Chapter 1:Introduction to Magnetic Circuits

Example 1.1, Page number: 9

In [1]:
from __future__ import division
from math import *
#Variable declaration:
Ac=9                            #Cross-sectional area of the core(cm**2)
Ag=9                            #Cross-sectional area of the air-gap(cm**2)
g=0.050                         #Air-gap length(cm)            
lc=30                           #Mean Length of the core(cm)
N=500                           #No. of windings
ur=70000                        #Relative permeability of the core material
Bc=1.0                          # Magnetic Flux Density of the core(T)
uo=4*pi*10**-7                  #Permeability of free space

#Calculation
Rc=lc*10**-2/((ur*uo*Ac)*10**-4)
Rg=g*10**-2/((uo*Ag)*10**-4)
Q=Bc*Ac*10**-4
i=Q*(Rc+Rg)/N

#Results
print "a.Reluctance of the core,Rc:",round(Rc,2), "A.turns/Wb" 
print "  Reluctance of the air-gap,Rg:", round(Rg,2), "A.turns/Wb"
print "b.The flux, Q:", round(Q,4), "Wb"
print "c.The current,i:", round(i,2), "A"
a.Reluctance of the core,Rc: 3789.4 A.turns/Wb
  Reluctance of the air-gap,Rg: 442097.06 A.turns/Wb
b.The flux, Q: 0.0009 Wb
c.The current,i: 0.8 A

Example 1.2, Page number: 10

In [2]:
from __future__ import division
from math import *
#Variable declaration:
I=10                            #Current in the coil(A)
N=1000                          #No of turns in the rotor
g=1                             #Air gap length(cm)
Ag=2000                         #Cross-section of the air-gap(cm**2)
uo=4*pi*10**-7                  #Permeability of free space

#Calculation:
Q=(N*I*uo*Ag*10**-4)/(2*g*10**-2)
Bg=round(Q,2)/(Ag*10**-4)

#Results
print "The air-gap flux, Q:", round(Q,2), "Wb"
print "The flux density, Bg:", round(Bg,4), "T"
The air-gap flux, Q: 0.13 Wb
The flux density, Bg: 0.65 T

Example 1.4, Page number: 13

In [3]:
from __future__ import division
from math import *
#Variable declaration
lc=0.3                                  #length of the core(cm)
ur1=72300                               #Relative permeablity for case(a)
ur2=2900                                #Relative permeablity for case(b)
Ac=9                                    #Cross-section of the core(cm**2)
Rg=4.42*10**5                           #Reluctance of the air-gap(A.turns/Wb)
N=500                                   #No of coil turns
uo=4*pi*10**-7                          #Permeability of free space(H/m)

#Calculations:
Rt1=(lc/(ur1*uo*Ac*10**-4))+Rg
L1=N**2/Rt1
Rt2=(lc/(ur2*uo*Ac*10**-4))+Rg
L2=N**2/Rt2



#Results:
print "(a)Inductance,L:",round(L1,2),"H"
print "(b)Inductance,L:",round(L2,2),"H"
(a)Inductance,L: 0.56 H
(b)Inductance,L: 0.47 H

Example 1.5, Page number: 15

In [1]:
from __future__ import division
from pylab import *
from matplotlib import *
from math import *
%matplotlib inline
#Variable declaration:
Ac=9e-4                                 #Cross-section of the core(m)
Ag=9e-4                                 #Cross-section of the air-gap(m)
g=5e-4                                  #Air-gap length(m)
lc=0.3                                  #Mean length of the core(m)
N=500                                   #No. of turns of the core(m)
uo=4*pi*10**-7                          #Permeability of free space(H/m)

#Calculations:
Rg=g/(uo*Ag)                            #Reluctance of the air-gap(A.turns/Wb)
ur=[0]*200                              #Initialising array
L=[0]*200

for n in range(1,101,1):
    ur[n-1]=100+(10000-100)*(n-1)/100
    Rc=lc/(ur[n-1]*uo*Ac)               #Reluctance of the core(A.turns/Wb)
    Rtot=Rg+Rc
    L[n-1]=(N**2)/Rtot                  #Inductance(H)
    

#Results:
print "The reqired plot is shown below:"
plot(ur, L,'g.')
xlabel('Core relative permeability, ur')
ylabel('Inductance,L (H) ')
title('plot of inductance vs. relative permeability for Example 1.5.')
show()
The reqired plot is shown below:

Example 1.6, Page number: 19

In [5]:
from __future__ import division
from sympy import *

#Variable declaration:
Bc=1.0                                 #Magnetic field induction in the core
w=377                                  #Angular frequency of magnetic field(rad/s)
Rc=3791.33                             #Reluctance of the core(A.turns/Wb)
Rg=442321.3                            #Reluctance of the air-gap(A.turns/Wb)
N=500                                  #No. of windings
i=0.80                                 #Current in the coil
Ac=9*10**-4                            #Cross-section of the core


#Calculations:
L=N**2/(Rc+Rg)
W=(1./2)*L*i**2
t = symbols('t')
Bc = 1.0*sin(w*t)
e=N*Ac*diff(Bc,t)

#Results:
print "The Inductance, L:", round(L,2), "H"
print "The magntic stored energy, W:", round(W,2), "J"
print "Induced voltage, e:",e,"V"
The Inductance, L: 0.56 H
The magntic stored energy, W: 0.18 J
Induced voltage, e: 169.65*cos(377*t) V

Example 1.7, Page number: 22

In [6]:
from __future__ import division
from math import *
#Variable declaration:
Bc=1                                    #Magnetic field in the core
Hc=11                                   #Magnetising force(A.turns/m)
lc=0.3                                  #length of the core(m)
N=500                                   #No of windings
g=0.050                                 #Air-gap length(cm)
uo=4*pi*10**-7                          #Permeability of free space(H/m)


#Calculation:
Fc=Hc*lc                                #mmf drop for the core path(A.turns)
Fg=Bc*g*10**-2/uo                       #mmf drop across the air gap(A.turns)
i=(Fc+Fg)/N


#Results:
print "The required current,i:" ,round(i,2) ,"A"
The required current,i: 0.8 A

Example 1.8, Page number: 28

In [8]:
from __future__ import division
from sympy import *

#Variable declaration:
N=200                               #No. of turns
Ac=4                                #Cross-section of the core(in**2)
w=377                               #Angular frequency of the magnetic field(rad/s)
Hm=36                               #Max value magnetising force(A.turns/m)
Pc=1.2                              #Core loss density(W/kg)


#Calculations:
t=symbols('t')
Bc=1.5*sin(w*t)
e=(round(N*Ac*0.94/(39.4**2),2)*diff(Bc,t))
Erms=275*0.707
lc=(6+6+8+8)/39.4                   #Mean length of the core(m)
I=Hm*lc/N
Vc=4*0.94*28                        #Core volume(m**3)
Wc=105.5*(2.54**3)*7.65*10**-3                  #Core weight(kg)
Pa=1.5*13.2                         #Watts per Kg
Irms=Pa/Erms                        #Current (A)
Pct=Pc*Wc                           #Total core loss(W)


#Results:
print "The applied voltage,e:", e, "V"
print "The peak current,I:", round(I,2), "A"
print "The total rms current. Irms:", round(Irms,2), "A"
print "Total Core loss, Pct:",round(Pct,2),"W"
The applied voltage,e: 271.44*cos(377*t) V
The peak current,I: 0.13 A
The total rms current. Irms: 0.1 A
Total Core loss, Pct: 15.87 W

Example 1.9, Page number: 32

In [10]:
from __future__ import division
from sympy import *
from math import *

#Variable Declaration:
g=0.2                               #air-gap length(cm)
lm=1.0                              #length of magnetic section(cm)
Am=4                                #Cross-section of the core(cm**2)
Ag=4                                #Cross-section of the air-gap(cm**2)

#Constants used:
uo=4*pi*10**-7                      #Permeability of free space(H/m)

#Calculations:
Hm=symbols('Hm')
def Bg(Hm):
    return -uo*Ag*lm*Hm/(Am*g)             

Hm1=-49*10**3                    #Coercivity of ALNICO 5 (A/m)
Hm2=-6                          #Coercivity of M-5 electrical steel (A/m)                     


#Results:
print "Flux Density of air gap:", round(Bg(Hm1),2),"T"
print "\nFlux Density of air gap:", round(Bg(Hm2*10**4),2),"gauss"
print "\nwhere value of Hm for different material."
Flux Density of air gap: 0.31 T

Flux Density of air gap: 0.38 gauss

where value of Hm for different material.

Example 1.10, Page number: 34

In [11]:
from __future__ import division
from sympy import *
from math import *

#Variable declaration:
Ag=2                                #Cross-section of air-gap(cm**2) 
Bg=0.8                              #Air-gap flux density(t)
Bm=1.0                              #Core-flux density(T)
Hm=-40                              #Magnetising force in the core(kA/m)
uo=4*pi*10**-7                      #permeability of free space(H/m)
g=0.2                               #Air-gap length(cm)

#Calculations:
Am=Ag*Bg/Bm
lm=-g*Bg/(Hm*uo*10**3)
Vm=Am*lm


#Results:
print "The minimum magnet volume,Vm:",round(Vm,2),"cm**3"
The minimum magnet volume,Vm: 5.09 cm**3

Example 1.11, Page number: 39

In [12]:
from __future__ import division
from sympy import *
from math import *

#Variable Declaration:
Am = 2                    #magnetic material cros-section(cm^2)
g=0.2                     #air gap length(cm)
uo=4*pi*10**-7            #permeability of free space(H/m)
N=100                     #No. of windings

#Calculations and results:
#for part (a)
Bma = 1.0                 #Tesla
Hma = - 4                 #kA/m
Ag1 = 2                    #cm**2
Ag2 = 4                   #cm**2

lm=g*(Am/Ag1)*(Bma/(-uo*Hma*10**4))
print "(a) The Requied magnet length = ",round(lm,2),"cm"


#for part (b):
i,Hm=symbols('i Hm')
Bm=-uo*(Ag1/Am)*(lm/g)*Hm+(uo*N/g)*(Ag1/Am)*i
H_max=200               #kA/m
B_max=2.1               #Tesla
i_max=(B_max+2.50*10**-5*H_max)/(6.28*10**-2)

print "(b) Thus with the air-gap area set to 2 cm^2,"
print "    increasing the current to i_max = 45.2 A and then reducing"
print "    it to zero will achieve the desired magnetization."

#for part (c):
Bm1=1.00               #Tesla
Bm2=1.08               #Tesla
Bg1=(Am/Ag1)*Bm1
Bg2=(Am/Ag2)*Bm2
print "(c) The flux densities when plunger moves at two extremes are:"
print "    Bg1 =",Bg1,"T and Bg2 =",Bg2,"T"
(a) The Requied magnet length =  3.98 cm
(b) Thus with the air-gap area set to 2 cm^2,
    increasing the current to i_max = 45.2 A and then reducing
    it to zero will achieve the desired magnetization.
(c) The flux densities when plunger moves at two extremes are:
    Bg1 = 1.0 T and Bg2 = 0.54 T