq.pylab.backend_inline].
For more information, type 'help(pylab)'.

  Result  



  (a) at Frequency = 50 Hz, hysteresis loss is 25.0  W and eddy current loss is 80.0  W

  (b) at Frequency = 60 Hz, hysteresis loss is 30.0  W and eddy current loss is 115.2  W

Example 10, page no. 703

In [3]:
from __future__ import division
import math
import cmath
#initializing  the  variables:
TL1  =  400;#  in  Watt
TL2  =  498;#  in  Watt
x  =  0.25;
y  =  0.4;
f1  =  50;#  in  Hz
n  =  1.7;#  Steinmetz  index
f2  =  60;#  in  Hz

 #calculation:  
 #if  volume  v  and  the  maximum  flux  density  are  constant
 #hysteresis  loss  Ph  =  kh*v*f*(Bm)**n  =  k1*f
 #(if  the  maximum  flux  density  and  the  lamination  thickness  are  constant)
 #eddy  current  loss,  Pe  =  ke*(Bm1*f1*t1)**2  =  k2*f**2
 #At  50  Hz  frequency,  TL1  =  k1*f1  +  k2*f1**2
 #At  60  Hz  frequency,  TL2  =  k1*f2  +  k2*f2**2
 #Solving  equations  gives  the  values  of  k1  and  k2.
k2  =  (5*TL2  -  6*TL1)/(5*(f2**2)  -  6*(f1**2))
k1  =  (TL1  -  k2*f1**2)/f1
 #hysteresis  loss  Ph  =  k1*f
Ph1  =  k1*f1
 #eddy  current  loss
Pe1  =  k2*f1**2
 #Since  at  50  Hz  the  flux  density  is  increased  by  25%,  the  new  hysteresis  loss  is
Ph2  =  Ph1*(1  +  x)**1.7
 #Since  at  50  Hz  the  flux  density  is  increased  by  25%,  and  the  lamination  thickness  is  increased  by  40%, 
    #the  new  eddy  current  loss  is
Pe2  =  Pe1*((1  +  x)**2)*(1  +  y)**3


#Results
print  "\n\n  Result  \n\n"
print  "\n  (a)the  hysteresis  and  eddy  current  losses  at  50  Hz  are  ",round(Ph1,2),"  W  and  ",round(  Pe1,2),"  W  resp."
print  "\n  (b)the  hysteresis  and  eddy  current  losses  at  50  Hz  after  increement  are  ",round(Ph2,2),"  W  and  ",round(  Pe2,2),"  W  resp."


  Result  



  (a)the  hysteresis  and  eddy  current  losses  at  50  Hz  are   325.0   W  and   75.0   W  resp.

  (b)the  hysteresis  and  eddy  current  losses  at  50  Hz  after  increement  are   474.93   W  and   321.56   W  resp.