Chapter 9: Transmission Lines

Example 9.1, page no. 237

In [2]:
# Variable Declaration
Zo = 75                       # Characteristic Impedance (Ohms) 
C  = 69.00*pow(10,-12)        # Nominal Capacitance (F/m)
Di = 0.584*pow(10,-3)         # Inner core diameter (m)
k  = 2.23                     # Dielectric Constant

# Calculation
import math                 # Math Library
L  = pow(Zo,2)*C            # Inductance per meter (H/m)
Do = Di*pow(10,Zo*math.sqrt(k)/138)  # Outer core diameter (m)

# Result
print "Inductance per meter, L =",round(L/pow(10,-6),3),"uH/m"
print "Outer Diameter, D =",round(Do/pow(10,-3),2),"mm"

Inductance per meter, L = 0.388 uH/m
Outer Diameter, D = 3.78 mm

Example 9.2, page no. 237

In [3]:
# Variable Declaration
s      = 1      # Assumed s (m)
d      = s      # Condition for minimum Zo (m)

# Calculation
import math                     # Math Library
Zo_min = 276*math.log10(2*s/d)  # Minimum value of characteristic impedance (Ohms)
 
# Result
print "The minimum value of characteristic impedance, Zo_min =",round(Zo_min),"Ohms"

The minimum value of characteristic impedance, Zo_min = 83.0 Ohms

Example 9.3, page no. 237

In [4]:
# Variable Declaration
Zo = 2000                     # Characteristic Impedance (Ohms) 
Di = 0.025*pow(10,-3)         # Inner cable diameter (m)
k  = 2.56                     # Dielectric Constant 

# Calculation
import math                         # Math Library
Do = Di*pow(10,Zo*math.sqrt(k)/138) # Outer conductor diameter (m)

# Result
print "Outer Diameter, D =",round(Do/pow(10,18),2),"* 10^(15) km or",round(Do/(9.44*pow(10,15))),"light years"

Outer Diameter, D = 3.86 * 10^(15) km or 409.0 light years

Example 9.4, page no. 243

In [5]:
# Variable Declaration
Zo = 200                # Characteristic Impedance of main line (Ohms) 
Zl = 300                # Load Impedance (Ohms)

# Calculation
import math             # Math Library
Zo1 = math.sqrt(Zo*Zl)  # Characteristic impedance of the quarter wave transformer (Ohms)

# Result
print "Characteristic impedance of the quarter wave transformer, Zo1 =",round(Zo1),"Ohms"

Characteristic impedance of the quarter wave transformer, Zo1 = 245.0 Ohms

Example 9.5, page no. 246

In [6]:
# Variable Declaration
Zl    = complex(200,75)      # Load Impedance (Ohms)
Zo    = 300                  # Characteristic Impedance (Ohms)

# Calculation
import math                  # Math Library
Yl    = 1/Zl                 # Admittance (Mho)
Bstub = 1/Yl.imag            # Reactance of the Stub (Ohms)
Gl    = Yl.real              # Real Part of Admittance (Mho)
Rl    = 1/Gl                 # Resistance (Ohms)
Zo1   = math.sqrt(Zo*Rl)     # Characteristic impedance of the quarter wave transformer (Ohms)

# Result
print "Reactance of the stub, Bstub =",round(Bstub,1),"Ohms"
print "Characteristic impedance of the quarter wave transformer, Zo1 =",round(Zo1),"Ohms"

Reactance of the stub, Bstub = -608.3 Ohms
Characteristic impedance of the quarter wave transformer, Zo1 = 262.0 Ohms

Example 9.6, page no. 250

In [7]:
# Variable Declaration
Y      = complex(0.004,0.002)  # Load Susceptance (Ohms)
Yo     = 0.0033                # Ohms - Characteristic Admittance (Ohms)
f      = 150*pow(10,6)         # Operating Frequency (Hz)
vc     = 3*pow(10,8)           # Speed of light in vacuum (m/s)

# Calculation
import math                    # Math Library
y      = Y/Yo                  # Normalized susceptance required to cancel loads normalized susceptance
Lambda = vc/f                  # Wavelength (m)
Length = 0.337*Lambda          # Length from Smith Chart (m)

# Result
print "Normalized susceptance required to cancel loads normalized susceptance = +j *",round(y.imag,2)
print "Length =",round(Length*100,1),"cm"

Normalized susceptance required to cancel loads normalized susceptance = +j * 0.61
Length = 67.4 cm

Example 9.7, page no. 250

In [8]:
# Variable Declaration
Z       = complex(100,50)    # Load Impedance (Ohms)
Zo      = 75                 # Characteristic Impedance (Ohms)

# Calculation
import math                  # Math Library
z       = Z/Zo               # Normalized Load Impedance (Ohms)
Zg      = 39.8               # Resistance at Distance = 0.184* Lambda, from Smith Chart (Ohms)
Zo_dash = math.sqrt(Zg*Zo)   # Impedance of the transformer (Ohms)

# Result
print "(a) From Smith Chart the Distance = 0.184 * Lambda"
print "(b) Zo_dash for the transformer, Zo' =",round(Zo_dash,1),"Ohms"

(a) From Smith Chart the Distance = 0.184 * Lambda
(b) Zo_dash for the transformer, Zo' = 54.6 Ohms

Example 9.8, page no. 253

In [9]:
# Variable Declaration
Z  = complex(450,-600)                          # Load Impedance (Ohms)
Zo = 300                                        # Characteristic Impedance (Ohms)

# Calculation
import math                                     # Math Library
z = Z/Zo                                        # Normalized Load Impedance (Ohms)
s = 4.6                                         # Standing Wave Ratio
L = 1/(2*math.pi)*math.atan(math.sqrt(s)/(s-1)) # (* Lambda) Stub Length (m)

# Result
print "Normalized load Impedance = ",z
print "From Smith Chart the Distance to the stub = 0.130 * Lambda"
print "Stub Length =",round(L,3),"* Lambda"

Normalized load Impedance =  (1.5-2j)
From Smith Chart the Distance to the stub = 0.130 * Lambda
Stub Length = 0.086 * Lambda

Example 9.9, page no. 254

In [10]:
# Variable Declaration
Z  = complex(450,-600)               # Load Impedance (Ohms)
Zo = 300                             # Characteristic Impedance (Ohms)
f1 = 10                              # Old frequency (MHz)
f2 = 12                              # New frequency (MHz)

# Calculation
import math                                            # Math Library
z  = Z/Zo                                              # Normalized Load Impedance (Ohms)
z1 = z.imag * f1/f2                                    # Intermediate Impedance (Ohms)
z  = complex(z.real,z1)                                # Normalized Load Impedance (Ohms)
s  = 4.6                                               # Standing Wave Ratio
L  = 1/(2*math.pi)*math.atan(math.sqrt(s)/(s-1))*f2/f1 # (* Lambda')Stub Length (m)

# Result
print "Normalized load Impedance = ",z
print "From Smith Chart the Distance to the stub = 0.156 * Lambda'"
print "Stub Length =",round(L,3),"* Lambda'"
print "From Smith chart SWR = 2.2"

Normalized load Impedance =  (1.5-1.66666666667j)
From Smith Chart the Distance to the stub = 0.156 * Lambda'
Stub Length = 0.103 * Lambda'
From Smith chart SWR = 2.2

Example 9.10, page no. 256

In [11]:
# Variable Declaration
Z             = 200.00             # Load Impedance (Ohms)
Zo            = 300.00             # Characteristic Impedance (Ohms)

# Calculation
import math                        # Math Library
z             = Z/Zo               # Normalized Load Impedance (Ohms)
L1_by_Lambda  = 0.311              # Ratio from Smith Chart
L2_by_Lambda1 = L1_by_Lambda*1.1   # Ratio

# Result
print "(a) Normalized load Impedance = ",round(z,2)
print "    From Smith Chart the Distance to the stub = 0.11 * Lambda"
print "    From Smith Chart the Length of stub = 0.311 * Lambda"
print "(b) New Length of stub =",round(L2_by_Lambda1,3),"* Lambda'"
print "    From Smith chart SWR = 1.3"

(a) Normalized load Impedance =  0.67
    From Smith Chart the Distance to the stub = 0.11 * Lambda
    From Smith Chart the Length of stub = 0.311 * Lambda
(b) New Length of stub = 0.342 * Lambda'
    From Smith chart SWR = 1.3