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
Problem:

Stripline and Microstrip

At low frequencies (below about 300 MHz), the characteristics of open and shorted lines have
little significance. At low frequencies the lines are just too long to be used as reactive
components or as filters and tuned circuits. However, at UHF (300 to 3000 MHz) and
microwave (1 GHz and greater) frequencies the length of one-half wavelength is less than 1 ft;
the values of inductance and capacitance become so small that it is difficult to realize them
physically with standard coils and capacitors. Special transmission lines constructed with
copper patterns on a printed circuit board (PCB), called microstrip or stripline, can be used as
tuned circuits, filters, phase shifters, reactive components, and impedance-matching circuits at
these high frequencies.
Microstrip.
Microstrip is a flat conductor separated by an insulating dielectric from a large conducting
ground plane [Fig. 13-26(a)]. The microstrip is usually one quarter or one-half wavelength
long. The ground plane is the circuit common. This type of microstrip is equivalent to an
unbalanced line. Shorted lines are usually preferred over open lines. Microstrip can also be
made in a two-line balanced version [Fig. 13-26(b)].
The characteristic impedance of microstrip, as with any transmission line, is dependent
on its physical characteristics. It can be calculated by using the formula

Problems:
Find the characteristic impedance of Microstrip with the dimensions h = 0.0625 in,
w = 0.1 in, t =0.003 in, and ε = 4.5

Solution:

Problems:
A microstrip transmission line is to be used as a capacitor of 4 pF at 800 MHz. The PCB
dielectric is 3.6. The Microstrip dimensions are h = 0.0625 in, w = 0.13 in, and t = 0.002 in.
What are (a) the characteristic impedance of the line and (b) the reactance of the capacitor?
Solution
Problems:
An RG-11/U foam coaxial cable has a maximum voltage standing wave of 52 V and
a minimum voltage of 17 V. Find (a) the SWR, (b) the reflection coefficient,

Problems

A Microstrip transmission line is to be used as a capacitor of 4 pF at 800 MHz. The

PCB dielectric is 3.6. The Microstrip dimensions are h = 0.0625 in, w =0.13 in, and
t = 0.002 in.
What are (a) the characteristic impedance of the line and
(b) the reactance of the capacitor?
Waveguides:
Any system of conductors and insulators for carrying electromagnetic waves could be called a
waveguide, but it is customary to reserve this name for specially constructed hollow metallic
pipes. They are used at microwave frequencies, for the same purposes as transmission lines
were used at lower frequencies. Hence, lines are considered as Low pass filter and Wave-guides
as High pass filters
 Waveguides are preferred to transmission lines because they are much less lossy at the
highest frequencies
 Stub: Piece of transmission lines use for impedance matching
RECTANGULAR WAVEGUIDES

FIGURE 10 · Waveguides. (a) Rectangular; (b) circular.

Due to skin effect the majority of the current flow (at very high frequencies) will occur along
the surface of the conductor and very little at the center. This phenomenon has led to the
development of hollow conductors known as waveguides.
 A rectangular waveguide is shown in Figure 10. [ as is a circular waveguide for
comparison]. In a typical system, there may be an antenna at one end of a waveguide and a
receiver or transmitter at the other end.
The antenna generates electromagnetic waves, which travel down the waveguide to be
eventually received by the load. The walls of the guide are conductors, and therefore reflections
from them take place. ,
In discussing the behaviour and properties of waveguides, it is necessary to speak of
electric and magnetic fields, as in wave propagation, instead of voltages and currents,
as in transmission lines.
Applications Because the cross-sectional dimensions of a waveguide must be of the same order
as those of a wavelength, use at frequencies below about 1 GHz is not normally practical
Advantages
A circular waveguide is that it looks like a coaxial line with the insides removed. This illustrates
the advantages that waveguides possess. Since it is easier to leave out the inner conductor than
to put it in, waveguides are simpler to manufacture than coaxial lines.
Similarly, because there is neither an inner conductor nor the supporting dielectric in a
waveguide, flashover is less likely. Therefore the power-handling ability of waveguides is
improved, and is about 10 times as high as for coaxial. air-dielectric rigid cables
Reflection of Waves from a Conducting Plane
In view of the way in which signals propagate in waveguides, it is now necessary to consider
what happens to electromagnetic waves when they encounter. a conducting Surface

Basic behaviour : An electromagnetic plane wave in space is transverse-electromagnetic, or

TEM. The electric field, the magnetic field and the direction of propagation are mutually
perpendicular. If such a wave were sent straight down a waveguide, it would not propagate in
it. This is because the electric field (no matter what its direction) would be short-circuited by
the walls, since the walls are assumed to be perfect conductors, and a potential cannot exist
across them.
 What must be found is some method of propagation which does not require an electric
field to exist near a wall and simultaneously be parallel to it. This is achieved by sending the
wave down the waveguide in a zigzag fashion (see Figure 10-3), bouncing it off the walls and
setting up a field that is maximum at or near the center of the guide, and zero at the walls.
In this case the walls have nothing to short-circuit, and they do not interfere with the wave
pattern set up between them. Thus propagation is not hindered.
Two major consequences of the zigzag propagation are apparent. The first is that the
velocity of propagation in a waveguide must be less than in free space, and the second is that
waves can no longer be TEM.
The second situation arises because propagation by reflection requires not only a normal
component but also a component in the direction of propagation for either the electric or the
·magnetic field, depending on the way in which waves are set up in the waveguide.
This extra component in the direction of propagation means that waves are no longer
transverse-electromagnetic, because there is now either an electric or a magnetic additional
component in the direction of propagation.'
Since there are two different basic methods of propagation, names must be given to the
resulting waves to distinguish them from each other.
The American system labels modes according to the field component that behaves as it did
in free space. Modes in which there is no component of electric field in the direction of
propagation are called transverse-electric (TE, see Figure I0-5b) modes, and modes with no
such component of magnetic· field are called transverse-magnetic (TM, see Figure 10-5a).
The British and European systems label the modes according to the component that has
behaviour different from that in free space, thus modes are called H instead of TE and E instead
of TM. [The American system will be used here exclusively]

Dominant mode of operation

The natural mode of operation for a waveguide is called the dominant mode. This mode is the
lowest possible frequency that can be propagated in a given waveguide. In Figure 10-6, half-
wave length is the lowest frequency where the waveguide will still present the properties
discussed-below.
The mode of operation of a waveguide is further divided into two submodes. They are as
follows:·
1. TE m,n for the transverse electric mode (electric field is perpendicular to the directionof wave
propagation)
2. TMm,n for the transverse magnetics mode (magnetic field is perpendicular to the direction of
wave propagation)

Waveguide Size and Frequency.

Fig. 16-21 shows the most important dimensions of a rectangular waveguide: the width a and
the height b. Note that these are the inside dimensions of the waveguide. The frequency of
operation of a waveguide is determined by the size of a. This dimension is usually made equal
to one-half wavelength, a bit below the lowest frequency of operation.
This frequency is known as the waveguide cutoff frequency. At its cutoff frequency and below,
a waveguide will not transmit energy.
At frequencies above the cutoff frequency, a waveguide will propagate electromagnetic
energy. A waveguide is essentially a high-pass filter with a cutoff frequency equal to
Where is in megahertz and a is in meters.

Cutoff frequency
The lower cutoff frequency for a mode may be calculated by using

Problem:5

A rectangular waveguide is 5.1 cm by 2.4 cm (inside measurements) Calculate the cutoff

frequency of the dominant .mode.
Solution:

Multiple Junctions:

When it is required to combine two or more signals (or split a signal into two or more parts)
in a waveguide system, some form of multiple junction must be used. For simpler
interconnections T-shaped junctions are used, whereas more complex junctions may be
hybrid T or hybrid rings.
T junctions Two examples of the T junction, or tee, are shown in Figure 10-25, together with
their transmission-line equivalents. Once again, they are referred to as E or H-plane trees,
depending on whether they are in the plane of the electric field or the magnetic field. All
three atoms of the H-plane tee lie in the plane of the magnetic field, which divides among the
arms
T junctions (particularly the E-plane tee) may themselves be used for impedance matching,
in a manner identical to the short-circuited transmission-line stub. The vertical arm is then
provided with a sliding piston to produce a short circuit at any desired point.
Hybrid junctions
If another arm is added to either of the T junctions, then a hybrid T junction. Or MAGIC
TEE, is obtained; it is shown in Figure 10-26. Such a junction is symmetrical about an
imaginary plane bisecting arms 3 and 4 and has some very useful and interesting properties

The basic property is that arms 3 and 4 are both connected to arms 1 and 2 but not to each
other. This applies for the dominant mode only, provided each arm is terminated in a correct
load.
If a signal is applied to arm 3 of the magic tee, it will be divided at the junction, with
some entering arm 1 and some entering arm 2, but none will enter arm 4.
 This is due to the electric field for the dominant mode is evenly symmetrical about the
plane A-B in arm 4 but is unevenly symmetrical about plane A-B in arm 3 (and also in arms
1and 2, as it happens). That is to say, the electric field in arm 4 on one side of A-B is a mirror
image of the electric field on the other side, but in arm 3 a phase change would be required to
give such even symmetry. Since nothing is there to provide such a phase change, no signal
applied to arm 3 can propagate in arm 4 except in a mode with uneven symmetry about the
plane A-B

The dimensions being such as to exclude the propagation of these higher modes, no
signal travels down arm 4. Because the arrangement is reciprocal, application of a signal into
arm 4 likewise results in no propagation down arm 3. Since arms 1 and 2 are symmetrically
disposed about the plane A-B, a signal entering either arm 3 or arm 4 divides evenly between
these two lateral arms if they are correctly terminated.
This means that it is possible to have two generators feeding signals, one into arm 3 and
the other into arm 4. Neither generator is coupled to the other, but both are coupled to the
load which, in Figure 10-28, is in arm 2, (while arm 1 has a matched termination connected to
it). The arrangement shown is one of a the applications of the magic tee
Cavity Resonators [ wave guide resonator]
A cavity resonator is a piece of waveguide closed off at both ends with metallic planes. Where
propagation in the longitudinal direction took place in the waveguide, standing waves exist in
the resonator, and oscillations can take place if the resonator is suitably excited.
Waveguides are used at the highest frequencies to transmit power and signals. Similarly,
cavity resonators are employed as tuned circuits at such frequencies. Their operation follows
directly from that of waveguides.
 Operation: Until now, waveguides have been considered from the point of view of standing
waves between the side walls and traveling waves in the longitudinal direction. If conducting
end walls are placed in the waveguide, then standing waves, or oscillations, will take place if a
source is located between the walls.
This assumes that the distance between the end walls is ( nλp/2) , where n is any integer.
The situation is illustrated in Figure 10-35.

FIGURE 10-35 Transformation from rectangular waveguide propagating TE1,0 mode to

cavity resonator oscillating in (a) TE1,0,1 mode; (b) TE1,0,2 mode.

As shown, here placement of the first wall ensures standing waves, and placement of the second
wall permits oscillations, provided that the second wall is placed so that the pattern due to the
first wall is left undisturbed. Thus, if the second wall is nλp/2 away from the first, as in Figure
10-36, oscillations between the two walls will take place.
They will then continue until all the applied energy is dissipated, or indefinitely if energy is
constantly supplied. This is identical to the behaviour of an LC tuned circuit.
It is thus seen that any space enclosed by conducting walls must have one (or more)
frequency at which the conditions just described are fulfilled. In other words, any such enclosed
space must have at least one resonant frequency. Indeed, the completely enclosed waveguide
has become a cavity resonator with its own system of modes, and therefore resonant
frequencies. The TE and TM mode-numbering' system breaks down unless the cavity has a
very simple shape, and it is preferable to speak of the resonant frequency rather than mode.

FIGURE 10-36 Reentrant cavity resonators.

Each cavity resonator has, an infinite number of resonant frequencies. This can be appreciated
if we consider that with the resonator of Figure I 0-35 oscillations would have been obtained
at twice the frequency, because every distance would now be λp instead of λp/2. Several other
resonant frequency series will also be present, based on other modes of propagation, all
permitting oscillations to take place within the cavity.
Naturally such behaviour is not really desired in a resonator, but it need not be especially
harmful. The fact that the cavity can oscillate at several frequencies does not mean that it will.
Such frequencies are not generated spontaneously; they must be fed in.
Types The simplest cavity resonators may be spheres, cylinders or rectangular prisms.
However, such cavities are not often used, because they all share a common defect; their
various resonant frequencies are harmonically related. This is a serious drawback in all those
situations in which pulses of energy are fed to a cavity. The cavity is supposed to maintain
sinusoidal oscillations through the flywheel effect, but because such pulses contain harmonics
and the cavity is able to oscillate at the harmonic frequencies, the output is still in the form of
pulses'. As a result, most practical· cavities have odd shapes to ensure that the various
oscillating frequencies are not harmonically related, and therefore that harmonics are
attenuated.
Some typical irregularly shaped resonators are illustrated. Those of Figure I0-36a might be
used with reflex klystrons, whereas the resonator of Figure I0-36b is popular for use with
magnetrons
They are known as reentrant resonators, that is, resonators so shaped that one of the walls re-
enters the resonator shape. The first two are figures of revolution about a central vertical axis,
and the third one is cylindrical. Apart from being useful as tuned circuits, they are also given
such shapes so that they can be integral parts of the above-named microwave devices, being
therefore doubly useful. However, because of their shapes, they have resonant frequencies that
are not at.all easy to calculate.
Applications Cavity resonators are employed for much the same purposes as tuned LC circuits
or resonant transmission lines, but naturally at much higher frequencies

Circulators A circulator is a ferrite device somewhat like a rat race. It is very often a four-port
(i.e., four-terminal) device, as shown in Figure1 a, although other forms also exist. It has the
property that each terminal is connected only to the next clockwise terminal. Thus port 1 is
connected to port 2, but not to 3 or 4; 2 is connected to 3, but not to 4 or l; and so on.
The main applications of such circulators are either the isolation of transmitters and receivers
connected to the same antenna (as in radar), or isolation of input and output in two-terminal
amplifying devices such as parametric amplifiers.
A four-port Faraday rotation circulator is shown in Figure 1. It is similar to the Faraday rotation
isolator already described. Power entering port 1 is converted to the TE1,1 mode in the circular
waveguide, passes port 3 unaffected because the electric field is not significantly cut, is rotated
through 45° by the ferrite insert (the magnet is omitted for simplicity), continues past port 4 for
the same reason that it passed port 3, and finally emerges from port 2, just as it did in the
isolator. Power fed to port 2 will

FIGURE 1: Ferrite circulator. (a) Schematic diagram; (b) Faraday rotation four-port
circulator.

Duplexers A duplexer is a circuit designed to allow the use of the same antenna for both
transmission and reception, with minimal interference between the transmitter and the receiver.
. An ordinary circulator is a duplexer, but the emphasis here is on a circuit using switching for
pulsed (not CW) transmission
The branch-type duplexer shown in Figure 2 is a type often used in radar. It has two switches,
the TR and the ATR (anti-TR), arranged in such a manner that the receiver and the transmitter
are alternately connected to the antenna, without ever being connected to each other. The
operation' is as follows. When the transmitter produces an RF impulse, both switches become
short-circuited either because of the presence of the pulse, as in TR cells, or because of an
external synchronized bias change.
The ATR switch reflects an open circuit across the main waveguide, through the quarter-wave
section connected to it, and so does the TR switch, for the same reason. Therefore, neither of
them affects the transmission, but the short-circuiting of the TR switch prevents RF power from
entering the receiver or at least reduces any such power down to a tolerable level. At the
termination of the transmitted pulse, both switches open-circuit by a reversal of the initial short-
circuiting process.
 FI.G: Branch-type duplexer for radar.
The ATR switch now throws a short circuit across the waveguide leading to the transmitter.
If this were not done, a significant loss of the received signal would be incurred. At the input
to the guide joining the TR branch to the main waveguide, this short circuit has now become
an open circuit and hence has no effect. Meanwhile, the guide leading through the TR switch
is now continuous and correctly matched. The signal from the antenna can thus go directly to
the receiver.
The branch-type duplexer is a narrowband device, because it relies on the length of the guides
connecting the switches to the main waveguide. Single-frequency operation is very often
sufficient, so that the branch-type duplexer is very common.

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.