# Chapter 4 : Microwaves Transmission Lines¶

## Example 4.1 Page No : 66¶

In [1]:
import math

#Given:
d = 0.49                #in cm
D = 1.1                 #in cm
e_r = 2.3

#calculations
c = 3*10**8                                  #in meter/second
L = 2*(10**-7)*math.log(D/d)                 #in Henry/meter
C = 55.56*(10**-12)*(e_r)/math.log(D/d)      #in farad/meter
R_o = (60/math.sqrt(e_r)) *math.log(D/d)     #in ohms
v = c/math.sqrt(e_r)                         #in meter/second

#---output---#
print 'Inducmath.tance per unit length(in H/m) =',round(L,4)
print 'Capacimath.tance per unit length(in F/m) =',round(C,4)
print 'Characteristic Impedance (in ohms) =',round(R_o,4)
print 'Velocity of propagation (in m/s)=',round(v,4)

Inducmath.tance per unit length(in H/m) = 0.0
Capacimath.tance per unit length(in F/m) = 0.0
Characteristic Impedance (in ohms) = 31.9929
Velocity of propagation (in m/s)= 197814142.019


## Example 4.2 Page No : 67¶

In [2]:
import math

R = 0.05                             #in ohms
G = 0
l = 50                               #in meter
e = 2.3                              #dielectric consmath.tant

#calculations
c = 3*10**8                          #in m/s
L = 2*(10**(-7))                     #from Exa:4.1
C = 1.58*(10**(-10))                 #from Exa:4.1
P_in = 480                           #in watts
f = 3*10**9                          #in hertz
Z_o = math.sqrt(L/C)
a = R/Z_o                            #in Np/m
b = 2*math.pi*f*math.sqrt(L*C)       #in rad/m
V_p = 1/math.sqrt(L*C)
e_r = (c/V_p)**2
P_loss = P_in*2*l

#---output---#
print 'Atteneuation (in Np/m) =',round(a,4)
print 'Phase consmath.tant (in rad/m) =',round(b,4)
print 'Phase velocity (in m/s) =',round(V_p,4)
print 'Relative permittivity =',round(e_r,4)
print 'Power loss (in watts) =',round(P_loss,4)

Atteneuation (in Np/m) = 0.0014
Phase consmath.tant (in rad/m) = 105.9607
Phase velocity (in m/s) = 177892016.741
Relative permittivity = 2.844
Power loss (in watts) = 48000.0


## Example 4.3 Page No : 69¶

In [3]:
import math

#Given
a = 2.42                                          #in cm
x = 2.3                                           #x=(b/a)

#calculation
P_bd = 3600*a**2*math.log(x)                      #in kilowatts

#---output---#
print 'Breakdown Power (in kW) =',round(P_bd,4)

#answer in book is wrongly written as 398 kW.

Breakdown Power (in kW) = 17560.2564


## Example 4.4 Page No : 74¶

In [4]:
import math

b = 0.3175                                     #in cm
d = 0.0539                                     #in cm
c = 3*10**8                                    #in m/s
e_r = 2.32

#calculations
Z_o = 60*math.log(4*b/(math.pi*d))/math.sqrt(e_r) #in ohms
V_p = c/math.sqrt(e_r)                            #in m/s

#---output---#
print 'Charcteristic impedance (in ohms) =',round(Z_o,4)
print 'Velocity of propagation (in m/s) =',round(V_p,4)

Charcteristic impedance (in ohms) = 79.3713
Velocity of propagation (in m/s) = 196959649.29


## Example 4.5 Page No : 79¶

In [7]:
import math

e_r = 9.7
c = 3*10**8           #in m/s
r_1 = 0.5             #when ratio: (W/h)=0.5
r_2 = 5               #when ratio: (W/h)=5
#For W/h ratio=0.5

#calculations
e_eff_1 = (e_r+1)/2+((e_r-1)/2)*(1/(math.sqrt(1+12*(1/r_1))+0.04*(1-r_1)))
Z_o_1 = 60*math.log(8/r_1+r_1/4)/math.sqrt(e_eff_1)
v_1 = c/math.sqrt(e_eff_1)
e_eff_2 = (e_r+1)/2+((e_r-1)/2)*(1/(math.sqrt(1+12*(1/r_2))))
Z_o_2 = 120*math.pi*(1/(r_2+1.393+0.667*math.log(1.444+r_2)))/math.sqrt(e_eff_2)
v_2 = c/math.sqrt(e_eff_2)

#---output---#
print "For W/h=0.5 ,"
print 'Effective dielectric consmath.tant =',round(e_eff_1,4)
print 'Charcteristic impedance (in ohms) =',round(Z_o_1,4)
print 'Velocity of propagation (in m/s) =',round(v_1)
#For W/h ratio=5
print "For W/h=5,";
print 'Effective dielectric consmath.tant =',round(e_eff_2,4)
print 'Charcteristic impedance (in ohms) =',round(Z_o_2,4)
print 'Velocity of propagation (in m/s) =',round(v_2,4)

For W/h=0.5 ,
Effective dielectric consmath.tant = 6.2165
Charcteristic impedance (in ohms) = 66.9083
Velocity of propagation (in m/s) = 120322571.0
For W/h=5,
Effective dielectric consmath.tant = 9.7
Charcteristic impedance (in ohms) = 15.8524
Velocity of propagation (in m/s) = 96324194.8602


## Example 4.6 Page No : 84¶

In [9]:
import math

#For TE Wave propagated:
#for Recmath.tangular , taking (a=2b)
r = 100                                    #assume
#for TE11, wavelength=2*pi*r/1.841
#for TE10, wavelength=2a
#calculations
a = (2*math.pi*r/1.841)/2
ar_rec_TE = (a)*(a/2)
ar_cir_TE = math.pi*r**2
ratio_TE = (ar_cir_TE)/(ar_rec_TE)
b = (2.6155*r)/1.78885
ar_rec_TM = (b)*(b)
ar_cir_TM = math.pi*r**2
ratio_TM = (ar_cir_TM)/(ar_rec_TM)

#---output---#
print 'Ratio of Circular & Rectangular coss-section area (in TE) =',round(ratio_TE,4)
#For TM Wave propagated:
#for Recmath.tangular , taking (a=2b)
#for TE01, wavelength=2.6155*r
#for TE11, wavelength=4b/math.sqrt(5)

print 'Ratio of Circular & Rectangular coss-section area (in TM) =',round(ratio_TM,4)

Ratio of Circular & Rectangular coss-section area (in TE) = 2.1577
Ratio of Circular & Rectangular coss-section area (in TM) = 1.4696


## Example 4.7 Page No : 89¶

In [10]:
import math

f = 9*10**9               #in Hz
c = 3*10**10              #in cm/s

#calculations
wl_g = 4                  #in m
wl_o = c/f
wl_c = (math.sqrt(1-((wl_o/wl_g)**2))/wl_o)**(-1)
b = wl_c/4

#---output---#
print 'Breadth of rectangular waveguide (in cm) =',round(b,4)

Breadth of rectangular waveguide (in cm) = 0.75


## Example 4.8 Page No : 96¶

In [11]:
import math

a = 10                               #in cm
c = 3*10**10                         #in cm/s

#calculations
wl_c = 2*a                           #in cm
f = 2.5*10**9                        #in Hz
wl_o = c/f
wl_g = wl_o/(math.sqrt(1-(wl_o/wl_c)**2))       #in cm
V_p = c/(math.sqrt(1-(wl_o/wl_c)**2))
V_g = c**2/V_p

#---output---#
print 'Cut-off wavelength (in cm) =',round(wl_c,4)
print 'Guide wavelength (in cm) =',round(wl_g,4)
print 'Phase velocity (in cm/s) =',round(V_p,4)
print 'Group velocity (in cm/s) =',round(V_g,4)

Cut-off wavelength (in cm) = 20.0
Guide wavelength (in cm) = 15.0
Phase velocity (in cm/s) = 37500000000.0
Group velocity (in cm/s) = 24000000000.0


## Example 4.9 Page No : 102¶

In [12]:
import math

#For TE mode:

a = 2.5                     #in cm
b = 1                       #in cm
f = 8.6*10**9               #in Hz
c = 3*10**10                #in cm/s

#calculations
wl_o = c/f
wl_c_1 = 2*b                #for TE01
wl_c_2 = 2*a                #for TE10
f_c = c/wl_c_2
wl_c_3 = 2*a*b/math.sqrt(a**2+b**2)                    #for TE11 & TM11
wl_g_TE10 = wl_o/(math.sqrt(1-(wl_o/wl_c_2)**2))       #for TE10
wl_c_TM11 = wl_c_3;
wl_g_TM11 = wl_o/(math.sqrt(1-(wl_o/wl_c_2)**2))       #for TM11

#---output---#
print 'Only TE10 mode is possible'
print 'Cut-off frequency(in Hz) =',round(f_c,4)
print wl_g_TE10,'Guide wavelength for TE10 (in cm) =',round(wl_g_TE10,4)
#For TM mode:
print 'TM11 also propagates'
print 'Guide wavelength for TM11 (in cm) =',round(wl_g_TM11,4)

Only TE10 mode is possible
Cut-off frequency(in Hz) = 6000000000.0
4.86920604871 Guide wavelength for TE10 (in cm) = 4.8692
TM11 also propagates
Guide wavelength for TM11 (in cm) = 4.8692


## Example 4.10 Page No : 105¶

In [13]:
import math

wl_c = 10                       #in cm
c = 3*10**10                    #in cm/s

#calculations
r = wl_c/(2*math.pi/1.841)      #in cm
area = math.pi*r**2             #in sq. cm
f_c = c/wl_c

#---output---#
print 'Radius of circular waveguide(in cm) =',round(r,4)
print 'Area of cross-section of circular waveguide(in cm) =',round(area,4)
print 'Frequency above',round(f_c,4),'can be propagated'

Radius of circular waveguide(in cm) = 2.93
Area of cross-section of circular waveguide(in cm) = 26.971
Frequency above 3000000000.0 can be propagated


## Example 4.11 Page No : 106¶

In [14]:
import math

a = 4                                    #in cm
b = 3                                    #in cm
f = 5*10**9                              #in Hz
c = 3*10**10                             #in cm/s

#calculations
wl_o = c/f

#For TE waves:
wl_c_TE01 = 2*b                          #for TE01
wl_c_TE10 = 2*a                          #for TE10
wl_c_TE11 = 2*a*b/math.sqrt(a**2+b**2)   #for TE11

#---logic---#
if(wl_c_TE01>wl_o):
print 'TE01 can propagate'
else:
print 'TE01 cannot propagate'

if(wl_c_TE10>wl_o):
print 'TE10 can propagate'
else:
print 'TE10 cannot propagate'

if(wl_c_TE11>wl_o):
print 'TE11 can propagate'
else:
print 'TE11 cannot propagate'

TE01 cannot propagate
TE10 can propagate
TE11 cannot propagate


## Example 4.12 Page No : 107¶

In [15]:
import math

c = 3*10**10                       #in cm/s
d = 4                              #in cm

#calculations
r = d/2                            #in cm
wl_c = 2*math.pi*r/1.841           #in cm
f_c = c/wl_c
f_signal = 5*10**9                 #in Hz
wl_o = c/f_signal
wl_g = wl_o/math.sqrt(1-(wl_o/wl_c)**2)

#---output---#
print 'Cut-off wavelength (in cm) =',round(wl_c,4)
print 'Cut-off frequency (in Hz) =',round(f_c,4)
print 'Guide wavelength (in cm) =',round(wl_g,4)

Cut-off wavelength (in cm) = 6.8258
Cut-off frequency (in Hz) = 4395063753.48
Guide wavelength (in cm) = 12.5839


## Example 4.14 Page No : 115¶

In [16]:
import math

c = 3*10**10                        #in cm/s
a = 5                               #in cm
b = 2.5                             #in cm
wl_o = 4.5                          #in cm
#For TE10 mode:

#calculations
wl_c = 2*a
wl_g = wl_o/math.sqrt(1-(wl_o/wl_c)**2)
V_p = c/math.sqrt(1-(wl_o/wl_c)**2)
w = 2*math.pi*c/wl_o
w_c = 2*math.pi*c/wl_c
b = math.sqrt(w**2-w_c**2)/c

#---output---#
print 'Guide wavelength (in cm) =',round(wl_g,4)
print 'Phase consmath.tant =',round(b,4)
print 'Phase velocity (in cm/s) =',round(V_p,4)

#answer in book is wrongly written as guide wavelength =7.803 cm
#answer in book is wrongly written as Phase velocity = 5.22*10**10 cm/s

Guide wavelength (in cm) = 5.039
Phase consmath.tant = 1.2469
Phase velocity (in cm/s) = 33593550657.4


## Example 4.15 Page No : 121¶

In [17]:
import math

c = 3*10**10                                   #in cm/s
wl_c_TE10 = 16                                 #Critical wavelength of TE10
wl_c_TM11 = 7.16                               #Critical wavelength of TM11
wl_c_TM21 = 5.6                                #Critical wavelength of TM21
#For (i): 10 cm
wl_o = 10                                      #in cm

wl_o=5                                         #in cm

#---logic---#
print 'For free space wavelength (in cm) =',round(wl_o,4)
if(wl_c_TE10>wl_o):
print '    TE10 can propagate'
else:
print '    TE10 cannot propagate'

if(wl_c_TM11>wl_o):
print '    TM11 can propagate'
else:
print '    TM11 cannot propagate'

if(wl_c_TM21>wl_o):
print '    TM21 can propagate'
else:
print '    TM21 cannot propagate'

#For (ii): 5 cm

print ('For free space wavelength (in cm) =',round(wl_o))
if(wl_c_TE10>wl_o):
print ('    TE10 can propagate')
else:
print ('    TE10 cannot propagate')

if(wl_c_TM11>wl_o):
print ('    TM11 can propagate')
else:
print ('    TM11 cannot propagate')

if(wl_c_TM21>wl_o):
print ('    TM21 can propagate')
else:
print ('    TM21 cannot propagate')

For free space wavelength (in cm) = 5.0
TE10 can propagate
TM11 can propagate
TM21 can propagate
('For free space wavelength (in cm) =', 5.0)
TE10 can propagate
TM11 can propagate
TM21 can propagate


## Example 4.16 Page No : 126¶

In [18]:
import math

c = 3*10**10                      #in cm/s
f = 10*10**9                      #in Hz
a = 3                             #in cm
b = 2                             #in cm

#calculations
n = 120 * math.pi
wl_o = c/f
wl_c = 2*a*b/math.sqrt(a**2+b**2)
Z_TM = round(n*math.sqrt(1-(wl_o/wl_c)**2),4)

#output
print 'Characteristic impedance (in ohms) =', Z_TM

#answer in book is wrongly written as 61.618 ohms

Characteristic impedance (in ohms) = 163.2419


## Example 4.17 Page No : 134¶

In [19]:
import math

c = 3*10**10                    #in cm/s
f = 6*10**9                     #in Hz

#calculations
f_c = 0.8*f
wl_c = c/f_c
D = round(1.841*(wl_c/math.pi),4)
wl_o = c/f
wl_g = round(wl_o/math.sqrt(1-(wl_o/wl_c)**2),4)

#output
print 'Diameter of waveguide (in cm) =', D
print 'Guide wavelength (in cm) =', wl_g

Diameter of waveguide (in cm) = 3.6626
Guide wavelength (in cm) = 8.3333


## Example 4.18 Page No : 142¶

In [20]:
import math

a = 1.5                       #in cm
b = 1                         #in cm
e_r = 4                       #dielectric
c = 3*10**10                  #in cm/s

#calculations
wl_c = 2*b
f_c = c/wl_c
f_imp = 6*10**9               #impressed frequency (in Hz)
wl_air = c/f_imp

#Inserting dielectric:
wl_dielec = wl_air/math.sqrt(e_r)

#---logic--#
if(wl_dielec > wl_c):
print '    TE01 can propagate'
else:
print '    TE01 cannot propagate'

    TE01 can propagate


## Example 4.19 Page No : 148¶

In [21]:
import math

u = 4*math.pi*10**-7
e = 8.85*10**-12
c = 3*10**10          #in cm/s
f = 6*10**9           #in Hz
a = 1.5               #in cm
b = 1                 #in cm
#For TE10 mode:
m = 1
n = 0

#calculations
wl_c = 2*a
f_c = c/wl_c
t_1 = (m*math.pi/a)**2
t_2 = (n*math.pi/b)**2
t_3 = (((2*math.pi*f)**2)*u*e)
a = math.sqrt(abs(t_1+t_2-t_3))              #in neper/m
# variable t_1+t_2-t_3 is negative. So I changed the sign to calculate sqrt.

#---output---#
print 'Attenuation (in dB/m) =', round(a*20 / math.log(10),4)

Attenuation (in dB/m) = 1091.8468


## Example 4.20 Page No : 149¶

In [22]:
import math

c = 3*10**10                           #in cm/s
f = 9*10**9                            #inHz
a = 3                                  #in cm
b = 1                                  #in cm
E_max = 3000                           #in V/cm

#calculations
wl_o = c/f
wl_c = 2*a                             #in TE10
wl_g = round(wl_o/math.sqrt(1-(wl_o/wl_c)**2))
P_max = (6.63*10**-4)*E_max**2*a*b*(wl_o/wl_g)

#---output---#
print 'Maximum power for rectangular waveguide (in kilowatts)=', round(P_max/1000, 4)

Maximum power for rectangular waveguide (in kilowatts)= 17.901


## Example 4.21 Page No : 150¶

In [23]:
import math

c = 3*10**10                #in cm/s
f = 9*10**9                 #inHz
E_max = 300                 #in V/cm
d = 5

#calculations
wl_o = c/f
#For TE11
wl_c = d*math.pi/1.841
wl_g = wl_o/math.sqrt(1-(wl_o/wl_c)**2)
P_max = 0.498*E_max**2*d**2*(wl_o/wl_g)

#---output---#
print 'Maximum power (in watts) =',round(P_max,4)

Maximum power (in watts) = 1048954.2981


## Example 4.22 Page No : 156¶

In [25]:
import math

c = 3.*10**10                      #in cm/s
f = 30.*10**9                      #inHz
a = 1.                             #in cm
b = 1.
P_max = 746.                       #in watts

#calculations
wl_o = c/f
wl_c = 2*a
Z = 120*math.pi/math.sqrt(1-(wl_o/wl_c)**2)
E_max = math.sqrt(P_max*4*Z/(a*b/10000))

#---output---#
print 'Peak value of electric field (in kV/m) =',round(E_max/1000,4)

Peak value of electric field (in kV/m) = 113.9724


## Example 4.23 Page No : 163¶

In [26]:
import math

#Given:
c = 3*10**10                      #in cm/s
a = 2.3                           #in cm
b = 1                             #in cm
f = 9.375*10**9                   #in Hz

#calculations
wl_o = c/f
P_bd_TE11 = 597 * 2.3 * 1 * (1-(wl_o/(2*a))**2)**0.5

#---output---#
print 'Breakdown power for dominant mode (in kW) =',round(P_bd_TE11,4)

Breakdown power for dominant mode (in kW) = 986.4059


## Example 4.24 Page No : 166¶

In [27]:
import math

#Given:
d = 5                                #in cm
c = 3*10**10                         #in cm/s
f = 9*10**9                          #inHz
#Dominant mode is TE11:
#calculations
wl_o = c/f
wl_c = math.pi*d/1.841
f_c = c/wl_c
P_bd_TE11 = 1790*(d/2)**2*(1-(f_c/f)**2)**0.5

#---output---#
print 'Breakdown power (in kW) =',round(P_bd_TE11/1000,4)

Breakdown power (in kW) = 6.591