TRANSMISSION LINES

subject code 19EC407

OBJECTIVES:
• To enable the student to understand the basic principles in low
frequency and radio frequency signal/energy transmission in a
close bound media.
• To enhance the student knowledge in the area of various
transmission line designs.
• To enhance the student knowledge in the area of low frequency
and radio frequency for practical applications.
OUTCOMES:
The student should be able to:
• Discuss the characteristics of signal propagation through
transmission lines and its losses.
• Deduce the standing wave ratio and input impedance in high
frequency transmission lines.
• Analyze impedance matching by stubs using smith charts.
• Categorize and explain radio propagation in guided systems.
• To learn low frequency and high frequency parameters.
UNIT II HIGH FREQUENCY TRANSMISSION LINES
Literature referred
1. David M. Pozar, "Microwave Engineering", Fourth Edition, Wiley
India, 2012.
2. R.E.Collin, "Foundations for Microwave Engineering", Second
edition, IEEE Press, 2001 .
3. G.S.N Raju, “Electromagnetic Field Theory and Transmission Lines ”
Pearson Education, First edition 2005.
4. John D Ryder, “Networks, lines and fields”, 2nd Edition, Prentice Hall
India, 2015.
UNIT II HIGH FREQUENCY TRANSMISSION LINES

• Transmission line equations at radio

frequencies. Line of Zero dissipation.
• Voltage and current on the dissipation-less
line. Standing Waves Nodes.
• Standing Wave Ratio.
• Input impedance of the
 dissipation-less line
 Open and short circuited lines.
• Power and impedance measurement on lines.
• Reflection losses.
• Measurement of VSWR and wavelength.
T transmission line used at radio frequency permits several simplifying approximation.
Hence the performance of the line may be analyzed simpler way.
Salient feature of a radio frequency line:
(i) The line has a considerable skin effect so that almost the entire current can be assumed to
floe through outer surface of the conductor. Thus the internal inductance of the wire can be
considered to be zero.
(ii) At high frequency reactance xx is much larger the resistance R. This happen because of
A.C. resistance of the wire increases proportional to square root of frequency due to skin
effect. While the inductance increases directly as the frequency f.
The skin depth, δ, is defined as the depth where the current density is just 1/e (about 37%) of the value
at the surface; it depends on the frequency of the current and the electrical and magnetic properties of
the conductor.

𝟏
Skin depth = 𝜹 =
𝝅𝒇𝝁𝝈 Cause of the skin effect. A current I flowing
through a conductor induces a magnetic field H. If
the current increases, as in this figure, the resulting
increase in H induces circulating eddy current
Distribution of current flow in a cylindrical IW which partially cancel the current flow in the
conductor, shown in cross section. center and reinforce it near the skin.
 (iii) The lines are so optimally constructed that shunt conductance G may be
considered to zero at the radio frequency
 Transmission line performance evaluated at radio frequency in fallowing cases:
 (a) Low dissipation line- Case in which R is small comparison with 𝝎𝑳 .
 (b) Zero dissipation line or Lossless line- Case in which R is negligible
comparison with 𝝎𝑳.
Parameters of Open-wire Line at Radio Frequencies
• At high frequencies, the current can be assumed to be flowing through the surface or
skin of the conductor.
• The skin depth is the depth upto which current flows.
• The skin effect reduces the effective cross-section area of the conductor through which
the current flows.
• This increases the resistance. At the same time, the internal inductance gets reduced to
zero almost since no current flows through the inner part of the conductor cross-section.
Accordingly Loop inductance for parallel-wire line becomes:
 (A) Open wire line. Consider an open wire line having two circular conductors parallel
to each other. Let the radius of each conductor be a and let them be separated by a
distance d as shown
 Cross-section of a parallel wire line.

• Then the self-inductance of the two-wires together is given by the equation,

𝒅
𝐋 = 𝝁𝒓 + 𝟗. 𝟐𝟏𝒍𝒐𝒈 𝟏𝟎−𝟕 𝒉𝒆𝒏𝒓𝒚𝒔/𝒍𝒐𝒐𝒑 meter (20.1)
𝒂
where is 𝝁𝒓 the relative magnetic permeability of the conductor material and
is unity for non-magnetic material. where d is the spacing between the conductors, and a is the
radius of each conductor both in meters
• The capacitance of open wire line is not effected by skin effect or the frequency of
signal. Hence the expression for capacitance:
𝟏𝟐.𝟎𝟕
𝑪= 𝒅 𝒑𝒇/𝒎𝒆𝒕𝒆𝒓 (20.2)
𝒍𝒐𝒈𝟏𝟎
𝒂
Parameters of Coaxial Lines at Radio Frequencies
• Coaxial cable. Fig. 17-4 shows the cross-section of a coaxial cable which consists of an
inner solid conductor of radius say a and an outer hollow tubular conductor of inner
surface radius say b.
• The outer conductor forms a conducting sheath.
• At high frequencies, the flow of current is confined within
the depth of penetration in both inner and outer conductor.
• Thus the outer conductor actually works as an electrostatic
shield to external interferences.
• Hence coaxial line forms an excellant transmission line for Fig 17.4: Cross section of
high frequency signals. coaxial line
• Neglecting the internal inductance, the inductance of a coaxial line at high frequencies is
given by:
−𝟕 𝒃
𝑳 = 𝟒. 𝟔 × 𝟏𝟎 𝝁𝒓 𝒍𝒐𝒈𝟏𝟎 𝒂 henrys/loop meter (20.10)
where 𝝁𝒓 may be taken as unity for air as the dielectric.
b is the inner radius of the outer conductor and a is the
outer radius of the inner conductor.
The capacitance of the coaxial cable dose not chance with frequency except for the
dielectric constant of spacer used to maintain the position of the center conductor.
Capacitance of the coaxial cable is given by the following equation :
𝟐𝟒.𝟏𝟒𝜺𝒓 𝐩𝐟
𝐂= 𝒃 𝐦𝐞𝐭𝐞𝐫 (20.11)
𝒍𝒐𝒈𝟏𝟎
𝒂
𝜺𝒓 is the relative dielectric constant of the dielectric material

Secondary Constant of a Dissipationless Line

• A line may be used for transmission of signals at very high frequencies in which case
the physical length of the line may be quite small say a few centimeters but
electrically the line may be several wavelengths long.
• In such case, the resistance of the line may be very small compared to its reactance.
Hence such high frequency lines may be considered as lossless with a=0 and G=0.
Such an assumption simplifies the analysis and performance of the line.
• The secondary constants of a lossless line can be determined by considering that
 𝒛 = 𝒋𝝎𝑳
and
𝒚 = 𝒋𝝎𝑪
so that the characteristic impedance Z, is given as,
𝒛 𝑳 (20.18)
𝒛𝟎 = = 𝒐𝒉𝒎𝒔
𝒚 𝑪
Eq. (2018) shows that the characteristic impedance of a dissipationless line is purely
resistive and may be called characteristic resistance, indicated by Ro.
Substituting the values of L and C for an open wire line at high frequencies from
Eqs. (20.1) and (20.2) into Eq. (20-18), Ro, may be written in terms of the dimensions of
the line.
Thus for open wire line, 𝟗. 𝟐 × 𝟏𝟎−𝟕 𝒅
𝑹𝟎 = 𝒍𝒐𝒈𝟏𝟎
𝟏𝟐. 𝟎𝟕 × 𝟏𝟎−𝟏𝟐 𝒂
𝒅
or 𝑹𝟎 = 𝟐𝟕𝟔𝒍𝒐𝒈𝟏𝟎
𝒂
𝒅
= 𝟏𝟐𝟎𝒍𝒏 (20.19)
𝒂
Similarly the characteristic impedance of a coaxial line in terms of its dimensions can
be written by substitution of Eq. (20.10) and (20.11) into Eq. (20-18) giving,
 𝟒.𝟔×𝟏𝟎−𝟕 𝒃
𝑹𝟎 = 𝒍𝒐𝒈 𝟏𝟎 𝒐𝒉𝒎𝒔
𝟐𝟒.𝟏𝟒×𝟏𝟎−𝟕 𝒂
𝟏𝟑𝟖 𝒃
= 𝒍𝒐𝒈𝟏𝟎 𝒐𝒉𝒎𝒔
′
𝜺𝒓 𝒂
𝟔𝟎 𝒃
= 𝒍𝒏 𝒐𝒉𝒎𝒔 (20.20)
𝜺𝒓 𝒂
′
𝟏
Where 𝜺′𝒓 is effective dielectric constant and given by 𝜺′𝒓 = 𝟏 + 𝜺𝒓 − 𝟏
𝒔
𝜺𝒓 is relative dielectric constant
S is the center to center distance between the spacer
The propagation constant is given by
𝜸 = 𝒛𝒚 = 𝜶 + 𝒋𝜷 = −𝝎𝟐 𝑳𝑪 = 𝒋𝝎 𝑳𝑪 (20.21)
Hence 𝜶 = 𝟎 ≠ 𝒂𝒏𝒅 𝜷 = 𝝎 𝑳𝑪 radian/meter (20.22)
where the values of L and C are for one meter length of line.
Substituting the values of L and C for open wire line as well as for Coaxial line, it is
found that 𝜷 is the same for both the lines and is directly proportional to frequency. The
phase velocity of propagation is,
𝝎 𝟏
𝒗𝒑 = = 𝒎𝒆𝒕𝒆𝒓/𝒔𝒆𝒄𝒐𝒏𝒅 (20.23)
𝜷 𝑳𝑪
 For open wire line phase velocity of propagation
𝒗𝒑 = 𝟑 × 𝟏𝟎𝟖 𝒎𝒆𝒕𝒆𝒓/𝒔𝒆𝒄𝒐𝒏𝒅
• Thus we see that the phase velocity of propagation in an air-spaced lossless open-
wire line is equal to the velocity of light in free space.
For coaxial line phase velocity of propagation
𝟑 × 𝟏𝟎𝟖 (20.24)
𝒗𝒑 =
𝜺′𝒓
• This equation shows that the effect of dielectric other than air, is to reduce the phase
velocity of propagation in a dissipationless line.
Voltages and Currents in a Lossless Line Here we look in to fallowing case
• Line with open circuit
• The voltage at any distance y from the receiving end • Line with short circuit

of a line terminated in impedance ZR, is given by, • Line terminated in pure resistance
𝑬𝑹 𝒁𝑹 +𝒁𝟎 grater then R0
.𝑬𝒙 = 𝒆𝜸𝒚 + 𝚪𝒆−𝜸𝒚 • Line terminated in R0 so that ZR = R0,
𝟐𝒁𝑹 𝑅
where ER is the voltage at the receiving end and 𝚪 is the ZR = 3R0, ZR = 0
3
reflection coefficient. • Line is terminated in a complex
impedance ZR
For lossless line 𝜶 is zero and 𝒁𝟎 = R0, so that
 𝑬𝑹 𝒁𝑹 + 𝑹𝟎 (20.25)
𝑬𝒙 = 𝒆𝜸𝒚 + 𝚪𝒆−𝜸𝒚
𝟐𝒁𝑹
• In Eq. (20.25), the term varying with 𝒆−𝜸𝒚 gives the incident , wave while the term
varying with 𝒆𝜸𝒚 gives the reflected wave.
• The magnitude of the reflected wave moving back toward the source is dependent on the
value of the reflection coefficient 𝜞.
• The magnitudes of rotating vectors corresponding to the sinusoidal signal input remain
constant along the entire length of the line.
• magnitude of the reflected wave remains constant at all points along the line.
• For an open circuited line, the instantaneous values of phasors for different instants of
time for one wavelength long line from the receiving end terminals are shown in Fig. 20.2
• For an open-circuit load, the reflection coefficient 𝜞 is unity and hence the magnitude of
the reflected wave is exactly the same as that of the incident wave at every point along the
line.
Fig. 20.2. Instantaneous values of voltage for open
circuit load in a lossless line
(a) at t=0 (b) at t=1/8f (c) at t=2/8f
(d) r.m.s. values of voltage along the line.
• The time t=0 is chosen such that the instantaneous value of voltage at the open-
circuited receiving end is zero as shown in Fig. 20.2 (a).
• The reflected wave has the same instantaneous values with a phase difference of
180°.
• Hence the total voltage at any point along the line, being the vector sum of the
two values, will be zero as shown by the horizontal line marked Resultant.
• As an instant corresponding to one-eighth of a cycle later, the phase relationship
between the incident and reflected waves has been shown in Fig. 20°.2 (b) along
with the resultant voltage.
• Similarly the waveforms of Fig. 20.2 (c) correspond to an instant one-eighth
cycle later than that of Fig. 20.2 (b).
• These plots show that the incident wave progress towards the load while the
reflected wave progress away from the load towards the source.
• It is seen that the resultant voltage is zero at all instants of time at the points
marked 2 and 6.
• Thus the resultant voltage has magnitude which oscillates with time and has
fixed positions of its maxima, minima and zero value points along the line. Such
a wave is known as a standing wave.
• If the voltage at different points along the line is measured with an r.m.s. voltmeter
and the values are plotted against the distance from. the receiving end, the magnitude
waveform as shown in Fig. 20.2 (d) is obtained.
• The r.m.s. values are always positive whether the instantaneous value is positive or
negative.
• This shows that the positions of nodes and anti-nodes of magnitude of voltage along
the length of an open circuited, lossless transmission line are fixed for a particular
frequency of the wave.
𝑬𝑹 𝒁𝑹 + 𝑹𝟎 𝒆𝒋𝒙 + 𝒆−𝒋𝒙
𝑬𝒙 = 𝒆𝜸𝒚 + 𝚪𝒆−𝜸𝒚 (20.25) ∵ 𝐜𝐨𝐬 𝒙 =
𝟐𝒁𝑹 𝟐
𝒆𝒋𝒙 − 𝒆−𝒋𝒙
This can be rearranged as, 𝒔𝒊𝒏 𝒙 =
𝟐𝒋
𝑬𝑹 𝒁𝑹 + 𝑹𝟎 𝒋𝜷𝒚 𝒁𝑹 − 𝑹𝟎 −𝒋𝜷𝒚
𝑬𝒙 = 𝒆 + 𝒆 (For lossless line 𝜶 is zero)
𝒁𝑹 𝟐 𝟐

𝑬𝑹 𝒆𝒋𝜷𝒚 + 𝒆−𝒋𝜷𝒚 𝒆𝒋𝜷𝒚 − 𝒆−𝒋𝜷𝒚

= 𝒁 + 𝒋𝑹𝟎
𝒁𝑹 𝑹 𝟐 𝟐𝒋

= 𝑬𝑹 𝐜𝐨𝐬 𝜷𝒚 + 𝒋𝑹𝟎 𝑰𝑹 𝐬𝐢𝐧 𝜷𝒚 (20.26)

Similarly, the current in a lossless line terminated in any impedance ZR other
than Z0 is given by,
𝑰𝑹 𝒁𝑹 + 𝑹𝟎 𝒋𝜷𝒚 𝒁𝑹 − 𝑹𝟎 −𝒋𝜷𝒚
𝑰𝒙 = 𝒆 − 𝒆
𝑹𝟎 𝟐 𝟐

𝑰𝑹 𝒆𝒋𝜷𝒚 − 𝒆−𝒋𝜷𝒚 𝒆𝒋𝜷𝒚 + 𝒆−𝒋𝜷𝒚

= 𝒁 + 𝑹𝟎
𝑹𝟎 𝑹 𝟐 𝟐
𝑬𝑹
= 𝑰𝑹 𝒄𝒐𝒔 𝜷𝒚 + 𝒋 𝐬𝐢𝐧 𝜷𝒚 (20.27)
𝑹𝟎
𝟐𝝅
Since 𝜷=
𝝀
, Equation (20.26) and (20.7) became
𝟐𝝅 𝟐𝝅
𝑬𝒙 = 𝑬𝑹 𝒄𝒐𝒔 𝒚 + 𝒋𝑰𝑹 𝑹𝟎 𝒔𝒊𝒏 𝒚 (20.28)
𝝀 𝝀
𝟐𝝅 𝑬𝑹 𝟐𝝅 (20.29)
𝑰𝒙 = 𝑰𝑹 𝒄𝒐𝒔 𝒚 +𝒋 𝒔𝒊𝒏 𝒚
𝝀 𝑹𝟎 𝝀

If the line id open circuited at receiving end IR became zero

𝟐𝝅 (20.30)
𝑬𝒙 = 𝑬𝑹 𝒄𝒐𝒔 𝒚
𝝀
𝑬𝑹 𝟐𝝅
𝑰𝒙 = 𝒋 𝒔𝒊𝒏 𝒚 (20.31)
𝑹𝟎 𝝀
 • The magnitudes of voltage and current given by Eqs. (20.30)
and (20.31) are plotted for different values of y and have been
shown in Fig. 20.3 (a).
• where the negative sign arising out of the cosine function has
been neglected since the measuring instrument does not
distinguish between positive and negative signs.
• Similarly with the receiving end short circuited, ER becomes
zero.
Thus Eqs. (20.28) and (20.29) can be written as,
𝟐𝝅
𝑬𝒙 = 𝒋𝑰𝑹 𝑹𝟎 𝒔𝒊𝒏 𝒚 (20.32)
𝝀
𝟐𝝅
𝑰𝒙 = 𝑰𝑹 𝒄𝒐𝒔 𝒚 (20.33)
𝝀
• These magnitudes have been plotted against distance y in
Fig 20.3 (b).
• It may be seen that in both these cases, the voltage end
Fig. 20.3. Magnitudes of voltage and current
current are always in phase quadrature along the entire
in a lossless line plotted against distance y
length of the line.
from the receiving end under open circuit and
short circuit conditions. • Hence no power gets transmitted by the line under the open
circuit or the short circuit condition at the receiving end.
 If the line is terminated in R0 so that ZR = R0, the refection coefficient
𝚪 becomes zero and Eq. (20.25) becomes,
𝑬𝒙 = 𝑬𝑹 𝒆𝒋𝜷𝒚 (20.34)
Similarly 𝑰𝒙 = 𝑰𝑹 𝒆𝒋𝜷𝒚 (20.35)
• Eqs. (20.34) and (20.35) show that the voltage and current magnitudes remain
constant and there is no attenuation with continuously increasing phage angle along
the line.
This magnitude has been shown in Fig. 19.4 (a) where straight
horizontal lines give the plots of ER and IR against distance y
from the receiving end.
• If the line is terminated in a pure resistance grater then R0, reflection coefficient 𝚪,
becomes positive Plots of magnitudes of ‘voltage and current with distance y are
then intermediate between those of open circuit and R0-terminated conditions.
• Thus if the line terminating resistance equals 3X, the reflection coefficient 𝚪 is
𝟏
𝟐
and the reflected wave has amplitude half of the incident wave.
• Hence the resultant voltage never becomes zero at any instant of time.
• The magnitude of voltage then has maxima and minima shown in Fig. 20 4 (b).
• The magnitude of current also has maxima and minima as shown in Fig. 20 4 (b).
 From Eq. (20.28), the magnitude of voltage
for termination ZR=3R0, is given by

𝟐𝝅 𝟐𝝅
𝑬𝒙 = 𝑬𝟐𝑹 𝒄𝒐𝒔𝟐 𝟐 𝟐
𝒚 + 𝑰𝑹 𝑹𝟎 𝒔𝒊𝒏 𝟐 𝒚
𝝀 𝝀
𝑬𝑹 𝑬𝑹
but 𝑰𝑹 = =
𝒁𝑹 𝟑𝑹𝟎

𝑬𝒙 𝟐𝝅 𝟏 𝟐𝝅
= 𝒄𝒐𝒔𝟐 𝒚 + 𝒔𝒊𝒏 𝟐 𝒚 (20.36)
𝑬𝑹 𝝀 𝟗 𝝀

Similarly from equ. (20.29) the magnitude of

current for ZR = 3R0 is given as

𝑰𝒙 𝟐𝝅 𝟐𝝅
Fig. 20.4. Magnitude of voltage and current in a lossless line = 𝒄𝒐𝒔𝟐 𝒚 + 𝟗 𝒔𝒊𝒏𝟐 𝒚 (20.37)
𝑰𝑹 𝝀 𝝀
plotted against distance y from the receiving end for
• 𝝀
The voltage maxima occur at integral multiples of or at every
𝟐
𝒏𝝀
𝟐
from the receiving
end
and the first voltage maximum occurs at the receiving end, where n=0.
• This is true since termination by a resistance much greater than R, tends to the condition
of 𝒁𝑹 → ∞ or towards the open circuit condition.
• Similarly the current minimum Occurs at receiving end and successive minima occur at
𝒏𝝀
every point from the receiving end.
𝟐
• 𝑹 𝟏
For 𝒁𝑹 = 𝟑𝟎 , the value of reflection coefficient 𝚪 is − 𝟐 .
• Hence the phase of the reflected wave is reversed in relation to that for ZR=3R0,.
• The variation of magnitude of voltage is now given by Eq. (20.37) while the variation of
magnitude of current is given by Eq. (20.36).
• Thus the maxima and minima points get interchanged between voltage and current as
compared to the case of ZR=3R0.
• Fig. 20.4c, shows the variation of magnitude of voltage and current.
• As the terminating resistance is reduced, the condition approaches the short circuit
condition and the magnitude of current increases while that the voltage decreases.
• In both the cases with finite terminating resistance, voltage and current at the receiving
terminals are finite. Hence certain amount of power is transmitted by the line and
transferred to the load resistance.
• The above considerations show that standing wave will always be present as long as 𝒁𝑹 ≠ 𝑹𝟎
• However, the amplitude of the standing wave depends upon the mismatch at the load; larger
the mismatch larger is the amplitude of the standing wave.
• If a lossless line is terminated in a complex impedance ZR, reflection coefficient 𝚪 is also
complex and the reflected wave is neither in phase or out of phase with the incident wave.
• Hence the points of maximum or minimum voltage will not fall at the receiving end but get
shifted.
Standing Wave Ratio
• An important quantity in the study of lossless lines is the standing wave ratio.
• Figs. 20.3 and 20.4 show the variation of magnitude of voltage and current along the
length of a lossless transmission line under various conditions of terminating
impedance.
• These magnitude plots do not change with time and are known as standing waves.
• A lossless line when not properly terminated in its characteristic impedance or
resistance is said to possess standing waves.
• The points of maximum voltage are called voltage antinodes while points of
minimum voltage are called voltage nodes.
• Similarly current antinodes and current nodes are the points of current
maximum and current minimum respectively.
• Obviously then voltage nodes coincide with current antinodes and voltage
antinodes coincide with current nodes.
• The standing wave ratio is defined as the ratio of the magnitude of a
standing wave at an antinode to the magnitude at a node.
 Now let us
• Assumes that the generator is matched, so that there is no re-reflection of the reflected
wave from z < 0.
• Average power flow is constant at any point on the line.
• that the total power delivered to the load (Pavg) is equal to the incident power
𝑽+𝟎
𝟐
/𝟐𝒁𝟎 minus the reflected power 𝑽 + 𝟐
𝟎 𝚪 𝟐
/𝟐𝒁𝟎 .
• If 𝚪 = 0, maximum power is delivered to the load, while no power is
delivered for | 𝚪 | = 1.
• When the load is mismatched, not all of the available power from the generator is
delivered to the load.
• This “loss” is called return loss (RL), and is defined (in dB) as
RL = −20 log | 𝚪 | dB, (2.38)
 A matched load (𝚪 = 0) has a return loss of ∞ dB (no reflected
power), for a open/short, total reflection (| 𝚪 | = 1) has a return loss of 0 dB
(all incident power is reflected).
• Return loss is a nonnegative number for reflection from a passive network.
• If the load is matched to the line, 𝚪 = 0 and the magnitude of the voltage on the line is
|V(z)| = |𝑽+
𝟎 |, which is a constant.
• When the load is mismatched, however, the presence of a reflected wave leads to standing
waves, and the magnitude of the voltage on the line is not constant.
Thus, The total voltage waves on the line can then be written as
(2.39a)

where ℓ = −z is the positive distance measured from the load at z = 0, and θ is the phase
of the reflection coefficient (𝚪 = | 𝚪 |ejθ).
This result shows that the voltage magnitude oscillates with position z along the line.
The maximum value occurs when the phase term ej(θ−2βℓ) = 1 and is given by

The minimum value occurs when the phase term e j (θ−2βℓ) = −1 and is given by
 Dividing the equation (2.40a) by equation (2.40b).
It gives the ratio of Vmax to Vmin and is called standing wave ration (SWR)

• SWR is also known as the voltage standing wave ratio and is sometimes identified as VSWR.
• From (2.41) it is seen that SWR is a real number such that 1 ≤ SWR ≤ ∞,
where SWR = 1 implies a matched load.
• From (2.39), it is seen that the distance between two successive voltage maxima (or minima) is
ℓ = 2π/2β = πλ/2π = λ/2,
• While the distance between a maximum and a minimum is
ℓ = π/2β = λ/4,
where λ is the wavelength on the transmission line.
• The reflection coefficient can be generalized to any point ℓ along the line as
follows.
 Under total mismatch condition there is no power flow.
 Forward propagating voltage equal to reflected voltage ie 𝑽− = 𝑽+, Vz = 0
Thus from (2.39a), with z = −ℓ, the ratio of the reflected component to the incident component is
 𝑽−
𝟎𝒆
−𝒋𝜷ℓ
𝜞 ℓ = + 𝒋𝜷ℓ
𝑽𝟎 𝒆
𝑽−
𝟎 𝒆
−𝒋𝜷ℓ
= + 𝒋𝜷ℓ
𝑽− 𝑽𝟎 𝒆
𝟎
∵ = 𝜞𝟎
𝑽+
𝟎 𝜞 ℓ = 𝜞𝟎 𝒆−𝒋𝜷ℓ 𝒆−𝒋𝜷ℓ
𝜞 ℓ = 𝜞𝟎 𝒆−𝟐𝒋𝜷ℓ (2.42)
where 𝜞(0) is the reflection coefficient at z = 0.
• This result is useful when transforming the effect of a load mismatch down the line.
• We have seen that the real power flow on the line is a constant (for a lossless line) but that
the voltage amplitude, at least for a mismatched line, is oscillatory with position on the
line.
At a distance ℓ = −z from the load, the input impedance seen looking toward the load is

Multiplying numerator and denominator by 𝒆−𝒋𝜷ℓ , we get

 We know that Inserting this value of 𝜞 in equation (2.43)

𝒆𝒋𝒙 + 𝒆−𝒋𝒙
∵ 𝐜𝐨𝐬 𝒙 =
𝟐
𝒆𝒋𝒙 − 𝒆−𝒋𝒙
𝒔𝒊𝒏 𝒙 =
𝟐𝒋

• This is an important result giving the input impedance of a length of

transmission line with an arbitrary load impedance.
INPUT IMPEDANCE OF A SHORT CIRCUITED LINE
• Consider the transmission line circuit shown in Figure 2.5, where a line is terminated in a
short circuit, ZL = 0.
• we know that the reflection coefficient for a short circuit load is 𝜞 = −1; and that the
standing wave ratio is infinite and the voltage and current on the line are

Using trigonometry identity

(2.45a) FIGURE 2.5 A transmission line terminated
in a s hort circuit.
(2.45b)

From figure 2.5, at z = −ℓ ,

As expected, for short circuited line, −ℓ =0, equation (2.45a) gives V=0
and equation (2.45a) gives I = 1 (maximum value)
Dividing equation (2.45a) with (2.45b)
 𝒁𝒊𝒏 = 𝒋𝒁𝟎 𝐭𝐚𝐧 𝜷ℓ (2.45c)

• Which is seen to be purely imaginary for any

length ℓ and to take on all values between + j∞
and − j∞.
• For example, when ℓ = 0 we have Zin = 0, but for
ℓ = λ/4 we have Zin = ∞ (open circuit).
• Equation (2.45c) also shows that the impedance
is periodic in ℓ, repeating for multiples of λ/2.
• The voltage, current, and input reactance for the
short circuited line are plotted in Figure 2.6

FIGURE 2.6 (a) Voltage, (b) current, and (c) impedance (Rin = 0 or ∞)
variation along a short-circuited transmission line.
INPUT IMPEDANCE OF A OPEN CIRCUITED LINE
consider the open-circuited line shown in Figure 2.7, where ZL = ∞.
Dividing the numerator and denominator of (2.35) by ZL and allowing ZL → ∞ shows
that the reflection coefficient for this case is 𝜞 = 1, and the standing wave ratio is again
infinite.

The voltage and current on the line are given as

FIGURE 2.7 A transmission line terminated in

an open circuit.
for this case is 𝜞 = 1 and using trigonometry identity

which shows that now I = 0 at the load, as expected for an open circuit, while
the voltage is a maximum. The input impedance is
• which is also purely imaginary for any length, ℓ. The
voltage, current, and input reactance of the open-
circuited line are in Figure 2.8.
plotted

• INPUT IMPEDANCE OF LINE LENGTH ℓ = λ/2,

TERMINATED TRANSMISSION LINES.
Keeping ℓ = λ/2, in equation (2.44) shows that

• This indicate a half-wavelength line (or any

multiple of λ/2) does not alter or transform the load
impedance, regardless of its characteristic
impedance.
If the line is a quarter-wavelength long or, more
generally, ℓ = λ/4 + nλ/2, for n =1, 2, 3, . . . , equ. (2.44)
shows that the input impedance is given by
FIGURE 2.8 (a) Voltage, (b) current, and (c)
impedance (Rin = 0 or ∞) variation along an
open-circuited transmission line.
 • Such a line is known as a quarter-wave transformer because it has the effect of
transforming the load impedance in an inverse manner, depending on the
characteristic impedance of the line.

---------------------------OOOO-------------------------------

• Standing wave ratio can be easily determined by measurements on actual lines.

• A voltmeter moved along the length of the line Measures the maximum and the minimum
values of voltage.
• The distance between two consecutive maxima or two consecutive minima corresponds to
one half wavelength and twice of this give wavetength of the operating frequency.
 Power and impedance measurement on transmission lines
Power measurement on transmission lines
voltage and current on the dissipation less transmission line is given as
𝑽𝑹 𝒁𝑹 + 𝒁𝟎
𝑽= 𝒆−𝒋𝜷𝒙 + 𝜞 ∙ 𝒆𝒋𝜷𝒙
𝟐𝒁𝑹
𝑽𝑹 𝑰𝑹 𝒁𝑹 + 𝒁𝟎
∵ = 𝑰𝑹 = 𝒆−𝒋𝜷𝒙 + 𝜞 ∙ 𝒆𝒋𝜷𝒙
𝒁𝑹 𝟐
𝑰𝑹 𝒁𝑹 + 𝒁𝟎
𝑰= 𝒆−𝒋𝜷𝒙 + 𝜞 ∙ 𝒆𝒋𝜷𝒙
𝟐𝒁𝟎
Maximum voltage and current is given as
For Vmax case: This will happen when forward(incident) voltage and reflected
voltage both are in same phase, under this condition 𝒆−𝒋𝜷𝒙 = 𝒆𝒋𝜷𝒙 = 𝟏

𝑰𝑹 𝒁𝑹 + 𝒁𝟎
Thus 𝑽𝒎𝒂𝒙 = 𝟏 + |𝜞|
𝟐
For Imax case: This will happen when forward(incident) current and reflected
current both are in same phase, under this condition 𝒆−𝒋𝜷𝒙 = 𝒆𝒋𝜷𝒙 = 𝟏
Thus 𝑰𝑹 𝒁𝑹 + 𝒁𝟎
𝑰𝒎𝒂𝒙 = 𝟏 + |𝜞|
𝟐𝑹𝟎
We will also find out Voltage minima and current minima on the line
For Vmin case: This will happen when forward(incident) voltage and reflected
voltage both are in out of phase or 1800.
𝑰𝑹 𝒁𝑹 + 𝒁𝟎
𝑽𝒎𝒊𝒏 = 𝟏 − |𝜞|
𝟐
This minus sign shows out of phase condition
For Imin case: This will happen when forward(incident) current and reflected
current both are in out of phase or 1800.
𝑰𝑹 𝒁𝑹 + 𝒁𝟎
𝑰𝒎𝒊𝒏 = 𝟏 − |𝜞|
𝟐𝑹𝟎
This minus sign shows out of phase condition
Now we will divide Vmax by Imax and Vmin by Imin respectively
𝑉𝑚𝑎𝑥
= 𝑅0
𝐼𝑚𝑎𝑥
𝑉𝑚𝑖𝑛
= 𝑅0
𝐼𝑚𝑖𝑛
 The current minimum and voltage maximum occur at same point on line and are in
same phase.
Now we divide the Vmax by Imin, this will give maximum impedance.
𝑽𝒎𝒂𝒙 𝟏+ 𝜞
= 𝑹𝟎 = 𝑺 𝑹𝟎 = 𝑹𝑴𝒂𝒙
𝑰𝒎𝒊𝒏 𝟏− 𝜞
Where S = Standing wave ratio (SWR)
Similarly we divide the Vmin by Imax, this will give minimum impedance.
𝑽𝒎𝒊𝒏 𝟏− 𝜞 𝑹𝑶
= 𝑹𝒎𝒊𝒏 = 𝒁𝟎 = = 𝑹𝑴𝒊𝒏
𝑰𝒎𝒂𝒙 𝟏+ 𝜞 𝑺
𝑽𝟐𝒎𝒂𝒙
Power flowing 𝑷=
𝑹𝒎𝒂𝒙
𝑽𝟐𝒎𝒊𝒏
𝑷=
𝑹𝒎𝒊𝒏

Multiplying above two Power 𝑽𝟐𝒎𝒂𝒙 𝑽𝟐𝒎𝒊𝒏

𝑷𝟐 = ,
𝑹𝒎𝒂𝒙 𝑹𝒎𝒊𝒏

𝑹𝟎
Inserting the value of Rmax and Rmin and they are S.R0 and respectively, is derived in previous steps.
𝑺
 𝑽𝒎𝒂𝒙 𝑽𝒎𝒊𝒏
𝑷=
𝑹𝟎
• From equation it is infer that we know the Vmax and Vmin in a transmission line and
now if we multiply the modules of 𝑽𝒎𝒂𝒙 𝒂𝒏𝒅 𝑽𝒎𝒊𝒏 and then, this product is divided
by characteristic impedance of line , we get the power in the line.
Power also can be given in term of current by fallowing same steps explained as
above 𝑷 = 𝑰𝟐 . 𝑹
Power flowing 𝒎𝒂𝒙 𝒎𝒂𝒙

𝑷 = 𝑰𝟐𝒎𝒊𝒏 . 𝑹𝒎𝒊𝒏

Multiplying above two Power 𝑷𝟐 = 𝑰𝟐𝒎𝒂𝒙 . 𝑹𝒎𝒂𝒙 . 𝑰𝟐𝒎𝒊𝒏 . 𝑹𝒎𝒊𝒏 ,

Inserting the value of Rmax and Rmi
Power flowing 𝑷 = 𝑰𝒎𝒂𝒙 𝑰𝒎𝒊𝒏 𝑹𝟎
Where 𝑹𝟎 = Characteristic impedance of line ie Z0

• From the above two derived equation we notice that, by measuring Vmax, Vmin, Imax Imin
on the transmission line and substituting the same in to above power equation we can
measure the power in the transmission line
 LOAD IMPEDANCE MEASUREMENT
• The unknown value of the load impedance ZL connected to a
transmission line can be determined by standing wave
measurement.
• Bridge circuit is used to measure the unknown impedance
• Consider the point of voltage minimum, distance from the load.
At this point resistance value also be minimum, Rmin since
Voltage is at minima point.
𝒁
This Rmin , we have already derived, 𝒁𝑳 = 𝑹𝒎𝒊𝒏 = 𝑺𝟎 (100)
s = SWR
Input impedance at any point on a line given as:
𝒁 +𝒋𝑹 𝐭𝐚𝐧 𝜷𝒙
𝒁𝑳 = 𝒁𝟎 𝑹𝑳 +𝑱𝒁𝟎 𝐭𝐚𝐧 𝜷𝒙
𝟎 𝟎
Transmission line connected with load ZL
Input impedance at distance from the load.
Put x = 𝓵 𝐦𝐞𝐭𝐞𝐫
𝒁 +𝒋𝑹 𝐭𝐚𝐧 𝜷𝓵
𝒁𝑳 = 𝒁𝟎 𝑹𝑳 +𝑱𝒁𝟎 𝐭𝐚𝐧 𝜷𝓵
𝟎 𝟎

Equating This ZL with ZL given by equation (100) as given above.

 𝒁𝟎 𝒁𝑳 +𝒋𝑹𝟎 𝐭𝐚𝐧 𝜷𝓵
𝑺
= 𝒁𝟎 𝑹 +𝑱𝒁 𝐭𝐚𝐧 𝜷𝓵
𝟎 𝑳
On cross multiplication and simplification we get
𝒁𝟎 𝑹𝟎 + 𝑱𝒁𝑳 𝐭𝐚𝐧 𝜷𝓵 = 𝑹𝟎 × 𝒔 𝒁𝑳 + 𝒋𝑹𝟎 𝐭𝐚𝐧 𝜷𝓵
𝒁𝑳(𝒔 − j tan𝜷𝓵) = 𝑹𝟎(𝟏 − 𝒋 𝒔 𝒕𝒂𝒏𝜷𝓵)
𝑹𝟎(𝟏 − 𝒋 𝒔 𝒕𝒂𝒏𝜷𝓵)
𝒁𝑳 =
(𝒔 − j tan𝜷𝓵)
(𝟏 − 𝒋 𝒔 𝒕𝒂𝒏𝜷𝓵)
𝒁𝑳 = 𝑹𝟎
(𝒔 − j tan𝜷𝓵)

• Load impedance for a transmission line can be evaluated by measuring

• s (SWR) at distance 𝓵 away from the load end and substituting in to
above equation. R0 is characteristic impedance of the line