Syllabus : Unit 2

Transmission Lines: Introduction, transmission lines equations and significance, termination

of line by infinite line, by characteristic impedance, short circuit line, open circuit line,
VSWR, problems Microstrip lines.
Qualitative Analysis of Waveguides: Rectangular and circular type, TE and TM waves in
wave guides, their transmission properties and attenuation., E-plane & H Plane Waveguides,
Magic Tee, Circulator, Duplexer and their S matrices, Wave guide resonator, loaded and
unloaded.
________________________________________________________________________

Transmission Lines:
Transmission lines in communication carry telephone signals, computer data in LANs, TV and
Internet signals in cable TV systems, and signals from a transmitter to an antenna or from an
antenna to a receiver. Transmission lines are also short cables connecting equipment or printed
circuit board copper traces that connect an embedded microcomputer to other circuits by way
of various interfaces. Transmission lines are critical links in any communication system. They
are more than pieces of wire or cable. Their electrical characteristics are critical and must be
matched to the equipment for successful communication to take place.
Transmission lines are also circuits. At very high frequencies where wavelengths are
short, transmission lines act as resonant circuits and even reactive components. At VHF, UHF,
and microwave frequencies, most tuned circuits and filters are implemented with transmission
lines.
The two primary requirements of a transmission line are that
(1) the line introduce minimum attenuation to the signal and
(2) the line not radiate any of the signal as radio energy.
All transmission lines and connectors are designed with these requirements in mind.

Types of Transmission Lines

Parallel-Wire Lines. Parallel-wire line is made of two parallel conductors separated by a

space of ½ in to several inches. Fig. 2-1(a) shows a two-wire balanced line in which insulating
spacers have been used to keep the wires separated. Such lines are rarely used today. A
variation of parallel line is the 300-V twin-lead type shown in Fig. 2-1(b), where the spacing
between the wires is maintained by a continuous plastic insulator. Parallel-wire lines are rarely
used today.
Coaxial Cable. The most widely used type of transmission line is coaxial cable, which
consists of a solid center conductor surrounded by a dielectric material, usually a plastic
insulator such as Teflon [see Fig. 2-1(c)].
An air or gas dielectric, in which the center conductor is held in place by periodic insulating
spacers, can also be used. Over the insulator is a second conductor, a tubular braid or shield
made of fine wires. An outer plastic sheath protects and insulates the braid. Coaxial cable
comes in a variety of sizes, from approximately 1Ú4 in to several inches in diameter.
The major benefit of coaxial cable is that it is completely shielded so that external noise has
little or no effect on it
Twisted-Pair Cable. Twisted-pair cable, as the name implies, uses two insulated solid copper
wires covered with insulation and loosely twisted together. See Fig. 2-1 (d). This type of cable
was originally used in telephone wiring and is still used for that today. But it is also used for
security system wiring of sensors and other equipment.

Equivalent circuit representation

Since each conductor has a certain length and diameter, it will have a resistance and an
inductance. Since there are two wires close to each other, there will be capacitance between
them. The wires are separated by a medium called the dielectric. which cannot be perfect in its
insulation; the current leakage through it can be represented by a shunt conductance. The
resulting equivalent circuit is as shown in Figure 7-2. and 7-3, are all measured per unit length,
e.g., per meter, because they occur periodically along the line. They are thus distributed
throughout the length of the line. Under no circumstances can they be assumed to be lumped
at anyone point.
Primary constants of a line:
A small length of the line can be represented by an equivalent L-section network with constant
parameters R, L, G, and C per unit length as shown in Fig. a,
where R is the resistance per unit length which takes into account the ohmic loss in the line
conductor
L is the inductance per unit length which takes into account the magnetic energy storage
around the conductor,
G is the conductance per unit length which takes into account the dielectric loss between the
line conductors, and
C is the capacitance per unit length which appears due to two conductors at different potentials
and represents the electric energy storage.
These parameters are independent of frequency and are called PRIMARY CONSTANTS of a
transmission lines.

Fig a: Equivalent circuit of a small sec on of a uniform transmission line a

Where Z = series impedance per unit length
And Y = shunt admittance per unit length.
Secondary constants of a Transmission lines:
1) characteristic impedance
The characteristic impedance of a transmission line is defined as it is the input impedance of an infinite
length transmission line as shown in fig 7.4

Or
If the transmission line terminates with its characteristic impedance then its input impedance is
called its characteristic impedance

Any circuit that consists of series and shunt impedances must have an input impedance. For
the transmission line this input impedance will depend on the type of line, its length and the
termination at the far end. To simplify description and calculation, the input impedance under
certain standard, simple and easily reproducible conditions is taken, as the reference and is
called the characteristic impedance of that line. By definition. the characteristic impedance of
a transmission line. Zo. is the impedance measured at the input of this line when its length is
infinite. Under these conditions the type of termination at the far end has no effect, and
consequently is not mentioned in the definition.
Methods of calculation
It can now be shown that the characteristic impedance of a line will be measured at its input
when the line, is terminated at the far end in an impedance equal to Zo (Zin = Zout max power
transfer), no matter what length the line has. This is important, because such a situation is far
easier to reproduce for measurement purposes than a line of infinite length.

The characteristic impedance of the line, Zo will be measured at the input of a transmission
line if the output is terminated in Z0 . Under these conditions Z0 is considered purely resistive.
It follows from filter theory that the characteristic impedance of an iterative circuit consisting
of series and shunt elements is giv6n by
are called the series impedance and shunt admittance per unit length of the line.

equation 7.2
From Equation (7-2) it follows that the characteristic impedance of a transmission line may
be complex, and indeed it very often is, especially in line communications, i.e., telephony at
voice frequencies. At radio frequencies the resistive components of the equivalent circuit
become insignificant, and the expression for Zo reduces to

equation 7.3
L is measured in henrys per meter and C in farads per meter; it follows that Equation (7-3)
shows the characteristic impedance of a line in ohms and is dimensionally correct. It also shows
that this characteristic impedance is resistive at radio frequencies.
2) propagation constant:
When the signal passes through the transmission line there are certain changes in the signal.
After passing through the line there is change in magnitude as well as phase of the signal.
The changes are given by propagation constant, γ, which is defined as

. γ= α + j β
γ is called the propagation constant,
α is called the attenuation constant (attenuation of signal per unit length of the line), and
β is the phase constant (change of phase of the signal per unit length of the line).
Attenuation constant α indicates change in magnitude and β phase constant indicated change
in phase.
The characteristic impedance Z0 and propagation constant γ are called SECOUNDARY
CONSTTANTS of a transmission lines
γ is called the propagation constant,
α is called the attenuation constant (attenuation of signal per unit length of the line), and
β is the phase constant (change of phase of the signal per unit length of the line).
Problem 1: A transmission line has the following parameters R = 2Ω/m G = 0.5mmho/m f =
1 GHz L= 8nH/m and C= 0.23 pF Calculate i) The characteristic impedance ii) the propagation
constant

Characteristic impedance Z0 of parallel-wire line and coaxial line

Physically; characteristic impedance is determined by the geometry, size and spacing of the
conductors, and by the dielectric constant of the ·insulator separating them. It may be calculated
from the following formulas, the various terms having meanings as shown in Figure 7-5:

For the parallel-wire line, we have

For the coaxial line, this is

where k = dielectric constant of the insulation.

The usual range of characteristic impedances for balanced lines is (150 to 600) Ω, and (40 to
150) Ω for coaxial lines

Problem2
A piece of coaxial cable has a 75Ω characteristic impedance and normal capacitance of
69 pF/m, what is the inductance per metre? If the diameter of the inner conductor is
0.584mm and the dielectric constant of the insulator is 2.23, what is the outer conductor
diameter?
Solution:

L = Z20C= (75)2 x 69 x 10-12 = 0.388μ H/m

D = d x antilog 0.81= 3.77 mm

Problem 3
What is the minimum value that the characteristic impedance of an air dielectric
parallel wire line can have?

Solution:
Minimum impedance will occur when the 2s/d is minimum and that is reached when the two
wires touch each other with the condition s=d

= 276 log (2 x1) =83Ω

Problem 4

A coaxial cable having an inner conductor of 0.025 mm and using an insulator with a
dielectric constant of 2,56 is to have characteristic impedances of 2000Ω. What must be
outer conductor diameter?
Solution:

D = d x antilog (23.1884
=3.86 x 1015 km
Note: A practically impossible case
Reflection Coefficients:

Since the transmission-line theory is described here in terms of propagation of voltage and
currents along the line, reflections of these signals occur at a point where the impedance is not
equal to the characteristic impedance Z0.

I) Voltage Reflection Coefficient

Voltage reflection coefficient of the load ZL at the load end is defined by the ratio of reflected
and incident voltages at the load:

=(reflected wave/ incident)

ii) Current Reflection Coefficient

Current reflection coefficient of the load is defined by the ratio of reflected and incident currents
through the load:

Fig. Short- and open-circuited lines

STANDING WAVES
When a line is terminated with a mismatched load, the incident and reflected signals interfere
with each other at different phases to produce a resultant wave called the standing wave along
the line.

In this at regular intervals of half wavelength resultant wave produces maxima where the incident
and reflected waves meet at same phase. Similarly it results in minima at regular intervals of half
wave length where they meet at opposite phase.

The magnitude of the standing waves is measured in terms of Standing Wave Ratio (SWR) defined
by

Scattering parameters
describes the input-output relationships between ports in an electrical system. Specifically at
high frequency it becomes essential to describe a given network in terms of waves rather than
voltage or current. Thus in S-parameters we use power waves.
In RF design, we cant use other parameters for analysis such as Z,Y,H parameters as we
can't do short circuit and open circuit analysis as it is not feasible.For a two port network, s-
parameters can be defined as

S11 is the input port voltage reflection coefficient

S12 is the reverse voltage gain
S21 is the forward voltage gain
S22 is the output port voltage reflection coefficient
The S-parameter matrix can be used to determine reflection coefficients and transmission
gains from both sides of a two port network. This concept can further be used to determine
s-parameters of a multi port network. These concepts can further be used in determining
Gain, Return loss, VSWR and Insertion Loss.

Additional problems

1) A telephone line has R = 6 ohms/km, L = 2.2 mH/km, C = 0.005 mF/km, and G =0.05
mmho/km. Determine Z0, α, β, at 1 kHz

2) A 50-ohm lossless line connects a matched signal of 100 kHz to a load of 100 ohms.

The load power is 100 mW. Calculate the

(a) Voltage reflection coefficient of the load

(b) VSWR of the load

3) An air-filled coaxial transmission line has outer and inner conductor radii equal to

6 cm and 3 cm, respectively. Calculate the values of (a) inductance per unit length,

(b) capacitance per unit length, and (c) characteristic impedance of the line.