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"
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"
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"
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()
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"
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"
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"
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."
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"
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"