Chapter 12: Waveguides, Resonators And Components

Example 12.1, page no. 344

In [2]:
# Variable Declaration
Theeta = 60                        # Angle of incidence (degrees)

# Calculation
import math                        # Math Library
vg     = math.sin(Theeta*math.pi/180)   # (X vc) Velocity of EM wave in parallel direction (m/S)
vn     = math.cos(Theeta*math.pi/180)   # (X vc) Velocity of EM wave in normal direction (m/s)

# Result
print "Velocity of EM wave in parallel direction, vg =",round(vg,2),"* vc"
print "Velocity of EM wave in normal direction, vn =",round(vn,2),"* vc"

Velocity of EM wave in parallel direction, vg = 0.87 * vc
Velocity of EM wave in normal direction, vn = 0.5 * vc

Example 12.2, page no. 345

In [9]:
# Variable Declaration
Theeta   = 60                             # Angle of incidence (degrees)

# Calculation
import math                               # Math Library
Lambda_g = 1/math.sin(Theeta*math.pi/180) # (* Lambda) Wavelength of EM wave in parallel direction (m)
Lambda_n = 1/math.cos(Theeta*math.pi/180) # (* Lambda) Wavelength of EM wave in normal direction (m)

# Result
print "Wavelength of EM wave in parallel direction, Lambda_g =",round(Lambda_g,2),"* Lambda"
print "Wavelength of EM wave in normal direction, Lambda_n =",round(Lambda_n,2),"* Lambda"

Wavelength of EM wave in parallel direction, Lambda_g = 1.15 * Lambda
Wavelength of EM wave in normal direction, Lambda_n = 2.0 * Lambda

Example 12.3, page no. 346

In [11]:
# Variable Declaration 
Theeta = 60                             # Angle of incidence (degrees)

# Calculation
import math                             # Math Library
vp     = 1/math.sin(Theeta*math.pi/180) # (* vc) Phase velocity of EM wave (m/s)

# Result
print "The phase velocity of EM wave, vp =",round(vp,2),"* vc"

The phase velocity of EM wave, vp = 1.15 * vc

Example 12.4, page no. 349

In [14]:
# Variable Declaration
a      = 5.1          # Dimension 1 of rectangular waveguide (cm)
b      = 2.4          # Dimension 2 of rectangular waveguide (cm)
m      = 2            # Number of half wavelengths

# Calculation
import math           # Math Library
Lambda = 2*a/m        # Cutoff wavelength of rectangular waveguide (cm)

# Result
print "The cutoff wavelength of rectangular waveguide, Lambda =",round(Lambda,1),"cm"

The cutoff wavelength of rectangular waveguide, Lambda = 5.1 cm

Example 12.5, page no. 349

In [16]:
# Variable Declaration
a  = 5.1                     # Dimension 1 of rectangular waveguide (cm)
b  = 2.4                     # Dimension 2 of rectangular waveguide (cm)
m  = 1                       # Constant m for TE10 mode
n  = 0                       # Constant n for TE10 mode

# Calculation
import math                                         # Math Library
fc = 1.5*pow(10,8)*math.sqrt(pow(m/a,2)+pow(n/b,2)) # Cutoff frequency of dominant mode (TE10) of rectangular waveguide (Hz)
                                                  
# Result
print "The cutoff frequency of dominant mode (TE10) of rectangular waveguide, fc =",round(fc/pow(10,7),2),"GHz"

The cutoff frequency of dominant mode (TE10) of rectangular waveguide, fc = 2.94 GHz

Example 12.6, page no. 350

In [17]:
# Variable Declaration
a  = 5.1               # Dimension 1 of rectangular waveguide (cm)
b  = 2.4               # Dimension 2 of rectangular waveguide (cm)
m1 = 0                 # Constant m for TE01 mode
n1 = 1                 # Constant n for TE01 mode
m2 = 2                 # Constant m for TE20 mode
n2 = 0                 # Constant n for TE20 mode
m3 = 0                 # Constant m for TE02 mode
n3 = 2                 # Constant n for TE02 mode

# Calculation
import math                       # Math Library
f1 = 1.5*pow(10,8)*math.sqrt(pow(m1/a,2)+pow(n1/b,2)) # Cutoff frequency of TE01 of rectangular waveguide (Hz)
f2 = 1.5*pow(10,8)*math.sqrt(pow(m2/a,2)+pow(n2/b,2)) # Cutoff frequency of TE20 of rectangular waveguide (Hz)
f3 = 1.5*pow(10,8)*math.sqrt(pow(m3/a,2)+pow(n3/b,2)) # Cutoff frequency of TE02 of rectangular waveguide (Hz)

# Result
print "The frequency of TE01 mode of rectangular waveguide, f1 =",round(f1/pow(10,7),2),"GHz"
print "The frequency of TE20 mode of rectangular waveguide, f2 =",round(f2/pow(10,7),2),"GHz"
print "The frequency of TE02 mode of rectangular waveguide, f3 =",round(f3/pow(10,7),2),"GHz"
print "Hence the lowest frequency except dominant mode is, f =",round(min(f1,f2,f3)/pow(10,7),2),"GHz"

The frequency of TE01 mode of rectangular waveguide, f1 = 6.25 GHz
The frequency of TE20 mode of rectangular waveguide, f2 = 5.88 GHz
The frequency of TE02 mode of rectangular waveguide, f3 = 12.5 GHz
Hence the lowest frequency except dominant mode is, f = 5.88 GHz

Example 12.7, page no. 351

In [18]:
# Variable Declaration
a        = 3.00                  # Plane Separation of rectangular waveguide (cm)
c        = 3.00*pow(10,8)        # Speed of light in vacuum (m/s)
f        = 6.00*pow(10,9)        # Operating frequency (Hz)
m        = 1.00                  # Constant m for dominant mode

# Calculation
import math                      # Math Library
Lambda   = c/f * 100                                  # Operating Wavelength (cm)
Lambda_o = 2*a/m                                      # Cutoff Wavelength (cm)
Lambda_p = Lambda/math.sqrt(1-pow(Lambda/Lambda_o,2)) # Guide Wavelength (cm)
vg       = c*math.sqrt(1-pow(Lambda/Lambda_o,2))      # Group velocity (m/s)
vp       = c/math.sqrt(1-pow(Lambda/Lambda_o,2))      # Phase velocity (m/s)

# Result
print "(a) Cutoff Wavelength, Lambda_o =",round(Lambda_o),"cm"
print "(b) Guide Wavelength, Lambda_p =",round(Lambda_p,2),"cm"
print "(c) Group Velocity, vg =",round(vg/pow(10,8),2),"*10^(8) m/s"
print "    Phase Velocity, vp =",round(vp/pow(10,8),2),"*10^(8) m/s"

(a) Cutoff Wavelength, Lambda_o = 6.0 cm
(b) Guide Wavelength, Lambda_p = 9.05 cm
(c) Group Velocity, vg = 1.66 *10^(8) m/s
    Phase Velocity, vp = 5.43 *10^(8) m/s

Example 12.8, page no. 352

In [20]:
# Variable Declaration
a         = 6.00               # Plane Separation of rectangular waveguide (cm)
c         = 3.00*pow(10,8)     # Speed of light in vacuum (m/s)
f         = 10.00*pow(10,9)    # Operating frequency (Hz)
m1        = 1# Dimensional Constant
m2        = 2# Dimensional Constant
m3        = 3# Dimensional Constant
m4        = 4# Dimensional Constant


# Calculation
import math                    # Math Library
Lambda    = c/f * 100          # Operating Wavelength (cm)
Lambda_o1 = 2*a/m1             # Wavelength (cm)
Lambda_o2 = 2*a/m2             # Wavelength (cm)
Lambda_o3 = 2*a/m3             # Wavelength (cm)
Lambda_o4 = 2*a/m4                                      # Wavelength (cm)
Lambda_p  = Lambda/math.sqrt(1-pow(Lambda/Lambda_o3,2)) # Guide Wavelength (cm)

# Result
print "(a) For m = 1, Lambda_o =",round(Lambda_o1),"cm"
print "    For m = 2, Lambda_o =",round(Lambda_o2),"cm"
print "    For m = 3, Lambda_o =",round(Lambda_o3),"cm"
print "    For m = 4, Lambda_o =",round(Lambda_o4),"cm"
print "    Hence The largest value of m = 3"
print "(b) Guide Wavelength, Lambda_p =",round(Lambda_p,2),"cm"

(a) For m = 1, Lambda_o = 12.0 cm
    For m = 2, Lambda_o = 6.0 cm
    For m = 3, Lambda_o = 4.0 cm
    For m = 4, Lambda_o = 3.0 cm
    Hence The largest value of m = 3
(b) Guide Wavelength, Lambda_p = 4.54 cm

Example 12.9, page no. 354

In [22]:
# Variable Declaration
Lambda   = 2.00                                    # Wavelength of travelling wave (cm)
Lambda_o = 4.00            # Cutoff wavelength (cm)

# Calculation
import math                    # Math Library
Zo       = 377/math.sqrt(1-pow(Lambda/Lambda_o,2)) # Characteristic impedance of the given waveguide (Ohms)

# Result
print "The characteristic impedance of the given waveguide, Zo =",round(Zo),"Ohms"

The characteristic impedance of the given waveguide, Zo = 435.0 Ohms

Example 12.10, page no. 355

In [24]:
# Variable Declaration
m  = 1             # Constant m for TM11 mode
n  = 1              # Constant n for TM11 mode

# Calculation
import math                                 # Math Library
Lambda_o = 2/math.sqrt(pow(m,2)+pow(2*n,2)) # (* a) Cutoff wavelength with b=a/2 (m)

# Result
print "Cutoff wavelength of standard rectangular waveguides, Lambda_o =",round(Lambda_o,3),"* a "

Cutoff wavelength of standard rectangular waveguides, Lambda_o = 0.894 * a

Example 12.11, page no. 356

In [26]:
# Variable Declaration
rho1 = 0.553                # Constant from Example 12.7
rho2 = 0.661                # Constant from Example 12.8

# Calculation
import math                 # Math Library
Zo1  = 120*math.pi/rho1     # Characteristic Wave Impedance (Ohms)
Zo2  = 120*math.pi/rho2     # Characteristic Wave Impedance (Ohms)

# Result
print "Ex.12.7 : Characteristic Wave Impedance, Zo1 =",round(Zo1),"Ohms"
print "Ex.12.8 : Characteristic Wave Impedance, Zo2 =",round(Zo2),"Ohms"

Ex.12.7 : Characteristic Wave Impedance, Zo1 = 682.0 Ohms
Ex.12.8 : Characteristic Wave Impedance, Zo2 = 570.0 Ohms

Example 12.12, page no. 357

In [28]:
# Variable Declaration
a   = 4.5                 # Dimension 1 of rectangular waveguide (cm)
b   = 3.0                 # Dimension 2 of rectangular waveguide (cm)
c   = 3.00*pow(10,8)      # Speed of light in vacuum (m/s)
f   = 9.00*pow(10,9)      # Operating frequency (Hz)
m1   = 1                   # Constant m for TE10 mode
n1   = 0                   # Constant n for TE10 mode
m2   = 1                   # Constant m for TM11 mode
n2   = 1               # Constant n for TM11 mode

# Calculation
import math                    # Math Library
Lambda    = c/f * 100          # Operating Wavelength (cm)
Lambda_o1 = 2*a/m1             # Cutoff Wavelength (cm)
Lambda_p1 = Lambda/math.sqrt(1-pow(Lambda/Lambda_o1,2))      # Guide Wavelength (cm)
vg1  = c*math.sqrt(1-pow(Lambda/Lambda_o1,2))           # Group velocity (m/s)
vp1  = c/math.sqrt(1-pow(Lambda/Lambda_o1,2))           # Phase velocity (m/s)
Zo1  = 120*math.pi/math.sqrt(1-pow(Lambda/Lambda_o1,2)) # Characteristic Wave Impedance (Ohms)
Lambda_o2 = 2/math.sqrt(pow(m2/a,2)+pow(n2/b,2))	     # Cutoff Wavelength (cm)
Lambda_p2 = Lambda/math.sqrt(1-pow(Lambda/Lambda_o2,2))	     # Guide Wavelength (cm)
vg2       = c*math.sqrt(1-pow(Lambda/Lambda_o2,2))	     # Group velocity (m/s)
vp2       = c/math.sqrt(1-pow(Lambda/Lambda_o2,2))	     # Phase velocity (m/s)
Zo2       = 120*math.pi*math.sqrt(1-pow(Lambda/Lambda_o2,2)) # Characteristic Wave Impedance (Ohms)

# Result
print "(a) For TE10 mode :"
print "Cutoff Wavelength, Lambda_o =",round(Lambda_o1),"cm"
print "Guide Wavelength, Lambda_p =",round(Lambda_p1,2),"cm"
print "Group Velocity, vg =",round(vg1/pow(10,8),2),"*10^(8) m/s"
print "Phase Velocity, vp =",round(vp1/pow(10,8),2),"*10^(8) m/s"
print "Characteristic Wave Impedance, Zo =",round(Zo1,1),"Ohms"
print "(b) For TM11 mode :"
print "Cutoff Wavelength, Lambda_o =",round(Lambda_o2),"cm"
print "Guide Wavelength, Lambda_p =",round(Lambda_p2,1),"cm"
print "Group Velocity, vg =",round(vg2/pow(10,8),2),"*10^(8) m/s"
print "Phase Velocity, vp =",round(vp2/pow(10,8),2),"*10^(8) m/s"
print "Characteristic Wave Impedance, Zo =",round(Zo2),"Ohms"

(a) For TE10 mode :
Cutoff Wavelength, Lambda_o = 9.0 cm
Guide Wavelength, Lambda_p = 3.59 cm
Group Velocity, vg = 2.79 *10^(8) m/s
Phase Velocity, vp = 3.23 *10^(8) m/s
Characteristic Wave Impedance, Zo = 405.9 Ohms
(b) For TM11 mode :
Cutoff Wavelength, Lambda_o = 5.0 cm
Guide Wavelength, Lambda_p = 4.5 cm
Group Velocity, vg = 2.23 *10^(8) m/s
Phase Velocity, vp = 4.03 *10^(8) m/s
Characteristic Wave Impedance, Zo = 281.0 Ohms

Example 12.13, page no. 357

In [30]:
# Variable Declaration
a = 3                 # Width of rectangular waveguide (cm)
c = 3.00*pow(10,8)    # Speed of light in vacuum (m/s)
m = 1                 # Constant m for dominant mode
Zo       = 500        # Characteristic wave impedance (Ohms)

# Calculation
import math                      # Math Library
Lambda_o = 2*a/m                                      # Cutoff Wavelength (cm)
Lambda   = pow(1-pow(120*math.pi/Zo,2),0.5)*Lambda_o  # Operating wavelength (cm)
f  = c/Lambda                                   # Operating Frequency (Hz)
 
# Result
print "Operating Frequency, f =",round(f/pow(10,7),2),"GHz"

Operating Frequency, f = 7.61 GHz

Example 12.14, page no. 360

In [32]:
# Example 12.14
# Calculate the cutoff wavelength,

# Variable Declaration
r  = 2.00              # Diameter of circular waveguide (cm)
c  = 3.00*pow(10,8)    # Speed of light in vacuum (m/s)
f  = 10.00*pow(10,9)   # Operating frequency (Hz)
kr = 1.84              # Constant from Table 12.2

# Calculation
import math                 # Math Library
Lambda   = c/f * 100                                       # Operating Wavelength (cm)
Lambda_o = 2*math.pi*r/kr                                  # Cutoff Wavelength (cm)
Lambda_p = Lambda/math.sqrt(1-pow(Lambda/Lambda_o,2))      # Guide Wavelength (cm)
Zo       = 120*math.pi/math.sqrt(1-pow(Lambda/Lambda_o,2)) # Characteristic Wave Impedance (Ohms)

# Result
print "Cutoff Wavelength, Lambda_o =",round(Lambda_o,2),"cm"
print "Guide Wavelength, Lambda_p =",round(Lambda_p,2),"cm"
print "Characteristic Wave Impedance, Zo =",round(Zo),"Ohms"

Cutoff Wavelength, Lambda_o = 6.83 cm
Guide Wavelength, Lambda_p = 3.34 cm
Characteristic Wave Impedance, Zo = 420.0 Ohms

Example 12.15, page no. 361

In [33]:
# Variable Declaration
r   = 1                      # Diameter(assumption) of circular waveguide (cm)
m   = 1                      # Constant m for dominant mode
kr  = 1.84                   # Constant from Table 12.2

# Calculation
import math                  # Math Library
Lambda_o1 = 2*math.pi*r/kr   # Cutoff Wavelength for circular waveguide (cm)
Lambda_o2 = Lambda_o1        # Cutoff Wavelength for rectangular waveguide (cm)
a         = Lambda_o2*m/2          # Dimensional Variable
Ac        = math.pi*pow(r,2)       # Cross Sectional Area of Circular Waveguide (m^2)
Ar        = pow(a,2)/2             # Cross Sectional Area of Rectangular Waveguide (m^2)
R 	  = Ac/Ar                  # Ratio
 
# Result
print "Ratio, Ac/Ar =",round(R,2)

Ratio, Ac/Ar = 2.16

Example 12.16, page no. 378

In [36]:
# Variable Declaration
a  = 1.00          # Dimension 1 of rectangular waveguide (cm)
b  = 0.50                # Dimension 2 of rectangular waveguide (cm)
m  = 1          # Constant m for dominant mode
del1     = 25.00              # Length of waveguide (cm)
c  = 3.00*pow(10,8)# Speed of light in vacuum (m/s) 
f        = 1.00*pow(10,9)# Operating frequency (Hz)

# Calculation
import math                     # Math Library
Lambda_o = 2 * a/m        # Cutoff Wavelength (cm)
Lambda   = c/f * 100            # Operating Wavelength (cm)
A_dB     = 54.5 * del1/Lambda_o # Voltage attenuation of waveguide in dominant mode (dB)

# Result
if Lambda_o<Lambda : print "The waveguide is operating below cutoff"
print "Voltage attenuation of waveguide in dominant mode, A_dB =",round(A_dB),"dB"

The waveguide is operating below cutoff
Voltage attenuation of waveguide in dominant mode, A_dB = 681.0 dB