# chapter03:Transmission Lines¶

## Example number 3.1, Page number 47¶

In [1]:
#calculate Terminating impedance
#Variable declaration
Zo = 100       #o/p impedance(Ohms)
s = 5          #VSWR

#Calculations
Zmax = Zo*s

#Results
print "Terminating impedance = ",Zmax,"Ohms"

Terminating impedance =  500 Ohms


## Example 3.2, Page number 47¶

In [2]:
#calculate Characteristic impedance,Attenuation constant,Phase constant,Power delivered to the load
import math
import cmath

#Varaible declaration
R = 8             #resistance(Ohms)
L = 2*10**-3      #inductance(H/km)
C = 0.002*10**-6  #capacitance(F)
G = 0.07*10**-6   #conductance(s/km)
f = 2*10**3       #frequency(Hz)
Vs = 2            #input signal(V)
l = 500.          #line length(km)

#Calculations
w = 2*math.pi*f
x = complex(R,w*L)
y = complex(G,w*C)
Zo = cmath.sqrt(x/y)
gamma = cmath.sqrt(x*y)
Is = Vs/Zo.real
Il = Is*cmath.exp(-1*gamma*l)
P = Il**2*Zo.real

#Results
print "Characteristic impedance =",Zo,"Ohms"
print "Attenuation constant =",round(gamma.real,6),"NP/km"
print "\ncalculation error in the textbook"
print "\nPower delivered to the load =", round((abs(P)/1E-6),1), "uW"

Characteristic impedance = (1012.50018135-155.813417548j) Ohms
Attenuation constant = 0.003987 NP/km

calculation error in the textbook

Power delivered to the load = 73.3 uW


## Example 3.3, Page number 48¶

In [3]:
#calculate Phase velocity
import math

#Varaible declaration
f = 2*10**3       #frequency(Hz)

#Calculations
w = 2*math.pi*f
Vp = w/B

#Results
print "Phase velocity =",round((Vp/1E+3),2),"km/sec"

Phase velocity = 494.16 km/sec


## Example 3.4, Page number 48¶

In [4]:
#calculate Current drawn from generator,Power delivered to the load,Current flowing through the load

import cmath
import math

#Variable declaration
f = 37.5*10**6       #frequency(Hz)
V = 200              #Voltage signal(Vrms)
r = 200              #internal resistance(Ohms)
Zo = 200             #characteristic impedance(Ohms)
l = 10               #line length(m)
c = 3*10**8          #velocity of propagation(m/s)

#Calculations
#Part a
lamda = c/f
Bl = (5*math.degrees(math.pi))/4
x = complex(Zl,(Zo*math.tan(Bl)))
y = complex(Zo,(Zl*math.tan(Bl)))
Zi = Zo*(x/y)
Vs = (Zi.real*Zo)/(Zi.real+Zo)
Is = Zo/(Zi.real+Zo)

#Part b
P = Vs*Is

#Part c
Il = math.sqrt(P/Zl)

#Results
print "Please note that the solution given in the textbook is incorrect.Hence the difference in answers\n"
print "Current drawn from generator is",round(Is,3),"A"
print "Power delivered to the load is",round(P,2),"W"
print "Current flowing through the load is",round(Il,3),"A"

Please note that the solution given in the textbook is incorrect.Hence the difference in answers

Current drawn from generator is 0.413 A
Power delivered to the load is 48.47 W
Current flowing through the load is 0.696 A


## Example 3.5, Page number 50¶

In [5]:
#calculate Reflection co-efficient, VSWR
import cmath
import math

#Variable declaration
zo = 50              #characteristic impedance(Ohms)
f = 300*10**6        #frequency(Hz)
c = 3*10**8          #velocity of propagation(m/s)

#Calculations
lamda = c/f
rho = (zl-zo)/(zl+zo)
phi = cmath.phase(rho)
s = (1+abs(rho))/(1-abs(rho))

#Results
print "Reflection co-efficient =",round(abs(rho),4),"with phase =",round(math.degrees(phi),1)
print "VSWR =",round(s,2)

Reflection co-efficient = 0.4472 with phase = 63.4
VSWR = 2.62


## Example 3.6, Page number 50¶

In [6]:
#calculate position of the stub,Length of stub
import math

#Variable declaration
Zo = 600.           #characteristic impedance(Ohms)
f = 100*10**6       #frequency(Hz)
c = 3*10**8         #velocity of propagation(m/s)

#Calculations
lamda = c/f
l = (lamda*math.atan(math.sqrt(Zl/Zo)))/(2*math.pi)
l_dash = (lamda*math.atan(math.sqrt((Zl*Zo)/(Zo-Zl))))/(2*math.pi)

#Results
print "The position of the stub is", round(l,3),"m\n"
print "Please note that the solution for l_dash given in the textbook is incorrect"
print "Length of stub is",round(l_dash,3),"m"

The position of the stub is 0.185 m

Please note that the solution for l_dash given in the textbook is incorrect
Length of stub is 0.707 m


## Example 3.7, Page number 50¶

In [7]:
#calculate Terminating impedance
import cmath
import math

#Variable declaration
s = 3.2            #VSWR
Xmin = 0.237       #minimum voltage(V)
Zo = 50            #characteristic impedance(Ohms)

#Calculations
q = math.tan(math.degrees(2*math.pi*Xmin))
x = complex(1,-(s*q))
y = complex(s, -q)
Zl = Zo*(x/y)

#Result
print "Please note that the solution given in the textbook is incorrect.Hence the difference in answers\n"
print "Terminating impedance =", Zl,"Ohms"

Please note that the solution given in the textbook is incorrect.Hence the difference in answers

Terminating impedance = (19.6572514629-23.7885950214j) Ohms


## Example 3.8, Page number 51¶

In [8]:
#calculate VSWR,First Vmax is loacted at load and first Vmin is located at,Vmin,Impedance at Vmin
import math

#Variable declaration
Zo = 50.               #characteristic impedance(Ohms)
f = 300*10**3          #frequency(Hz)
c = 3*10**8            #velocity of propagation(m/s)

#Calculations
lamda = c/f

#Part a
rho = (Zl-Zo)/(Zl+Zo)
s = (1+abs(rho))/(1-abs(rho))

#Part b
#Since real Zl>Zo, first Vmax is located at the load
Vmin_pos = lamda/4

#Part c
Vmax = math.sqrt(Pl*Zl)
Vmin = Vmax/s

#Part d
Zin_at_Vmin = Zo/s
Zin_at_Vmax = Zo*s

#Results
print "VSWR = ", s
print "First Vmax is loacted at load and first Vmin is located at=", Vmin_pos,"m from the load"
print "Vmax = ",round(Vmax,2),"V","\nVmin = ",round(Vmin,2),"V"
print "Impedance at Vmin is ", Zin_at_Vmin,"Ohm and impedance at Vmax is",Zin_at_Vmax,"Ohm"

VSWR =  2.0
First Vmax is loacted at load and first Vmin is located at= 250 m from the load
Vmax =  2.24 V
Vmin =  1.12 V
Impedance at Vmin is  25.0 Ohm and impedance at Vmax is 100.0 Ohm


## Example 3.9, Page number 52¶

In [9]:
#calculate Reflection loss, transmission loss, return loss

import math

#Variable declaration
Zo = 600.           #characteristic impedance(Ohms)
Zs = 50             #source impedance(Ohms)
l = 200             #length of line(m)

#Calculations
rho = (Zl-Zo)/(Zl+Zo)

#Part a
ref_l = math.log10(1/(1-((abs(rho))**2)))

#Part b
#Since, the line is lossless,
att_l = 0
trans_l = ref_l+att_l

#Part c
ret_l = math.log10(abs(rho))

#Results
print "Reflection loss =",round(ref_l,4),"dB"
print "Transmission loss =",round(trans_l,4),"dB"
print "Return loss =",round(ret_l,3),"dB (Calculation error in the textbook)"

Reflection loss = 0.0036 dB
Transmission loss = 0.0036 dB
Return loss = -1.041 dB (Calculation error in the textbook)


## Example 3.10, Page number 52¶

In [10]:
#calculate Characteristic impedance,Phase velocity

import cmath
import math

#Variable declaration
l = 10 #length of line(km)
zsc = complex(1895.47,2234.29)
zoc = complex(216.99,-143.37)
f = 1*10**3  #frequency(Hz)

#Calculations
zo = cmath.sqrt(zsc*zoc)
x = cmath.sqrt(zsc/zoc)
t = (1+x)/(1-x)
gamma = cmath.log(t)/(l*2)
B = gamma.imag
w = 2*math.pi*f
Vp = w/B

#Results
print "There is calculation mistake throughout the problem in the textbook\n"
print "Characteristic impedance =",zo,"Ohms"
print "Phase velocity =",round((Vp/1E+3),3),"*10^3 m/sec"

There is calculation mistake throughout the problem in the textbook

Characteristic impedance = (864.190238563+123.274392427j) Ohms
Phase velocity = 45.994 *10^3 m/sec