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EC8701 - ANTENNAS & MICROWAVE ENGG. - STUCOR APP

UNIT I INTRODUCTION TO MICROWAVE SYSTEMS AND ANTENNAS
Electromagnetic spectrum:

Microwave Frequency Bands:

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Physical Concept of Radiation ( Radiation Mechanism)

One of the first questions that may be asked concerning antennas would be “how is
radiation accomplished?”
In other words, how are the electromagnetic fields generated by the source, contained and
guided within the transmission line and antenna, and finally “detached” from the antenna
to form a free-space wave?
Let us first examine some basic sources of radiation.
Radiation from Single Wire
Conducting wires are material whose prominent characteristic is the motion of electric
charges and the creation of current flow.
Let us assume that an electric volume charge density, represented by qv (coulombs/m3), is
distributed uniformly in a circular wire of cross-sectional area A and volume V, as shown
in Figure.

The total charge Q with in volume V is moving in the z direction with a uniform velocity
vz (meters/sec). It can be shown that the current density Jz (amperes/m2) over the cross
section of the wire is given by

Jz = qvvz (1a)

If the wire is made of an ideal electric conductor, the current density Js (amperes/m) resides
on the surface of the wire and it is given by

Js = qsvz (1b)

where qs (coulombs/m2) is the surface charge density.

If the wire is very thin (ideally zero radius), then the current in the wire can be represented
by

Iz = qlvz (1c)

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where ql (coulombs/m) is the charge per unit length. Instead of examining all three current
densities, we will primarily concentrate on the very thin wire. The conclusions apply to all
three.
If the current is time varying, then the derivative of the current of (1c) can be written as

dIz /dt = ql dvz /dt = qlaz (2)

where dvz/dt = az (meters/sec2) is the acceleration. If the wire is of length l, then (2) can
be written as

l dIz /dt = lql dvz /dt = lqlaz (3)

Equation (3) is the basic relation between current and charge, and it also serves as the
fundamental relation of electromagnetic radiation.
It simply states that to create radiation, there must be a time-varying current or an
acceleration (or deceleration) of charge. We usually refer to currents in time-harmonic
applications while charge is most often mentioned in transients. To create charge
acceleration (or deceleration) the wire must be curved, bent, discontinuous, or terminated.
Periodic charge acceleration (or deceleration) or time-varying current is also created when
charge is oscillating in a time-harmonic motion.
Important Conclusions:
(i)If a charge is not moving, current is not created and there is no radiation.
(ii)If charge is moving with a uniform velocity:
(a)There is no radiation if the wire is straight, and infinite in extent.
(b)There is radiation if the wire is curved, bent, discontinuous, terminated, or
truncated, as shown in Figure
(iii)If charge is oscillating in a time-motion, it radiates even if the wire is straight.

A qualitative understanding of the radiation mechanism may be obtained by

considering a pulse source attached to an open-ended conducting wire, which may be
connected to the ground through a discrete load at its open end, as shown in Figure (d).
When the wire is initially energized, the charges (free electrons) in the wire are set in
motion by the electrical lines of force created by the source. When charges are accelerated
in the source-end of the wire and decelerated (negative acceleration with respect to original
motion) during reflection from its end, it is suggested that radiated fields are produced at
each end and along the remaining part of the wire.
Stronger radiation with a more broad frequency spectrum occurs if the pulses are of
shorter or more compact duration while continuous time-harmonic oscillating charge
produces, ideally, radiation of single frequency determined by the frequency of oscillation.

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The acceleration of the charges is accomplished by the external source in which forces
set the charges in motion and produce the associated field radiated. The deceleration of the
charges at the end of the wire is accomplished by the internal (self) forces associated with
the induced field due to the build up of charge concentration at the ends of the wire. The
internal forces receive energy from the charge build up as its velocity is reduced to zero at
the ends of the wire. Therefore, charge acceleration due to an exciting electric field and
deceleration due to impedance discontinuities or smooth curves of the wire are
mechanisms responsible for electromagnetic radiation.

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Radiation from Two-Wires

Let us consider a voltage source connected to a two-conductor transmission line which
is connected to an antenna. This is shown in Figure. Applying a voltage across the two-
conductor transmission line creates an electric field between the conductors. The electric
field has associated with it electric lines of force which are tangent to the electric field at
each point and their strength is proportional to the electric field intensity. The electric lines
of force have a tendency to act on the free electrons (easily detachable from the atoms)
associated with each conductor and force them to be displaced. The movement of the
charges creates a current that in turn creates a magnetic field intensity. Associated with the
magnetic field intensity are magnetic lines of force which are tangent to the magnetic field.
We have accepted that electric field lines start on positive charges and end on negative
charges. They also can start on a positive charge and end at infinity, start at infinity and end
on a negative charge, or form closed loops neither starting or ending on any charge.
Magnetic field lines always form closed loops encircling current-carrying conductors
because physically there are no magnetic charges.

The electric field lines drawn between the two conductors help to exhibit the distribution
of charge. If we assume that the voltage source is sinusoidal, we expect the electric field
between the conductors to also be sinusoidal with a period equal to that of the applied source.
The relative magnitude of the electric field intensity is indicated by the density (bunching) of

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the lines of force with the arrows showing the relative direction (positive or negative). The
creation of time-varying electric and magnetic fields between the conductors forms
electromagnetic waves which travel along the transmission line, as shown in Figure (a).
The electromagnetic waves enter the antenna and have associated with them electric
charges and corresponding currents. If we remove part of the antenna structure, as shown in
Figure (b), free-space waves can be formed by “connecting” the open ends of the electric lines
(shown dashed).
The free-space waves are also periodic but a constant phase point P0 moves outwardly with
the speed of light and travels a distance of λ/2 (to P1) in the time of one-half of a period. It has
been shown that close to the antenna the constant phase point P0 moves faster than the speed
of light but approaches the speed of light at points far away from the antenna (analogous to
phase velocity inside a rectangular waveguide).
free-space waves and water waves -analogy
The question still unanswered is how the guided waves are detached from the antenna to
create the free-space waves that are indicated as closed loops. Before we attempt to explain
that, let us draw a parallel between the guided and free-space waves, and water waves created
by the dropping of a pebble in a calm body of water or initiated in some other manner.
Once the disturbance in the water has been initiated, water waves are created which begin
to travel outwardly. If the disturbance has been removed, the waves do not stop or extinguish
themselves but continue their course of travel. If the disturbance persists, new waves are
continuously created which lag in their travel behind the others. The same is true with the
electromagnetic waves created by an electric disturbance.
If the initial electric disturbance by the source is of a short duration, the created
electromagnetic waves travel inside the transmission line, then into the antenna, and finally are
radiated as free-space waves, even if the electric source has ceased to exist (as was with the
water waves and their generating disturbance). If the electric disturbance is of a continuous
nature, electromagnetic waves exist continuously and follow in their travel behind the
others.This is shown in Figure for a biconical antenna. When the electromagnetic waves are
within the transmission line and antenna, their existence is associated with the presence of the
charges inside the conductors. However, when the waves are radiated, they form closed loops
and there are no charges to sustain their existence. This leads us to conclude that electric
charges are required to excite the fields but are not needed to sustain them and may exist in
their absence. This is in direct analogy the water waves.

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Isotropic, Directional, and Omnidirectional Patterns

An isotropic radiator is defined as “a hypothetical lossless antenna having equal
radiation in all directions.” Although it is ideal and not physically realizable, it is often taken
as a reference for expressing the directive properties of actual antennas.
A directional antenna is one “having the property of radiating or receiving
electromagnetic waves more effectively in some directions than in others”. Example of antenna
with directional radiation patterns is shown in Figure.

It is seen that the pattern in Figure below is nondirectional in the azimuth plane [f (φ),
θ = π/2] and directional in the elevation plane [g(θ ), φ = constant]. This type of a pattern is
designated as omnidirectional, and it is defined as one “having an essentially nondirectional
pattern in a given plane (in this case in azimuth) and a directional pattern in any orthogonal
plane (in this case in elevation).” An omnidirectional pattern is then a special type of a
directional pattern.

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Antenna near and far field(Field Regions of Antenna)

The space surrounding an antenna is usually subdivided into three regions:
(a) reactive near-field,
(b) radiating near-field (Fresnel) and
(c) far-field (Fraunhofer) regions as shown in Figure .

These regions are so designated to identify the field structure in each. Although no abrupt
changes in the field configurations are noted as the boundaries are crossed, there are distinct
differences among them.
Reactive near-field region
Reactive near-field region is defined as “that portion of the near-field region
immediately surrounding the antenna wherein the reactive field predominates.” For most
antennas, the outer boundary of this region is commonly taken to exist at a distance 𝑅 <
3
0.62√𝐷 ⁄𝜆 from the antenna surface, where λ is the wavelength and D is the largest dimension
of the antenna. “For a very short dipole, or equivalent radiator, the outer boundary is commonly
taken to exist at a distance λ/2π from the antenna surface.”
Radiating near-field (Fresnel) region
Radiating near-field (Fresnel) region is defined as “that region of the field of an antenna
between the reactive near-field region and the far-field region wherein radiation fields
predominate and wherein the angular field distribution is dependent upon the distance from the
antenna.
If the antenna has a maximum dimension that is not large compared to the wavelength,
this region may not exist. For an antenna focused at infinity, the radiating near-field region is
sometimes referred to as the Fresnel region on the basis of analogy to optical terminology. If
the antenna has a maximum overall dimension which is very small compared to the wavelength,
3
this field region may not exist.” The inner boundary is taken to be the distance 𝑅 ≥ 0.62√𝐷 ⁄𝜆
and the outer boundary the distance R < 2D2/λ where D is the largest∗ dimension of the antenna.
This criterion is based ona maximum phase error of π/8. In this region the field pattern is, in
general, a function of the radial distance and the radial field component may be appreciable.

∗To be valid, D must also be large compared to the wavelength (D > λ)

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Far-field (Fraunhofer) region

Far-field (Fraunhofer) region is defined as “that region of the field of an antenna where
the angular field distribution is essentially independent of the distance from the antenna. If the
antenna has a maximum∗ overall dimension D, the far-field region is commonly taken to exist
at distances greater than 2D2/λ from the antenna, λ being the wavelength.
The far-field patterns of certain antennas, such as multibeam reflector antennas, are
sensitive to variations in phase over their apertures. For these antennas 2D2/λ may be
inadequate. In physical media, if the antenna has a maximum overall dimension, D, which is
large compared to π/|γ |, the far-field region can be taken to begin approximately at a distance
equal to |γ |D2/π from the antenna, γ being the propagation constant in the medium.
For an antenna focused at infinity, the far-field region is sometimes referred to as the
Fraunhofer region on the basis of analogy to optical terminology.” In this region, the field
components are essentially transverse and the angular distribution is independent of the radial
distance where the measurements are made. The inner boundary is taken to be the radial
distance R = 2D2/λ and the outer one at infinity.
The amplitude pattern of an antenna in different regions
The amplitude pattern of an antenna, as the observation distance is varied from the
reactive near field to the far field, changes in shape because of variations of the fields, both
magnitude and phase.
A typical progression of the shape of an antenna, with the largest dimension D, is shown
in Figure. It is apparent that in the reactive near field region the pattern is more spread out and
nearly uniform, with slight variations. As the observation is moved to the radiating near-field
region(Fresnel), the pattern begins to smooth and form lobes. In the far-field region
(Fraunhofer), the pattern is well formed, usually consisting of few minor lobes and one, or
more, major lobes.

Figure: Typical changes of antenna amplitude pattern shape from reactive near field
toward the far field

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Antenna Parameters

(a)Radiation pattern.
The radiation pattern of an antenna is a plot of the magnitude of the far-zone field
strength versus position around the antenna, at a fixed distance from the antenna.
Thus the radiation pattern can be plotted from the pattern function Fθ (θ,φ) or Fφ(θ,φ),
versus either the angle θ (for an elevation plane pattern) or the angle φ (for an azimuthal plane
pattern). The choice of plotting either Fθ or Fφ is dependent on the polarization of the antenna.

(b)main lobe, side lobe, minor lobe and back lobe with reference to antenna radiation
pattern.
Major Lobe: Major lobe is also called as main beam and is defined as “the radiation lobe
containing the direction of maximum radiation”. In some antennas, there may be more than
one major lobe.
Minor lobe: All the lobes except the major lobes are called minor lobe.
Side lobe: A side lobe is adjacent to the main lobe.
Back lobe: Normally refers to a minor lobe that occupies the hemisphere in a direction
opposite to that of the major(main) lobe .
• Minor lobes normally represents radiation in undesired directions and they should be
minimized.

(c) Half Power Beam Width (HPBW) of an antenna.

Half Power Beam Width is a measure of directivity of an antenna. It is an angular
width in degrees, measured on the radiation pattern (main lobe) between points where the
radiated power has fallen to half its maximum value.

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(d) beam solid angle

The beam area or beam solid angle  A for antenna is given by integral of the
normalized power pattern over a sphere.
2 
A =   P ( , ) d
0 0
n steradian

Pn ( ,  ) = Normalized power pattern

Beam solid angle is also given approximately by
 A =  HP  HP steradian
 HP = HPBW in E − plane or  plane
 HP = HPBW in H − plane or  plane
(e) Beam Width between First Null
Beam width between first null (BWFN) is the angular width in degrees, measured on
the radiation pattern between first null points on either side of the main lobe.

(f)Radiation Intensity
Radiation Intensity 𝑈(𝜃, ∅) in given direction is defined as the power per unit solid
angle in that direction.
• The power radiated per unit area in any direction is given by pointing vector P.
• For distant field for which E and H are orthogonal in a plane normal to the radius
vector,
E2
The power flow per unit area is given by P = watts / sqm
v
• 2
There are r square meters of surface area per unit solid angle( or steradian).
𝑟 2𝐸2
• 𝑈(𝜃, ∅) = 𝑟 2 𝑃 = 𝑤𝑎𝑡𝑡𝑠/𝑢𝑛𝑖𝑡 𝑠𝑜𝑙𝑖𝑑 𝑎𝑛𝑔𝑙𝑒
𝜂𝑣
The radiation intensity gives the variation in radiated power versus position around the
antenna. We can find the total power radiated by the antenna by integrating the Poynting vector
over the surface of a sphere that encloses the antenna. This is equivalent to integrating the
radiation intensity over a unit sphere.
2𝜋 𝜋

𝑃𝑟𝑎𝑑 = 𝑃𝑜𝑤𝑒𝑟 𝑟𝑎𝑑𝑖𝑎𝑡𝑒𝑑 = ∫ ∫ 𝑈(𝜃, ∅)𝑠𝑖𝑛𝜃𝑑𝜃𝑑∅

∅=0 𝜃=0
(g) Directivity of an antenna
The directivity(D) of an antenna is defined as the ratio of the maximum value of the
power radiated per unit solid angle to the average power radiated per unit solid angle.
That is, directivity is ratio of the maximum radiation intensity in the main beam to the average
radiation intensity over all space.

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𝑈𝑚𝑎𝑥 𝑈𝑚𝑎𝑥 4𝜋𝑈𝑚𝑎𝑥

𝐷= = = 2𝜋 𝜋
𝑈𝑎𝑣𝑔 𝑃𝑟𝑎𝑑⁄ 𝑈(𝜃, ∅)𝑠𝑖𝑛𝜃𝑑𝜃𝑑∅
4𝜋 ∫∅=0 ∫𝜃=0
Thus, the directivity measures how intensely the antenna radiates in its preferred direction than
an isotropic radiator would when fed with the same total power.
Directivity is a dimensionless ratio of power, and is usually expressed in dB as D(dB) = 10
log(D)
directivity of isotropic radiator:
An isotropic radiator is a hypothetical loss less radiator having equal radiation in all
directions.
2𝜋 𝜋
U(θ,φ) = 1 for isotropic antenna. Applying the integral identity, ∫∅=0 ∫𝜃=0 𝑠𝑖𝑛𝜃𝑑𝜃𝑑∅ = 4𝜋,
we have,

4𝜋𝑈𝑚𝑎𝑥
𝐷= 2𝜋 𝜋 =1
∫∅=0 ∫𝜃=0 𝑈(𝜃, ∅)𝑠𝑖𝑛𝜃𝑑𝜃𝑑∅

The directivity of an isotropic antenna is D = 1, or 0 dB.

Relationship between Directivity and beamwidth

Beamwidth and directivity are both measures of the focusing ability of an antenna: an antenna
pattern with a narrow main beam will have a high directivity, while a pattern with a wide beam
will have a lower directivity.
Approximate relation between beam width and directivity that apply with reasonable
accuracy for antennas with pencil beam patterns is the following:
32,000
𝐷≅ where θ1 and θ2 are the beam widths in two orthogonal planes of the main
𝜃1 𝜃2
beam, in degrees. This approximation does not work well for omnidirectional patterns because
there is a well-defined main beam in only one plane for such patterns.
(h) radiation efficiency of antenna
Radiation efficiency of an antenna is defined as the ratio of the radiated output power to the
supplied input power.
𝑃𝑟𝑎𝑑 𝑃𝑖𝑛 − 𝑃𝑙𝑜𝑠𝑠 𝑃𝑙𝑜𝑠𝑠
𝜂𝑟𝑎𝑑 = = =1−
𝑃𝑖𝑛 𝑃𝑖𝑛 𝑃𝑖𝑛
where Prad is the power radiated by the antenna, Pin is the power supplied to the input of the
antenna, and Ploss is the power lost in the antenna(dissipative losses) due to metal conductivity
or dielectric loss with in the antenna.

(i)Gain of an antenna
The gain of the antenna is closely related to the directivity, it is a measure that takes into
account the efficiency of the antenna as well as its directional capabilities.
Antenna gain is defined as the product of directivity and efficiency:
𝐺𝑎𝑖𝑛 = 𝐺 = 𝜂𝑟𝑎𝑑 × 𝐷.
Thus, gain is always less than or equal to directivity.
(j) Aperture efficiency
Aperture efficiency is defined as the ratio of the actual directivity of an aperture antenna to the
maximum directivity of aperture antenna.

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The maximum directivity that can be obtained from an electrically large aperture of area A is
4𝜋𝐴
given as, 𝐷𝑚𝑎𝑥 = 𝜆2
𝐷
𝜂𝑎𝑝 = 𝑎𝑝𝑒𝑟𝑡𝑢𝑟𝑒 𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑦 = 𝐷
𝑚𝑎𝑥

(k) Effective aperture area

Received power is proportional to the power density, or Poynting vector, of the incident wave.
Since the Poynting vector has dimensions of W/m2, and the received power, Pr, has dimensions
of W, the proportionality constant must have units of area.
We have, 𝑃𝑟 = 𝐴𝑒 × 𝑆𝑎𝑣𝑔
where Ae is defined as the effective aperture area of the receive antenna. The effective aperture
area has dimensions of m2, and can be interpreted as the “capture area” of a receive antenna,
intercepting part of the incident power density radiated toward the receive antenna.
relation between effective aperture area and Directivity(gain)
The maximum effective aperture area of an antenna is related to the directivity of the antenna
as,
𝐷𝜆2
𝐴𝑒 =
4𝜋
The maximum effective aperture area as defined above does not include the effect of losses in
the antenna, which can be accounted for by replacing D with G, the gain, of the antenna.
𝐺𝜆2
𝐴𝑒 = 4𝜋

(l)Antenna Brightness temperature

When the antenna beam width is broad enough that different parts of the antenna pattern see
different background temperatures, the effective brightness temperature seen by the antenna
can be found by weighting the spatial distribution of background temperature by the pattern
function of the antenna.
Mathematically we can write the brightness temperature Tb seen by the antenna as
2𝜋 𝜋
∫∅=0 ∫𝜃=0 𝑇𝐵 (𝜃, ∅)𝐷(𝜃, ∅)𝑠𝑖𝑛𝜃𝑑𝜃𝑑∅
𝑇𝑏 = 2𝜋 𝜋
∫∅=0 ∫𝜃=0 𝐷(𝜃, ∅)𝑠𝑖𝑛𝜃𝑑𝜃𝑑∅
Where 𝑇𝐵 (𝜃, ∅) is the distribution of the background temperature, and 𝐷(𝜃, ∅) is the
directivity (or the power pattern function) of the antenna. Antenna brightness temperature is
referenced at the terminals of the antenna. Observe that when TB is a constant, Tb = TB
(m) Antenna Noise Temperature
If a receiving antenna has dissipative loss, so that its radiation efficiency ηrad is less than
unity, the power available at the terminals of the antenna is reduced by the factor ηrad from that
intercepted by the antenna (the definition of radiation efficiency is the ratio of output to input
power).
This reduction applies to received noise power, as well as received signal power, so the
noise temperature of the antenna will be reduced from the brightness temperature by the factor
ηrad.
In addition, thermal noise will be generated internally by resistive losses in the antenna,
and this will increase the noise temperature of the antenna. We can find the resulting noise
temperature seen at the antenna terminals as,
TA = ηradTb + (1 − ηrad)Tp.

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The equivalent temperature TA is called the antenna noise temperature, and is a

combination of the external brightness temperature seen by the antenna and the thermal noise
generated by the antenna.
Note: This temperature is referenced at the output terminals of the antenna.
TA = Tb for a lossless antenna with ηrad = 1.
If the radiation efficiency is zero, meaning that the antenna appears as a matched load and does
not see any external background noise, then TA = Tp , due to the thermal noise generated by the
losses.
(n)G/T ratio
Useful figure of merit for receive antennas is the G/T ratio, defined as 10 log( G/ TA)
dB/K, where G is the gain of the antenna, and TA is the antenna noise temperature.
This quantity is important because, the signal-to-noise ratio (SNR) at the input to a
receiver is proportional to G/TA. The ratio G/T can often be maximized by increasing the gain
of the antenna, since this increases the numerator and usually minimizes reception of noise
from hot sources at low elevation angles. Of course, higher gain requires a larger and more
expensive antenna, and high gain may not be desirable for applications requiring
omnidirectional coverage (e.g., cellular telephones or mobile data networks), so often a
compromise must be made.
Note: that the dimensions given for 10 log(G/T ) are not actually decibels per degree kelvin,
but this is the nomenclature that is commonly used for this quantity.

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Friis Transmission Formula

This formula gives the power received over a radio communication link.

Let the transmitter feed a power Pt to a transmitting antenna of effective aperture Aet.At a
distance r a receiving antenna of effective aperture Aer, intercepts some of the power radiated
by the transmitting antenna and delivers it to the receiver.
Assuming that the transmitting antenna is isotropic, the power per unit area at the receiving
antenna is
𝑃
𝑆𝑟 = 4𝜋𝑟𝑡 2 (W) ----- 1

If the transmitting antenna has gain Gt, the power per unit area at the receiving antenna will be
increased in proportion as given by,
𝑃𝐺
𝑡 𝑡
𝑆𝑟 = 4𝜋𝑟 2 (w) ------- 2

Now, the power collected by the receiving antenna of effective aperture Aer is,
𝐴𝑒𝑟 𝑃𝑡 𝐺𝑡
𝑃𝑟 = 𝐴𝑒𝑟 𝑆𝑟 = (w) --------3
4𝜋𝑟 2

The gain of the transmitting antenna can be expressed as,

4𝜋𝐴𝑒𝑡
𝐺𝑡 = ----- 4a
𝜆2
4𝜋𝐴𝑒𝑟
𝐺𝑟 = ------ 4b
𝜆2

Substituting for gain in equation 3, we have,

𝐴𝑒𝑟 𝑃𝑡 𝐴𝑒𝑡 4𝜋 𝐴𝑒𝑟 𝑃𝑡 𝐴𝑒𝑡
𝑃𝑟 = = ------- 5a
4𝜋𝑟 2 𝜆2 𝑟 2 𝜆2

In terms of antenna gain, received power can be expressed as,

𝐺 𝑃 𝐺 𝜆2
𝑟 𝑡 𝑡 𝐺𝑟 𝑃𝑡 𝐺𝑡 𝜆2
𝑃𝑟 = 4𝜋×4𝜋𝑟 2 = (4𝜋𝑟)2
-------- 5b

Equation 5 is Friis transmission formula

𝐺𝑟 𝑃𝑡 𝐺𝑡 𝜆2
𝑃𝑟 =
(4𝜋𝑟)2
Pr = Received power ( antenna matched) in W
Pt = power in to transmitting antenna in W

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Aet = Effective aperture of transmitting antenna, m2

Aer = Effective aperture of Receiving antenna, m2
r = distance between transmitting and receiving antenna, m
λ = wave length, m
EIRP and it’s significance
The product PtGt is defined as the Effective Isotropic Radiated Power (EIRP).
EIRP = PtGt W
For a given frequency, range, and receiver antenna gain, the received power is proportional to
the EIRP of the transmitter and received power can only be increased by increasing the EIRP.
This can be done by increasing the transmit power, or the transmit antenna gain, or both.
Path Loss
Path loss is the quantity that account for the free-space reduction in signal strength with
distance between the transmitter and receiver.
Path loss = Transmitted power- Received power=Pt - Pr
Assuming unity gain antennas, path loss is given as( using Friis formula)
4𝜋𝑟
𝑝𝑎𝑡ℎ 𝑙𝑜𝑠𝑠 (𝑑𝐵) = 20 𝑙𝑜𝑔 ( )
𝜆

Link Budget and Link Margin

The various terms in the Friis formula are often tabulated separately in a link budget, where
each of the factors can be individually considered in terms of its net effect on the received
power.
Additional loss factors, such as line losses or impedance mismatch at the antennas, atmospheric
attenuation, and polarization mismatch can also be added to the link budget.

One of the terms in a link budget is the path loss, accounting for the free-space reduction in
signal strength with distance between the transmitter and receiver.

Path loss= Transmitted power- Received power=Pt-Pr

Assuming unity gain antennas, path loss is given as( using Friis formula)
4𝜋𝑟
𝑝𝑎𝑡ℎ 𝑙𝑜𝑠𝑠 (𝑑𝐵) = 20 𝑙𝑜𝑔 ( )
𝜆
We can write the budget as shown in the following link budget:

Transmit power Pt
Transmit antenna line loss (−)Lt
Transmit antenna gain Gt
Path loss (free-space) (−)L0
Atmospheric attenuation (−)LA
Receive antenna gain Gr
Receive antenna line loss (−)Lr

Receive power Pr

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We have also included loss terms for atmospheric attenuation and line attenuation.
Assuming that all of the above quantities are expressed in dB (or dBm, in the case of Pt), we
can write the receive power as
Pr(dBm) = Pt − Lt + Gt − L0 − LA + Gr − Lr

If the transmit and/or receive antenna is not impedance matched to the transmitter/
receiver (or to their connecting lines), impedance mismatch will reduce the received power by
the factor (1 − |Γ|2 ) where Γ is the appropriate reflection coefficient.

The resulting impedance mismatch loss,

Limp(dB) = −10 log(1 − | Γ |2) ≥ 0,
can be included in the link budget to account for the reduction in received power.
Another possible entry in the link budget relates to the polarization matching of the
transmit and receive antennas, as maximum power transmission between transmitter and
receiver requires both antennas to be polarized in the same manner.
If a transmit antenna is vertically polarized, for example, maximum power will only be
delivered to a vertically polarized receiving antenna, while zero power would be delivered to a
horizontally polarized receive antenna, and half the available power would be delivered to a
circularly polarized antenna.
Link Margin
In practical communications systems it is usually desired to have the received power
level greater than the threshold level required for the minimum acceptable quality of service
(usually expressed as the minimum carrier-to-noise ratio (CNR), or minimum SNR).
This design allowance for received power is referred to as the link margin, and can be
expressed as the difference between the design value of received power and the minimum
threshold value of receive power:

Link margin (dB) = LM = Pr − Pr(min) > 0, where all quantities are in dB.

Link margin should be a positive number; typical values may range from 3 to 20
dB.Having a reasonable link margin provides a level of robustness to the system to account for
variables such as signal fading due to weather, movement of a mobile user, multipath
propagation problems, and other unpredictable effects that can degrade system performance.
Link margin for a given communication system can be improved by increasing the
received power (by increasing transmit power or antenna gains), or by reducing the minimum
threshold power (by improving the design of the receiver, changing the modulation method, or
by other means)
Fade margin.
Signal fading occur due to weather, movement of a mobile user, multipath propagation
problems, and other unpredictable effects that can degrade system performance and quality of
service. Link margin that is used to account for fading effects is sometimes referred to as fade
margin.

Noise Characterization of a Microwave Receiver

(i)NOISE FIGURE and EQUIVALENT NOISE TEMPERATURE of a SYSTEM
(General concepts)

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The signal-to-noise ratio is the ratio of desired signal power to undesired noise power, and so
is dependent on the signal power.
When noise and a desired signal are applied to the input of a noiseless network, both noise and
signal will be attenuated or amplified by the same factor, so that the signal-to-noise ratio will
be unchanged.
However, if the network is noisy, the output noise power will be increased more than the output
signal power, so that the output signal-to-noise ratio will be reduced.
The noise figure, F, is a measure of this reduction in signal-to-noise ratio, and is defined as,
𝑆𝑖
⁄𝑁
𝑖
𝐹= 𝑆𝑜 ≥1 --------------- (1)
⁄𝑁
𝑜

where Si, Ni are the input signal and noise powers, and So, No are the output signal and noise
powers. By definition, the input noise power is assumed to be the noise power resulting from
a matched resistor at T0 = 290 K; that is, Ni = kT0B.

Consider Figure shown above, which shows noise power Ni and signal power Si being fed into
a noisy two-port network.
The network is characterized by a gain, G, a bandwidth, B, and an equivalent noise temperature,
Te.
The input noise power is Ni = kT0B, and the output noise power is a sum of the amplified input
noise and the internally generated noise: No = kGB(T0 + Te).
The output signal power is So = GSi . Using these results in (1) gives the noise figure as,
𝑆 𝑘𝐺𝐵(𝑇𝑜 +𝑇𝑒 ) 𝑇
𝐹 = 𝑘𝑇𝑖 𝐵 × = 1 + 𝑇𝑒 ≥ 1 -----------(2)
𝑜 𝐺𝑆𝑖 𝑜

𝑇𝑒 = (𝐹 − 1)𝑇𝑜 -----------(3)
It is important to keep in mind two things concerning the definition of noise figure: noise figure
is defined for a matched input source, and for a noise source equivalent to a matched load at
temperature T0 = 290 K. Noise figure and equivalent noise temperatures are interchangeable
characterizations of the noise properties of a component.

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An important special case occurs in practice for a two-port network consisting of a passive,
lossy component, such as an attenuator or lossy transmission line, held at a physical
temperature T . Consider such a network with a matched source resistor that is also at
temperature T , as shown in Figure.

The power gain, G, of a lossy network is less than unity; the loss factor, L, can be defined as L
= 1/G > 1. Because the entire system is in thermal equilibrium at the temperature T, and has a
driving point impedance of R, the output noise power must be No = kTB. However, we can
also think of this power as coming from the source resistor (attenuated by the lossy line), and
from the noise generated by the line itself. Thus we also have that
No = kTB = GkTB + G Nadded ----------(4)
Where Nadded is the noise generated by the line, as if it appeared at the input terminals of the
line. Solving (4) for this power gives
(1−𝐺)
𝑁𝑎𝑑𝑑𝑒𝑑 = 𝑘𝑇𝐵 = (𝐿 − 1)𝑘𝑇𝐵 ---------- (5)
𝐺

Then (5) shows that the lossy line has an equivalent noise temperature (referred to the input)
given by,
𝑇𝑒 = (𝐿 − 1)𝑇 ----------- (6)
Noise figure is,
(𝐿−1)𝑇
𝐹 =1+ ≥ 1 -------(7)
𝑇𝑜

Noise Figure of a Cascaded System

In a typical microwave system the input signal travels through a cascade of many different
components, each of which may degrade the signal-to-noise ratio to some degree. If we know
the noise figure (or noise temperature) of the individual stages, we can determine the noise
figure (or noise temperature) of the cascade connection of stages.
We will see that the noise performance of the first stage is usually the most critical, an
interesting result that is very important in practice.
Consider the cascade of two components, having gains G1, G2, noise figures F1, F2, and
equivalent noise temperatures Te1, Te2, as shown in Figure.

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We wish to find the overall noise figure and equivalent noise temperature of the cascade, as if
it were a single component. The overall gain of the cascade is G1G2.
Using noise temperatures, we can write the noise power at the output of the first stage as
N1 = G1kT0B + G1kTe1B --------------- (8)
since Ni = kT0B for noise figure calculations. The noise power at the output of the second stage
is
No = G2N1 + G2kTe2B
𝑁𝑜 = 𝐺1 𝐺2 k𝑇𝑜 B + 𝐺1 𝐺2 k𝑇𝑒1 B + 𝐺2 k𝑇𝑒2 B
𝑇𝑒2
𝑁𝑜 = 𝐺1 𝐺2 𝑘𝐵 (𝑇𝑜 + 𝑇𝑒1 + ) ------- (9)
𝐺1

For the equivalent system we have,

𝑁𝑜 = 𝐺1 𝐺2 𝑘𝐵(𝑇𝑜 + 𝑇𝑐𝑎𝑠 ) ------ (10)
Where,
𝑇𝑒2
𝑇𝑐𝑎𝑠 = 𝑇𝑒1 + --------- (11)
𝐺1

Using (3) to convert the temperatures in (11) to noise figures yields the noise figure of the
cascade system as,
(𝐹2 −1)
𝐹𝑐𝑎𝑠 = 𝐹1 + -------- (12)
𝐺1

Equations (11) and (12) show that the noise characteristics of a cascaded system are dominated
by the characteristics of the first stage since the effect of the second stage is reduced by the
gain of the first (assuming G1 > 1).
Thus, for the best overall system noise performance, the first stage should have a low noise
figure and at least moderate gain. Expense and effort should be devoted primarily to the first
stage, as opposed to later stages, since later stages have a diminished impact on the overall
noise performance.
Equations (11) and (12) can be generalized to an arbitrary number of stages, as

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𝑇𝑒2 𝑇
𝑇𝑐𝑎𝑠 = 𝑇𝑒1 + + 𝐺 𝑒3𝐺 + … … --------(13)
𝐺1 1 2

(𝐹2 −1) (𝐹3 −1)

𝐹𝑐𝑎𝑠 = 𝐹1 + + + … … --------(14)
𝐺1 𝐺1 𝐺2

(ii) Noise Characterization of Receiver

We can now analyze the noise characteristics of a complete antenna–transmission line–
receiver front end, as shown in Figure. In this system the total noise power at the output of the
receiver, No, will be due to contributions from the antenna pattern, the loss in the antenna, the
loss in the transmission line, and the receiver components.
This noise power will determine the minimum detectable signal level for the receiver and, for
a given transmitter power, the maximum range of the communication link.

The receiver components in Figure consist of an RF amplifier with gain GRF and noise
temperature TRF, a mixer with an RF-to-IF conversion loss factor LM and noise temperature TM
, and an IF amplifier with gain GIF and noise temperature TIF.
The noise effects of later stages can usually be ignored since the overall noise figure is
dominated by the characteristics of the first few stages.
The component noise temperatures can be related to noise figures as T = (F − 1)T0.
The equivalent noise temperature of the receiver can be found as
𝑇 𝑇𝐼𝐹 𝐿𝑀
𝑇𝑅𝐸𝐶 = 𝑇𝑅𝐹 + 𝐺 𝑀 + -------------- (1)
𝑅𝐹 𝐺𝑅𝐹

The transmission line connecting the antenna to the receiver has a loss LT , and is at a physical
temperature Tp. So, its equivalent noise temperature is
𝑇𝑇𝐿 = (𝐿𝑇 − 1)𝑇𝑝 --------- (2)

We can find that the noise temperature of the transmission line (TL) and receiver (REC)
cascade is
𝑇𝑇𝐿+𝑅𝐸𝐶 = 𝑇𝑇𝐿 + 𝐿𝑇 𝑇𝑅𝐸𝐶 = (𝐿𝑇 − 1)𝑇𝑝 + 𝐿𝑇 𝑇𝑅𝐸𝐶 --------- (3)

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This noise temperature is defined at the antenna terminals (the input to the transmission line).
The entire antenna pattern can collect noise power. If the antenna has a reasonably high gain
with relatively low sidelobes, we can assume that all noise power comes via the main beam, so
that the noise temperature of the antenna is given by,
𝑇𝐴 = 𝜂𝑟𝑎𝑑 𝑇𝑏 + (1 − 𝜂𝑟𝑎𝑑 )𝑇𝑝 -------- (4)

where ηrad is the efficiency of the antenna, Tp is its physical temperature, and Tb is the
equivalent brightness temperature of the background seen by the main beam.
The noise power at the antenna terminals, which is also the noise power delivered to the
transmission line, is
𝑁𝑖 = 𝑘𝐵𝑇𝐴 = 𝑘𝐵[𝜂𝑟𝑎𝑑 𝑇𝑏 + (1 − 𝜂𝑟𝑎𝑑 )𝑇𝑝 ] -----------(5)

where B is the system bandwidth. If Si is the received power at the antenna terminals, then the
input SNR at the antenna terminals is Si /Ni .
The output signal power is,
𝑆𝑖 𝐺𝑅𝐹 𝐺𝐼𝐹
𝑆𝑜 = = 𝑆𝑖 𝐺𝑆𝑌𝑆 ------ (6)
𝐿𝑇 𝐿𝑀

where GSYS has been defined as a system power gain.

The output noise power is,
𝑁𝑜 = (𝑁𝑖 + 𝑘𝐵𝑇𝑇𝐿+𝑅𝐸𝐶 )𝐺𝑆𝑌𝑆
𝑁𝑜 = (𝑘𝐵𝑇𝐴 + 𝑘𝐵𝑇𝑇𝐿+𝑅𝐸𝐶 )𝐺𝑆𝑌𝑆
𝑁𝑜 = 𝑘𝐵(𝑇𝐴 + 𝑇𝑇𝐿+𝑅𝐸𝐶 )𝐺𝑆𝑌𝑆 = 𝑘𝐵𝑇𝑆𝑌𝑆 𝐺𝑆𝑌𝑆 ----------- (7)
where TSYS has been defined as the overall system noise temperature.
The output SNR is,
𝑆𝑜 𝑆
= 𝑘𝐵𝑇𝑖 --------- (8)
𝑁𝑜 𝑆𝑌𝑆

𝑆𝑜 𝑆𝑖
= 𝑘𝐵[𝜂
𝑁𝑜 𝑟𝑎𝑑 𝑇𝑏 +(1−𝜂𝑟𝑎𝑑 )𝑇𝑝 +(𝐿𝑇 −1)𝑇𝑝 +𝐿𝑇 𝑇𝑅𝐸𝐶 ]

It may be possible to improve this output SNR by various signal processing techniques.

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Impedance Matching
• Impedance matching, is often an important part of a larger design process
for a microwave component or system.
• The basic idea of impedance matching is illustrated in Figure, which shows
an impedance matching network placed between a load impedance and a
transmission line.
• The matching network is ideally lossless, to avoid unnecessary loss of power,
and is usually designed so that the impedance seen looking into the
matching network is Z0.
• Then reflections will be eliminated on the transmission line to the left of the
matching network, although there will usually be multiple reflections
between the matching network and the load.
• This procedure is sometimes referred to as tuning. Impedance matching or
tuning is important for the following reasons:
1
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Impedance Matching

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Impedance matching or tuning is important for the following reasons:

1) Maximum power is delivered when the load is matched to the line

(assuming the generator is matched), and power loss in the feed
line is minimized.
2) Impedance matching sensitive receiver components (antenna, low-
noise amplifier, etc.) may improve the signal-to-noise ratio of the
system.
3) Impedance matching in a power distribution network (such as an
antenna array feed network) may reduce amplitude and phase
errors.

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• As long as the load impedance, ZL, has a positive real part, a
matching network can always be found.
• Many choices are available.
Factors that may be important in the selection of a particular matching
network include the following:
• Complexity
• Bandwidth
• Implementation
• Adjustability

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Complexity
• The simplest design that satisfies the required specifications is
generally preferable.
• A simpler matching network is usually cheaper, smaller, more
reliable, and less lossy than a more complex design.
Bandwidth
• Any type of matching network can ideally give a perfect match (zero
reflection) at a single frequency.
• In many applications, however, it is desirable to match a load over a
band of frequencies.
• There are several ways of doing this, with, of course, a
corresponding increase in complexity.
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Implementation
• Depending on the type of transmission line or waveguide being used,
one type of matching network may be preferable to another.
• For example, tuning stubs are much easier to implement in
waveguide than are multi section quarter-wave transformers.

Adjustability
• In some applications the matching network may require adjustment
to match a variable load impedance.
• It is possible in some types of matching networks.

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Impedance matching techniques:

• MATCHING WITH LUMPED ELEMENTS (L NETWORKS)

• SINGLE-STUB TUNING
• DOUBLE-STUB TUNING
• THE QUARTER-WAVE TRANSFORMER
• TAPERED LINES

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MATCHING WITH LUMPED ELEMENTS (L NETWORKS)
• The simplest type of matching network is the L-section, which uses two
reactive elements to match an arbitrary load impedance to a transmission line.
• There are two possible configurations for this network, as shown in Figure:

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• If the normalized load impedance, zL = ZL /Z0, is inside the 1 + jx
circle on the Smith chart, then the circuit of Figure (a) should be
used.
• If the normalized load impedance is outside the 1 + jx circle on the
Smith chart, the circuit of Figure (b) should be used.
• The 1 + jx circle is the resistance circle on the impedance Smith
chart for which r = 1.
• In either of the configurations of Figure, the reactive elements may
be either inductors or capacitors, depending on the load impedance.

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• Thus, there are eight distinct possibilities for the matching circuit
for various load impedances.

• If the frequency is low enough and/or the circuit size is small

enough, actual lumped-element capacitors and inductors can be
used. This may be feasible for frequencies up to about 1 GHz or so.

• Modern microwave integrated circuits may be small enough such

that lumped elements can be used at higher frequencies as well.

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Determination of component values of matching network:
consider the circuit of Figure (b)

This circuit is used when zL is outside the 1 + jx circle on the Smith chart, which
implies that RL < Z0.

The admittance seen looking into the matching network, followed by the load
impedance, must be equal to 1/Z0 for an impedance-matched condition:

1 1
= 𝑗𝐵 +
𝑍0 𝑅𝐿 + 𝑗(𝑋 + 𝑋𝐿 )

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For circuit in fig (a), component values can be determined as follows:

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SINGLE-STUB TUNING
• Another popular matching technique uses a single open-circuited
or short-circuited length of transmission line (a stub) connected
either in parallel or in series with the transmission feed line at a
certain distance from the load, as shown in Figure.

• Such a single-stub tuning circuit is often very convenient because

the stub can be fabricated as part of the transmission line media of
the circuit, and lumped elements are avoided.

• Shunt stubs are preferred for microstrip line or stripline, while

series stubs are preferred for slotline or coplanar waveguide.
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• In single-stub tuning the two adjustable parameters are the distance,
d, from the load to the stub position, and the value of susceptance or
reactance provided by the stub.

• For the shunt-stub case, the basic idea is to select d so that the
admittance, Y , seen looking into the line at distance d from the load
is of the form Y0 + j B. Then the stub susceptance is chosen as − j B,
resulting in a matched condition.

• For the series-stub case, the distance d is selected so that the

impedance, Z, seen looking into the line at a distance d from the
load is of the form Z0 + j X. Then the stub reactance is chosen as − j
X, resulting in a matched condition.
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DOUBLE-STUB TUNING
• The single-stub tuner is able to match any load impedance (having a
positive real part) to a transmission line, but suffers from the
disadvantage of requiring a variable length of line between the load
and the stub.

• This may not be a problem for a fixed matching circuit, but would
probably pose some difficulty if an adjustable tuner is desired.

• In this case, the double-stub tuner, which uses two tuning stubs in
fixed positions, can be used. Such tuners are often fabricated in
coaxial line with adjustable stubs connected in shunt to the main
coaxial line.
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THE QUARTER-WAVE TRANSFORMER
• The quarter-wave transformer is a simple and useful circuit for
matching a real load impedance to a transmission line.
• An additional feature of the quarter-wave transformer is that it can
be extended to multi section designs in a methodical manner to
provide broader bandwidth.
• If only a narrow band impedance match is required, a single-section
transformer may suffice.
• One drawback of the quarter-wave transformer is that it can only
match a real load impedance.

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UNIT II
RADIATION MECHANISMS AND DESIGN ASPECTS
▪ RADIATION MECHANISMS OF LINEAR WIRE ANTENNAS
➢ Alternating Current Element (Oscillating Dipole/ Hertzian dipole)
➢ Half-wave Dipole Antenna∗
▪ LOOP ANTENNAS∗
▪ APERTURE ANTENNAS
➢ Wire Antennas Vs Aperture Antennas
➢ Field Equivalence Principle⋕
➢ Horn Antennas∗
• Design Principle
• Rectangular Horn Antennas and Solved Problem
• Conical Horn Antennas
• Ridge Horns
• Septum Horns
• Corrugated Horns
• Aperture-Matched Horn
➢ Slot Antennas
• Methods of Feeding∗
1. Coaxial feed
2. Offset feed
3. Boxed-in Slot Antenna
4. Waveguide-fed Slot
5. Broadside Array of Slots in a Waveguide
• Babinet’s Principle⋕
• Booker’s Extension of Babinet’s Principle⋕
• Impedance of Slot Antenna and Solved Problem⋕
▪ REFLECTOR ANTENNAS
➢ Reflectors of various shapes
➢ Parabolic Reflector∗
• f/d ratio
• Feed systems for Parabolic Reflectors
1. Axial/Front Feed
2. Offset Feed
3. Cassegrain Feed
4. Gregorian Feed

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▪ MICROSTRIP ANTENNAS
➢ Basic Characteristics of Microstrip antennas∗
➢ Feeding Methods∗
• Microstrip Line Feed
• Coaxial Probe Feed
• Aperture Coupled Feed
• Proximity Coupled Feed
➢ Methods of Analysis
• Transmission-Line Model
• Cavity Model
▪ FREQUENCY INDEPENDENT ANTENNAS
➢ Rumsey’s principle⋕
➢ Frequency-Independent Planar Log Spiral Antenna∗
➢ Frequency Independent Conical Spiral Antenna
➢ Log-Periodic Antenna∗
• Basic concept
• Regions of LPDA
• Radiation pattern
• Log-periodic behaviour
• Design equations for LPDA
• Solved Problem

⋕ Sections requiring additional attention in the perspective of Part-A.

∗ Sections requiring additional attention in the perspective of Part-B.

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Far-field due to an Alternating Current Element (Oscillating Dipole/ Hertzian dipole)

Consider that a time varying current I is flowing in a very short and very thin wire of length dl
in the z-direction. This current is given by 𝐼𝑑𝑙 cos 𝜔𝑡. Since the current is in the z-direction,
the current density J will have only a z-component.
𝐽 = 𝑎𝑧 𝐽𝑧 (1)
The vector magnetic potential A will also have only a z-component.
𝐴 = 𝑎𝑧 𝐴𝑧 (2)

Configuration of filamentary current carrying conductor

For a line charge density,
𝜇 𝐼 𝑑𝑙 𝜇 𝐼 𝑑𝑙 cos 𝜔(𝑡 − 𝑟⁄𝑣 )
𝐴𝑧 = ∫ = (3)
4𝜋𝑅 4𝜋𝑟
We know that 𝐴𝑥 = 0, 𝐴𝑦 = 0 𝑎𝑛𝑑 𝐴𝑧 ≠ 0. Since the three-dimensional radiation problem
needs to be tackled in a spherical co-ordinate system, 𝐴𝑧 needs to be transformed into the
spherical co-ordinate system.
𝐴𝑟 𝑠𝑖𝑛𝜃𝑐𝑜𝑠𝜙 𝑠𝑖𝑛𝜃𝑠𝑖𝑛𝜙 𝑐𝑜𝑠𝜃 𝐴𝑥
[ 𝐴𝜃 ] = [𝑐𝑜𝑠𝜃𝑐𝑜𝑠𝜙 𝑐𝑜𝑠𝜃𝑠𝑖𝑛𝜙 −𝑠𝑖𝑛𝜃] [𝐴𝑦 ]
𝐴𝜙 −𝑠𝑖𝑛𝜙 𝑐𝑜𝑠𝜙 0 𝐴𝑧

𝐴𝑟 𝑠𝑖𝑛𝜃𝑐𝑜𝑠𝜙 𝑠𝑖𝑛𝜃𝑠𝑖𝑛𝜙 𝑐𝑜𝑠𝜃 0

[ 𝐴𝜃 ] = [𝑐𝑜𝑠𝜃𝑐𝑜𝑠𝜙 𝑐𝑜𝑠𝜃𝑠𝑖𝑛𝜙 −𝑠𝑖𝑛𝜃 ] [ 0 ]
𝐴𝜙 −𝑠𝑖𝑛𝜙 𝑐𝑜𝑠𝜙 0 𝐴𝑧

𝐴𝑟 = 𝐴𝑧 𝑐𝑜𝑠𝜃 ; 𝐴𝜃 = −𝐴𝑧 𝑠𝑖𝑛𝜃 ; 𝐴𝜙 = 0 (4)

𝜇 𝐼 𝑑𝑙 cos 𝜔(𝑡 − 𝑟⁄𝑣)

𝐴𝑟 = 𝑐𝑜𝑠𝜃 (5)
4𝜋𝑟
𝜇 𝐼 𝑑𝑙 cos 𝜔(𝑡 − 𝑟⁄𝑣)
𝐴𝜃 = − 𝑠𝑖𝑛𝜃 (6)
4𝜋𝑟

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Further from the relation 𝐵 = ∇ × 𝐴, the components of ∇ × 𝐴 are obtained as below.

𝑎𝑟 𝑟𝑎𝜃 𝑟𝑠𝑖𝑛𝜃𝑎𝜙
1 𝜕 𝜕 𝜕
𝐵 = ∇×𝐴 = 2 || |
𝑟 𝑠𝑖𝑛𝜃 𝜕𝑟 𝜕𝜃 𝜕𝜙 |
𝐴𝑟 𝑟𝐴𝜃 𝑟𝑠𝑖𝑛𝜃𝐴𝜙
𝜕
We know that 𝐴𝜙 = 0 and 𝐴𝑟 & 𝐴𝜃 are independent of 𝜙. So 𝜕𝜙 = 0.

𝑎𝑟 𝑟𝑎𝜃 𝑟𝑠𝑖𝑛𝜃𝑎𝜙
1 𝜕 𝜕
𝐵 =∇×𝐴= 2 | 0 |
𝑟 𝑠𝑖𝑛𝜃 𝜕𝑟 𝜕𝜃
𝐴𝑟 𝑟𝐴𝜃 0
1 𝜕(𝑟𝐴𝜃 ) 𝜕𝐴𝑟
𝐵 = ∇×𝐴 = [𝑎 𝑟 (0) − 𝑟𝑎 𝜃 (0) + 𝑟𝑠𝑖𝑛𝜃𝑎 𝜙 ( − )]
𝑟 2 𝑠𝑖𝑛𝜃 𝜕𝑟 𝜕𝜃
(∇ × 𝐴)𝑟 = 𝐵𝑟 = 𝜇𝐻𝑟 = 0 ⇒ 𝐻𝑟 = 0 (7)
(∇ × 𝐴)𝜃 = 𝐵𝜃 = 𝜇𝐻𝜃 = 0 ⇒ 𝐻𝜃 = 0 (8)
1 𝜕(𝑟𝐴𝜃 ) 𝜕𝐴𝑟
(∇ × 𝐴)𝜙 = 𝐵𝜙 = 𝜇𝐻𝜙 = {𝑟𝑠𝑖𝑛𝜃 ( − )}
𝑟 2 𝑠𝑖𝑛𝜃 𝜕𝑟 𝜕𝜃

𝟏 𝝏(𝒓𝑨𝜽 ) 𝝏𝑨𝒓
⇒ 𝑯𝝓 = [ − ] (𝟗)
𝝁𝒓 𝝏𝒓 𝝏𝜽

To find 𝑯𝝓

Using eq(6),
𝜕(𝑟𝐴𝜃 ) 𝜕 𝜇 𝐼 𝑑𝑙 cos 𝜔(𝑡 − 𝑟⁄𝑣 ) 𝜇 𝐼 𝑑𝑙𝑠𝑖𝑛𝜃 𝜕
= (−𝑟 𝑠𝑖𝑛𝜃) = − [cos 𝜔(𝑡 − 𝑟⁄𝑣)]
𝜕𝑟 𝜕𝑟 4𝜋𝑟 4𝜋 𝜕𝑟
𝜇 𝐼 𝑑𝑙𝑠𝑖𝑛𝜃 −𝜔
=− ( ) [−𝑠𝑖𝑛𝜔(𝑡 − 𝑟⁄𝑣 )]
4𝜋 𝑣
𝜕(𝑟𝐴𝜃 ) 𝜇 𝐼 𝑑𝑙𝑠𝑖𝑛𝜃 𝜔
=− ( ) 𝑠𝑖𝑛𝜔(𝑡 − 𝑟⁄𝑣 ) (10)
𝜕𝑟 4𝜋 𝑣
Using eq(5)
𝜕𝐴𝑟 𝜕 𝜇 𝐼 𝑑𝑙 cos 𝜔(𝑡 − 𝑟⁄𝑣) 𝜇 𝐼 𝑑𝑙 cos 𝜔(𝑡 − 𝑟⁄𝑣) 𝜕
= ( 𝑐𝑜𝑠𝜃) = [𝑐𝑜𝑠𝜃]
𝜕𝜃 𝜕𝜃 4𝜋𝑟 4𝜋𝑟 𝜕𝜃
𝜕𝐴𝑟 𝜇 𝐼 𝑑𝑙 cos 𝜔(𝑡 − 𝑟⁄𝑣 )
= − 𝑠𝑖𝑛𝜃 (11)
𝜕𝜃 4𝜋𝑟
Using eq(10) and eq(11) in eq(9),
𝟏 𝝏(𝒓𝑨𝜽 ) 𝝏𝑨𝒓
𝑯𝝓 = [ − ]
𝝁𝒓 𝝏𝒓 𝝏𝜽

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1 𝜇 𝐼 𝑑𝑙𝑠𝑖𝑛𝜃 𝜔 𝜇 𝐼 𝑑𝑙 cos 𝜔(𝑡 − 𝑟⁄𝑣 )

𝐻𝜙 = [(− ( ) 𝑠𝑖𝑛𝜔(𝑡 − 𝑟⁄𝑣) ) − (− 𝑠𝑖𝑛𝜃)]
𝜇𝑟 4𝜋 𝑣 4𝜋𝑟

1 𝜇 𝐼 𝑑𝑙𝑠𝑖𝑛𝜃 𝜔 𝜇 𝐼 𝑑𝑙 cos 𝜔(𝑡 − 𝑟⁄𝑣)

𝐻𝜙 = [− ( ) 𝑠𝑖𝑛𝜔(𝑡 − 𝑟⁄𝑣 ) + 𝑠𝑖𝑛𝜃]
𝜇𝑟 4𝜋 𝑣 4𝜋𝑟

𝐼 𝑑𝑙𝑠𝑖𝑛𝜃 𝜇 𝜔 𝜇 cos 𝜔(𝑡 − 𝑟⁄𝑣 )

𝐻𝜙 = [− ( ) 𝑠𝑖𝑛𝜔(𝑡 − 𝑟⁄𝑣) + ]
4𝜋 𝜇𝑟 𝑣 𝜇𝑟 2

𝑰 𝒅𝒍𝒔𝒊𝒏𝜽 𝝎 𝐜𝐨𝐬 𝝎(𝒕 − 𝒓⁄𝒗)

𝑯𝝓 = [− ( ) 𝒔𝒊𝒏𝝎(𝒕 − 𝒓⁄𝒗) + ] (𝟏𝟐)
𝟒𝝅 𝒓𝒗 𝒓𝟐

Now our objective is to find the electric field components from the magnetic field strength
relations.
From Maxwell’s Equation,
∇ × 𝐻 = 𝐽 + 𝜕𝐷⁄𝜕𝑡 = 𝜎𝐸 + 𝜀 𝜕𝐸 ⁄𝜕𝑡
The observation point P lies at a distance r away from the antenna. Moreover, the medium
surrounding the antenna element is air. Hence 𝜎 = 0.
∇ × 𝐻 = 𝜀 𝜕𝐸 ⁄𝜕𝑡
1
⇒𝐸= ∫(∇ × 𝐻) 𝑑𝑡
𝜀
𝑎𝑟 𝑟𝑎𝜃 𝑟𝑠𝑖𝑛𝜃𝑎𝜙
1 𝜕 𝜕 𝜕
∇×𝐻 = 2 || |
𝑟 𝑠𝑖𝑛𝜃 𝜕𝑟 𝜕𝜃 𝜕𝜙 |
𝐻𝑟 𝑟𝐻𝜃 𝑟𝑠𝑖𝑛𝜃𝐻𝜙

Using eq(7), eq(8) and eq(9), 𝐻𝑟 = 0, 𝐻𝜃 = 0 & 𝐻𝜙 ≠ 0

𝜕𝐻𝜙
From eq(12), we can observe that 𝐻𝜙 is independent of 𝜙 ⇒ =0
𝜕𝜙

𝑎𝑟 𝑟𝑎𝜃 𝑟𝑠𝑖𝑛𝜃𝑎𝜙
1 𝜕 𝜕
∇×𝐻 = 2 | 0 |
𝑟 𝑠𝑖𝑛𝜃 𝜕𝑟 𝜕𝜃
0 0 𝑟𝑠𝑖𝑛𝜃𝐻𝜙
1 𝜕 𝜕
∇×𝐻 = [𝑎 𝑟 (𝑟𝑠𝑖𝑛𝜃𝐻𝜙 ) − 𝑟𝑎 𝜃 (𝑟𝑠𝑖𝑛𝜃𝐻𝜙 )]
𝑟 2 𝑠𝑖𝑛𝜃 𝜕𝜃 𝜕𝑟
1 𝜕
(∇ × 𝐻)𝑟 = [ (𝑟𝑠𝑖𝑛𝜃𝐻𝜙 )] (13)
𝑟 2 𝑠𝑖𝑛𝜃 𝜕𝜃
1 𝜕 −1 𝜕
(∇ × 𝐻)𝜃 = [−𝑟 (𝑟𝑠𝑖𝑛𝜃𝐻𝜙 )] = [ (𝑟𝑠𝑖𝑛𝜃𝐻𝜙 )] (14)
𝑟 2 𝑠𝑖𝑛𝜃 𝜕𝑟 𝑟𝑠𝑖𝑛𝜃 𝜕𝑟
(∇ × 𝐻)𝜙 = 0 ⇒ 𝑬𝝓 = 𝟎 (𝟏𝟓)

Now we have to find 𝐸𝑟 and 𝐸𝜃

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To find 𝑬𝒓
1
𝐸𝑟 = ∫(∇ × 𝐻)𝑟 𝑑𝑡
𝜀
1 𝜕
(∇ × 𝐻)𝑟 = [ (𝑟𝑠𝑖𝑛𝜃𝐻𝜙 )]
𝑟 2 𝑠𝑖𝑛𝜃 𝜕𝜃
Using eq(12),
1 𝜕 𝐼 𝑑𝑙𝑠𝑖𝑛𝜃 𝜔 cos 𝜔(𝑡 − 𝑟⁄𝑣 )
(∇ × 𝐻)𝑟 = [ {𝑟𝑠𝑖𝑛𝜃 ( − ( ) 𝑠𝑖𝑛𝜔(𝑡 − 𝑟 ⁄𝑣 ) + )} ]
𝑟 2 𝑠𝑖𝑛𝜃 𝜕𝜃 4𝜋 𝑟𝑣 𝑟2

1 𝑟 𝐼 𝑑𝑙 𝜔 cos 𝜔(𝑡 − 𝑟⁄𝑣 ) 𝜕

= ( − ( ) 𝑠𝑖𝑛𝜔(𝑡 − 𝑟⁄𝑣) + )[ (𝑠𝑖𝑛2 𝜃) ]
𝑟 2 𝑠𝑖𝑛𝜃 4𝜋 𝑟𝑣 𝑟2 𝜕𝜃

1 𝐼 𝑑𝑙 𝜔 cos 𝜔(𝑡 − 𝑟⁄𝑣 )

= ( − ( ) 𝑠𝑖𝑛𝜔(𝑡 − 𝑟⁄𝑣) + ) [2𝑠𝑖𝑛𝜃𝑐𝑜𝑠𝜃 ]
𝑟𝑠𝑖𝑛𝜃 4𝜋 𝑟𝑣 𝑟2

1 𝐼 𝑑𝑙 𝜔 cos 𝜔(𝑡 − 𝑟⁄𝑣 )

(∇ × 𝐻)𝑟 = ( − ( ) 𝑠𝑖𝑛𝜔(𝑡 − 𝑟⁄𝑣) + ) (2𝑐𝑜𝑠𝜃)
𝑟 4𝜋 𝑟𝑣 𝑟2

2 𝐼 𝑑𝑙𝑐𝑜𝑠𝜃 𝜔 cos 𝜔(𝑡 − 𝑟⁄𝑣)

(∇ × 𝐻)𝑟 = ( − ( 2 ) 𝑠𝑖𝑛𝜔(𝑡 − 𝑟⁄𝑣) + )
4𝜋 𝑟 𝑣 𝑟3

Using above equation,

1
𝐸𝑟 = ∫(∇ × 𝐻)𝑟 𝑑𝑡
𝜀
2 𝐼 𝑑𝑙𝑐𝑜𝑠𝜃 𝜔 cos 𝜔(𝑡 − 𝑟⁄𝑣 )
𝐸𝑟 = ∫ [− ( 2 ) 𝑠𝑖𝑛𝜔(𝑡 − 𝑟⁄𝑣) + ] 𝑑𝑡
4𝜋𝜀 𝑟 𝑣 𝑟3
2 𝐼 𝑑𝑙𝑐𝑜𝑠𝜃 𝜔 −𝑐𝑜𝑠𝜔(𝑡 − 𝑟⁄𝑣) sin 𝜔(𝑡 − 𝑟⁄𝑣)
= [− ( 2 ) + ]
4𝜋𝜀 𝑟 𝑣 𝜔 𝜔𝑟 3

𝟐 𝑰 𝒅𝒍𝒄𝒐𝒔𝜽 𝒄𝒐𝒔𝝎(𝒕 − 𝒓⁄𝒗) 𝐬𝐢𝐧 𝝎(𝒕 − 𝒓⁄𝒗)

𝑬𝒓 = [ 𝟐
+ ] (𝟏𝟔)
𝟒𝝅𝜺 𝒓 𝒗 𝝎𝒓𝟑

To find 𝑬𝜽
1
𝐸𝜃 = ∫(∇ × 𝐻)𝜃 𝑑𝑡
𝜀
−1 𝜕
(∇ × 𝐻)𝜃 = [ (𝑟𝑠𝑖𝑛𝜃𝐻𝜙 )]
𝑟𝑠𝑖𝑛𝜃 𝜕𝑟

Using eq(12),
−1 𝜕 𝐼 𝑑𝑙𝑠𝑖𝑛𝜃 𝜔 cos 𝜔(𝑡 − 𝑟⁄𝑣)
(∇ × 𝐻)𝜃 = { 𝑟𝑠𝑖𝑛𝜃 [− ( ) 𝑠𝑖𝑛𝜔(𝑡 − 𝑟⁄𝑣) + ]}
𝑟𝑠𝑖𝑛𝜃 𝜕𝑟 4𝜋 𝑟𝑣 𝑟2

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−1 𝐼 𝑑𝑙𝑠𝑖𝑛𝜃 𝜕 𝑟𝜔 𝑟 cos 𝜔(𝑡 − 𝑟⁄𝑣 )

(∇ × 𝐻)𝜃 = 𝑠𝑖𝑛𝜃 [− ( ) 𝑠𝑖𝑛𝜔(𝑡 − 𝑟⁄𝑣) + ]
𝑟𝑠𝑖𝑛𝜃 4𝜋 𝜕𝑟 𝑟𝑣 𝑟2

−𝐼 𝑑𝑙𝑠𝑖𝑛𝜃 𝜕 𝜔 cos 𝜔(𝑡 − 𝑟⁄𝑣)

= [− ( ) 𝑠𝑖𝑛𝜔(𝑡 − 𝑟⁄𝑣 ) + ]
4𝜋𝑟 𝜕𝑟 𝑣 𝑟
−𝜔
−𝐼 𝑑𝑙𝑠𝑖𝑛𝜃 𝜔 2 −𝑟 ( 𝑣 ) 𝑠𝑖𝑛 𝜔(𝑡 − 𝑟⁄𝑣) − cos 𝜔(𝑡 − 𝑟⁄𝑣)
= [( ) 𝑐𝑜𝑠𝜔(𝑡 − 𝑟⁄𝑣 ) + ]
4𝜋𝑟 𝑣 𝑟2

𝑟𝜔
−𝐼 𝑑𝑙𝑠𝑖𝑛𝜃 𝜔 2 ( 𝑣 ) 𝑠𝑖𝑛 𝜔(𝑡 − 𝑟⁄𝑣 ) cos 𝜔(𝑡 − 𝑟⁄𝑣 )
= [( ) 𝑐𝑜𝑠𝜔(𝑡 − 𝑟⁄𝑣 ) + − ]
4𝜋𝑟 𝑣 𝑟2 𝑟2

−𝐼 𝑑𝑙𝑠𝑖𝑛𝜃 𝜔2 𝑐𝑜𝑠𝜔(𝑡 − 𝑟⁄𝑣) 𝜔𝑠𝑖𝑛 𝜔(𝑡 − 𝑟⁄𝑣) cos 𝜔(𝑡 − 𝑟⁄𝑣)

(∇ × 𝐻)𝜃 = [ + − ]
4𝜋𝑟 𝑣2 𝑟𝑣 𝑟2
1
𝐸𝜃 = ∫(∇ × 𝐻)𝜃 𝑑𝑡
𝜀
1 −𝐼 𝑑𝑙𝑠𝑖𝑛𝜃 𝜔2 𝑐𝑜𝑠𝜔(𝑡 − 𝑟⁄𝑣 ) 𝜔𝑠𝑖𝑛 𝜔(𝑡 − 𝑟⁄𝑣) cos 𝜔(𝑡 − 𝑟⁄𝑣 )
𝐸𝜃 = ∫ [ + − ] 𝑑𝑡
𝜀 4𝜋𝑟 𝑣2 𝑟𝑣 𝑟2

−𝐼 𝑑𝑙𝑠𝑖𝑛𝜃 𝜔2 𝑠𝑖𝑛𝜔(𝑡 − 𝑟⁄𝑣) 𝜔𝑐𝑜𝑠 𝜔(𝑡 − 𝑟⁄𝑣 ) sin 𝜔(𝑡 − 𝑟⁄𝑣)

= [ − − ]
4𝜋𝜀𝑟 𝑣 2𝜔 𝑟𝑣𝜔 𝜔𝑟 2
−𝐼 𝑑𝑙𝑠𝑖𝑛𝜃 𝜔𝑠𝑖𝑛𝜔(𝑡 − 𝑟⁄𝑣) 𝑐𝑜𝑠 𝜔(𝑡 − 𝑟⁄𝑣 ) sin 𝜔(𝑡 − 𝑟⁄𝑣)
= [ − − ]
4𝜋𝜀𝑟 𝑣2 𝑟𝑣 𝜔𝑟 2

𝐼 𝑑𝑙𝑠𝑖𝑛𝜃 𝜔𝑠𝑖𝑛𝜔(𝑡 − 𝑟⁄𝑣 ) 𝑐𝑜𝑠 𝜔(𝑡 − 𝑟⁄𝑣) sin 𝜔(𝑡 − 𝑟⁄𝑣 )

𝐸𝜃 = [− + + ]
4𝜋𝜀 𝑟𝑣 2 𝑟 2𝑣 𝜔𝑟 3

Putting 𝑡 ′ = 𝑡 − 𝑟⁄𝑣,
𝑰 𝒅𝒍𝒔𝒊𝒏𝜽 𝝎𝒔𝒊𝒏𝝎𝒕′ 𝒄𝒐𝒔 𝝎𝒕′ 𝐬𝐢𝐧 𝝎𝒕′
𝑬𝜽 = [− + + ] (𝟏𝟕)
𝟒𝝅𝜺 𝒓𝒗𝟐 𝒓𝟐 𝒗 𝝎𝒓𝟑

Putting 𝑡 ′ = 𝑡 − 𝑟⁄𝑣, eq(16) becomes

𝟐 𝑰 𝒅𝒍𝒄𝒐𝒔𝜽 𝒄𝒐𝒔𝝎𝒕′ 𝐬𝐢𝐧 𝝎𝒕′
𝑬𝒓 = [ + ] (𝟏𝟖)
𝟒𝝅𝜺 𝒓𝟐 𝒗 𝝎𝒓𝟑

Putting 𝑡 ′ = 𝑡 − 𝑟⁄𝑣, eq(12) becomes

𝑰 𝒅𝒍𝒔𝒊𝒏𝜽 𝝎 ′
𝐜𝐨𝐬 𝝎𝒕′
𝑯𝝓 = [− ( ) 𝒔𝒊𝒏𝝎𝒕 + ] (𝟏𝟗)
𝟒𝝅 𝒓𝒗 𝒓𝟐

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Inference
The expressions of 𝐸𝑟 , 𝐸𝜃 and 𝐻𝜙 involve three types of terms:

1. The terms inversely proportional to 𝑟 3 represent electrostatic field.

2. The terms inversely proportional to 𝑟 2 represent induction or near-field.
3. The terms which are inversely proportional to r represent radiation field (distant or
far-field)
It can be noted the magnitudes of the two bracketed terms in eq(19) will become equal if the
following relation is satisfied:
𝜔 1 𝑣 𝑓𝜆 𝜆 𝝀
= 2 ⇒ 𝑟= = = ⟹ 𝒓≈
𝑟𝑣 𝑟 𝜔 2𝜋𝑓 2𝜋 𝟔
Far-field Region or Radiation Zone
When the distance between the antenna and observation point (r) is very large, the terms
1⁄ and 1⁄ in eq(17), eq(18) and eq(19) can be neglected in favour of terms of 1⁄ .
𝑟2 𝑟3 𝑟
𝑰 𝒅𝒍𝒔𝒊𝒏𝜽 𝝎𝒔𝒊𝒏𝝎𝒕′
𝑬𝜽 ≈ [− ] (𝟐𝟎)
𝟒𝝅𝜺 𝒓𝒗𝟐

𝑬𝒓 ≈ 𝟎 (𝟐𝟏)
𝑰 𝒅𝒍𝒔𝒊𝒏𝜽 𝝎
𝑯𝝓 ≈ [− ( ) 𝒔𝒊𝒏𝝎𝒕′ ] (𝟐𝟐)
𝟒𝝅 𝒓𝒗
The amplitudes in the far-field will be
𝐼 𝑑𝑙𝑠𝑖𝑛𝜃 𝜔
|𝐸𝜃 | = ( 2) (23)
4𝜋𝜀 𝑟𝑣
𝐼 𝑑𝑙𝑠𝑖𝑛𝜃 𝜔 𝐼 𝑑𝑙𝑠𝑖𝑛𝜃 2𝜋𝑓 𝐼 𝑑𝑙𝑠𝑖𝑛𝜃 𝑓 𝐼 𝑑𝑙𝑠𝑖𝑛𝜃
|𝐻𝜙 | = ( )= ( )= ( )= (24)
4𝜋 𝑟𝑣 4𝜋 𝑟𝑣 2𝑟 𝑣 2𝜆𝑟
Using above two equations,
𝐼 𝑑𝑙𝑠𝑖𝑛𝜃 𝜔
|𝐸𝜃 | 4𝜋𝜀 (𝑟𝑣 2 ) 1
= =
|𝐻𝜙 | 𝐼 𝑑𝑙𝑠𝑖𝑛𝜃 𝜔 𝑣𝜀
(𝑟𝑣)
4𝜋
We know that velocity of a wave in a medium is given by
1
𝑣=
√𝜇𝜀
|𝐸𝜃 | 1 1 √𝜇𝜀 𝜇
= = = = √ = 𝜂 = 120𝜋 (𝑜𝑟) 377Ω (25)
|𝐻𝜙 | 𝑣𝜀 ( 1 ) 𝜀 𝜀 𝜀
√𝜇𝜀
Where 𝜂 is the intrinsic impedance of the medium.

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Power Radiated and Radiation Resistance

It can be noted that E has no 𝜙 component and H contains only a 𝜙 component.
𝐸 = 𝐸𝑟 𝑎𝑟 + 𝐸𝜃 𝑎𝜃
𝐻 = 𝐻𝜙 𝑎𝜙

The power flow can be given by the poynting vector

1 1 1
𝑆= 𝑅𝑒 (𝐸 × 𝐻 ∗ ) = 𝑅𝑒 {(𝐸𝑟 𝑎𝑟 + 𝐸𝜃 𝑎𝜃 ) × 𝑎𝜙 𝐻𝜙∗ } = 𝑅𝑒 {−𝑎𝜃 𝐸𝑟 𝐻𝜙∗ + 𝑎𝑟 𝐸𝜃 𝐻𝜙∗ }
2 2 2
The total radiated power is given by
𝜙=2𝜋 𝜃=𝜋
𝑃𝑟𝑎𝑑 = ∯ 𝑊. 𝑑𝑠 = ∫ ∫ 𝑆. (𝑎𝑟 𝑟 2 𝑠𝑖𝑛𝜃 𝑑𝜃 𝑑𝜙)
𝜙=0 𝜃=0

𝜙=2𝜋 𝜃=𝜋
1
𝑃𝑟𝑎𝑑 = ∫ ∫ 𝑅𝑒 {−𝑎𝜃 𝐸𝑟 𝐻𝜙∗ + 𝑎𝑟 𝐸𝜃 𝐻𝜙∗ }. (𝑎𝑟 𝑟 2 𝑠𝑖𝑛𝜃 𝑑𝜃 𝑑𝜙)
𝜙=0 𝜃=0 2
𝜙=2𝜋 𝜃=𝜋
1 ∗ 1 𝜙=2𝜋 𝜃=𝜋
=∫ ∫ 2
𝑅𝑒 {𝐸𝜃 𝐻𝜙 } 𝑟 𝑠𝑖𝑛𝜃 𝑑𝜃 𝑑𝜙 = ∫ ∫ |𝐸𝜃 ||𝐻𝜙∗ | 𝑟 2 𝑠𝑖𝑛𝜃 𝑑𝜃 𝑑𝜙
𝜙=0 𝜃=0 2 2 𝜙=0 𝜃=0

|𝐸𝜃 |
Since ⁄ = 𝜂 ⟹ |𝐸𝜃 | = 𝜂|𝐻𝜙 |
|𝐻𝜙 |

1 𝜙=2𝜋 𝜃=𝜋
𝑃𝑟𝑎𝑑 = ∫ ∫ 𝜂 |𝐻𝜙 | |𝐻𝜙∗ | 𝑟 2 𝑠𝑖𝑛𝜃 𝑑𝜃 𝑑𝜙
2 𝜙=0 𝜃=0

1 𝜙=2𝜋 𝜃=𝜋
= ∫ ∫ 𝜂 |𝐻𝜙 | 2 𝑟 2 𝑠𝑖𝑛𝜃 𝑑𝜃 𝑑𝜙
2 𝜙=0 𝜃=0

1 𝜙=2𝜋 𝜃=𝜋 𝐼 𝑑𝑙𝑠𝑖𝑛𝜃 2 2

= ∫ ∫ 𝜂 ( ) 𝑟 𝑠𝑖𝑛𝜃 𝑑𝜃 𝑑𝜙
2 𝜙=0 𝜃=0 2𝜆𝑟

1 𝜙=2𝜋 𝜃=𝜋 𝐼 2 𝑑𝑙 2 𝑠𝑖𝑛2 𝜃 2

= ∫ ∫ 𝜂 𝑟 𝑠𝑖𝑛𝜃 𝑑𝜃 𝑑𝜙
2 𝜙=0 𝜃=0 4𝜆2 𝑟 2

𝜂 𝐼 2 𝑑𝑙 2 𝜙=2𝜋 𝜃=𝜋 3
= ∫ ∫ 𝑠𝑖𝑛 𝜃 𝑑𝜃 𝑑𝜙
2 4𝜆2 𝜙=0 𝜃=0

𝜂 𝐼 2 𝑑𝑙 2 𝜙=2𝜋 𝜃=𝜋
= ( ) ∫ 𝑑𝜙 ∫ 𝑠𝑖𝑛3 𝜃 𝑑𝜃
2 4 𝜆 𝜙=0 𝜃=0

𝜂 𝐼 2 𝑑𝑙 2 𝜃=𝜋
= ( ) (2𝜋) ∫ 𝑠𝑖𝑛3 𝜃 𝑑𝜃
2 4 𝜆 𝜃=0

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𝜂 𝐼 2 𝑑𝑙 2 𝜃=𝜋
𝑃𝑟𝑎𝑑 = ( ) (2𝜋) ∫ 𝑠𝑖𝑛3 𝜃 𝑑𝜃
2 4 𝜆 𝜃=0

Using the following identity, above equation can be re-written as

3𝑠𝑖𝑛𝜃 − 𝑠𝑖𝑛3𝜃
𝑠𝑖𝑛3 𝜃 =
4
𝜂 𝐼 2 𝑑𝑙 2 𝜃=𝜋 3𝑠𝑖𝑛𝜃 − 𝑠𝑖𝑛3𝜃
𝑃𝑟𝑎𝑑 = ( ) (2𝜋) ∫ ( ) 𝑑𝜃
2 4 𝜆 𝜃=0
4

𝜂 𝐼 2 𝑑𝑙 2 (2𝜋) 𝑐𝑜𝑠3𝜃 𝜋
= ( ) [−3𝑐𝑜𝑠𝜃 + ]
2 4 𝜆 4 3 0

𝜂 𝐼 2 𝑑𝑙 2 (2𝜋) 1 1
= ( ) [3 − + 3 − ]
2 4 𝜆 4 3 3
𝜂 𝐼 2 𝑑𝑙 2 (2𝜋) 16
= ( ) [ ]
2 4 𝜆 4 3
𝝅 𝟐 𝒅𝒍 𝟐
𝑷𝒓𝒂𝒅 = 𝜼 𝑰 ( )
𝟑 𝝀
Assuming the antenna is lossless, the power radiated by the dipole will be equal to the power
delivered to the dipole.
𝑃𝑟𝑎𝑑 = 𝑃𝑑𝑒𝑙𝑖𝑣𝑒𝑟𝑒𝑑

𝜋 𝑑𝑙 2
𝜂 𝐼 2 ( ) = 𝐼𝑟𝑚𝑠
2
𝑅𝑟
3 𝜆
We know that
𝐼
𝐼𝑟𝑚𝑠 =
√2
𝜋 2 𝑑𝑙 2 𝐼 2
𝜂 𝐼 ( ) = ( ) 𝑅𝑟
3 𝜆 √2
𝜋 2 𝑑𝑙 2 𝐼 2
120𝜋 𝐼 ( ) = 𝑅
3 𝜆 2 𝑟
𝒅𝒍 𝟐
𝟐
𝒅𝒍 𝟐
𝑹𝒓 = 𝟖𝟎 𝝅 ( ) = 𝟕𝟗𝟎 ( )
𝝀 𝝀

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Directivity
4𝜋
𝐷=
∫ ∫ 𝑃𝑛 (𝜃, 𝜙) 𝑑Ω
4𝜋
𝐷= 𝜙=2𝜋 𝜃=𝜋
∫𝜙=0 ∫𝜃=0 𝑃𝑛 (𝜃, 𝜙) 𝑠𝑖𝑛𝜃 𝑑𝜃 𝑑𝜙

4𝜋
𝐷= 𝜙=2𝜋 𝜃=𝜋
∫𝜙=0 ∫𝜃=0 (𝑠𝑖𝑛2 𝜃) 𝑠𝑖𝑛𝜃 𝑑𝜃 𝑑𝜙

4𝜋
= 𝜙=2𝜋 𝜃=𝜋
∫𝜙=0 ∫𝜃=0 𝑠𝑖𝑛3 𝜃 𝑑𝜃 𝑑𝜙

4𝜋
= 𝜙=2𝜋 𝜃=𝜋
∫𝜙=0 𝑑𝜙 ∫𝜃=0 𝑠𝑖𝑛3 𝜃 𝑑𝜃

4𝜋
= 𝜙=2𝜋 𝜃=𝜋
∫𝜙=0 𝑑𝜙 ∫𝜃=0 𝑠𝑖𝑛3 𝜃 𝑑𝜃

4𝜋
= 𝜃=𝜋
(2𝜋) ∫𝜃=0 𝑠𝑖𝑛3 𝜃 𝑑𝜃

𝟒𝝅 𝟑
𝑫 = = = 𝟏. 𝟓
𝟒 𝟐
(𝟐𝝅) ( )
𝟑

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EC8701

ANTENNAS AND MICROWAVE

ENGINEERING
Course Instructor: M.Lingeshwaran M.E.,(Ph.D.)

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UNIT II
RADIATION MECHANISMS AND DESIGN ASPECTS

Design considerations and applications

▪ Radiation Mechanisms of Linear Wire and Loop antennas
▪ Aperture antennas
▪ Reflector antennas
▪ Microstrip antennas
▪ Frequency independent antennas

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Loop Antenna
▪ Let us assume a loop antenna of radius 𝑎 which is large enough i.e., the
circumference is almost equal to one wavelength.
▪ The center of the large loop is at the origin of the spherical co-ordinate system.
▪ It is assumed that the current in the loop is uniform and in-phase.
▪ The far-field components can be derived from vector magnetic potential which is a
function of current distribution over the loop.
▪ As shown in the following figure, we assume two small current carrying elements
( or short dipoles) located diametrically opposite on the loop.
▪ If these dipoles are too short and makes an angle of 𝑑𝜙 at the center of the loop
then the length of each short dipole is given as 𝑎 𝑑𝜙.
▪ As the loop is lying in the 𝑥𝑦-plane only (current is confined to the loop) , hence,
the vector magnetic potential will have only 𝜙 component 𝐴𝜙 and other
components are zero 𝐴𝑟 = 𝐴𝜃 = 0 .
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Loop Antenna

Loop in spherical co-ordinate system

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Loop Antenna
▪ The observation point P is at distance 𝑟 from origin, hence, the infinitesimal
component of the vector magnetic potential in 𝜙 direction due to diametrically
opposite short dipoles is given as
𝜇 𝑑𝑀
𝑑𝐴𝜙 =
4𝜋𝑟
where 𝑑𝑀 is the current moment produced by short dipoles of length 𝑎 𝑑𝜙
▪ In 𝑥𝑧-plane or 𝜙 = 0° 𝑝𝑙𝑎𝑛𝑒 , the 𝜙 component of retarded current moment due
to one short dipole is defined as
𝑑𝑀1 = 𝐼 𝑎 𝑑𝜙 𝑐𝑜𝑠𝜙
where 𝐼 = 𝐼0 𝑒 𝑗𝜔 𝑡− 𝑟 Τ𝑐 and 𝐼0 is the peak current on the loop.

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Loop Antenna

Cross section of the loop in 𝑥𝑧-plane

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Loop Antenna
▪ The resultant current moment due to two short dipoles is
𝜓
𝑑𝑀 = 2 𝑗 𝐼 𝑎 𝑑𝜙 𝑐𝑜𝑠𝜙 𝑠𝑖𝑛
2
2𝜋
where 𝜓 = 2𝛽𝑎 𝑐𝑜𝑠𝜙 𝑠𝑖𝑛𝜃 and 𝛽 =
𝜆
▪ By substituting the value of 𝜓 in above expression, the resultant current moment
comes out as
𝑑𝑀 = 2 𝑗 𝐼 𝑎 𝑐𝑜𝑠𝜙 sin 𝛽𝑎 𝑐𝑜𝑠𝜙 𝑠𝑖𝑛𝜃 𝑑𝜙
▪ This expression can be substituted in 𝑑𝐴𝜙 ,
𝜇 𝑑𝑀 𝜇 2 𝑗 𝐼 𝑎 𝑐𝑜𝑠𝜙 sin 𝛽𝑎 𝑐𝑜𝑠𝜙 𝑠𝑖𝑛𝜃 𝑑𝜙
𝑑𝐴𝜙 = =
4𝜋𝑟 4𝜋𝑟
𝑗𝜇 𝐼 𝑎
𝑑𝐴𝜙 = 𝑐𝑜𝑠𝜙 sin 𝛽𝑎 𝑐𝑜𝑠𝜙 𝑠𝑖𝑛𝜃 𝑑𝜙
2𝜋𝑟

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Loop Antenna
▪ Integrating 𝑑𝐴𝜙 yields,
𝑗𝜇 𝐼 𝑎 𝜙=𝜋
𝐴𝜙 = න sin 𝛽𝑎 𝑠𝑖𝑛𝜃 𝑐𝑜𝑠𝜙 𝑐𝑜𝑠𝜙 𝑑𝜙
2𝜋𝑟 𝜙=0

𝒋𝝁 𝑰 𝒂
⇒ 𝑨𝝓 = 𝑱𝟏 𝜷𝒂 𝒔𝒊𝒏𝜽
𝟐𝒓
where 𝐽1 is a Bessel function of the first order and of argument 𝛽𝑎 𝑠𝑖𝑛𝜃 .
▪ Note
1 𝜙=𝜋
𝐽1 𝛽𝑎 𝑠𝑖𝑛𝜃 = න sin 𝛽𝑎 𝑠𝑖𝑛𝜃 𝑐𝑜𝑠𝜙 𝑐𝑜𝑠𝜙 𝑑𝜙
𝜋 𝜙=0

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Loop Antenna
▪ The far electric field of the loop has only 𝜙 component given by
𝐸𝜙 = −𝑗𝜔𝐴𝜙
▪ Substituting 𝐴𝜙 in above expression yields
𝑗𝜇 𝐼 𝑎 𝝁𝝎 𝑰 𝒂
𝐸𝜙 = −𝑗𝜔𝐴𝜙 = −𝑗𝜔 𝐽1 𝛽𝑎 𝑠𝑖𝑛𝜃 ⟹ 𝑬𝝓 = 𝑱𝟏 𝜷𝒂 𝒔𝒊𝒏𝜽
2𝑟 𝟐𝒓
2𝜋
Since 𝜔 = 2𝜋𝑓, 𝑐 = 𝑓𝜆 and 𝛽 =
𝜆
𝜇 2𝜋𝑓 𝐼 𝑎 𝜆 2𝜋 𝐼𝑎
𝐸𝜙 = × × 𝐽1 𝛽𝑎 𝑠𝑖𝑛𝜃 = × 𝑓𝜆 × 𝜇 × 𝐽1 𝛽𝑎 𝑠𝑖𝑛𝜃
2𝑟 𝜆 𝜆 2𝑟
𝐼𝑎
𝐸𝜙 = 𝛽𝑐𝜇 𝐽1 𝛽𝑎 𝑠𝑖𝑛𝜃
2𝑟

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Loop Antenna
𝐼𝑎
𝐸𝜙 = 𝛽𝑐𝜇 𝐽1 𝛽𝑎 𝑠𝑖𝑛𝜃
2𝑟
▪ Substituting the values of 𝑐 and 𝜇 in above expression yields
8 −7
𝛽𝑎 𝐼
𝐸𝜙 = 3 × 10 × 4𝜋 × 10 𝐽1 𝛽𝑎 𝑠𝑖𝑛𝜃
2𝑟
𝟔𝟎𝝅𝜷𝒂 𝑰
𝑬𝝓 = 𝑱𝟏 𝜷𝒂 𝒔𝒊𝒏𝜽
𝒓
▪ This expression gives the instantaneous electric field at a distance 𝑟 from a loop of
any radius 𝑎.
▪ The magnetic field 𝐻𝜃 at a large distance is related to 𝐸𝜙 by intrinsic impedance
of the free space 𝜂. Thus,
𝑬𝝓 𝑬𝝓 𝜷𝒂 𝑰
𝑯𝜽 = = = 𝑱𝟏 𝜷𝒂 𝒔𝒊𝒏𝜽
𝜼 𝟏𝟐𝟎 𝝅 𝟐𝒓
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Far-field Patterns of Circular Loop Antenna
▪ The far-field patterns for a loop of any size is given by
60𝜋𝛽𝑎 𝐼 𝛽𝑎 𝐼
𝐸𝜙 = 𝐽1 𝛽𝑎 𝑠𝑖𝑛𝜃 and 𝐻𝜃 = 𝐽1 𝛽𝑎 𝑠𝑖𝑛𝜃
𝑟 2𝑟
▪ For a loop of a given size, 𝛽𝑎 is constant and shape of the far-field pattern is given
as a function of 𝜃 by
𝑱𝟏 𝑪𝝀 𝐬𝐢𝐧 𝜽
where 𝐶𝜆 is the circumference of the loop in wavelengths. That is,
2𝜋
𝐶𝜆 = 𝑎 = 𝛽𝑎
𝜆
▪ 0 ≤ sin 𝜃 ≤ 1
▪ When 𝜃 = 90° , the relative field is 𝐽1 𝐶𝜆
▪ As 𝜃 decreases to zero, the values of the relative field vary in accordance with the
𝐽1 curve from 𝐽1 𝐶𝜆 to zero.
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Far-field Patterns of Circular Loop Antenna

Rectified first-order Bessel curve for patterns of loops

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Far-field Patterns of Circular Loop Antenna

Far-field patterns of loops of 0.1, 1, 1.5, 5 𝑎𝑛𝑑 8𝜆 diameter.

Uniform in-phase current is assumed on the loops
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Small Loop as a Special Case
▪ The far-field patterns for a loop of any size is given by
60𝜋𝛽𝑎 𝐼
𝐸𝜙 = 𝐽1 𝛽𝑎 𝑠𝑖𝑛𝜃
𝑟
𝛽𝑎 𝐼
𝐻𝜃 = 𝐽1 𝛽𝑎 𝑠𝑖𝑛𝜃
2𝑟

▪ For small arguments of the first-order Bessel function, the following approximate
relation can be used
𝑥
𝐽1 𝑥 =
2
1
▪ When 𝑥 = , the approximation is about 1% in error.
3
▪ The relation becomes exact as 𝑥 approaches to zero.

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Small Loop as a Special Case
𝜆
▪ Thus, if the perimeter of the loop is 𝜆Τ3 or less 𝐶𝜆 < , far-field equations for a
3
small loop is
2
2𝜋
60𝜋𝛽𝑎 𝐼 𝛽𝑎 𝑠𝑖𝑛𝜃 60𝜋𝛽2 𝑎2 𝐼 60 𝜋𝑎2 𝐼
𝐸𝜙 = = 𝑠𝑖𝑛𝜃 = 𝜆 𝑠𝑖𝑛𝜃
𝑟 2 2𝑟 2𝑟
𝟏𝟐𝟎 𝝅𝟐 𝑰 𝐬𝐢𝐧 𝜽 𝑨
𝑬𝝓 =
𝒓 𝝀𝟐
2
2𝜋
𝛽𝑎 𝐼 𝛽𝑎 𝑠𝑖𝑛𝜃 2 2
𝛽 𝑎 𝐼 𝑠𝑖𝑛𝜃 𝑎2 𝐼 𝑠𝑖𝑛𝜃
𝐻𝜃 = = = 𝜆
2𝑟 2 4𝑟 4𝑟
𝝅 𝑰 𝒔𝒊𝒏𝜽 𝑨
𝑯𝜽 =
𝒓 𝝀𝟐
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Aperture Antennas
▪ The term aperture refers to an opening in an otherwise closed surface.
▪ As applied to antennas, aperture antennas represent a class of antennas that are
generally analyzed considering the antenna as an opening in a surface.
▪ Typical antennas that fall in this category are the slot, horn, reflector, and lens
antennas.
▪ Aperture antennas are most common at microwave frequencies.
▪ Aperture antennas are very practical for space applications, because they can be
flush mounted on the surface of the spacecraft or aircraft.
▪ Their opening can be covered with a dielectric material to protect them from
environmental conditions. This type of mounting does not disturb the aerodynamic
profile of the craft, which in high-speed applications is critical.

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Wire Antennas Vs Aperture Antennas
▪ The radiation characteristics of wire antennas can be determined once the current
distribution on the wire is known.
▪ For many configurations, however, the current distribution is not known exactly
and only physical intuition or experimental measurements can provide a
reasonable approximation to it. This is even more evident in aperture antennas
(slits, slots, waveguides, horns, reflectors, lenses).
▪ The central idea used in the analysis of aperture type antennas is the conversion of
the original antenna geometry into an equivalent geometry which can be looked at
as radiation through an aperture in a closed surface.
▪ In case of wire antennas, the fields of an antenna are expressed in terms of the
vector potential which, in turn, is an integral over the current distribution on the
antenna surface. The major difficulty in computing the fields of an aperture type
antenna is the integration over a complex surface of the antenna.

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Wire Antennas Vs Aperture Antennas
▪ This issue is somewhat simplified by converting the antenna into an aperture
problem, using the field equivalence principle.

▪ The aperture geometry is conveniently chosen as some regular surface so that the
integration can be carried out with much less effort.

▪ All that is needed is the knowledge of the tangential E or H fields in the aperture
to compute the far-fields of the antenna

▪ Obviously, some approximation is involved in determining the tangential fields in

the aperture, but in general, the computed far-fields are fairly accurate for all
practical purposes, if sufficient care is taken in arriving at this approximation.

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Field Equivalence Principle
▪ The field equivalence is a principle by which actual sources, such as an antenna
and transmitter, are replaced by equivalent sources. The fictitious sources are said
to be equivalent within a region because they produce the same fields within that
region.

▪ FEP is based on
▪ Huygens’ Principle
▪ Uniqueness Theorem

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Huygens’ Principle
▪ “Each point on a primary wavefront can be considered to be a new source of a
secondary spherical wave and that a secondary wave front can be constructed as
the envelope of these secondary waves”.

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Uniqueness Theorem
▪ “A field in a lossy region is uniquely specified by the sources within the region
plus the tangential components of the electric field over the boundary, or the
tangential components of the magnetic field over the boundary, or the former over
part of the boundary and the latter over the rest of the boundary”.
▪ “For a given set of sources and boundary conditions in a lossy medium, the
solution to Maxwell’s equations is unique”.
▪ Consider a source-free volume V in an isotropic homogeneous medium bounded
by a surface S, and let (E1, H1) be the fields inside it produced by a set of sources
external to the volume.
▪ Now, let (E2, H2) be another possible set of fields in the volume V.
▪ It can be shown that if either the tangential E or the tangential H is the same on the
surface S for the two sets of solutions, the fields are identical everywhere in the
volume. This is known as the uniqueness theorem.

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Uniqueness Theorem
▪ It is important to note that it is sufficient to equate either the tangential E or the
tangential H on S for the solution to be unique.

▪ In other words, in a source-free region the fields are completely determined by the
tangential E or the tangential H on the bounding surface.

▪ Although the uniqueness theorem is derived for a dissipative medium, one can
prove the theorem for a lossless medium by a limiting process as loss tends to
zero.

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Love’s Field Equivalence Principle
▪ Consider a set of current sources in a homogeneous isotropic medium producing
electromagnetic fields E and H everywhere.
▪ Enclose all the sources by a closed surface S, separating the entire space into two
parts, volume V1containing the sources and the volume V2 being source-free.
▪ Let the surface S be chosen such that it is also source-free.
▪ Let 𝒏
ෝ be a unit normal to the surface drawn from V1 into V2.

Fields and Sources

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Love’s Field Equivalence Principle
▪ According to the field equivalence principle, the fields in V2 due to the sources in
volume V1 can also be generated by an equivalent set of virtual sources on surface
S, given by
ෝ×𝐇
𝐉𝐒 = 𝐧
𝐌𝐒 = −ෝ 𝐧×𝐄=𝐄×𝐧 ෝ
where E and H are the fields on the surface S produced by the original set of sources
in volume V1.
▪ Further the set of virtual sources produce null fields everywhere in V1.
▪ Here MS represents the magnetic surface current density and JS the electric surface
current density.

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Love’s Field Equivalence Principle

Three forms of the field equivalence principle: (a) surface current densities JS and
MS on the surface S, (b) surface current density MS alone on the surface S, which
is a conducting surface, and (c) surface current density JS alone on the surface S,
which is a magnetic conductor surface
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Love’s Field Equivalence Principle
▪ The proof of this principle makes use of the uniqueness theorem.
▪ Consider a situation where the fields in volume V2 are the same as before, (E,H),
but we delete all the sources in V1 and assume the fields to be identically zero
everywhere in V1.
▪ At the boundary surface, S, the fields are discontinuous and, hence, cannot be
supported unless we introduce sources on the discontinuity surface.
▪ Specifically, we introduce surface current sheets on S, such that 𝐉𝐒 = 𝐧
ෝ × 𝐇 and
𝐌𝐒 = 𝐄 × 𝐧 ෝ, so that the boundary conditions are satisfied.
▪ Since the tangential E and H satisfy the boundary conditions, it is a solution of
Maxwell’s equations, and from the uniqueness theorem, it is the only solution.
▪ Thus, the original sources in V1 and the new set of surface current sources
produce the same fields (E,H) in the volume V2. Therefore, these are equivalent
problems as far as the fields in the volume V2 are concerned.
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Love’s Field Equivalence Principle
▪ Let us assume that the medium inside S is replaced by a perfect electric conductor
(σ =∞). The introduction of the perfect electric conductor shorts out electric
current density (JS=0) and there exists only a magnetic current density MS as
shown in Fig.(b).

▪ Let us assume that instead of placing a perfect electric conductor within S, we

introduce a perfect magnetic conductor which will short out the magnetic current
density and reduce the equivalent problem to that shown in Fig.(c).

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Horn Antennas
▪ Microwave Antenna
▪ Used as a feed element for large radio astronomy, satellite tracking and
communication dishes.
▪ Used as a feed for reflectors and lenses
▪ It is a common element of phased arrays and serves as a universal standard for
calibration and gain measurements of other high-gain antennas.
▪ Its widespread applicability stems from its
▪ simplicity in construction
▪ ease of excitation
▪ versatility
▪ large gain and preferred overall performance.

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Horn Antennas
▪ A horn antenna may be regarded as a flared-out (or opened-out) waveguide.
▪ Function of the horn is to produce a uniform phase front with a large aperture than
that of the waveguide and hence greater directivity.
▪ Jagadis Chandra Bose constructed a pyramidal horn in 1897.
▪ Horn antennas (Rectangular/Circular) are energized from waveguides.
▪ To minimize the reflections of the guided wave, the transition region or horn
between the waveguide at the throat and free space at the aperture could be given a
gradual exponential taper as in fig(a) or fig(e).
▪ Assuming that the rectangular waveguide is energized with a TE10 mode wave
electric field (E in the y direction), the horn in fig(b) is flared out in a plane
perpendicular to E. This is the plane of the magnetic field H. Hence, this type of
horn is called a sectoral horn flared in H plane or simply an H-plane sectoral
horn.
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Rectangular Horn Antennas

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Horn Antennas
▪ The horn in fig(c) is flared out in the plane of the electric field E, and hence is
called an E-plane sectoral horn.
▪ A rectangular horn with flare in both planes, as in fig(d), is called a pyramidal
horn.
▪ Horn shown in fig(f) is a conical type.
▪ When excited with a circular waveguide carrying a TE11 mode wave, the electric
field distribution at the aperture is as shown by the arrows.
▪ Horn in fig(g) and fig(h) are biconical types.
▪ TEM Biconical antenna is excited in the TEM mode by a vertical radiator.
▪ TE01 Biconical antenna is excited in the TE01 mode by a small horizontal loop
antenna
▪ Biconical horns are non-directional in the horizontal plane.

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Circular Horn Antennas

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Horn Antennas – Design Principle
▪ The principle of equality of path length(Fermat’s principle) is applicable to the
horn design but with a different emphasis.
▪ Instead of requiring a constant phase across the horn mouth, the requirement is
relaxed to a one where the phase may deviate, but by less than a specified amount
𝛿, equal to the path length difference between a ray traveling along the side and
along the axis of the horn.
▪ Design parameters:
▪ E-plane : 𝜃𝐸 is the flare angle (deg) and 𝑎𝐸 is the aperture dimension (m)
▪ H-pane: 𝜃𝐻 is the flare angle (deg) and 𝑎𝐻 is the aperture dimension (m)
▪ L = Horn Length (m)
▪ 𝛿 = path length difference (m)

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Horn Antennas – Design Principle

(a) Pyramidal horn antenna

(b) Cross section with dimensions

used in analysis

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Horn Antennas – Design Equations
▪ From the figure,
𝜃 𝐿
cos =
2 𝐿+𝛿 𝐿+𝛿
𝜃 𝑎 Τ2 𝑎 𝑎 Τ2
sin = =
2 𝐿+𝛿 2 𝐿+𝛿
𝑎 𝜃 Τ2
𝜃 2 𝐿+𝛿 𝑎
tan = = 𝐿
2 𝐿 2𝐿
𝐿+𝛿
▪ From the geometry, we have also that
𝑎 2 𝑎 2 𝑎 2
𝐿 + 𝛿 2 = 𝐿2 + ⇒ 𝐿2 + 𝛿 2 + 2𝐿𝛿 = 𝐿2 + ⇒ 𝛿 2 + 2𝐿𝛿 =
2 4 4
𝒂𝟐 −𝟏
𝒂 −𝟏
𝑳
⇒𝑳= 𝛿 ≪ 𝐿 𝑎𝑛𝑑 𝜽 = 𝟐 𝐭𝐚𝐧 = 𝟐 𝐜𝐨𝐬
𝟖𝜹 𝟐𝑳 𝑳+𝜹
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Horn Antennas – Design Constraints
▪ In E-plane of horn, 𝛿 ≤ 0.25𝜆
▪ In H-plane, 𝛿 ≈ 0.4𝜆 (Since E goes to zero at the horn edges)
▪ To obtain as uniform an aperture distribution as possible, a very long horn with a
small flare angle is required. Practically, the horn should be as short as possible.
▪ If 𝛿 is a sufficiently small fraction of a wavelength, the field has nearly uniform
phase over the entire aperture.
▪ For a constant length L, the directivity of the horn increases (beamwidth
decreases) as the aperture 𝑎 and flare angle 𝜃 are increased.
▪ However, if the aperture and flare angle become so large that 𝛿 ≅ 180° , the field
at the edge of the aperture is in phase opposition to the field on the axis.
▪ For all but very large flare angles, the ratio 𝐿Τ 𝐿 + 𝛿 is so nearly unity that the
effect of the additional path length 𝛿 on the distribution of the field magnitude can
be neglected.
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Horn Antennas – Design Constraints
▪ However, when 𝛿 = 180° , the phase reversal at the edges of the aperture reduces
the directivity (increases sidelobes).
▪ It follows that the maximum directivity occurs at the largest flare angle for which
𝛿 does not exceed a certain value 𝛿0 .
▪ Thus, the optimum horn dimensions can be related by

𝐿
𝛿0 = − 𝐿 = 𝑜𝑝𝑡𝑖𝑚𝑢𝑚 𝛿
Τ
cos 𝜃 2

𝛿0 𝑐𝑜𝑠 𝜃Τ2
𝐿= = 𝑜𝑝𝑡𝑖𝑚𝑢𝑚 𝑙𝑒𝑛𝑔𝑡ℎ
1 − 𝑐𝑜𝑠 𝜃Τ2

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Rectangular Horn Antennas
▪ Referring to the following figure, the total flare angle in the E plane is 𝜃𝐸 and the
total flare angle in the H plane is 𝜃𝐻 . The axial length of the horn from throat to
aperture is 𝐿 and the radial length is 𝑅.

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Rectangular Horn Antennas

Measured E-plane and

H-plane field patterns of
rectangular horns as a
function of flare angle
and horn length

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Rectangular Horn Antennas
▪ Patterns measured by Donald Rhodes are shown in above figure.

▪ In (a), the patterns in the E plane and H plane are compared as a function of R.
Both sets are for a flare angle of 20° . The E-plane patterns have minor lobes
whereas the H-plane patterns have practically none.

▪ In (b), measured patterns for horns with 𝑅 = 8𝜆 are compared as a function of

flare angle. In the upper row E-plane patterns are given as a function of the E-
plane flare angle 𝜃𝐸 and in the lower row H-plane patterns are shown as a function
of the H-plane flare angle 𝜃𝐻 .

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Rectangular Horn Antennas
▪ For a flare angle 𝜃𝐸 = 50° , the E-plane pattern is split, whereas for 𝜃𝐻 = 50° , the
H-plane pattern is not.
▪ This is because a given phase shift at the aperture in the E-plane horn has more
effect on the pattern than the same phase shift in the H-plane horn.
▪ In the H-plane horn, the field goes to zero at the edges of the aperture, so the
phase near the edge is relatively less important.
▪ Accordingly, we should expect the value of 𝛿0 for the H plane to be larger than
for the E plane.
▪ From Rhodes's experimental patterns, optimum dimensions were selected for both
E- and H-plane flare as a function of flare angle and horn length L.
▪ These optimum dimensions are indicated by the solid lines in the following figure.
The corresponding half-power beam widths and apertures in wavelengths are also
indicated.
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Rectangular Horn Antennas

Experimentally determined optimum dimensions for rectangular horn antennas. Solid curves give relation of flare angle 𝜃𝐸 in
E plane and flare angle 𝜃𝐻 in H plane to horn length. The corresponding half-power beamwidths and apertures in wavelengths
are indicated along the curves. Dashed curves show calculated dimensions for 𝛿0 = 0.25𝜆 and 𝛿0 = 0.4𝜆.

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Rectangular Horn Antennas
▪ The dashed curves show the calculated dimensions for a path length 𝛿0 = 0.25𝜆
and 𝛿0 = 0.4𝜆.
▪ The value of 0.25𝜆 gives a curve close to the experimental curve for E-plane flare,
while the value of 0.4𝜆 gives a curve close to the experimental one for H-plane
flare over a considerable range of horn length.
▪ Thus, the tolerance in path length is greater for H-plane flare than for E-plane
flare, as indicated above.

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Rectangular Horn Antennas
▪ The directivity (or gain, assuming no loss) of a horn antenna can be expressed in
terms of its effective aperture. Thus,
4𝜋𝐴𝑒 4𝜋𝜀𝑎𝑝 𝐴𝑝
𝐷= 2 =
𝜆 𝜆2
where 𝐴𝑒 = effective aperture, 𝑚2 ; 𝐴𝑝 = physical aperture, 𝑚2 ; 𝜀𝑎𝑝 = aperture
efficiency; 𝜆 = wavelength, 𝑚
▪ For a rectangular horn 𝐴𝑝 = 𝑎𝐸 𝑎𝐻 and for a conical horn 𝐴𝑝 = 𝜋𝑟 2 , where
r = aperture radius. It is assumed that 𝑎𝐸 , 𝑎𝐻 or 𝑟 are all at least 1𝜆.
▪ Taking 𝜀𝑎𝑝 ≃ 0.6,
7.5 𝐴𝑝 7.5 𝐴𝑝
𝐷≃ 2
⇒ 𝐷 ≃ 10 log 𝑑𝐵𝑖 ⇒ 𝑫 ≃ 𝟏𝟎 𝐥𝐨𝐠 𝟕. 𝟓𝒂𝑬𝝀 𝒂𝑯𝝀
𝜆 𝜆2
where 𝑎𝐸𝜆 = E-plane aperture in 𝜆 ; 𝑎𝐻𝜆 = H-plane aperture in 𝜆

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Rectangular Horn Antennas - Example
(a) Determine the length 𝐿, H-plane aperture and flare angles 𝜃𝐸 and 𝜃𝐻 (in the E
and H planes respectively) of a pyramidal horn for which the E-plane aperture
𝑎𝐸 = 10𝜆. The horn is fed by a rectangular waveguide with 𝑇𝐸10 mode. Let
𝛿 = 0.2𝜆 in the E-plane and 0.375𝜆 in the H-plane. (b) What are the beamwidths?
(c) What is the directivity?
Solution:
Taking 𝛿 = 0.2𝜆 in the E-plane, the required horn length
𝑎2 10𝜆 2 100𝜆
𝐿= = = = 62.5𝜆
8𝛿 8 × 0.2𝜆 1.6
Flare angle in E-plane is
𝑎
−1 𝐸 −1
10𝜆 −1
10
𝜃𝐸 = 2 tan = 2 tan = 2 tan = 9.1°
2𝐿 2 × 62.5𝜆 125

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Rectangular Horn Antennas - Example
Taking 𝛿 = 0.375 𝜆 in the H-plane, the flare angle in the H-plane
−1
𝐿 −1
62.5𝜆 −1
62.5
𝜃𝐻 = 2 cos = 2 cos = 2 cos = 12.52°
𝐿+𝛿 62.5𝜆 + 0.375𝜆 62.875
H-plane aperture is
𝜃𝐻 12.52°
𝑎𝐻 = 2𝐿 tan = 2 × 62.5𝜆 tan = 125𝜆 × 0.109 = 13.7𝜆
2 2
56° 56°
𝐻𝑃𝐵𝑊 𝐸 − 𝑃𝑙𝑎𝑛𝑒 = = = 5.6°
𝑎𝐸𝜆 10
67° 67°
𝐻𝑃𝐵𝑊 𝐻 − 𝑃𝑙𝑎𝑛𝑒 = = = 4.9°
𝑎𝐻𝜆 13.7
𝐷 ≃ 10 log 7.5 𝑎𝐸𝜆 𝑎𝐻𝜆 = 10 log 7.5 × 10 × 13.7 = 30.1 𝑑𝐵𝑖

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Conical Horn Antennas
▪ Conical Horn can be directly excited from a circular waveguide.
▪ Dimensions can be determined from the following expressions by taking
𝛿0 = 0.32𝜆,
𝐿
𝛿0 = − 𝐿 = 𝑜𝑝𝑡𝑖𝑚𝑢𝑚 𝛿
cos 𝜃Τ2
𝛿0 𝑐𝑜𝑠 𝜃Τ2
𝐿= = 𝑜𝑝𝑡𝑖𝑚𝑢𝑚 𝑙𝑒𝑛𝑔𝑡ℎ
1 − 𝑐𝑜𝑠 𝜃Τ2

▪ For optimum conical horns,

𝐻𝑃𝐵𝑊 𝐸 − 𝑝𝑙𝑎𝑛𝑒 = 60Τ𝑎𝐸𝜆
𝐻𝑃𝐵𝑊 𝐻 − 𝑝𝑙𝑎𝑛𝑒 = 70Τ𝑎𝐻𝜆
These values are about 6% more than the values for a rectangular horn.

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Conical Horn Antennas
▪ Biconical horns have patterns that are non-directional in the horizontal plane (axis
of horns vertical).
▪ These horns may be regarded as modified pyramidal horns with a 360° flare angle
in the horizontal plane.
▪ The optimum vertical-plane flare angle is about the same as for a sectoral horn of
the same cross section excited in the same mode.

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Ridge Horns
▪ A central ridge loads a waveguide and increases its useful bandwidth by lowering
the cutoff frequency of the dominant mode.
▪ Rectangular guides with single and double ridges are shown in the following
figures(a) and (b)
▪ A very thin ridge or fin is also effective in producing the loading of a central ridge.
It may consist of a metal-clad ceramic sheet which facilitates the installation of
shunt circuit elements as suggested in figure(c).

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Ridge Horns
▪ Cutoff frequency can be lowered by placing dielectric material in the waveguide,
but this does not increase the bandwidth and it may increase losses.
▪ By continuing a double-ridge structure from a waveguide into a pyramidal horn as
suggested in the following figure, the useful bandwidth of the horn can be
increased manyfold.

Double-ridge or Vivaldi horn with coaxial feed.

The view at (a) is a cross section at the feed point.

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Septum Horns
▪ Although the electric field in the H plane of a pyramidal horn tends to zero at the
edges, resulting in a tapered distribution and reduced sidelobes, the electric field in
the E plane may be close to uniform in amplitude to the edges, resulting in
significant sidelobes.
▪ By introducing septum plates bonded to the horn walls, a stepped-amplitude
distribution can be achieved in the E plane with a reduction in E-plane sidelobes.
▪ Typically, the first sidelobes of a uniform amplitude distribution are down about
13 dB.
▪ A cosine field distribution is approximated with a 1:2:1 stepped amplitude
distribution with apertures also in the ratio 1:2:1 as suggested in the following
figure. To achieve this distribution, the septums must be appropriately space at the
throat of the horn.

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Septum Horns

Two-septum horn with 1:2:1 stepped amplitude distribution in field intensity at

mouth of horn
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Corrugated Horns
▪ Corrugated horns can provide reduced edge diffraction, improved pattern
symmetry and reduced cross-polarization (less E field in the H plane).
▪ Corrugations on the horn walls acting as 𝜆Τ4 chokes are used to reduce E to very
low values at all horn edges for all polarizations. These prevent waves from
diffracting around the edges of the horn (or surface currents flowing around the
edge and over the outside).
▪ The reactance at open end of corrugation is
2𝜋𝑑
𝑋 ≃ 377 tan Ω
𝜆
▪ Corrugations with depth 𝑑 = 𝜆Τ2 act as conducting surface 𝑋 = 0
▪ Corrugations with depth 𝑑 = 𝜆Τ4 present a high impedance 𝑋 = ∞

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Corrugated Horn

Cross section of circular waveguide-fed corrugated horn with corrugated transition

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Aperture-Matched Horn
▪ By attaching a smooth curved (or
rolled) surface section to the
outside of the aperture edge of a
horn, a significant improvement in
the pattern, impedance and
bandwidth characteristics can be
achieved.
▪ This arrangement is an attractive
alternative to a corrugated horn.
▪ The shape of the rolled edge is not
critical but its radius of curvature
should be at least 𝜆Τ4.

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Slot Antennas
▪ Slot antennas are useful in many applications, especially where low-profile or
flush installations are required as, for example, on high-speed aircraft.

▪ The antenna shown in above figure, consisting of two resonant 𝜆Τ4 stubs
connected to a 2-wire transmission line, forms an inefficient radiator. The long
wires are closely spaced (𝑤 ≪ 𝜆) and carry currents of opposite phase so that
their fields tend to cancel. The end wires carry currents in the same phase, but they
are too short to radiate efficiently. Hence, enormous currents may be required to
radiate appreciable amounts of power.
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Slot Antennas
▪ The antenna shown in the following figure is a very efficient radiator. In this
arrangement a 𝜆Τ2 slot is cut in a flat metal sheet. Although the width of the slot is
small (𝑤 ≪ 𝜆), the currents are not confined to the edges of the slot but spread out
over the sheet. This is a simple type of slot antenna. Radiation occurs equally from
both sides of the sheet. If the slot is horizontal, as shown, the radiation normal to
the sheet is vertically polarized.

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Slot Antennas - Methods of Feeding
▪ A slot antenna may be conveniently energized with a coaxial transmission line as
in the following figure(a).
▪ The outer conductor of the cable is bonded to the metal sheet. Since the terminal
resistance at the center of a resonant 𝜆Τ2 slot in a large sheet is about 500Ω and
the characteristic impedance of coaxial transmission lines is usually much less, an
offset feed as shown in figure(b) may be used to provide a better impedance
match.
▪ For a 50Ω coaxial cable, the distance 𝑠 ≈ 𝜆Τ20.

Slot Orientation Horizontal Slot Vertical Slot

as shown in figure (c) as shown in figure (d)
Wave Polarization Vertical Polarization Horizontal Polarization

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Slot antennas fed by coaxial transmission lines

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Boxed-in Slot Antenna
▪ A flat sheet with a 𝜆Τ2 slot radiates equally on both sides of the sheet. However, if
the sheet is very large (ideally infinite) and boxed in as shown in the following
figure, radiation occurs only from one side.
▪ If the depth 𝑑 of the box is appropriate 𝑑~ 𝜆Τ4 𝑓𝑜𝑟 𝑎 𝑡ℎ𝑖𝑛 𝑠𝑙𝑜𝑡 , no appreciable
shunt susceptance appears across the terminals. With such a zero susceptance box,
the terminal impedance of the resonant 𝜆Τ2 slot is nonreactive and approximately
twice its value without the box, or about 1000Ω.

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Boxed-in Slot Antenna as Flush Radiator
▪ The boxed-in slot antenna might be applied even at relatively long wavelengths by
using the ground as the flat conducting sheet and excavating a trench 𝜆Τ2 long by
𝜆Τ4 deep as shown in the following figure.

▪ Radiation is maximum in all directions at right angles to the slot and is zero along
the ground in the directions of the ends of the slot. The radiation along the ground
is vertically polarized.
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Waveguide-fed Slot
▪ Radiation from only one side of a large flat sheet may also be achieved by a slot
fed with a waveguide as shown in the figure. With the transmission in the guide in
the TE10 mode, the direction of the electric field E is as shown.
▪ The width L of the guide must be more than 𝜆Τ2 to transmit energy, but it should
be less than 1𝜆 to suppress higher-order transmission modes. With the horizontal
slot, the radiation normal to sheet is vertically polarized.

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Broadside Array of Slots in a Waveguide
▪ An array of slots may be cut in the waveguide as shown in the figure so as to
produce a directional radiation pattern.
▪ By cutting inclined slots as shown at intervals of 𝜆𝑔 Τ2 (where 𝜆𝑔 is the guide
wavelength), the slots are energized in phase and produce a directional pattern
with maximum radiation broadside to the guide.

▪ If the guide is horizontal and E inside the guide is vertical, the radiated field is
horizontally polarized.
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Slot Antenna Vs Complementary Dipole

A 𝜆Τ2 slot in an infinite sheet (a) and a complementary 𝜆Τ2 dipole antenna (b)

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Slot Antenna Vs Complementary Dipole

Radiation-field patterns of slot in an infinite sheet (a) and of complementary dipole

antenna (b). The patterns have the same shape but with E and H interchanged.
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Babinet’s Principle
▪ Babinet’s principle in optics states that when the field behind a screen with an
opening is added to the field of a complementary structure, the sum is equal to the
field when there is no screen .
▪ Let a source and 2 imaginary planes, plane of screens A and plane of observation
B, be arranged as in the following figure.
▪ Case 1: Let a perfectly absorbing screen be placed in plane A. Then in plane B
there is a region of shadow as indicated. Let the field behind this screen be some
function 𝑓1 of 𝑥, 𝑦 and 𝑧. Thus,
𝐹𝑆 = 𝑓1 𝑥, 𝑦, 𝑧
▪ Case 2: Let the first screen be replaced by its complementary screen and the field
behind it be given by
𝐹𝐶𝑆 = 𝑓2 𝑥, 𝑦, 𝑧

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Babinet’s Principle
▪ Case 3: With no screen, the field is
𝐹0 = 𝑓3 𝑥, 𝑦, 𝑧
▪ Babinet’s principle asserts that at the same point 𝑥, 𝑦, 𝑧,
𝐹𝑆 + 𝐹𝐶𝑆 = 𝐹0

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Babinet’s Principle

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Babinet’s Principle

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Babinet’s Principle

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Booker’s Extension of Babinet’s Principle
▪ Babinet’s principle has been extended and generalized by Booker to take into
account the vector nature of the electromagnetic field.
▪ In this extension, it is assumed that the screen is plane, perfectly conducting and
infinitesimally thin.
▪ Furthermore, if one screen is perfectly conducting 𝜎 = ∞ , the complementary
screen must have infinite permeability 𝜇 = ∞ .
▪ Thus, if one screen is a perfect conductor of electricity, the complementary screen
is a perfect conductor of magnetism.
▪ No infinitely permeable material exists, but the equivalent effect may be obtained
by making both the original and complementary screens of perfectly conducting
material and interchanging electric and magnetic quantities everywhere.

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Booker’s Extension of Babinet’s Principle
▪ Case 1: The dipole is horizontal and the original screen is an infinite, perfectly
conducting, plane, infinitesimally thin sheet with a vertically cut out slot as
indicated. At a point P behind the screen the field is 𝐸1 .
▪ Case 2: The original screen is replaced by the complementary screen consisting of
a perfectly conducting, plane, infinitesimally thin strip of the same dimensions as
the slot in the original screen. In addition, the dipole source is turned vertical so as
to interchange E and H. At the same point P behind the screen the field is 𝐸2 .
▪ Case 2 Alternative: The dipole source is horizontal and the strip is also turned
horizontal.
▪ Case 3: No screen is present and the field at point P is 𝐸0 . Then, by Babinet’s
principle,
𝐸1 𝐸2
𝐸1 + 𝐸2 = 𝐸0 ⟹ + =1
𝐸0 𝐸0
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Booker’s Extension of Babinet’s Principle

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Impedance of Slot Antenna
▪ Knowing the impedance 𝑍𝑑 of the complementary dipole antenna, the impedance
𝑍𝑠 of the slot antenna can be determined.
▪ Consider the slot antenna and the complementary dipole antenna as shown in the
following figure. The terminals of each antenna are indicated by FF, and it is
assumed that they are separated by an infinitesimal distance. It is assumed that the
dipole and slot are cut from an infinitesimally thin, plane, perfectly conducting
sheet.
▪ Let a generator be connected to the terminals of the slot antenna. The driving point
impedance 𝑍𝑠 at the terminals is the ratio of the terminal voltage 𝑉𝑠 to the terminal
current 𝐼𝑠 .
▪ Let 𝐸𝑠 and 𝐻𝑠 be the electric and magnetic fields of the slot at any point P.

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Impedance of Slot Antenna

Slot antenna (a) and complementary dipole antenna (b)

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Impedance of Slot Antenna
▪ The voltage 𝑉𝑠 at the terminals FF of the slot is given by the line integral of 𝐸𝑠
over the path 𝐶1 as 𝐶1 approaches zero. Thus,

𝑉𝑠 = lim න 𝐸𝑠 ∙ 𝑑𝑙 (1)
𝐶1 → 0 𝐶
1

where 𝑑𝑙 = infinitesimal vector element of length 𝑑𝑙 along the contour or path 𝐶1 .

▪ The current 𝐼𝑠 at the terminals of the slot is

𝐼𝑠 = 2 lim න 𝐻𝑠 ∙ 𝑑𝑙 (2)
𝐶2 → 0 𝐶
2

▪ The path 𝐶2 is just outside the metal sheet and parallel to its surface. The factor 2
1
enters because only of the closed line integral is taken, the line integral over the
2
other side of the sheet being equal by symmetry.

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Impedance of Slot Antenna
▪ Let a generator be connected to the terminals of the dipole. The driving-point
impedance 𝑍𝑑 at the terminals is the ratio of the terminal voltage 𝑉𝑑 to the
terminal current 𝐼𝑑 . Let 𝐸𝑑 and 𝐻𝑑 be the electric and magnetic fields of the dipole
at any point P. Then the voltage at the dipole terminals is

𝑉𝑑 = lim න 𝐸𝑑 ∙ 𝑑𝑙 (3)
𝐶2 → 0 𝐶
2

▪ Current at the dipole terminals is

𝐼𝑑 = 2 lim න 𝐻𝑑 ∙ 𝑑𝑙 (4)
𝐶1 → 0 𝐶
1

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Impedance of Slot Antenna
▪ However,

lim න 𝐸𝑑 ∙ 𝑑𝑙 = 𝑍0 lim න 𝐻𝑠 ∙ 𝑑𝑙 5
𝐶2 → 0 𝐶 𝐶2 → 0 𝐶
2 2

1
lim න 𝐻𝑑 ∙ 𝑑𝑙 = lim න 𝐸𝑠 ∙ 𝑑𝑙 (6)
𝐶1 → 0 𝐶
1
𝑍0 𝐶1→ 0 𝐶1
where 𝑍0 is the intrinsic impedance of the surrounding medium.
▪ Substituting (3) and (2) in (5) yields,
𝑍0
𝑉𝑑 = 𝐼𝑠 7
2
▪ Substituting (4) and (1) in (6) gives,
𝑍0
𝑉𝑠 = 𝐼𝑑 8
2
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Impedance of Slot Antenna
▪ Multiplying (7) and (8) we have
𝑉𝑠 𝑉𝑑 𝑍02
= 9
𝐼𝑠 𝐼𝑑 4
𝑍02 𝑍02
⟹ 𝑍𝑠 𝑍𝑑 = ⟹ 𝑍𝑠 =
4 4𝑍𝑑
▪ For free space 𝑍0 = 376.7 Ω,
𝑍02 35476
𝑍𝑠 = = Ω
4𝑍𝑑 𝑍𝑑

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Impedance of Slot Antenna-Example Calculation
The impedance of an infinitesimally thin 𝜆Τ2 antenna 𝐿 = 0.5𝜆, 𝐿Τ𝐷 = ∞ is
73 + 𝑗42.5 Ω. Calculate the terminal impedance of an infinitesimally thin 𝜆Τ2 slot
antenna.
Given:
𝑍𝑑 = 73 + 𝑗 42.5 Ω
Solution:
𝑍02 35476 35476
𝑍𝑠 = = = = 363 − 𝑗 211 Ω
4𝑍𝑑 𝑍𝑑 73 + 𝑗 42.5

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Reflector Antennas
▪ Reflectors are widely used to modify the radiation pattern of a radiating element.
▪ For example, the backward radiation from an antenna may be eliminated with a
plane sheet reflector of large enough dimensions.
▪ In the more general case, a beam of pre-determined characteristics may be
produced by means of a large, suitably shaped, and illuminated reflector surface.
▪ Reflector antenna is basically a combination of feed and reflecting surface to
achieve high directivity at microwave frequencies.
▪ The feed is known as primary antenna and it can be slot, dipole or horn antenna.
▪ The reflector is also known as secondary antenna and it is a metallic plate which
may be curved or flat and used to direct the incident energy in a specific direction
to achieve high directivity
▪ Applications: Radars, radio astronomy, microwave communication, and satellite
tracking
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Reflectors of various shapes

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Reflectors of various shapes
▪ Figure(a) has a large, flat sheet reflector near a linear dipole antenna to reduce the
backward radiation.
▪ With small spacings between the antenna and sheet, this arrangement also yields a
substantial gain in the forward radiation.
▪ The desirable properties of the sheet reflector may be largely preserved with the
reflector reduced in size as in figure(b) and even in the limiting case of figure(c).
▪ Here the sheet has degenerated into a thin reflector element.
▪ Whereas the properties of the large sheet are relatively insensitive to small
frequency changes, the thin reflector element is highly sensitive to frequency
changes.

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Reflectors of various shapes

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Reflectors of various shapes
▪ With two flat sheets intersecting at an angle 𝛼 (< 180° ) as in figure(d), a sharper
radiation pattern than from a flat sheet reflector (𝛼 = 180° ) can be obtained. This
arrangement, called an active corner reflector antenna, is most practical where
apertures of 1 or 2𝜆 are of convenient size.
▪ A corner reflector without an exciting antenna can be used as a passive reflector
or target or radar waves. In this application the aperture may be many
wavelengths, and the corner angle is always 90° . Reflectors with this angle have
the property that an incident wave is reflected back toward its source as in
figure(e), the corner acting as a retroreflector.
▪ Parabolic reflectors can be used to provide highly directional antennas as shown in
figure(f). The parabola reflects the waves originating from a source at the focus
into a parallel beam, the parabola transforming the curved wave front from the
feed antenna at the focus into a plane wave front.

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Reflectors of various shapes

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Reflectors of various shapes
▪ Many other shapes of reflectors can be employed for special applications.
▪ For instance, with an antenna at one focus, the elliptical reflector as shown in
figure (g) produces a diverging beam with all reflected waves passing through the
second focus of the ellipse.
▪ Examples of reflectors of other shapes are the hyperbolic and the circular
reflectors as shown in figure(h) and (i).

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Parabolic Reflector
▪ The surface generated by the revolution of a parabola around its axis is called a
paraboloid or a parabola of revolution.
▪ If an isotropic source is placed at the focus of a paraboloidal reflector as in
figure(a), the portion A of the source radiation that is intercepted by the paraboloid
is reflected as a plane wave of circular cross section provided that the reflector
surface deviates from a true parabolic surface by no more than a small fraction of
a wavelength.
▪ If the distance L between the focus and vertex of the paraboloid is an even number
of 𝜆Τ4, the direct radiation in the axial direction from the source will be in
opposite phase and will tend to cancel the central region of the reflected wave.
▪ However, if 𝐿 = 𝑛𝜆Τ4 where 𝑛 = 1,3,5, …, the direct radiation in the axial
direction from the source will be in the same phase and will tend to reinforce the
central region of the reflected wave.

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Parabolic Reflector

Parabolic reflectors of different focal lengths (L) with sources of different patterns

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Parabolic Reflector
▪ Direct radiation from the source can be eliminated by means of a directional
source or primary antenna as in figure(b) and (c).
▪ A primary antenna with the idealized hemispherical pattern is shown in figure(b)
results in a wave of uniform phase over the reflector aperture. However, the
amplitude is tapered as indicated.
▪ To obtain a more uniform aperture field distribution or illumination, it is necessary
to make 𝜃1 small, as suggested in figure(c), by increasing the focal length L while
keeping the reflector diameter D constant.
▪ A typical pattern for a directional source as indicated by the dashed curve at (c)
(left) gives a more tapered aperture distribution as shown by the dashed curve at
(c) (right). The greater amount of taper with resultant reduction in edge
illumination may be desirable in order to reduce the minor-lobe level, this being
achieved, however, at some sacrifice in directivity.

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Parabolic Reflector

Parabolic reflectors of different focal lengths (L) with sources of different patterns

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Parabolic Reflector

Parabolic reflectors of different focal lengths (L) with sources of different patterns

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Parabolic Reflector-(f/d) ratio
▪ The surface of a paraboloidal reflector is formed by rotating a parabola about its
axis. Its surface must be a paraboloid of revolution so that rays emanating from
the focus of the reflector are transformed into plane waves. The design is based on
optical techniques, and it does not take into account any deformations
(diffractions) from the rim of the reflector.
▪ Referring to the following figure and choosing a plane perpendicular to the axis of
the reflector through the focus, it follows that
𝑂𝑃 + 𝑃𝑄 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 = 2𝑓 1
▪ From the figure, P Q
𝑂𝑃 = 𝑟 ′ ′
𝑃𝑄 𝑃𝑄 𝜃
′
cos 𝜃 = = ′ ⟹ 𝑃𝑄 = 𝑟 ′ cos 𝜃 ′
𝑂𝑃 𝑟 𝑟′

O 93
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Parabolic Reflector-(f/d) ratio

Two-dimensional configuration of a paraboloidal reflector

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Parabolic Reflector-(f/d) ratio
▪ Eq.(1) can be written as
2𝑓 𝜃 ′
𝑟 ′ 1 + cos 𝜃 ′ = 2𝑓 ⟹ 𝑟 ′ = = 𝑓 𝑠𝑒𝑐 2 𝜃 ≤ 𝜃0
1 + cos 𝜃 ′ 2
▪ Since a paraboloid is a parabola of revolution (about its axis), above equation is
also the equation of a paraboloid in terms of the spherical coordinates 𝑟 ′ , 𝜃 ′ , 𝜙 ′ .
Because of its rotational symmetry, there are no variations with respect to 𝜙 ′ .
Above equation can also be written in terms of the rectangular coordinates
𝑥 ′ , 𝑦 ′ , 𝑧 ′ . That is,
𝑟 ′ + 𝑟 ′ cos 𝜃 ′ = 𝑥′ 2 + 𝑦′ 2 + 𝑧′ 2 + 𝑧 ′ = 2𝑓

⟹ 𝑥′ 2 + 𝑦′ 2 + 𝑧′ 2 = 2𝑓 − 𝑧 ′

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Parabolic Reflector-(f/d) ratio
⟹ 𝑥 ′ 2 + 𝑦 ′ 2 + 𝑧 ′ 2 = 2𝑓 − 𝑧 ′
▪ Squaring on both sides,
𝑥 ′ 2 + 𝑦 ′ 2 + 𝑧 ′ 2 = 2𝑓 − 𝑧 ′ 2
⟹ 𝑥 ′ 2 + 𝑦 ′ 2 + 𝑧 ′ 2 = 4𝑓 2 + 𝑧 ′ 2 − 4𝑓𝑧 ′
𝑥 ′ 2 + 𝑦 ′ 2 = 4𝑓 𝑓 − 𝑧 ′ with 𝑥 ′ 2 + 𝑦 ′ 2 ≤ 𝑑Τ2 2
▪ Another expression that is usually very prominent in the analysis of reflectors is
that relating the subtended angle 𝜃0 to the 𝑓Τ𝑑 ratio. From the geometry of
previous figure,
−1
𝑑 Τ2
𝜃0 = tan
𝑧0
where 𝑧0 is the distance along the axis of the reflector from the focal point to the
edge of the rim.

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Parabolic Reflector-(f/d) ratio
▪ We can similarly have for point R 𝑥0 , 𝑦0 , 𝑧0 as
𝑥0 2
+ 𝑦0 2
2 2
𝑥0 + 𝑦0 = 4𝑓 𝑓 − 𝑧0 ⟹ = 𝑓 − 𝑧0
4𝑓
𝑥0 2 + 𝑦0 2 𝑑 Τ2 2 𝑑2
𝑧0 = 𝑓 − ⟹ 𝑧0 = 𝑓 − ⟹ 𝑧0 = 𝑓 −
4𝑓 4𝑓 16𝑓
𝑑 Τ2
▪ Substituting above expression in 𝜃0 = tan−1 reduces it to
𝑧0
𝑑
−1
𝑑 Τ2 2
𝜃0 = tan = tan−1
𝑧0 𝑑2
𝑓−
16𝑓
𝑑
⟹ 𝜃0 = tan−1 2
𝑑2 𝑓 2 1
−
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𝑓 𝑑 STUCOR APP 97
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Parabolic Reflector-(f/d) ratio
𝑑 1 𝑓
2 2 𝑑
𝜃0 = tan−1 ⟹ 𝜃0 = tan−1
𝑑2 𝑓 2
1 𝑓 2 1
− −
𝑓 𝑑 16 𝑑 16
▪ It can also be shown that another form of above expression is
𝑑 𝜃0
𝑓 = cot
4 2

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Feed systems for Parabolic Reflectors
Axial/Front Feed:
▪ The overall radiation characteristics (antenna pattern, antenna efficiency,
polarization discrimination, etc.) of a reflector can be improved if the structural
configuration of its surface is upgraded.
▪ It has been shown by geometrical optics that if a beam of parallel rays is incident
upon a reflector whose geometrical shape is a parabola, the radiation will converge
(focus) at a spot which is known as the focal point.
▪ In the same manner, if a point source is placed at the focal point, the rays
reflected by a parabolic reflector will emerge as a parallel beam.
▪ This is one form of the principle of reciprocity, and it is demonstrated
geometrically in the following figure.
▪ The symmetrical point on the parabolic surface is known as the vertex. Rays that
emerge in a parallel formation are usually said to be collimated.
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Feed systems for Parabolic Reflectors
Axial/Front Feed:

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Feed systems for Parabolic Reflectors
Axial/Front Feed:
▪ In practice, collimation is often used to describe the highly directional
characteristics of an antenna even though the emanating rays are not exactly
parallel. Since the transmitter (receiver) is placed at the focal point of the parabola,
the configuration is usually known as front fed.
▪ A 𝜆Τ2 antenna with small ground plane is shown in figure(a) and a small horn
antenna in figure(b).
▪ Drawbacks: The Presence of primary antenna in the path of reflected wave has 2
principal disadvantages:
▪ Waves reflected from the parabola back to the primary antenna produce
interaction and mismatching.
▪ Primary antenna acts as an obstruction, blocking out the central portion of the
aperture and increasing the minor lobes. (Aperture Blockage).
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Feed systems for Parabolic Reflectors
Axial/Front Feed:

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Feed systems for Parabolic Reflectors
Axial/Front Feed:
▪ The disadvantage of the front-fed arrangement is that the transmission line from
the feed must usually be long enough to reach the transmitting or the receiving
equipment, which is usually placed behind or below the reflector.
▪ This may necessitate the use of long transmission lines whose losses may not be
tolerable in many applications, especially in low-noise receiving systems.
▪ In some applications, the transmitting or receiving equipment is placed at the focal
point to avoid the need for long transmission lines.
▪ However, in some of these applications, especially for transmission that may
require large amplifiers and for low-noise receiving systems where cooling and
weatherproofing may be necessary, the equipment may be too heavy and bulky
and will provide undesirable blockage.

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Feed systems for Parabolic Reflectors
Offset Feed:
▪ To eliminate some of the deficiencies of the symmetric configurations, offset-
parabolic reflector designs have been developed for single- and dual-reflector
systems.
▪ Offset-reflector designs reduce aperture blocking and VSWR. In addition, they
lead to the use of larger f/d ratios while maintaining acceptable structural rigidity,
which provide an opportunity for improved feed pattern shaping and better
suppression of cross-polarized radiation emanating from the feed.
▪ However, offset-reflector configurations generate cross-polarized antenna
radiation when illuminated by a linearly polarized primary feed. Circularly
polarized feeds eliminate depolarization, but they lead to squinting of the main
beam from boresight.
▪ In addition, the structural asymmetry of the system is usually considered a major
drawback
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Feed systems for Parabolic Reflectors
Offset Feed:

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Feed systems for Parabolic Reflectors
Cassegrain Feed:
▪ Another arrangement that avoids placing the feed (transmitter and/or receiver) at
the focal point is that shown in the following figure, and it is known as the
Cassegrain feed.
▪ Through geometrical optics, Cassegrain, a famous astronomer (hence its name),
showed that incident parallel rays can be focused to a point by utilizing two
reflectors.
▪ To accomplish this, the main (primary) reflector must be a parabola, the secondary
reflector (subreflector) a hyperbola, and the feed placed along the axis of the
parabola usually at or near the vertex.
▪ Cassegrain used this scheme to construct optical telescopes, and then its design
was copied for use in radio frequency systems.

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Feed systems for Parabolic Reflectors
Cassegrain Feed:

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Feed systems for Parabolic Reflectors
Cassegrain Feed:
▪ For this arrangement, the rays that emanate from the feed illuminate the
subreflector and are reflected by it in the direction of the primary reflector, as if
they originated at the focal point of the parabola (primary reflector).
▪ The rays are then reflected by the primary reflector and are converted to parallel
rays, provided the primary reflector is a parabola and the subreflector is a
hyperbola.
▪ Diffractions occur at the edges of the subreflector and primary reflector, and they
must be taken into account to accurately predict the overall system pattern,
especially in regions of low intensity.
▪ Even in regions of high intensity, diffractions must be included if an accurate
formation of the fine ripple structure of the pattern is desired.

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Feed systems for Parabolic Reflectors
Cassegrain Feed:
▪ With the Cassegrain-feed arrangement, the transmitting and/or receiving
equipment can be placed behind the primary reflector. This scheme makes the
system relatively more accessible for servicing and adjustments.
▪ In general, the Cassegrain arrangement provides a variety of benefits, such as the
▪ ability to place the feed in a convenient location
▪ reduction of spillover and minor lobe radiation
▪ ability to obtain an equivalent focal length much greater than the physical
length
▪ capability for scanning and/or broadening of the beam by moving one of the
reflecting surfaces

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Feed systems for Parabolic Reflectors
Cassegrain Feed:
▪ To achieve good radiation characteristics, the subreflector or subdish must be
several, at least a few, wavelengths in diameter.
▪ However, its presence introduces shadowing which is the principal limitation of its
use as a microwave antenna.
▪ The shadowing can significantly degrade the gain of the system, unless the main
reflector is several wavelengths in diameter.
▪ Therefore the Cassegrain is usually attractive for applications that require gains of
40 dB or greater.

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Feed systems for Parabolic Reflectors
Gregorian Feed:
▪ One of the other reflector arrangements is the classical Gregorian design as shown
in the following figure, where the main reflector is a parabola while the
subreflector is a concave ellipse.
▪ The focal point is between the main reflector and subreflector. Its equivalent
parabola is shown dashed in the figure.
▪ When the overall size and the feed beamwidth of the classical Gregorian are
identical to those of the classical Cassegrain, the Gregorian requires a shorter focal
length for the main dish.

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Feed systems for Parabolic Reflectors
Gregorian Feed:

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Microstrip antennas
▪ In high-performance aircraft, spacecraft, satellite, and missile applications, where
size, weight, cost, performance, ease of installation, and aerodynamic profile are
constraints, low-profile antennas may be required.
▪ Microstrip antennas are
➢low profile
➢conformable to planar and nonplanar surfaces
➢simple and inexpensive to manufacture using modern printed-circuit
technology
➢mechanically robust when mounted on rigid surfaces
➢compatible with MMIC designs and
➢when the particular patch shape and mode are selected, they are very versatile
in terms of resonant frequency, polarization, pattern, and impedance.

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Microstrip antennas
▪ Major operational disadvantages of microstrip antennas are their
➢low efficiency
➢low power
➢High Q (sometimes in excess of 100)
➢poor polarization purity
➢poor scan performance
➢spurious feed radiation and
➢very narrow frequency bandwidth, which is typically only a fraction of a
percent or at most a few percent.

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Basic Characteristics of Microstrip antennas
▪ Microstrip antennas received considerable attention starting in the 1970s, although
the idea of a microstrip antenna can be traced to 1953 and a patent in 1955.
▪ Microstrip antennas, as shown in Figure (a), consist of a very thin (t ≪ λ0, where
λ0 is the free-space wavelength) metallic strip (patch) placed a small fraction of a
wavelength (h ≪ λ0, usually 0.003λ0 ≤ h ≤ 0.05λ0) above a ground plane.
▪ The microstrip patch is designed so its pattern maximum is normal to the patch
(broadside radiator). This is accomplished by properly choosing the mode (field
configuration) of excitation beneath the patch.
▪ End-fire radiation can also be accomplished by judicious mode selection.
▪ For a rectangular patch, the length L of the element is usually λ0/3 < L < λ0/2.
▪ The strip(patch) and the ground plane are separated by a dielectric sheet (referred
to as the substrate), as shown in Figure (a).

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Basic Characteristics of Microstrip antennas

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Basic Characteristics of Microstrip antennas
▪ There are numerous substrates that can be used for the design of microstrip
antennas, and their dielectric constants are usually in the range of 2.2 ≤ 𝜀r ≤ 12.
▪ The ones that are most desirable for good antenna performance are thick
substrates whose dielectric constant is in the lower end of the range because they
provide better efficiency, larger bandwidth, loosely bound fields for radiation into
space, but at the expense of larger element size.
▪ Thin substrates with higher dielectric constants are desirable for microwave
circuitry because they require tightly bound fields to minimize undesired radiation
and coupling, and lead to smaller element sizes; however, because of their greater
losses, they are less efficient and have relatively smaller bandwidths.
▪ Since microstrip antennas are often integrated with other microwave circuitry, a
compromise has to be reached between good antenna performance and circuit
design.

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Basic Characteristics of Microstrip antennas

Typical Substrates and Their Parameters

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Basic Characteristics of Microstrip antennas
▪ Often microstrip antennas are also referred to as patch antennas.
▪ The radiating elements and the feed lines are usually photoetched on the dielectric
substrate.
▪ The radiating patch may be square, rectangular, thin strip (dipole), circular,
elliptical, triangular, or any other configuration.
▪ Square, rectangular, dipole (strip), and circular are the most common because of
ease of analysis and fabrication, and their attractive radiation characteristics,
especially low cross-polarization radiation.
▪ Microstrip dipoles are attractive because they inherently possess a large
bandwidth and occupy less space, which makes them attractive for arrays.
▪ Linear and circular polarizations can be achieved with either single elements or
arrays of microstrip antennas.

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Basic Characteristics of Microstrip antennas

Representative shapes of microstrip patch elements

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Microstrip antennas - Feeding Methods
▪ The four most popular configurations that can be used to feed microstrip antennas
are
➢Microstrip Line Feed

➢Coaxial Probe Feed

➢Aperture Coupled Feed

➢Proximity Coupled Feed

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Microstrip antennas - Feeding Methods
1. Microstrip Line Feed:

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Microstrip antennas - Feeding Methods
1. Microstrip Line Feed:
▪ Easy to fabricate

▪ Simple to match by controlling inset position

▪ Simple to model

▪ As substrate thickness increases, surface waves and spurious feed radiation

increase
▪ This limits the bandwidth

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Microstrip antennas - Feeding Methods
2. Coaxial line feed:

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Microstrip antennas - Feeding Methods
2. Coaxial line feed:
▪ Inner conductor attached to the patch and outer conductor to ground plane

▪ Easy to fabricate and to match

▪ Low spurious radiation

▪ Narrow bandwidth

▪ Difficult to model, especially for thick substrates

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Microstrip antennas - Feeding Methods
3. Aperture-coupled feed:

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Microstrip antennas - Feeding Methods
3. Aperture-coupled feed:
▪ Non-contacting feed

▪ Most difficult to fabricate

▪ Narrow bandwidth

▪ Somewhat easier to model

▪ Moderate spurious radiation

▪ Independent optimization of feed mechanism and radiating element possible

▪ Feed line width and slot length used to control matching

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Microstrip antennas - Feeding Methods
4. Proximity-coupled feed:

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Microstrip antennas - Feeding Methods
4. Proximity-coupled feed:
▪ Non-contacting feed

▪ Largest bandwidth

▪ Somewhat easy to model

▪ Low spurious radiation

▪ Fabrication somewhat more difficult

▪ Length of feeding stub and width-to-line ratio of patch can be used to control
matching

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Microstrip antennas - Feeding Methods

Equivalent circuits for typical feeds

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Microstrip antennas - Methods of Analysis
▪ There are many methods of analysis for microstrip antennas. The most popular
models are the transmission-line, cavity and full wave (which include primarily
integral equations/Moment Method).

▪ The transmission-line model is the easiest of all, it gives good physical insight,
but is less accurate and it is more difficult to model coupling.

▪ Compared to the transmission-line model, the cavity model is more accurate but at
the same time more complex. However, it also gives good physical insight and is
rather difficult to model coupling, although it has been used successfully.

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Microstrip antennas - Transmission-Line Model
▪ It was indicated earlier that the transmission-line model is the easiest of all but it
yields the least accurate results and it lacks the versatility. However, it does shed
some physical insight.
▪ Basically the transmission-line model represents the microstrip antenna by two
slots, separated by a low-impedance Zc transmission line of length L.
Fringing Effects:
▪ Because the dimensions of the patch are finite along the length and width, the
fields at the edges of the patch undergo fringing.
▪ This is illustrated along the length in Figures (a,b) for the two radiating slots of the
microstrip antenna.
▪ The same applies along the width.
▪ The amount of fringing is a function of the dimensions of the patch and the height
of the substrate.
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Microstrip antennas - Transmission-Line Model
Fringing Effects:
▪ For the principal E-plane (xy-plane) fringing is a function of the ratio of the length
of the patch L to the height h of the substrate (L/h) and the dielectric constant 𝜀r of
the substrate.
▪ Since for microstrip antennas L/h ≫ 1, fringing is reduced; however, it must be
taken into account because it influences the resonant frequency of the antenna.
The same applies for the width.
▪ For a microstrip line shown in Figure (a), typical electric field lines are shown in
Figure (b).
▪ This is a nonhomogeneous line of two dielectrics; typically the substrate and air.
▪ As can be seen, most of the electric field lines reside in the substrate and parts of
some lines exist in air.

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Microstrip antennas - Transmission-Line Model
Fringing Effects:
▪ As W∕h ≫ 1 and 𝜀r ≫ 1, the electric field lines concentrate mostly in the substrate.
▪ Fringing in this case makes the microstrip line look wider electrically compared
to its physical dimensions.
▪ Since some of the waves travel in the substrate and some in air, an effective
dielectric constant 𝜀reff is introduced to account for fringing and the wave
propagation in the line.
▪ For a line with air above the substrate, the effective dielectric constant has values
in the range of 1 < 𝜀reff < 𝜀r.
−1Τ2
𝜀𝑟 + 1 𝜀𝑟 − 1 12ℎ
𝜀𝑟𝑒𝑓𝑓 = + 1+
2 2 𝑊

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Microstrip antennas - Transmission-Line Model

Microstrip line and its electric field lines, and effective dielectric constant geometry
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Microstrip antennas - Transmission-Line Model

Physical and effective lengths of rectangular microstrip patch

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Microstrip antennas - Transmission-Line Model
Effective Length, Resonant Frequency, and Effective Width:
▪ Because of the fringing effects, electrically the patch of the microstrip antenna
looks greater than its physical dimensions.
𝑊
Δ𝐿 (𝜀𝑟𝑒𝑓𝑓 + 0.3)( + 0.264)
= 0.412 ℎ
ℎ 𝑊
(𝜀𝑟𝑒𝑓𝑓 − 0.258)( + 0.8)
ℎ

𝐿𝑒𝑓𝑓 = 𝐿 + 2Δ𝐿
▪ L = λ∕2 for dominant TM010 mode with no fringing.
1
𝑓𝑟𝑐 010 =
2𝐿𝑒𝑓𝑓 𝜀𝑟𝑒𝑓𝑓 𝜇0 𝜀0

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Microstrip antennas - Cavity Model
▪ Using the cavity model, a rectangular microstrip antenna can be represented as an
array of two radiating narrow apertures (slots), each of width W and height h,
separated by a distance L.
▪ Excitation establishes charge distribution
▪ This distribution controlled by two mechanisms:
▪ An attractive one between the corresponding opposite charges on the bottom
side of the patch and the ground plane; and
▪ A repulsive one between like charges on the bottom side of the patch
▪ While the attraction tends to keep charge distribution on the bottom of the patch,
the repulsion tends to push some charges from the patch bottom to the top surface,
around the edges.
▪ The consequent movement of charge carriers gives rise to Jb and Jt

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Microstrip antennas - Cavity Model

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Microstrip antennas - Cavity Model
▪ The height-to-width ration generally being very small, attractive mechanism
dominates, thus restricting current flow to bottom surface.
▪ Thus the tangential magnetic field to the edges is insignificant
▪ Consequently, the four side walls can be modeled as PMC which ideally would
not disturb the H-field, and in turn, the E-field beneath the patch.
▪ This model produces good E and H field distributions beneath the patch.
▪ With the assumption of lossless cavity walls and the filling dielectric, the cavity
would not radiate and its input impedance would be purely reactive.
▪ To account for radiation, therefore, a loss mechanism needs to be introduced.
▪ This is done by introducing an effective loss tangent 𝛿𝑒𝑓𝑓 .

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Microstrip antennas - Cavity Model
▪ The top and bottom walls are PEC
▪ The four side walls are PMC
▪ Tangential magnetic fields vanish along these four walls
▪ In summary, the cavity method is based on the following physical model:
▪ The electric field is mainly localized in the cylinder of height h between the
patch and the ground plane;
▪ The radiation is the result of leakage from the cylindrical cavity via its lateral
walls (as the cavity ends are perfectly conducting)

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FREQUENCY INDEPENDENT ANTENNAS

CONTENTS

▪ Principle of frequency independent antennas

➢ Spiral antenna
➢ Log-Periodic antenna

FREQUENCY INDEPENDENT ANTENNA

▪ A true frequency independent antenna is physically fixed in size and operates on

an instantaneous basis over a wide bandwidth with relatively constant
impedance, pattern, polarization and gain.
▪ Their geometries are specified in angles.
▪ E.g.: Bi-conical antenna, Spiral antenna and Helical antenna

RUMSEY’S PRINCIPLE

Rumsey’s principle is that the impedance and pattern properties of an antenna will be
frequency independent if the antenna shape is specified only in terms of angles.

We begin by assuming that an antenna, whose geometry is best described by the

spherical coordinates (𝑟, 𝜃, 𝜙), has both terminals infinitely close to the origin and
each is symmetrically disposed along the 𝜃 = 0, 𝜋-axes. It is assumed that the antenna
is perfectly conducting, it is surrounded by an infinite homogeneous and isotropic
medium, and its surface or an edge on its surface is described by a curve

𝑟 = 𝐹(𝜃, 𝜙) (1)

where r represents the distance along the surface or edge. If the antenna is to be scaled
to a frequency that is K times lower than the original frequency, the antenna’s physical
surface must be made K times greater to maintain the same electrical dimensions.
Thus, the new surface is described by

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𝑟 ′ = 𝐾𝐹(𝜃, 𝜙) (2)

The new and old surfaces are identical; that is, not only are they similar but they are
also congruent (if both surfaces are infinite). Congruence can be established only by
rotation in𝜙. Translation is not allowed because the terminals of both surfaces are at
the origin. Rotation in 𝜃 is prohibited because both terminals are symmetrically
disposed along the 𝜃 = 0, 𝜋-axes.

For the second antenna to achieve congruence with the first, it must be rotated by an
angle C so that

𝐾𝐹(𝜃, 𝜙) = 𝐹(𝜃, 𝜙 + 𝐶) (3)

The angle of rotation C depends on K but neither depends on 𝜃 or 𝜙. Physical

congruence implies that the original antenna electrically would behave the same at
both frequencies. However, the radiation pattern will be rotated azimuthally through
an angle C. For unrestricted values of K(0 ≤ 𝐾 ≤ ∞), the pattern will rotate by C in 𝜙
with frequency, because C depends on K but its shape will be unaltered. Thus, the
impedance and pattern will be frequency independent.

To obtain the functional representation of 𝐹(𝜃, 𝜙), both sides of (3) are differentiated
with respect to C to yield

𝑑 𝑑
[𝐾𝐹(𝜃, 𝜙)] = [𝐹(𝜃, 𝜙 + 𝐶)] (4)
𝑑𝐶 𝑑𝐶

𝑑𝐾 𝜕
𝐹(𝜃, 𝜙) = [𝐹(𝜃, 𝜙 + 𝐶)] (5)
𝑑𝐶 𝜕(𝜙 + 𝐶)

Differentiate (3) relative to 𝜙, we get

𝜕 𝜕
[𝐾𝐹(𝜃, 𝜙)] = [𝐹(𝜃, 𝜙 + 𝐶)] (6)
𝜕𝜙 𝜕𝜙

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𝜕 𝜕
𝐾 [𝐹(𝜃, 𝜙)] = [𝐹(𝜃, 𝜙 + 𝐶)] (7)
𝜕𝜙 𝜕(𝜙 + 𝐶)

Comparing (5) and (7), we get

𝑑𝐾 𝜕𝐹(𝜃, 𝜙)
𝐹(𝜃, 𝜙) = 𝐾 (8)
𝑑𝐶 𝜕𝜙

By substituting 𝑟 = 𝐹(𝜃, 𝜙),

𝑑𝐾 𝜕𝑟
𝑟 = 𝐾 (9)
𝑑𝐶 𝜕𝜙

1 𝑑𝐾 1 𝜕𝑟
= (10)
𝐾 𝑑𝐶 𝑟 𝜕𝜙

LHS of above equation is independent of 𝜃 and 𝜙, therefore the general solution is

given as

𝑟 = 𝐹(𝜃, 𝜙) = 𝑒 𝑎𝜙 𝑓(𝜃) (11)

1 𝑑𝐾
where 𝑎 = and 𝑓(𝜃) is completely arbitrary function
𝐾 𝑑𝐶

Thus, for any antenna to have frequency independent characteristics, its surface must
be described by (11).

FREQUENCY-INDEPENDENT PLANAR LOG SPIRAL ANTENNA

The equation for a logarithmic or log spiral is given by

𝑟 = 𝑎𝜃 (12)

ln 𝑟 = ln 𝑎𝜃 = 𝜃 ln 𝑎 (13)

where,
𝑟 = radial distance to point P on spiral
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𝜃= angle with respect to x-axis

a=constant

Logarithmic spiral or log spiral

From (12), the rate of change of radius with angle is

𝑑𝑟
= 𝑎𝜃 ln 𝑎 = 𝑟 ln 𝑎 (14)
𝑑𝜃

The constant a in (14) is related to the angle 𝛽 between the spiral and a radial line
from the origin as given by

𝑑𝑟 1
ln 𝑎 = = (15)
𝑟𝑑𝜃 𝑡𝑎𝑛𝛽

Thus, from (14) and (15),

ln 𝑟
𝜃= = 𝑡𝑎𝑛𝛽 ln 𝑟 (16)
ln 𝑎

The log spiral was constructed so as to make 𝑟 = 1 at 𝜃 = 0 and 𝑟 = 2 at 𝜃 = 𝜋.

These conditions determine the value of the constants a and 𝛽. Thus, from (15) and
(16),

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𝜃 𝜃
𝑡𝑎𝑛𝛽 = ⇒ 𝛽 = tan−1 [ ] (17)
ln 𝑟 ln 𝑟

For 𝑟 = 2 at 𝜃 = 𝜋,

𝜋
𝛽 = tan−1 [ ] = 77.60
ln 2

1 1
ln 𝑎 = ⇒ 𝑎 = exp { } = 1.247
𝑡𝑎𝑛𝛽 𝑡𝑎𝑛77.60

Thus, the shape of the spiral is determined by the angle 𝛽 which is same for all points
on the spiral.

Let a second log spiral, identical in form to the one in figure shown below, be
generated by an angular rotation of the spiral by a factor 𝛿 so that (12) becomes

𝑟2 = 𝑎𝜃−𝛿 (18)

and a third spiral and fourth spiral given by

𝑟3 = 𝑎𝜃−𝜋 (19)

and

𝑟4 = 𝑎𝜃−𝜋−𝛿 (20)

𝜋
Then, for a rotation 𝛿 = , we have 4 spirals at 900 angles. Metalizing the areas
2

between the spirals 1 & 4 and 2 & 3, with the other areas open, self-complementary
and congruence conditions are satisfied. Connecting a generator or receiver across the
inner terminals, we obtain Dyson’s frequency independent planar spiral antenna.

▪ Polarization: RHCP radiation from the page and LHCP radiation into the page
▪ High frequency limit of operation is determined by the spacing d of the input
terminal and the low frequency limit by overall diameter D. The ratio D/d for
the antenna is about 25 to 1.

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𝜆 𝜆
▪ If we take 𝑑 = at the high frequency limit and 𝐷 = at the low frequency
10 2

limit, the antenna bandwidth is 5 to 1.

Frequency Independent Planar Spiral Antenna

▪ Spiral slot antenna: Spiral-shaped slots are cut from a large ground plane and
the antenna is fed with a co-axial cable bonded to one of the spiral arms. A
dummy cable may be bonded to the other arm of symmetry.
▪ Radiation Pattern for spiral antenna: Bi-directional broadside to the plane of
the spiral. The patterns in both directions have a single broad lobe so that the
gain is only a few dBi.
▪ Input Impedance depends on 𝛿 and a and terminal separation (≃ 50 𝑡𝑜 100 Ω)
▪ Ratio K of the radii across any arm, such as between spiral 2 and 3 is given by
𝑟3 𝑎𝜃−𝜋
𝐾 = = 𝜃−𝛿 = 𝑎−𝜋+𝛿
𝑟2 𝑎

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𝜋
For 𝛿 =
2
𝑟3 −
𝜋
−
𝜋 1
𝐾 = = 𝑎 = (1.247) 2 = 0.707 =
2
𝑟2 √2

FREQUENCY INDEPENDENT CONICAL SPIRAL ANTENNA

▪ A tapered helix is a conical spiral antenna in which pitch angle is constant with
diameter and turn spacing variable.

Tapered Helix or Conical Spiral (Forward-fire)

▪ Two arms of the conical spiral are fed at the center point or apex from a co-axial
cable bonded to one of the arms, the spiral acting as a balun.
▪ For symmetry, a dummy cable may be bonded to the other arm.
▪ According to Dyson, input impedance is between 100 to 150 Ω for pitch angle
𝛼 = 170 and full cone angles of 200 and 600 . Smaller cone angles (less than
300 ) have high F/B ratio.
▪ Radiation Pattern: Uni-directional with maxima towards the apex.
𝜆
𝐵𝑎𝑠𝑒 𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟 (∼ 𝑎𝑡 𝑙𝑜𝑤 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦)
▪ 𝐵𝑎𝑛𝑑𝑤𝑖𝑑𝑡ℎ ∝ 2
𝜆
𝑇𝑟𝑢𝑛𝑐𝑎𝑡𝑒𝑑 𝑎𝑝𝑒𝑥 𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟 (∼ 𝑎𝑡 ℎ𝑖𝑔ℎ 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦)
4

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Dyson 2-arm balanced conical

spiral (backward-fire antenna).
Polarization is RCP. Inner
conductor of coax connects to
dummy at apex.

LOG-PERIODIC ANTENNA

▪ Log-Periodic antenna operates over broadband and its size varies with the operating
frequency or wavelength. Although, LPDA is not specified in terms of angles yet its
geometry is adjusted such that all the electrical properties of the antenna are repeated
periodically with the logarithm of the frequency.
▪ Dwight Isbell demonstrated first LPDA (1960).
▪ Basic concept: A gradually expanding periodic structure array radiates more
effectively when the array elements(dipoles) are near resonance so that with change in
frequency, the active region moves along the array.
▪ LPDA– a number of dipole antennas of different lengths are arranged at different
spacings, used in array form.
▪ Dipole lengths increase along the antenna so that the included angle 𝛼 is constant, and
the lengths (𝑙) and spacing(𝑆) of the adjacent elements are scaled so that
𝑙𝑛+1 𝑆𝑛+1 1
= =𝑘= (21)
𝑙𝑛 𝑆𝑛 𝜏
where
𝑘 = constant (𝑘 > 1)
𝜏 = scale factor or design ratio or geometric ratio or periodicity factor (𝜏 > 1)
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▪ Antenna is fed through a balanced twin line in zig-zag form which means the alternate
dipole arms are fed through common line.
▪ The apex angle is formed by the two imaginary straight lines passing through the
edges of the dipole arms located on either side.
▪ Moreover, the spacing between the dipoles near to the apex is smaller as compared to
the spacing at the base.
▪ There are three important regions of LPDA namely
i. Inactive region (Transmission line)
ii. Active region
iii. Inactive region (Stop)

Log-Periodic Dipole Array (LPDA)

𝝀
i. Transmission line region (𝑳 ≤ ): At the middle of the operating range, the antenna
𝟐

elements are short with the resonant length, therefore the elements offer large
capacitive reactance to the line. Hence, currents in these elements (1, 2, 3, 4, 5) are
small and radiation is small.
𝝀
ii. Active region (𝑳 = ): At a wavelength near the middle of the operating range,
𝟐

radiation occurs primarily from the central region of the antenna. This region offers

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resistive impedance. Currents in this region have large values and maximum radiation
takes place from this region. The current and input RF voltage is in phase. The spacing
between the elements are now sufficiently large, causing the phase in the particular
element to lead approximately by 900 . For example, by the time the field radiated
from the element 𝑙𝑛+1 reaches 𝑙𝑛 , the phase of the element 𝑙𝑛 advances by 900 and the
field from element 𝑙𝑛 add to the field of 𝑙𝑛+1 element , in phase producing a large
resultant field towards left. Hence, there is a strong radiation towards right.
𝝀
iii. Stop region (𝑳 ≤ ): Elements 9,10&11 are almost one wavelength long and carry
𝟐

only small currents (they present a large inductive reactance to the line). Small
currents in 9,10 & 11 mean that the antenna is effectively truncated at the right of the
active region. Any fields (smaller magnitude) from elements 9,10&11 tend to cancel in
both forward and backward directions. However, some radiation may occur broadside
since the currents are approximately in phase.

Radiation pattern: Thus, at a wavelength(𝜆), the radiation occurs from the middle
𝝀
portion where the dipole elements are long. When the wavelength is increased, the
𝟐

radiation zone moves towards the right and when the wavelength is decreased, it moves
to the left with maximum radiation toward the apex or feed point of the array.

Log-periodic behaviour:

▪ If the input impedance of a LPDA is plotted as a function of frequency, it will be

repetitive.
▪ However, if the input impedance is plotted as a function of logarithm of frequency,
it will be periodic with each cycle being exactly identical to the preceding one.
Hence the name log-periodic, because the variations are periodic with respect to
the logarithm of frequency.
▪ Other parameters that undergo the similar variations are the pattern, directivity,
beamwidth and sidelobe level.
▪ The relationship between two consecutive maxima frequencies and logarithmic
frequency period is
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𝑓2 1 𝑓2 1
log = log ⇒ = ⇒ 𝑓1 = 𝜏𝑓2
𝑓1 𝜏 𝑓1 𝜏
Because 𝜏 > 1, 𝑓1 < 𝑓2 .

Input impedance of LPDA as function of logarithm of frequency

Design equations for LPDA

LPDA geometry or determining the relation of parameters

From the above figure,

𝑙𝑛+1 − 𝑙𝑛 𝑙𝑛+1 𝑙
[ ] [1 − 𝑛 ]
2 2 𝑙𝑛+1
tan 𝛼 = = (22)
𝑆 𝑆

𝑙𝑛+1 1
[1 − ]
tan 𝛼 = 2 𝑘 (23)
𝑆
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where 𝛼 =apex angle

𝑘 = scale-factor

𝝀
For 𝑙𝑛+1 = (when active)
𝟐

1
[1 − ]
tan 𝛼 = 𝑘 (24)
𝑆
4( )
𝜆

1
[1 − ]
tan 𝛼 = 𝑘 (25)
4𝑆𝜆

From (24),
1
[1 − ]
tan 𝛼 = 𝑘 =1−𝜏 (26)
𝑆 4𝜎
4( )
𝜆

Where

1
𝜏=
𝑘
𝑆
𝜎 = ( ) = spacing factor
𝜆

Hence from (26),

1−𝜏
𝛼 = tan−1 [ ] (27)
4𝜎
𝑙𝑛+1 1
▪ Bandwidth of LPDA (frequency ratio) is 𝐹 = = 𝑘𝑛 = . The length 𝑙 and
𝑙1 𝜏𝑛

spacing 𝑆 for element 𝑛 + 1 is 𝑘 𝑛 greater than for element 1.

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Example:

Design a log periodic dipole array operating from 50 MHz to 200 MHz which has 𝜏 = 0.822
and 𝜎 = 0.149. Find out the number of dipoles required to cover this bandwidth.

Solution:

Given: 𝜏 = 0.822; 𝜎 = 0.149; 𝑓𝑚𝑖𝑛 = 50 MHz; 𝑓𝑚𝑎𝑥 = 200 MHz

The apex angle (𝛼) is defined as

1−𝜏 1 − 0.822
𝛼 = tan−1 [ ] = tan−1 [ ] = 16.62°
4𝜎 4 × 0.149

The lowest and highest wavelength can be obtained as

𝑐 3 × 108
𝜆𝐿 = = = 1.5 m
𝑓𝑚𝑎𝑥 200 × 106

𝑐 3 × 108
𝜆𝐻 = = =6m
𝑓𝑚𝑖𝑛 50 × 106

The maximum and minimum lengths required for dipole antennas are defined as follows:

𝜆𝐻 6
𝐿𝑑 𝑚𝑎𝑥 = = =3m
2 2

𝜆𝐿 1.5
𝐿𝑑 𝑚𝑖𝑛 = = = 0.75 m
2 2

The dipole lengths to cover 50 MHz to 200 MHz can be obtained using the following
relation:

𝐿𝑛 = 𝜏𝐿𝑛+1

𝐿𝑛
𝐿𝑛+1 =
𝜏
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Now, let us start from the highest frequency of operation, i.e., 200 MHz for which the
𝜆𝐿 1.5
minimum size of dipole is 𝐿𝑑 𝑚𝑖𝑛 = = = 0.75 m.
2 2

Therefore, to start with minimum size, we have

𝐿1 = 𝐿𝑑 𝑚𝑖𝑛 = 0.75 m

The other dipole lengths are

𝐿1 0.75
𝐿2 = = = 0.912 m
𝜏 0.822

𝐿2 0.912
𝐿3 = = = 1.109 m
𝜏 0.822

𝐿3 1.109
𝐿4 = = = 1.350 m
𝜏 0.822

𝐿4 1.350
𝐿5 = = = 1.642 m
𝜏 0.822

𝐿5 1.642
𝐿6 = = =2m
𝜏 0.822

𝐿6 2
𝐿7 = = = 2.433 m
𝜏 0.822

𝐿7 2.433
𝐿8 = = = 2.96 m
𝜏 0.822

𝐿8 2.96
𝐿9 = = = 3.6 m
𝜏 0.822

It should be noted that the highest length of the dipole should be greater than or equal to the
𝜆𝐻 6
maximum half wavelength, i.e., 𝐿𝑑 𝑚𝑎𝑥 = = =3m
2 2

The spacing between 𝑛th and (𝑛 + 1)th dipole can be derived from the following
relationship:

𝑆𝑛 = 2𝜎𝐿𝑛

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Therefore, the spacing between all sections is given as:

𝑆1 = 2𝜎𝐿1 = 2 × 0.149 × 0.75 = 0.224

𝑆2 = 2𝜎𝐿2 = 2 × 0.149 × 0.912 = 0.272

𝑆3 = 2𝜎𝐿3 = 2 × 0.149 × 1.109 = 0.33

𝑆4 = 2𝜎𝐿4 = 2 × 0.149 × 1.350 = 0.402

𝑆5 = 2𝜎𝐿5 = 2 × 0.149 × 1.642 = 0.489

𝑆6 = 2𝜎𝐿6 = 2 × 0.149 × 2 = 0.596

𝑆7 = 2𝜎𝐿7 = 2 × 0.149 × 2.433 = 0.725

𝑆8 = 2𝜎𝐿8 = 2 × 0.149 × 2.96 = 0.882

To cover the entire frequency range, nine dipoles are required.

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Wave Propagation APPof ECE UNIT 3

UNIT-III
ANTENNA ARRAYS

 Antenna array is system of a similar antennas oriented similarly to get greater directivity
in a desired direction.
 Antenna array is a radiating system consisting of several spaced and properly phased
(current phase) radiators.

Linear Array:
 An antenna array is said to be linear if the individual antennas of the array are equally
spaced along a straight line.
 Individual elements of the array are termed as Elements.

Uniform Linear Array:

 Uniform linear array is one in which the elements are fed with a current of equal
amplitude(magnitude) with uniform progressive phase shift along the line.

 Elements in a multi-element array is generally a dipole antenna.
2

Factors that shape the radiation pattern of antenna array:

 The geometrical configuration of the array
 The spacing between the elements
 The excitation amplitude of the individual elements
 The excitation phase of the individual elements
 The radiation pattern of the individual elements

Two element array:

 Simplest array is an array of two isotropic point sources separated by a distance d.
 Two isotropic point sources symmetrically situated w.r.t the origin in the Cartesian co-
ordinate system is shown in figure

 Consider that the two isotropic point sources are fed with current of equal amplitude and
phase
 The fields at a greater distant point at distant R from the origin O can be calculated as
follows:
St.Joseph’ s College of Engineering/ St.Joseph’ s Institute of Technology 1
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EC6602 Antenna and FROM STUCORDepartment
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 Origin is taken as reference point for phase calculation. The waves from source1 reaches
the point P at a later time than the waves from source 2 because of path difference
between the two waves.
Path difference between the two waves is, d cos 
2 2
  phase angle   path difference  d cos 
 
  d cos  radians

j
2
Field component due to source1 ( field lags) = E1e

j
Field component due to source2( field leads) = E 2 e 2
Two isotropic point sources are fed with current of equal amplitude and phase.
E1  E2  E0
Total electric field at point P = E  E1  E2
 
j j    d cos  
E  E1e 2
 E2 e 2
 2 E0 cos   2 E0 cos 
2  2 

E max  2 E 0
E
E nor 
2 E0
  d cos  
E nor  cos 
 2 


For the case d  ,
2
 2  
  cos  
E nor  cos  2   cos  cos  
 2  2 
 
 

Calculation of maximum, minimum and half power direction of the field pattern:
Maxima directions
 
Normalized total field is maximum when cos cos    1
2 
 
 cos  max   n where n  0,1,2......
2 
 
 cos  max   0 when n  0
2 
 max  90 and 270 0
0

The field is maximum in the directions where   90 0 and 270 0

Minima directions
 
Normalized total field is minimum when cos cos    0
2 
  
 cos  min   (2n  1) where n  0,1,2......
2  2

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  
 cos  min    when n  0
2  2
cos  min   1
 min  0 0 and 180 0
The field is minimum in the directions where   0 0 and 180 0
Half Power Directions:
1
At half power points, power is half the maximum and voltage or current is times the
2
maximum.
  1
Normalized total field is cos cos    
2  2
  
 cos  HPPD   (2n  1) where n  0,1,2......
2  4
  
 cos  HPPD    when n  0
2  4

cos  HPPD    1
2
 HPPD  60 and 120 0
0

1
The field is times the maximum in the directions where   60 0 and 120 0
2

 The radiation pattern is perpendicular to the array axis. This array is referred to as
Broadside array.
Note: Broadside array is defined as an array in which the principal direction is perpendicular
to the array axis

Array of two point source with equal amplitude and opposite phase:

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 Consider that the two isotropic point sources are fed with current of equal amplitude and
opposite phase
 The fields at a greater distant point at distant R from the origin O can be calculated as
follows:
 Origin is taken as reference point for phase calculation. The waves from source1 reaches
the point P at a later time than the waves from source 2 because of path difference
between the two waves.
Path difference between the two waves is, d cos 
2 2
  phase angle   path difference  d cos 
 
  d cos  radians

j
2
Field component due to source1 ( field lags) = - E1e

j
Field component due to source2( field leads) = E 2 e 2
Two isotropic point sources are fed with current of equal amplitude
E1  E2  E0
Total electric field at point P
 
j j    d cos  
= E   E1e 2  E 2 e 2  2 j E0 sin   2 jE 0 sin 
2  2 

E max  2 jE 0
E
E nor 
2 jE 0
  d cos  
E nor  sin  
 2 


For the case d  ,
2
 2  
  cos  
E nor  sin   2   sin   cos  
 2  2 
 
 

Calculation of maximum, minimum and half power direction of the field pattern:

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Maxima directions
 
Normalized total field is maximum when sin cos    1
2 
  (2n  1)
 cos  max    where n  0,1,2......
2  2
  
 cos  max    when n  0
2  2
 max  0 0 and 180 0
The field is maximum in the directions where   0 0 and 180 0
Minima directions
 
Normalized total field is minimum when sin cos    0
2 
 
 cos  min   n where n  0,1,2......
2 
 
 cos  min   0 when n  0
2 
cos  min   0
 min  90 0 and 270 0
The field is minimum in the directions where   90 0 and 270 0
Half Power Directions:
1
At half power points, power is half the maximum and voltage or current is times the
2
maximum.
  1
Normalized total field is sin cos    
2  2
  
 cos  HPPD   (2n  1) where n  0,1,2......
2  4
  
 cos  HPPD    when n  0
2  4

cos  HPPD    1
2
 HPPD  60 and 120 0
0

1
The field is times the maximum in the directions where   60 0 and 120 0
2

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 Maximum radiation is along the axis of the array. This array is referred to as END FIRE
array.

Arrays of point sources with unequal amplitude and any phase

 Let  be the phase difference between the currents.

 The total phase difference between radiations of two sources at a distant point P is given
by
2
 d cos   

2
d cos   phase difference due to path difference



E  E1e j 0  E 2 e j  E1 1  Ke j 
E2
K
E1
| E | E1 1  K cos    K sin  
2 2

 K sin  
phase angle  tan 1  
 1  K cos   
Linear Array with n isotropic point sources of equal amplitude
and spacing

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 The point sources are fed with currents of equal amplitude and having an uniform
progressive phase shift along the line
 Field at a distant point P is given by,
Et  E0 e j 0  E0 e j  E0 e j 2   E0 e j 3  .........  E0 e j ( n1) 
Et  E0 {e j 0  e j  e j 2   e j 3  .........  e j ( n1)  }        1
2
 d cos   

2
d cos   phase difference due to path difference

between two po int sources
  phase shift between two po int sources

Et  E0 e j 0  E0 e j  E0 e j 2   E0 e j 3  .........  E0 e j ( n1) 
Et  E0 {e j 0  e j  e j 2   e j 3  .........  e j ( n1)  }        1

Et e j  E0 {e j  e j 2  e j 3  .........  e jn }        2
1 2 gives
Et (1  e j )  E0 {1  e jn }

Et  E0
1  e  jn

(1  e j )
 j n2 j
n

n e  e 2 
j  
  
2
e
Et  E0 
j  j2 j 

e 2 e  e 2 
 
 

 n 
( n 1)  sin  
Et  E 0 e
j
2  2 

sin  
2

 n 
sin  
 2  (n  1)
E t  E 0 e j where  
 2
sin  
2

In this derivation, point source 1 is taken as reference point. In case the reference point is
(n  1)
shifted to the center of the array,   is automatically eliminated .
2

 n 
sin  
Et  E 0  2 

sin  
2

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 n  n  n 
sin   cos 
Et max  lt E0  2 
 lt E0
2  2   nE
 
0
 0  0 1
sin   cos 
2 2 2

Radiation pattern of two nondirectional radiators fed with equal currents at

the phasings shown.

Radiation pattern of four nondirectional radiators

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Principle of pattern multiplication

The field pattern of an array of non-isotropic but similar sources is the product of the
pattern of the individual source and the pattern of an array of isotropic point sources having the
same locations, relative amplitudes, and phase as the non- isotropic sources.
The total field pattern of an array of non-isotropic but similar sources is the product of
individual source pattern and the pattern of an array of isotropic point sources each located at the
phase center of the individual source and having the same relative amplitude and phase, while the
total phase pattern is the sum of the phase patterns of the individual source and the array of
isotropic point sources.
E  f  ,  F  ,   f p  ,   F p  ,  
f  ,  F  ,    field pattern of array
 f  ,  F  ,   phase pattern of array
p p

f  ,    field pattern of individual source

f p  ,    phase pattern of individual source
F  ,    field pattern of array of isotropic sources
F p  ,    phase pattern of array of isotropic sources
Multiplication of Patterns
 Simple method of obtaining radiation pattern.
 Makes it possible to sketch rapidly, almost by inspection, the patterns of
complicated arrays.
 Useful tool in the design of arrays
Example:
Determination of radiation pattern of a four element array in which the spacing between units is

and the currents are in phase (   0 )
2

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Determination of radiation pattern by multiplication of patterns is illustrated below:

(a) -- unit pattern

(b) -- Group pattern
(c) – resultant pattern

 This procedure provides a means for rapidly determining the radiation pattern of a
complicated array without making length calculations.
 The width of the principle lobe (between nulls) is the same as the width of the
corresponding lobe of the group pattern.

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 The number of secondary lobes can be determined from the number of nulls in the
resultant pattern, which is just the sum of the nulls in the unit and group
pattern(assuming none of the nulls are coincident).
 Point by point multiplication of patterns yields the exact pattern .

Example 2

Linear Array with uniform spacing, non uniform amplitude:

(i) Binomial array
(ii) Dolph-Tschebysheff array ( also referred to as Chebyshev array or
Tschbyscheff array)
Binomial array:
 Current distribution follows the binomial series.
 The current amplitudes are proportional to the coefficients of the
successive terms of the Binomial series.

 Binomial array possess the smallest side lobes. If the spacing is or less
2

than , Binomial array has no side lobes.
2
 Binomial series :

1  x m1  1  m  1x  m  1m  2 x 2  m  1m  2m  3 x 3  ............

2! 3!
 For m =3
1  x 31  1  3  1x  3  13  2 x 2  3  13  23  3 x 3  ............
2! 3!
 1  2x  x 2

Current distribution for 3 element binomial array is 1:2:1

 Current distribution for 4 element binomial array is 1:3:3:1
So on…..
 Current distribution can be determined easily from Pascal’ s Triangle
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 Principle of pattern multiplication can be used to determine the resultant radiation

pattern.
 For 3 element Binomial array, the resultant pattern is shown below:

 Here antenna 2 and 3 coincide , and so they would be replaced by with a

single antenna carrying double the current.
 Four element Binomial array is shown in figure below:

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Note:
 No side lobes in the radiation pattern of Binomial array
 Half Power Beam width is more

Disadvantages of Binomial array:

(i) HPBW increases and hence the directivity decreases
(ii) For design of large array, larger amplitude ratio of sources required.
Adaptive array (smart antennas)
 Smart antennas (also known as adaptive array antennas, digital antenna arrays)
are antenna arrays with smart signal processing algorithms used to identify spatial signal
signatures such as the direction of arrival (DOA) of the signal, and use them to
calculate beamforming vectors which are used to track and locate the antenna beam on the
mobile/target.
 Smart antenna techniques are used notably in acoustic signal processing, track and
scan radar, radio astronomy and radio telescopes, and mostly in cellular systems like W-
CDMA, UMTS, and LTE.
 Smart antennas have many functions: DOA(Direction of Arrival) estimation,
beamforming, interference nulling, and constant modulus preservation.
Beam forming
 Beamforming is the method used to create the radiation pattern of the antenna array by
adding constructively the phases of the signals in the direction of the targets/mobiles
desired, and nulling the pattern of the targets/mobiles that are undesired/interfering
targets.
 This can be done with a simple Finite Impulse Response (FIR) tapped delay line filter.
The weights of the FIR filter may also be changed adaptively, and used to provide optimal
beamforming, in the sense that it reduces the Minimum Mean Square Error between the
desired and actual beam pattern formed.
 Typical algorithms are the steepest descent, and Least Mean Squares algorithms. In digital
antenna arrays with multi channels use the digital beamforming, usually by DFT or FFT.
Types of Smart antennas
Two main types of smart antennas:
(i) Switched beam smart antennas
(ii) Adaptive array smart antennas
 Switched beam systems have several available fixed beam patterns. A decision is made as
to which beam to access, at any given point in time, based upon the requirements of the
system.

Switched beam smart antennas

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 Adaptive arrays allow the antenna to steer the beam to any direction of interest while
simultaneously nulling interfering signals.

Adaptive array smart antenna

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N-ELEMENT LINEAR ARRAY: UNIFORM SPACING, NONUNIFORM AMPLITUDE

• Of the three distributions (uniform, binomial, and Tschebyscheff), a uniform amplitude
array yields the smallest half-power beamwidth. It is followed, in order, by the Dolph-
Tschebyscheff and binomial arrays.
• In contrast, binomial arrays usually possess the smallest side lobes followed, in order,
by the Dolph-Tschebyscheff and uniform arrays.
• Binomial arrays with element spacing equal or less than λ/2 have no side lobes.
• It is apparent that the designer must compromise between side lobe level and
beamwidth.
• A criterion that can be used to judge the relative beamwidth and side lobe level of one
design to another is the amplitude distribution (tapering) along the source.
• It has been shown analytically that for a given side lobe level the Dolph-Tschebyscheff
array produces the smallest beamwidth between the first nulls. Conversely, for a given
beamwidth between the first nulls, the Dolph-Tschebyscheff design leads to the
smallest possible side lobe level.
Array Factor of non uniform array
• An array of an even number of isotropic elements 2M (where M is an integer) is
positioned symmetrically along the z-axis, as shown in Figure (a). The separation
between the elements is d, and M elements are placed on each side of the origin.
• If the total number of isotropic elements of the array is odd 2M + 1 (where M is an
integer), the arrangement is as shown in Figure (b)

An array of an even number of isotropic elements 2M (where M is an integer) is

positioned symmetrically along the z-axis, as shown in Figure (a). The separation between the

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elements is d, and M elements are placed on each side of the origin. Assuming that the
amplitude excitation is symmetrical about the origin, the array factor for a nonuniform
amplitude broadside array can be written as,
1 3 (2𝑀−1)
+𝑗( )𝑘𝑑𝑐𝑜𝑠𝜃
(𝐴𝐹)2𝑀 = 𝑎1 𝑒 +𝑗(2 )𝑘𝑑𝑐𝑜𝑠𝜃 + 𝑎2 𝑒 +𝑗(2 )𝑘𝑑𝑐𝑜𝑠𝜃 + ⋯ + 𝑎𝑀 𝑒 2

1 3 (2𝑀−1)
+𝑎1 𝑒 −𝑗(2 )𝑘𝑑𝑐𝑜𝑠𝜃 + 𝑎2 𝑒 −𝑗(2 )𝑘𝑑𝑐𝑜𝑠𝜃 + ⋯ + 𝑎𝑀 𝑒 −𝑗( )𝑘𝑑𝑐𝑜𝑠𝜃
2

(𝐴𝐹)2𝑀 (𝑒𝑣𝑒𝑛) = ∑ 𝑎𝑛 cos [(2𝑛 − 1)𝑢]

𝑛=1
𝜋𝑑
Where 𝑢 = 𝑐𝑜𝑠𝜃
𝜆

𝑎𝑛 is excitation coefficient
If the total number of isotropic elements of the array is odd 2M + 1 (where M is an integer),
as shown in Figure (b), the array factor can be written as
(𝐴𝐹)2𝑀+1 = 2𝑎1 + 𝑎2 𝑒 +𝑗𝑘𝑑𝑐𝑜𝑠𝜃 + ⋯ + 𝑎𝑀+1 𝑒 +𝑗(𝑀)𝑘𝑑𝑐𝑜𝑠𝜃

+ 𝑎2 𝑒 −𝑗𝑘𝑑𝑐𝑜𝑠𝜃 + ⋯ + 𝑎𝑀+1 𝑒 −𝑗(𝑀)𝑘𝑑𝑐𝑜𝑠𝜃

𝑀+1

(𝐴𝐹)2𝑀+1 (𝑜𝑑𝑑) = ∑ 𝑎𝑛 cos [2(𝑛 − 1)𝑢]

𝑛=1
𝜋𝑑
Where 𝑢 = 𝑐𝑜𝑠𝜃
𝜆

𝑎𝑛 is excitation coefficient

I. Dolph-Tschebyscheff Array: Broadside

• Array, with many practical applications, is the Dolph-Tschebyscheff array.
• The method was originally introduced by Dolph. It is primarily a compromise between
uniform and binomial arrays.
• Its excitation coefficients are related to Tschebyscheff polynomials.
• A Dolph-Tschebyscheff array with no side lobes (or side lobes of −∞ dB) reduces to
the binomial design. The excitation coefficients for this case, as obtained by both
methods, would be identical.
Array Factor( for odd and even number of elements)
𝑀

(𝐴𝐹)2𝑀 (𝑒𝑣𝑒𝑛) = ∑ 𝑎𝑛 cos [(2𝑛 − 1)𝑢]

𝑛=1

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𝑀+1

(𝐴𝐹)2𝑀+1 (𝑜𝑑𝑑) = ∑ 𝑎𝑛 cos [2(𝑛 − 1)𝑢]

𝑛=1
𝜋𝑑
Where 𝑢 = 𝑐𝑜𝑠𝜃
𝜆

𝑎𝑛 is excitation coefficient
• The array factor of an array of even or odd number of elements with symmetric
amplitude excitation is nothing more than a summation of M or M + 1 cosine terms.
• The largest harmonic of the cosine terms is one less than the total number of elements
of the array. Each cosine term, whose argument is an integer times a fundamental
frequency, can be rewritten as a series of cosine functions with the fundamental
frequency as the argument.
That is,
m=0 cos(mu) =1
m=1 cos(mu) = cosu
m=2 cos(mu) = cos(2u) = 2 cos2u-1
m=3 cos(mu) = cos(3u) = 4cos3u-3cosu
m=4 cos(mu) = cos(4u) = 8cos4u-8cos2u+1
m=5 cos(mu) = cos(5u) = 16cos5u-20cos3u+5cosu
m=6 cos(mu) = cos(6u) = 32cos6u-48cos4u+18cos2u-1
m=7 cos(mu) = cos(7u) = 64cos7u-112cos5u+56cos3u-7cosu
m=8 cos(mu) = cos(8u) = 128cos8u-256cos6u+160cos4u-32cos2u+1
m=9 cos(mu) = cos(9u) = 256cos9u-576cos7u+432cos5u-120cos3u+9cosu
The above equations are obtained using Euler’s formula:
𝑚
[𝑒 𝑗𝑢 ] = (𝑐𝑜𝑠𝑢 + 𝑗𝑠𝑖𝑛𝑢)𝑚 = 𝑒 𝑗𝑚𝑢 = cos(mu)+jsin(mu)

And sin2u=1-cos2u
If we let z = cosu
The above equations can be written as
m=0 cos(mu) =1=T0(z)
m=1 cos(mu) = cosu = z = T1(z)
m=2 cos(mu) = cos(2u) = 2z2-1 = T2(z)
m=3 cos(mu) = cos(3u) = 4z3-3z = T3(z)
m=4 cos(mu) = cos(4u) = 8z4-8z2+1= T4(z)

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m=5 cos(mu) = cos(5u) = 16z5-20z3+5z= T5(z)

m=6 cos(mu) = cos(6u) = 32z6-48z4+18z2-1= T6(z)
m=7 cos(mu) = cos(7u) = 64z7-112z5+56z3-7z= T7(z)
m=8 cos(mu) = cos(8u) = 128z8-256z6+160z4-32z2+1 = T8(z)
m=9 cos(mu) = cos(9u) = 256z9-576z7+432z5-120z3+9z= T9(z)
and each is related to a Tschebyscheff (Chebyshev) polynomial Tm(z). These relations between
the cosine functions and the Tschebyscheff polynomials are valid only in the −1 ≤ z ≤ +1 range.
• Because | cos(mu)| ≤ 1, each Tschebyscheff polynomial is|Tm(z)| ≤ 1 for −1 ≤ z ≤ +1.
For |z| > 1, the Tschebyscheff polynomials are related to the hyperbolic cosine
functions.
• The recursion formula for Tschebyscheff polynomials is
𝑇𝑚 (𝑧) = 2𝑧𝑇𝑚−1 (𝑧) − 𝑇𝑚−2 (𝑧)
• Each polynomial can also be computed using
Tm(z) = cos[m cos−1(z)] − 1 ≤ z ≤ +1
Tm(z) = cosh[m cosh−1(z)] z < −1,z > +1
In Figure below, the first six Tschebyscheff polynomials have been plotted.

The following properties of the polynomials are of interest:

1. All polynomials, of any order, pass through the point (1, 1).
2. Within the range −1 ≤ z ≤ 1, the polynomials have values within −1 to +1.
3. All roots occur within −1 ≤ z ≤ 1, and all maxima and minima have values of +1 and −1,
respectively.

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Since the array factor of an even or odd number of elements is a summation of cosine
terms whose form is the same as the Tschebyscheff polynomials, the unknown coefficients of
the array factor can be determined by equating the series representing the cosine terms of the
array factor to the appropriate Tschebyscheff polynomial. The order of the polynomial should
be one less than the total number of elements of the array.

Outline of the design procedure

Assumptions:
Number of elements, spacing between the elements, and ratio of major-to-minor lobe intensity
(R0) are known. The requirements will be to determine the excitation coefficients and the array
factor of a Dolph-Tschebyscheff array.
Problem Statement
Design a broadside Dolph-Tschebyscheff array of 2M or 2M + 1 elements with spacing d
between the elements. The side lobes are R0 dB below the maximum of the major lobe. Find
the excitation coefficients and form the array factor.
Procedure
a. Select the appropriate array factor ( for odd or even number of array elements).
b. Expand the array factor. Replace each cos(mu) function (m = 0, 1, 2, 3,…) by its appropriate
series expansion.
c. Determine the point z = z0 such that Tm(z0) = R0 (voltage ratio). The order m of the
Tschebyscheff polynomial is always one less than the total number of elements. The design
procedure requires that the Tschebyscheff polynomial in the −1 ≤ z ≤ z1, where z1 is the null
nearest to z = +1, be used to represent the minor lobes of the array. The major lobe of the
pattern is formed from the remaining part of the polynomial up to point z0(z1 < z ≤ z0).
d. Substitute cos(u) = z /z0 in the array factor of step b. The cos(u) is replaced by z∕z0, and not
by z, so that the equation, cos(u) = z /z0 would be valid for |z| ≤ |z0|.At |z| = |z0|, the equation,
cos(u)=z /z0 attains its maximum value of unity.
e. Equate the array factor from step b, after substitution of cos(u) = z /z0, to a Tm(z). The Tm(z)
chosen should be of order m where m is an integer equal to one less than the total number of
elements of the designed array. This will allow the determination of the excitation coefficients
an’s.
f. Write the array factor using the coefficients found in step e.
Example problem
Design a broadside Dolph-Tschebyscheff array of 10 elements with spacing d between the
elements and with a major-to-minor lobe ratio of 26 dB. Find the excitation coefficients
and form the array factor.
Solution:
1.The array factor is given by
𝑀

(𝐴𝐹)2𝑀 (𝑒𝑣𝑒𝑛) = ∑ 𝑎𝑛 cos [(2𝑛 − 1)𝑢]

𝑛=1
𝜋𝑑
Where, 𝑢= 𝑐𝑜𝑠𝜃 and 𝑎𝑛 is excitation coefficient
𝜆

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2. When expanded, the array factor can be written as

(AF) 10= a1 cos(u) + a2 cos(3u) + a3 cos(5u) + a4 cos(7u) + a5 cos(9u)
Replace cos(u), cos(3u), cos(5u), cos(7u), and cos(9u) by their series expansions
cosu = z
cos(3u) = 4z3-3z
cos(5u) = 16z5-20z3+5z
cos(7u) = 64z7-112z5+56z3-7z
cos(9u) = 256z9-576z7+432z5-120z3+9z
(AF) 10= a1 [z] + a2 [4z3-3z ] + a3 [16z5-20z3+5z] + a4 [64z7-112z5+56z3-7z]
+ a5 [256z9-576z7+432z5-120z3+9z ]
(AF)10 = z[(a1 − 3a2 + 5a3 − 7a4 + 9a5)] + z3[(4a2 − 20a3 + 56a4 − 120a5)]
+ z5[(16a3 − 112a4 + 432a5)] + z7[(64a4 − 576a5)] + z9[(256a5)]

3. R0 (dB) = 26 = 20 log10(R0) or R0 (voltage ratio) = 20.

Determine z0 by equating R0 to T9(z0).
Thus R0 = 20 = T9(z0) = cosh[9 cosh−1(z0)]
or z0 = cosh[ (1 /9) cosh−1(20)] = 1.0851

Note:Another equation which can, in general, be used to find z0 and does not require hyperbolic
functions is
1 1
𝑃 𝑃
1
𝑧0 = 2 [(𝑅0 + √𝑅0 2 − 1 ) + (𝑅0 − √𝑅0 2 − 1 ) ]

where P is an integer equal to one less than the number of array elements (in this case P=9)

4.Substitute cos(u) = z/z0 = z /1.0851 in the array factor found in step 2.

5.Equate the array factor of step 2, after the substitution from step 4, to T9(z).
(AF)10 = z[(a1 − 3a2 + 5a3 − 7a4 + 9a5)/z0]
+ z3[(4a2 − 20a3 + 56a4 − 120a5)/z03] + z5[(16a3 − 112a4 + 432a5)/z05]
+ z7[(64a4 − 576a5)/z07] + z9[(256a5)/z09] = 256z9-576z7+432z5-120z3+9z = T9(z)
Matching similar terms allows the determination of the an’s.
That is,
(256a5)/z09 = 256 → a5 = 2.0856
(64a4 − 576a5)/z07 = -576 → a4 = 2.8308
(16a3 − 112a4 + 432a5)/z05 = 432 → a3 = 4.1184
(4a2 − 20a3 + 56a4 − 120a5)/z03 = -120 → a2 = 5.2073
(a1 − 3a2 + 5a3 − 7a4 + 9a5)/z0 = 9 → a1 = 5.8377
In normalized form, the an coefficients can be written as
a5 = 1 , a4 = 1.357, a3 = 1.974, a2 = 2.496 ,a1 = 2.798 normalized with respect to the amplitude
of the elements at the edge.
(The values can also be normalized with respect to the amplitude of the center element)

6. Using the set of normalized coefficients, the array factor can be written as
(AF)10 = 2.798 cos(u) + 2.496 cos(3u) + 1.974 cos(5u) + 1.357 cos(7u) + cos(9u)
where u = [(𝜋d∕λ) cos 𝜃].

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Binomial array:
• Current distribution follows the binomial series.
• The current amplitudes are proportional to the coefficients of the
successive terms of the Binomial series.

• Binomial array possess the smallest side lobes. If the spacing is or
2

less than , Binomial array has no side lobes.
2
• Binomial series :

(1 + x )m−1 = 1 + (m − 1)x + (m − 1)(m − 2) x 2 + (m − 1)(m − 2)(m − 3) x 3 + ............

2! 3!
• For m =3
(1 + x )3−1 = 1 + (3 − 1)x + (3 − 1)(3 − 2) x 2 + (3 − 1)(3 − 2)(3 − 3) x 3 + ............
2! 3!
= 1 + 2x + x 2

Current distribution for 3 element binomial array is 1:2:1

• Current distribution for 4 element binomial array is 1:3:3:1
So on…..
• Current distribution can be determined easily from Pascal’s Triangle

• Principle of pattern multiplication can be used to determine the resultant

radiation pattern.
• For 3 element Binomial array, the resultant pattern is shown below:

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• Here antenna 2 and 3 coincide , and so they would be replaced by with

a single antenna carrying double the current.
• Four element Binomial array is shown in figure below:

Comparison of Uniform array and Binomial array

Note:
• No side lobes in the radiation pattern of Binomial array
• Half Power Beam width is more
Disadvantages of Binomial array:
(i) HPBW increases and hence the directivity decreases
(ii) For design of large array, larger amplitude ratio of sources required.

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Smart antennas
Smart antenna systems combine:
(i)Antenna arrays with
(ii)Digital signal processing algorithms to make the antenna systems smart.

• Smart antennas integrate antenna array technology and Digital Signal Processing(DSP)
Techniques to enhance communication system performance including
(a) Capacity improvement
(b) Range increase
(c) Link quality improvement
(d) Mitigation of fading
• These are accomplished by
(1)Beam steering
Placing Beam maxima toward Signals of Interest (SOI)
(2)Null Steering
Placing Beam minima, ideally nulls, toward Interfering signals; Signals Not of Interest
(SNOI)
(3) Spatially separate signals
Allowing different users to share the same spectral resources(SDMA)

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Beam forming in Smart antenna systems

By means of an internal feed back control, smart antenna can generate a customized radiation
pattern to each remote user. In general, they form a main lobe toward a desired signal and
rejects interference outside the main lobe.
There are two types of smart antenna system
(i) Switched – beam systems
(ii) Adaptive antenna systems

(a)Switched Beam Systems

• They use a number of fixed beams at the base station. The base station selects one of
the pre- determined fixed beam that provides the greatest output power for the desired
user.

Figure: Concept of Switched Beam systems

• This concept is obviously an extension of cell sectoring as each sector is subdivided
into smaller sectors. As the mobile unit moves throughout the cell, the switched-beam
system detects the signal strength, chooses the appropriate predefined beam pattern.
Advantages over Adaptive antenna systems:
Low cost
Less complex and easier to retrofit to existing wireless technologies
Dis advantages over Adaptive antenna systems:
Beam Resolution is lower

(b) Adaptive array systems

Adaptive array systems, provide more degrees of freedom since they have the ability to
adapt in real time the radiation pattern to the RF signal environment. In other words, they can
direct the main beam toward the pilot signal or SOI while suppressing the antenna pattern in

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the direction of the interferers or SNOIs. To put it simply, adaptive array systems can customize
an appropriate radiation pattern for each individual user.
Adaptive array systems can locate and track signals (users and interferers) and
dynamically adjust the antenna pattern to enhance reception while minimizing interference
using signal-processing algorithms. A functional block diagram of such a system is shown in
Figure.This figure shows that after the system down converts the received signals to baseband
and digitizes them, it locates the SOI using the direction-of-arrival (DOA) algorithm, and it
continuously tracks the SOI and SNOIs by dynamically changing the weights (amplitudes and
phases of the signals).
Basically, the DOA computes the direction of arrival of all signals by computing the
time delays between the antenna elements, and afterward the adaptive algorithm, using a cost
function, computes the appropriate weights that result in an optimum radiation pattern.

Figure Functional block diagram of an adaptive array system

Optimal Beam forming techniques

In optimal beamforming techniques, a weight vector that minimizes a cost function is
determined. Typically, this cost function, related with a performance measure, is inversely
associated with the quality of the signal at the array output, so that when the cost function is
minimized, the quality of the signal is maximized at the array output.

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The most commonly used optimally beamforming techniques or performance measures

are the Minimum Mean Square Error (MMSE), Maximum Signal-to-Noise Ratio (MSNR), and
Minimum (noise) Variance (MV).
Adaptive algorithm
In practice, the signal environment is dynamic or time varying, and therefore, the
weights need to be computed with adaptive methods. One of the simplest algorithms that is
commonly used to adapt the weights is the Least Mean Square (LMS) algorithm. The LMS
algorithm is a low complexity algorithm that requires no direct matrix inversion and no
memory.

Comparison of switched-beam scheme and adaptive array scheme

(i)Minimizing Interference
Adaptive array systems can customize an appropriate radiation pattern for each
individual user. This is far superior to the performance of a switched-beam system, as shown
in Figure. This figure shows that not only the switched-beam system may not able to place the
desired signal at the maximum of the main lobe but also it exhibits the inability to fully reject
the interferers.

Figure Comparison of (a) switched-beam scheme, and (b) adaptive array scheme.
(ii)Coverage area comparison
Figure shows a comparison, in terms of relative coverage area, of conventional
sectorized, switched-beam and adaptive arrays. In the presence of a low-level interference, both
types of smart antennas provide significant gains over the conventional sectored systems.
However, when a high-level interference is present, the interference rejection capability of the

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adaptive systems provides significantly more coverage than either the conventional or
switched-beam system.

Figure: Relative coverage area comparison among sectorized systems, switched-

beam systems, and adaptive array systems in (a) low interference environment,
and (b) high interference environment

SMART ANTENNAS’ BENEFITS

(i) capacity increase
In densely populated areas, mobile systems are usually interference-limited, meaning
that the interference from other users is the main source of noise in the system. This means that
the signal-to interference ratio (SIR) is much smaller than the signal-to-noise ratio (SNR).
In general, smart antennas will, by simultaneously increasing the useful received signal
level and lowering the interference level, increase the SIR. (there by increase capacity)

(ii) range increase

Another benefit that smart-antenna systems provide is range increase. Because smart
antennas are more directional than omnidirectional and sectorized antennas, a range increase
potential is available.
In other words, smart antennas are able to focus their energy toward the intended users,
instead of directing it in other unnecessary directions (wasting) like omnidirectional antennas
do. This means that base stations can be placed further apart, leading to a more cost-efficient
development.
Therefore, in rural and sparsely populated areas, where radio coverage rather than
capacity is more important, smart-antenna systems are well suited.

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(iii) Provide security

Another added advantage of smart-antenna systems is security. Smart antennas make
it more difficult to tap a connection, because the intruder must be positioned in the same
direction as the user as seen from the base station to successfully tap a connection.

(iv) Helps in location finding

Because of the spatial detection nature of smart-antenna systems, they can be used to
locate humans in emergencies or for any other location-specific service.

SMART ANTENNAS’ DRAWBACKS

While smart antennas provide many benefits, they do suffer from certain drawbacks.

• Their transceivers are much more complex than traditional base station transceivers.
The antenna needs separate transceiver chains for each array antenna element and
accurate real-time calibration for each of them.

• The antenna beamforming is computationally intensive, which means that smart-

antenna base stations must be equipped with very powerful digital signal processors.
This tends to increase the system costs in the short term, but since the benefits outweigh
the costs, it will be less expensive in the long run.

• For a smart antenna to have pattern-adaptive capabilities and reasonable gain, an array
of antenna elements is necessary.

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EC8701

ANTENNAS AND MICROWAVE

ENGINEERING

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UNIT IV
PASSIVE AND ACTIVE MICROWAVE DEVICES

1. Microwave Passive components:

▪ Directional Coupler, Power Divider, Magic Tee, Attenuator, Resonator

2. Principles of Microwave Semiconductor Devices:

▪ Gunn Diodes, IMPATT diodes, Schottky Barrier diodes, PIN diodes

3. Microwave tubes:
▪ Klystron, TWT, Magnetron

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Directional Couplers
▪ A directional coupler is a four-port microwave junction as shown below:

Two commonly used symbols for directional couplers, and power flow conventions
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Directional Couplers
▪ Power supplied to port 1 is coupled to port 3 (the coupled port) with the coupling
factor 𝑆13 2 = 𝛽2 , while the remainder of the input power is delivered to port 2
(the through port) with the coefficient 𝑆12 2 = 𝛼 2 = 1 − 𝛽2 . In an ideal
directional coupler, no power is delivered to port 4 (the isolated port).
▪ The following quantities are commonly used to characterize a directional coupler:
𝑃1
𝐶𝑜𝑢𝑝𝑙𝑖𝑛𝑔 = 𝐶 = 10 log = −20 log 𝛽 dB
𝑃3
𝑃3 𝛽
𝐷𝑖𝑟𝑒𝑐𝑡𝑖𝑣𝑖𝑡𝑦 = 𝐷 = 10 log = 20 log dB
𝑃4 𝑆14
𝑃1
𝐼𝑠𝑜𝑙𝑎𝑡𝑖𝑜𝑛 = 𝐼 = 10 log = −20 log 𝑆14 dB
𝑃4
𝑃1
𝐼𝑛𝑠𝑒𝑟𝑡𝑖𝑜𝑛 𝑙𝑜𝑠𝑠 = 𝐿 = 10 log = −20 log 𝑆12 dB
𝑃2
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Directional Couplers
▪ The coupling factor indicates the fraction of the input power that is coupled to the
output port.
▪ The directivity is a measure of the coupler’s ability to isolate forward and
backward waves (or the coupled and uncoupled ports).
▪ The isolation is a measure of the power delivered to the uncoupled port.
▪ These quantities are related as
𝐼 = 𝐷 + 𝐶 dB
▪ The insertion loss accounts for the input power delivered to the through port,
diminished by power delivered to the coupled and isolated ports.
▪ The ideal coupler has infinite directivity and isolation 𝑆14 = 0 . Then both 𝛼 and
𝛽 can be determined from the coupling factor, C.

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Scattering Matrix of Directional Couplers
▪ The scattering matrix of a four-port network has the following form:
𝑆11 𝑆12 𝑆13 𝑆14
𝑆21 𝑆22 𝑆23 𝑆24
𝑆 =
𝑆31 𝑆32 𝑆33 𝑆34
𝑆41 𝑆42 𝑆43 𝑆44
▪ The scattering matrix of a reciprocal four-port network matched at all ports has the
following form:
0 𝑆12 𝑆13 𝑆14
𝑆12 0 𝑆23 𝑆24
𝑆 =
𝑆13 𝑆23 0 𝑆34
𝑆14 𝑆24 𝑆34 0
where 𝑆11 = 𝑆22 = 𝑆33 = 𝑆44 = 0 and
𝑆21 = 𝑆12 ; 𝑆31 = 𝑆13 ; 𝑆41 = 𝑆14 ; 𝑆32 = 𝑆23 ; 𝑆42 = 𝑆24 ; 𝑆43 = 𝑆34

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Scattering Matrix of Directional Couplers
Unitary Property of S-matrix:
𝑁
∗
𝛿𝑖𝑗 = 1 if 𝑖 = 𝑗
෍ 𝑆𝑘𝑖 𝑆𝑘𝑗 = 𝛿𝑖𝑗 for all 𝑖, 𝑗 where ൝
𝛿𝑖𝑗 = 0 if 𝑖 ≠ 𝑗
𝑘=1
▪ The dot product of any column of 𝑆 with the conjugate of that same column
gives unity. For 𝑖 = 𝑗,
𝑁
∗
෍ 𝑆𝑘𝑖 𝑆𝑘𝑖 =1
𝑘=1
▪ The dot product of any column of 𝑆 with the conjugate of a different column
gives zero (the columns are orthonormal).
𝑁
∗
෍ 𝑆𝑘𝑖 𝑆𝑘𝑗 = 0 for 𝑖 ≠ 𝑗
𝑘=1
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Scattering Matrix of Directional Couplers
0 𝑆12 𝑆13 𝑆14
𝑆12 0 𝑆23 𝑆24
𝑆 =
𝑆13 𝑆23 0 𝑆34
𝑆14 𝑆24 𝑆34 0
▪ If the network is lossless, 10 equations result from the unitary, or energy
conservation, condition. Consider the multiplication of row 1 and row 2, and the
multiplication of row 4 and row 3:
∗ ∗
𝑆13 𝑆23 + 𝑆14 𝑆24 = 0 1
∗ ∗
𝑆14 𝑆13 + 𝑆24 𝑆23 = 0 2
∗ ∗
▪ Multiply 1 by 𝑆24 , and 2 by 𝑆13 ,
∗ ∗ ∗ ∗ ∗ ∗ ∗
𝑆13 𝑆23 𝑆24 + 𝑆14 𝑆24 𝑆24 = 0 ⟹ 𝑆13 𝑆23 𝑆24 + 𝑆14 𝑆24 2 = 0 3
∗ ∗ ∗ ∗ ∗ ∗ ∗
𝑆14 𝑆13 𝑆13 + 𝑆24 𝑆23 𝑆13 = 0 ⟹ 𝑆14 𝑆13 2 + 𝑆24 𝑆23 𝑆13 =0 4

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Scattering Matrix of Directional Couplers
∗ ∗ ∗ ∗ ∗ ∗ ∗
𝑆13 𝑆23 𝑆24 + 𝑆14 𝑆24 𝑆24 = 0 ⟹ 𝑆13 𝑆23 𝑆24 + 𝑆14 𝑆24 2 = 0 3
∗ ∗ ∗ ∗ ∗ ∗ ∗
𝑆14 𝑆13 𝑆13 + 𝑆24 𝑆23 𝑆13 = 0 ⟹ 𝑆14 𝑆13 2 + 𝑆24 𝑆23 𝑆13 =0 4
▪ Subtract 3 from 4 ,
∗
𝑆14 𝑆13 2 − 𝑆24 2 = 0 5
▪ Similarly, the multiplication of row 1 and row 3, and the multiplication of row 4
and row 2, gives
∗ ∗
𝑆12 𝑆23 + 𝑆14 𝑆34 = 0 6
∗ ∗
𝑆14 𝑆12 + 𝑆34 𝑆23 = 0 7
▪ Multiply 6 by 𝑆12 , and 7 by 𝑆34 ,
∗ ∗ ∗
𝑆12 𝑆12 𝑆23 + 𝑆12 𝑆14 𝑆34 = 0 ⟹ 𝑆12 2 𝑆23 + 𝑆12 𝑆14 𝑆34 = 0 8
∗ ∗ ∗
𝑆34 𝑆14 𝑆12 + 𝑆34 𝑆34 𝑆23 = 0 ⟹ 𝑆34 𝑆14 𝑆12 + 𝑆34 2 𝑆23 = 0 9

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Scattering Matrix of Directional Couplers
∗ ∗ ∗
𝑆12 𝑆12 𝑆23 + 𝑆12 𝑆14 𝑆34 = 0 ⟹ 𝑆12 2 𝑆23 + 𝑆12 𝑆14 𝑆34 = 0 8
∗ ∗ ∗
𝑆34 𝑆14 𝑆12 + 𝑆34 𝑆34 𝑆23 = 0 ⟹ 𝑆34 𝑆14 𝑆12 + 𝑆34 2 𝑆23 = 0 9
▪ Subtract 9 from 8 ,
𝑆23 𝑆12 2 − 𝑆34 2 = 0 10
▪ One way for 5 and 10 to be satisfied is if 𝑆14 = 𝑆23 = 0, which results in a
directional coupler.

0 𝑆12 𝑆13 0
𝑆 0 0 𝑆24
𝑆 = 12
𝑆13 0 0 𝑆34
0 𝑆24 𝑆34 0

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Scattering Matrix of Directional Couplers
0 𝑆12 𝑆13 0
𝑆12 0 0 𝑆24
𝑆 =
𝑆13 0 0 𝑆34
0 𝑆24 𝑆34 0
▪ Then the self-products of the rows of the unitary scattering matrix yield the
following equations:
𝑆12 2 + 𝑆13 2 = 1 11
𝑆12 2 + 𝑆24 2 = 1 12
𝑆13 2 + 𝑆34 2 = 1 13
𝑆24 2 + 𝑆34 2 = 1 14
11 − 12 ⟹ 𝑆13 = 𝑆24
12 − 14 ⟹ 𝑆12 = 𝑆34

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Scattering Matrix of Directional Couplers
▪ Further simplification can be made by choosing the phase references on three of
the four ports. Thus, we choose 𝑆12 = 𝑆34 = 𝛼, 𝑆13 = 𝛽𝑒 𝑗𝜃 , and 𝑆24 = 𝛽𝑒 𝑗𝜙 ,
where 𝛼 and 𝛽 are real, and 𝜃 and 𝜙 are phase constants to be determined (one of
which we are still free to choose). The dot product of rows 2 and 3 gives
∗ ∗
𝑆12 𝑆13 + 𝑆24 𝑆34 = 0
which yields a relation between the remaining phase constants as
𝜃 + 𝜙 = 𝜋 ± 2𝑛𝜋
If we ignore integer multiples of 2π, there are two particular choices that commonly
occur in practice:
1. Symmetric Coupler: 𝜃 = 𝜙 = 𝜋Τ2
2. Antisymmetric Coupler: 𝜃 = 0, 𝜙 = 𝜋

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Scattering Matrix of Directional Couplers
1. Symmetric Coupler: 𝜃 = 𝜙 = 𝜋Τ2,
0 𝛼 𝑗𝛽 0
𝛼 0 0 𝑗𝛽
𝑆 =
𝑗𝛽 0 0 𝛼
0 𝑗𝛽 𝛼 0
2. Antisymmetric Coupler: 𝜃 = 0, 𝜙 = 𝜋,
0 𝛼 𝛽 0
𝛼 0 0 −𝛽
𝑆 =
𝛽 0 0 𝛼
0 −𝛽 𝛼 0
Note that these two couplers differ only in the choice of reference planes. In
addition, the amplitudes α and β are not independent, as 11 requires that
𝛼 2 + 𝛽2 = 1

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Types of Directional Couplers
1. Bethe hole directional coupler:
▪ The directional property of all directional couplers is produced through the use of
two separate waves or wave components, which add in phase at the coupled port
and are canceled at the isolated port.
▪ One of the simplest ways of doing this is to couple one waveguide to another
through a single small hole in the common broad wall between the two
waveguides. Such a coupler is known as a Bethe hole coupler, two versions of
which are shown in the following figures.
▪ From the small-aperture coupling theory, an aperture can be replaced with
equivalent sources consisting of electric and magnetic dipole moments.
▪ An incident 𝑇𝐸10 mode in guide 1, with an amplitude A, produces a normal
electric dipole in the aperture plus a tangential magnetic dipole proportional and in
the same direction as the magnetic field of the incident wave.

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Types of Directional Couplers
Two versions of the Bethe hole directional coupler:

(a) Parallel waveguides

(b) Skewed waveguides

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Types of Directional Couplers
1. Bethe hole directional coupler:
▪ In the upper guide, the normal electric dipole and the axial component of the
magnetic dipole radiate symmetrically in both directions.
▪ The transverse component of the magnetic dipole radiates antisymmetrically.
▪ The normal electric dipole moment and the axial magnetic dipole moment radiate
with even symmetry in the coupled guide, while the transverse magnetic dipole
moment radiates with odd symmetry.
▪ Thus, by adjusting the relative amplitudes of these two equivalent sources, we can
cancel the radiation in the direction of the isolated port, while enhancing the
radiation in the direction of the coupled port.
▪ Above figure shows two ways in which these wave amplitudes can be controlled;
in the coupler shown in figure(a), the two waveguides are parallel and the
coupling is controlled by 𝑠, the aperture offset from the sidewall of the waveguide.
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Types of Directional Couplers
1. Bethe hole directional coupler:
▪ For the coupler of figure(b), the wave amplitudes are controlled by the angle,𝜃,
between the two waveguides.
▪ First consider the configuration of figure (a), with an incident 𝑇𝐸10 mode into
port 1. These fields can be written as
𝜋𝑥 −𝑗𝛽𝑧
𝐸𝑦 = 𝐴 sin 𝑒
𝑎
−𝐴 𝜋𝑥 −𝑗𝛽𝑧
𝐻𝑥 = sin 𝑒
𝑍10 𝑎
𝑗𝜋𝐴 𝜋𝑥 −𝑗𝛽𝑧
𝐻𝑧 = cos 𝑒
𝛽𝑎𝑍10 𝑎
where 𝑍10 = 𝑘0 𝜂0 Τ𝛽 is the wave impedance of the 𝑇𝐸10 mode .

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Types of Directional Couplers
1. Bethe hole directional coupler:
▪ The amplitudes of the forward and reverse traveling waves in the top guide are
−𝑗𝜔𝐴 𝜋𝑠 𝜇 𝛼 𝜋𝑠 𝜋 2 𝜋𝑠
+ 2 0 𝑚 2 2
𝐴10 = 𝜖0 𝛼𝑒 𝑠𝑖𝑛 − 2 𝑠𝑖𝑛 + 2 2 𝑐𝑜𝑠
𝑃10 𝑎 𝑍10 𝑎 𝛽 𝑎 𝑎
2
−
−𝑗𝜔𝐴 2
𝜋𝑠 𝜇 𝛼
0 𝑚 2
𝜋𝑠 𝜋 2
𝜋𝑠
𝐴10 = 𝜖0 𝛼𝑒 𝑠𝑖𝑛 + 2 𝑠𝑖𝑛 − 2 2 𝑐𝑜𝑠
𝑃10 𝑎 𝑍10 𝑎 𝛽 𝑎 𝑎
where 𝑃10 = 𝑎𝑏/𝑍10 is the power normalization constant.

▪ Note from above expressions that the amplitude of the wave excited toward port 4
+ −
𝐴10 is generally different from that excited toward port 3 𝐴10 , so we can cancel
+
the power delivered to port 4 by setting 𝐴10 = 0.

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Types of Directional Couplers
1. Bethe hole directional coupler:
▪ If we assume that the aperture is round, then the polarizabilities as 𝛼𝑒 = 2𝑟03 /3
and 𝛼𝑚 = 4𝑟03 /3, where 𝑟0 is the radius of the aperture. Then we obtain the
+
following condition 𝐴10 = 0.
𝜋𝑠 𝜆0
sin =
𝑎
2 𝜆20 − 𝑎2
▪ The coupling factor is then given by
𝐴
𝐶 = 20 log − dB
𝐴10
and the directivity by
−
𝐴10
𝐷 = 20 log + dB
𝐴10
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Types of Directional Couplers
1. Bethe hole directional coupler:
▪ For the skewed geometry of figure(b), the aperture may be centered at 𝑠 = 𝑎/2,
and the skew angle 𝜃 adjusted for cancellation at port 4. In this case, the normal
electric field does not change with 𝜃 , but the transverse magnetic field
components are reduced by cos θ. We can account for the skew angle by replacing
𝛼𝑚 in the previous derivation by 𝛼𝑚 cos 𝜃. The wave amplitudes, then become,
for 𝑠 = 𝑎/2,
+
−𝑗𝜔𝐴 𝜇0 𝛼𝑚
𝐴10 = 𝜖0 𝛼𝑒 − 2 cos 𝜃
𝑃10 𝑍10
−
−𝑗𝜔𝐴 𝜇0 𝛼𝑚
𝐴10 = 𝜖0 𝛼𝑒 + 2 cos 𝜃
𝑃10 𝑍10

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Types of Directional Couplers
1. Bethe hole directional coupler:
+
▪ Setting 𝐴10 = 0 results in the following condition for the angle 𝜃,
𝑘02
cos 𝜃 = 2
2𝛽
▪ The coupling factor then simplifies to
𝐴 4𝑘02 𝑟03
𝐶 = 20 log − = −20 log dB
𝐴10 3𝑎𝑏𝛽
▪ The angular geometry of the skewed Bethe hole coupler is often a disadvantage in
terms of fabrication and application. In addition, both coupler designs operate
properly only at the design frequency; deviation from this frequency will alter the
coupling level and the directivity

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Types of Directional Couplers
2. Multihole Couplers:
▪ A single-hole coupler has a relatively narrow bandwidth, at least in terms of its
directivity. However, if the coupler is designed with a series of coupling holes, the
extra degrees of freedom can be used to increase this bandwidth.
▪ First let us consider the operation of the two-hole coupler shown in figure. Two
parallel waveguides sharing a common broad wall are shown, although the same
type of structure could be made in microstrip line or stripline form.
▪ Two small apertures are spaced 𝜆𝑔 Τ4 apart and couple the two guides. A wave
entering at port 1 is mostly transmitted through to port 2, but some power is
coupled through the two apertures.
▪ If a phase reference is taken at the first aperture, then the phase of the wave
incident at the second aperture will be −90°.

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Types of Directional Couplers
2. Multihole Couplers:

Basic operation of a two-hole directional coupler

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Types of Directional Couplers
2. Multihole Couplers:
▪ Each aperture will radiate a forward wave component and a backward wave
component into the upper guide; in general, the forward and backward amplitudes
are different.
▪ In the direction of port 3, both wave components are in phase because both have
traveled 𝜆𝑔 Τ4 to the second aperture.
▪ However, we obtain a cancellation in the direction of port 4 because the wave
coming through the second aperture travels 𝜆𝑔 Τ2 further than the wave component
coming through the first aperture.
▪ Clearly, this cancellation is frequency sensitive, making the directivity a sensitive
function of frequency.
▪ The coupling is less frequency dependent since the path lengths from port 1 to
port 3 are always the same.
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Types of Directional Couplers
2. Multihole Couplers:

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T-Junction Power Divider/ Waveguide Tees
▪ The T-junction power divider is a simple three-port network that can be used for
power division or power combining, and it can be implemented in virtually any
type of transmission line medium.

(a) E-plane waveguide T (b) H-plane waveguide T

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Scattering Matrix of Three-Port Networks (T-Junctions)
▪ The simplest type of power divider is a T-junction, which is a three-port network
with two inputs and one output. The scattering matrix of an arbitrary three-port
network has nine independent elements:
𝑆11 𝑆12 𝑆13
𝑆 = 𝑆21 𝑆22 𝑆23
𝑆31 𝑆32 𝑆33
▪ If the device is passive and contains no anisotropic materials, then it must be
reciprocal and its scattering matrix will be symmetric 𝑆𝑖𝑗 = 𝑆𝑗𝑖 .
▪ Usually, to avoid power loss, we would like to have a junction that is lossless and
matched at all ports. We can easily show, however, that it is impossible to
construct such a three-port lossless reciprocal network that is matched at all
ports.

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Scattering Matrix of Three-Port Networks (T-Junctions)
▪ If all ports are matched, then 𝑆𝑖𝑖 = 0, and if the network is reciprocal, the
scattering matrix reduces to
0 𝑆12 𝑆13
𝑆 = 𝑆12 0 𝑆23
𝑆13 𝑆23 0
▪ If the network is also lossless, then energy conservation requires that the scattering
matrix satisfy the unitary properties, which leads to the following conditions:
𝑆12 2 + 𝑆13 2 = 1 1
𝑆12 2 + 𝑆23 2 = 1 2
𝑆13 2 + 𝑆23 2 = 1 3
∗
𝑆13 𝑆23 = 0 4
∗
𝑆23 𝑆12 = 0 5
∗
𝑆12 𝑆13 = 0 6
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Scattering Matrix of Three-Port Networks (T-Junctions)
𝑆12 2
+ 𝑆13 2 = 1 1
𝑆122+ 𝑆 2 =1 2
23
𝑆132+ 𝑆 2 =1 3
23
∗
𝑆13 𝑆23 = 0 4
∗
𝑆23 𝑆12 = 0 5
∗
𝑆12 𝑆13 = 0 6
▪ Equations (4)-(5) show that at least two of the three parameters 𝑆12 , 𝑆13 , 𝑆23 must
be zero.
▪ However, this condition will always be inconsistent with one of equations (1)–(3),
implying that a three-port network cannot be simultaneously lossless, reciprocal,
and matched at all ports.
▪ If any one of these three conditions is relaxed, then a physically realizable device
is possible.
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E-Plane Tee
▪ An E-plane tee is a waveguide tee in which the axis of its side arm is parallel to
the E field of the main guide.

E-plane Tee or Series-Tee

▪ If the collinear arms are symmetric about the side arm, there are two different
transmission characteristics as shown in the following figure.

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E-Plane Tee

(a)Input through main arm

(b)Input from side arm

Two way transmission of E-plane Tee

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E-Plane Tee
▪ When waves are fed into the side arm (port 3), the waves appearing at port 1 and
port 2 of the collinear arm will be in opposite phase and in the same magnitude.
Therefore,
𝑆13 = −𝑆23
▪ In general, when an E-plane tee is constructed of an empty waveguide, it is poorly
matched at the tee junction. Hence 𝑆𝑖𝑗 ≠ 0 if 𝑖 = 𝑗. However, since the collinear
arm is usually symmetric about the side arm, 𝑆13 = 𝑆23 and 𝑆22 = 𝑆11 . Hence
the S-matrix is simplified as
𝑆11 𝑆12 𝑆13
𝑆 = 𝑆12 𝑆11 −𝑆13
𝑆13 −𝑆13 𝑆33

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E-Plane Tee
▪ If two in-phase waves are fed into port1 and 2 of the collinear arm, the output
waves at port 3 will be opposite in phase and subtractive. Sometimes, this third
port is called the difference arm.
▪ By analogy with the voltage relationship in a series circuit, E-plane tee is also
called Series-T .
▪ By suitable matching elements, we can make 𝑆33 = 0 so that 𝑆13 = −𝑆23

𝑆11 𝑆12 𝑆13

𝑆 = 𝑆12 𝑆11 −𝑆13
𝑆13 −𝑆13 0

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E-Plane Tee
▪ Based on power consideration,
1 1
𝑆13 = ⟹ 𝑆23 = −
2 2
▪ It can be shown that
1 1
𝑆11 = 𝑆22 = and 𝑆12 = 𝑆21 =
2 2
▪ S-matrix of E-plane Tee is
1 1 1
2 2 2
𝑆11 𝑆12 𝑆13 1 1 1 1 1 1 2
𝑆 = 𝑆12 𝑆11 −𝑆13 = − = 1 1 − 2
2 2 2 2
𝑆13 −𝑆13 0 2 − 2 0
1 1
− 0
2 2
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H-Plane Tee
▪ In an H-plane tee, if two in-phase input waves are fed into ports 1 and 2 of the
collinear arm, the output waves at Port 3 will be in-phase and additive. Because of
this, the third port is called the sum arm.

H-Tee or Shunt-T

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H-Plane Tee
▪ Conversely, an input wave at Port 3 will be equally divided into ports 1 and 2 in
phase. Because the magnetic field loops get divided into two arms 1 and 2 in a
manner similar to currents between branches in the parallel circuit, an H-plane
junction is also called a shunt junction.
▪ For a symmetrical and lossless junction, in absence of non-linear elements at the
H-plane junction, the S-parameters are obtained in a similar manner as in the case
of E-plane junction: Since it is a three-port junction, the scattering matrix can be
derived as follows:
𝑆11 𝑆12 𝑆13
𝑆 = 𝑆21 𝑆22 𝑆23
𝑆31 𝑆32 𝑆33

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H-Plane Tee
▪ Because of plane of symmetry of the junction, the Scattering coefficients are
𝑆23 = 𝑆13
▪ If Port 3 is perfectly matched to the junction, 𝑆33 = 0
▪ For symmetric property, 𝑆𝑖𝑗 = 𝑆𝑗𝑖
𝑆21 = 𝑆12 ; 𝑆31 = 𝑆13 ; 𝑆32 = 𝑆23 = 𝑆13
▪ With the above properties, 𝑆 becomes,
𝑆11 𝑆12 𝑆13 𝑆11 𝑆12 𝑆13
𝑆 = 𝑆21 𝑆22 𝑆23 = 𝑆12 𝑆22 𝑆13
𝑆31 𝑆32 𝑆33 𝑆13 𝑆13 0
▪ For the symmetry and lossless properties,
𝑆 𝑆 ∗𝑡 = 𝑈

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H-Plane Tee
𝑆11 2
+ 𝑆12 2 + 𝑆13 2 =1 1
𝑆12 2+ 𝑆 2 2
22 + 𝑆13 =1 2
𝑆13 2 + 𝑆13 2 =1 3
∗ ∗
𝑆13 𝑆11 + 𝑆13 𝑆12 =0 4
▪ From first two equations, we get 𝑆22 = 𝑆11
1
▪ From third equation, we get 𝑆13 =
2
▪ From last equation, we get
∗ ∗
𝑆13 𝑆11 + 𝑆12 =0
∗ ∗
⟹ 𝑆13 = 0 or 𝑆11 + 𝑆12 =0
∗ ∗
𝑆13 ≠ 0 ⟹ 𝑆11 + 𝑆12 = 0 ⟹ 𝑆12 = −𝑆11
1 1
▪ Substituting these values in the first equation, 𝑆11 = = 𝑆22 ⟹ 𝑆12 = −
2 2

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H-Plane Tee
1 1
𝑆11 = = 𝑆22 ⟹ 𝑆12 = −
2 2
1
𝑆13 =
2
▪ Substituting these values in S-matrix, we get S-matrix of the H-plane Tee
1 1 1
−
2 2 2
𝑆11 𝑆12 𝑆13 𝑆11 𝑆12 𝑆13 1 1 1
𝑆 = 𝑆21 𝑆22 𝑆23 = 𝑆12 𝑆22 𝑆13 = −
2 2 2
𝑆31 𝑆32 𝑆33 𝑆13 𝑆13 0
1 1
0
2 2
1 1 −1 2
𝑆 = −1 1 2
2
2
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Magic-Tee
▪ A hybrid tee is formed with the combination of the E-plane and H-plane tees and
is called a magic-T. It has four ports as shown in figure:

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Magic-Tee
▪ The magic-T has the following characteristics when all the ports are terminated
with a matched loads:
1. If two waves of equal magnitude and equal phase are fed into ports 1 and 2, the
output at Port 3 is subtractive and becomes zero and total output will appear
additively at the port 4. Hence, Port 3 is called the difference or E-arm and 4, the
sum or H-arm.
2. A wave incident at Port 3 (E-arm) divides equally between ports 1 and 2 but is
opposite in phase with no coupling to Port 4 (H-arm). Thus,
𝑆13 = −𝑆23 , 𝑆43 = 0
3. A wave incident at Port 4 (H-arm) divides equally between ports 1 and 2 in
phase with no coupling to port 3 (E-arm). Thus,
1
𝑆14 = 𝑆12 = = 𝑆24 = 𝑆42 𝑎𝑛𝑑 𝑆34 = 0
2
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Magic-Tee
4. A wave fed into one collinear port, 1 or 2, will not appear in the other collinear
Ports, 2 or 1, respectively. Hence, two collinear ports 1 and 2 are isolated from
each other, making
𝑆12 = 𝑆21 = 0
▪ A magic-T can be matched by putting tuning screws suitably in the E and H-arms
without destroying the symmetry of the junctions. Therefore, for an ideal lossless
magic-T matched at ports 3 and 4, 𝑆33 = 𝑆44 = 0. Therefore, the S-matrix for a
magic-T, matched at ports 3 and 4 given by
𝑆11 𝑆12 𝑆13 𝑆14 𝑆11 𝑆12 𝑆13 𝑆14
𝑆21 𝑆22 𝑆23 𝑆24 𝑆12 𝑆22 −𝑆13 𝑆14
𝑆 = =
𝑆31 𝑆32 𝑆33 𝑆34 𝑆13 −𝑆13 0 0
𝑆41 𝑆42 𝑆43 𝑆44 𝑆14 𝑆14 0 0

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Magic-Tee
𝑆11 𝑆12 𝑆13 𝑆14 𝑆11 𝑆12 𝑆13 𝑆14
𝑆21 𝑆22 𝑆23 𝑆24 𝑆12 𝑆22 −𝑆13 𝑆14
𝑆 = =
𝑆31 𝑆32 𝑆33 𝑆34 𝑆13 −𝑆13 0 0
𝑆41 𝑆42 𝑆43 𝑆44 𝑆14 𝑆14 0 0
▪ From the unitary property applied to rows 1 and 2, we get
𝑆11 2 + 𝑆12 2 + 𝑆13 2 + 𝑆14 2 = 1 1
𝑆12 2 + 𝑆22 2 + 𝑆13 2 + 𝑆14 2 = 1 2
▪ Subtracting these two equations:
𝑆11 2 + 𝑆22 2 = 0 ⟹ 𝑆11 = 𝑆22
1 1
▪ Form the unitary property applied to rows 3 and 4, 𝑆13 = and 𝑆14 =
2 2
2 2
▪ Substituting these values in equation(1), 𝑆11 + 𝑆12 = 0 ⟹ 𝑆11 = 𝑆12 = 0

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Magic-Tee
𝑆11 𝑆12 𝑆13 𝑆14 0 0 1Τ 2 1Τ 2
𝑆12 𝑆22 −𝑆13 𝑆14 0 0 −1Τ 2 1Τ 2
𝑆 = =
𝑆13 −𝑆13 0 0 1Τ 2 −1Τ 2 0 0
𝑆14 𝑆14 0 0 0 0
1Τ 2 1Τ 2

0 0 1 1
1 0 0 −1 1
𝑆 =
2 1 −1 0 0
1 1 0 0

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Attenuator
▪ Attenuators are passive devices used to control power levels in a microwave
system by partially absorbing the transmitted signal wave. Both fixed and variable
attenuators are designed using resistive films (aquadag coated dielectric sheet).
▪ A coaxial fixed attenuator uses a film with losses on the center conductor to
absorb some of the power as shown in figure(a).
▪ The fixed waveguide type [figure (b)] consists of a thin dielectric strip coated with
resistive film and placed at the center of the waveguide parallel to the maximum E
field.
▪ Induced current on the resistive film due to the incident wave results in power
dissipation, leading to attenuation of microwave energy.
▪ The dielectric strip is tapered at both ends up to a length of more than half
wavelength to reduce reflections. The resistive vane is supported by two dielectric
rods separated by an odd multiple of quarter wavelength and perpendicular to the
electric field [figure (b)].
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Attenuator

Microwave attenuator:
(a) coaxial line fixed attenuator (b) and (c) waveguide attenuators
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Attenuator
Variable-type Attenuator:
▪ A variable-type attenuator can be constructed
▪ by moving the resistive vane by means of micrometer screw from one side of
the narrow wall to the center where the E-field is maximum [figure(b)] or
▪ by changing the depth of insertion of a resistive vane at an E-field maximum
through a longitudinal slot at the middle of the broad wall as shown in
figure(c).
▪ A maximum of 90 dB attenuation is possible with VSWR of 1.05.
▪ The resistance card can be shaped to give a linear variation of attenuation with the
depth of insertion.

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Attenuator
Precision-type Variable Attenuator:
▪ A precision-type variable attenuator makes use of a circular waveguide section (C)
containing a very thin tapered resistive card (R2), to both sides of which are
connected axisymmetric sections of circular-to-rectangular waveguide tapered
transitions (RC1 and RC2) as shown in figure(a).
▪ The center circular section with the resistive card can be precisely rotated by 360°
with respect to the two fixed sections of circular to rectangular waveguide
transitions.
▪ The induced current on the resistive card R2 due to the incident signal is dissipated
as heat and produces attenuation of the transmitted signal.
▪ The incident TE10 dominant wave in the rectangular waveguide is converted into a
dominant TE11 mode in the circular waveguide.

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Attenuator
Precision-type Variable Attenuator:

R1,R2,R3:Tapered resistive cards; RC1 and RC2: Rectangular-to-circular waveguide

transitions; C: Circular waveguide section
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Attenuator
Precision-type Variable Attenuator:
▪ A very thin tapered resistive card is placed perpendicular to the E-field at the
circular end of each transition section so that it has a negligible effect on the field
perpendicular to it but absorbs any component parallel to it.
▪ Therefore, a pure TE11 mode is excited in the middle section.
▪ With reference to figure(b), if the resistive card in the center section is kept at an
angle 𝜃 relative to the E-field direction of the TE11 mode, the component 𝐸 cos 𝜃
parallel to the card gets absorbed while the component 𝐸 sin 𝜃 is transmitted
without attenuation. This later component finally appears as electric field
component 𝐸 𝑠𝑖𝑛2 𝜃 in the rectangular output guide.
▪ Therefore, the attenuation of the incident wave is
𝛼 = −20 log 𝑆21 = −20 log 𝑠𝑖𝑛2 𝜃 = −40 log sin 𝜃

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Attenuator
Electronically-Controlled Attenuators:
▪ For applications in various microwave systems, it is desirable to have an
attenuator whose attenuation can be controlled by the application of a suitable
signal, such as dc voltage or a bias current.
▪ Two devices that are suitable for use in an electronically controlled attenuator are
the PIN diode and a field-effect transistor.
▪ These devices can be used as variable resistors whose resistance is controlled by
the applied signal.
▪ The basic attenuator network is a symmetric T or 𝜋 network as shown in figures(a)
and (b).
▪ Resistor values 𝑅1 and 𝑅2 are chosen so that when the attenuator is terminated in a
resistance equal to the transmission-line characteristic impedance 𝑍𝑐 , the input is
matched, that is, 𝑍𝑖𝑛 = 𝑍𝑐 , and to provide an output voltage reduction by factor K.
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Attenuator
Electronically-Controlled Attenuators:

𝑅1 𝑅1 𝑅2

𝑍𝑐 𝑍𝑐
𝑅2 𝑍𝑐 𝑅1 𝑅1 𝑍𝑐
𝑉𝑔 𝑉𝑔

Two basic attenuator networks

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Attenuator
Electronically-Controlled Attenuators:
▪ For the T-network , we have
𝑅2 𝑅1 + 𝑍𝑐
𝑅𝑖𝑛 = 𝑅1 +
𝑅1 + 𝑅2 + 𝑍𝑐
▪ For 𝑅𝑖𝑛 = 𝑍𝑐 , We get the equation
𝑅2 𝑅1 + 𝑍𝑐
𝑍𝑐 = 𝑅1 +
𝑅1 + 𝑅2 + 𝑍𝑐
𝑍𝑐 𝑅1 + 𝑅2 + 𝑍𝑐 = 𝑅1 𝑅1 + 𝑅2 + 𝑍𝑐 + 𝑅2 𝑅1 + 𝑍𝑐
𝑍𝑐 𝑅1 + 𝑅2 + 𝑍𝑐2 = 𝑍𝑐 𝑅1 + 𝑅2 + 𝑅12 + 2𝑅1 𝑅2
⟹ 𝑍𝑐2 = 𝑅12 + 2𝑅1 𝑅2
▪ For 𝑅𝑖𝑛 = 𝑍𝑐 , the Thevenin impedance seen by the load equals 𝑍𝑐 .

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Attenuator
Electronically-Controlled Attenuators:
▪ The Thevenin open-circuit voltage across 𝑅2 is
𝑅2
𝑉𝑇𝐻 = 𝑉𝑔
𝑅1 + 𝑅2 + 𝑍𝑐
▪ The power delivered to the load is
2 2 2
1 𝑉𝑇𝐻 𝑅2 𝑉𝑔
𝑃𝐿 = 𝑍𝑐 =
2 2𝑍𝑐 𝑅1 + 𝑅2 + 𝑍𝑐 8𝑍𝑐
2
▪ The available power is 𝑉𝑔 ൗ8𝑍𝑐 , so that the power attenuation 𝐾 2 is given by
2
2
𝑅2
𝐾 =
𝑅1 + 𝑅2 + 𝑍𝑐

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Attenuator
Electronically-Controlled Attenuators:
▪ When K has been specified, and also requiring that 𝑅𝑖𝑛 = 𝑍𝑐 , we have two
equations that are readily solved for the required values of 𝑅1 and 𝑅2 . Thus, we
find that
1−𝐾
𝑅1 = 𝑍𝑐
1+𝐾
2𝐾
𝑅2 = 2
𝑍𝑐
1−𝐾
▪ For 10-dB attenuation in a 50 Ω system, 𝐾 = 0.1 which results in
𝑅1 = 25.97 Ω
𝑅2 = 35.14 Ω

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Attenuator
Electronically-Controlled Attenuators:
▪ For 3-dB attenuation in a 50 Ω system, 𝐾 = 0.5 which results in
𝑅1 = 8. 58Ω
𝑅2 = 141.4 Ω
▪ For a 𝜋-network, 𝑅1 and 𝑅2 are given by
1+𝐾
𝑅1 = 𝑍𝑐
1−𝐾
1 − 𝐾2
𝑅2 = 𝑍𝑐
2𝐾

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Microwave Resonators

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Contents
• Series and Parallel Resonant Circuits
• Q-factor (unloaded and loaded)
• Bandwidth
• Transmission Line Resonators
• Waveguide resonators

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Introduction
A resonator is a device or circuit that • Filters
exhibits resonance • Oscillators
• Tuned amplifiers
In an electrical circuit, resonance • Frequency meters
condition occurs at a frequency
when capacitive and inductive
reactances become equal in At frequencies near resonance, a
magnitude and electric energy microwave resonator can be
oscillates between electric field of a modeled as series or parallel RLC
capacitor and magnetic field of an lumped element electric circuit
inductor.
The basic properties of series and
Microwave resonators are used in a parallel RLC circuits are reviewed
variety of applications: first.
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Series RLC Circuit
For the series RLC circuit shown in the
𝐼 𝑅 𝐿
figure:
1
𝑍𝑖𝑛 = 𝑅 + 𝑗𝜔𝐿 − 𝑗 +
𝜔𝐶 𝑉 𝐶
1
= 𝑅 + 𝑗𝜔𝐿 1 − 2 −
𝜔 𝐿𝐶
𝑍𝑖𝑛
𝜔02
𝑍𝑖𝑛 = 𝑅 + 𝑗𝜔𝐿 1 − 2
𝜔 𝑅𝑒(𝑍𝑖𝑛 ) 𝐼𝑚(𝑍𝑖𝑛 )

1 𝑅
where 𝜔0 =
𝐿𝐶
0 𝜔 0 𝜔0 𝜔

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Series RLC Circuit
𝑍𝑖𝑛 (𝜔) = 𝑅 2 + 𝜔 2 𝐿2 1 − 𝜔02 Τ𝜔 2 2 𝑍𝑖𝑛 𝜔

An important parameter of the resonant circuit

is its 𝑄 which is defined as:
𝑅Τ0.707
average energy stored 𝑅
𝑄=𝜔
average power dissipated
1 𝜔 Τ 𝜔0
𝑊𝑚 + 𝑊𝑒
𝑄=𝜔
𝑃loss
At resonance, 𝑊𝑚 = 𝑊𝑒
The 𝑄 of a resonator itself disregarding the external loading effect is called the
unloaded 𝑄 and denoted by 𝑄0
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Series RLC Circuit
2𝑊𝑚
Therefore, 𝑄0 = 𝜔0 𝜔2 − 𝜔02
𝑃loss
2𝐿 𝑍𝑖𝑛 = 𝑅 + 𝑗𝜔𝐿
𝐼 𝐿 𝜔2
𝑄0 = 𝜔0 2 = 𝜔0
𝐼 𝑅 𝑅 In the vicinity of resonance,
1
Since 𝜔0 2 = 𝜔2 − 𝜔02 =
𝐿𝐶 𝜔 − 𝜔0 𝜔 + 𝜔0 ≈2𝜔∆𝜔
1
𝑄0 = For small ∆𝜔,
𝜔0 𝑅𝐶
2𝜔∆𝜔
𝑍𝑖𝑛 ≈ 𝑅 + 𝑗𝜔𝐿 ≈ 𝑅 + 𝑗2∆𝜔𝐿
𝜔2
Let us now study the behavior of the
input impedance of a series RLC 𝑗2∆𝜔𝑅𝑄0
resonator near its resonance 𝑍𝑖𝑛 ≈𝑅+
𝜔0

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Series RLC Circuit
Let us now consider half power fractional bandwidth of the resonator
1 1 𝑉 2 1 𝑉 2
We have 𝑃𝑖𝑛 = 𝑉𝐼∗ = 𝑍 Therefore, 𝑅𝑒 𝑃𝑖𝑛 = R
2 2 𝑖𝑛 𝑍𝑖𝑛 2 𝑍𝑖𝑛
𝑉2
When 𝜔 = 𝜔0 , 𝑍𝑖𝑛 = 𝑅 and 𝑅𝑒 𝑃𝑖𝑛 ‫𝜔=𝜔ۂ‬0 =
2𝑅
2 𝑅 1
When 𝑍𝑖𝑛 = 2𝑅2 that is 𝑍𝑖𝑛 = 𝑅𝑒 𝑃𝑖𝑛 = 𝑅𝑒 𝑃𝑖𝑛 ‫𝜔=𝜔ۂ‬0
0.707 2
𝑗2∆𝜔𝑅𝑄0 2 2 4∆𝜔2 𝑅 2 𝑄02
From 𝑍𝑖𝑛 ≈ 𝑅 + , 𝑍𝑖𝑛 =𝑅 + = 2𝑅2
𝜔0 𝜔02

2∆𝜔 2 1 2∆𝜔 1
⇒ = Therefore, fractional bandwidth =
𝜔0 𝑄02 𝜔0 𝑄0

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Parallel RLC Circuit
For the parallel RLC circuit shown in the figure: 𝐼
1 1 +
𝑍𝑖𝑛 = =
1
+
1
+ 𝑗𝜔𝐶 1 𝑗 𝑉 𝑅 𝐿 𝐶
𝑅 𝑗𝜔𝐿 − 1 − 𝜔 2 𝐿𝐶
𝑅 𝜔𝐿
−
2
1 1 − 𝜔 𝐿𝐶
+ 𝑗
𝑅 𝜔𝐿 𝑍𝑖𝑛
𝑍𝑖𝑛 = 2 2
1 1 − 𝜔 2 𝐿𝐶 𝑅𝑒 𝑍𝑖𝑛
+
𝑅 𝜔𝐿 𝐼𝑚 𝑍𝑖𝑛
𝑅
𝑅𝑒 𝑍𝑖𝑛 attains its maximum value 𝑅 at the
resonance frequency
𝜔
0 𝜔0 𝜔 0 𝜔0
1
𝜔0 = 𝐿𝐶

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Parallel RLC Circuit
For such parallel RLC circuit
𝑅 𝑍𝑖𝑛 𝜔
𝑄0 = 𝜔0 𝑅𝐶 =
𝜔0 𝐿
Near resonance 𝜔 = 𝜔0 + ∆𝜔 𝑅
−1
1 1
𝑍𝑖𝑛 = + 𝑗𝜔𝐶 + 0.707𝑅
𝑅 𝑗𝜔𝐿
−1
1 1
= + 𝑗𝜔0 𝐶 + 𝑗∆𝜔𝐶 +
𝑅 𝑗 𝜔0 + ∆𝜔 𝐿 1 𝜔 Τ 𝜔0
−1
1 1
𝑍𝑖𝑛 = + 𝑗𝜔0 𝐶 + 𝑗∆𝜔𝐶 +
𝑅 𝑗𝜔0 𝐿 1 + ∆𝜔Τ𝜔0

When ∆𝜔Τ𝜔0 ≪ 1 we can use the approximation 1Τ 1 + ∆𝜔Τ𝜔0 ≈ 1 − ∆𝜔Τ𝜔0

−1
1 1 ∆𝜔
𝑍𝑖𝑛 ≈ + 𝑗𝜔0 𝐶 + 𝑗∆𝜔𝐶 + −
𝑅 𝑗𝜔0 𝐿 𝑗𝜔02 𝐿
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Parallel RLC Circuit
−1
1 1 ∆𝜔
𝑍𝑖𝑛 ≈ + 𝑗𝜔0 𝐶 + 𝑗∆𝜔𝐶 + − 2
𝑅 𝑗𝜔0 𝐿 𝑗𝜔0 𝐿
−1
1 𝑗∆𝜔
𝑍𝑖𝑛 = + 𝑗∆𝜔𝐶 + 2
𝑅 𝜔0 𝐿
−1
1 𝑅
𝑍𝑖𝑛 = + 𝑗∆𝜔𝐶 + 𝑗∆𝜔𝐶 =
𝑅 1 + 𝑗2∆𝜔𝑅𝐶
Since 𝑄0 = 𝜔0 𝑅𝐶,
𝑅
𝑍𝑖𝑛 =
1 + 𝑗2∆𝜔 𝑄0 Τ𝜔0

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Parallel RLC Circuit
∗
1 𝑉 1 𝑉2 1 2 2
1
𝑅𝑒 𝑃𝑖𝑛 = 𝑉 = = 𝐼 𝑍𝑖𝑛
2 𝑅 2 𝑅 2 𝑅
1
At resonance 𝑅𝑒 𝑃𝑖𝑛 ȁ𝜔=𝜔0 = 𝐼 2𝑅
2
𝑅𝑒 𝑃𝑖𝑛 𝑍𝑖𝑛 2
Therefore, =
𝑅𝑒 𝑃𝑖𝑛 ȁ𝜔=𝜔0 𝑅2
𝑅𝑒 𝑃𝑖𝑛 1 𝑅2 2
For to become , = 𝑍𝑖𝑛
𝑅𝑒 𝑃𝑖𝑛 ȁ𝜔=𝜔0 2 2
𝑅
From 𝑍𝑖𝑛 = , 2∆𝜔 𝑄0 Τ𝜔0 = 1
1+𝑗2∆𝜔𝑄0 Τ𝜔0
Therefore, fractional bandwidth 2∆𝜔Τ𝜔0 = 1Τ𝑄0

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Loaded Q
The unloaded 𝑄 of a circuit 𝑄0 is the quality factor of the circuit without any
external loading
In practice, external circuitry connected to the resonator will produce loading
effect.
Let the loading of the external circuit be represented by a load resistance 𝑅𝐿
and the Q of the external circuit by 𝑄𝑒 .
Let 𝑄𝐿 be the 𝑄of the loaded circuit.
𝐿 1 𝑅+𝑅𝐿 1 1
For series RLC circuit 𝑄𝐿 = 𝜔0 Therefore, = = +
𝑅+𝑅𝐿 𝑄𝐿 𝜔0 𝐿 𝑄0 𝑄𝑒
Similarly, for a parallel RLC circuit 𝑅 and 𝑅𝐿 are in parallel and
𝑅𝑅𝐿 1 𝜔0 𝑅+𝑅𝐿 𝐿 1 1
𝑄𝐿 = Therefore, = = +
𝜔0 𝑅+𝑅𝐿 𝐿 𝑄𝐿 𝑅𝑅𝐿 𝑄𝑒 𝑄0

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Transmission Line Resonators
Transmission line sections of various lengths and
terminations (open or short) can be used as a
resonator. 𝑙
Let us consider a lossy transmission line of length 𝑙 𝛼 𝛽 𝑍0
terminated to a short circuit at one end. The
transmission line is low loss with very small value of
attenuation constant 𝛼
𝑍𝑖𝑛
The transmission line has characteristic impedance 𝑍0
At 𝜔 = 𝜔0 𝑙 = 𝜆Τ2
𝑍𝐿 + 𝑍0 tanh 𝛾𝑙
𝑍𝑖𝑛 = 𝑍0
𝑍0 + 𝑍𝐿 tanh 𝛾𝑙

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Transmission Line Resonators
For 𝑍𝐿 = 0 𝑍𝑖𝑛 = 𝑍0 tanh 𝛾𝑙 = 𝑍0 tanh 𝛼 + 𝑗𝛽 𝑙
tanh 𝛼𝑙+𝑗 tan 𝛽𝑙
Therefore, 𝑍𝑖𝑛 = 𝑍0
1+𝑗 tan 𝛽𝑙 tanh 𝛼𝑙
Since we have considered a low loss line, 𝛼𝑙 ≪ 1
tanh 𝛼𝑙 ≈ 𝛼𝑙
𝜔𝑙 𝜔0 𝑙 ∆𝜔𝑙
𝛽𝑙 = = +
𝑣𝑝 𝑣𝑝 𝑣𝑝
𝜆 𝜔0 𝑙 2𝜋𝑓0 𝜆
Since 𝑙 = at 𝜔 = 𝜔0 , = =𝜋
2 𝑣𝑝 𝜆𝑓0 2

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Transmission Line Resonators
𝜔0 𝑙 ∆𝜔𝑙 𝜔0 𝑙
Now, 𝛽𝑙 = + and =𝜋
𝑣𝑝 𝑣𝑝 𝑣𝑝
∆𝜔𝜋 ∆𝜔𝜋 ∆𝜔𝜋
Therefore, 𝛽𝑙 = 𝜋 + and tan 𝛽𝑙 = tan 𝜋 + ≈
𝜔0 𝜔0 𝜔0
∆𝜔𝜋
tanh 𝛼𝑙+𝑗 tan 𝛽𝑙 𝛼𝑙+𝑗 𝜔
0
Hence, 𝑍𝑖𝑛 = 𝑍0 ≈ 𝑍0 ∆𝜔𝜋
1+𝑗 tan 𝛽𝑙 tanh 𝛼𝑙 1+𝑗𝛼𝑙
𝜔0
∆𝜔𝜋
Therefore, 𝑍𝑖𝑛 ≈ 𝑍0 𝛼𝑙 + 𝑗
𝜔0
Comparing with a series resonant circuit for which
𝑍𝑖𝑛 ≈ 𝑅 + 𝑗2∆𝜔𝐿
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Transmission Line Resonators
𝜋𝑍0
𝑅 = 𝑍0 𝛼𝑙 and 𝐿 =
2𝜔0
1 2
Capacitance 𝐶 can be found from 𝐶 = =
𝜔02 𝐿 𝜋𝜔0 𝑍0

𝜔0 𝐿 𝜋
Unloaded 𝑄 of the resonator 𝑄0 = =
𝑅 2𝛼𝑙

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Transmission Line Resonators
Let us now consider another transmission line resonator which
consists of a short-circuited transmission line of length 𝜆Τ4 .
𝜆
𝑙= at 𝜔 = 𝜔0
4
tanh 𝛼𝑙+𝑗 tan 𝛽𝑙
We have 𝑍𝑖𝑛 = 𝑍0
1+𝑗 tan 𝛽𝑙 tanh 𝛼𝑙
Multiplying the numerator and denominator by −𝑗 cot 𝛽𝑙
1 − 𝑗tanh 𝛼𝑙 cot 𝛽𝑙
𝑍𝑖𝑛 = 𝑍0
tanh 𝛼𝑙 − 𝑗 cot 𝛽𝑙

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Transmission Line Resonators
Let 𝜔 = 𝜔0 + ∆𝜔
𝜔0 𝑙 ∆𝜔𝑙 𝜋 𝜋∆𝜔
𝛽𝑙 = + = +
𝑣𝑝 𝑣𝑝 2 2𝜔0
𝜋∆𝜔 𝜋∆𝜔
Therefore, cot 𝛽𝑙 = − tan ≈ −
2𝜔0 2𝜔0
1−𝑗tanh 𝛼𝑙 cot 𝛽𝑙
We have 𝑍𝑖𝑛 = 𝑍0
tanh 𝛼𝑙−𝑗 cot 𝛽𝑙
𝜋∆𝜔
1+𝑗 𝛼𝑙 2𝜔 𝑍0
0
Therefore, 𝑍𝑖𝑛 = 𝑍0 𝜋∆𝜔 ≈ 𝜋∆𝜔
𝛼𝑙+𝑗 2𝜔 𝛼𝑙+𝑗 2𝜔
0 0

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Transmission Line Resonators
𝑍0 1
𝑍𝑖𝑛 ≈ =
𝜋∆𝜔 𝛼𝑙 𝜋∆𝜔
𝛼𝑙 + 𝑗
2𝜔0 𝑍0 + 𝑗 2𝜔0 𝑍0
For a parallel RLC circuit near resonance,
𝑅 1
𝑍𝑖𝑛 ≈ = 1
1+𝑗2∆𝜔𝑅𝐶 +𝑗2∆𝜔𝐶
𝑅
𝑍0 𝜋
Therefore, 𝑅 = and 𝐶 =
𝛼𝑙 4𝜔0 𝑍0
𝜋
𝑄0 = 𝜔0 𝑅𝐶 =
4𝛼𝑙

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Waveguide Resonators
At higher microwave frequencies transmission line
resonators have relatively low value of 𝑄.

Since open ended waveguide can radiate

significantly, waveguide resonators are usually short
circuited at both ends forming a cavity.
Rectangular cavity

Electric and magnetic energy is stored within the

cavity enclosed.

Cylindrical cavity
Dissipation of power takes place on the waveguide
walls as well as in the dielectric material filling the
cavity if the dielectric is lossy.
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Waveguide Resonators
Coupling to cavity resonator may be done using a small aperture or a probe or a
loop.

Aperture coupling
Probe coupling

Loop coupling
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Resonant Frequencies of Rectangular Cavity
We first find the resonant frequencies 𝑌
of the cavity assuming it to be 𝑏
lossless.
0
𝑎 𝑋
The unloaded 𝑄 of the cavity is then
determined considering small amount
of loss on the waveguide walls as well 𝑑
as in the dielectric material.
𝑍

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Resonant Frequencies of Rectangular Cavity
For TEmn or TMmn mode 𝑌
𝑏

𝐸𝑡 𝑥, 𝑦, 𝑧 = 𝑒Ԧ 𝑥, 𝑦 𝐴+ 𝑒 −𝑗𝛽𝑚𝑛 𝑧 + 𝐴− 𝑒 𝑗𝛽𝑚𝑛 𝑧 0
𝑎 𝑋

Transverse variation Amplitudes of forward

and backward wave
𝑑
where,
𝑚𝜋 2 𝑛𝜋 2 𝑍
𝛽𝑚𝑛 = 𝑘2 − −
𝑎 𝑏

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Resonant Frequencies of Rectangular Cavity
𝑌
𝐸𝑡 = 0 at 𝑧 = 0 ⇒ 𝐴+ = −𝐴−
𝑏
𝐸𝑡 = 0 at 𝑧 = 𝑑
0
𝑎 𝑋
∴ 𝐸𝑡 𝑥, 𝑦, 𝑑 = −𝑒Ԧ 𝑥, 𝑦 𝐴+ 2𝑗 sin 𝛽𝑚𝑛 𝑑 = 0

For 𝐴+ ≠ 0
𝑑
𝛽𝑚𝑛 𝑑 = 𝑙𝜋 where 𝑙 = 1,2,3 …
∴ For a rectangular cavity, the wave number 𝑍
2 For 𝑏<𝑎<𝑑 , the dominant
𝑚𝜋 2 𝑛𝜋 2 𝑙𝜋 𝜆𝑔
𝑘𝑚𝑛𝑙 = + + resonant mode is TE101 and 𝑑 =
2
𝑎 𝑏 𝑑
for TE10 mode.
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For TE10𝑙 mode we can write the field We have seen that for TE10 mode
components as follows:
𝜋𝑥 −𝑗𝛽𝑧 𝜋𝑥
𝐸𝑦 = 𝐴+ sin 𝑒 − 𝑒 𝑗𝛽𝑧 𝐻𝑧 = 𝐴10 cos 𝑒 −𝑗𝛽𝑧
𝑎 𝑎
𝐴+ 𝜋𝑥 −𝑗𝛽𝑧 −𝑗𝜔𝜇𝑎 𝜋𝑥 −𝑗𝛽𝑧
𝐻𝑥 = − sin 𝑒 + 𝑒 𝑗𝛽𝑧 𝐸𝑦 = 𝐴10 sin 𝑒
𝑍TE 𝑎 𝜋 𝑎
𝑗𝜋𝐴+ 𝜋𝑥 −𝑗𝛽𝑧 𝑗𝛽𝑎 𝜋𝑥 −𝑗𝛽𝑧
𝐻𝑧 = cos 𝑒 − 𝑒 𝑗𝛽𝑧 𝐻𝑥 = 𝐴 sin 𝑒
𝑘𝜂𝑎 𝑎 𝜋 10 𝑎
𝐻𝑦 = 𝐸𝑥 = 0
−𝑗𝜔𝜇𝑎
∴ 𝐴10 = 𝐴+
𝜋 ∵ 𝜔𝜇 = 𝑘𝜂
+𝜋
𝜋 𝑗𝐴
⇒ 𝐴10 = 𝑗𝐴+ =
𝜔𝜇𝑎 𝑘𝜂𝑎
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𝜋𝑥 −𝑗𝛽𝑧
𝐸𝑦 = 𝐴+ sin 𝑒 − 𝑒 𝑗𝛽𝑧
𝑎
+
𝐴 𝜋𝑥 −𝑗𝛽𝑧
𝐻𝑥 = − sin 𝑒 + 𝑒 𝑗𝛽𝑧
𝑍TE 𝑎
+
𝑗𝜋𝐴 𝜋𝑥 −𝑗𝛽𝑧
𝐻𝑧 = cos 𝑒 − 𝑒 𝑗𝛽𝑧
𝑘𝜂𝑎 𝑎

2𝐴+ 𝜖 𝜖𝑎𝑏𝑑
Substituting 𝐸0 = ,
we get
𝑗 𝑊𝑒 = න 𝐸𝑦 𝐸𝑦∗ 𝑑𝑣 = 𝐸0 2
𝜋𝑥 𝑙𝜋𝑧 4 𝑉 16
𝐸𝑦 = 𝐸0 sin sin
𝑎 𝑑
𝑗𝐸0 𝜋𝑥 𝑙𝜋𝑧
𝐻𝑥 = − sin cos At resonance,
𝑍TE 𝑎 𝑑
𝑗𝜋𝐸0 𝜋𝑥 𝑙𝜋𝑧
𝐻𝑧 = cos sin 𝑊𝑒 = 𝑊𝑚
𝑘𝜂𝑎 𝑎 𝑑
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Case I. The dielectric is perfect but cavity walls are slightly lossy

The power loss on the conducting walls can be found as

𝑅𝑠 Surface resistivity
𝑃𝑐 = න 𝐻𝑡 2 𝑑𝑠 of metallic walls
2 𝑤𝑎𝑙𝑙𝑠
𝜔𝜇0
𝑅𝑠 =
2𝜎
The conductor loss can be found as
𝑅𝑠 𝐸02 𝜆2 𝑙 2 𝑎𝑏 𝑏𝑑 𝑙 2 𝑎 𝑑
𝑃𝑐 = + 2+ +
8𝜂2 𝑑2 𝑎 2𝑑 2𝑎
2𝜔0 𝑊𝑒
𝑄𝑐 =
𝑃𝑐
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Case II. The dielectric is lossy but cavity 𝑄𝑑 with lossy dielectric but perfectly
walls are perfectly conducting. conducting wall is

𝜖 = 𝜖 ′ − 𝑗𝜖 ′′ = 𝜖0 𝜖𝑟 1 − 𝑗 tan 𝛿 𝜖 ′ 𝑎𝑏𝑑
2𝜔 𝐸0 2 𝜖 ′ 1
𝑄𝑑 = 16 = ′′ =
𝑎𝑏𝑑𝜔𝜖 ′′ 𝐸0 2 𝜖 tan 𝛿
Power dissipated within the dielectric volume 8
is
1 𝜔𝜖 ′′
𝑃𝑑 = න 𝐽Ԧ . 𝐸 𝑑𝑣 =
∗ න 𝐸 2 𝑑𝑣 Unloaded Q of the cavity is
2 𝑉 2 𝑉
−1
1 1
𝑎𝑏𝑑𝜔𝜖 ′′ 𝐸0 2 𝑄0 = +
= 𝑄𝑐 𝑄𝑑
8
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Circular Waveguide Cavity Resonator
Since the dominant mode of circular waveguide
is TE11 , the dominant mode of the circular
waveguide cavity is TE111 .
For TE𝑛𝑚 mode
′ 2
For TM modes, the mode with the lowest cut 𝑝𝑛𝑚
off frequency is TM01 mode. 𝛽𝑛𝑚 = 𝑘 2 −
𝑎
For TM𝑛𝑚 mode
The resonant frequencies of TE𝑛𝑚𝑙 and TM𝑛𝑚𝑙
𝑝𝑛𝑚 2
modes of the circular waveguide cavities can be 𝛽𝑛𝑚 = 𝑘2 −
found as follows: 𝑎

𝐸𝑡 𝜌, ∅, 𝑧 = 𝑒Ԧ 𝜌, ∅ 𝐴+ 𝑒 −𝑗𝛽𝑛𝑚 𝑧 + 𝐴− 𝑒 𝑗𝛽𝑛𝑚 𝑧
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Circular Waveguide Cavity Resonator

𝐸𝑡 = 0 at 𝑧 = 0 For the resonant TE𝑛𝑚𝑙 mode

We have
𝐴+ = −𝐴− ′ 2 2
𝑐 𝑝𝑛𝑚 𝑙𝜋
𝐸𝑡 = 0 at 𝑧 = 𝑑 𝑓𝑛𝑚𝑙 = +
2𝜋 𝜇𝑟 𝜖𝑟 𝑎 𝑑
We have
sin 𝛽𝑛𝑚 𝑑 = 0 For the resonant TM𝑛𝑚𝑙 mode

𝛽𝑛𝑚 𝑑 = 𝑙𝜋 where 𝑙 = 1,2,3 … 2

𝑐 𝑝𝑛𝑚 2 𝑙𝜋
𝑓𝑛𝑚𝑙 = +
2𝜋 𝜇𝑟 𝜖𝑟 𝑎 𝑑

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Circular Waveguide Cavity Resonator

𝑄 factor for the cylindrical cavities can be found

in the same manner as in rectangular cavities.

Cylindrical cavity operating at TE011 mode is

often used for frequency meters because of its
higher 𝑄

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EC8701

ANTENNAS AND MICROWAVE

ENGINEERING

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UNIT V
MICROWAVE DESIGN PRINCIPLES

▪ Impedance transformation
▪ Impedance Matching
▪ Microwave Filter Design
▪ RF and Microwave Amplifier Design
▪ Microwave Power amplifier Design
▪ Low Noise Amplifier Design
▪ Microwave Mixer Design
▪ Microwave Oscillator Design

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 L-section impedance matching

 Single and double stub matching
 Quarter wave transformer

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Matching Network
An impedance matching network
is placed between a load
impedance and a transmission 𝑍0 Matching
line. Network 𝑍𝐿

The matching network is ideally

lossless, to avoid any loss of
power A matching network can be found as long
as 𝑍𝐿 has positive real part
It is designed in such a way that Factors that are considered while selecting
the impedance seen looking into a particular matching network are:
the matching network is 𝑍0 . Complexity, bandwidth, Implementation
and adjustibility

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L-section impedance matching network
Uses two reactive elements to match an arbitrary load to a
transmission line

𝑗𝑋 𝑗𝑋

𝑍0 𝑗𝐵 𝑍𝐿 𝑍0 𝑗𝐵 𝑍𝐿

Used when 𝓏𝐿 = 𝑍𝐿 Τ𝑍0 is inside the Used when 𝓏𝐿 = 𝑍𝐿 Τ𝑍0 is outside the
1 + 𝑗𝑥 circle in the smith chart 1 + 𝑗𝑥 circle in the smith chart

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L-section impedance matching network
For this case 𝑅𝐿 > 𝑍0
𝑗𝑋
For impedance matching, we must have
1
𝑍0 = 𝑗𝑋 + 𝑍0 𝑗𝐵 𝑍𝐿
𝑗𝐵 + 1Τ 𝑅𝐿 + 𝑗𝑋𝐿
𝑅𝐿 + 𝑗𝑋𝐿
𝑍0 = 𝑗𝑋 +
𝑗𝐵𝑅𝐿 − 𝐵𝑋𝐿 + 1
𝑍0 𝑗𝐵𝑅𝐿 − 𝐵𝑋𝐿 + 1
Used when 𝓏𝐿 = 𝑍𝐿 Τ𝑍0 is inside the
= 𝑗𝑋 𝑗𝐵𝑅𝐿 − 𝐵𝑋𝐿 + 1 + 𝑅𝐿 + 𝑗𝑋𝐿
1 + 𝑗𝑥 circle in the smith chart
Equating the real part from both sides
−𝑍0 𝐵𝑋𝐿 + 𝑍0 = −𝑋𝐵𝑅𝐿 + 𝑅𝐿
𝐵 𝑋𝑅𝐿 − 𝑍0 𝑋𝐿 = 𝑅𝐿 − 𝑍0
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L-section impedance matching network
𝑍0 𝑗𝐵𝑅𝐿 − 𝐵𝑋𝐿 + 1 = 𝑗𝑋 𝑗𝐵𝑅𝐿 − 𝐵𝑋𝐿 + 1 + 𝑅𝐿 + 𝑗𝑋𝐿
Equating the imaginary parts we can write
𝑍0 𝐵𝑅𝐿 = −𝑋𝐵𝑋𝐿 + X + 𝑋𝐿
⇒ 𝑋 1 − 𝐵𝑋𝐿 = 𝑍0 𝐵𝑅𝐿 − 𝑋𝐿
We therefore have a set of two equations
𝐵 𝑋𝑅𝐿 − 𝑍0 𝑋𝐿 = 𝑅𝐿 − 𝑍0
and
𝑋 1 − 𝐵𝑋𝐿 = 𝑍0 𝐵𝑅𝐿 − 𝑋𝐿
from which 𝐵and 𝑋 are to be determined

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L-section impedance matching network
From 𝑋 1 − 𝐵𝑋𝐿 = 𝑍0 𝐵𝑅𝐿 − 𝑋𝐿 ,

𝑍0 𝐵𝑅𝐿 − 𝑋𝐿
𝑋=
1 − 𝐵𝑋𝐿

Substituting 𝑋 in 𝐵 𝑋𝑅𝐿 − 𝑍0 𝑋𝐿 = 𝑅𝐿 − 𝑍0

𝑍0 𝐵2 𝑅𝐿2 − 𝐵𝑅𝐿 𝑋𝐿 − 𝐵𝑍0 𝑋𝐿 + 𝐵2 𝑍0 𝑋𝐿2 = 𝑅𝐿 − 𝑍0 − 𝑅𝐿 − 𝑍0 𝐵𝑋𝐿

𝐵2 𝑍0 (𝑅𝐿2 + 𝑋𝐿2 ) − 2𝐵𝑍0 𝑋𝐿 − 𝑅𝐿 − 𝑍0 = 0

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L-section impedance matching network
𝐵2 𝑍0 (𝑅𝐿2 + 𝑋𝐿2 ) − 2𝐵𝑍0 𝑋𝐿 − 𝑅𝐿 − 𝑍0 = 0

2𝑍0 𝑋𝐿 ± 4𝑍02 𝑋𝐿2 + 4𝑍0 (𝑅𝐿2 + 𝑋𝐿2 ) 𝑅𝐿 − 𝑍0

𝐵=
2𝑍0 (𝑅𝐿2 + 𝑋𝐿2 )

𝑋𝐿 ± 𝑋𝐿2 + (𝑅𝐿2 + 𝑋𝐿2 ) 𝑅𝐿 Τ𝑍0 − 1

𝐵=
(𝑅𝐿2 + 𝑋𝐿2 )

𝑋𝐿 ± 𝑅𝐿 Τ𝑍0 𝑅𝐿2 + 𝑋𝐿2 − 𝑅𝐿 𝑍0

𝐵=
(𝑅𝐿2 + 𝑋𝐿2 )

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L-section impedance matching network
Since we use the matching network for 𝑅𝐿 > 𝑍0 , the term 𝑅𝐿2 +
𝑋𝐿2 − 𝑅𝐿 𝑍0 is always positive and therefore there exist a real
valued solution for 𝐵.

Once 𝐵 is calculated, 𝑋 can be calculated from

𝑍0 𝐵𝑅𝐿 − 𝑋𝐿
𝑋=
1 − 𝐵𝑋𝐿

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Example: L-section matching
Let an impedance of 𝑍𝐿 = 100 − 𝑗50 Ω is to be matched to a 50Ω line using
a L-section matching network at an operating frequency of 500 MHz. Let us
design the matching network.
We have
𝑋𝐿 ± 𝑅𝐿 Τ𝑍0 𝑅𝐿2 + 𝑋𝐿2 − 𝑅𝐿 𝑍0
𝐵=
(𝑅𝐿2 + 𝑋𝐿2 )
−50 ± 100Τ50 1002 + 502 − 100 × 50 0.0058 Ω −1
𝐵= =ቊ
1002 + 502 −0.0138 Ω−1
𝑍0 𝐵𝑅𝐿 − 𝑋𝐿 61.2372 Ω
𝑋= =ቊ
1 − 𝐵𝑋𝐿 −61.2372 Ω

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Example: L-section matching
We have two solutions which are as follows:
Solution 1
𝐵
𝐶= = 1.85 pF
2𝜋𝑓
𝑋
𝐿= = 19.49 nH
2𝜋𝑓 Network for solution 1
Solution 2
1
𝐶=− = 5.2 pF
2𝜋𝑓𝑋
1
𝐿=− = 23.1 nH
2𝜋𝑓𝐵
Network for solution 2

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L-section matching
For the matching network as shown, we have
𝑅𝐿 < 𝑍0 𝑗𝑋
1 1
= 𝑗𝐵 +
𝑍0 𝑗𝑋 + 𝑅𝐿 + 𝑗𝑋𝐿 𝑍0 𝑗𝐵 𝑍𝐿
1 1
= 𝑗𝐵 +
𝑍0 𝑅𝐿 + 𝑗(𝑋 + 𝑋𝐿 )
𝑋 and 𝐵 can be found as Used when 𝓏𝐿 = 𝑍𝐿 Τ𝑍0 is outside
the 1 + 𝑗𝑥 circle in the smith chart
𝑋 = ± 𝑅𝐿 𝑍0 − 𝑅𝐿 − 𝑋𝐿
𝑍0 − 𝑅𝐿 /𝑅𝐿
𝐵=±
𝑍0

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Example: L-section matching with Smith chart
Let us now consider an example how L-
impedance can be done using a Smith chart. Since 𝑅𝐿 > 𝑍0 , we use
the following circuit

Let us discuss the matching of a 100 Ω load

with a transmission line of characteristic
impedance 50 Ω at 100 MHz. We use Smith 𝑗𝑋
chart to do this matching.
𝑍0 𝑗𝐵 𝑍𝐿

When we consider the matching in Smith chart

our starting point is normalized 𝓏𝐿 = 2 + 𝑗0
and after matching we reach 𝓏 = 1 + 𝑗0

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DOWNLOADED FROM STUCOR APP We mark 𝓏𝐿 = 2 + 𝑗0 on the smith chart.
Since we need to add an admittance first, we
first find 𝑦 = 0.5 + 𝑗0 and add a mirror of
𝑟 = 1 circle.
Rotated 𝑟 = 1 circle We add 𝑗𝑏 = 𝑗0.5 to reach rotated r = 1
circle
VSWR circle Therefore,
0.5
= 2𝜋 × 108 × 𝐶
50
𝑦 = 0.5 𝓏=2 𝐶 = 16 pF
Having determined we come back to 𝑟 = 1
circle and we land at the point (1 − 𝑗1 ).
We add a reactance of jx = 𝑗1 to move to the
centre of the Smith chart.
Therefore,
50 = 2𝜋 × 108 × 𝐿
𝐿 = 80 nH

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Stub Matching
A stub is a short section of transmission line which is either short circuited or
open circuited at one end.
A single stub matching circuit consists of a series or shunt stub as shown in the
figures below:
The design parameters are the distance of the stub 𝑑 from the load and length
of the stub 𝑙
Open or Short
−𝑗𝐵 𝑑

𝑙 𝑍0 𝑍0 𝑍0 𝑍𝐿
−𝑗𝑋

𝑍0 𝑑 𝑍0 𝑍𝐿 Open 𝑍0
or 𝑙
𝑌𝑖𝑛 = 𝑌0 + 𝑗𝐵
𝑍𝑖𝑛 = 𝑍0 + 𝑗𝑋 Short
SeriesALSO
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Series Stub Matching
Analytical solution
Open or Short
The distance of the stub location 𝑑 is so
chosen that 𝑍𝑖𝑛 = 𝑍0 + 𝑗𝑋 𝑙 𝑍0
−𝑗𝑋
The stub length 𝑙 is then so chosen for a short
or open stub that input impedance of the stub 𝑍0 𝑑 𝑍0 𝑍𝐿
is −𝑗𝑋. This results in matching.
𝑍𝑖𝑛 = 𝑍0 + 𝑗𝑋
𝑍𝐿 + 𝑗𝑍0 tan 𝛽𝑑
𝑍𝑖𝑛 = 𝑍0
𝑍0 + 𝑗𝑍𝐿 tan 𝛽𝑑
We equate 𝑅𝑒 𝑍𝑖𝑛 to 𝑍0 and find solution
for 𝑑.
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Series Stub Matching
Analytical solution
For the computed value of 𝑑 we calculate 𝑋.
The stub length 𝑙 is then found out for a short
Open or Short
or open stub to provide − 𝑗𝑋.
𝑙 𝑍0
−𝑗𝑋
Let us now derive the closed form
expressions 𝑍0 𝑑 𝑍0 𝑍𝐿
Let 𝑍𝐿 = 𝑅𝐿 + 𝑗𝑋𝐿
𝑍𝑖𝑛 = 𝑍0 + 𝑗𝑋
1
𝑌𝐿 = = 𝐺𝐿 + 𝑗𝐵𝐿
𝑍𝐿

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Series Stub Matching
Let 𝑡 = tan 𝛽𝑑 1
𝑍𝑖𝑛 = 𝑅 + 𝑗𝑋 =
𝑌𝑖𝑛
1 𝑌𝐿 + 𝑗𝑌0 tan 𝛽𝑑
𝑌𝑖𝑛 = = 𝑌0
𝑍𝑖𝑛 𝑌0 + 𝑗𝑌𝐿 tan 𝛽𝑑
𝐺𝐿 1 + 𝑡 2
𝑅= 2
𝐺𝐿 + 𝑗𝐵𝐿 + 𝑗𝑌0 𝑡 𝐺𝐿 + 𝐵𝐿 + 𝑌0 𝑡 2
= 𝑌0
𝑌0 + 𝑗 𝐺𝐿 + 𝑗𝐵𝐿 𝑡
𝐺𝐿2 𝑡 − 𝑌0 − 𝑡𝐵𝐿 𝐵𝐿 + 𝑡𝑌0
𝑋=
𝐺𝐿 + 𝑗 𝐵𝐿 + 𝑌0 𝑡 𝑌0 𝐺𝐿2 + 𝐵𝐿 + 𝑌0 𝑡 2
𝑌𝑖𝑛 = 𝑌0
𝑌0 − 𝐵𝐿 𝑡 + 𝑗𝐺𝐿 𝑡

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Series Stub Matching
𝐺𝐿 1+𝑡 2
From 𝑅 =
𝐺𝐿2 + 𝐵𝐿 +𝑌0 𝑡 2

𝑌0 𝐺𝐿 − 𝑌0 𝑡 2 − 2𝐵𝐿 𝑌0 𝑡 + 𝐺𝐿 𝑌0 − 𝐺𝐿2 − 𝐵𝐿2 = 0

If 𝐺𝐿 = 𝑌0 , 𝑡 = −𝐵𝐿 Τ 2𝑌0
else
𝐵𝐿 ± 𝐺𝐿 𝑌0 − 𝐺𝐿 2 + 𝐵𝐿2 Τ𝑌0
𝑡=
𝐺𝐿 − 𝑌0
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Series Stub Matching
We get two solutions for 𝑑 which are given by
1 −1
𝑑 tan 𝑡 𝑡≥0
= 2𝜋
𝜆 1
𝜋 + tan−1 𝑡 𝑡 < 0
2𝜋
With the values of 𝑡 calculated, we calculate the values of 𝑋. Necessary stub
reactance 𝑋𝑆 = −𝑋.
If 𝑙𝑜 and 𝑙𝑠 respectively denote the lengths for the open and short circuited stubs,
then
𝑙𝑠 1 𝑋 1 𝑋 𝑙𝑜 1 𝑍 1 𝑍0
= 2𝜋 tan−1 𝑍𝑆 = − 2𝜋 tan−1 𝑍 and = − 2𝜋 tan−1 𝑋0 = 2𝜋 tan−1
𝜆 0 0 𝜆 𝑆 𝑋
If any of the lengths comes out to be negative, 𝜆Τ2 is added.

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Example: Impedance Matching Series Stub
Let us consider an example where 𝑍𝐿 = 100 + 𝑗50 Ω is to be matched to a 50
Ω line. By applying the analytical solutions we get:
𝑡 = −0.333 = 𝑡1 and 𝑡 = 1.0 = 𝑡2 . We get two solutions for 𝑑
𝑑1 1
= 𝜋 + tan−1 𝑡1 = 0.45
𝜆 2𝜋
𝑑2 1
= tan−1 𝑡2 = 0.125
𝜆 2𝜋
We get two solutions for 𝑋 as 𝑋1 = 50 and 𝑋2 = −50
Let us now find the lengths of the open circuited stubs to complete the solution
𝑙𝑜1 1 𝑍 𝑙𝑜2 1 𝑍
= tan−1 0 =0.125 and = 0.5 + tan−1 0 =0.375
𝜆 2𝜋 𝑋1 𝜆 2𝜋 𝑋2

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Solution: 1 FROM STUCOR APP 𝑙1
𝑑 1 + 𝑗1
= 0.338 − 0.213 = 0.125
𝜆

𝑥 = −1
𝑥𝑆 = 1
𝑍𝐿 = 2 + 𝑗1
𝑙0
= 0.375
𝜆
Solution: 2
𝑑
= 0.5 − 0.213 − 0.164
𝜆
= 0.451

𝑥=1
𝑥𝑆 = −1
𝑙0 1 − 𝑗1
= 0.125
𝜆 𝑙2 IN STUCOR APP
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Shunt Stub Matching
Analytical solution
The distance of the stub location 𝑑 is so chosen that
𝑌𝑖𝑛 = 𝑌0 + 𝑗B −𝑗𝐵 𝑑
The stub length 𝑙 is then so chosen for a short or open
stub that input susceptance of the stub is −𝑗B. This
results in matching. 𝑍0 𝑍0 𝑍𝐿

𝑍𝐿 + 𝑗𝑍0 tan 𝛽𝑑
𝑍𝑖𝑛 = 𝑍0 𝑍0
𝑍0 + 𝑗𝑍𝐿 tan 𝛽𝑑 Open 𝑙
or 𝑌𝑖𝑛 = 𝑌0 + 𝑗𝐵
𝑌𝑖𝑛 = 𝐺 + 𝑗𝐵 Short

We equate 𝑅𝑒 𝑌𝑖𝑛 = G to 𝑌0 and find solution for 𝑑.

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Shunt Stub Matching
Analytical solution

−𝑗𝐵 𝑑
For the computed value of 𝑑 we calculate 𝐵.
The stub length 𝑙 is then found out for a short 𝑍0 𝑍0 𝑍𝐿
or open stub to provide − 𝑗𝐵.
Open 𝑍0
Let us now derive the closed form or 𝑙
𝑌𝑖𝑛 = 𝑌0 + 𝑗𝐵
expressions Short
Let
1
𝑍𝐿 = = 𝑅𝐿 + 𝑗𝑋𝐿
𝑌𝐿
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Shunt Stub Matching
Similar to series stub matching, let 1
𝑡 = tan 𝛽𝑑 𝑌𝑖𝑛 = 𝐺 + 𝑗𝐵 =
𝑍𝑖𝑛

𝑍𝐿 + 𝑗𝑍0 tan 𝛽𝑑
𝑍𝑖𝑛 = 𝑍0
𝑍0 + 𝑗𝑍𝐿 tan 𝛽𝑑 𝑅𝐿 1 + 𝑡 2
𝐺= 2 2
𝑅𝐿 + 𝑋𝐿 + 𝑍0 𝑡
𝑅𝐿 + 𝑗𝑋𝐿 + 𝑗𝑍0 𝑡
= 𝑍0
𝑍0 + 𝑗 𝑅𝐿 + 𝑗𝑋𝐿 𝑡 𝑅𝐿2 𝑡 − 𝑍0 − 𝑡𝑋𝐿 𝑋𝐿 + 𝑡𝑍0
𝐵=
𝑍0 𝑅𝐿2 + 𝑋𝐿 + 𝑍0 𝑡 2
𝑅𝐿 + 𝑗 𝑋𝐿 + 𝑍0 𝑡
𝑍𝑖𝑛 = 𝑍0
𝑍0 − 𝑋𝐿 𝑡 + 𝑗𝑅𝐿 𝑡
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Shunt Stub Matching
𝑅𝐿 1+𝑡 2
From 𝐺 =
𝑅𝐿2 + 𝑋𝐿 +𝑍0 𝑡 2

𝑍0 𝑅𝐿 − 𝑍0 𝑡 2 − 2𝑋𝐿 𝑍0 𝑡 + 𝑅𝐿 𝑍0 − 𝑅𝐿2 − 𝑋𝐿2 = 0

If 𝑅𝐿 = 𝑍0 , 𝑡 = −𝑋𝐿 Τ 2𝑍0
else
𝑋𝐿 ± 𝑅𝐿 𝑍0 − 𝑅𝐿 2 + 𝑋𝐿2 Τ𝑍0
𝑡=
𝑅𝐿 − 𝑍0
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Shunt Stub Matching
We get two solutions for 𝑑 which are given by
1 −1
𝑑 tan 𝑡 𝑡≥0
= 2𝜋
𝜆 1
𝜋 + tan−1 𝑡 𝑡 < 0
2𝜋
With the values of 𝑡 calculated, we calculate the values of 𝐵. Necessary stub
reactance 𝐵𝑆 = −𝐵.
If 𝑙𝑜 and 𝑙𝑠 respectively denote the lengths for the open and short circuited stubs,
then
𝑙𝑜 1 𝐵𝑆 1 𝐵 𝑙𝑠 1 𝑌 1 𝑌0
= 2𝜋 tan−1 = − 2𝜋 tan−1 𝑌 and = − 2𝜋 tan−1 𝐵0 = 2𝜋 tan−1
𝜆 𝑌0 0 𝜆 𝑆 𝐵
If any of the lengths comes out to be negative, 𝜆Τ2 is added.

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Example: Impedance Matching- Shunt Stub
Let us consider an example where 𝑍𝐿 = 100 + 𝑗60 Ω is to be matched to a 50
Ω line. By applying the analytical solutions we get:
𝑡 = 3.4091 = 𝑡1 and 𝑡 = −1.0091 = 𝑡2 . We get two solutions for 𝑑
𝑑1 1
= tan−1 𝑡1 = 0.2046
𝜆 2𝜋
𝑑2 1
= 𝜋 + tan−1 𝑡2 = 0.3743
𝜆 2𝜋
We get two solutions for 𝐵 as 𝐵1 = 0.0221and 𝐵2 = −0.0221
Let us now find the lengths of the open circuited stubs to complete the solution
𝑙𝑜1 1 𝐵 𝑙𝑜2 1 𝐵
= tan−1 1 =0.3671 and = 0.5 + tan−1 2 =0.1329
𝜆 2𝜋 𝑌0 𝜆 2𝜋 𝑌0

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Shunt Stub Matching using Open Stub

From Smith Chart

1 + 𝑗1.1
𝑑1
= 0.5 − 0.46 + 0.164 = 0.204
𝜆
𝑑2 2 + 𝑗1.2
= 0.5 − 0.46 + 0.336 = 0.376
𝜆

𝑙𝑜1 /𝜆 = 0.132
𝑙𝑜2 /𝜆 = 0.368

𝑍𝐿 = 100 + 𝑗60 𝑦𝐿
Analytical
𝑡1 = 3.4091 1 − 𝑗1.1
𝑡2 = −1.0091
𝑑1 /𝜆 = 0.20459
𝑑2 /𝜆 = 0.37428
𝑙𝑜1 /𝜆 = 0.36710
𝑙𝑜2 /𝜆 = 0.13290
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Double Stub Matching
As shown in Fig.1, the load is at an
𝑑 arbitrary distance from the first
stub. 𝑑
𝑌0 𝑗𝐵2 𝑌0 𝑗𝐵1 𝑌0 𝑌𝐿′

𝑌0 𝑗𝐵2 𝑌0 𝑗𝐵1 𝑌𝐿

𝑙2 𝑙1 In Fig.2, the load 𝑌𝐿′ is

transformed to the position of 𝑙2 𝑙1

Open Open
the first stub as 𝑌𝐿
Or short Or short
Open Open
or short Or short
Fig.1
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Double Stub Matching
Analytical solution
𝑑
From the figure, we have
𝑌1 = 𝑌𝐿 + 𝑗𝐵1 = 𝐺𝐿 + 𝑗𝐵𝐿 + 𝑗𝐵1
𝑌0 𝑗𝐵2 𝑌0 𝑗𝐵1 𝑌𝐿
= 𝐺𝐿 + 𝑗 𝐵𝐿 + 𝐵1

𝑌1 + 𝑗𝑌0 tan 𝛽𝑑
𝑌2 = 𝑌0 𝑙2 𝑙1
𝑌0 + 𝑗𝑌1 tan 𝛽𝑑
We equate 𝑅𝑒 𝑌2 to 𝑌0 and find solution
for 𝑑. Open Open
or short Or short

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Double Stub Matching
Similar to earlier assumptions, let 𝑡 = tan 𝛽𝑑

𝑌1 + 𝑗𝑌0 tan 𝛽𝑑
𝑌2 = 𝑌0
𝑌0 + 𝑗𝑌1 tan 𝛽𝑑

𝐺𝐿 + 𝑗 𝐵𝐿 + 𝐵1 + 𝑗𝑌0 𝑡
= 𝑌0
𝑌0 + 𝑗 𝐺𝐿 + 𝑗 𝐵𝐿 + 𝐵1 𝑡

𝐺𝐿 + 𝑗 𝐵𝐿 + 𝐵1 + 𝑌0 𝑡
𝑌2 = 𝑌0
𝑌0 − 𝐵𝐿 𝑡 − 𝐵1 𝑡 + 𝑗𝐺𝐿 𝑡

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Double Stub Matching
On equating 𝑅𝑒 𝑌2 to 𝑌0
2 2
2
1 + 𝑡 𝑌0 − 𝐵𝐿 𝑡 − 𝐵1 𝑡
𝐺𝐿 + 𝐺𝐿 𝑌0 + =0
𝑡2 𝑡2
1 + 𝑡2 4𝑡 2 𝑌0 − 𝐵𝐿 𝑡 − 𝐵1 𝑡 2
𝐺𝐿 = 𝑌0 1± 1−
𝑡2 𝑌2 1 + 𝑡 2 2

∵ 𝐺𝐿 is real,
4𝑡 2 𝑌0 − 𝐵𝐿 𝑡 − 𝐵1 𝑡 2
0≤ ≤1
𝑌2 1 + 𝑡2 2

1 + 𝑡2 1 + tan2 𝛽𝑑 𝑌0
0 ≤ 𝐺𝐿 ≤ 𝑌0 2
= 𝑌0 2
=
𝑡 tan 𝛽𝑑 sin2 𝛽𝑑
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Double Stub Matching
𝑌0 ± 1 + 𝑡 2 𝐺𝐿 𝑌0 − 𝐺𝐿2 𝑡 2
𝐵1 = −𝐵𝐿 +
𝑡
𝑌0 1 + 𝑡 2 𝐺𝐿 𝑌0 − 𝐺𝐿2 𝑡 2 + 𝐺𝐿 𝑌0
𝐵2 = ±
𝐺𝐿 𝑡
If 𝑙𝑜 and 𝑙𝑠 respectively denote the lengths for the open and short
circuited stubs
𝑙𝑜 1 −1
𝐵
=− tan
𝜆 2𝜋 𝑌0
and
𝑙𝑠 1 −1
𝑌0
= tan
𝜆 2𝜋 𝐵
𝐵 = 𝐵1 or 𝐵2
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Double Stub Matching Using Smith Chart
𝑌𝑖 = 𝑌𝐵 + 𝑌𝑠𝐵 = 𝑌0
In normalized form, 1 = 𝑦𝐵 + 𝑦𝑠𝐵
Since 𝑦𝑠𝐵 is purely imaginary we must have, 𝐵 𝐴
𝑦𝐵 = 1 + 𝑗𝑏𝐵 and 𝑦𝑠𝐵 = −𝑗𝑏𝐵 𝑑
Therefore, in the Smith chart 𝑦𝐵 must lie in the 𝑔 = 1
𝑌𝑖 𝑌𝐴
circle. 𝑌𝐵 𝑌𝐿
To meet this requirement 𝑦𝐴 at 𝐴𝐴′ must lie on the 𝑌0 𝑌𝑠𝐵 𝑌𝑠𝐴
4𝜋𝑑
𝑔 = 1 circle rotated by 𝜆 counter clockwise
𝑌0 𝑌0
direction. 𝑙𝐵 𝑙𝐴
Since 𝑦𝑠𝐴 is purely imaginary, the real part of 𝑦𝐴 must
be contributed solely by real part of 𝑦𝐿 i.e. 𝑔𝐿 .
The solution of double stub matching is then Open
𝐵′ Open 𝐴′
determined by the intersection of 𝑔𝐿 circle with or short Or short
rotated 𝑔 = 1 circle .
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Procedure

Plot 𝑔 = 1 circle. 𝑦𝐵 should be located on

this circle
𝑦𝐴2
Plot the rotated circle where 𝑦𝐴 should be
located
𝑦𝐴1 𝑦𝐵1
Plot 𝑦𝐿
Find intersection of 𝑔 = 𝑔𝐿 circle with
rotated 𝑔 = 1 circle at 𝑦𝐴1 & 𝑦𝐴2
Find 𝑦𝐵 points on 𝑔 = 1 circle:
𝑦𝐵1 and 𝑦𝐵2 𝑦𝐵2
𝑦𝐿
Determine the stub lengths 𝑙𝐴 and 𝑙𝐵

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The shaded
region is the
forbidden range
of load
admittances that
can not be
matched with the
given double
stub tuner

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Quarter-wave Transformer
𝜆
A quarter-wave transformer is transmission line section of length having
4
characteristic impedance 𝑍1 and used to match a real load 𝑅𝐿 to a
transmission line of characteristic impedance 𝑍0 , as shown in the figure.
We know that
𝑅𝐿 + 𝑗𝑍1 tan 𝛽𝑙
𝑍𝑖𝑛 = 𝑍1
𝑍1 + 𝑗𝑅𝐿 tan 𝛽𝑙 𝑙 = 𝜆Τ4

Dividing the numerator and

denominator by tan 𝛽𝑙 and take 𝑍0 𝑍1 𝑍𝐿 = 𝑅𝐿
the limit as 𝛽𝑙 → 𝜋Τ2, we can
𝑍12
write 𝑍𝑖𝑛 = . Equating 𝑍𝑖𝑛 to
𝑅𝐿
𝑍0 we get 𝑍1 = 𝑅𝐿 𝑍0 𝑍𝑖𝑛

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Quarter-wave Transformer
We note that matching is obtained at the frequency at which the
transformer is quarter wavelength long and at all odd harmonics
where the length corresponds to 2𝑛 + 1 𝜆Τ4.
The fractional bandwidth of such quarter-wave transformer can be
found as:

∆𝑓 4 Γ𝑚 2 𝑍1 𝑅𝐿
=2− cos −1
𝑓0 𝜋 1 − Γ𝑚2 𝑅𝐿 − 𝑍0
Γ𝑚 is the magnitude of the acceptable value of reflection coefficient

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Use of Quarter-wave Transformer
Quarter-wave transformers can also be used in design of
matching network for matching complex load impedance to a
transmission line. The examples of such networks are shown:

𝑙 = 𝜆Τ4 𝑑 𝑙 = 𝜆Τ4

𝑍0 𝑍1 𝑍0 𝑍𝐿 𝑍0 𝑍1 𝑍𝐿

𝑍𝑖𝑛 𝑍2
𝑍𝑖𝑛 𝑑
Short or Open
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Microwave Filter Design
▪ A filter is a two-port network used to control the frequency response at a certain
point in an RF or microwave system by providing transmission at frequencies
within the passband of the filter and attenuation in the stopband of the filter.
▪ Typical frequency responses include low-pass, high-pass, bandpass, and band-
reject characteristics.
▪ Applications can be found in virtually any type of RF or microwave
communication, radar, or test and measurement system.
▪ Filters designed using the image parameter method consist of a cascade of
simpler two-port filter sections to provide the desired cutoff frequencies and
attenuation characteristics but do not allow the specification of a particular
frequency response over the complete operating range. Thus, although the
procedure is relatively simple, the design of filters by the image parameter method
often must be iterated many times to achieve the desired results.

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Microwave Filter Design
▪ A more modern procedure, called the insertion loss method, uses network
synthesis techniques to design filters with a completely specified frequency
response. The design is simplified by beginning with low-pass filter prototypes
that are normalized in terms of impedance and frequency. Transformations are
then applied to convert the prototype designs to the desired frequency range and
impedance level.
▪ Both the image parameter and insertion loss methods of filter design lead to
circuits using lumped elements (capacitors and inductors).
▪ For microwave applications such designs usually must be modified to employ
distributed elements consisting of transmission line sections. The Richards
transformation and the Kuroda identities provide this step.

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Image Parameters for T- and π-Networks

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Filter Design by the Image Parameter Method
Constant-k Filter Sections:
▪ First consider the T-network shown in figure. Intuitively, we can see that this is a
low-pass filter network because the series inductors and shunt capacitor tend to
block high-frequency signals while passing low-frequency signals.

Low-pass constant-k filter sections in T and π forms. (a) T-section. (b) π-section.

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Constant-k Filter Sections
▪ We have 𝑍1 = 𝑗𝜔𝐿 and 𝑍2 = 1ൗ𝑗𝜔𝐶 , so the image impedance is

𝑍1 𝑗𝜔𝐿 𝑗𝜔𝐿 𝐿 𝜔 2 𝐿𝐶
𝑍𝑖𝑇 = 𝑍1 𝑍2 1+ = 1+ = 1−
4𝑍2 𝑗𝜔𝐶 1 𝐶 4
4
𝑗𝜔𝐶

▪ If we define a cutoff frequency 𝜔𝑐 as 𝜔𝑐 = 2Τ 𝐿𝐶 and a nominal characteristic

impedance, 𝑅0 as 𝑅0 = 𝐿Τ𝐶 = 𝑘, where k is a constant, then we can rewrite 𝑍𝑖𝑇
as
𝜔2
𝑍𝑖𝑇 = 𝑅0 1− 2
𝜔𝑐
▪ Then 𝑍𝑖𝑇 = 𝑅0 for 𝜔 = 0.

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Constant-k Filter Sections
▪ The propagation factor is
2
𝛾
𝑍1 𝑍1 𝑍1 𝑗𝜔𝐿 𝑗𝜔𝐿 𝑗𝜔𝐿 2
𝑒 =1+ + + 2 =1+ + + 2
2𝑍2 𝑍2 4𝑍2 2 1ൗ𝑗𝜔𝐶 1ൗ
𝑗𝜔𝐶 4 1ൗ𝑗𝜔𝐶

𝜔 2 𝐿𝐶 𝜔 2 𝐿𝐶 2 𝜔2 𝐿𝐶 𝜔 2 𝐿𝐶
𝑒𝛾 = 1 − + −𝜔 2 𝐿𝐶 + =1− + 𝜔 2 𝐿𝐶 −1
2 4 2 4
▪ Since 𝜔𝑐 = 2Τ 𝐿𝐶,
2𝜔 2 2𝜔 𝜔2
𝛾
𝑒 =1− 2 +
𝜔𝑐 2−1
𝜔𝑐 𝜔𝑐

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Constant-k Filter Sections

Typical passband and stopband characteristics of the low-pass constant-k sections

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Constant-k Filter Sections
Now consider two frequency regions:
▪ For 𝜔 < 𝜔𝑐 : This is the passband of the filter section. 𝑍𝑖𝑇 is real, 𝛾 is imaginary,
𝜔2
since − 1 is negative and 𝑒 𝛾 = 1
𝜔𝑐2
▪ For 𝜔 > 𝜔𝑐 : This is the stopband of the filter section. 𝑍𝑖𝑇 is imaginary, 𝑒 𝛾 is real
and −1 < 𝑒 𝛾 < 0 (as seen from the limits as 𝜔 ⟶ 𝜔𝑐 and 𝜔 ⟶ ∞). The
attenuation rate for 𝜔 ≫ 𝜔𝑐 is 40 dB/decade.
▪ Typical phase and attenuation constants are sketched in previous figure. Observe
that the attenuation, α, is zero or relatively small near the cutoff frequency,
although α→∞ as ω→∞. This type of filter is known as a constant-k low-pass
prototype. There are only two parameters to choose (L and C), which are
determined by 𝜔𝑐 , the cutoff frequency, and 𝑅0 , the image impedance at zero
frequency.

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Constant-k Filter Sections
▪ The above results are valid only when the filter section is terminated in its image
impedance at both ports. This is a major weakness of the design because the
image impedance is a function of frequency, and is not likely to match a given
source or load impedance. This disadvantage, as well as the fact that the
attenuation is rather low near cutoff, can be remedied with the modified m-
derived sections.

High-pass constant-k filter sections in T and π forms. (a) T-section. (b) π-section.
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m-Derived Filter Sections
▪ We have seen that the constant-k filter section suffers from the disadvantages of a
relatively slow attenuation rate past cutoff, and a nonconstant image impedance.
▪ The m-derived filter section is a modification of the constant-k section designed to
overcome these problems.

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Development of an m-derived filter section from a constant-k section

(a) Constant-k section. (b) General m-derived section. (c) Final m-derived section
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Microwave Filters
An ideal filter provides:
- Perfect transmission for all frequencies in certain passband region.
- Infinite attenuation in stopband region.
Typical filter responses are:
a. Low pass : Transmits all signals between zero frequency and some upper limit
and attenuates all frequencies above the cut off value 𝜔𝑐 .
b. High pass : Transmits all frequencies above some lower cut off frequency and
attenuates all frequencies below the cut off value 𝜔𝑐 .
c. Band pass : Transmits all frequencies in the range 𝜔1 and 𝜔2 and attenuates all
frequencies outside the range.
d. Band reject : Attenuates signals over a band of frequencies

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Microwave Filters
Filter design problems at microwave frequencies where distributed parameters must be
used is quite complicated.

Two commonly used low frequency filter synthesis techniques are:

a. The image parameter method: Filter with the required passband and stopband
characteristics can be synthesized, but without exact frequency characteristics over
each region.

b. The insertion loss method: A systematic way to synthesize the desired response
with a higher degree of control over the passband and stopband amplitude and
phase characteristics.

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Design of Microwave Filters by Insertion Loss
Some of the design trade-offs for microwave filter synthesis using the
insertion loss method are:

1. A binomial response is used when obtaining a minimum insertion

loss is the priority.

2. A Chebyshev response satisfies the requirement for the sharp

cutoff.

3. A linear phase filter design is used in cases where the attenuation

rate can be sacrificed for a better phase response.

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Characterization by Power Loss Ratio
The insertion loss or the power loss ratio
2
𝑀 𝜔2
in a filter network can be defined as: Γ 𝜔 =
𝑀 𝜔2 + 𝑁 𝜔2
Power available from the source where, 𝑀 and 𝑁 are real polynomials in
𝑃𝐿𝑅 = 𝜔2 .
Power delivered to the load
𝑃𝑖𝑛 1
= = 1
𝑃𝐿𝑜𝑎𝑑 1 − Γ 𝜔 2 ∴ 𝑃𝐿𝑅 =
𝑀 𝜔2
1−
The insertion loss in dB is given by 𝑀 𝜔2 + 𝑁 𝜔2
𝑀 𝜔2 + 𝑁 𝜔2
=
𝐼𝐿 = 10 log 𝑃𝐿𝑅 𝑁 𝜔2
Since Γ 𝜔 2 is an even function of 𝜔. 𝑀 𝜔2
∴ 𝑃𝐿𝑅 = 1 +
We can express Γ 𝜔 2 as a polynomial in 𝑁 𝜔2
𝜔2 .
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Some Practical Filter Responses

Maximally flat: Such filters are also known

For 𝜔 > 𝜔𝑐 , the attenuation increases
as binomial or Butterworth filters. For a low
monotonically with frequency.
pass filter, power loss ratio is specified as
2𝑁
𝜔 For 𝜔 ≫ 𝜔𝑐 ,
𝑃𝐿𝑅 = 1 + 𝑘 2
𝜔𝑐 𝜔
2𝑁
𝑃𝐿𝑅 ≃ 𝑘 2
𝜔𝑐
where, 𝑁 is the order of the filter and 𝜔𝑐 is
The insertion loss increases at the rate
the cut off frequency.
20N dB/decade.
The power loss at the band edge is 1 + 𝑘 2 .

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Some Practical Filter Responses
Equal ripple: Such filter response is also For 𝜔 ≫ 𝜔𝑐 ,
2𝑁
known as Chebyshev response. For a low 𝜔 1 2𝜔
pass filter power loss ratio is given by 𝑇𝑁2 ≃
𝜔𝑐 2 𝜔𝑐
𝜔 2𝑁
𝑃𝐿𝑅 = 1 + 𝑘 2 𝑇𝑁2 𝑘 2 2𝜔
𝜔𝑐 ∴ 𝑃𝐿𝑅 ≃
4 𝜔𝑐
𝜔 The insertion loss increases at the rate
For 𝜔 < 𝜔𝑐 , 𝑇𝑁2 (𝜔 ) will oscillate between
𝑐 20N dB/decade similar to binomial.
± 1. However, the insertion loss in
(22𝑁 )
Chebyshev response is greater
The passband response has ripples of 4
amplitude 1 + 𝑘 2 . than binomial response at 𝜔 ≫ 𝜔𝑐 .
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Some Practical Filter Responses
𝑃𝐿𝑅
𝑁=3
Equal Ripple

Maximally flat

1 + 𝑘2

0.5 1 1.5 𝜔/𝜔𝑐

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Some Practical Filter Responses
Linear phase: In some applications, a linear
phase response is desirable in the passband.
The group delay is defined as
A linear phase response can be achieved
with the following phase response 𝑑ϕ
𝜏𝑑 =
𝑑𝜔
2𝑁 2𝑁
𝜔 𝜔
ϕ(ω) = 𝐴𝜔 1 + 𝑝 = 𝐴 1 + 𝑝 2𝑁 + 1
𝜔𝑐 𝜔𝑐
Group delay is a maximally flat
where, ϕ(ω) is the phase of the voltage response
transfer function of the filter and 𝑝 is a
constant.
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Steps involved in filter design by insertion loss method

Filter specifications

Low pass prototype design

Scaling and Conversion

Implementation

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Maximally Flat Low-Pass Filter Prototype

Let the source impedance be 1Ω and

𝜔𝑐 = 1 rad/sec. 1 𝐿
For 𝑁 = 2,
𝑃𝐿𝑅 = 1 + 𝜔4
and C 𝑅
𝑅 1 − 𝑗𝜔𝑅𝐶
𝑍𝑖𝑛 = 𝑗𝜔𝐿 +
1 + 𝜔 2 𝑅2 𝐶 2

𝑍𝑖𝑛 − 1
Γ= 𝑍𝑖𝑛
𝑍𝑖𝑛 + 1
2
1 1 𝑍𝑖𝑛 + 1
∴ 𝑃𝐿𝑅 = 2
= ∗
= ∗
1− Γ 𝑍𝑖𝑛 − 1 𝑍𝑖𝑛 −1 2 𝑍𝑖𝑛 + 𝑍𝑖𝑛
1−
𝑍𝑖𝑛 + 1 ∗
𝑍𝑖𝑛 +1
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Maximally Flat Low-Pass Filter Prototype
Now
∗ 𝑅 1 − 𝑗𝜔𝑅𝐶 𝑅 1 + 𝑗𝜔𝑅𝐶 2𝑅
𝑍𝑖𝑛 + 𝑍𝑖𝑛 = 𝑗𝜔𝐿 + − 𝑗𝜔𝐿 + =
1 + 𝜔 2 𝑅2 𝐶 2 1 + 𝜔 2 𝑅2 𝐶 2 1 + 𝜔 2 𝑅2 𝐶 2

and
2
2
𝑅 1 − 𝑗𝜔𝑅𝐶
𝑍𝑖𝑛 + 1 = 𝑗𝜔𝐿 + +1
1 + 𝜔 2 𝑅2 𝐶 2
2
𝑅 𝜔𝑅2 𝐶
= + 1 + 𝑗 𝜔𝐿 −
1 + 𝜔2 𝑅2 𝐶 2 1 + 𝜔 2 𝑅2 𝐶 2
2 2
𝑅 𝜔𝑅2 𝐶
= +1 + 𝜔𝐿 −
1 + 𝜔 2 𝑅2 𝐶 2 1 + 𝜔 2 𝑅2 𝐶 2

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Maximally Flat Low-Pass Filter Prototype
2 2 2
𝑍𝑖𝑛 + 1 1 + 𝜔2 𝑅 2 𝐶 2 𝑅 𝜔𝑅2 𝐶
∴ 𝑃𝐿𝑅 = = +1 + 𝜔𝐿 −
∗
2 𝑍𝑖𝑛 + 𝑍𝑖𝑛 4𝑅 1 + 𝜔 2 𝑅2 𝐶 2 1 + 𝜔 2 𝑅2 𝐶 2
1
= 𝑅2 + 2𝑅 + 1 + 𝜔2 𝑅2 𝐶 2 + 𝜔2 𝐿2 + 𝜔4 𝐿2 𝑅2 𝐶 2 − 2𝜔2 𝐿𝐶𝑅 2
4𝑅
1
=1+ 1 − 𝑅 2 + 𝜔2 𝑅2 𝐶 2 + 𝐿2 − 2𝐿𝐶𝑅2 + 𝜔4 𝐿2 𝑅2 𝐶 2
4𝑅
On comparing with 𝑃𝐿𝑅 = 1 + 0 𝜔2 + (1)𝜔4 , we get

1 − 𝑅 = 0 ⇒ 𝑅 = 1,
𝐶 2 + 𝐿2 − 2𝐿𝐶 = 0 ⇒ 𝐿 − 𝐶 2 = 0 ⇒ 𝐿 = 𝐶
and
𝐿2 𝑅2 𝐶 2 𝐿2 𝐶 2 𝐶 2𝐶 2
=1⇒ =1⇒ =1⇒𝐿=𝐶= 2
4𝑅 4 4
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Ladder circuit for a lowpass prototype
𝑅0 = 𝑔0 = 1 𝐿2 = 𝑔2
Prototype beginning
with a shunt element
𝐶1 = 𝑔1 𝐶3 = 𝑔3
𝑔𝑁+1

𝐿1 = 𝑔1 𝐿3 = 𝑔3
Prototype beginning
with a series element

𝐺0 = 𝑔0 = 1 𝐶1 = 𝑔2
𝑔𝑁+1

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Element Values for Maximally Flat Low-Pass Filter Prototypes
𝑔0 = 1, 𝜔𝑐 = 1

N 𝑔1 𝑔2 𝑔3 𝑔4 𝑔5 𝑔6
1 2.0000 1.0000
2 1.4142 1.4142 1.0000
3 1.0000 2.0000 1.0000 1.0000
4 0.7654 1.8478 1.8478 0.7654 1.0000
5 0.6180 1.6180 2.0000 1.6180 0.6180 1.0000

For practical filter, it will be necessary to determine the order of the filter. This is
usually dictated by a specification on the insertion loss at some frequency in the
stop-band of the filter.
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Impedance and frequency scaling
The prototype filter has 𝑅𝑠 = 1and 𝜔𝑐 = 1. Also for a maximally flat
response, the prototype has 𝑅𝐿 = 1
A source resistance of 𝑅0 can be obtained by multiplying all the impedances
of the prototype design by 𝑅0
The change of cutoff frequency from unity to 𝜔𝑐 requires scaling of frequency
dependence of filter which is accomplished by replacing 𝜔 by 𝜔Τ𝜔𝑐
Therefore, when both impedance and frequency scaling is applied,
𝑅0 𝐿𝑘 𝐶𝑘
𝐿′𝑘 = 𝐶𝑘′ =
𝜔𝑐 𝑅0 𝜔𝑐
Note that with impedance scaling, the scaled values of source and load
resistances become 𝑅0 and 𝑅0 𝑅𝐿

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Example: Design of a Low Pass Butterworth Filter
Let us consider a maximally flat filter that has cutoff frequency of 2 GHz and
the filter provides at least 15 attenuation at 4 GHz. The source and load
impedances are 50Ω
First we need to determine the order of the filter. From the expression of 𝑃𝐿𝑅 ,
we find that at angular frequency 𝜔, the attenuation of the filter in dB is
𝜔 2𝑁
10 log10 1 +
𝜔𝑐

Therefore, 1.5 = log10 1 + 22𝑁 . So we get 𝑁 = 2.47 i.e. we use 𝑁 = 3

From the table, 𝑔1 = 1 𝑔2 = 2 𝑔3 = 1

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Example: Continued
50Ω 𝐿2 After impedance and frequency scaling

𝐶1 =1.5915 pF
𝐶1 𝐶3 50Ω
𝐿2 =7.9577 nH
𝐶3 =1.5915 pF

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Chebyshev Polynomials
𝑛th order Chebyshev polynomial is a
polynomial of degree n and denoted 𝑛=4
by 𝑇𝑛 𝑥
𝑇𝑛 (x)

𝑇1 𝑥 = 𝑥 𝑛=2
𝑇2 𝑥 = 2𝑥 2 − 1 𝑛=1

𝑇𝑛 𝑥 = 2𝑥𝑇𝑛−1 𝑥 − 𝑇𝑛−2 (𝑥) 𝑛=3

𝑥
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Equal Ripple Low-Pass Filter Prototype

Let the cutoff frequency be 𝜔𝑐 = 1

rad/sec. As
𝑃𝐿𝑅 = 1 + 𝑘 2 𝑇𝑁2 𝜔 𝑇2 𝑥 = 2𝑥 2 − 1
Since, For 𝑁 = 2,
0 for N odd 𝑃𝐿𝑅 = 1 + 𝑘 2 𝑇22 𝜔
𝑇𝑁 0 = ቊ
±1 for N even
= 1 + 𝑘 2 2𝜔2 − 1 2

Therefore, at 𝜔 = 0,
= 1 + 𝑘 2 4𝜔4 − 4𝜔2 + 1
1 for N odd
𝑃𝐿𝑅 = ቊ 2
1 + 𝑘 for N even

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Equal Ripple Low-Pass Filter Prototype
1 𝐿
For the two element network,
C 𝑅
we have seen

1 2
𝑃𝐿𝑅 = 1 + 1−𝑅 + 𝜔2 𝑅2 𝐶 2 + 𝐿2 − 2𝐿𝐶𝑅 2 + 𝜔4 𝐿2 𝑅2 𝐶 2 𝑍𝑖𝑛
4𝑅

Also 𝑃𝐿𝑅 = 1 + 𝑘 2 𝑇22 𝜔 = 1 + 𝑘 2 4𝜔4 − 4𝜔2 + 1

For 𝜔 = 0,
1
𝑃𝐿𝑅 = 1 + 𝑘 2 = 1 + 1−𝑅 2
4𝑅
1−𝑅 2
4𝑘 2 = ⇒ 𝑅 = 1 + 2𝑘 2 ± 2𝑘 1 + 𝑘 2 (for 𝑁 even)
4𝑅

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Equal Ripple Low-Pass Filter Prototype
Equating the two expressions for 𝑃𝐿𝑅
1
1 + 𝑘 2 4𝜔4 − 4𝜔2 + 1 = 1 + 1 − 𝑅 2 + 𝜔2 𝑅2 𝐶 2 + 𝐿2 − 2𝐿𝐶𝑅2 + 𝜔4 𝐿2 𝑅2 𝐶 2
4𝑅
We get
𝑅 2 𝐶 2 + 𝐿2 − 2𝐿𝐶𝑅 2
−4𝑘 2 = The values of 𝐿
4𝑅 and 𝐶 can be
and obtained by
𝐿2 𝑅2 𝐶 2
4𝑘 2 = solving these
4𝑅 equations.
Note that value for R (for N even) is not unity, so there will be an impedance mismatch if
the load has a unity (normalized) impedance; this can be corrected with a quarter-wave
transformer, or by using an additional filter element to make N odd. For odd N, it can be
shown that R = 1.
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Element Values for Equi-ripple Low-Pass Filter Prototypes
𝑔0 = 1, 𝜔𝑐 = 1 , 0.5 dB ripple

N 𝑔1 𝑔2 𝑔3 𝑔4 𝑔5 𝑔6
1 0.6986 1.0000
2 1.4029 0.7071 1.9841
3 1.5963 1.0967 1.5963 1.0000
4 1.6703 1.1926 2.3661 0.8491 1.9841
5 1.7058 1.2296 2.5408 1.2296 1.7058 1.0000

For practical filter, it will be necessary to determine the order of the filter. This is
usually dictated by a specification on the insertion loss at some frequency in the
stop-band of the filter.
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Low Pass to High Pass Transformation
The low pass prototype filter designs can be
transformed to high pass, band pass or band
𝑃𝐿𝑅 𝑃𝐿𝑅
reject response.
Loss pass to high pass transformation is
achieved by the frequency substitution:
−𝜔𝑐 −1 0 1 𝜔 −𝜔𝑐 𝜔𝑐
for 𝜔
𝜔

When this transformation is applied After performing impedance scaling

𝜔 1 1 1
𝑗𝜔𝐿𝑘 becomes −𝑗 𝜔𝑐 𝐿𝑘 = 𝑗𝜔𝐶 ′ , where 𝐶𝑘′ = 𝜔 𝐿 𝐶𝑘′ =
𝑘 𝑐 𝑘 𝑅0 𝜔𝑐 𝐿𝑘
and and
𝜔 1 1
𝑗𝜔𝐶𝑘 becomes −𝑗 𝑐 𝐶𝑘 = , where 𝐿′𝑘 = 𝑅0
𝜔 𝑗𝜔𝐿′𝑘 𝜔𝑐 𝐶𝑘 𝐿′𝑘 =
𝜔𝑐 𝐶𝑘
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Low Pass to Band Pass and Band Stop Transformation
𝑃𝐿𝑅
For LPF to BPF
𝜔0 𝜔 𝜔0
𝜔← −
𝜔2 − 𝜔1 𝜔0 𝜔
−𝜔0 𝜔0 1 𝜔 𝜔0
−1 0 1 𝜔 −𝜔2 −𝜔1 𝜔1 𝜔2 𝜔 = −
∆ 𝜔0 𝜔
Band Pass where,
𝜔2 − 𝜔1
𝑃𝐿𝑅
∆=
𝜔0
Simpler equations are
obtained when
−𝜔0 𝜔0 𝜔0 = 𝜔1 𝜔2
−𝜔2 −𝜔1 𝜔1 𝜔2 𝜔
Band
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Low Pass to Band Pass and Band Stop Transformation
When this transformation is applied,
𝑃𝐿𝑅
A series inductor 𝐿𝑘 is transformed to a
series LC circuit with
−𝜔0 𝜔0
𝐿𝑘 ∆
−1 0 1 𝜔 −𝜔2 −𝜔1 𝜔1 𝜔2 𝜔 𝐿′𝑘 = 𝐶𝑘′ =
∆𝜔0 𝜔 0 𝐿𝑘
Band Pass
𝑃𝐿𝑅
And the shunt capacitor 𝐶𝑘 is transformed to
a shunt LC circuit with elements
−𝜔0 𝜔0

−𝜔2 −𝜔1 𝜔1 𝜔2 𝜔 ∆ 𝐶𝑘
𝐿′𝑘 = 𝐶𝑘′ =
𝜔0 𝐶𝑘 ∆𝜔0
Band Stop
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Low Pass to Band Pass and Band Stop Transformation

For LPF to Band Stop

𝑃𝐿𝑅 −1
𝜔 𝜔0
𝜔←∆ −
𝜔0 𝜔
−𝜔0 𝜔0
−1 0 1 𝜔 𝜔1 𝜔2 A series inductors are transformed to parallel
−𝜔2 -𝜔1 𝜔
LC circuit with
Band Pass ∆𝐿𝑘 1
𝑃𝐿𝑅 𝐿′𝑘 = 𝐶𝑘′ =
𝜔0 𝜔0 ∆𝐿𝑘
−𝜔0 𝜔0
And the shunt capacitors are transformed to a
series LC circuit with elements
−𝜔1 𝜔1 𝜔2
−𝜔2 𝜔 1 ∆𝐶𝑘
𝐿′𝑘 = 𝐶𝑘′ =
Band Stop 𝜔0 ∆𝐶𝑘 𝜔0

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Summary of prototype filter transformations
Low Pass High Pass Band Pass Band Stop

𝐿
L 1 𝜔0 ∆ 𝐿∆ 1
𝜔𝑐 C 𝜔0 𝜔0 𝐿∆
∆
𝜔0 𝐿

1
1 ∆ 𝐶 𝜔0 𝐶∆
𝜔𝑐 L 𝜔0 𝐶 𝜔0 ∆
C
𝐶∆
𝜔0

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Filter Implementation
 Lumped element filter design discussed so far works well at lower frequencies.
 At higher RF frequencies lumped element inductors and capacitors are generally
available for limited range of values.
 At higher microwave frequencies, such elements are difficult to implement.
 Distributed elements such as stubs are often used to approximate the lumped
elements.
 Conversion of lumped element to equivalent transmission line sections can be
done using Richards’s transformation.
 Moreover, at microwave frequencies the spacing between the filter elements are
also to be considered.
 Kuroda’s identities are used to physically separate filter elements by various
transmission line sections.

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Filter Implementation
Tutorial

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Steps to design a Microstrip Filter

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Design Goal

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1.Find the order of Filter (N)

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2.Apply Richard’s Transformation

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2.Apply Richard’s Transformation

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3.Apply Kuroda’s Identities

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3.Apply Kuroda’s Identities

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3.Apply Kuroda’s Identities

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3.Apply Kuroda’s Identities

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3.Apply Kuroda’s Identities

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3.Apply Kuroda’s Identities

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4.Denormalization to 50 ohms

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Attenuation Profile

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CST MWS Simulation
Front View

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CST MWS Simulation
Back View

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CST MWS Simulation
Transmission Co-efficient S(2,1)(Linear)

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CST MWS Simulation
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BASIC CHARACTERISTICS OF MIXERS
Mixers are commonly used to multiply signals of different frequencies in an effort to achieve frequency
translation. The motivation for this translation stems from the fact that filtering out a particular RF signal
channel centered among many densely populated, narrowly spaced neighboring channels would require
extremely high Q-filters. This task , however, becomes much more manageable if the Rf signal carrier
frequency can be reduced or down-converted within the communication system. Perhaps one of the best
known systems is the down-conversion in a heterodyne receiver, schematically depicted in the following
figure.

Heterodyne receiver system incorporating a mixer

Here the received RF signal is, after preamplification in a low-noise amplifier (LNA), supplied to a mixer
whose task is to multiply the input signal of center frequency fRF with a local oscillator (LO) frequency fLO.
The signal obtained after the mixer contains the frequencies fRF ± fLO , of which after low-pass(LP) filtering,
the low frequency component fRF − fLO, known as the intermediate frequency (IF), is selected for further
processing. The two key ingredients constituting a mixer are the combiner and detector. The combiner can
be implemented through the use of 900 or 1800 directional coupler. The detector may employ non-linear
devices like diode or BJT or MESFET.

Basic mixer concept: two input frequencies are used to create new frequencies at the output of the system
Above figure depicts the basic system arrangement of a mixer connected to an RF signal, V RF(t), and local
oscillator signal, VLO(t), which is also known as the pump signal. It is seen that the RF input voltage signal is
combined with the LO signal and supplied to a semiconductor device with a nonlinear characteristic at its
output side driving a current into the load.

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Both diode and BJT have an exponential transfer characteristic, as expressed for instance by the Shockley’s
diode equation as follows:
𝑉⁄
𝐼 = 𝐼0 (𝑒 𝑉𝑇 − 1)

Alternatively, for a MESFET we have approximately a square behavior:

2
𝐼(𝑉) = 𝐼𝐷𝑆𝑆 (1 − 𝑉⁄𝑉 )
𝑇0

where the subscripts denoting drain current and gate-source voltage are omitted for simplicity. The input
voltage is represented as sum of RF signal 𝑣𝑅𝐹 = 𝑉𝑅𝐹 cos (𝜔𝑅𝐹 𝑡) and the LO signal 𝑣𝐿𝑂 = 𝑉𝐿𝑂 cos (𝜔𝐿𝑂 𝑡)
and a bias VQ; that is
𝑉 = 𝑉𝑄 + 𝑉𝑅𝐹 cos(𝜔𝑅𝐹 𝑡) + 𝑉𝐿𝑂 cos (𝜔𝐿𝑂 𝑡)

This voltage is applied to the nonlinear device whose current output characteristic can be found via a Taylor
series expansion around the Q-point:
𝑓 ′ (𝑎) 𝑓 ′′ (𝑎) 2
𝑓 ′′′ (𝑎)
𝑓(𝑥) = 𝑓(𝑎) + (𝑥 − 𝑎) + (𝑥 − 𝑎) + (𝑥 − 𝑎)3 + ⋯
1! 2! 3!
𝑑𝐼 1 𝑑2𝐼
𝐼(𝑉) = 𝐼𝑄 + 𝑉 ( )| + 𝑉 2 ( 2 )| + ⋯ = 𝐼𝑄 + 𝑉𝐴 + 𝑉 2 𝐵 + ⋯
𝑑𝑉 𝑉𝑄 2 𝑑𝑉 𝑉
𝑄

𝑑𝐼 𝑑2 𝐼
Where the constants A and B refer to (𝑑𝑉)| and (𝑑𝑉 2 )| , respectively. Neglecting the constant bias VQ
𝑉𝑄 𝑉𝑄
and IQ,

𝐼(𝑉) = 𝑉𝐴 + 𝑉 2 𝐵 + ⋯
𝐼(𝑉) = 𝐴 {𝑉𝑅𝐹 cos(𝜔𝑅𝐹 𝑡) + 𝑉𝐿𝑂 cos(𝜔𝐿𝑂 𝑡)} + 𝐵 {𝑉𝑅𝐹 cos(𝜔𝑅𝐹 𝑡) + 𝑉𝐿𝑂 cos (𝜔𝐿𝑂 𝑡)}2 + ⋯

𝐼(𝑉) = 𝐴 {𝑉𝑅𝐹 cos(𝜔𝑅𝐹 𝑡) + 𝑉𝐿𝑂 cos(𝜔𝐿𝑂 𝑡)}

2 2
+ 𝐵 {𝑉𝑅𝐹 cos2 (𝜔𝑅𝐹 𝑡) + 𝑉𝐿𝑂 cos 2 (𝜔𝐿𝑂 𝑡) + 2 𝑉𝑅𝐹 cos(𝜔𝑅𝐹 𝑡) 𝑉𝐿𝑂 cos(𝜔𝐿𝑂 𝑡)}
The key lies in the last term
𝐼(𝑉) = ⋯ + 2𝐵 𝑉𝑅𝐹 𝑉𝐿𝑂 cos(𝜔𝑅𝐹 𝑡) cos(𝜔𝐿𝑂 𝑡)
𝐼(𝑉) = ⋯ + 𝐵 𝑉𝑅𝐹 𝑉𝐿𝑂 {cos[(𝜔𝑅𝐹 +𝜔𝐿𝑂 )𝑡] + cos[(𝜔𝑅𝐹 −𝜔𝐿𝑂 )𝑡]}
This expression makes clear that the non-linear action of the diode or transistor can generate new frequency
components of the form 𝜔𝑅𝐹 ± 𝜔𝐿𝑂 . It is also noted that the amplitudes are multiplied by 𝑉𝑅𝐹 𝑉𝐿𝑂 , and B is
a device dependent factor.

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Spectral representation of mixing process

Problem of image frequency mapping

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TYPES OF MIXERS
Single-Ended Mixer
▪ RF and LO sources are supplied to an appropriately biased diode followed by a resonator circuited
tuned to the desired IF.

▪ Since LO and RF signals are not electrically isolated. There is a potential danger that the LO signal can
interfere with the RF reception, possibly even reradiating portions of the LO energy through the
receiving antenna.
▪ Following figure shows an improved design involving a FET, which, unlike the diode, is able to provide
gain to the incoming RF and LO signals.

▪ FET realization allows not only allows for LO and RF isolation but also provides a signal gain and thus
minimizes conversion loss
▪ Conversion Loss (CL) of a mixer is generally defined in dB as the ratio of supplied input power PRF
over the obtained IF power PIF. When dealing with BJTs and FETs, it is preferable to specify a
conversion gain (CG) defined as the inverse of the power ratio.
𝑃𝑅𝐹
CL = 10log ( )
𝑃𝐼𝐹
▪ Noise Figure(F) of a mixer is defined as
𝑃𝑛 𝑜𝑢𝑡
𝐹=
𝐶𝐺 𝑃𝑛 𝑖𝑛

Where CG being the conversion gain and 𝑃𝑛 𝑜𝑢𝑡 , 𝑃𝑛 𝑖𝑛 the noise power at the output due to the RF
signal input (at RF) and the total noise power at the output (at IF).
▪ FET generally has a lower noise figure than a BJT, and because of a nearly quadratic transfer
characteristic, the influence of higher-order nonlinear terms is minimized.

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▪ BJT finds application when high conversion gain and low voltage bias conditions are needed.
Single-Balanced Mixer
▪ Single-ended mixers are rather easy to construct circuits and the main disadvantage of these designs
is the difficulty associated with providing LO energy while maintaining separation between LO, RF,
and IF signals for broadband applications
▪ Balanced dual-diode or dual transistor mixer in conjunction with a hybrid coupler offers the ability to
conduct such broadband operations. Moreover, it provides further advantages related to noise
suppression and spurious mode rejection
▪ Spurs arise in oscillators and amplifiers due to parasitic resonances and non-linearities and are only
suppressed by the front-end. Thermal noise can critically raise the noise floor in the receiver.
▪ Following figure shows the basic mixer design featuring a quadratic coupler and a dual-diode detector
followed by a capacitor acting as summation point.

▪ This design provides excellent VSWR and is capable of suppressing a considerable amount of noise
because the opposite diode arrangement in conjunction with the 900 phase shift provides a good
degree of noise cancellation.
MICROWAVE NETWORKS
A microwave network is formed when several microwave devices and components such as sources,
attenuators, resonators, filters, amplifiers, etc., are coupled by transmission lines or waveguides for the
desired transmission of a microwave signal. The point of interconnection of two or more devices is called a
junction.
For a low-frequency network, a port is a pair of terminals whereas for a microwave network, a port is a
reference plane transverse to the length of the microwave transmission line or waveguide. At low
frequencies, the physical length of the network is much smaller than the wavelength of the signal
transmitted. Therefore, the measurable input and output variables are voltage and current which can be
related in terms of the impedance Z-parameters, or admittance Y-parameters, or hybrid h-parameters, or
ABCD parameters. For a two-port network as shown schematically in following figure, these relationships
are given by

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Basic voltage and current definitions for a two-port network

At microwave frequencies, the physical length of the component or line is comparable to or much larger
than the wavelength. Furthermore, the voltage and current cannot be uniquely defined at a given point in a
single conductor waveguide.

Besides this constraint, measurement of Z, Y, h and ABCD parameters is difficult at microwave frequencies
due to the following reasons:
1. Non-availability of terminal voltage and current measuring equipment even in the cases of TEM lines
(coaxial, strip and microstrip lines) where voltage and current can be uniquely defined.
2. Short-circuit and open-circuit conditions are not easily achieved over a wide range of frequencies.
3. Presence of active devices makes the circuit unstable or open and short-circuit.

Therefore, microwave circuits are analysed using Scattering or S-parameters which linearly relate the
amplitudes of scattered (reflected or transmitted) waves with those of incident waves. However, many of

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the circuit-analysis techniques and circuit properties that are valid at low frequencies are also valid for
microwave circuits.

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Meaning of S-parameters
S11 represents Input Reflection Coefficient; S21 indicates Forward Voltage Gain; S12 represents Reverse
Voltage Gain; S22 indicates Output Reflection Coefficient
SIGNAL FLOW CHART MODELING
The analysis of RF networks and their overall interconnection is greatly facilitated through signal flow charts
as commonly used in system and control theory. Even complicated networks are easily reduced to input
output relations in which the reflection and transmission coefficients play integral parts.

Terminated transmission line segment with incident and reflected power wave description. (a)
Conventional form, and (b) Signal flow form

Generic source node (a), receiver node (b), and the associated branch connection (c).

Terminated transmission line with source. (a) conventional form, (b) signal flow form, and (c) simplified
signal flow form

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A self-loop that collapses to a single branch

Table: Signal Flowgraph Building Blocks

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AMPLIFIER POWER RELATIONS
RF SOURCE

Generic amplifier system

Let us examine the above figure in terms of its power flow relations under the assumptions that the two
matching networks are included in the source and load impedances. This simplifies our system to the
configuration shown below:

Source and load connected to a single-stage amplifier network

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The starting point of our power analysis is the RF source connected to the amplifier network. The source
voltage is written as
𝑏𝑆 = 𝑏1′ (1 − Γ𝑖𝑛 Γ𝑆 ) (1)
The incident power wave associated with 𝑏1′ is given as
|𝑏1′ |2 1 |𝑏𝑆 |2
𝑃𝑖𝑛𝑐 = = (2)
2 2 |1 − Γ𝑖𝑛 Γ𝑆 |2
which is the power launched toward the amplifier. The actual input power 𝑃𝑖𝑛 observed at the input
terminal of the amplifier is composed of the incident and reflected power waves. With the aid of the input
reflection coefficient Γ𝑖𝑛 we can therefore write:
1 |𝑏𝑆 |2
𝑃𝑖𝑛 = 𝑃𝑖𝑛𝑐 (1 − |Γ𝑖𝑛 |2 ) = (1 − |Γ𝑖𝑛 |2 ) (3)
2 |1 − Γ𝑖𝑛 Γ𝑆 |2
The maximum power transfer from source to the amplifier is achieved if the input impedance is complex
conjugate matched 𝑍𝑖𝑛 = 𝑍𝑆∗ or in terms of the reflection coefficients , if Γ𝑖𝑛 = Γ𝑆∗ . Under maximum power
transfer condition, we define available power 𝑃𝐴 as

1 |𝑏𝑆 |2 1 |𝑏𝑆 |2
𝑃𝐴 = 𝑃𝑖𝑛 |Γ𝑖𝑛=Γ∗𝑆 = (1 − |Γ𝑖𝑛 |2 ) | = (4)
2 |1 − Γ𝑖𝑛 Γ𝑆 |2 Γ ∗ 2 1 − |Γ𝑆 |2
𝑖𝑛 =Γ𝑆

Transducer Power Gain

The Transducer Power Gain quantifies the gain of the amplifier placed between source and load.
𝑃𝑜𝑤𝑒𝑟 𝑑𝑒𝑙𝑖𝑣𝑒𝑟𝑒𝑑 𝑡𝑜 𝑡ℎ𝑒 𝑙𝑜𝑎𝑑 𝑃𝐿
𝐺𝑇 = =
𝐴𝑣𝑎𝑖𝑙𝑎𝑏𝑙𝑒 𝑝𝑜𝑤𝑒𝑟 𝑓𝑟𝑜𝑚 𝑡ℎ𝑒 𝑠𝑜𝑢𝑟𝑐𝑒 𝑃𝐴
1
Or with 𝑃𝐿 = 2 |𝑏2 |2 (1 − |Γ𝐿 |2 ) we obtain

𝑃𝐿 |𝑏2 |2
𝐺𝑇 = = (1 − |Γ𝐿 |2 ) (1 − |Γ𝑆 |2 ) (5)
𝑃𝐴 |𝑏𝑆 |2
In this expression, the ratio 𝑏2 ⁄𝑏𝑆 has to be determined. With the help of our signal flow graph discussion
in above section and based on the following figure, we establish
𝑆21 𝑎1
𝑏2 = (6𝑎)
1 − 𝑆22 Γ𝐿
𝑆21 𝑆12 Γ𝐿
𝑏𝑆 = [1 − (𝑆11 + )Γ ] 𝑎 (6𝑏)
1 − 𝑆22 Γ𝐿 𝑆 1
The required ratio is therefore given by
𝑏2 𝑆21
= (7)
𝑏𝑆 (1 − 𝑆22 Γ𝐿 )(1 − 𝑆11 Γ𝑆 ) − 𝑆21 𝑆12 Γ𝐿 Γ𝑆

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𝒃𝟐
Step-by-step simplification to determine the ratio 𝒃𝑺

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Inserting equation (7) into equation (5) results in

(1 − |Γ𝐿 |2 ) |𝑆21 |2 (1 − |Γ𝑆 |2 )

𝐺𝑇 = (8)
|(1 − 𝑆22 Γ𝐿 )(1 − 𝑆11 Γ𝑆 ) − 𝑆21 𝑆12 Γ𝐿 Γ𝑆 |2
Which can be rearranged by defining the input and output reflection coefficients
𝑆21 𝑆12 Γ𝐿
Γ𝑖𝑛 = 𝑆11 + (9𝑎)
1 − 𝑆22 Γ𝐿
𝑆12 𝑆21 Γ𝑆
Γ𝑜𝑢𝑡 = 𝑆22 + (9𝑏)
1 − 𝑆11 Γ𝑆
With these two definitions, two more transducer power gain expressions can be derived. First, by incorporating
equation (9a) into equation (8), it is seen that

(1 − |Γ𝐿 |2 ) |𝑆21 |2 (1 − |Γ𝑆 |2 )

𝐺𝑇 = (10)
|1 − Γ𝑆 Γ𝑖𝑛 |2 |1 − 𝑆22 Γ𝐿 |2
Second, using equation(9b) into equation (8), it is seen that

(1 − |Γ𝐿 |2 ) |𝑆21 |2 (1 − |Γ𝑆 |2 )

𝐺𝑇 = (11)
|1 − Γ𝐿 Γ𝑜𝑢𝑡 |2 |1 − 𝑆11 Γ𝑆 |2
An often employed approximation for the transducer power gain is the so-called unilateral power gain, 𝐺𝑇𝑈 , which
neglects the feedback effect of the amplifier(𝑆21 = 0). This simplifies equation (11) to
(1 − |Γ𝐿 |2 ) |𝑆21 |2 (1 − |Γ𝑆 |2 )
𝐺𝑇𝑈 = (12)
|1 − Γ𝐿 𝑆22 |2 |1 − 𝑆11 Γ𝑆 |2

Additional Power Relations

Available power gain for load side matching is defined as

𝑝𝑜𝑤𝑒𝑟 𝑎𝑣𝑎𝑖𝑙𝑎𝑏𝑙𝑒 𝑓𝑟𝑜𝑚 𝑡ℎ𝑒 𝑎𝑚𝑝𝑙𝑖𝑓𝑖𝑒𝑟
𝐺𝐴 = 𝐺𝑇 |Γ𝐿 =Γ∗𝑜𝑢𝑡 =
𝑝𝑜𝑤𝑒𝑟 𝑎𝑣𝑎𝑖𝑙𝑎𝑏𝑙𝑒 𝑓𝑟𝑜𝑚 𝑡ℎ𝑒 𝑠𝑜𝑢𝑟𝑐𝑒
With the aid of equation (11),

|𝑆21 |2 (1 − |Γ𝑆 |2 )
𝐺𝐴 = (13)
(1 − |Γ𝑜𝑢𝑡 |2 ) |1 − 𝑆11 Γ𝑆 |2
Power gain (operating power gain) is defined as
𝑝𝑜𝑤𝑒𝑟 𝑑𝑒𝑙𝑖𝑣𝑒𝑟𝑒𝑑 𝑡𝑜 𝑡ℎ𝑒 𝑙𝑜𝑎𝑑 𝑃𝐿 𝑃𝐿 𝑃𝐴 𝑃𝐴
𝐺= = = . = 𝐺𝑇 .
𝑝𝑜𝑤𝑒𝑟 𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑑 𝑡𝑜 𝑡ℎ𝑒 𝑎𝑚𝑝𝑙𝑖𝑓𝑖𝑒𝑟 𝑃𝑖𝑛 𝑃𝐴 𝑃𝑖𝑛 𝑃𝑖𝑛
Using equations (3), (4) and (10)

(1 − |Γ𝐿 |2 ) |𝑆21 |2
𝐺= (14)
(1 − |Γ𝑖𝑛 |2 ) |1 − 𝑆22 Γ𝐿 |2

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Example: Power relations or an RF amplifier

An RF Amplifier has the following S-parameters:

𝑆11 = 0.3∠−700 , 𝑆12 = 0.2∠−100 , 𝑆21 = 3.5∠850 𝑎𝑛𝑑 𝑆22 = 0.4∠−450 . Furthermore, the input side of
the amplifier is connected to a voltage source with 𝑉𝑆 = 5𝑉∠00 and source impedance 𝑍𝑆 = 40Ω. The
output is utilized to drive an antenna which has an impedance of 𝑍𝐿 = 73Ω. Assuming that the S-parameters
of the amplifier are measured with reference to a 𝑍0 = 50Ω characteristic impedance, find the following
quantities:
(a) Transducer gain 𝐺𝑇 , Unilateral transducer gain 𝐺𝑇𝑈 , available gain 𝐺𝐴 , operating power gain G and
(b) Power delivered to the load 𝑃𝐿 , available power 𝑃𝐴 and incident power to the amplifier 𝑃𝑖𝑛𝑐

Solution:
𝑍𝑆 − 𝑍0
Γ𝑆 = = −0.111
𝑍𝑆 + 𝑍0
𝑍𝐿 − 𝑍0
Γ𝐿 = = 0.187
𝑍𝐿 + 𝑍0
𝑆21 𝑆12 Γ𝐿
Γ𝑖𝑛 = 𝑆11 + = 0.146 − 𝑗0.151
1 − 𝑆22 Γ𝐿
𝑆12 𝑆21 Γ𝑆
Γ𝑜𝑢𝑡 = 𝑆22 + = 0.265 − 𝑗0.358
1 − 𝑆11 Γ𝑆
(1 − |Γ𝐿 |2 ) |𝑆21 |2 (1 − |Γ𝑆 |2 )
𝐺𝑇 = = 12.56 𝑜𝑟 10.99𝑑𝐵
|1 − Γ𝐿 Γ𝑜𝑢𝑡 |2 |1 − 𝑆11 Γ𝑆 |2
(1 − |Γ𝐿 |2 ) |𝑆21 |2 (1 − |Γ𝑆 |2 )
𝐺𝑇𝑈 = = 12.67 𝑜𝑟 11.03𝑑𝐵
|1 − Γ𝐿 𝑆22 |2 |1 − 𝑆11 Γ𝑆 |2
|𝑆21 |2 (1 − |Γ𝑆 |2 )
𝐺𝐴 = = 14.74 𝑜𝑟 11.68 𝑑𝐵
(1 − |Γ𝑜𝑢𝑡 |2 ) |1 − 𝑆11 Γ𝑆 |2
(1 − |Γ𝐿 |2 ) |𝑆21 |2
𝐺= = 13.74 𝑜𝑟 11.38 𝑑𝐵
(1 − |Γ𝑖𝑛 |2 ) |1 − 𝑆22 Γ𝐿 |2
1 |𝑏𝑆 |2 1 𝑍0 |𝑉𝑆 |2
𝑃𝑖𝑛𝑐 = = = 74.7 𝑚𝑊
2 |1 − Γ𝑖𝑛 Γ𝑆 |2 2 (𝑍𝑆 + 𝑍0 )2 |1 − Γ𝑖𝑛 Γ𝑆 |2

𝑃𝑖𝑛𝑐
𝑃𝑖𝑛𝑐 (𝑑𝐵𝑚) = 10 log [ ] = 18.73 𝑑𝐵𝑚
1𝑚𝑊
1 |𝑏𝑆 |2 1 𝑍0 |𝑉𝑆 |2
𝑃𝐴 = = = 78.1 𝑚𝑊 𝑜𝑟 18.93 𝑑𝐵𝑚
2 1 − |Γ𝑆 |2 2 (𝑍𝑆 + 𝑍0 )2 1 − |Γ𝑆 |2

𝑃𝐿 (𝑑𝐵𝑚) = 𝑃𝐴 (𝑑𝐵𝑚) + 𝐺𝑇 (𝑑𝐵𝑚) = 29.92 𝑑𝐵𝑚

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STABILITY of Amplifier

A single-stage microwave transistor amplifier can be modelled by the circuit shown in

Figure, where matching networks are used on both sides of the transistor to transform the input
and output impedance Z0 to the source and load impedances ZS and ZL .
In the circuit shown in Figure, oscillation is possible if either the input or output port
impedance has a negative real part; this would then imply that |Γ𝑖𝑛 | > 1 or |Γ𝑜𝑢𝑡 | > 1. Because
Γ𝑖𝑛 and Γ𝑜𝑢𝑡 depend on the source and load matching networks, the stability of the amplifier
depends on Γ𝑆 and Γ𝐿 as presented by the matching networks.

Two types of stability:

(i)Unconditional stability:
The network is unconditionally stable if |Γ𝑖𝑛 | < 1 and |Γ𝑜𝑢𝑡 | < 1 for all passive source and load
impedances (i.e., |Γ𝑆 | < 1 or |Γ𝐿 | < 1 ).
(ii)Conditional stability:
The network is conditionally stable if |Γ𝑖𝑛 | < 1 and |Γ𝑜𝑢𝑡 | < 1 only for a certain range of
passive source and load impedances. This case is also referred to as potentially unstable.

Note that the stability condition of an amplifier circuit is usually frequency dependent
since the input and output matching networks generally depend on frequency. It is therefore
possible for an amplifier to be stable at its design frequency but unstable at other frequencies.
Careful amplifier design should consider this possibility.

Requirements for unconditional stability

The following conditions must be satisfied by Γ𝑆 and Γ𝐿 ,if the amplifier is to be unconditionally
stable:

𝑆21 𝑆12 Γ𝐿
|Γ𝑖𝑛 | = |𝑆11 + |< 1 ------ (1)
1−𝑆22 Γ𝐿
𝑆12 𝑆21 Γ𝑆
|Γ𝑜𝑢𝑡 | = |𝑆22 + |< 1 -------- (2)
1−𝑆11 Γ𝑆

If the device is unilateral (S12 = 0), these conditions reduce to the simple results that |S11| < 1
and |S22| < 1 are sufficient for unconditional stability. Otherwise, the inequalities define a range
of values for Γ𝑆 and Γ𝐿 where the amplifier will be stable. Finding this range for Γ𝑆 and Γ𝐿 can
be facilitated by using the Smith chart and plotting the input and output stability circles.
The stability circles are defined as the loci in the Γ𝐿 (or Γ𝑆 ) plane for which |Γ𝑖𝑛 | = 1 (or
|Γ𝑜𝑢𝑡 | = 1). The stability circles then define the boundaries between stable and potentially

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unstable regions of Γ𝑆 and Γ𝐿 . Γ𝑆 and Γ𝐿 must lie on the Smith chart (|Γ𝑆 | < 1, |Γ𝐿 | < 1 for passive
matching networks).
We can derive the equation for the output stability circle as follows:
Express the condition that |Γ𝑖𝑛 | = 1 using equation (1)
𝑆21 𝑆12 Γ𝐿
|Γ𝑖𝑛 | = |𝑆11 + |= 1 ------- (3)
1−𝑆22 Γ𝐿
|𝑆11 (1 − 𝑆22 Γ𝐿 ) + 𝑆21 𝑆12 Γ𝐿 | = |1 − 𝑆22 Γ𝐿 | ------- (4)
Now define ∆ as the determinant of the scattering matrix:
∆ = S11 S22 − S12 S21
We can write the above equation as,
|𝑆11 − ∆Γ𝐿 | = |1 − 𝑆22 Γ𝐿 | -------- (5)

In the complex plane, an equation of the form |Γ − 𝐶| = 𝑅 represents a circle having a center
at C ( a complex number) and a radius R ( a real number). The above resultant equation defines
the output stability circle with a center CL and radius RL where,
(𝑆22 −Δ𝑆11 ∗ )∗
𝐶𝐿 = |𝑆22 |2 −|∆|2
(𝑐𝑒𝑛𝑡𝑒𝑟) -------- (6a)

𝑆12 𝑆21
𝑅𝐿 = ||𝑆 2 2|
(𝑟𝑎𝑑𝑖𝑢𝑠) -------- (6b)
22 | −|∆|
Similar results can be obtained for the input stability circle by interchanging S11 and S22.
(𝑆11 −Δ𝑆22∗ )∗
𝐶𝑆 = |𝑆11 | −|∆|2
2
(𝑐𝑒𝑛𝑡𝑒𝑟) -------- (7a)

𝑆12 𝑆21
𝑅𝑆 = ||𝑆 2 −|∆|2
| (𝑟𝑎𝑑𝑖𝑢𝑠) -------- (7b)
11 |

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Given the scattering parameters of the transistor, we can plot the input and output
stability circles to define where |Γ𝑖𝑛 | = 1 and |Γ𝑜𝑢𝑡 | = 1. On one side of the input stability circle
we will have |Γ𝑜𝑢𝑡 | < 1, while on the other side we will have |Γ𝑜𝑢𝑡 | > 1.
Similarly, we will have |Γ𝑖𝑛 | < 1 on one side of the output stability circle, and |Γ𝑖𝑛 | > 1
on the other side. We need to determine which areas on the Smith chart represent the stable
region, for which |Γ𝑖𝑛 | < 1 and |Γ𝑜𝑢𝑡 | < 1.

Consider the output stability circles plotted in the Γ𝐿 plane for |S11| < 1 and |S11| > 1, as
shown in Figure. If we set ZL = Z0, then Γ𝐿 = 0, and (1) shows that |Γ𝑖𝑛 | =|S11|. Now if |S11| <
1, then |Γ𝑖𝑛 |< 1, so Γ𝐿 = 0 must be in a stable region. This means that the center of the Smith
chart (Γ𝐿 = 0) is in the stable region, so all of the Smith chart (|Γ𝐿 | < 1) that is exterior to the
stability circle defines the stable range for Γ𝐿 . This region is shaded in Figure (a). Alternatively,
if we set ZL = Z0 but have |S11| > 1, then |Γ𝑖𝑛 | > 1 for Γ𝐿 = 0, and the center of the Smith chart
must be in an unstable region. In this case the stable region is the inside region of the stability
circle that intersects the Smith chart, as illustrated in Figure (b). Similar results apply to the
input stability circle.
If the device is unconditionally stable, the stability circles must be completely outside (or
totally enclose ) the smith chart. We can state this mathematically as,
||𝐶𝐿 | − 𝑅𝐿 | > 1 𝑓𝑜𝑟 |𝑆11 | < 1 ----- (8a)
||𝐶𝑆 | − 𝑅𝑆 | > 1 𝑓𝑜𝑟 |𝑆22 | < 1 ----- (8a)

If |S11| > 1 or |S22| > 1, the amplifier cannot be unconditionally stable because we can always
have a source or load impedance of Z0 leading to Γ𝑆 = 0 or Γ𝐿 = 0, thus causing |Γ𝑖𝑛 | > 1 or |Γ𝑜𝑢𝑡 |
> 1. If the device is only conditionally stable, operating points for Γ𝑆 and Γ𝐿 must be chosen
in stable regions, and it is good practice to check stability at several frequencies over the range
where the device operates. Also note that the scattering parameters of a transistor depend on
the bias conditions, and so stability will also depend on bias conditions.

Tests for Unconditional Stability

The stability circles discussed above can be used to determine regions for Γ𝑆 and Γ𝐿
where the amplifier circuit will be conditionally stable, but simpler tests can be used to
determine unconditional stability. One of these is the K − ∆ test, where it can be shown that a
device will be unconditionally stable if Rollet’s condition, defined as,

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1−|𝑆11 |2 − |𝑆22 |2 +|∆|2

𝐾= > 1 ------- (9)
2|𝑆12 𝑆21 |

Along with the auxiliary condition that

|∆| = |𝑆11 𝑆22 − 𝑆12 𝑆21 | < 1 --------(10)

are simultaneously satisfied. These two conditions are necessary and sufficient for
unconditional stability, and are easily evaluated. If the device scattering parameters do not
satisfy the K −∆ test, the device is not unconditionally stable, and stability circles must be used
to determine if there are values of Γ𝑆 and Γ𝐿 for which the device will be conditionally stable.
Also recall that we must have |S11| < 1 and |S22| < 1 if the device is to be unconditionally stable.
While the K − ∆ test is a mathematically rigorous condition for unconditional stability, it
cannot be used to compare the relative stability of two or more devices because it involves
constraints on two separate parameters.

𝝁 -test
combines the scattering parameters in a test involving only a single parameter, 𝜇 .
𝜇 is defined as,
1−|𝑆11 |2
𝜇 = |𝑆 ∗ >1 -------- (11)
22 −∆𝑆11 |+|𝑆12 𝑆21 |

If 𝜇 > 1, the device is unconditionally stable. In addition, the larger values of 𝜇 imply greater
stability.

Development of 𝝁 -test
𝑆12 𝑆21 Γ𝑆 𝑆 −∆Γ𝑆
22
Γ𝑜𝑢𝑡 = 𝑆22 + = 1−𝑆 --------(12)
1−𝑆11 Γ𝑆 11 Γ𝑆
Where, ∆ is the determinant of the scattering matrix. Unconditional stability implies that |Γ𝑜𝑢𝑡 |
< 1 for any passive source termination, Γ𝑆 . The reflection coefficient for a passive source
impedance must lie within the unit circle on a Smith chart, and the outer boundary of this circle
can be written as Γ𝑆 = e jφ. The expression given in (12) maps this circle into another circle in
the Γ𝑜𝑢𝑡 plane.
Substituting Γ𝑆 = e jφ into (12) and solving for e jφ:
𝑆22 −Γ𝑜𝑢𝑡
𝑒 𝑗∅ =
∆ − 𝑆11 Γ𝑜𝑢𝑡

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This equation is of the form | Γ𝑜𝑢𝑡 − C| = R, representing a circle with center C and radius R in
the Γ𝑜𝑢𝑡 plane. Thus the center and radius of the mapped |Γ𝑆 | = 1 circle are given by,

𝑆22 −∆𝑆11∗
𝐶= -------- (13a)
1−|𝑆11 |2
|𝑆12 𝑆21 |
𝑅 = 1−|𝑆 2
--------- (13b)
11 |

If points within this circular region are to satisfy |Γ𝑜𝑢𝑡 | < 1, then we must have that ,
|C| + R < 1 -------- (14)
Substituting equation (13) in equation (14) gives,
∗ |
|𝑆22 − ∆𝑆11 + |𝑆12 𝑆21 | < 1 − |𝑆11 |2
Rearranging the above equation yields the 𝜇- test.
1−|𝑆11 |2
∗ |+ |𝑆 𝑆 |
|𝑆22 −∆𝑆11
>1
12 21

Development of K − ∆ test
The K − ∆ test can be derived more simply from the µ-test.
1−|𝑆11 |2
∗ |+ |𝑆 𝑆 | >
|𝑆22 −∆𝑆11
1
12 21
|𝑆22 − ∆𝑆11 ∗ | |𝑆11 |2 − |𝑆12 𝑆21 | -------- (15)
< 1−
Rearranging and squaring gives,

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Which yields the Rollet’s condition

1−|𝑆11 |2 − |𝑆22 |2 +|∆|2

𝐾= > 1
2|𝑆12 𝑆21 |

The squaring of equation (15) introduces an ambiguity in the sign of the right hand side, thus
requiring an additional condition. The right hand side of equation (15) should be positive before
squaring. Thus,

Which is the required additional condition.

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SINGLE-STAGE TRANSISTOR AMPLIFIER DESIGN for maximum gain

A single-stage microwave transistor amplifier can be modelled by the circuit shown in

Figure, where matching networks are used on both sides of the transistor to transform the input
and output impedance Z0 to the source and load impedances ZS and ZL . The most useful gain
definition for amplifier design is the transducer power gain, which accounts for both source
and load mismatch. We can define separate effective gain factors for the input (source)
matching network, the transistor itself, and the output (load) matching network as follows:
1−|Γ𝑆 |2
𝐺𝑆 = |1−Γ 2
------- (1a)
𝑖𝑛 Γ𝑆 |
|S21 | 2
𝐺0 = -------(1b)
1−|Γ𝐿 |2
𝐺𝐿 = |1−S 2
------(1c)
22 Γ𝐿 |
The overall transducer gain is then GT = GSG0GL . The effective gains GS and GL of the
matching networks may be greater than unity. This is because the unmatched transistor would
incur power loss due to reflections at the input and output of the transistor, and the matching
sections can reduce these losses.
Design for Maximum Gain (Conjugate Matching)
After the stability of the transistor has been determined and the stable regions for Γ𝑆 and
Γ𝐿 have been located on the Smith chart, the input and output matching sections can be
designed. Since G0 is fixed for a given transistor, the overall transducer gain of the amplifier
will be controlled by the gains, GS and GL , of the matching sections.
Maximum gain will be realized when these sections provide a conjugate match
between the amplifier source or load impedance and the transistor.
We know that maximum power transfer from the input matching network to the
transistor will occur when
Γ𝑖𝑛 = Γ𝑆∗ ------(2a)
and that maximum power transfer from the transistor to the output matching network will
occur when
Γ𝑜𝑢𝑡 = Γ𝐿∗ -------- (2b)
With the assumption of lossless matching sections, these conditions will maximize the overall
transducer gain.
The maximum gain is given by,
1 1−|Γ𝐿 |2
𝐺𝑇𝑚𝑎𝑥 = 1−|Γ 2
|S21 |2 -------- (3)
𝑆| |1−S22 Γ𝐿 |2

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In the general case with a bilateral (S12 = 0) transistor, Γ𝑖𝑛 is affected by Γ𝑜𝑢𝑡 and vice versa, so
the input and output sections must be matched simultaneously. The necessary equations are:
𝑆21 𝑆12 Γ𝐿
Γ𝑖𝑛 = 𝑆11 + = Γ𝑆∗ --------(4)
1−𝑆22 Γ𝐿
𝑆12 𝑆21 Γ𝑆
Γ𝑜𝑢𝑡 = 𝑆22 + = Γ𝐿∗ --------(5)
1−𝑆11 Γ𝑆

Solutions to Γ𝑆 and Γ𝐿 are only possible if the quantity within the square root is positive, and
it can be shown that this is equivalent to requiring K > 1. Thus, unconditionally stable devices
can always be conjugately matched for maximum gain, and potentially unstable devices can be
conjugately matched if K > 1 and |∆| < 1.
∗
The results are much simpler for the unilateral case. When S12 = 0, Γ𝑆 = 𝑆11 and Γ𝐿 =
∗
𝑆22 , and the maximum unilateral transducer gain is

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1 1
𝐺𝑇𝑈𝑚𝑎𝑥 = 2
|S21 |2
1 − |𝑆11 | 1 − |S22 |2

The maximum transducer power gain occurs when the source and load are conjugately matched
to the transistor. If the transistor is unconditionally stable, so that K > 1, the maximum
transducer power gain can be simply rewritten as follows:
|S21 |
𝐺𝑇𝑚𝑎𝑥 = (𝐾 − √𝐾 2 − 1)
|S12 |

The maximum transducer power gain is also sometimes referred to as the matched gain. The
maximum gain does not provide a meaningful result if the device is only conditionally stable
since simultaneous conjugate matching of the source and load is not possible if K < 1. In this
case a useful figure of merit is the maximum stable gain, defined as the maximum transducer
power gain with K = 1.
|S |
Thus, 𝐺𝑚𝑠𝑔 = |S21|
12
The maximum stable gain is easy to compute and offers a convenient way to compare the gain
of various devices under stable operating conditions.

Single stage amplifier design

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Low-Noise Amplifier Design

Besides stability and gain, another important design consideration for a microwave
amplifier is its noise figure. In receiver applications especially it is often required to have a
preamplifier with as low a noise figure as possible since, the first stage of a receiver front end
has the dominant effect on the noise performance of the overall system.
Generally it is not possible to obtain both minimum noise figure and maximum gain for
an amplifier, so some sort of compromise must be made. This can be done by using constant-
gain circles and circles of constant noise figure to select a usable trade-off between noise figure
and gain. Here we will derive the equations for constant–noise figure circles and show how
they are used in transistor amplifier design.

The noise figure of a two-port amplifier can be expressed as

𝑅𝑁 2
𝐹 = 𝐹𝑚𝑖𝑛 + |𝑌𝑆 − 𝑌𝑜𝑝𝑡 | ------ (1)
𝐺𝑆
where the following definitions apply:
YS = GS + j BS = source admittance presented to transistor.
Yopt = optimum source admittance that results in minimum noise figure.
Fmin = minimum noise figure of transistor, attained when YS = Yopt.
RN = equivalent noise resistance of transistor.
GS = real part of source admittance.
Instead of the admittance YS and Yopt, we can use the reflection coefficients Γ𝑆 and Γ𝑜𝑝𝑡 ,
1 1−Γ𝑆
𝑌𝑆 = 𝑍 ---- (2a)
0 1+Γ𝑆

1 1−Γ𝑜𝑝𝑡
𝑌𝑜𝑝𝑡 = 𝑍 ---- (2b)
0 1+Γ𝑜𝑝𝑡

Γ𝑆 is the source reflection coefficient. The quantities Fmin, Γ𝑜𝑝𝑡 and RN are characteristics of
the particular transistor being used, and are called the noise parameters of the device; they may
be given by the manufacturer or measured.

2
Using equations for YS and Yopt , we can express the quantity |𝑌𝑆 − 𝑌𝑜𝑝𝑡 | interms of Γ𝑆 and
Γ𝑜𝑝𝑡 as,
2
2 4 |Γ𝑆 −Γ𝑜𝑝𝑡 |
|𝑌𝑆 − 𝑌𝑜𝑝𝑡 | = 𝑍 2 2 --------(3)
0 |1+Γ𝑆 |2 |1+Γ𝑜𝑝𝑡 |

In addition,
1 1−Γ 1−Γ∗ 1 1−|Γ𝑆 |2
𝐺𝑆 = 𝑅𝑒{𝑌𝑆 } = 2𝑍 (1+Γ𝑆 + 1+Γ∗𝑆 ) = 𝑍 |1+Γ𝑆 |2
-----(4)
0 𝑆 𝑆 0

Using these results, Noise figure F is given as,

𝑅 2
𝐹 = 𝐹𝑚𝑖𝑛 + 𝐺𝑁 |𝑌𝑆 − 𝑌𝑜𝑝𝑡 |
𝑆
2
4𝑅𝑁 |Γ𝑆 −Γ𝑜𝑝𝑡 |
𝐹 = 𝐹𝑚𝑖𝑛 + -------(5)
𝑍0 (1−|Γ𝑆 |2 )|1+Γ𝑜𝑝𝑡 |2

For a fixed Noise figure F, this result defines a circle in the Γ𝑆 plane.

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Noise figure parameter N is defined as,

2
|Γ𝑆 −Γ𝑜𝑝𝑡 |
𝑁= ------- (6)
(1−|Γ𝑆 |2 )

Using equation (5), N can be written as,

𝐹−𝐹𝑚𝑖𝑛 2
𝑁= 4𝑅𝑁 |1 + Γ𝑜𝑝𝑡 | -----(7)
⁄𝑍
0

N is a constant for a given noise figure and set of noise parameters. Equation (6) can be
rewritten as,

Generally it is not possible to obtain both minimum noise figure and maximum gain for
an amplifier, so some sort of compromise must be made. This can be done by using constant-
gain circles and circles of constant noise figure to select a usable trade-off between noise figure
and gain.

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