# Chapter 11: Transmission Lines

### Example 11.1, Page number: 482

In [1]:


import scipy

#Variable Declaration

R=0
G=0
a=0
Ro=70            #characteristic impedence in ohms
B=3              #phase constant in rad/sec
f=100*10**6      #frequency in Hz
w=2*scipy.pi*f   #omega in rad/sec

#Calculations

C=B/(w*Ro)       #capacitance in F/m
L=Ro*Ro*C        #inductance in H/m

#Results

print 'inductance per meter =',round(L*10**9,1),'nH/m'
print 'capacitance per meter =',round(C*10**12,1),'pF/m'

inductance per meter = 334.2 nH/m
capacitance per meter = 68.2 pF/m


### Example 11.2, Page number: 483

In [2]:

#Variable Declaration

Zo=60           #in ohms
a=20*10**-3     #in Np/m
u=0.6*3*10**8   #in m/sec
f=100*10**6     #in Hz

#Calculations

R=a*Zo          #resistance in ohms/m
L=Zo/u          #inductance in H/m
G=a*a/R         #conductivity in S/m
C=1/(u*Zo)      #capacitance in F/m
lam=u/f         #wavelentgh in m

#Results

print 'R =',R,'ohm/m'
print 'L =',round(L*10**9,0),'nH/m'
print 'G =',round(G*10**6,0),'micro S/m'
print 'C =',round(C*10**12,2),'pF/m'
print 'lambda =',lam,'m'

R = 1.2 ohm/m
L = 333.0 nH/m
G = 333.0 micro S/m
C = 92.59 pF/m
lambda = 1.8 m


### Example 11.3, Page number: 490

In [3]:

import scipy
import cmath
from numpy import *

#Variable Declaration

w=10**6                     #omega in rad/sec
B=1                         #phase factor in rad/m
a=8.0/8.686                 #alpha in Np/m
Y=a+1j                      #in m^-1
l=2                         #length in m
Vg=10                       #source voltage in volts
Zo=60+40j                   #in ohms
Zg=40                       #in ohms
Zl=20+50j                   #load impedance in ohms

#Calculations

s=scipy.tanh(Y*l)
Zin=Zo*(Zl+Zo*s)/(Zo+Zl*s)  #input impedance in ohms
Zinr=round(Zin.real,2)      #real part of Zin rounded to 2 decimal places
Zini=round(Zin.imag,2)      #imaginary part of Zin rounded to 2 decimal places
Io=Vg/(Zin+Zg)              #in A
absIo=round(abs(Io),6)      #absolute value of Io rounded to 6 decimal place
Ior=Io.real                 #real part of Io
Ioi=Io.imag                 #imaginary part of Io
angIo=scipy.arctan(Ioi/Ior)*180/scipy.pi
#in degrees
Vo=Zin*Io
Vop=(Vo+Zo*Io)/2
Vom =(Vo-Zo*Io)/2
Im=((Vop*scipy.e**(-Y)/Zo))-((Vom*scipy.e**Y)/Zo)
#current at the middle in A
absIm=round(abs(Im),5)      #absolute value of Im rounded to 6 decimal place
Imr=Im.real                 #real part of Im
Imi=Im.imag                 #imaginary part of Im
angIm=360+scipy.arctan(Imi/Imr)*180/scipy.pi
#in degrees

#Results

print 'The input impedance  =',Zinr,'+',Zini,'j ohms'
print 'The sending-end current  is'
print 'mod =',absIo*10**3,'mA, angle =',round(angIo,2),'degrees'
print 'The current at the middle is'
print 'mod =',absIm*10**3,'mA, angle =',round(angIm,0),'degrees'

The input impedance  = 60.25 + 38.79 j ohms
The sending-end current  is
mod = 93.03 mA, angle = -21.15 degrees
The current at the middle is
mod = 34.92 mA, angle = 281.0 degrees


### Example 11.4, Page number: 499

In [4]:

import scipy
import cmath
from numpy import *

#Variable Declaration

l=30                #length in m
Zo=50               #in ohms
f=2*10**6           #frequency in Hz
Zl=60+40j           #load impedence in ohms
u=0.6*3*10**8       #in m/s
w=2*scipy.pi*f      #omega in rad/sec

#Calculations

T=(Zl-Zo)/(Zl+Zo)                             #reflection coefficient
ang=scipy.arctan(T.imag/T.real)*180/scipy.pi  #argument of T is degrees
s=(1+abs(T))/(1-abs(T))                       #standing wave ratio
B=w/u                                         #propogation vector in m^-1
Zin=Zo*(Zl+Zo*scipy.tan(B*l)*1j)/(Zo+Zl*scipy.tan(B*l)*1j)
Zinr=round(Zin.real,2)      #real part of Zin rounded to 2 decimal places
Zini=round(Zin.imag,2)      #imaginary part of Zin rounded to 2 decimal places

#Results

print 'The reflection coefficient is'
print 'mod =',round(abs(T),4),'angle =',round(ang,0),'degrees'
print 'The standing wave ratio s =',round(s,3)
print 'The input impedance  =',Zinr,'+',Zini,'j ohms'

The reflection coefficient is
mod = 0.3523 angle = 56.0 degrees
The standing wave ratio s = 2.088
The input impedance  = 23.97 + 1.35 j ohms


### Example 11.5, Page number: 501

In [5]:


import scipy
import cmath
from numpy import *

#Variable Declaration

Zl=100+150j                #load impedance in ohms
Zo=75                      #impedance of line in ohms
B=2*scipy.pi

#Calculations

T=(Zl-Zo)/(Zl+Zo)
angT=scipy.arctan(T.imag/T.real)*180/scipy.pi
s=(1+abs(T))/(1-abs(T))
Yl=(1/Zl)*10**3             #admittance in mS
Ylr=round(Yl.real,2)        #real part of Yl rounded to 2 decimal places
Yli=round(Yl.imag,2)        #imaginary part of Yl rounded to 2 decimal places
l1=0.4
Zin=Zo*(Zl+Zo*scipy.tan(B*l1)*1j)/(Zo+Zl*scipy.tan(B*l1)*1j)
Zinr=round(Zin.real,2)      #real part of Zin rounded to 2 decimal places
Zini=round(Zin.imag,2)      #imaginary part of Zin rounded to 2 decimal places

#Results

print 'r is mod =',round(abs(T),3),',angle =',round(angT,0),'degrees'
print 's =',round(s,2)
print 'The load admittance Yl =',Ylr,'+',Yli,'j mS'
print 'Zin at O.4 lambda from the load =',Zinr,'+',Zini,'j ohms'
#part (e) and (f) don't require computations

r is mod = 0.66 ,angle = 40.0 degrees
s = 4.88
The load admittance Yl = 3.08 + -4.62 j mS
Zin at O.4 lambda from the load = 21.96 + 47.61 j ohms


### Example 11.6, Page number: 509

In [6]:

import scipy
import cmath
from numpy import *

#Variable Declaration

s=2
l1=11
l2=19
ma=24
mi=16
u=3*10**8        #speed of wave in m/s
Zo=50            #in ohms

#Calculations

l=(l2-l1)*2      #lambda in cm
f=(u/l)*10**-7   #frequency in GHz
L=(24-19)/l      #Let us assume load is at 24cm
zl=1.4+0.75j     # by smith chart
Zl=Zo*zl         #ZL in ohms

#Results

print 'lambda =',l,'cm'
print 'f =',f,'GHz'
print 'ZL =',Zl,'ohms'

lambda = 16 cm
f = 1.875 GHz
ZL = (70+37.5j) ohms


### Example 11.7, Page number: 510

In [7]:

import scipy
import cmath
from numpy import *

#Variable Declaration

Zo=100      #in ohms
Zl=40+30j   #in ohms

#Calculations

Yo=1.0/Zo                          #in S
yl=Zo/Zl
ys1=1.04j                          #By smith chart
ys2=-1.04j                         #By smith chart
Ys1=Yo*ys1                         #in S
Ys2=Yo*ys2                         #in S
la=round(0.5-(62-(-39))/720.0,2)   #in units of lambda
lb=round((62-39)/720.0,3)          #in units of lambda
da=round(88/720.0,4)               #in units of lambda
db=round(272/720.0,4)              #in units of lambda

#Results

print 'The required stub admittance values in mS are',Ys1*1000,'and',Ys2*1000
print 'The distance between stub and antenna at A =',la,'in units of lambda'
print 'The distance between stub and antenna at B =',lb,'in units of lambda'
print 'The stub lengths =',da,'and',db,'in units of lambda'
print 'Part (d) is done using smith chart'

The required stub admittance values in mS are 10.4j and -10.4j
The distance between stub and antenna at A = 0.36 in units of lambda
The distance between stub and antenna at B = 0.032 in units of lambda
The stub lengths = 0.1222 and 0.3778 in units of lambda
Part (d) is done using smith chart


### Example 11.9, Page number: 521¶

In [15]:

import scipy
import matplotlib.pyplot as plt

#Variable Declarataion

zo=75     #in ohms
zg=25     #in ohms
zl=100    #in ohms
vg=4      #in volts
l=60      #in m
c=3*10**8 #speed of light in m/s
u=0.1*c   #in m/s

#Calculations

gammag=(zg-zo)/(zg+zo)
gammal=(zl-zo)/(zl+zo)
Vo=zo*vg/(zo+zg)        #in V
t1=l/u                  #in micro sec
Io=vg/(zo+zg)           #in mA

#Results

t1=[0,4,5,8,9,12,13,15]
I1=[40,31.43,-8.571,-7.959,0.6123,0.5685,-0.0438,-0.438]
fig = plt.figure()
ax = fig.add_subplot(111)
ax.step(t1,I1,where='post')
ax.set_xlabel('Time (micro s)')
ax.set_ylabel(r'I(0,t) in mA')
plt.show()

t2=[0,2,6,7,10,11,14]
I2=[0,34.3,31.9,-2.46,-2.28,0.176,0.176]
fig = plt.figure()
ax = fig.add_subplot(111)
ax.step(t2,I2,where='post')
ax.set_xlabel('Time (micro s)')
ax.set_ylabel(r'I(l,t) in mA')
plt.show()


### Example 11.10, Page number: 527

In [8]:

import scipy

#Variable Declaration

Er=3.8          #relative permittivity
c=3*10**8       #speed of wave in m/s
r=4.5           #ratio of line width to substrate thickness

#Calculations

Eeff=((Er+1)/2)+((Er-1)/(2*(1+12/r)**0.5))
Zo=(120*scipy.pi)/((r+1.393+(0.667*scipy.log(r+1.444)))*((Eeff)**0.5))
f=10**10
l=c/(f*scipy.sqrt(Eeff))

#Results

print 'The effective relative permittivity of the substrate =',round(Eeff,3)
print 'The characteristic impedance of the line =',round(Zo,2),'ohms'
print 'The wavelength of the line at 10 GHz =',round(l*1000,2),'mm'

The effective relative permittivity of the substrate = 3.131
The characteristic impedance of the line = 30.08 ohms
The wavelength of the line at 10 GHz = 16.95 mm


### Example 11.11, Page number: 527

In [9]:


import scipy

#Variable Declaration

h=1                     #in mm
w=0.8                   #in mm
Er=6.6                  #relative permittivity
P=scipy.arctan(0.0001)
c=5.8*10**7             #conductivity in S/m
f=10**10                #frequency in Hz
mu=4*scipy.pi*10**-7    #permeability of free space
C=3*10**8               #speed of wave in m/s
r=w/h

#Calculations

Ee=((Er+1)/2.0)+((Er-1)/(2.0*(1+12/r)**0.5))
Zo=(120.0*scipy.pi)/((r+1.393+(0.667*scipy.log(r+1.444)))*((Ee)**0.5))
Rs=scipy.sqrt((scipy.pi*f*mu)/c)
ac=8.686*Rs/(w*(10**-3)*Zo)
l=C/(f*(Ee)**0.5)
ad=27.3*(Ee-1)*Er*scipy.tan(P)/((Er-1)*Ee*l)

#Results

print 'attenuation due to conduction loss =',round(ac,2),'dB/m'
print 'attenuation due to dielectric loss =',round(ad,3),'dB/m'

attenuation due to conduction loss = 4.35 dB/m
attenuation due to dielectric loss = 0.177 dB/m