# Chapter 3:Transmission Media,Transmission Lines and Waveguides¶

## Example 2,Page No:65¶

In :
import math

# Variable Declaration
Lr      = 18;       # return loss in db

# Calculations
# Lr   = 20*math.log(1/float(p));
p       = 1/10**(Lr/float(20));     # reflection co-efficient
swr     = (1 + p)/(1 - p);      # standing wave ratio

# Result
print'Reflection co-efficient is %3.3f\n'%p,'SWR = %3.2f'%(swr);

Reflection co-efficient is 0.126
SWR = 1.29


## Example 3,Page No:66¶

In :
import math

# Variable Declaration
PW  = 30*10**-6;     # pulse width in sec
ips = 20*10**-6;     # inter pulse separation
v   = 3*10**8;       # propagation speed in m/s

# Calculations
T   = PW+ips+PW+ips+PW      # time duration of the pulse train for having 3 pulses on the line at a time
l   = v*T;                  # minimum length of cable required

# Result
print'Minimum length of cable required = %d km' %(l/float(1000));

Minimum length of cable required = 39 km


## Example 4,Page No:66¶

In :
import math

# Variable Declaration
RmsVmax    = 100;      # max value of RMS vtg
RmsVmin    = 25;       # min value of RMS vtg
Zl         = 300;      # load impedance in ohm

# Calculations
VSWR       = RmsVmax/float(RmsVmin);
# wkt VSWR  = Zl/float(Zo); assuming Zl > Zo
Zo         = Zl/float(VSWR);   # charecteristic impedance in ohm
p          = (Zl - Zo)/float((Zl + Zo));   # reflection co-efficient

# Result
print'Reflection Co-efficient = %3.1f\n'%p, 'Charecteristic impedance = %d ohm' %Zo;

Reflection Co-efficient = 0.6
Charecteristic impedance = 75 ohm


## Example 5,Page No:66¶

In :
import math

# Variable Declaration
Zo      = 75;       # charecteristic impedance in ohm
Vref    = 100;      # reflected voltage
Pref    = 100;      # reflected power in watts

# Calculations
Zl      = (Vref)**2 /float(Pref);       # load impedance
p       = (Zl - Zo)/float((Zl + Zo));   # reflection co-efficient
Pinc    = Pref/float(p);                 # incident power
Pobs    = Pinc - Pref            # power obsorbed

# Result
print'Load Resistance = %d ohm\n'%Zl,'Reflection Co-efficient = %3.3f\n'%p,'incident power = %d watts\n'%Pinc,'power obsorbed = %d watts'%Pobs;

Load Resistance = 100 ohm
Reflection Co-efficient = 0.143
incident power = 700 watts
power obsorbed = 600 watts


## Example 6,Page No:67¶

In :
import math

# Variable Declaration
c      = 3*10**8;        # velocity in m/s
f      = 100*10**6       # operating frequency in hz
Zin    = 100;
Zl     = 25;

# Calculations

lamda  = c/float(f);            # wavelength in m
Lreq   = lamda/float(4);       # required length in m
Zo     = math.sqrt(Zin*Zl);  #charecteristic impedance in ohm

# Result
print'Length of line required = %d cm\n'%(Lreq*10**2),'Charecteristic impedance = %d ohm',%Zo;

Length of line required = 75 cm
Charecteristic impedance = 50 ohm


## Example 7,Page No:67¶

In :
import math

#Variable Declaration
# in the first case when the line is lamda/2 long, the i/p impedance is same as the load resistance
Zl      = 300;      # load resistance in ohm
Zo      = 75;       # charecteristic impedance in ohm

# calculations
#Zi      = Zo*((Zl + iZotanβl)/float((Zo + iZltanβl));
#        = Zo*(((Zl/tanβl) + iZo))/((Zl/tanβl) + iZo)))
# for l = lamda/2 βl = (2* π/lamda)*(lamda/float(2)) =  π
# therefore tanβl = 0 which gives Zi = Zl
# in the second case when the operating frequency is halved, the wavelength is douβled which means the same line is now lamda/4 long
# for l = lamda/4 ,βl = (2* π/lamda)*(lamda/4) =  π/2
# therefore tanβl = ∞

Zi         = (Zo**2)/float(Zl);  # input impedance

#Result
print'Input impedance = %3.2f ohm'%Zi;

Input impedance = 18.75 ohm


## Example 8,Page No:68¶

In :
import math
# Given data
f       = 100*10**6;             # operating frequency in Hz
v       = 2*10**8 ;              # propagation velocity in m/s
Zo      = 300;                  # charecteristic impedance in ohm
Zin     = 300;                  # input impedance in ohm
l       = 1;                    # length in m
V       = 100;

# Calculations
lamda   = v/float(f);                   # wavelength in m
if(lamda/float(2) == l):
Zl  = Zin;

k       = (V*Zin)/float((Zin+Zl));
#Vin   = k*cos(2*%pi*f*t)
# since the line is lamda/2 long ,the signal undergoes a phase delay of  βl = (2*π)/lamda *(lamda/2) = π
# Result

print 'Vin = %dcos(2π'%k,'%3.0et)'%f
print 'Vl  = %dcos(2π'%k,'%3.0et-π)'%f

Vin = 50cos(2π 1e+08t)
Vl  = 50cos(2π 1e+08t-π)


## Example 9,Page No:68¶

In :
import math

# Variable Declaration
VSWR    = 3;         # voltage standing wave ratio
d       = 20*10**-2  # separation b/w 2 successive minimas
er      = 2.25;      #dielectric constant
v       = 3*10**8;   # velocity in m/s

# Calculations
# VSWR = (1 + p)/float((1 - p));
p       = (VSWR -1)/float((VSWR + 1));     # reflection co-efficient
lamda   = 2*d;                      # wavelength of tx line
lamda_fr= lamda*math.sqrt(er);           # free space wavelength
f       = v/float(lamda_fr);               # operating frequency in Hz

#Result
print'Magnitude of Reflection Co-efficient = %3.1f\n'%p,'Frequency of Operation = %3.0f Mhz'%(f/10**6);

Magnitude of Reflection Co-efficient = 0.5
Frequency of Operation = 500 Mhz


## Example 10,Page No:69¶

In :
import math
# Given data
C   = 30;           # per unit capacitance in pF/m
Vp  = 260;          # velocity of propagation in m/us
f   = 500*10**6      # freq in Hz
Zl  = 50;           # terminating load impedance in ohm

# calculations
v   = Vp/float(10**-6);      # conversion from m/us to m/s
C1  = C*10**-12      # conversion from pF/m to F/m
# 1/math.sqrt(LC) = Vp
L   = 1/float((v**2 * C1));  # per unit inductance
Zo  = math.sqrt(L/float(C1));     # charecteristic impedance in ohm
lamda = v/float(f);          # wavelength
b    = (2*math.pi)/lamda  # phase shift constant
p    = (Zl - Zo)/float((Zl + Zo));     # Reflection coefficient

# Result
print'Per Unit inductance = %d nH/m\n'%(L*10**9);
print'Charecteristic Impedance = %d ohm\n'%Zo;
print'Phase shift Constant = %d rad/m\n'%b;
print'Reflection co-efficient = %3.3f'%abs(p);

Per Unit inductance = 493 nH/m

Charecteristic Impedance = 128 ohm

Phase shift Constant = 12 rad/m

Reflection co-efficient = 0.439


## Example 11,Page No:77¶

In :
import math

# Variable Declaration
a   = 1.5*10**-2;        # width of waveguide
b   = 1*10**-2           #narrow dimension of waveguide
er  = 4;                 #dielectric constant
f   = 8*10**9;           #frequency in Hz
c   = 3*10**8;           #velocity in m/s

#calculations
lamda_c     = 2*a;      #cut-off wavelength for TE10 mode
lamda       = c/f       #wavelength corresponding to given freq.
lamda_d     = lamda/math.sqrt(er);   #wavelength when waveguide filled with dielectric
if lamda_d < lamda_c :
print('8 Ghz frequency will pass through the guide');

8 Ghz frequency will pass through the guide


## Example 12,Page No:77¶

In :
# Variable Declaration
a   = 4*10**-2;          # width of waveguide
b   = 2*10**-2           # narrow dimension of waveguide
er  = 4;                # dielectric constant
c   = 3*10**8            # velocity in m/s

# Calculations
lamda_c     = 2*a;      # max cut-off wavelength
fcmin       = c/lamda_c # min freq
lamda_d     = lamda_c/math.sqrt(er);     # wavelength if we insert dielectric
fc          = c/lamda_d        # min frequency in presence of dielectric

# Result
print'Minimum Frequency that can be passed with dielectric in waveguide is %3.1f Ghz'%(fc/float(10**9));


Minimum Frequency that can be passed with dielectric in waveguide is 7.5 Ghz


## Example 13,Page No:78¶

In :
import math

# Variable Declaration
f   = 1*10**9;       # frequency in Hz
a   = 5*10**-2;      # wall separation
c   = 3*10**8;       # velocity of EM wave in m/s
m   = 1;            # for TE10
n   = 0;            # for TE10

# Calculations
# lamda0   = 2/math.sqrt((m/a)**2 + (n/b)**2)
lamda0      = (2*a)/m
lamda_frspc = c/float(f);
if lamda_frspc > lamda0:
print'1 Ghz signal cannot propagate in TE10 mode'
else:
print'1 Ghz signal can propagate in TE10 mode';


1 Ghz signal cannot propagate in TE10 mode


## Example 14,Page No:78¶

In :
import math

# Variable Declaration
a   = 30;            # width of waveguide
b   = 20;           # narrow dimension of waveguide
c   = 3*10**8;       # velocity of EM wave in m/s
m   = 1;            # for TE10
n   = 0;            # for TE10

#Calculations
#lamda0   = 2/math.sqrt((m/a)**2 + (n/b)**2)
lamda0      = (2*a)/m;      # longest cut-off wavelength in dominant mode TE10

# Result
print'longest cut-off wavelength = %d mm'%lamda0;

longest cut-off wavelength = 60 mm


## Example 15,Page No:107¶

In :
import math

# Variable Declaration
a   = 4*10**-2;       # width of waveguide
b   = 2*10**-2;       #narrow dimension of waveguide
c   = 3*10**8;        # velocity of EM wave in m/s
m1  = 1;              # for TE10
m2  = 2;              # for TE20
n   = 0;              # for TE10

# Calculations
lamda_c     = 2*a       # cutoff wavelength for TE10 mode
f1          = c/lamda_c # frequency in Hz
# the frequency range for single mode operation is the range of frequencies corresponding to the dominant mode and the second highest cutoff wavelength
lamda_c_2   = 2/math.sqrt((m2/a)**2 + (n/b)**2)
f2          = c/lamda_c_2;   # freq at second largest cutoff wavelength

# Result
print'Therefore,single mode operating range = %3.2f Ghz'%(f1/float(10**9)),'to %3.1f Ghz\n' %(f2/float(10**9));
print'Note: instead of 3.75,3.5 is printed in textbook';

Therefore,single mode operating range = 3.75 Ghz to 7.5 Ghz

Note: instead of 3.75,3.5 is printed in textbook


## Example 16,Page No:108¶

In :
import math

# Variable Declaration
a   = 7.2;            # width of waveguide in cm
b   = 3.4;            # narrow dimension of waveguide in cm
c   = 3*10**10;       # free space velocity of EM wave in cm/s
f   = 2.4*10**9;      #frequency in Hz

#Calculation
lamda   = c/f       # free space wavelength in cm
lamda_c = 2*a       # cutoff wavelength in cm
lamda_g = lamda/math.sqrt(1 - (lamda/lamda_c)**2); # guide wavelength in cm
vp      = (lamda_g * c)/lamda                      # phase velocity in cm/s
vg      = c**2/float(vp);                          # group velocity in cm/s

# Result
print'Group velocity = %3.1e cm/s\n'%vg, 'Phase Velocity = %3.1e cm/s'%vp;

Group velocity = 1.5e+10 cm/s
Phase Velocity = 6.0e+10 cm/s


## Example 18,Page No:109¶

In :
import math
# Given data
# let 'a' and 'b' be the broad and narrow dimensions of the rectangular guide and 'r' be internal radius of circular guide
# Dominant mode in rectangular guide =TE10
# cutoff wavelength = 2a
# dominant mode in circular guide = TE11
# cut-off wavelength = 2*pi*r/1.841  = 3.41r
# for the two cut-off wavelengths to equal
# 2a  = 3.41r
# a = 1.705r
# now area of cross section of rectangular guide = a*b
# assuming a= 2b,which is very reasonable assumption ,we get
# area of cross section of rectangular waveguide = a*a/2 = ((1.705^2)*r*r)/2 = 1.453r**2
# area of cross-section of circular guide = pi*r*r = 3.14r**2
# ratio of two cross sectional areas = (3.14r**2)/(1.453r**2) = 2.16
# output
print'Circular guide is 2.16 times larger in cross section as compared to rectangular guide';

Circular guide is 2.16 times larger in cross section as compared to rectangular guide


## Example 19,Page No:110¶

In :
import math

# Variable Declaration
a   = 4*10**-2;      # width of waveguide
b   = 2*10**-2;      # narrow dimension of waveguide
c   = 3*10**8;      # velocity of EM wave in m/s
f   = 5*10**9       # operating frequency in Hz
m0  = 0;            # for TE01
m1  = 1;            # for TE10 / TE11 /TM11
n0  = 0;            # for TE10
n1  = 1;            # for TE11 or TM11

# Calculations
lamda        = c/f;                                # operating wavelength
lamda_TE01   = 2/math.sqrt((m0/a)**2 + (n1/b)**2)  # cutoff wavelength for TE01
lamda_TE10   = 2/math.sqrt((m1/a)**2 + (n0/b)**2)  # cutoff wavelength for TE10
lamda_TE11   = 2/math.sqrt((m1/a)**2 + (n1/b)**2)       # cutoff wavelength for TE11 or TM11

# Result
if lamda_TE01 >lamda:
print'TE01 propagates in the given guide at the given operating frequency';

elif lamda_TE10 >lamda:
print'TE10 propagates in the given guide at the given operating frequency';
elif lamda_TE11 >lamda:
print'TE11 propagates in the given guide at the given operating frequency';

TE01 propagates in the given guide at the given operating frequency


## Example 20,Page No:110¶

In :
import math

# Variable Declaration
a   = 4*10**-2;      # width of waveguide
b   = 2*10**-2;      # narrow dimension of waveguide
c   = 3*10**8;       # velocity of EM wave in m/s
d   = 4*10**-2;      # distance b/w field maxima and minima

# Calculations
lamda_c     = 2*a;      # cut-off wavelength in dominant mode
lamda_g     = 4*d;      # guide wavelength
# lamda_g  = lamda0/float((math.sqrt(1 -(lamda0/lamda_c)**2)))
lamda0      = math.sqrt(((lamda_c * lamda_g)**2) / float((lamda_c**2 )+ (lamda_g**2)));
f0          = c/float(lamda0);     # frequency of the wave

# Result
print'Frequency of the wave = %3.3f Ghz'%(f0/float(10**9));

Frequency of the wave = 4.193 Ghz


## Example 21,Page No:111¶

In :
import math

# Variable Declaration
a   = 6;      # width of waveguide in cm
b   = 3;      # narrow dimension of waveguide in cm
lamda = 4;    # operating wavelength in cm
c   = 3*10**8;       # velocity of EM wave in cm/s

# Calculations
lamda_c  = 2*a;      # cut-off wavelength in dominant mode
lamda_g  = lamda/(float((math.sqrt(1 - ((lamda/float(lamda_c))**2))))); # guide wavelength
Vp       = (lamda_g/float(lamda))*c;
b        = (2*math.pi)/float(lamda_g);     # phase shift constant

#Result
print'Guide wavelength = %3.2f cm\n'%lamda_g;
print'Phase velocity = %3.2e m/s\n'%Vp;
print'Phase shift constant = %3.2f radians/cm'%b;

Guide wavelength = 4.24 cm

Phase velocity = 3.18e+08 m/s

Phase shift constant = 1.48 radians/cm


## Example 22,Page No:111¶

In :
import math

# variable Declaration
er      = 9;          # relative permittivity
c       = 3*10**10;   # velocity of EM wave in free space
f       = 2*10**9;    # operating frequency in Ghz
a       = 7;          # width of waveguide in cm
b       = 3.5;        # narrow dimension of waveguide in cm

# calculations
lamda_c  = 2*a;                 # cut-off wavelength in dominant mode
fc       = c/float(lamda_c );           # cut-off frequency in Hz
lamda    = c/float((math.sqrt(er)*f));       # operating wavelength
lamda_g  = lamda/(math.sqrt(1 - ((lamda/float(lamda_c))**2))); # guide wavelength
Vp       = (lamda_g/float(lamda))*c

# Result
print'Cut-off frequency = %3.3f Ghz\n'%(fc/float(10**9));
print'Phase velocity = %3.2e m/s\n'%(Vp/float(10**2));
print'Guide wavelength = %3.2f cm'%lamda_g;

Cut-off frequency = 2.143 Ghz

Phase velocity = 3.21e+08 m/s

Guide wavelength = 5.35 cm