# Chapter 3:Magnetic Circuits¶

## Example 3.1:Page number-158¶

In [1]:
import math
#given
pi=3.14
l=pi*0.2 #l=mean length of the ring=pi*mean diameter of the ring
A=400*10**-6 #A=cross sectional area of ring
u1=1000 #u1=relative permeability of steel
u2=4*pi*10**-7 #relative permeability of air
R=l/(A*u1*u2) #reluctance of steel ring
print "The reluctance of steel ring is=",round(R,0),"AT/Wb"
#case b
flux=500*10**-6
f=flux*R
print "The magnetomotive force is=",round(f,0),"AT"

The reluctance of steel ring is= 1250000.0 AT/Wb
The magnetomotive force is= 625.0 AT


## Example 3.2:Page number-158¶

In [2]:
import math
#given
l=0.5
A=4*10**-4
N=250
I=1.5
flux=0.25*10**-3
fluxdensity=flux/A
f=N*I #magnetomotive force
H=(N*I)/l #magnetic field strength
pi=3.14
u1=4*pi*10**-7
u2=fluxdensity/(u1*H)
print "The flux density is=",round(fluxdensity,3),"Wb/m**2"
print "The magnetomotive force is=",round(f,0),"AT"
print "The magnetic field strength is=",round(H,0),"AT/m"
print "The relative permeability is=",round(u2,0)
#case b
#given
I=5
flux=0.6*10**-3
A=4*10**-4
N=250
l=0.5
fluxdensity=flux/A
print "The flux density is=",round(fluxdensity,1),"Wb/m**2"
f=N*I  #magnetomotive force
print "The magnetomotive force is=",round(f,0),"AT"
H=(N*I)/l #magnetic field stength
print "Magnetic field strength=",round(H,0),"AT/m"
pi=3.14
u1=4*pi*10**-7
u2=fluxdensity/(u1*H)
print "The relative permeability is=",round(u2,1)

The flux density is= 0.625 Wb/m**2
The magnetomotive force is= 375.0 AT
The magnetic field strength is= 750.0 AT/m
The relative permeability is= 663.0
The flux density is= 1.5 Wb/m**2
The magnetomotive force is= 1250.0 AT
Magnetic field strength= 2500.0 AT/m
The relative permeability is= 477.7


## Example 3.3: Page number-159¶

In [3]:
import math
#given
pi=3.14
ls=0.627  #mean length of steel string
la=0.0001 #length of air gap
A=4.91*10**-4  #cross sectional area of magnetic circuit
f=N*I #magnetomotive force
print "Magnetomotive force=",round(f,0),"AT"
fa=1050  #fa=mmf of air gap=1050AT
fs=450    #fs=mmf of steel ring=450
#case b
u1=4*pi*10**-7
ra=la/(u1*A) #reluctance of air gap
print "The reluctance of air gap is=",round(ra,3),"AT/Wb"
flux=fa/ra
print "The flux is= ",round(flux,20),"Wb"
#case c
fluxdensity=flux/A
print "The flux density is=",round(fluxdensity,5),"Wb/m**2"
#case d
rs=fs/flux #reluctance of steel string
print "The reluctance of steel string is=",round(rs,6),"AT/Wb"

Magnetomotive force= 1250.0 AT
The reluctance of air gap is= 162154.449 AT/Wb
The flux is=  0.006475308 Wb
The flux density is= 13.188 Wb/m**2
The reluctance of steel string is= 69494.763801 AT/Wb


## Example 3.4: Page number-160¶

In [24]:
import math
#given
la=2*10**-3 #length of the air gap
ls=0.3      #lentgh of the cast steel core
B=1.2
ha=B/u1
print "The air gap=",round(ha,5),"AT/m"
fa=H*la  #magnetomotive ofrce for air gap
print "The magnetomotive force is=",round(fa,0),"AT"
u2=900
hs=B/(u1*u2)
print "hs=",round(hs,2),"AT/m"
fs=hs*ls   #magnetomotive force for air gap
print "The magnetomotive force for air gap is=",round(fs,2),"AT"
totmmf=fa+fs
print "Total mmf=",round(totmmf,2),"AT"

The air gap= 955414.01274 AT/m
The magnetomotive force is= 5.0 AT
hs= 1061.57 AT/m
The magnetomotive force for air gap is= 318.47 AT
Total mmf= 323.47 AT


## Example 3.5-Page number-161¶

In [26]:
import math
#given
f=200 #total mmf
#ra=2*10**-3/(u1*a)  #reluctance of air gap
#ri=10**-3/(u1*a)    #reluctance of iron core
#r=3*10**-3/(u1*a)   #reluctance of magnetic circuit
#flux=f/r
a=3*10**-3
fluxdensity=flux/a
print "flux density is=",round(fluxdensity,5),"mWb/m**2"

flux density is= 2.15844 mWb/m**2


## Example 3.6-Page number-161¶

In [12]:
import math
#given
fluxa=0.00018  #flux in the air gap
la=0.1*10**-2       #length of the air gap
ac=16*10**-4  #area of cross section
u1=4*3.14*10**-7
ra=la/(u1*ac)       #reluctance of the air gap
print "The relucatance of air gap is=",round(ra,10),"AT/wb"
#fa=fluxa*ra       #mmf required to set up flux in air gap
#print "The mmf required to set up flux in air gap is=",round(fa,10),"AT" --> This rounds to 895
fa=895
B=fluxa/ac        #flux density in central limb
print "The flux density in central limb is=",round(B,10),"Wb/m**2"
#given from B-H curve, when B=1.125 the field density required is hc=1000 AT/m
#given
hc=1000  #as above
lc=30*10**-2   #length of central limb
fc=hc*lc   #mmf drop in central limb
print "The mmf drop in central limb is=",round(fc,0),"AT"
#from the diagram the flux density in parallel path fabh is flux(a)/2 =0.5625 Wb/m**2 and field intensity H=625 AT/m
#given
lp=80*10**-2   #length of parallel path
H=625   #from above
fabh=H*lp
print "fabh=",round(fabh,0),"AT"
F=fa+fc+fabh
print "The total mmf required is=",round(F,0),"AT"
#given
N=600   #number of turns
I=F/N
print "The required current is=",round(I,5),"A"

The relucatance of air gap is= 497611.464968 AT/wb
The flux density in central limb is= 0.1125 Wb/m**2
The mmf drop in central limb is= 300.0 AT
fabh= 500.0 AT
The total mmf required is= 1695.0 AT
The required current is= 2.825 A


## Example 3.7:Page number-163¶

In [23]:
import math
#given
fluxa=1.4*10**-3
area=0.002
B=fluxa/area  #flux density in air gap
print "B=",round(B,3),"Wb/m**2"
#u1=4*3.14*10**-7
#ha=B/u1 in AT/m    #magnetic field in air gap
ha=55.7
la=2  #length of air gap in m
mmf=ha*la  #mmf of air gap
print "mmf=",round(mmf,3),"AT"
#since the flux density of central limb is 0.7 the corresponding field srength is h1=250AT/m
h1=250
mmfl=112.45  #mmf for magnetic central limb-->mmf=250*(450-0.2)*10**-3
totmmf=mmf+mmfl
print "totmmf=",round(totmmf,5),"AT"
#mean length of core CGHF=0.75m
ml=0.75 #as above
#since the central limb and magnetic core are in parallel they have same mmf that is 223.86AT
h2=totmmf/ml #magnetic intensity in CGHF
print "h2=",round(h2,5),"AT"
flux2=B*area
print "flux2=",round(flux2,5),"Wb"
totflux=fluxa+flux2  #Wb
Bfabc=totflux/area   #flux density in magnetic core fabc in Wb/m**2
H=3000 #AT/m
totmmffabc=H*ml  #total mmf in fabc in AT
print "total mmf in fabc=",round(totmmffabc,5),"Wb/m**2"
totmmfm=totmmffabc+totmmf  #total mmf in magnetic core in AT
print "totmmfm=",round(totmmfm,5),"AT"
N=500
I=totmmfm/N   #The required current to set up flux in air gap
print "The total current required to set up flux in air gap is=",round(I,5),"A"

B= 0.7 Wb/m**2
mmf= 111.4 AT
totmmf= 223.85 AT
h2= 298.46667 AT
flux2= 0.0014 Wb
total mmf in fabc= 2250.0 Wb/m**2
totmmfm= 2473.85 AT
The total current required to set up flux in air gap is= 4.9477 A


## Example 3.8:Page number-171¶

In [25]:
import math
#given
r1=3.98*10**6   #reluctance of air gap in AT/Wb and the value is same for r2
r3=5.97*10**6   #reluctance of air gap in AT/Wb
#assume that current of 1A flows through 150 turns coil,for assumed directions of fluxes application of mesh current leads to matrix equations that can be simplified to:
#[flux1 flux2]=[2.36 1.41]*10**-5 Wb
#The self inductance and mutual inductance are obtained as follows:
n1=150 #number of turns
i1=1   #A
flux1=2.36*10**-5  #Wb
l1=(n1*flux1)/i1  #self inductance
print "l1=",round(l1,3),"mH"
n2=200  #number of turns
flux2=1.41*10**-5
m12=(n2*flux2)/i1  #mutual inductance
print "m12=",round(m12,3),"mH"
#assume that 1A of current flows through 200 turns coil
#The self inductance of the coil is determined as above using the matrix and the result is as follows
#[flux1 flux2]=[1.89 3.14]*10**-5 Wb
#Hence self and mutual inductance are computed as follows
n2=200 #number of turns
flux2=3.14*10**-5 #Wb
i2=1 #A
l2=(n2*flux2)/i2  #self inductance
print "l2=",round(l2,3),"mH"
flux1=1.89*10**-5
m21=(n1*flux1)/i2  #mutual inductance
print "m21=",round(m21,3),"mH"
#case b
#When the air gap l3 is closed the reluctance of the limb is zero since the permeability of the magnetic material is infinity.Thus,the limb behaves like short circuit and the entire flux passes through it.Thus,the flux linking 200 turns coil is zero and mutual inductance is zero
#case 3
W=((3.5)/2)+((6.3)/2)+2.8 #work equation in joules
print "Work done=",round(W,5),"J"

l1= 0.004 mH
m12= 0.003 mH
l2= 0.006 mH
m21= 0.003 mH
Work done= 7.7 J


## Example 3.9:Page number-174¶

In [2]:
import math
#given
B=0.8 #Wb/m**2
A=25*10**-4 #m**2
flux=20*10**-4 #Wb
l=3.14*40*10**-2 #m
f=2000*3.14 #AT
n=800 #number of turns
#case a
i=f/n  #A exciting current
print "i=",round(i,3),"A"
l=(n*flux)/i   #self inductance in H
print "l=",round(l,5),"H"
#case b
fluxa=20*10**-4  #Wb
gap=1*10**-2
u1=4*3.14*10**-7
rair=gap/(u1*A) #reluctance of air in AT/Wb
print "rair=",round(rair,5),"AT/Wb"
fair=rair*flux  #mmf for air gap in AT
print "fair=",round(fair,5),"AT"
fcore=6233.18 #AT--> 5000*((0.4*3.14)-0.01)=6233.18
totmmf=fcore+fair
print "total mmf=",round(totmmf,5),"AT"
I=totmmf/n #A exciting current
#self inductance
L=(n*flux)/I
print "L=",round(L,5),"H"

i= 7.85 A
l= 0.20382 H
rair= 3184713.3758 AT/Wb
fair= 6369.42675 AT
total mmf= 12602.60675 AT
L= 0.10157 H


## Example 3.10:Page number-175¶

In [7]:
import math
#given
n=2000 #number of turns
flux=0.05*10**-3 #Wb
i=10 #A
lx=(n*flux)/i   #self inductance in X
print "lx=",round(lx,5),"H"
#since coils are identical self inductance in Y=self inductance in x
print "m=",round(m,5),"H"
#The rate of change in current di/dt=2000A/sec --> di/dt=(10-(-10))/0.01
rate=2000
ey=m*rate
print "The induced emf in coil Y=",round(ey,0),"V"

lx= 0.01 H
m= 0.015 H
The induced emf in coil Y= 30.0 V


## Example 3.11:Page number-175¶

In [8]:
import math
#given
#when currents are in same direction the total induction is:
#lt=l1+l2+2m
#when currents are in opposite direction the total emf is:
#lt=l1+l2-2m
#According to this problem
#l1+l2+2m=1.2
#l1+l2-2m=0.2
#Solving the above equations we get l1=0.4H M=0.25H
#on substituting we get l2=0.3H
#k=m/squareroot(l1*l2)
print "k=0.72168"

k=0.72168


## Example 3.12:Page number-176¶

In [13]:
import math
#given
#case a
B=1 #Wb/m**2
A=10**-4 #cm**2
per=800 #permeability
n=250 #number of turns
flux=B*A
print "flux",round(flux,5),"Wb"
r=781250   #AT/Wb calculated using formula for reluctance
mmf=flux*r #AT
i=mmf/n  #exciting current required in A
print "i",round(i,5),"A"
l=(n*flux)/i  #self inductance of the coil
print "l=",round(l,5),"H"
w=(l*i*i)/2  #energy stored
print "w=",round(w,5),"J"
#case b
airgap=1*10**-3   #air gap is assumed
rair=airgap/(u1*A) #reluctance of air gap in AT/Wb
mmfa=flux*rair     #mmf of air in AT
print mmfa
#rcore=((2.5*3.14)-0.1)/(32*3.14*10**-6)  #reluctance of core
#mmfc=flux*rcore
mmfc=780 #AT
F=mmfc+mmfa
I=F/n  #A
print "exciting current=",round(I,2),"A"
n=250 #number of turns
L=(n*flux)/I   #self inductanc eof coil with air gap
print "l=",round(L,5),"H"
e=(L*I*I)/2  #energy stored in coil
print "e=",round(e,5),"J"

flux 0.0001 Wb
i 0.3125 A
l= 0.08 H
w= 0.00391 J
796.178343949
exciting current= 6.3 A
l= 0.00397 H
e= 0.07881 J


## Example 3.13:Page number:178¶

In [16]:
import math
#given
A=10**-1  #area
flux=0.1 #Wb
#case a
B=flux/A  #flux density Wb/m**2
u1=4*3.14*10**-7
F=(B*B*A)/(2*u1) #force in N
print "force=",round(F,5),"N"
#case b
l=10**-2  #length of the air gap
w=(B*B*A*l*2)/(2*u1)   #energy stored in two airgaps, 2=air gaps
print "W=",round(w,5),"J"

force= 39808.9172 N
W= 796.17834 J

In [ ]: