Microwave Book

Table of Contents Fahim Aziz Umrani (2KES23)

TABLE OF CONTENTS

Table of Contents............................................................................................... i

Acknowledgments .......................................................................................... iii

Unit – I: Transmission Line Theory .......................................................1-10

1.1 Microwave............................................................................................. 2 1.2 Transmission Line................................................................................ 2

1.2.1 Characteristic Impedance .................................................... 3 1.2.2 Equivalent Circuit for Transmission Line ......................... 3

1.3 Traveling Waves................................................................................... 4 1.4 Impedance & Admittance .................................................................. 5 1.5 Transmission Line Parameters .......................................................... 5 1.6 Incident and Reflected Waves............................................................................ 6 1.7 Transmission Modes ............................................................................................ 7 1.8 Discontinuity in Transmission Line.................................................. 9

Unit – II: Propagation Characteristics..................................................11-18

2.1 Propagation Characteristics ................................................................... 12 2.2 Sources of Attenuation............................................................................ 12

2.2.1 Conductor losses ................................................................ 12 2.2.2 Dielectric losses .................................................................. 13 2.2.3 Hysteresis losses ................................................................. 13 2.2.4 Mismatch losses .................................................................. 13 2.2.5 Losses due to radiation...................................................... 14

2.3 Reflection Coefficient .............................................................................. 14 2.4 Standing Waves ....................................................................................... 15 2.5 Smith Chart .............................................................................................. 16

Unit – III: Transmission Lines...............................................................19-26

3.1 Coaxial Transmission Line ............................................................... 20 3.1.1 Defining Equivalent Circuit Components....................... 20 3.1.2 Attenuation ........................................................................... 21

3.2 Field Configurations on Coaxial Transmission Lines ................. 22 3.2.1 Higher-Order Modes........................................................... 23

3.3 Waveguide Transmission Line........................................................ 23

Unit – IV: Microwave Sources & Detectors........................................27-33

4.1 Microwave Sources.................................................................................. 28 4.2 Klystron...................................................................................................... 28 4.3 Multi-Cavity Klystron Amplifiers ........................................................ 29 4.4 Reflex Klystron ......................................................................................... 29

Microwave Engineering

Table of Contents Fahim Aziz Umrani (2KES23)

4.5 Backward Wave Oscillator (BWO)........................................................ 30 4.6 Detection of Microwave Signals ............................................................ 31 4.7 Detector ...................................................................................................... 31

4.7.1 Crystal Detectors.................................................................. 31 4.7.2 Square Law of Crystal Detectors....................................... 32

4.8 Indicators ................................................................................................... 33

Unit – V: Microwave Mixing .................................................................34-43

5.1 Introduction to Mixers ...................................................................... 35 5.2 Theory of Mixing................................................................................ 35 5.3 Conversion Loss ................................................................................. 37 5.4 Parametric Amplifier......................................................................... 37 5.5 Parametric Up-Converter ................................................................. 38 5.6 Parametric Down-Converter............................................................ 39 5.7 Manely – Rowe Power Relation ...................................................... 39 5.8 Negative Resistance Parametric Amplifier ................................... 40 5.9 Harmonic Frequency Conversion................................................... 41

References ......................................................................................................... iv

Microwave Engineering

Unit – 01

Transmission Line Theory

1.1 Microwave

1.2 Transmission Line 1.2.1 Characteristic Impedance

1.2.2 Equivalent Circuit for Transmission Line

1.3 Traveling Waves

1.4 Impedance & Admittance

1.5 Transmission Line Parameters

1.6 Incident and Reflected Waves

1.7 Transmission Modes

1.8 Discontinuity in Transmission Line

Microwave Engineering Fahim Aziz Umrani (2KES23)

1.1 Microwave

Microwave is a descriptive term used to identify the electromagnetic waves in the frequency spectrum, ranging from 30 MHz to 3000 GHz. This corresponds to the wavelength of 10 mm to 1 m (or 3 mm to 1.3m). This means that microwave have very short wavelength, and high frequencies. The microwave fills the part of electromagnetic frequency spectrum between conventional radio wave and optical wave or infrared waves. Microwave engineering is also called engineering of information and applied electromagnetic of electronics.

Microwave signals propagate in straight lines and are affected very little by the troposphere. They are not refracted or reflected by ionized regions in the upper atmosphere. Microwave beams do not readily diffract around barriers such as hills, mountains, and large human-made structures. Some attenuation occurs when microwave energy passes through trees and frame houses. Radio-frequency (RF) energy at longer wavelengths is affected to a lesser degree by such obstacles.

The microwave band is well suited for wireless transmission of signals having large bandwidth. In communications, a large allowable bandwidth translates into high data speed. The short wavelengths allow the use of dish antennas having manageable diameters. These antennas produce high power gain in transmitting applications, and have excellent sensitivity and directional characteristics for reception of signals.

S.No Name Abbreviation Frequency Wavelength (λ) 1 Metric wave VHF 30 – 300 MHz 10 m – 1 m 2 Decimetric wave UHF 300 – 3000 MHz 1 m – 10 cm 3 Centimetric wave SHF 3 – 30 GHz 10 cm – 1 cm 4 Millmetric wave EHF 30 – 300 GHz 1 cm – 1 mm 5 Decimillimetric wave EHF 300 GHz – 3000 GHz 1 mm – 0.1 mm

Characteristics:

Increased bandwidth Ability to use high gain directive antennas It gives direct signal transmission (as in Radar) In comparison to radio and infrared waves, microwave ranging from 1 MHz – 10 GHz are

acceptable to propagate freely through inside sphere of the layer surrounding the earth (ionized)

Short wavelength simplifies the design and installation of high dielectric antenna. Antenna directivity depends upon antenna aperture and wavelength.

1.2 Transmission Line

The material medium or structure that forms all or part of a path from one place to another for directing the transmission of energy, such as electric currents, magnetic fields, acoustic waves, or electromagnetic waves is called transmission line. Examples of transmission lines include wires, optical fibers, coaxial cables, rectangular closed waveguides, and dielectric slabs. When analyzing a transmission line it is generally assumed that the cross-sectional geometry is constant, forming a uniform transmission line. If there is a change in the geometry at any point, there will be a “discontinuity” in the line.

As the uses of electromagnetic spectra increases, telecommunication bandwidth requirements increase, and equipment must be designed for higher frequencies. As the frequency increases, the value of components used in networks keep decreasing. As one approaches ultrahigh frequencies, the values of inductors and capacitors become so small that the ordinary techniques are not usable anymore.

Unit – 01: Transmission Line Theory

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On a transmission line carrying alternating current signals, the current and voltage vary sinusoidally along the line as well as in time at a fixed point on the line. The repetition time is called the period, and the repetition distance is called the wavelength. The velocity of waves on the line is given by the expression velocity = (wavelength)/ (period). The waves travel a distance of one wavelength during a time of one period.

Transmission lines store energy, and convey it from one place to another. The lossless transmission lines absorb energy or power from a generator, convey it elsewhere, but don't dissipate it. What goes in must come out. If it can't come out anywhere else it must come back to the source ("generator" end). This is the fundamental property of lossless transmission line.

The velocity of electromagnetic waves on transmission line is equal to LC/1 where the inductance and capacitance are taken for unit distance only (Henries per meter and Farads per meter). Using these SI units for inductance and capacitance, the velocity is also expressed in SI units; meters/sec.

To slow down the waves on the transmission line, all you need to do is to increase either or both of the inductance/meter or the capacitance/meter. The capacitance/meter is increased most easily by encasing the conductors in a dielectric having permittivity greater than unity. The inductance/meter can also be increased by enclosing the conductors in a lossless non-conducting magnetic material (maybe ferrite) but this is more difficult. Another way to slow the waves down is to coil up one of the conductors.

1.2.1 Characteristic Impedance

The ratio of voltage (between the wires) to current (along one wire and back along the other) has dimensions of impedance or resistance. At a single frequency, on a lossless line, the current is in phase with the voltage and the impedance is real. It is called the Characteristic Impedance (Usually denoted by Zo.) It does not depend on what is connected to the ends of the line, but only on the line geometry and material construction.

The Characteristic Impedance, although real and looking like a resistance, is actually “lossless, non-dissipative impedance”. Nothing gets hot as a result of supplying energy to this resistance. All that happens is that energy is transferred from the generator and stored temporarily in the transmission line. At some later time, possibly a great many transit times later, it can be extracted and returned to the generator, or used to make a real resistive dissipative load get hot. The normalized impedance, a dimensionless number ( ), is the ratio of the actual impedance Z in ohms to the characteristic impedance in ohms. Similarly, the "characteristic admittance" Yo = 1/Zo Siemens, and the "normalized admittance" = 1/z = (Zo)/Z.

oZZz /=

1.2.2 Equivalent Circuit for Transmission Line

To understand completely the behavior of signal propagation on transmission line, it is not enough to understand the voltages between conductor and currents carried in the conductor. If a signal is applied to a uniformly long transmission line, electromagnetic waves will be carried down its path. Voltage exists between the conductor and current flows through them. Electric and magnetic fields are formed between and around the conductors, respectively, and their behavior and their field configuration are also very important.

A small section of such type of transmission path can be analyzed by using lumped circuits. For example, a unit length long piece of parallel wire transmission line is shown in Figure 1-1 above.

This circuit contains a series inductance. An inductance is defined current carrying conductor forming a magnetic field around itself that delays a voltage. Since a piece of wire does establish a magnetic field around itself, according to Biot- Savart law it does have inductance. Since the conductor can have finite


3


resistance, a series resistor will define it adequately. The two conductors are a finite distance apart, and they from some parallel capacitance. A dielectric medium keeping the two conductors a constant distance apart can have dielectric losses, so parallel capacitance would describe this effect sufficiently. In Giorgi (MKSA) system, inductance is measured in Henries/unit length; capacitance in farads/unit length, resistance is measured in ohms/unit length, and conductance in mhos/unit length. It can be imagined that a transmission line is built of an infinite number of infinitely short lengths of this type of “two port” networks cascaded one after another or connected in a tandem situation.

Figure 1-1: Unit-length piece of parallel wire and its equivalent lumped-circuit model.

1.3 Traveling Waves

When a sine wave is applied to an infinitely long transmission line, the wave will propagate along the line. Figure 1-2, shows this wave at three successive instants in time. (Note that the crest of the wave progresses down the transmission line.) The voltage wave on a uniform, lossless transmission line is always accompanied by a current wave of similar shape, and, regardless of their shape, the two waves will be propagated without any change in magnitude or shape. These waves have different electrical characteristics. The length of the wave λ is defined as the distance between successive points which have the same electrical phase. This wavelength depends upon the frequency of variation of the wave and dielectric constant of the medium through which the wave is traveling. In free space a wave will travel with a velocity of approximately m/s. however, in a medium other than free space, the velocity will

be reduced by the factor

8103×

rε/1 , where rε is the relative dielectric constant of the medium.

Figure 1-2: Traveling wave.

The following formula shows the relationship between the various factors which determine wavelength:

fv

r

.1ε

λ =

where v = velocity of propagation in free space, f = frequency of oscillation rε = relative dielectric constant of the medium the wave is traveling in.

Wavelength can also be defined as the distance in which the phase changes by 2π radians, where 2π = 360º.


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1.4 Impedance & Admittance

In transmission systems, impedance relationship takes a leading role in defining propagation characteristics. It is desirable to analyze the series and parallel elements of the equivalent circuit separately. Kirchhoff’s law allows us to add impedance in series and admittance in parallel configurations.

Figure 1-3: Impedance of the equivalent circuit.

The impedance of the circuit can be measured as shown in Figure 1-3, with output shorted out and input open-circuited. The parallel circuit components are shorted; only series components are measured. Impedance can be expressed as

LfjRjwLRZ 02π+=+=

Figure 1-4: Admittance of the equivalent circuit.

Admittance information can be gained by measuring from the other end when the input is open-circuited and output is short-circuited. Since the series elements are left open, only parallel components will be measured.

CfjGjwCGY 02π+=+=

1.5 Transmission Line Parameters

The four components of equivalent circuit of a transmission line are divided into series and parallel groups defining the impedance and admittance of transmission line, respectively. Two parameters can be derived using the impedance and admittance expressions. It is convenient to define propagation constant as

))(( jwCGjwLRYZ ++=×=γ

Since the square root of the product of two complex numbers is also a complex, the propagation constant is generally expressed as

βαγ j+=


5


Where ‘α’ is the attenuation constant in nepers/unit length (if the circuit components are given in MKSA system) and β is the phase constant in terms of radians/unit length. By definition, the other parameters like length characteristic parameters. Then

jwCGjwLR

YZZ

++

==0

If R and G are negligible in size, that is, if there is no absorptive loss on the transmission line, then

ohmsCLZ =0

It is the characteristic equation of impedance. The reciprocal of characteristic equation is called Admittance.

1.6 Incident and Reflected Waves

Voltage applied to a transmission line can be written in exponential form as

jwtpVV ε=1

Where Vp stands for peak-voltage. The current resulting from the applied voltage can be written as

jwtpII ε=1

These voltages and currents are periodical waves. If that voltage is applied to a transmission line, a voltage wave will proceed along that line. The voltage wave may be written in the exponential form as:

lVV γε1=

The associated current wave flowing in the line is

lII γε1=

If the transmission line is not infinitely long, it is terminated with an impedance ZL as shown in Figure 1-5.

Figure 1-5: transmission line transmitted with impedance not equal to the characteristic impedance.


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Since that the load impedance is not equal to the characteristic impedance, all the energy is propagated down the transmission line will be absorbed, and part of the signal is reflected because it is mismatched. This signal is traveling in the opposite direction from the “incident” signal. The voltage and current waves are:

ll

ll

eIeIIeVeVVγγ

γγ

21

21

+=+=

Where I1 and I2 are periodical current waves and V1 and V2 are periodical voltage waves. Thus, the voltage across load impedance will be

21 VVVL += The current flowing through the load is

0

2

0

121 Z

VZVIII L −=−=

Then the load impedance is given by

L

LL I

VZ =

Two wave trains are traveling opposite to each other: the incident wave and reflected wave. Since both are really traveling on the same line, which has a characteristic impedance of Z0, then the equation becomes

1

1

2

20 I

VIVZ ==

The following equation can be derived from the preceding equation.

OL

OL

ZZZZ

IV

+−

==2

2

ageInput voltVoltageOutput

This equation shows that the relative amplitudes and phases of both waves are determined by the terminating impedance only. The absolute magnitudes of the waves are dependent of the impedance of source.

1.7 Transmission Modes

Associated electric and magnetic fields form voltage and current waves travel down a transmission line. Since these fields are the result of current and voltage waves, which are periodical, the electric and magnetic fields also vary in periodic manner. As propagation frequency increases, an appreciable portion of wavelength of that propagation signal becomes comparable to the cross-sectional geometry of transmission line; more than one kind of electromagnetic field configurations can be imagined. As frequency increases, more and more different types of propagation modes can exist on a certain transmission line. If propagation frequency increases to infinity, infinite number of propagation mode can exist. These modes are called high-order modes of propagation. The principle mode is that which can carry the energy at all the frequencies. Higher order modes are those modes that propagate only above the definite frequency range. The point at which these frequencies start to propagate is called cut-off frequency for particular mode.

Figure 1-6: rice propagating down a blowgun.


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The following analogy shows how high-order modes are established. Figure 1-6 shows rice being blown down the inside of the blowgun. As the figure clearly shows that rice can fit in only one way. If the rice is continuously blown through the tube with a constant velocity, certain propagation will exist. This is analogous to single mode propagation. If one either increases the inside diameter of blowgun or decreases the particle size of the rice, the rice could propagate down the tube in different modes are shown in Figure 1-7.

Figure 1-7: Rice propagating down a blowgun when the particle size of rice is small compared to the cross-sectional geometry of the tube.

In Figure 1-7, particle size of the rice is small as compared to cross-sectional geometry of tube. Consequently, the rice will not be required to move down in the tube in predetermined way. It can tumble around and move all over inside the tube, showing down propagation of each particle and at the same time increasing it to a certain extent as rotational velocity may be added to the motion. This is analogous to some high-order mode of propagation on transmission lines.

Similarly, if rice size decreases again one can imagine more and more types of pattern that are analogous again to some even higher-order modes. This clearly shows that, the certain patterns can occur only when a definite size change occurs either in transmission line or in the propagation frequency.

The transmission line in Figure 1-8 shows the principle mode of propagation and the electric and magnetic field configuration of the pattern on the parallel wire. Since there is a difference in potential between the wires, an electric field is established between them. The solid lines in the figure show the electric field configuration. Since current flows in the conductors, magnetic fields are established around them. At any point in space, the electric and magnetic field lines are perpendicular to each other. The figure also clearly shows that these fields are all transverse to the direction of propagation. That is why these waves, in the principle mode, are called transverse electromagnetic waves, abbreviated as the TEM mode of propagation.

Figure 1-8: electric and magnetic field configuration of the parallel wire transmission line

Now, if propagating frequency increases so much that the length of the wave traveling down the transmission line is comparable in size to the cross-sectional geometry of that transmission line, higher order modes can propagate. These higher-order modes will have at least one of their field components in the direction of propagation. Depending on which component shows in the direction of propagation, it will


8


be called the H- or the E-wave of propagation. The H-wave is that in which at least one component of the magnetic field shows in the direction of propagation. This mode is called the transverse electric TE mode. The E-wave is that in which the electric field will have at least one component showing in the direction of propagation. This is called the transverse magnetic TM mode. Although each TE and TM mode can be infinite in number, at a certain frequency there can be only a finite number of higher-order modes propagating. The number of these modes is dependent entirely upon the geometry of the transmission line.

The velocity of propagation of TE and TM mode is different from each other and from a TEM mode. In fact, two types of velocities can be imagined: group velocity and phase velocity. Group velocity means the velocity of the entire group moving down in transmission line, and phase velocity includes all rotations and turns of the individual moving particles. Generally, in the TEM or principle mode, phase and group velocities are identical to each other. In a standard transmission line with no dielectric material around it, they would move with exactly the velocity of light.

In higher-order modes, group and phase velocity are related to each other by the following equation:

pg vvc =

Where is group velocity and is a phase velocity. The geometric means of the phase and group velocity are equal to the velocity of light.

gv pv

1.8 Discontinuity in Transmission Line

When standing waves are traveling from source to destination, then a sudden change in geometry occurs over transmission line and when uniform transmission line exists before and after the plane of that discontinuity, the problem can be handled as two transmission lines joined together. The only question is what happens at the plane or near the plane of the discontinuity. Figure 1-9 shows the discontinuity formed at the plane where tow uniform lines are joined together. The electric field between the conductors is drawn. As is apparent, the electric field lines are bent in the region near the discontinuity; but after some distance the lines are straighten out again. When either electric or magnetic field components, are aligned in the direction of propagation, higher order modes are launched. Although it is assumed that higher order modes cannot be propagated on this particular transmission line, this does not mean that they cannot be launched. Discontinuities in transmission line will effectively launch certain higher order modes and energy will be stored when they do. It is known from the lumped circuit theory that the energy storage will occur where either the capacitance or inductance and both are present. Discontinuities can be understood as reactive components on a transmission.

Figure 1-9: discontinuity on a transmission line.

Another effect can be observed from Figure 1-9, the electric field distortion occurs only at right (widest) side of the discontinuity, then number of field distortions occurs immediately left of the plane of discontinuity. Whether the discontinuity bents the electric or magnetic fields determines the equivalent circuits. If there are more discontinuities then there are more steps one after another on the transmission line. If they are close enough to each other they might interact with each other as shown in Figure 1-10. As can be seen, interference will occur when the fields lines due to discontinuity are not straighten before


9


another discontinuity occurs. Some discontinuities are close to each other (as shown in Figure 1-10(b)), but as they do not distort the field in common direction, and they do not interfere with each other. If they interfere, a third effect will occur. This modifies their signal and simple effect by a mutual coupled effect. This field distortion is very similar to the effect in capacitance due to fringing field effect. From this it is clear the very same term is used for these field distortions as in magnetic and electric fields.

Figure 1-10: multiple discontinuities (a) interfering (b) not interfering with each other.


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Unit – 02

Propagation Characteristics

2.1 Propagation Characteristics

2.2 Sources of Attenuation 2.2.1 Conductor losses

2.2.2 Dielectric losses

2.2.3 Hysteresis losses

2.2.4 Mismatch losses

2.2.5 Losses due to radiation

2.3 Reflection Coefficient

2.4 Standing Waves

2.5 Smith Chart


2.1 Propagation Characteristics

2.1.1 Attenuation constant and phase constant

As an electromagnetic wave is applied down a transmission line it is continuously attenuated by the lossy element of the line. Then the propagation constant is

βαγ jjwCGjwLR +=++= ))((

This equation clearly shows that it is composed of attenuation and phase constant (α is negative and it is not shown here since it is not gain but attenuation). As we know that incident waves are

lV γε=1

which can be further expressed as

ljlljV βαβα εεε ×== + )(1

The first part of this equations shows that voltage gets attenuated exponentially as the wave travels down the line. It was also seen that attenuation constant α will be expressed in terms of nepers per unit length, if it is calculated in MKSA system. To convert nepers to the more commonly used decibel per unit length, multiply by 8.69: 1 neper = 8.69 dB.

2.2 Sources of Attenuation

Attenuation can be contributed by many factors such as the following:

1. Conductor losses (skin effect) 2. dielectric losses 3. hysteresis losses 4. mismatch losses 5. losses due to radiation

The first three losses are absorptive losses by nature, since they dissipate energy. Mismatch loss and losses due to radiation reflect and guide the energy away from the transmission line, respectively.

2.2.1 Conductor losses (Skin Effect)

This loss is caused by series resistance of conducting medium. This loss is an absorptive type by nature meaning it absorbs energy and also dissipate it in the form of heat. Existence of skin effect is known from lower frequency techniques. As frequency increases, skin effect becomes more critical. In other words we can say, in higher frequencies in the transmission line, the current is restricted to travel in only surface layer conductor. The penetration of the current flow is defined by skin depth (δ). The skin depth is the thickness of the layer where the current density drops to 1/ε the value on the surface. Skin depth can be calculated by following equation

)(21 cm

f rµρ

πδ =

Unit – 02: Propagation Characteristics

12


where ρ = specific resistivity of conductor (Ω-cm) µr = relative permeability of conductor (this is considered only when the material is

ferromagnetic and has a relative permeability different from non-ferrous materials, in other words its value is equal to 1)

f = operating frequency (GHz)

Since skin depth at microwave frequencies is a very small fraction of the conductor thickness, it is common practice to plate the surfaces of conductors. Platting with a highly conductive metal a couple of thousands of an inch thick completely masks off the effect of the base material since all the current is flowing in the platting.

When very low losses are described, microwave transmission components are usually highly polished so that all the machining marks are essentially polished out. Even microscopic scratches crossing the current flow can appreciably increase the equivalent resistance of the conductor. However, if machining operations operation is planned so that the machining marks will be in line with the current flow, the effect of these mask on resistance will be negligible.

2.2.2 Dielectric Losses

Dielectric losses are absorptive losses in nature and are caused due to dielectric material in transmission line. Propagation velocity is slowed down if the dielectric insulator is placed around and between the conductors. It is also known that the dielectric material in a capacitor increases the effective capacitance between said conductors; however, most dielectric material has losses associated with this space-saving effort. These losses can be taken into account if the dielectric constant is handled as a complex value, as in the formula

''' εεε j−=

where ε’ is the real part of the dielectric constant and ε’’ is the imaginary part of the dielectric constant. Losses of the dielectric material are usually expressed by the loss tangent. Using the complex expression of the dielectric constant, the loss tangent can be defined as

'''tan

εεδ =

Since, the loss tangent of commonly used dielectric material is very small, it is approximately equal to the power factor of the capacitor. It is true that the power factor is defined by θcos , where . Since very small angles, tangents and sines are approximately equal, power factor and loss tangent can also be taken as equal.

δθ ο −= 90

2.2.3 Hysteresis Losses

Hysteresis losses are due to the permeability of magnetic material. For most practical purpose, hysteresis losses are included in skin effect formula. If the material with permeability differing from non-magnetic material is used, platting is usually applied and the effect of hysteresis losses is negligible or entirely alleviated. So that hysteresis losses are considerable for ferro-materials.

2.2.4 Mismatch Losses

Mismatch losses occur when a discontinuity appears in a transmission line or when a termination (load impedance) of transmission line is not equal to the characteristic impedance. This loss is not absorptive, but it reflects and guides the energy away from line. Consequently, not all the power available at that


13


point on transmission line is transmitted or propagated, part of it is reflected. So as far as the transmission line and the terminating load are concerned, not all the energy is delivered into the load. The losses from the available signals and the dissipated signals are due to mismatch. Discontinuities or opening on transmission line at higher frequencies are more serious then at low frequencies. For microwave transmissions care should be for such discontinuities or openings in transmission line.

2.2.5 Losses Due to Radiation

This loss also guides the energy away from the transmission line, so that, the power cannot be transmitted correctly. Radiation is caused due to imperfect shielding, dissimilarities, and openings on transmission line.

For all practical purposes, this type of loss is not intentional and cracks should be looked and repaired. At high microwave frequencies, some cables have to be double and triple shielded to alleviate losses due to radiation.

Since propagation constant (γ) is defined as

βαγ j+=

The term α’ in this expression includes all the losses discussed above. Usually all the losses first calculate separately and can be added to give total attenuation.

2.3 Reflection Coefficient

If a signal is applied to a uniform, practically lossless, transmission line, and if the transmission line is terminated with impedance not equal to the characteristic impedance of the line, that impedance will not be able to absorb all the energy. Some part of the signal will be reflected as shown in Figure 2-1.

Figure shows a transmission line not terminated in its characteristic impedance. Ei is the incident signal traveling towards the transmission line whereas Er is the reflective signal traveling in the opposite direction. The ratio of these two voltages, the reflective signal over the incident signal is called reflection coefficient.

Figure 2-1: a transmission line terminated with impedance not equal to characteristic impedance will reflect part of the incident singal.

τ=i

r

EE

The reflection coefficientτ is a vector since it has a magnitude and phase information. Both the incident and reflected waves are traveling on same transmission line but in opposite direction. Their relative phases are dependent on the terminating impedance and the distance from the termination to the point of measurement. The magnitude of the reflected voltage depends on how much terminating impedance is mismatched. That’s why the reflection coefficient serves as a figure of merit for the termination at the end of any particular transmission line. Now the absolute value of the reflection coefficient,


14


ρτ ==i

r

EE

2.4 Standing Waves

If two waves of same amplitude and frequency are traveling on a transmission line in opposite directions, they will alternately add to and subtract from each other. The result is known as standing waves as shown in Figure 2-2. The Figure shows how two traveling waves combine to form standing waves (note that the maximum and zero voltage points do not shift with respect to the time. This is the difference between traveling and standing waves). The zero crossings are called nodes, and the positions of maximum amplitudes are called antinodes.

Figure 2-2: formation of standing waves.

The wave having the same length but not necessarily the same magnitude (amplitude) will form an interference pattern. This is called standing wave pattern. The bottom line of figure shows the standing wave as it is from the transmission line. In practice, this pattern has to be deducted to unable one to plot, and only envelope will be shown as in Figure 2-3. It is worthwhile to mention that these standing waves were built of total reflection. Figure 2-4 shows standing waves built of known total reflection. The reflection coefficient will be less than 1.


15


Figure 2-3: detected standing wave pattern of total reflection.

The peak value of standing wave pattern will be called E max. The smallest value will be called E main. Now the standing wave voltage ratio is defined as the ratio of two voltages

min

max

EE

VSWR == σ

Figure 2-4: standing-wave pattern of a load not forming a total reflection.

2.5 Smith Chart

It is a polar plot of the complex reflection coefficient. It was devised by Philip H. Smith in the late 1930’s. This special chart considerably reduces the work involved by the calculations of transmission line characteristics. The chart is based on the relationship given by:

⎟⎠⎞

⎜⎝⎛

−+

=ττ

11

0ZZ

In terms of normalized impedance the above equation can be written as

⎟⎠⎞

⎜⎝⎛

−+

==ττ

11

0ZZz

Smith chart consists of two sets of circles or arcs of circles called X-circle and R-circles. X being reactance can be positive or negative. Whenever X is positive, the circle lies above the horizontal line i.e., Kx = 0. On the other hand when X is negative the circle lies below the axis Kx = 0. When X = 0, the circle degenerates into a straight line Kx = 0 because straight line is a circle whose radius is infinity and for X = 0, the radius 1/X will be infinity. All circles touch the point (1, 0).

2.5.1 Properties of Smith Chart

It will be advantageous to study properties of Smith chart before going into any further details:

Normalizing impedance


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The two circles of smith chart are in fact, circles and circles because before starting the

mathematical analysis of the chart was divided by . The process of dividing impedance by is

called normalizing impedance. is the characteristic impedance of the line, for which a lossless line is

a pure resistance. If however, is not given in a problem, a suitable value is assumed for the purpose of normalizing the impedance.

oZR / oZjX /

RZ oZ oZ

oZ

oZ

Figure 2-5: Smith Chart

The process of normalization is reversed if a certain impedance is takes from the Smith chart i.e., this impedance will be multiplied by . oZ

Plotting of an impedance

Any complex impedance can be shown by a single point on the Smith chart. This point will be the point of intersection circles and circle. oZR / oZjX /


17


Determination of SWR

After having located the point of given impedance, the S-circle can be drawn with 0 as center OP as radius shown in Figure 2-5. In order to find the VSWR from the chart we follow the S-circle around to its right hand intersection with the horizontal axis AB, i.e., point M. The normalized resistance at the point M is numerically equal to the voltage standing wave ratio.

Determination of τ in magnitude and direction

If OP is produced till it cuts the angle of reflection coefficient circle at N, the reading of this will give the angle (i.e., the direction) of reflection coefficientτ . The line ON, in fact is the τ -scale giving the magnitude ofτ . Plastic smith chart available in the market provides this scale. This scale is termed as the radical scale and is graduated from 0 to 1. The radical scale can be rotated about the center of Smith chart. However, in the ordinary Smith chart, this scale is provided on the side of the chart. The measure of OP on this scale from 0 onward will give the magnitude ofτ .

Location of voltage maximum and minimum

The intersection of the S-circles with horizontal axis AB that is on the left of the chart center represents voltage minima; intersection with the horizontal axis on the right of the center corresponds to voltage maxima. Line impedance at and can be directly read from the chart. Since they are on the horizontal axis the reactive components are zero.

maxV minV


18

Unit – 03

Transmission Lines

3.1 Coaxial Transmission Line

3.1.1 Defining Equivalent Circuit Components 3.1.2 Attenuation

3.2 Field Configurations on Coaxial Transmission Lines

3.2.1 Higher-Order Modes 3.3 Waveguide Transmission Line


3.1 Coaxial Transmission Line

The preceding discussions were based on parallel wire transmission; another name for this Lecher Noise. The lecher wire has very serious limitations as far as radiation and losses are concerned. Coaxial transmission line is far superior to lecher wire and is preferred at high frequencies. A coaxial transmission line consists of a center conductor with another conductor around it. Since it is a two wire system, it carries the TEM waves of propagation (the principle mode of transmission). The operation of the coaxial transmission line is limited to the principle mode. Higher order modes are not wanted. Therefore, the useful range of coaxial transmission is restricted to the principle mode, which are under the first higher order mode cut-off frequencies.

3.1.1 Defining Equivalent Circuit Components

To analyze a coaxial transmission line the equivalence of parameters has to be determined. The series inductance in terms of Henries per unit length (H/cm), the series resistance in terms of ohm per unit length, the parallel capacitance in terms of farads per unit length, and the parallel conductance in terms of mho per unit length. Centimeters will be used as a unit length in this procedure so that the following equations describe these parameters

cmHabL r /10log4605.0 8−×⎟⎠⎞

⎜⎝⎛= µ Equation 1

This equation neglects current penetration into the conductor. rµ stands for relative permeability factor.

cmF

ab

C r /10log

214.0 12−×=ε Equation 2

rε stands for relative dielectric constant compared to the vacuum or air. In resistance

cmba

fba

R r /1110

112 9 Ω⎟

⎠⎞

⎜⎝⎛ +=⎟

⎠⎞

⎜⎝⎛ +=

ρµπδρ

δ stands for skin depth in cm, is the frequency in Hz, f ρ is the resistivity in terms of . It can be seen that the resistance is proportional to the square root of frequency. This is due to the skin depth. If only copper conductors are considered then this can be expressed as

cm/Ω

cmba

fR /1110414 8 Ω⎟⎠⎞

⎜⎝⎛ +×= −

Where ‘a’ and ‘b’ are not diameters but are the radii in terms of cm in these equations. Now following equation shows the effective line resistance if the inner and outer conductors are made of different metals.

cmba

fR bbaa /109 Ω⎟

⎟⎠

⎞⎜⎜⎝

⎛+=

µρµρ Equation 3

Where aρ and aµ are the specific resistance and relative permeability factor of the inner conductor

respectively. And bρ , bµ are the specific resistance and relative permeability factor of the outer

Unit – 03: Transmission Lines

20


conductor. As µ is a permeability factor it provides information on Hysteresis losses of the ferromagnetic materials that are used.

3.1.2 Attenuation

The real part of the propagation constant provides information on attenuation in terms of nepers per unit length. If losses are small, they can be neglected in most cases. Furthermore, where the characteristic impedance can be assumed to be a real value not having imaginary components, the attenuation due to conductor losses can be calculated by

oc Z

R2

=α Equation 4

and attenuation due to dielectric losses by

od Y

G2

=α Equation 5

The total attenuation for above equations is

oodc Y

GZR

22+=+= ααα Equation 6

The formula for the equivalent circuit components will have to be calculated to get R and G values. The characteristic impedance also has to be calculated to use these expressions.

Using geometrical parameters and the parameters for the materials comprising the coaxial line in equations, more direct expression can be given for gaining information on attenuation.

Attenuation due to Conductor Losses

unitlengthdB

abbarr

c /ln

116.13ε

λδµ

α ⎟⎠⎞

⎜⎝⎛ += Equation 7

Where the wavelength (λ) and the inner (a) and the outer (b) conductor radii are given in terms of unit length. δ is the skin depth in same nit. For copper conductor the expression will be

cmdB

abba

f rc /

ln

111098.2 9 εα ⎟

⎠⎞

⎜⎝⎛ +×= − Equation 8

If the inner and outer conductor are made of different materials, or if they are platted with different metal then the expression can be written as

lengthdB

abbarbbaar

c /ln

86.13εµδµδ

λµ

α ⎟⎠⎞

⎜⎝⎛ += Equation 9


21


The subscripts refer to the particular conductor. If you would like to make a less lossy coaxial transmission line for a given inner and outer conductor dimension and optimum ratio of radii can be derived from an air-insulated line. Then the ratio will be

6.3=ab

which gives a characteristic impedance of 77 Ω.

Attenuation due to Dielectric Losses

Attenuation due to dielectric losses depends upon the properties of dielectric material being used between the conductors. As in ordinary transmission line theory the dielectric constant of a dielectric material is expressed as a complex quantity

''' εεε j−= Equation 10

where the real part of the expression is relative dielectric constant (εr) and the imaginary part contain the information about the shunt losses of that material. The loss tangent is customary way in which manufacturers of dielectric material provides information about dielectric losses. Then loss tangent is

'''tan

εεδ = Equation 11

Dielectric losses of coaxial transmission line can be calculated as

δλε

α tan3.27 rd = dB / unit length Equation 12

3.2 Field Configurations on Coaxial Transmission Lines

Since coaxial transmission lines have two conductors, according to transmission line theory they are capable of carrying the principal TEM mode. TEM mode is a special type of TM waves in which electric field component along the direction of propagation is also zero so that electromagnetic field is entirely transverse. It is normally abbreviated as TEM waves and is often referred to as principal waves. Transmission line theory states that both electric and magnetic fields can only be perpendicular to each other. Consequently there is only one way the electric and magnetic fields can exist in a coaxial structure. This is shown in Figure 3-1.

Figure 3-1: Field distribution for the principal mode in a coaxial line

As the electric field is being established only between two conductors in the principal mode, it has to be radial. The magnetic field is placed around the center conductor between it and the outer conductor. On the longitudinal cross-sectional part in Figure 3-1, the electric field distribution is periodical according to wavelength. There are high intensity and low intensity-planes periodically changing place as the traveling wave goes down the transmission line. If total reflection occurs on a line, it is obvious that, where the electric field intensity is high, the magnetic field intensity will be low, and the vice versa.

3.2.1 Higher-Order Modes


22


As the propagation frequency increases, the wavelength decreases. After a definite limit where the cross-sectional dimensions of the coax are comparable to the boundary conditions of higher-order modes are established. In other words, high-order modes will be able to propagate beside the principal mode. In any transmission line, an infinite number of modes can exist but at a certain frequency. Below the first higher-order mode cutoff frequency only the principal mode can be propagated; but above that frequency the first high-order mode will be able to exist, and it will carry energy.

Figure 3-2: cross section of a coaxial line showing the midradius and its circle determining the cutoff wavelength of the TE11 mode

For the lowest cutoff frequency higher-order mode the cutoff wavelength is approximately equal to the length of the circle drawn between the inner and outer conductors. The approximate cutoff wavelength for the first high-order mode is given as:

( )baaabc +=⎟

⎠⎞

⎜⎝⎛ +

−= ππλ 2

2 Equation 13

If coaxial line is filled with dielectric material, then this equation must be multiplied by the square root of the relative dielectric constant, mathematically,

( ) rc ba επλ += Equation 14

This formula will provide the information to an accuracy of about 8%. The higher-order modes are denoted depending upon their H-waves (TM) or E-waves (TE). Two indices after this notation are given depending upon their field configurations. It is a good practice not to approach the cutoff frequency too closely. Furthermore, discontinuities on the transmission line excite the higher-order modes, although it is true that they do not carry energy and they attenuate exponentially. Attenuation of any of higher-order modes near cutoff is

216.54

⎟⎠⎞

⎜⎝⎛−=λλ

λα c

cdB / cm Equation 15

3.3 Waveguide Transmission Line

Waveguides generally are defined as transmission lines that cannot carry the principle mode of transmission. Only the higher-order modes can exist on them. These are non-conductive type transmission lines. Waveguides can be arbitrarily shaped hollow pipe with or without conductive, metallic boundaries.

The lowest possible higher-order mode in a waveguide is called the dominant mode. It is the only type od mode in waveguide which can exist and can propagate without any interference from other higher-order modes.

Figure 3-3: wavelength relationships in any air-filled waveguide for any mode of propagation.

The useful bandwidth in a waveguide is called the range of frequencies where single-mode propagation exists – in other words, the bandwidth where only the dominant mode can propagate. The useful bandwidth of the dominant mode starts usually about 20% higher in frequency from cutoff.


23


In waveguides where only higher-order modes can exist, the group and phase velocities differ from each other. In fact, in air- or vacuum-insulated waveguide, the geometric means of the phase velocity and group velocity are equal to the velocity of light.

cvv gp = Equation 16

which means that the phase velocity could be higher than the velocity of light, in which case the

group velocity would be slower that the velocity of light. The waveguide wavelength relates to the cutoff frequency and to free-space wavelength by

pv

gv

2

1 ⎟⎟⎠

⎞⎜⎜⎝

⎛−

=

c

g

λλ

λλ Equation 17

where gλ stands for the guide wavelength, λ is the free-space wavelength, and cλ is the cutoff wavelength. It can be seen that, if the free-space wavelength approaches the cutoff wavelength, the guide wavelength gets infinitely long. Furthermore, if one goes higher and higher in frequency, one would find that the waveguide wavelength gets closer and closer to the free-space wavelength. When the frequency becomes very high and tends to go to infinity, then the waveguides wavelength and free-space wavelength becomes the same. The relationship shown in Equation 17 can be plotted in terms of cλλ / versus gλλ / as in Figure 3-3.

If one takes waveguide completely filler with very-low-loss dielectric material where the dielectric constant is εr, the waveguide wavelength expression will be modified:

2

⎟⎟⎠

⎞⎜⎜⎝

⎛−

=

cor

g

λλε

λλ Equation 18

where coλ is the cutoff wavelength of the empty waveguide, not the dielectric filled guides.

Rectangular Waveguide

Figure 3-4 shows a rectangular waveguide in a rectangular coordinate system where ‘a’ is the wider dimension and ‘b’ is the narrower inside dimension. The rectangular waveguide, being a waveguide and a one-conductor propagating transmission line, can propagate only higher-order modes. The modes of transmission can consequently be only in TE and TM modes. The general symbols of TEmn or TMmn are used to describe transverse electric or transverse magnetic waves. The subscript m indicates the number of half-wave variations of the electric field along the wide dimensions of the waveguide, and n indicates the number of half-wave variations of electric or magnetic field in the narrow dimension of the guide. The TE10 mode, which has the longest operating wavelength, is designated as the dominant mode. If a dimension is less than ½ wavelengths, no propagation will occur. Therefore, waveguide acts as a high-pass filter. For a rectangular piece of waveguide, the cutoff frequency can be found from

22

21

⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛=

bn

amcfc Equation 19


24


Figure 3-4: Rectangular waveguide.

where ‘c’ is the velocity of light, m and n are the subscripts of the particular TE or TM mode, and a and b are the wide and narrow inside dimensions of the rectangular guide, respectively. The cutoff wave length is then

22

2

⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

=

bn

am

cλ Equation 20

The cutoff for the dominant mode can be calculated easily. The dominant mode being the TE10, substituting 1 and 0 in place of m and n, respectively,

221001

2

⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

=

ba

cTEλ

yields, acTE 210 =λ

Figure 3-5: fields in a rectangular guide (dominant TE10 mode).

The field configuration in a rectangular waveguide propagating the commonly used TE10 mode is shown in Figure 3-5. Rectangular waveguides were originally chosen so that the dominant mode would exist over a certain frequency range. This frequency determines the ‘a’ dimension. The ‘b’ dimension is chosen on the basis of the following criteria:

The attenuation loss is greater than as the b dimension is made smaller; and


25


The b dimension determines the voltage breakdown characteristics therefore determine the maximum power handling capacity.


26

Unit – 04

Microwave Sources & Detectors

4.1 Microwave Sources

4.2 Klystron

4.3 Multi-Cavity Klystron Amplifiers

4.4 Reflex Klystron

4.5 Backward Wave Oscillator (BWO)

4.6 Detection of Microwave Signals 4.7 Detector

4.7.1 Crystal Detectors 4.7.2 Square Law of Crystal Detectors

4.8 Indicators


4.1 Microwave Sources

Vacuum tubes were the devices used before the evolution of transistor. These tubes were used for controlling a large signal through a smaller signal to produce desired amplification, oscillation and other applications. Through, now tubes are used rarely but they are still found in microwave equipment specifically in microwave transmitters used for producing high output power. They are called microwave sources.

4.2 Klystron

A klystron is a microwave vacuum tube using cavity resonators to produce velocity modulation of the electron beam and to produce amplification. Figure 4-1 shows the concept of a two-cavity klystron. The vacuum tube contains a cathode that is heated by a filament at a very high temperature to emit electrons. A positive plate or collector attracts these negative particles. Thus, current flow is established between the cathode and the collector inside the evacuated tube.

Figure 4-1: Klystron amplifier

Operation: in the first cavity (buncher cavity), where the RF energy is coupled in, and the electron beam is velocity modulated, and the second cavity is tuned to the same frequency. In the second cavity (catcher cavity) the RF energy is coupled through the electron beam by placing the second cavity into the proper position at an optimum distance. The RF interacting with the electron beam causes a kinetic energy loss from the beam that result in gain. This was the first use of velocity modulation of an electron beam. Several cavities can be placed one after another to achieve higher gain and narrower bandwidth.

This technique met the first requirement for an oscillator: the gain. Positive feedback could be established by connecting the second cavity back into the first cavity. To assure a wide tuning capability of this dual cavity requires more consideration about phase relations involved. Using the klystron as an oscillator would require a variable-phase shifter between the first and second cavities to assure positive feedback at all frequencies.

Practical Consideration and Amplifications

Klystrons are also constructed with additional cavities between the buncher and catcher cavity. These intermediate cavities produce further bunching which causes increased amplification of the signal if the buncher cavities are tuned off the frequencies from input and output cavities, they have the effect of broadening the bandwidth of the tube. The frequency of operation of a klystron is set by the sizes of the input and output cavities. Klystrons are available in a wide range of sizes. Small band-held units produce only milli-watts of power amplification. While large size klystrons produce many thousands of watts of power. Klystrons are used at frequencies as low as UHF and as high as 300 GHz in the microwave region.

Unit – 04: Microwave Sources & Detectors

28


Current klystron developments are aimed at improving efficiency providing longer life and reducing life typical efficiencies of 30 –50 %. To improve reliability and MTBF (mean time between failures), tungsten-ridium cathodes are used to reduce cathode temperature and provide longer life.

Since the time taken by a given electron to pass through influence of a klystron is influenced by collector voltage, this voltage must be regulated, typically 9 KV at 750 mA. Two practical microwave applications are a multi-cavity power oscillator. Aside from these, single cavity reflex klystrons are also used.

4.3 Multi-Cavity Klystron Amplifiers

The bunching process in a two-cavity klystron is by no means complete; since there are large number of out of phase electrons arriving at the catcher cavity between bunches. Consequently, more than two cavities are always employed in practical klystron amplifiers. Partially bunched current pulses will now also excite oscillations in the intermediate cavities, and these cavities in turn set up voltages which help to produce more complete bunching. The extra cavities help to improve the efficiency and power gain. Considerably, the cavities may all be tuned to the same frequency, such synchronous tuning being employed for narrowband operation. For broadband work, e.g., with UHF klystrons used as TV transmitter output tubes or GHz tubes used as power amplifiers in same satellite station transmitter stagger tuning is used. Here, the intermediate carriers are tuned to either side of the center frequency, improving the bandwidth very significantly.

The multi-cavity klystron is used as a medium for very high power amplifiers in the UHF and microwave ranges, for either continuous or pulsed operation. The frequency range covered is from about 250 MHz to over 95 GHz and power available is much higher then currently needed.

Multi-cavity klystron amplifiers suffer from noise, because bunching is never complete and so electrons arrive at random at the cathode cavity. This makes them too noisy for use in receivers, however, for transmitters their typically noise figure of 35 dB is more than adequate.

Two-Cavity Klystron Oscillation

If the portion of the signal in the catcher cavity is coupled back to the buncher cavity oscillation will take place. As with other oscillator, the feedback must have the correct polarity and sufficient amplitude.

The schematic diagram for two cavity klystron amplifiers, accept for the addition of a (permanent) feedback loop. Oscillations in the two-cavity klystron behave as in any other feedback oscillator. Having been started by a switching transient or noise impulses, they continue as long as the dc power is present.

The two-cavity klystron oscillator has fallen out of favor, having been displaced by continuous wave (CW). Magnetron, semiconductor deiced and high gain of klystron and TWT amplifiers.

4.4 Reflex Klystron

A special variation of the basic klystron tube is known as a reflex klystron, in which oscillation can be achieved within a single cavity. There are three basic regions of a reflex klystron: the cathode-anode region, or the electron gun; the RF structure where interaction takes place, causing electron bunching; and the drift space. The repeller, an electrode with a more negative potential, is placed at this point where the electrons are returned. After they turn around, they may arrive back at the interaction space when bunches are being formed. If there travel time in drift space, coming back into the RF structure, is such that they are in phase (in other words, when the electrons are being bunched), they will provide the necessary positive feedback to provide oscillation. Oscillation can be obtained as the repeller voltage is varied. The transit time from interaction space into the drift space and back to the interaction space is varied. If bunches returning from the drift space are not in phase with bunching being formed, oscillation will not occur. The bunches will be out of phase. Variations of the transmit


29


time by the adjustment of repeller voltage will make the returning bunch meet in phase with another bunch that is being formed. Then oscillation will occur again. This means that oscillation can occur only in one repeller voltage at each frequency; several repeller voltages will provide oscillation. These are called different repeller modes. The higher the repeller voltage, the shorter the transit time from interaction space out into the drift space an d back into interaction space.

Figure 4-2: Reflex klystron

Because of the nature of the reflex klystrons, amplitude modulation is impractical. Any voltage variation on the reflex klystron will result in the frequency modulation. If the repeller voltage is varied, frequency change will occur. If the electron beam voltage is varied, of course, the velocity of the electron beam will be changing, which will also result in frequency variation. Only frequency modulation is possible on a reflex klystron; amplitude modulation will result in unwanted frequency modulation.

4.5 Backward Wave Oscillator (BWO)

A BWO is a microwave cathode wave oscillator with enormous tuning and overall frequency range. It operates on TWT principle of electron beam RF field interaction generally uses a helix slow wave structure. In general appearance a BWO looks like a shorter thicker TWT. The unique features of B”WO is that it can be electronically tuned over wide bandwidth. The BWO contains an electron gun focusing magnetic collector and helical interaction structure used exactly like the TWT.

Operation: if the presence of starting oscillation may be assumed the operation of BWO becomes very similar to that of TWT. Electrons are injected from the electron gun, cathode focused by an excel magnetic field and collected at the far end of the glass tube. They have meanwhile travel through helix slow wave structure and bunching has taken place. With bunching increasing in competence from cathode to collector. In interchanging of energy occurs exactly as in the TWT with RF around the helix growing. The signal towards the collector and of the helix. Unlike the TWT the BWO does not have an attenuation along the tube. As a simplification oscillation may be thought of as occurring simply because of reflection from imperfectly terminated collector end of the helix. This is feedback and is collected from the cathode of the helix towards which reflection tool place. Because helix is essentially a non-resonant structure bandwidth is very high and the operating frequency is determined by collector voltage with associated cavity system.

Practical aspect:

1. BWO are used as signal sources in instruments and transmitters. 2. They can also be made broadband noise whose output is amplified equally wideband TWT is

transmitted as a means of energy radar confusion. 3. The frequency spectrum over which BWO can be made to operate is, vast structuring from

one to well over 1000 GHz. 4. permanent magnets are normally used for focusing twice these results in simplest meagnetics

and smallest tube


30


5. However, the solenoids are used as the highest frequency since it has been found that they give the high penetration and distribution for axil magnetic field. A recent development in this respect has been those of samaricum cobalt permanent magnet to reduce weight size.

4.6 Detection of Microwave Signals

Measuring any unknown quantity of a certain parameter is always done by comparison of the unknown quantity to a known, which is taken at that time as a standard. In other words, whenever measurement is performed, some kind of substitution or comparison of a value of an unknown with a known is done. These requirements must be met to make measurements on a transmission system. A signal source is needed, plus a device that will direct or modify the signal and connect it to an instrument which provides indication of the presence or behavior of the signal.

4.7 Detector

Any device used to detect the presence of a physical property or phenomenon, such as radiation. A detector rectifies the RF signal subjected to it.

Microwave crystal detectors are usually the point-contact type of semiconductor devices. These diodes have low capacitance across the point contact junction; consequently, they are suitable for microwave rectification. The semiconductor used is usually doped Silicon or Germanium.

The diode operates because a contact potential is established between two dissimilar conductors. To understand how this can happen, consider two metals joined as shown in Figure 4-3.

Figure 4-3: two metals contact with dissimilar conductivity

If metal B has more free electrons (is more conductive) upon contact, an electron flow takes place predominantly from B to A. After equilibrium is reached, A will apparently be charged more negatively and B will be more positive. Essentially, a potential barrier will be formed at the junction, which effectively provides rectification.

4.7.1 Crystal Detectors

Crystal detectors are widely used in the microwave field because of their sensitivity and simplicity. They are used as video detectors to provide either a dc output when unmodulated microwave energy is applied or a low frequency ac output up to tens of MHz or higher when the microwave signal is modulated. They are also used as mixers in superhetrodyne systems specifically at microwave frequencies where other mixers, such as vacuum tubes, are insufficient or inefficient.

The essential parts of the crystal detector are a semiconducting chip and a metal whisker, which contacts the chip. A typical microwave crystal detector uses a silicon chip about 1/16-inch square and a pointed tungsten whisker wire about 3/1,000 inch in diameter. The other part of the crystal detector or mount is needed simply to support the chip and the whisker and to couple electrical energy to a


31


detector. Crystal detectors are successful at microwave frequencies partly because of their extremely small size, these dimensions also limit their power handling capability; just 100 mW is sufficient to damage crystals.

4.7.2 Square Law of Crystal Detectors

Square law of crystal detectors states that the output voltage is proportional to the square of the input voltage. Relatively large variations in output voltage result from minor variations inn the input voltage and the sensitivity of this type of detector is therefore relatively high.

Figure 4-4: idealized crystal detector circuit.

Figure 4-5: square characteristics of a crystal detector.

Consider the idealized circuit shown in Figure 4-4, in which a sinusoidal microwave voltage is applied to flow to the milliammeter. A typical crystal detector has a current voltage characteristic similar to that shown in Figure 4-5. Any such curve can always be approximated by a Taylor series consisting of terms involving powers of , that is, v

...33

2210 ++++= vavavaai Equation 1

If the operating point is the origin ( 0,0 == iv ), then 00 =a . Let

tAv ωcos= Equation 2

where A is the amplitude, ω is equal to 2πf, and f is the microwave frequency. Substituting in Equation 1 yields

...)cos()cos()cos( 33

221 +++= tAatAatAai ωωω Equation 3

For extremely small signals, all terms in Equation 1 except the first are negligible. And, )cos(1 tAai ω= . The current is simply proportional to the applied voltage, and the crystal behaves as a simple resistor with negligible dc current flowing through the milliammeter. However, for somewhat larger signals, the second term must be included to obtain reasonable accuracy.

)2cos1(2

)cos()cos()cos(2

21

211 tAatAatAatAai ωωωω ++=+= Equation 4

The current now includes the dc component , which flows through the milliammeter, and

the second harmonic component , which flows through C. Thus, the milliammeter indication is proportional to the square of the amplitude A of the microwave voltage.

2/)( 22 Aa

tAa 2cos2/)( 22 ×


32


4.8 Indicators

Indicators are used primarily for visual presentation of the presence or behavior of the detected signal. Such devices can be oscilloscopes, galvanometers, or special indicator devices like the standing-wave indicator.

Figure 4-6: block diagram of a standing-wave indicator.

A standing-wave indicator is a high-gain, tuned amplifier which takes an input from a crystal detector or any audio source. It also has a built-in bias supply to provide bias current for devices like bolometer. The input signal is amplified and applied to a meter calibrated for use with square-law detector. Figure 4-6 shows a block diagram of a standing-wave indicator. Input voltage first goes to input switching to provide either the bias supply or an impedance match for the right input characteristics needed. Then the signal is fed to the first section of a range switch and then to the input of an amplifier. The second section of the switch is located between the first amplifier and the second amplifier. Switch positions are in 10-dB steps. In the input amplifier, the gain and vernier controls associated with the amplifier and vary gain over a range of more than 10:1. The gain control is a coarse control to adjust the negative feedback in this amplifier. Vernier is a fine gain control and changes gain in series with the output signal. An ac feedback is provided in the second amplifier for stability and high-input impedance. The output of this amplifier is applied to the expand attenuator and then to the following amplifier. The expand function allows any signal level to be measured on the expanded scale. Expansion is accomplished by applying a precise amount of dc offset current to the meter and simultaneously increasing the signal to the third amplifier. This increased gain allows a certain decibel change in signal level to deflect the meter across its full scale. The offset current places the zero signal indication off-scale to the left.


33

Unit – 05

Microwave Mixing

5.1 Introduction to Mixers

5.2 Theory of Mixing

5.3 Conversion Loss

5.4 Parametric Amplifier

5.5 Parametric Up-Converter

5.6 Parametric Down-Converter

5.7 Manely – Rowe Power Relation

5.8 Negative Resistance Parametric Amplifier

5.9 Harmonic Frequency Conversion


5.1 Introduction to Mixers

Mixers are used for frequency conversion and are critical components in modern radio frequency (RF) systems. A mixer converts RF power at one frequency into power at another frequency to make signal processing easier and also inexpensive. A fundamental reason for frequency conversion is to allow amplification of the received signal at a frequency other than the RF, or the audio, frequency.

Figure 5-1: Circuit symbol for a mixer

The ideal mixer, represented by figure 1, is a device which multiplies two input signals. If the inputs are sinusoids, the ideal mixer output is the sum and difference frequencies given by

Typically, either the sum, or the difference, frequency is removed with a filter.

5.2 Theory of Mixing

Radio communication requires that we shift a base band information signal to a frequency or frequencies suitable for electromagnetic propagation to the desired destination. At the destination, we reverse this process, shifting the received radiofrequency signal back to base band to allow the recovery of the information it contains. This frequency-shifting function is traditionally known as mixing; the stages that perform it are termed mixers. Any device that exhibits amplitude-nonlinear behavior can serve as a mixer, as nonlinear distortion results in the production, from the signals present at the input of a device, of signals at new frequencies.

Although mixers are equally important in wireless transmission and reception, traditional mixer terminology favors the receiving case. Thus, the signal to be frequency-shifted is applied to the mixer’s RF port, and the frequency-shifting power or voltage (from a local oscillator [LO]) is applied to the mixer’s LO port, resulting in two outputs at the mixer’s intermediate frequency (IF) port. If the wanted IF is lower in frequency than the RF signal, the mixer is a down converter; if the wanted IF is higher than the RF, the mixer is an up converter. Converter may also be used as a term for a single stage that simultaneously acts as mixer and LO.

The simultaneous generation of LO+RF and LO-RF outputs result not from a departure of mixer performance from the ideal, but from the mathematics of mixing itself. Just as a given RF/LO combination produces two IF outputs (LO+RF and LO-RF, the IF and IF image), the mixer will produce output at the desired IF (LO+RF or LO-RF) in response to two possible RF inputs: one at LO+IF and another at LO-IF (Figure 5-2). The undesired response, the RF image (traditionally referred to merely as the image), is 2fIF removed from the desired response. Even if no manmade signals exist at the RF image frequency, reducing a mixer’s RF image response can be important because noise at that frequency, including that produced by circuitry between the mixer and antenna, will still be mixed to the desired IF, degrading the signal-to-noise ratio.

Filtering and phasing techniques can be used to reduce the RF or IF image responses —filtering if the image is sufficiently removed from the desired response for filtering to provide the necessary rejection, phasing if the desired and image responses are insufficiently spaced for filtering to work, as

Unit – 05: Microwave Mixing

35


in the case of a double-conversion receiver in which signals at a high first IF (for example, 50 to 70 MHz), must be converted to a very low first IF, such as 25 kHz.

Figure 5-2: relationship between a mixer’s image and desired signal response. The image is 2fIF away from the desired signal.

All mixers are multipliers in the sense that the various new outputs they produce can be described mathematically as the multiplicative products of their inputs. Let us now consider the basic theory of mixers. Mixing is achieved by the application of two signals to a nonlinear device. Depending upon the particular device, the nonlinear characteristic may differ. However, it can generally be expressed in the form:

I = K (v1+v2+V)n Equation 1

The exponent n is not necessarily an integer, V may be a dc offset voltage, and the signal voltages v1 and v2 may be expressed as

v1 = V1 sin ( ω1t) and v2 = V2 sin ( ω2t)

When n = 2, Equation (1) may then be written as:

22211 )sinsin( tVtVVKI ωω ++= Equation 2

This assumes the use of a device with a square-law characteristic. A different exponent will result in the generation of other mixing products, but this is not relevant for a basic understanding of the process. Expanding (2)

)]sin()sin(2)sin(2 )sin(2)(sin)(sin[(

211222

112

222

2122

12

ttVVtVVtVVtVtVVKI

ωωωωωω

+++++= Equation 3

The output comprises a direct current and a number of alternating current contributions. We are interested only in that portion of the current that generates the IF; so, if we neglect those terms that do not include both V1 and V2, we may write:

)sin()sin(2 2112 ttVKVI IF ωω=

[ ] [ ] ttVKVI IF )cos()cos(2 121212 ωωωω +−−= Equation 4

This means that at the output, we have the sum and difference signals available, and the one of interest may be selected by the IF filter.


36


5.3 Conversion Loss

It is the ratio of the output signal level to the input signal level expressed in dB. In a single sideband system, only one sideband is used; therefore 3 dB of loss is theoretical. The additional loss is diode and transformer loss. These losses can be minimized by driving the diodes with sufficient current and operating in the best portions of the frequency band and are generally between 5 - 9 dB for passive mixers. Conversion loss is specified in a 50 Ω system with an LO drive level of +7 dBm. High level mixers are specified with more LO drive power.

5.4 Parametric Amplifier

Normal amplifiers convert power from dc source (power supply or battery) into power at same signal frequency i.e., time varying or alternating power. On other hand a parametric amplifier convert power at one frequency (from a source pump) into a power at another frequency i.e., signal frequency. Parametric amplifier is a device that uses non-linear reactance (capacitance or inductance) or time-varying reactance. The word parametric is derived from the word “excitation”. Parametric excitation may be subdivided into parametric amplification and oscillation.

The solid state varactor diode is most widely used parametric amplifier. Unlike microwave tubes, transistors, and lasers, the parametric amplifier diode is of the reactive nature and thus generates very small amount of thermal noise (Johnson noise). One of the distinguishing features of a parametric amplifier is that it uses an ac rather sc power supply.

Parametric Circuit Analysis

In a superhetrodyne receiver, RF-signal can be mixed with a signal from the local oscillator in a nonlinear circuit to generate the sum and difference of the frequencies. In a parametric amplifier, the local oscillator is replaced by a pumping generator and the non-linear element of a time varying capacitor as shown in Figure 5-3.

Figure 5-3: equivalent circuit of parametric amplifier

Operation: The signal frequency fs and the pump frequency fp are mixed in the nonlinear capacitor C. Consequently, a voltage of fundamental frequencies fs and fp, as well as the sum and difference frequencies mfp ± nfs (where m and n are integers from zero to infinity); will appear across C. If a resistive load is connected across the terminal of Idler circuit, an output voltage can be generated across the load at the output frequency fo. The output circuit which does not require any external excitation is called Idler circuit. The output (or idler) frequency fo, in Idler circuit is expressed as the sum and difference frequencies of the signal frequency fs and pump frequency fp.


37


5.5 Parametric Up-Converter

A parametric up-converter is a device in which frequency of oscillation fs is greater than the signal frequency fs (i.e., fo > fs). In a parametric up-converter Output frequency is equal to the sum of the signal frequency and the pump frequency i.e., pso fff += . There is no power flow in parametric device at frequencies other than signal, pump, and output frequencies.

Power Gain:

If above tow conditions are satisfied then the maximum power gain of parametric amplifier is given as:

( )

( )

as

o

s

s

o

CRfQand

rQff

xwhere

x

xff

Gain

π21,

,

11

2

2

=

=

++×=

Ra is the series resistance of a pn-junction diode and rQ is the figure of merit for the nonlinear

capacitor. The quantity ( 211/ xx ++ ) may be regarded as a gain degradation factor. As Rd

approaches zero, the figure of merit goes to infinity and degradation factor becomes equal to the infinity. As a result power gain of a parametric up-converter for a lossless diode is equal to fo/fs.

Noise Figure

One advantage of parametric amplifiers over a transistor amplifier is its low noise figure, because pure reactance does not contribute thermal noise to the circuit. The noise figure F, for parametric up-converter is given as:

( ) ⎥⎥⎦

⎤

⎢⎢⎣

⎡++= 2

1121rQrQT

TF

o

d

where, Td = diode temperature (K) To = 300 K (ambient temperature) rQ = figure of merit for nonlinear capacitance.

In a typical microwave diode rQ = 10, then fo/fs = 10. So that, minimum noise figure will be F = 0.86 dB

Bandwidth

The bandwidth of a parametric amplifier is related to the figure of merit and the ratio of signal frequency fo and the output frequency. The bandwidth is given by,

s

off

rBW 2=


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5.6 Parametric Down-Converter

A parametric up-converter is a device in which frequency of oscillation fs is less than the signal frequency fs (i.e., fo < fs). If a mode of down converter for a parametric amplifier is desirable, the signal frequency fs must be equal to sum of the pump frequency fp and the output frequency fo. In other words, the input power must feed into the idler circuit and the output circuit must move out from the signal. The down-conversion gain (actually a loss) is given by

( )211 x

xff

Gains

o

++×=

5.7 Manely – Rowe Power Relation

Manely and Rowe have derived a set of general energy relations regarding power flowing into and out of an ideal nonlinear reactance. These relations are useful in predicting whether power gain is possible in a parametric amplifier. Figure 5-4 shows an equivalent circuit for the Manely-Rowe observation.

Figure 5-4: equivalent circuit for Manely-Rowe observation.

In Figure 5-4 one signal generator and one pump generator at their respective frequencies fs and fp together with their associated series resistances and band filters are applied to nonlinear capacitance c(t). These resonating circuits of filters are designed to reject power at all frequencies other than their respective signal frequencies. An infinite number of resonant frequencies mfp ± nfs are generated in the presence of two applied frequencies of fs and fp, where m and n are any integers from zero to infinity.

Each of the resonant circuit is assumed to be ideal. The power loss by the nonlinear resistor is negligible, the power entering the nonlinear capacitor at the pump frequency is equal to nonlinear power leaving the capacitor at the other frequency through the nonlinear interaction. Manely and Rowe established the power relations between the input power at the frequencies fs and fp and the output power at the other frequencies.

Equation from the voltage across the nonlinear capacitor c (t) can be expressed as exponential form as

( )tsjwtsjwstpjwtpjwpsp ee

Vee

VvvV −−

++⎟⎠⎞⎜

⎝⎛ +=+=

22


39


The general expression of the charge Q deposited on the capacitor is given as

)( tsnwtpmwj

nmn

meQQ +∞

−∞=

∞

−∞=∑∑=

In order for the charge Q to be real, it is necessary that nmmn QQ −−=

The total voltage V can be expressed as a function of the charge Q. a similar Taylor series expansion of V(Q) shows that

)( tsnwtpmwj

nmn

meVV +∞

−∞=

∞

−∞=∑∑=

In order for the voltage V to be real, it is necessary that nmmn VV −−=

The current flowing through c(t) is the total derivative of Equation (ii) with respect to time. That is,

)( tsnwtpmwj

nmn

meI

dtdQI +∞

−∞=

∞

−∞=∑∑==

Because the capacitor c(t) is assumed to be pure reactance, the average power at the frequencies mfp + nfs is

nmnmnmnmnmnmnmnmnm IVIVIVIVP −−−−−−−− +=+= ,,,,,,,,, ****

Then, conservation of power can be expressed as

∑∑∞

−∞=

∞

−∞==

nnm

mP 0,

5.8 Negative Resistance Parametric Amplifier

If a significant portion of power flows only at signal frequency fs, pump frequency fp, and idler frequency fi, then a negative condition with the possibility with the oscillation at both the signal frequency and the idler frequency will occur. The idler frequency is defined as the difference between the pump frequency and the signal frequency spi fff −= . When the mode operates below oscillation threshold, the device behaves as the bilateral negative resistance parametric amplifier.

Power gain

Output power is taken from the resistance Ri at the frequency fi and the conversion gain from fs to fi is given by

( )21..4

aa

RRRR

ff

GainTiTs

ig

s

i

−=

where fs = signal frequency


40


fp = pump frequency fi = fp – fs = idler frequency a = R/Ts Rg = output resistance of a signal generator RTs = total series resistance at signal frequency RTi = total series resistance at idler frequency

Noise figure

The optimum noise figure of a negative resistance parametric amplifier is expressed as

( ) ⎥⎥⎦

⎤

⎢⎢⎣

⎡++= 2

1121rQrQT

TF

o

d

where, Td = diode temperature (K) To = 300 K (ambient temperature) rQ = figure of merit for nonlinear capacitor.

Bandwidth

The maximum gain bandwidth of a negative resistance parametric amplifier is given by

GainffrBWs

i ×=2

5.9 Degenerate Parametric Amplifier

Degenerate parametric amplifier or oscillator is defined as a negative resistance amplifier with the signal frequency equal to the idler frequency. Because the idler frequency fi is the difference between fp and fs, th signal frequency is just one-half of the pump frequency.

Power Gain and Bandwidth

The power gain and bandwidth characteristics of degenerate parametric amplifier are exactly same as for the parametric up=converter with fs = fi and fp = 2fs, the power transferred from the pump to the idler frequency. As high gain, the total power at the singal frequency is almost equal to the total power at the signal frequency. So the total power in the pass band will have 3 dB more gain.

Noise Figure

The noise figure for the single sideband and he double sideband degenerate parametric amplifiers are given respectively by

go

dddsb

go

ddssb

RTRT

F

RTRT

F

+=

+=

1

22

where Td = average diode temperature in 0K To = 300 K is the ambient noise temperature in 0K Rd = diode series resistance in ohms, and Rg = external output resistance if the signal generator in ohms.


41


5.9 Harmonic Frequency Conversion

A mixer will convert an input signal of an IF by taking the sum or difference between the local oscillator and the input signal. Many mixers will also convert an input signal with harmonics of the local oscillator.

Figure 5-5 Simple series mixer capable of mixing by harmonics.

An example of a simple harmonic mixer is shown in Figure 5-5. In this example a single diode is used to mix an input RF signal with the third harmonic of the local oscillator. If the level of the local oscillator is sufficiently high, the diode can be thought of as a switch being switched at the rate of the local oscillator. Mixing is essentially multiplying two signals together, and the switch action of the diode can be thought of as multiplying a square wave with an amplitude of 1 with the input waveform. Because a square wave is made of the summation of the fundamental and all the odd harmonics as well. In a practical circuit, because the duty cycle is not an exact 50 percent, more than just the odd harmonics are present, and the example diode mixer will mix the input RF signal with all harmonics of the local oscillator.

Harmonic mixing is used to extend the frequency range of the swept super heterodyne spectrum analyzer. Figure 5-6 is a plot of the transfer function A (t) of the input mixer, which shows the diode gate being switched off and on by the first local oscillator drive signal. This transfer function of time is shown expressed as a Fourier series function of frequency. Bias is applied to the drive waveform to unbalance the duty cycle so that even as well as odd harmonics are present. An input signal is multiplied by the transfer function, which produce sum and difference frequency output signals with each term of the Fourier series. In other words, the input signal can be heterodyned with the fundamental, second harmonic, third harmonic, etc., of the first local-oscillator frequency. Any output signals produced at 2 GHz will pass through the system and be displayed as already described. So the analyzer will respond to signals which differ not only from the fundamental by 2 GHz but also by 2 GHz from the second harmonic, third harmonic, etc.

Figure 5-6 Transfer function of input mixer of a spectrum analyzer, A(t)=A0+A1sin(ω0t)+A2sin(2 ω0t+φ2)+ A3sin(3 ω0t+φ3)…….

Harmonic mixers respond to several input frequencies simultaneously, but preselection filters can be used to eliminate the confusion. Bandpass broadband filters can help with higher-order harmonic mixing modes, but the most affective solution is a tracking narrow-band filter which can be adjusted to


42


track a desired harmonic mixing mode as the spectrum analyzer is tuned and scanned. Such a filter is the YIG filter, consisting of one or more coupled yittrium-iron-garnet resonators whose resonant frequency is proportional to the strength of the field from an electromagnet tin which they are placed. This filter can be biased and electrically swept to allow only the signal matching a desired mixing mode to enter the input mixer.


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