Chapter 10: Electromagnetic Wave Propagation

Example 10.1, Page number: 416

In [1]:


import scipy
from pylab import *

#Variable Declaration

w=10**8
c=3.0*10**8

#Calculations

T=2*scipy.pi/w            #timeperiod of the wave in sec
lam=2*scipy.pi/B          #wavelength in m
t1=lam*10**9/(2*c)        #time taken to travel half the wavelength in ns

x=arange(-6*scipy.pi,6*scipy.pi,0.1)

t=0
E=50*scipy.cos(10**8*t+x*w/c)

subplot(3,1,1)
xlabel("x")
ylabel("E for t=0")
plot(x,E,'r')

subplot(3,1,2)
t=T/4
E=50*scipy.cos(10**8*t+x*w/c)
xlabel("x")
ylabel("E for t=T/4")
plot(x,E)

subplot(3,1,3)
t=T/2
E=50*scipy.cos(10**8*t+x*w/c)
xlabel("x")
ylabel("E for t=T/2")
plot(x,E,'g')
show()

#Results

print 'Since the argument of cosine function is positive, '
print 'the wave is propagating in the negative x direction.'
print 'Time taken to travel a distance of lambda/2 =',round(t1,2),'n sec'

Since the argument of cosine function is positive,
the wave is propagating in the negative x direction.
Time taken to travel a distance of lambda/2 = 31.42 n sec


Example 10.2, Page number: 428

In [2]:

import cmath
import scipy

#Variable Declaration

Ho=10
n=200*scipy.exp(1)**(1j*scipy.pi/6)
b=0.5

#Calclations

Eo=n*Ho                         #amplitude of electric field in kV/m
P=scipy.arctan(scipy.sqrt(3))
a=b*((scipy.sqrt(((1+(scipy.tan(P))**2)**0.5)-1))/(scipy.sqrt(((1+(scipy.tan(P)
)**2)**0.5)+1)))
delta=1/a

#Results

print 'E has the same form as H except for amplitude and phase.'
print 'The amplitude and phase of E =',Eo,'kV/m'
print '= magnitude of 2000 and angle of pi/6'
print 'a =',round(a,4),'Np/m'
print 'Skin depth =',round(delta,3),'m'
print 'The polarization of wave is in z direction since it has an z component.'

E has the same form as H except for amplitude and phase.
The amplitude and phase of E = (1732.05080757+1000j) kV/m
= magnitude of 2000 and angle of pi/6
a = 0.2887 Np/m
Skin depth = 3.464 m
The polarization of wave is in z direction since it has an z component.


Example 10.3, Page number: 430

In [3]:


import scipy

#Variable Declaration

B=1
n=60*scipy.pi
Ur=1                     #relative permeability
Eo=10**-9/(36*scipy.pi)  #permittivity of free space
Uo=4*scipy.pi*10**-7     #permeability of free space

#Calculations

Er=Uo*Ur/(n**2*Eo)           #relative permittivity
eps=Eo*Er                    #permittivity of the medium in Farad/m
H1o=-0.1
H2o=0.5
Ex=H2o/(eps*w)           #amplitude of x component of E in V/m
Ey=H1o/(eps*w)           #amplitude of y component of E in V/m

#Results

print 'er =',Er
print 'E =',round(Ex,2),'sin(wt-z)ax +',round(-Ey,2),'cos(wt-z)ay V/m'

er = 4.0
E = 94.25 sin(wt-z)ax + 18.85 cos(wt-z)ay V/m


Example 10.4, Page number: 432

In [4]:

import scipy

#Variable Declaration

E=2                         #amplitude of E in V/m
sigma=3                     #in mhos/m
Ur=20                       #relative permeability
Eo=10**-9/(36*scipy.pi)     #permittivity of free space in Farad/m
Er=1                        #relative permittivity
Uo=4*scipy.pi*10**-7        #permeability of free space

#Calculations

a=round(scipy.sqrt(Uo*Ur*w*sigma/2),1)  #in Np/m
H=E/(scipy.sqrt(Uo*Ur*w/sigma))*10**3   #amplitude of H in mA/m

#Results

print 'alpha =',a,'Np/m'
print 'H =',round(H,1),'e^ (',a,'z ) sin(wt - Bz -',thetad,') mA/m'

alpha = 61.4 Np/m
H = 69.1 e^ ( 61.4 z ) sin(wt - Bz - 45.0 ) mA/m


Example 10.6, Page number: 434

In [5]:

import scipy

#Variable Declaration

a=2*10**-3              #in m
b=6*10**-3              #in m
t=10**-3                #in m
l=2                     #in m
c=5.8*10**7             #conductivity in seimens
f=100*10**6             #frequency in Hz
mu=4*scipy.pi*10**-7    #permeability of free space

#Calculations

Ri=l/(c*scipy.pi*a*a)               #dc resistance of inner cable in ohms
Ro=l/(c*scipy.pi*((b+t)**2-b**2))   #dc resistance of outer cable in ohms
Rdc=Ro+Ri                           #total dc resistance in ohms

Ria=round(l/(2*scipy.pi*a)*scipy.sqrt(scipy.pi*f*mu/c),1)
Roa=round(l/(2*scipy.pi*b)*scipy.sqrt(scipy.pi*f*mu/c),4)
Rac=Ria+Roa                         #ac resistance in ohms

#Results

print 'Rdc =',round(Rdc*10**3,3),'m ohms'
print 'Rac =',round(Rac,4),'ohms'

Rdc = 3.588 m ohms
Rac = 0.5384 ohms


Example 10.7, Page number: 439

In [6]:


import scipy
from numpy import *

#Variable Declaration

ax=array([1,0,0])               #Unit vector along x direction
ay=array([0,1,0])               #Unit vector along y direction
az=array([0,0,1])               #Unit vector along z direction
a=0                             #alpha in m^-1
b=0.8                           #beta in m^-1
Eo=10**-9/(36*scipy.pi)         #permittivity of free space in farad/m
Uo=4*scipy.pi*10**-7            #permeability of free space
Ur=1                            #relative permeability of medium
Eamp=4                          #amplitude of the field in V/m

#Calculations

Er=b**2/(Uo*Eo*w*w)             #relative permittivity of the medium
n=scipy.sqrt(Uo/(Eo*Er))        #eta in ohms
Pav=Eamp**2/(2*n)*ax            #average power in W/m^2
an=(2*ax+ay)/scipy.sqrt(5)      #normal to the plane
S=100*10**-4*an                 #area in m^2
P=dot(Pav,S)*10**6              #power through the plane in micro W

#Results

print 'Er=',round(Er,2)
print 'eta= ',round(n,1),'ohms'
print 'The time-average power =',round(dot(Pav,ax)*10**3,0),'ax mW/m^2'
print 'The total power crossing 100 cm^2 of the plane =',round(P,2),'micro W'

Er= 14.59
eta=  98.7 ohms
The time-average power = 81.0 ax mW/m^2
The total power crossing 100 cm^2 of the plane = 725.0 micro W


Example 10.10, Page number: 458

In [7]:

import scipy
from numpy import *

#Variable Declaration

ax=array([1,0,0])               #Unit vector along x direction
ay=array([0,1,0])               #Unit vector along y direction
az=array([0,0,1])               #Unit vector along z direction
kx=0                            #in m^-1
ky=0.866                        #in m^-1
kz=0.5                          #in m^-1
Eo=10**-9/(36*scipy.pi)         #permittivity of free space in farad/m
Uo=4*scipy.pi*10**-7            #permeability of free space
c=1/(scipy.sqrt(Uo*Eo))         #speed of light in m/s
kvect=kx*ax+ky*ay+kz*az         #propogation vector in m^-1
Eo=100                          #amplitude of electric field

#Calculations

k=round(scipy.sqrt(kx*kx+ky*ky+kz*kz),0)  #magnitude of k in m^-1
lam=2*scipy.pi/k                          #wavelength in m
Ho=cross(kvect,Eo*ax*10)/(Uo*w)           #amplitude of magnetic field in mA/m
Hoy=round(dot(Ho,ay),2)                   #y component of Ho
Hoz=round(dot(Ho,az),1)                   #z component of Ho
Hr=array([0,Hoy,Hoz])                     #Ho with components rounded off
P=Eo**2/(2*120*scipy.pi)*kvect            #average power in W/m^2
Py=round(dot(P,ay),2)                     #y component of P
Pz=round(dot(P,az),3)                     #z component of P
Pr=array([0,Py,Pz])                       #P with components rounded off

#Results

print 'lambda =',round(lam,3),'m'
print 'The magnetic field component =',Hr,'e^j(0.866x-0.5z) mA/m'
print 'The time average power in the wave =',Pr,'W/m^2'

w = 300000000.0 rad/sec
lambda = 6.283 m
The magnetic field component = [ 0.    1.33 -2.3 ] e^j(0.866x-0.5z) mA/m
The time average power in the wave = [  0.     11.49    6.631] W/m^2


Example 10.11, Page number: 459¶

In [25]:

import scipy

#Variable Declaration

ax=array([1,0,0])               #Unit vector along x direction
ay=array([0,1,0])               #Unit vector along y direction
az=array([0,0,1])               #Unit vector along z direction
Ei=8                            #incident wave amplitude
k=5                             #propogation constant
Eo=10**-9/36*scipy.pi           #permittivity of free space
Erel=2.5                        #relative permittivity
muo=4*scipy.pi*10**-7           #permeability of free space
mur=1                           #relative permeability
c=3*10**8                       #speed of light
etao=377

#Calculations

theta=scipy.arctan(4/3.0)       #angle of incidence in rad
eta1=etao
eta2=377/scipy.sqrt(2.5)
thetai=scipy.arcsin(sin(theta)/scipy.sqrt(2.5))
gamm=(eta2*cos(theta)-eta1*cos(thetai))/(eta2*cos(theta)+eta1*cos(thetai))
Er=Ei*gamm                      #reflected E field amplitude in V/m
kt=w*scipy.sqrt(mur*Erel)/c
tao=2*eta2*cos(theta)/((eta2*cos(theta)+eta1*cos(thetai)))
Et=tao*Ei*ay
Ht=cross((4*ax+6.819*az)/(eta2*kt),Et)*10**3
Htx=round(dot(Ht,ax),2)
Hty=round(dot(Ht,ay),2)
Htz=round(dot(Ht,az),2)
Htc=array([Htx,Hty,Htz])        #transmitted H field amplitude

#Results

print 'Polarisation is perpendicular polarization'
print 'Angle of incidence is ',round(180*theta/scipy.pi,2),'degrees'
print 'Er =',round(Er,3),'cos(',w,'t - 4x + 3z) V/m'
print 'Ht =',Htc,'cos(',w,'t - 4x - 6.819z) mA/m'

Polarisation is perpendicular polarization
Angle of incidence is  53.13 degrees
Er = -3.112 cos( 1500000000 t - 4x + 3z) V/m
Ht = [-17.68   0.    10.37] cos( 1500000000 t - 4x - 6.819z) mA/m