MICRO WAVE ENGINEERING (R18A0420)

LECTURE NOTES

B.TECH (IV YEAR – I SEM) (2021-22)

Prepared by: Mrs.N.Saritha, Assistant Professor Mrs.S.Aruna Kumari, Assistant Professor

Department of Electronics and Communication Engineering

MALLA REDDY COLLEGE OF ENGINEERING & TECHNOLOGY (Autonomous Institution – UGC, Govt. of India) Recognized under 2(f) and 12 (B) of UGC ACT 1956 (Affiliated to JNTUH, Hyderabad, Approved by AICTE - Accredited by NBA & NAAC – ‘A’ Grade - ISO 9001:2015 Certified) Maisammaguda, Dhulapally (Post Via. Kompally), Secunderabad – 500100, Telangana State, India

B.Tech (ECE) R-18

MALLA REDDY COLLEGE OF ENGINEERING AND TECHNOLOGY IV Year B.Tech ECE-I Sem L T/P/D C 3 -/-/- 3

(R18A0420) MICROWAVE ENGINEERING COURSE OBJECTIVES 1. To analyze micro-wave circuits incorporating hollow, dielectric and planar waveguides, transmission lines, filters and other passive components, active devices. 2. To Use S-parameter terminology to describe circuits. 3. To explain how microwave devices and circuits are characterized in terms of their “S” Parameters. 4. To give students an understanding of microwave transmission lines. 5. To Use microwave components such as isolators, Couplers, Circulators, Tees, Gyrators etc.. 6. To give students an understanding of basic microwave devices (both amplifiers and oscillators). 7. To expose the students to the basic methods of microwave measurements. UNIT I: Waveguides & Resonators: Introduction, Microwave spectrum and bands, applications of Microwaves, Rectangular Waveguides-Solution of Wave Equation in Rectangular Coordinates, TE/TM mode analysis, Expressions for fields, Cutoff frequencies, filter characteristics, dominant and degenerate modes, sketches of TE and TM mode fields in the cross-section, Mode characteristics - Phase and Group velocities, wavelengths and impedance relations, Rectangular Waveguides – Power Transmission and Power Losses, Impossibility of TEM Modes, losses, Q- factor, Cavity resonators-introduction, Rectangular and cylindrical cavities, dominant modes and resonant frequencies, Q-factor and coupling coefficients, Illustrative Problems. UNIT II: Waveguide Components-I: Scattering Matrix - Significance, Formulation and properties, Wave guide multiport junctions - E plane and H plane Tees, Magic Tee, 2-hole Directional coupler, S Matrix calculations for E plane and H plane Tees, Magic Tee, Directional coupler, Coupling mechanisms - Probe, Loop, Aperture types, Wave guide discontinuities - Waveguide Windows, tuning screws and posts, Irises, Transitions, Twists, Bends, Corners and matched loads, Illustrative Problems. Waveguide Components-II: Ferrites composition and characteristics, Faraday rotation, Ferrite components - Gyrator, Isolator, Circulator. UNIT III: Linear beam Tubes: Limitations and losses of conventional tubes at microwave frequencies, Classification of Microwave tubes, O type tubes - 2 cavity klystrons-structure, Reentrant cavities, velocity modulation process and Applegate diagram, bunching process and small signal theory Expressions for o/p power and efficiency, Reflex Klystrons-structure, Velocity Modulation, Applegate diagram, mathematical theory of bunching, power output, efficiency, oscillating modes and o/p characteristics, Effect of Repeller Voltage on Power o/p, Significance, types and characteristics of slow wave structures, structure of TWT and amplification process (qualitative treatment), Suppression of oscillations, Gain considerations.

Malla Reddy College of Engineering and Technology (MRCET)

170

B.Tech (ECE) R-18

UNIT IV: Cross-field Tubes: Introduction, Cross field effects, Magnetrons-different types, cylindrical travelling wave magnetron-Hull cutoff and Hartree conditions, modes of resonance and PI- mode operation, separation of PI-mode, O/P characteristics. Microwave Semiconductor Devices: Introduction to Microwave semiconductor devices, classification, applications, Transfer Electronic Devices, Gunn diode - principles, RWH theory, Characteristics, Basic modes of operation - Gunn oscillation modes, LSA Mode, Varactor diode, Parametric amplifier, Introduction to Avalanche Transit time devices (brief treatment only), Illustrative Problems. UNIT V: Microwave Measurements: Description of Microwave Bench – Different Blocks and their Features, Precautions; Waveguide Attenuators – Resistive Card, Rotary Vane types; Waveguide Phase Shifters – Dielectric, Rotary Vane types. Network Analyzer, Power Meter, Spectrum Analyzer, Microwave Power Measurement – Bolometer Method. Measurement of Attenuation, Frequency, VSWR, Cavity Q. Impedance Measurements.

TEXT BOOKS: 1. Microwave Devices and Circuits – Samuel Y. Liao, PHI, 3rd Edition,1994. 2. Microwave and Radar Engineering- M.Kulkarni, Umesh Publications,1998. REFERENCES : 1. Foundations for Microwave Engineering – R.E. Collin, IEEE Press, John Wiley, 2nd Edition, 2002. 2. Microwave Circuits and Passive Devices – M.L. Sisodia and G.S.Raghuvanshi, Wiley Eastern Ltd., New Age International Publishers Ltd., 1995. 3. Microwave Engineering Passive Circuits – Peter A. Rizzi, PHI, 1999. 4. Electronic and Radio Engineering – F.E. Terman, McGraw-Hill, 4th ed., 1955. 5. Elements of Microwave Engineering – R. Chatterjee, Affiliated East-West Press Pvt. Ltd., New Delhi,1988.

COURSE OUTCOMES 1. Understand the significance of microwaves and microwave transmission lines 2. Analyze the characteristics of microwave tubes and compare them 3. Be able to list and explain the various microwave solid state devices 4. Can set up a microwave bench for measuring microwave parameters

Malla Reddy College of Engineering and Technology (MRCET)

171

UNIT-I Waveguides & Resonators Contents:  Introduction  Microwave spectrum and bands  Applications of Microwaves  Rectangular Waveguides-Solution of Wave Equation in Rectangular Coordinates, TE/TM mode analysis, Expressions for fields, Cutoff frequencies, filter characteristics, dominant and degenerate modes, sketches of TE and TM mode fields in the cross-section, Mode characteristics - Phase and Group velocities, wavelengths and impedance relations, Rectangular Waveguides – Power Transmission and Power Losses, Impossibility of TEM Modes, losses

Cavity resonators:

 Introduction  Rectangular and cylindrical cavities  Dominant modes and resonant frequencies  Q-factor and coupling coefficients  Illustrative Problems.

UNIT- I Wave Guides and Resonators INTRODUCTION: Microwaves are electromagnetic waves with frequencies between 300MHz (0.3GHz) and 300GHz in the electromagnetic spectrum. Radio waves are electromagnetic waves within the frequencies 30 KHz - 300GHz, and include microwaves. Microwaves are at the higher frequency end of the radio wave band and low frequency radio waves are at the lower frequency end. Mobile phones, phone mast antennas (base stations), DECT cordless phones, Wi-Fi,WLAN, WiMAX and Bluetooth have carrier wave frequencies within the microwave band of the electromagnetic spectrum, and are pulsed/modulated. Most Wi-Fi computers in schools use 2.45GHz (carrier wave), the same frequency as microwave ovens. Information about the frequencies can be found in Wi-Fi exposures and guidelines. It is worth noting that the electromagnetic spectrum is divided into different bands frequency. But the effects of electromagnetic radiation do not necessarily fit into these artificial divisions. A waveguide consists of a hollow metallic tube of either rectangular or circular cross section used to guide electromagnetic wave. Rectangular waveguide is most commonly used as waveguide. Waveguides are used at frequenciesMicrowave in the Engineering microwave range. At microwave frequencies (above 1GHz to 100 GHz) the losses in the two line transmission system will be very high and hence it cannot be used at those frequencies. Hence microwave signals are propagated through the waveguides in order to minimize the losses. Microwave Spectrum and bands: Electromagnetic Spectrum consists of entire range of electromagnetic radiation. Radiation is the energy that travels and spreads out as it propagates. The type of electromagnetic radiation that makes the electromagnetic spectrum is depicted in the following screenshot.

Properties of Microwaves Following are the main properties of Microwaves.  Microwaves are the waves that radiate electromagnetic energy with shorter wavelength.  Microwaves are not reflected by Ionosphere.  Microwaves travel in a straight line and are reflected by the conducting surfaces.  Microwaves are easily attenuated within shorter distances.  Microwave currents can flow through a thin layer of a cable. Advantages of Microwaves There are many advantages of Microwaves such as the following:

 Supports larger bandwidth and hence more information is transmitted. For this reason, microwaves are used for point-to-point communications.  More antenna gain is possible.  Higher data rates are transmitted as the bandwidth is more.  Antenna size gets reduced, as the frequencies are higher.  Low power consumption as the signals are of higher frequencies.  Effect of fading gets reduced by using line of sight propagation.  Provides effective reflection area in the radar systems.  Satellite and terrestrial communications with high capacities are possible.  Low-cost miniature microwave components can be developed.

 Effective spectrum usage with wide variety of applications in all available frequency ranges of operation.

Disadvantages of Microwaves There are a few disadvantages of Microwaves such as the following:  Cost of equipment or installation cost is high.  They are hefty and occupy more space.  Electromagnetic interference may occur.  Variations in dielectric properties with temperatures may occur.  Inherent inefficiency of electric power.

Applications of Microwaves There are a wide variety of applications for Microwaves, which are not possible for other radiations. They are -

Wireless Communications  For long distance telephone calls  Bluetooth  WIMAX operations  Outdoor broadcasting transmissions  Broadcast auxiliary services  Remote pickup unit  Studio/transmitter link  Direct Broadcast Satellite (DBS)  Personal Communication Systems (PCSs)  Wireless Local Area Networks (WLANs)  Cellular Video (CV) systems  Automobile collision avoidance system

Electronics  Fast jitter-free switches  Phase shifters  HF generation  Tuning elements  ECM/ECCM (Electronic Counter Measure) systems  Spread spectrum systems

Commercial Uses  Burglar alarms  Garage door openers

 Police speed detectors  Identification by non-contact methods  Cell phones, pagers, wireless LANs  Satellite television, XM radio  Motion detectors  Remote sensing

Navigation  Global navigation satellite systems  Global Positioning System (GPS)

Military and Radar  Radars to detect the range and speed of the target.  SONAR applications  Air traffic control  Weather forecasting  Navigation of ships  Minesweeping applications  Speed limit enforcement  Military uses microwave frequencies for communications and for the above mentioned applications.

Research Applications  Atomic resonances  Nuclear resonances

Radio Astronomy  Mark cosmic microwave background radiation  Detection of powerful waves in the universe  Detection of many radiations in the universe and earth’s atmosphere

Food Industry  Microwave ovens used for reheating and cooking  Food processing applications  Pre-heating applications  Pre-cooking  Roasting food grains/beans  Drying potato chips  Moisture levelling  Absorbing water molecules Industrial Uses  Vulcanizing rubber

 Analytical chemistry applications  Drying and reaction processes  Processing ceramics  Polymer matrix  Surface modification  Chemical vapor processing  Powder processing  Sterilizing pharmaceuticals  Chemical synthesis  Waste remediation  Power transmission  Tunnel boring  Breaking rock/concrete  Breaking up coal seams  Curing of cement  RF Lighting  Fusion reactors  Active denial systems

Semiconductor Processing Techniques  Reactive ion etching  Chemical vapor deposition

Spectroscopy  Electron Paramagnetic Resonance (EPR or ESR) Spectroscopy  To know about unpaired electrons in chemicals  To know the free radicals in materials  Electron chemistry

Medical Applications  Monitoring heartbeat  Lung water detection  Tumor detection  Regional hyperthermia  Therapeutic applications  Local heating  Angioplasty  Microwave tomography  Microwave Acoustic imaging

Types of waveguides

Waveguides are majorly classified as rectangular or circular but these are basically of 5 different types:

Modes of propagation in a Waveguide

When an electromagnetic wave is transmitted through a waveguide. Then it has two field components that oscillate mutually perpendicular to each other. Out of the two one is electric field and the other is a magnetic field.

The figure below represents the propagation of an electromagnetic wave in the z- direction with the two field components:

The propagation of wave inside the waveguide originates basically 2 modes. However, overall basically 3 modes exist, which are as follows:

 Transverse Electric wave: In this mode of wave propagation, the electric field component is totally transverse to the direction of wave propagation whereas the magnetic field is not totally transverse to the direction of wave propagation. It is abbreviated as TE mode.

 Transverse Magnetic wave: In this mode of wave propagation, the magnetic field component is totally transverse to the direction of wave propagation while the electric field is not totally transverse to the direction of wave propagation. It is abbreviated as TM mode.

 Transverse electromagnetic wave: In this mode of wave propagation, both the field components i.e., electric and magnetic fields are totally transverse to the direction of wave propagation. It is abbreviated as TEM mode.

It is to be noted here that, TEM mode is not supported in waveguides. As for the TEM mode, there is a need for the presence of two conductors and we already know that a waveguide is a single hollow conductor.

Parameters of a Waveguide:

 Cut-off wavelength: It the maximum signal wavelength of the transmitted signal that can be propagated within the waveguide without any attenuation. This means up to cut-off wavelength, a microwave signal can be easily transmitted through the waveguide. It is denoted by λc.  Group velocity: Group velocity is the velocity with which wave propagates inside the waveguide. If the transmitted carrier is modulated, then the velocity of the modulation envelope is somewhat less as compared to the carrier signal. This velocity of the envelope is termed as group velocity. It is represented by Vg.  Phase velocity: It is the velocity with which the transmitted wave changes its phase during propagation. Or we can say it is basically the velocity of a particular phase of the propagating wave. It is denoted by Vp.  Wave Impedance: It is also known as the characteristic impedance. It is defined as the ratio of the transverse electric field to that of the transverse magnetic field during wave propagation at any point inside the waveguide. It is denoted by Zg.

Advantages of waveguides

1. In waveguides, the power loss during propagation is almost negligible. 2. Waveguides have the ability to manage large-signal power. 3. As waveguides possess a simple structure thus their installation is somewhat easy.

Disadvantages of waveguides

1. Its installation and manufacturing cost is high. 2. Waveguides are generally rigid in nature and hence sometimes causes difficulty in applications where tube flexibility is required. 3. It is somewhat large in size and bulkier as compared to other transmission lines. It is noteworthy in the case of waveguides that their diameter must have some certain value in order to have proper signal propagation. This is so because if its diameter is very small and the wavelength of the signal to be propagated is large (or signal frequency is small) then it will not be propagated properly.

So, the signal frequency must be greater than the cutoff frequency in order to have a proper signal transmission.

wall. This process results in a component of either electric or magnetic field in the direction of propagation of the resultant wave. Therefore the wave is no longer a transverse electromagnetic wave. Any uniform plane wave in a lossless guide may be resolved into TE and TM waves.

In rectangular guide the modes are designed TEmn or TMmn.

Propagation of waves in Rectangular waveguides Consider a rectangular waveguide situated in the rectangular coordinate system with its breadth along x-axis, width along y-axis and the wave is assume to propagate along the z-direction. Waveguide is filled with air. In a waveguide no TEM wave is exists.

TEM(Transverse Electromagnetic wave): in TEM both electric and magnetic fields are purely transverse to the direction of propagation and consequens have no ‘z’ directed E & H components. TE(Transverse Electric Wave) In TE wave only the E field is purely transverse to the direction of propagation and the magnetic field is not purely transverse i.e. Ez=0,Hz#0 TM(Transverse Magnetic Wave) In TE wave only the H field is purely transverse to the direction of propagation and the Electric field is not purely transverse i.e. Ez#0,Hz=0 HE(Hybrid wave) In this neither electric nor magnetic fields are purely transverse to the direction of propagation. i.e. Ez#0, Hz#0

WAVE EQUATIONS Since we assumed that the wave direction is along z-direction then the wave equation are 2 2 ∇ Ez= -ω μϵEz for TM wave------(1) 2 2 ∇ Hz= -ω μϵHz for TE wave ------(2) −훾푧 −훾푧 Where Ez=E0푒 , Hz=H0푒 ------(3) The condition for wave propagation is that γ must be imaginary. Differentiating eqn(3) w.r.t ‘z’ we get −훾푧 휕퐸z/휕푧= E0푒 (-γ)= -γEz------(4) Hence we can define operator 휕/휕푧= -γ------(5) By differentiating eqn(4) w.r.t ‘z’ we get 2 2 2 휕 Ez/휕z = γ Ez We can define the operator 휕2/휕z2 = γ2------(6) From eqn(1) we can write

2 2 ∇ Ez= -ω μϵEz 2 By expanding ∇ Ez in rectangular coordinate system

휕2퐸푧 휕2퐸푧 2 + +h EZ=0 for TM wave------(7) 휕푥2 휕푦2

휕2퐻푧 휕2퐻푧 2 Similarly + +h HZ=0 for TE wave------(8) 휕푥2 휕푦2

By solving above two partial differential equations we get solutions for Ez and Hz. Using Maxwell’s equations. it is possible to find the various components along x an y-directions. From Maxwell’s first equation, we have ∇XH= jωϵE

푎푥 푎푦 푎푧

휕/휕푥 휕/휕푦 휕/휕푧 = jωϵ[Exax+Eyay+Ezaz] 퐻푥 퐻푦 퐻푧

ax→ γHy+휕퐻z/휕푦 = jωϵEx______(9) ay→ γHx+휕퐻z/휕푥 = -jωϵEy------(10) az → 휕퐻y/휕푥+휕퐻푥/휕푦 = jωϵEz------(11) similarly from Maxwell’s 2nd equation we have ∇푋퐸 = −푗휔휇H By expanding ax ay az

∂/ ∂x ∂/ ∂y ∂/ ∂z = -jω[Hxax+Hyay+Hzaz] Ex Ey Ez Since ∂/ ∂z =γ ax ay az

∂/ ∂x ∂/ ∂y −γ = -jω[Hxax+Hyay+Hzaz] Ex Ey Ez

By comparing ax,ay,az components ax→ γEy+∂Ez/ ∂y = -jωμHx------(12) ∂Ez γEx+ = jωμHy------(13) ∂x

휕퐸y/휕푥-휕퐸푥/휕푦 = -jωϵHz------(14) From eqn(13)

∂Ez Hy= [ γEx+ ]/ jωμ ------(15) ∂x By substituting eqn(15) in eqn(9) we get

2 ∂Ez γ /jωμEx+γ/jωϵ + 휕퐻z/휕푦= jωϵEx ∂x since γ2+ω2μϵ= h2 by dividing the above equation with h2 we get

2∂Ez 2 Ex= -γ/h - jωμ/h 휕퐻z/휕푦------(15) ∂x Similarly

2∂Ez 2 Ey = -γ/h +jωϵ/h 휕퐸z/휕푦------(16) ∂x And

2∂Hz 2 Hx= -γ/h + jωμ/h 휕퐸z/휕푦------(17) ∂x

2∂Hz 2 Hy= -γ/h - jωμ/h 휕퐸z/휕푥------(18) ∂y These equations give a general relationship for field components with in a waveguide. Propagation of TEM Waves: For TEM wave

Ez=0 and Hz=0

Substituting these values in equns (15) to (18) all the field components along x and y directions Ex,Ey,Hx,Hy vanish and have a TEM wave cannot exist inside a waveguide.

Modes The electromagnetic wave inside a waveguide can have an infinite number of patterns which are called modes. The electric field cannot have a component parallel to the surface i.e. the electric field must always be perpendicular to the surface at the conductor.

The magnetic field on the other hand always parallel to the surface of the conductor and cannot have a component perpendicular to it at the surface.

TE Mode Analysis

The TEmn modes in a rectangular waveguide are characterized by EZ=0. The z component of the magnetic field,HZ must exist in order to have energy transmission in the guide. The wave equation for TE wave is given by ∇2Hz = −ω2μϵHz------(1) i.e. 휕2퐻푧 + 휕2퐻푧 + 휕2퐻푧 = −휔2휇휖퐻푧 휕푥2 휕푦2 휕푧2

휕2퐻푧 휕2퐻푧 2 + +γ2HZ+ω μϵHZ=0 휕푥2 휕푦2

휕2퐻푧 휕2퐻푧 2 2 2 + +(γ2+ω μϵ)HZ=0 γ2+ω μϵ=h 휕푥2 휕푦2

휕2퐻푧 휕2퐻푧 2 + +h HZ=0------(2) 휕푥2 휕푦2 This is a partial differential equation whose solution can be assumed. Assume a solution

HZ=XY Where X=pure function of x only Y= pure function of y only From equation 2

휕2[푋푌] + 휕2[푋푌]+h2 XY=0 휕푥2 휕푦2

Y휕2푋 + 푋휕2푌/휕푦2+h2 XY=0 휕푥2 Dividing above equation with XY on both sides

1/X휕2푋 + 1/푌휕2푌/휕푦2+h2 =0------(3) 휕푥2 Here 1/X휕2푋 is purely a function of x and 1/푌휕2푌/휕푦2 is purely a function of y 휕푥2 Let 1/X휕2푋 = -B2 & 1/푌휕2푌/휕푦2 = -A2 휕푥2 i.e. from equation (3) -B2-A2+h2=0 i.e. h2=A2+B2------(4)

X=c1cosBx+c2sinBx

Y=c3cosAy+c4sinAy i.e. the complete solution for Hz=XY is HZ= ( c1cosBx+c2sinBx)( c3cosAy+c4sinAy)----(5)

Where c1,c2,c3 and c4 are constants which can be evaluated by applying boundary conditions. Boundary Conditions

Since we consider a TE wave propagating along z direction. So EZ=0 but we have components along x and y direction.

EX=0 waves along bottom and top walls of the waveguide

Ey=0 waves along left and right walls of the waveguide 1st Boundary condition:

EX=0 at y=0 ∀ x→ 0 to a(bottom wall) 2nd Boundary condition

EX=0 at y=b ∀ x→ 0 to a (top wall) 3rd Boundary condition

Ey=0 at x=0 ∀ y→ 0 to b (left side wall) 4th Boundary condition

Ey=0 at x=a ∀ y→ 0 to b (right side wall) i) Substituting 1st Boundary condition in eqn(5) Since we have 2 2 EX= -γ/h 휕퐸Z/휕푥-jωμ/h 휕퐻Z/휕푦-----(6) 2 Since EZ=0 → EX= -jωμ/h 휕[( c1cosBx + c2sinBx)( c3cosAy + c4sinAy)/휕푦] 2 EX= -jωμ/h 휕[( c1cosBx + c2sinBx)(−A c3sinAy + Ac4cosAy)/휕푦] From the first boundary condition we get 0= -jωμ/h2휕[( c1cosBx + c2sinBx) ≠0,A≠0 c4=0 Substituting the value of c4 in eqn (5), the solution reduces to

HZ= (c1cosBx+c2sinBx)(c3cosAy)------(7) ii) from third boundary condition

Ey=0 at x=0 ∀ y→ 0 to b Since we have 2 2 Ey= -γ/h 휕퐸z/휕푦+jωμ/h 휕퐻z/휕푥------(8)

Since Ez=0 and substituting the value of Hz in eqn(7), we get 2 Ey= jωμ/h 휕[(c1cosBx + c2sinBx)(c3cosAy)]/휕푥 2 Ey= jωμ/h [(−Bc1sinBx + Bc2sinBx)(c3cosAy)] From third condition, 0=jωμ/h2(0+Bc2)c3cosAy Since cosAy≠0,B≠ 0, c3#0 c2=0 from eq (7)

Hz=c1c3cosBxcosAy------(9) iii) 2nd Boundary condition since we have 2 2 Ex= -γ/h 휕퐸Z/휕푥-jωμ/h 휕퐻Z/휕푦 2 = -jωμ/h 휕/휕푦[ c1c3cosBxcosAy] [EZ=0] 2 Ex = jωμ/h c1c3cosBxsinAy From the second boundary condition,

Ex=0 at y=b ∀ x→ 0 to a 0= jωμ/h2 c1c3cosBxsinAb cosBx#0,c1c3#0 sinAb=0 or Ab=nπ where n=0,1,2----

A=nπ/b------(10) iv) 4th Boundary condition since 2 2 Ey= -γ/h 휕퐸z/휕푦+jωμ/h 휕퐻z/휕푥 2 Ey= -jωμ/h 휕/휕푥[c1c3cosBxcosAy] 2 Ey= -jωμ/h c1c3sinBx.BcosAy From the 4th Boundary condition

Ey=0 at x=a ∀ y→ 0 to b 0= -jωμ/h2Bc1c3sinBx.cosAy ∀ y→ 0 to b cosAy#0,c1c3#0 sinBa=0 B=mπ/a------(11) From eq(9)

Hz= c1c3cos(mπ/a)xcos(nπ/b)y Let c1c3=c (푗휔푡−훾푧) HZ=ccos(mπ/a)xcos(nπ/b)y푒 ------(12) Field Components 2 2 Ex= -γ/h 휕퐸Z/휕푥-jωμ/h 휕퐻Z/휕푦

Since Ez=0 for TE wave

2 (푗휔푡−훾푧 Ex=jωμ/h c(nπ/b)cos(mπ/a)xsin(nπ/b)y푒 ------(13) 2 2 Ey= -γ/h 휕퐸z/휕푦+jωμ/h 휕퐻z/휕푥

Since Ez=0 for TE wave 2 Ey= jωμ/h 휕퐻z/휕푥

2 (푗휔푡−훾푧) Ey= -jωμ/h c[mπ/a]sin(mπ/a)xcos(nπ/b)y푒 ------(14) Similarly 2 2 Hx= -γ/h 휕퐻Z/휕푥-jωϵ/h 휕퐸Z/휕푦

2 (푗휔푡−훾푧) Hx= γ/h c(mπ/a)sin(mπ/a)xcos(nπ/b)y푒 ------(15) 2 2 Hy= -γ/h 휕퐻z/휕푦-jωϵ/h 휕퐸z/휕푥

2 2 푗휔푡−훾푧 Hy= -γ/h c(nπ/b) cos(mπ/a)x.sin(nπ/b)y푒 ------(16) TM Mode Analysis

For TM wave Hz=0 Ez#0 2 2 2 2 2 휕 Ez/휕x +휕 Ez/휕y +h Ez=0------(1)

This is a partial differential equation which can be solved to get the different field components Ex,Ey,Hx and Hy by variable separable method. Let us assume a solution

Ez=XY------(2) Using these two equations from eqn(1) we get

Y휕2푋 + 푋휕2푌/휕푦2+h2 XY=0-----(3) 휕푥2 Dividing above equation with XY on both sides

1/X휕2푋 + 1/푌휕2푌/휕푦2+h2 =0------(4) 휕푥2 Here 1/X휕2푋 is purely a function of x and 1/푌휕2푌/휕푦2 is purely a function of y 휕푥2 Let 1/X휕2푋 = -B2------(5) 휕푥2 1/푌휕2푌/휕푦2 = -A2------(6) i.e. from equation (4),(5) and(6) -B2-A2+h2=0 i.e. h2=A2+B2------(7) the solution of eqn(5) and(6) are

X=c1cosBx+c2sinBx

Y=c3cosAy+c4sinAy

Where c1,c2,c3 and c4 are constants which can be evaluated by applying boundary conditions From eqn(1)

EZ= XY

EZ= ( c1cosBx+c2sinBx)( c3cosAy+c4sinAy)----(10) Boundary Conditions

Since we consider a TE wave propagating along z direction. So EZ=0 but we have components along x and y direction.

EX=0 waves along bottom and top walls of the waveguide

Ey=0 waves along left and right walls of the waveguide 1st Boundary condition:

EX=0 at y=0 ∀ x→ 0 to a(bottom wall) 2nd Boundary condition

EX=0 at y=b ∀ x→ 0 to a (top wall) 3rd Boundary condition

Ey=0 at x=0 ∀ y→ 0 to b (left side wall) 4th Boundary condition

Ey=0 at x=a ∀ y→ 0 to b (right side wall) i) Substituting 1st Boundary condition in eqn(10) Since we have

0=EZ= [c1cosBx+c2sinBx][c3cosA0+c4sinA0] [c1cosBx+c2sinBx]c3=0 c1cosBx+c2sinBx#0 c3=0 i.e. Ez=[c1cosBx+c2sinBx]c4sinAy------(11) ii) Substituting 2nd Boundary condition in eqn(11), we get

Ez=c2c4sinBxsinAy------(12) iii) Substituting 3rd Boundary condition in eqn(12), we get sinAb=0 A=nπ/b------(13) iv) Substituting 4th Boundary condition in eqn(12), we get sinBa=0 B=mπ/a------(14) From (12),(13),(14)

푗(휔푡−훾푧) Ez=csin(mπ/a)xsin(nπ/b)y푒 ------(15) 2 Ex= -γ/h 휕Ez/휕x 2 푗휔푡−훾푧 Ex= -γ/h c(mπ/a)cos(mπ/a)xsin(nπ/b)y푒 ------(16)

2 푗휔푡−훾푧 Ey= -γ/h c(nπ/b)sin(mπ/a)xcos(nπ/b)y푒 ------(17) 2 푗휔푡−훾푧 Hx=jωϵ/h c(nπ/b)sin(mπ/a)xcos(nπ/b)y푒 ------(18)

2 푗휔푡−훾푧 Hy= jωϵ/h c[mπ/a]cos(mπ/a)xsin(nπ/b)y푒 ------(19)

Cut-off Frequency of a Waveguide Since we have γ2+ω2μϵ=h2=A2+B2 A=nπ/b,B=mπ/a γ2=(mπ/a)2+(nπ/b)2-ω2μϵ γ=√(푚휋)2+(nπ/b)2-ω2μϵ=α+jβ 푎 At lower frequencies γ> 0

√(푚휋)2+(nπ/b)2-ω2μϵ> 0 푎 γ then becomes real and positive and equal to the attenuation constant α i.e. the wave is completely attenuated and there is no phase change. Hence the wave cannot propagate. However at higher frequencies, γ<0 √(푚휋)2+(nπ/b)2-ω2μϵ<0 푎 γ becomes imaginary there will be phase change β and hence the wave propagates. At the transition γ becomes zero and the propagation starts. The frequency at which γ just becomes zero is defined as the cut-off frequency fc

At f=fc,γ=0 2 2 2 0=(mπ/a) +(nπ/b) -ωc μϵ or

2 2 1/2 fc=1/2π√휇휖[(mπ/a) +(nπ/b) ] 2 2 1/2 fc=c/2[(mπ/a) +(nπ/b) ]

The cut-off wavelength(λc) is 2 2 1/2 λc=c/fc=c/c/2[(mπ/a) +(nπ/b) ] 2 2 2 2 1/2 λcm,n=2ab/[m b +n a ]

All wavelengths greater than λc are attenuated and these less than λc are allowed to propagate inside the waveguide.

Guided Wavelength (λg) It is defined as the distance travelled by the wave in order to undergo a phase shift of 2π radians. It is related to phase constant by the relation

λg=2π/β the wavelength in the waveguide is different from the wavelength in free space. Guide wavelength is related to free space wavelength λ0 and cut-off wavelength λc by 2 2 2 1/λg =1/λ0 -1/λc The above equation is true for any mode in a waveguide of any cross section

Phase Velocity(vp)

Wave propagates in the waveguide when guide wavelength λg is grater than the free space wavelength λ0.

In a waveguide, vp= λgf where vp is the phase velocity. But the speed of light is equal to product of λ0 and f.This vp is greater then the speed of light since λg> λ0.

The wavelength in the guide is the length of the cycle and vp represents the velocity of the phase. It is defined as the rate at which the wave changes its phase interms of the guide wavelength.

Vp=ω/β 2 1/2 Vp=c/[1-(λ0/λc) ]

Group Velocity(vg) The rate at which the wave propagates through the waveguide and is given by

Vg=dω/dβ

2 2 1/2 Since β=[μϵ(ω -ωc )] Now differentiating β w.r.t ω we get 2 1/2 Vg= c[1-(λ0/λc) ]

Consider the product of Vp and Vg 2 VP.Vg=c Dominant Mode The mode for which the cut-off wavelength assumes a maximum value.

λcmn= 2푎푏 √푚2푏2+푛2푎2

Dominant mode in TE

For TE01 mode λc01=2b

TE10 mode λc10=2a

Among all λc10 has the maximum value since ‘a’ is the larger dimensions than ‘b’. Hence TE10 mode is the dominant mode in rectangular waveguide. Dominant Mode in TM

Minimum possible mode is TM11. Higher modes than this also exist.

Degenerate Modes Two or more modes having the same cut-off frequency are called ‘Degenerate modes’

For a rectangular waveguide TEmn/TMmn modes for which both m#0,n#0 will always be degenerate modes. Wavelengths and Impedance Relations[TE &TM WAVES]

Guide Wavelength(λg) It is defined as the distance travelled by the wave in order to under go a phase shift of 2π radians. 2 2 2 1/λg =1/λ0 -1/λc Wave impedance is defined as the ratio of the strength of electric field in one transverse direction to the strength of the magnetic field along the other transverse direction.

ZZ=Ex/Hy 1) Wave impedance for a TM wave in rectangular waveguide 2 2 2 ZZ= -γ/h 휕퐸z/휕x-jωμ휕Hz/휕y/-γ/h 휕Hz/휕y-jωϵ/h 휕Ez/휕x For a TM wave Hz=0

ZTM= γ/jωϵ = β/ωϵ 2 2μϵ 1/2 Since we have β=[ω μϵ-ωc ] 2 1/2 ZTM= ἠ[1-(λ0/λc) ]

Since ‘λ0 is always less than λc for wave propagation ZTM<ἠ 2) Wave impedance of TE waves in rectangular waveguide

2 1/2 ZTE= ἠ/[1-(λ0/λc) ]

Therefore ZTE> ἠ

For TEM waves between parallel planes the cut-off frequency is zero and wave impedance for TEM wave is the free space impedance itself

ZTEM= ἠ Microstrip Line Microstrip Line is an unsymmetrical stripline that is nothing but a parallel plate transmission line having dielectric substrate, the one face of which is metallised ground and the other face has a thin conducting strip of certain width ‘w’ and thickness ‘t’ some times a cover plate is used for shielding purposes but it is kept much farther away than the ground plane so as not to affect the microstrip field lines.

LOSSES IN STRIP LINES: For low-loss dielectric substrate, the attenuation factor in the strip line arises from conductor losses and is given by

The attenuation constant of a microstrip line depends on frequency of operation, electrical properties of substrate and the conductors and the geometry of mounting of strip on the dielectric. When the dielectric substrate of dielectric constant is purely non-magnetic then three types of losses occur in microstrip lines. They are 1. Dielectric losses in substrate

1.Ohmic losses in strip conductor and ground plane 1.Radiation loss

Dielectric losses in substrate:

All dielectric materials possess some conductivity but it will be small, but when it is not negligible, then the displacement current density leads the conduction current density by 90 degrees, introducing loss tangent for a lossy dielectric.

1. Ohmic losses in strip conductor and ground plane

In a microstrip line the major contribution to losses at micro frequencies is from finite conductivity of microstrip conductor placed on a low loss dielectric substrate. Due to current flowing through the strip, there will be ohmic losses and hence attenuation of the microwave signal takes place. The current distribution in the transverse plane is fairly uniform with minimum value at the central axis and shooting up to maximum values at the edge of the strip.

2. Radiation losses: At microwave frequencies, the microstrip line acts as an antenna radiating a small amount of power resulting in radiation losses. This loss depends on the thickness of the substrate, the characteristic impedance Z, effective dielectric constant and the frequency of operation.

For low-loss dielectric substrate, the attenuation factor in the strip line arises from conductor losses and is given by

Cavity Resonators

A cavity resonator is a metallic enclosure that confines the electromagnetic energy i.e. when one end of the waveguide is terminated in a shorting plate there will be reflections and hence standing waves. When another shorting plate is kept at a distance of a multiple of λg/2 than the hollow space so form can support a signal which bounces back and forth between the two shorting plates. This results in resonance and hence the hollow space is called “cavity” and the resonator as the ‘cavity resonator’

The waveguide section can be rectangular or circular.

The microwave cavity resonator is similar to a tuned circuit at low frequencies having a resonant frequency 1 f0= 2휋√√퐿퐶 The cavity resonator can resonate at only one particular frequency like a parallel resonant circuit.

From the figure

d=3λg/2

The stored electric and magnetic energies inside the cavity determine it’s equivalent inductance and capacitance.

The energy dissipated by the finite conductivity of the cavity walls determines it’s equivalent resistance.

A given resonator has an infinite number of resonant modes and each mode corresponds to a definite resonant frequency.

When the frequency of an impressed signal is equal to a resonant frequency a maximum amplitude of the standing wave occurs and the peak energies stored in the electric and magnetic fields are equal.

The mode having the lowest resonant frequency is called as the ‘Dominant mode’

Rectangular cavity Resonator The electromagnetic field inside the cavity should satisfy Maxwell’s equations subject to the boundary conditions that the electric field tangential to and the magnetic field normal to the metal walls must vanish. The wave equations in the rectangular resonator should satisfy the boundary condition of the zero tangential ‘E’ At four of the walls.

(1) TE waves

For a TE wave Ez=0,Hz#0 From Maxwell’s equation 2 2 ∇ Hz= -ω μϵ Hz 2 2 2 2 2 2 2 휕 Hz/휕푥 + 휕 Hz/휕푦 + 휕 Hz/휕푧 = - ω μϵ Hz Since 휕2 /휕푧2 = γ2 2 2 2 2 2 2 휕 Hz/휕푥 + 휕 Hz/휕푦 +(γ + ω μϵ) Hz=0 Let γ2+ ω2μϵ=h2

2 2 2 2 2 휕 Hz/휕푥 + 휕 Hz/휕푦 + h Hz=0------(1) This is a partial differential equation of 2nd order

Let Hz= XY------(2) Where X is a function of ‘x’ alone, Y is a function of ‘y’ alone Y 휕2 X/휕푥2+X 휕2 Y/휕푦2+ h2XY=0 1/X 휕2 X/휕푥2+1/Y 휕2 Y/휕푦2+ h2=0------(3) Where h2 is a constant since γ2 and ω2μϵ are constants. So to satisfies the above equation sum of functions of ‘X’ and ‘Y’ must be equalent to a constant. It is possible when individual one must be a constant. Let 1/X 휕2 X/휕푥2= -B2 , 1/Y 휕2 Y/휕푦2= -A2------(4) Where A2 and B2 are constants -A2-B2+ h2=0 h2= A2+B2------(5) solutions of equation (4) are X=c1cosBx+c2sinBx------(6) Y=c3cosAy+c4sinAy------(7) Where c1,c2,c3 and c4 are constants Which are determined by applying boundary conditions i) Boundary condition(Bottom wall)

Ex=0 for y=0 and all values of x varying from 0 to a we know 2 2 Ex= -γ/h 휕퐸z/휕푥-jωμ/h 휕퐻z/휕푦

Since Ez=0 2 Ex= -jωμ/h 휕퐻z/휕푦 2 Ex= -jωμ/h 휕/휕푦[(c1cosBx + c2sinBx)(c3cosAy + c4sinAy)] = -jωμ/h2[(c2sinBx+c1cosBx)(-Ac3sinAy+Ac4cosAy)] From the boundary condition (i) 0= -jωμ/h2[c1cosBx+c2sinBx]Ac4 [c1cosBx+c2sinBx]Ac4=0 A#0, c1cosBx+c2sinBx#0 c4=0

Hz=( c1cosBx+c2sinBx)(c3cosAy)

Hz= c3(c1cosBx+c2sinBx)cosAy ii) 2nd Boundary condition[left side wall]

Ey=0 for x=0 and y varying from 0 to b. 2 Since Ey= jωμ/h 휕퐻z/휕푥 2 Ey= jωμ/h 휕/휕푥[(c1cosBx+c2sinBx)c3cosAy] 2 Ey= jωμ/h [-Bc1sinBx+Bc2cosBx]c3cosAy From the boundary condition 0= jωμ/h2[Bc2c3cosBxcosAy] c3cosAy#0, c2=0

Hz= c1cosBx.c3cosAy iii) 3rd Boundary condition[Top wall]

Ex=0 for y=b and x varies from 0 to a 2 Ex= -jωμ/h 휕퐻z/휕푦 2 2 Ex= -jωμ/h 휕[푐1푐3푐표푠퐵푥푐표푠퐴푦]/휕푦 = jωμ/h c1c3AcosBxsinAy c1c3cosBxsinAyb=0 Ab=nπ where n=0,1,2,3---- A=nπ/b iv) 4th Boundary condition[Right side wall]

Ey=0 for x=a and y varying from 0 to a 2 2 Ey= jωμ/h 휕/휕푥[c1c3cosBxcosAy]= jωμ/h c1c3BsinBxcosAy 2 Ey= jωμ/h c1c3BsinBxcosAy

From the boundary condition Ey=0 for x=a and y varying from 0 to a c1c3sinBacosAy=0 sinBa=0 Ba=mπ B=mπ/a c1c3=c 푗(휔푡−훽푧) Hz= ccos(mπ/a)xcos(nπ/b)y푒

The amplitude constant along the positive ‘z’ direction is represents by A+ , and that along the negative ‘z’ direction by A-. Adding the two travelling waves to obtain the fields of standing wave when we have from above equation.

+ −푗훽푧 - 푗훽푧 푗휔푡 Hz= (A 푒 +A 푒 )cos(mπ/a)xcos(nπ/b)y푒 + - To make Ey vanish at z=0 and z=d we must A = A 2 2 Ey= -γ/h 휕퐸z/휕푦 +jωμ/h 휕퐻z/휕푥 since Ez=0

2 −푗훽푧 푗훽푧 푗휔푡 Ey= jωμ/h 휕(A + 푒 + A − 푒 )cos(mπ/a)xcos(nπ/b)y푒 ]/휕푥

0= [(A+푒−푗훽푧+A-푒푗훽푧) − (mπ) sin (mπ) xcos (nπ) y] 푒푗휔푡 a a b But sin (mπ) xcos (nπ) y#0 a b Therefore A+푒−푗훽푧 +A-푒푗훽푧 = 0 + - To make Ey=0, A = -A A+[푒−푗훽푧-푒푗훽푧]=0 -2jsinβz.A+=0 A+#0 only sinβd=0 with z=d βd=pπ β=pπ/d

푗(휔푡−훽푧) Hz=ccos(mπ/a)xcos(nπ/b)ysin(pπ/d)z푒

Resonant Frequency(f0) Since we know that for a rectangular waveguide h2=γ2+ω2μϵ= A2+B2= (mπ/a)2+(nπ/b)2 ω2μϵ=(mπ/a)2+(nπ/b)2- γ2 for a wave propagation γ= jβ or γ2= -β2 ω2μϵ=(mπ/a)2+(nπ/b)2+ β2 if a wave has to exist in a cavity resonator there must be a phase change corresponding to a given guide wavelength. The condition for the resonator to resonate is β=pπ/d. where p=a constant 1,2,3------∞, that indicates half wave variation of either electric or magnetic field along that z- direction d=length of the resonator when β=pπ/d,f=fc ω=ω0 2 2 2 2 ω0 μϵ= (mπ/a) +(nπ/b) + (pπ/d) 2 2 2 2 (2πf0) μϵ= (mπ/a) +(nπ/b) + (pπ/d)

1 2 2 2 1/2 f0= [(mπ/a) +(nπ/b) + (pπ/d) ] 2휋√휇휖

퐶 2 2 2 1/2 f0= [(m/a) +(n/b) + (p/d) ] 2

General mode of propagation in a cavity resonator is TEmnp or TMmnp. For both TE and TM the resonant frequency is the same in a rectangular cavity resonator.

Dominant Mode The mode for which cut-off wavelength is large is called Dominant Mode.

For a>b

Q-Factor The quality factor ‘Q’ is a measure of the frequency selectivity of a resonator or antiresonant circuit and it is defined as

Q=2π 푀푎푥푖푚푢푚 푒푛푒푟푔푦 푠푡표푟푒푑 푝푒푟 푐푦푐푙푒= ωw/P 퐸푛푒푟푔푦 푑푖푠푠푖푝푎푡푒푑 푝푒푟 푐푦푐푙푒

The Q of a perfect or ideal cavity resonator is infinite since in a perfect conductor forming the cavity P would be zero and also once energized it would resonate for ever.

If there is more than one resonate frequency there will be different values of Q for the various values of frequencies.

Coupling loops are used to couple the energy in and out of a cavity resonator. This coupling changes the value of Q. that to into account the coupling between the cavity and coupling path is known as the loaded QL. So,QL can be given by

1/QL=1/Q0+1/Qext There are three types of couplings they are critically coupling,under coupling,over coupling.

An unloaded resonator can be represented by a series or parallel resonant circuit. The unloaded Q is then given by

Q0= ω0L/R

Then Qext can be written as Qext= Q0/k =ω0L/Kr

QL= Q0/1+k For critical coupled cavity is given by

QL= 1/2Qext=1/2Q0

For under coupled cavity k<1 the cavity terminals are at a viltage minimum and the input impedance is the reciprocal of standing wave ratio. i.e. k=1/휌 휌 i.e. QL= 1+휌 For over coupled cavity k>1 cavity terminals are at a voltage maximum and the impedance is standing wave ratio. i.e. k= 휌

i.e. QL= Q0/1+ 휌

UNIT-II WAVEGUIDE COMPONENTS Contents:

 Waveguide Components-I:  Scattering Matrix – Significance Formulation and properties  Wave guide multiport junctions - E plane and H plane Tees, Magic Tees Matrix calculations for E , H plane and Magic Tee  2-hole Directional coupler and its s-matrix  Coupling mechanisms - Probe, Loop, Aperture types  Wave guide discontinuities - Waveguide Windows, tuning screws and posts, Irises, Transitions, Twists, Bends, Corners and matched loads, Illustrative Problems.  Waveguide Components-II: Ferrites composition and characteristics, Faraday rotation, Ferrite components - Gyrator, Isolator, Circulator

SCATTERING PARAMETERS

 Linear two-port (and multi-port) networks are characterized by a number of equivalent circuit parameters, such as their transfer matrix, impedance matrix, admittance matrix, and scattering matrix. Fig. shows a typical two-port network.

 The transfer matrix, also known as the ABCD matrix, relates the voltage and current at port 1 to those at port 2, whereas the impedance matrix relates the two voltages V1,V2 to the two currents I1, I2.

 Thus, the transfer and impedance matrices are the 2×2 matrices:  The admittance matrix is simply the inverse of the impedance matrix, Y = Z−1. The scattering matrix relates the outgoing waves b1, b2 to the incoming waves a1, a2 that are incident on the two-port:

 The matrix elements S11, S12, S21, S22 are referred to as the scattering parameters or the S- parameters. The parameters S11, S22 have the meaning of reflection coefficients, and S21, S12, the meaning of transmission coefficients.

S- THE SCATTERING MATRIX

 The scattering matrix is defined as the relationship between the forward and backward moving waves. For a two-port network, like any other set of two-port parameters, the scattering matrix is a 2| matrix.

PROPERTIES OF SMATRIX:

In general the scattering parameters are complex quantities having the following Properties: Property (1)  When any Z port is perfectly matched to the junction, then there are no reflections from that S = 0. If all the ports are perfectly matched, then the leading diagonal II elements will all be zero. Property (2)  Symmetric Property of S-matrix: If a microwave junction satisfies reciprocity condition and if there are no active devices, then S parameters are equal to their corresponding transposes. i.e., Sij=Sji Property (3)  Unitary property for a lossless junction - This property states that for any lossless network, the sum of the products of each term of anyone row or anyone column of the [SJ matrix with its complex conjugate is unity Property (4) Phase - Shift Property:

 Complex S-parameters of a network are defined with respect to the positions of the port or reference planes. For a two-port network with unprimed reference planes 1 and 2 as shown in figure 4.6, the S- parameters have definite values.

WAVEGUIDE MULTIPORT JUNCTIONS:

T-JUNCTION POWER DIVIDER USING WAVEGUIDE:

 The T-junction power divider is a 3-port network that can be constructed either from a transmission line or from the waveguide depending upon the frequency of operation.

For very high frequency, power divider using waveguide is of 4 types  H-Plane Tee  E-Plane Tee  E-H Plane Tee/Magic Tee  Rat Race Tee

H-Plane Tee

An H-Plane Tee junction is designed by bestowing a simple waveguide to a rectangular waveguide which previously has two ports. The arms of rectangular waveguides make two ports called collinear ports i.e., Port1 and Port2, while the new one, Port3 is called as Side arm or H-arm. This H-plane Tee is also called as Shunt Tee.

As the axis of the side arm is similar to the magnetic field, this junction is called H- Plane Tee junction. This is also called as Current junction, as the magnetic field splits itself into arms. The cross-sectional details of H-plane tee can be agreed by the resulting figure.

The following figure shows the connection made by the sidearm to the bi-directional waveguide to form the serial port.

Properties of H-Plane Tee

The properties of H-Plane Tee can be defined by its [S]3×3matrix. 1.It is a 3×3 matrix as there are 3 possible inputs and 3 possible outputs.

2.Scattering coefficients S13 and S23 are equal here as the junction is symmetrical in plane

3. The port is perfectly matched to the junction.

4.We can say that we have four unknowns, considering the symmetry property.

From the Unitary property

From the Equation 6,

We know that [b] = [s][a]

This is the scattering matrix for H-Plane Tee, which explains its scattering properties.

E-Plane Tee An E-Plane Tee junction is formed by attaching a simple waveguide to the broader dimension of a rectangular waveguide, which already has two ports. The arms of rectangular waveguides make two ports called collinear ports i.e., Port1 and Port2, while the new one, Port3 is called as Side arm or E-arm. T his E-plane Tee is also called as Series Tee. As the axis of the side arm is parallel to the electric field, this junction is called E- Plane Tee junction. This is also called as Voltage or Series junction. The ports 1 and 2 are 180° out of phase with each other. The cross-sectional details of E-plane tee can be understood by the following figure. An E-Plane Tee junction is designed by assigning a simple waveguide to the broader dimension of a rectangular waveguide, which previously has two ports. The arms of rectangular waveguides create two ports called collinear ports i.e. Port1 and Port2, while the new one, Port3 is called as Side arm or E-arm. T his E- plane Tee is also called as Series Tee. As the axis of the side arm is similar to the electric field, this junction is called E- Plane Tee junction. This is also called as Voltage or Series junction. The ports 1 and 2 are 180° out of phase with each other. The cross-sectional details of E-plane tee can be assumed by the resulting figure.

The resulting figure displays the connection made by the sidearm to the bi-directional waveguide to form the parallel port.

Properties of E-Plane Tee

The properties of E-Plane Tee can be defined by its [S]3x3[S]3x3 matrix. 1.It is a 3×3 matrix as there are 3 possible inputs and 3 possible outputs.

2.Scattering coefficients S13 and S23 are out of phase by 180° with an input at port 3

3.The port is perfectly matched to the junction.

4.From the symmetric property,

Considering equations 3 & 4, the [S] matrix can be written as,

We can say that we have four unknowns, considering the symmetry property.

5.From the Unitary property

Multiplying we get,

(Noting R as row and C as column)

Equating the equations 6 & 7, we get

From Equation 8,

From Equation 9,

Using the equations 10, 11, and 12 in the equation 6, we get,

Substituting the values from the above equations in [S][S] matrix,

We get,

We know that [b]= [S][a]

This is the scattering matrix for E-Plane Tee, which explains its scattering properties.

E-H-Plane

The resulting figure shows the assembly made by the side arms to the bi-directional waveguide to form both parallel and serial ports.

Characteristics of E-H Plane Tee

 If a signal of equal phase and magnitude is sent to port 1 and port 2, then the output at port 4 is zero and the output at port 3 will be the additive of both the ports 1 and 2.  If a signal is sent to port 4, (E-arm) then the power is divided between port 1 and 2 equally but in opposite phase, while there would be no output at port 3. Hence, S34 = 0.  If a signal is fed at port 3, then the power is divided between port 1 and 2 equally, while there would be no output at port 4. Hence, S43 = 0.  If a signal is fed at one of the collinear ports, then there appears no output at the other collinear port, as the E-arm produces a phase delay and the H-arm produces a phase advance. So, S12 = S21 = 0.

Properties of E-H Plane Tee

The properties of E-H Plane Tee can be defined by its [S]4×4matrix. 1.It is a 4×4 matrix as there are 4 possible inputs and 4 possible outputs.

2.As it has H-Plane Tee section As it has H-Plane Tee section

3.As it has E-Plane Tee section

4.The E-Arm port and H-Arm port are so isolated that the other won't deliver an output, if an input is applied at one of them. Hence, this can be noted as

5.From the symmetry property, we have

6.If the ports 3 and 4 are perfectly matched to the junction, then

Substituting all the above equations in equation 1, to obtain the [S][S] matrix,

7.From Unitary property, [S][S]∗=[I]

From the equations 10 and 11, we get

Comparing the equations 8 and 9, we have

Using these values from the equations 12 and 13, we get

From equation 9, we get

S22 = 0 ………Equation 16

Now we understand that ports 1 and 2 are perfectly matched to the junction. As this is a 4 port junction, whenever two ports are perfectly matched, the other two ports are also perfectly matched to the junction. The junction where all the four ports are perfectly matched is called as Magic Tee Junction.

By substituting the equations from 12 to 16, in the [S][S] matrix of equation 7, we obtain the scattering matrix of Magic Tee as

We already know that, [b] = [S][a] Rewriting the above, we get

Applications of E-H Plane Tee

Some of the greatest mutual applications of E-H Plane Tee are as follows :

 E-H Plane junction is used to amount the impedance − A null detector is linked to E- Arm port while the Microwave source is linked to H-Arm port. The collinear ports composed with these ports make a bridge and the impedance measurement is done by balancing the bridge.  E-H Plane Tee is used as a duplexer − A duplexer is a circuit which mechanisms as both the transmitter and the receiver, by means of a single antenna for both drives. Port 1 and 2 are used as receiver and transmitter where they are inaccessible and hence will not interfere. Antenna is connected to E-Arm port. A matched load is connected to H- Arm port, which provides no reflections. Currently, there exists transmission or reception without any problem.  E-H Plane Tee is used as a mixer − E-Arm port is connected with antenna and the H- Arm port is connected with local oscillator. Port 2 has a matched load which has no reflections and port 1 has the mixer circuit, which gets half of the signal power and half of the oscillator power to produce IF frequency.  In addition to the above applications, an E-H Plane Tee junction is also used as Microwave bridge, Microwave discriminator, etc.  If we need to association two signals with no phase modification and to avoid the signals with a path difference then we need microwave device. A usual three-port Tee junction is taken and a fourth port is added to it, to make it a ratrace junction. All of these ports are linked in angular ring forms at equal intervals using series or parallel junctions.  The mean circumference of total race is 1.5λ and each of the four ports is detached by a distance of λ/4. The resulting figure shows the image of a Rat-race junction.

Let us study a few cases to appreciate the operation of a Rat-race junction. Case 1 If the input power is applied at port 1, it gets similarly split into two ports, but in clockwise direction for port 2 and anti-clockwise direction for port 4. Port 3 has unconditionally no output.The reason being, at ports 2 and 4, the powers combine in phase, whereas at port 3, cancellation occurs due to λ/2 path difference. Case 2 If the input power is applied at port 3, the power gets similarly separated between port 2 and port 4. But there will be no output at port 1. Case 3 If two unequal signals are applied at port 1 itself, then the output will be relative to the sum of the two input signals, which is separated between port 2 and 4. Now at port 3, the differential output appears. The Scattering Matrix for Rat-race junction is represented as

Applications:

Rat-race junction is used for uniting two signals and separating a signal into two halves.

Directional coupler

 A Directional coupler is a device that trials a minor amount of Microwave power for measurement tenacities. The power measurements comprise incident power, reflected power, VSWR values, etc.

 Directional Coupler is a 4-port waveguide junction comprising of a primary main waveguide and a secondary supporting waveguide. The resulting figure shows the image of a directional coupler.

 Directional coupler is used to couple the Microwave power which may be unidirectional or bi-directional.

Properties of Directional Couplers:

The properties of an ideal directional coupler are as follows.  All the finishes are matched to the ports.  When the power travels from Port 1 to Port 2, some portion of it gets coupled to Port 4 but not to Port 3.  As it is also a bi-directional coupler, when the power travels from Port 2 to Port 1, some portion of it gets coupled to Port 3 but not to Port 4.  If the power is incident through Port 3, a portion of it is coupled to Port 2, but not to Port 1.  If the power is incident through Port 4, a portion of it is coupled to Port 1, but not to Port 2.

 Port 1 and 3 are decoupled as are Port 2 and Port 4.

 Preferably, the output of Port 3 should be zero. Though, almost, a small amount of power called back power is practical at Port 3. The resulting figure specifies the power flow in a directional coupler.

Where

 Pi = Incident power at Port 1  Pr = Received power at Port 2  Pf= Forward coupled power at Port 4  Pb = Back power at Port 3 Resulting are the parameters used to define the performance of a directional coupler.

Coupling Factor (C) The Coupling factor of a directional coupler is the ratio of incident power to the forward power, measured in dB.

Directivity (D) The Directivity of a directional coupler is the ratio of forward power to the back power, measured in dB.

Isolation It defines the directive properties of a directional coupler. It is the ratio of incident power to the back power, measured in dB.

Isolation in dB = Coupling factor + Directivity

Two-Hole Directional Coupler This is a directional coupler with same main and auxiliary waveguides, but with two small holes that are common between them. These holes are λg/4 distance apart where λg is the guide wavelength. The following figure shows the image of a two-hole directional coupler.

A two-hole directional coupler is planned to see the ideal condition of directional coupler, which is to evade back power. Some of the power while travelling between Port 1 and Port 2, escapes through the holes 1 and 2. The greatness of the power depends upon the dimensions of the holes. This leakage power at both the holes are in phase at hole 2, adding up the power causal to the forward power Pf. Though, it is out of phase at hole 1, stopping each other and stopping the back power to occur. Therefore, the directivity of a directional coupler improves. The general S matrix of a directional coupler is,

1.Since all ports in a directional coupler are matched.

S11 = S22 = S33 = S44 = 0 ------(2)

2.Since there is no coupling between ports 1 & 3 and ports 2 & 4

S13 = S31 = S24 = S42 = 0 ------(3)

Apply equation (2) & (3) in (1)

3.By unitary property, [S][S]* = I

2 2 R1C1 => |S12| + |S14| = 1 ------(4) 2 2 R2C2 => |S12| + |S23| = 1 ------(5) 2 2 R3C3 => |S23| + |S34| = 1 ------(6)

R1C3 => S12 S23* + S14 S34* = 0 ------(7)

Comparing equations (4) and (5)

2 2 2 2 |S12| + |S14| = |S12| + |S23| S14 = S23 ------(8)

Comparing equations (5) and (6)

2 2 2 2 |S12| + |S23| = |S34| + |S23| S12 = S34 ------(9)

Let, S12 be real and positive, i.e, S12 = S34 = p ------(10) applying equation (10) in (7)

Therefore, p S23* + S14 p = 0 p [S23* + S14] = 0 p [S23* + S23] = 0

S23* + S23 = 0

To satisfy the above condition, S23 should be a complex value.

Let S23 = jq

Therefore, the S matrix of directional coupler is,

COUPLING MECHANISMS: PROBE, LOOP, APERTURE TYPES:

 The three devices used to inject or remove energy from waveguides are PROBES, LOOPS, and SLOTS. Slots may also be called APERTURES or WINDOWS.  As previously discussed, when a small probe is inserted into a waveguide and supplied with microwave energy, it acts as a quarter-wave antenna. Current flows in the probe and sets up an E field such as the one shown in figure 1-39, view (A). The E lines detach themselves from the probe. When the probe is located at the point of highest efficiency, the E lines set up an E field of considerable intensity.

Fig: Probe coupling in a rectangular waveguide.

Fig: Probe coupling in a rectangular waveguide.

 The most efficient place to locate the probe is in the center of the "a" wall, parallel to the "b" wall, and one quarter-wavelength from the shorted end of the waveguide, as shown in figure 1-39, views (B) and (C). This is the point at which the E field is maximum in the dominant mode. Therefore, energy transfer (coupling) is maximum at this point. Note that the quarter-wavelength spacing is at the frequency required to propagate the dominant mode.  In many applications a lesser degree of energy transfer, called loose coupling, is desirable. The amount of energy transfer can be reduced by decreasing the length of the probe, by moving it out of the center of the E field, or by shielding it. Where the degree of coupling must be varied frequently, the probe is made retractable so the length can be easily changed.  The size and shape of the probe determines its frequency, bandwidth, and power- handling capability. As the diameter of a probe increases, the bandwidth increases. A probe similar in shape to a door knob is capable of handling much higher power and a larger bandwidth than a conventional probe. The greater power-handling capability is directly related to the increased surface area. Two examples of broad- bandwidth probes are illustrated in figure 1-39, view (D). Removal of energy from a waveguide is simply a reversal of the injection process using the same type of probe.  Another way of injecting energy into a waveguide is by setting up an H field in the waveguide. This can be accomplished by inserting a small loop which carries a high current into the waveguide, as shown in figure 1-40, view (A). A magnetic field builds up around the loop and expands to fit the waveguide, as shown in view (B). If the frequency of the current in the loop is within the bandwidth of the waveguide, energy will be transferred to the waveguide.  For the most efficient coupling to the waveguide, the loop is inserted at one of several points where the magnetic field will be of greatest strength. Four of those points are shown in figure 1-40, view (C).

Figure - Loop coupling in a rectangular waveguide.

 When less efficient coupling is desired, you can rotate or move the loop until it encircles a smaller number of H lines. When the diameter of the loop is increased, its power-handling capability also increases. The bandwidth can be increased by increasing the size of the wire used to make the loop.

 When a loop is introduced into a waveguide in which an H field is present, a current is induced in the loop. When this condition exists, energy is removed from the waveguide.

 Slots or apertures are sometimes used when very loose (inefficient) coupling is desired, as shown in figure. In this method energy enters through a small slot in the waveguide and the E field expands into the waveguide. The E lines expand first across the slot and then across the interior of the waveguide. Minimum reflections occur when energy is injected or removed if the size of the slot is properly proportioned to the frequency of the energy.

 After learning how energy is coupled into and out of a waveguide with slots, you might think that leaving the end open is the most simple way of injecting or removing energy in a waveguide. This is not the case, however, because when energy leaves a waveguide, fields form around the end of the waveguide. These fields cause an impedance mismatch which, in turn, causes the development of standing waves and a drastic loss in efficiency. Impedance matching using a waveguide iris

 Irises are effectively obstructions within the waveguide that provide a capacitive or inductive element within the waveguide to provide the impedance matching.

 The obstruction or waveguide iris is located in either the transverse plane of the magnetic or electric field. A waveguide iris places a shunt capacitance or inductance across the waveguide and it is directly proportional to the size of the waveguide iris.

 An inductive waveguide iris is placed within the magnetic field, and a capacitive waveguide iris is placed within the electric field. These can be susceptible to breakdown under high power conditions - particularly the electric plane irises as they concentrate the electric field. Accordingly the use of a waveguide iris or screw / post can limit the power handling capacity.

Fig: Impedance matching using a waveguide iris

 The waveguide iris may either be on only one side of the waveguide, or there may be a waveguide iris on both sides to balance the system. A single waveguide iris is often referred to as an asymmetric waveguide iris or diaphragm and one where there are two, one either side is known as a symmetrical waveguide iris.

Fig :Symmetrical and asymmetrical waveguide iris implementations

 A combination of both E and H plane waveguide irises can be used to provide both inductive and capacitive reactance. This forms a tuned circuit. At resonance, the iris acts as a high impedance shunt. Above or below resonance, the iris acts as a capacitive or inductive reactance.

Impedance matching using a waveguide post or screw

 In addition to using a waveguide iris, post or screw can also be used to give a similar effect and thereby provide waveguide impedance matching.

 The waveguide post or screw is made from a conductive material. To make the post or screw inductive, it should extend through the waveguide completely making contact with both top and bottom walls. For a capacitive reactance the post or screw should only extend part of the way through.

 When a screw is used, the level can be varied to adjust the waveguide to the right conditions.

 The diagram shows the electric field across the cross section of the waveguide. The lowest frequency that can be propagated by a mode equates to that were the wave can "fit into" the waveguide.

 As seen by the diagram, it is possible for a number of modes to be active and this can cause significant problems and issues. All the modes propagate in slightly different ways and therefore if a number of modes are active, signal issues occur.

 It is therefore best to select the waveguide dimensions so that, for a given input signal, only the energy of the dominant mode can be transmitted by the waveguide. For example: for a given frequency, the width of a rectangular guide may be too large: this would cause the TE20 mode to propagate.

 As a result, for low aspect ratio rectangular waveguides the TE20 mode is the next higher order mode and it is harmonically related to the cutoff frequency of the TE10 mode. This relationship and attenuation and propagation characteristics that determine the normal operating frequency range of rectangular waveguide

Waveguide Bends

 Details of RF waveguide bends allowing changes in the direction of the transmission line waveguide E bend and waveguide H bend.

 Waveguide is normally rigid, except for flexible waveguide, and therefore it is often necessary to direct the waveguide in a particular direction. Using waveguide bends and twists it is possible to arrange the waveguide into the positions required.

 When using waveguide bends and waveguide twists, it is necessary to ensure the bending and twisting is accomplished in the correct manner otherwise the electric and magnetic fields will be unduly distorted and the signal will not propagate in the manner required causing loss and reflections. Accordingly waveguide bend and waveguide twist sections are manufactured specifically to allow the waveguide direction to be altered without unduly destroying the field patterns and introducing loss.

Types of waveguide bend

There are several ways in which waveguide bends can be accomplished. They may be used according to the applications and the requirements.

1. Waveguide E bend

2. Waveguide H bend 3. Waveguide sharp E bend 4. Waveguide sharp H bend

 Each type of bend is achieved in a way that enables the signal to propagate correctly and with the minimum of disruption to the fields and hence to the overall signal.  Ideally the waveguide should be bent very gradually, but this is normally not viable and therefore specific waveguide bends are used.  Most proprietary waveguide bends are common angles - 90° waveguide bends are the most common by far.

Waveguide H bend

 This form of waveguide bend is very similar to the E bend, except that it distorts the H or magnetic field. It creates the bend around the thinner side of the waveguide

Waveguide sharp E bend

 In some circumstances a much shorter or sharper bend may be required. This can be accomplished in a slightly different manner. The techniques is to use a 45° bend in the waveguide. Effectively the signal is reflected, and using a 45° surface the reflections occur in such a way that the fields are left undisturbed, although the phase is inverted and in some applications this may need accounting for or correcting.

Waveguide sharp H bend

This waveguide bend is the same as the sharp E bend, except that the waveguide bend affects the H field rather than the E field.

WAVEGUIDE TWISTS

 There are also instances where the waveguide may require twisting. This too, can be accomplished. A gradual twist in the waveguide is used to turn the polarisation of the waveguide and hence the waveform.

 In order to prevent undue distortion on the waveform a 90° twist should be undertaken over a distance greater than two wavelengths of the frequency in use. If a complete inversion is required, e.g. for phasing requirements, the overall inversion or 180° twist should be undertaken over a four wavelength distance.

 Waveguide bends and waveguide twists are very useful items to have when building a waveguide system. Using waveguide E bends and waveguide H bends and their srap bend counterparts allows the waveguide to be turned through the required angle to meet the mechanical constraints of the overall waveguide system. Waveguide twists are also useful in many applications to ensure the polarisation is correct.

ATTENUATORS:

 In order to control power levels in a microwave system by partially absorbing the transmitted microwave signal, attenuators are employed. Resistive films (dielectric glass slab coated with aquadag) are used in the design of both fixed and variable attenuators.

 A co-axial fixed attenuator uses the dielectric lossy material inside the centre conductor of the co-axial line to absorb some of the centre conductor microwave po wer propagating through it dielectric rod decides the amount of attenuation introduced. The microwave power absorbed by the lossy material is dissipated as heat.  In waveguides, the dielectric slab coated with aduadag is placed at the centre of the waveguide parallel to the maximum E-field for dominant TEIO mode. Inducedcurrent on the lossy material due to incoming microwave signal, results in power dissipation, leading to attenuation of the signal. The dielectric slab is tapered at both ends upto a length of more than half wavelength to reduce reflections as shown in figure 5.7. The dielectric slab may be made movable along the breadth of the waveguide by supporting it with two dielectric rods separated by an odd multiple of quarter guidewavelength and perpendicular to electricfield.

 When the slab is at the centre, then the attenuation is maximum (since the electric field is concentrated at the centre for TEIO mode) and when it is moved towards one side-wall, the attenuation goes on decreasing thereby controlling the microwave power corning outof the other port.

 Above figure shows a flap attenuator which is also a variable attenuator. A semi-circular flap made of lossy dielectric is made to descend into the longitude in slot cut at the center of the top wall of rectangular waveguide. When the flap is completely outside the slot, then the attenuation is zero and when it is completely inside, the attenuation is maximum. A maximum direction of 90 dB attenuation is possible with this attenuator with a VSWR of 1.05. The dielectric slab can be properly shaped according to convenience to get a linear variation of attenuation within the depth of insertion.

 A precision type variable attenuator consists of a rectangular to circular transition (ReT), a piece of circular waveguide (CW) and a circular-to- rectangular transition (CRT) as shown in below figure . Resistive cards Ra, Rb and Rc are placed inside these sections as shown. The centre circular section containing the resistive card Rb can be precisely rotated by 3600 with respect to the two fixed resistive cards. The induced current on the resistive card R due to the incident signal is dissipated as heat producing attenuation of the transmitted signal. TE mode in RCT is converted into TE in circular waveguide. The resistive cards R and R a kept perpendicular to the electric field of TEIO mode so that it does not absorb the energy. But any component parallel to its plane will be readily abs orbed. Hence, pure TE mode is excited in circular waveguide section II.

 If the resistive card in the centre section is kept at an angle 8 relative to the E-field direction of the TEll mode, the component Ecos(θ) parallel to the card get absorbed while the component E sin θis transmitted without attenuation. This component finally comes out as E sin2θ as shown in figure below.

PHASE SHIFTERS:

 A microwave phase shifter is a two port device which produces a variable shift in phase of the incoming microwave signal. A lossless dielectric slab when placed inside the rectangular waveguide produces a phase shift.

PRECISION PHASE SHIFTER

 The rotary type of precision phase shifter is shown in figure below which consists of a circular waveguide containing a lossless dielectric plate of length 2lcalled "half-wave section", a section of rectangular-to-circular transition containing a lossless dielectric plate of length l, called "quarter- wave section", oriented at an angle of 45° to the broader wall of the rectangular waveguide and a circular-to-rectangular transition again containing a lossless dielectric plate of same length 1 (quarter wave section) oriented at an angle 45°.

 The incident TE10 mode becomes TEll mode in circular waveguide section. The half-wave section produces a phase shift equal to twice that produced by the quarter wave section. The dielectric plates are tapered at both ends to reduce reflections due to discontinuity.

 When TE10 mode is propagated through the input rectangular waveguide of the rectangular to circular transition, then it is converted into TEll in the circular waveguide section. Let E; be the maximum elelectric field strength of this mode which is resolved into components, EI parallel to the plate and E2 perpendicular to El as shown in figure 5.12 (b). After propagation through the plate these components are given by the length I is adjusted such that these two components E1 and Ez have equal amplitude but differing in phase by = 90.  The quarter wave sections convert a linearly polarized TEll wave into a circularly polarized wave and vice-versa. After emerging out of the half- wave section,the electric field components parallel and perpendicular to the half-wave plate After emerging out of the half-wave section, the field components E3 and E4 as given in above equations, may again be resolved into two TEll modes, polarized parallel and perpendicular to the output quarter-wave plate. At the output end of this quarter-wave plate, the field components parallel and perpendicular to the quarter wave plate, by referring to figure above.

Waveguide Joints As a waveguide system cannot be made in a single piece continuously, occasionally it is essential to join dissimilar waveguides. This joining must be sensibly done to stop problems such as − Reflection effects, creation of standing waves, and increasing the attenuation, etc.The waveguide joints also evading irregularities should also take care of E and H field patterns by not affecting them. There are many types of waveguide joints such as bolted flange, flange joint, choke joint, etc.

UNIT III Linear beam Tubes

Contents:

 Limitations and losses of conventional tubes at microwave frequencies  Classification of Microwave tubes  O type tubes - 2 cavity klystrons, structure, Reentrant cavities, velocity modulation process Applegate diagram, bunching process ,small signal theory Expressions for o/p power and efficiency  Reflex Klystrons-structure, Velocity Modulation, Applegate diagram, mathematical theory of bunching, power output, efficiency, oscillating modes and o/p characteristics, Effect of Repeller Voltage on Power o/p  Significance, types and characteristics of slow wave structures, structure of TWT and amplification process (qualitative treatment), Suppression of oscillations, Gain considerations. Unit-V Microwave Engineering

UNIT-V

MICROWAVE TUBES

Limitations and losses of conventional Tubes at Microwave Frequencies

Conventional vacuum triodes, tetrodes and pentodes are less useful signal sources at frequencies above 1 G Hz because of

 lead inductance  Inter-electrode capacitance effects,  Transit angle effects  Gain bandwidth product limitations.  Power losses

Lead inductance and inter-electrode capacitance effects

At frequencies above 1 GHz conventional vacuum tubes are impaired by parasitic circuit reactance because the circuit capacitances between tube electrodes and the circuit inductance of the lead wire are too large for a microwave resonant circuit. Further as the frequency increases the real part of the input admittance may be large enough to cause a serious over load of the input circuit and thereby reduce the operating efficiency of the tube. Transit angle effects

Another limitation in the application of conventional tubes at microwave frequencies is the electron transit angle between electrodes. The electron transit angle is defines as

ωd Θg = ω = 푣표

d Where = is the transit time across the gap 푣표 d = separation between cathode and grid 5 푣표 = Velocity of the electron 0.593 x 10 푉표 Vo = DC voltage

When frequencies are below microwave range, the transit angle is negligible. At microwave frequencies, however the transit time is large compared to the period of the microwave signal, and the potential between the cathode and the grid may alternate from 10 to 100 times during the electron transit. The grid potential during the negative half cycle thus removes energy that was given to the electron during the positive half cycle. Consequently, the electrons may oscillate back and forth in the cathode-grid space or return to the cathode. The overall result of transit angle effects

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Unit-V Microwave Engineering is to reduce the operating efficiency of the vacuum tube. The degenerate effect becomes more serious when frequencies are well above 1 GHz. Gain bandwidth product limitations The gain-bandwidth product is independent of frequency. For a given tube, a higher gain can be achieved only at the expense of a narrower bandwidth. This restriction is applicable to a resonant circuit only. In microwave devices either reentrant cavities or slow-wave structures are used to obtain a possible overall high gain over a bandwidth. Power losses The use of conventional tubes at higher frequencies also increases in power losses resulting from skin effect, I2R losses resulting from capacitance charging currents, losses due to radiation from the circuit and dielectric losses. Classification of Microwave tubes.

Microwave Tubes

O-Type Tubes M-Type Tubes (Linear beam tubes) (Cross field tubes)

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Unit-V Microwave Engineering

Energy Transfer Mechanism

In most of the microwave tubes, the signal is placed in a cavity gap and electrons are forced to cross the gap at time when they face maximum opposition. Crosssing the gap under opposition lead to transfer of energy to the cavity gap signal. When the gap voltage is sinusoidal time-varying and the charge corssing is continuous and uniform, which is usually the case, no net transfer of energy takes place between cavity and the charge crossing the gap. It is because the energy transfer is equal and opposite in direction during a half cycle when compalred to previous half cycle resulting in no net transfer of energy in a cycle. To have net energy transfer, preferabley maximum, from electron beam to gap signal voltage the disributed charge is compressed into a thin sheet or bunch, so that it requires less time to cross the gap and it is arranged such hat the bunch corssing is at peak gap voltage so that the bunch faces maximum opposition and retardation from the signal voltage. When the gap voltage is simusoidal and bunch corssing is at a uniform and constant rate, for maximum unidirectional flow of energy, there is only one instant, either at positive peak or negative peak, for the bunch to cross the gap. The bunch crossing hence mus be once per cycle of the gap voltage. In cse of bunch corssing at a uniform rate of f, transfer of maximum energy can take place only with a a component of grid gap field whose frequency is also f. Other components of the grid gap voltge like 2f, 4f, 8f, ets., do not involve in the energy transfer, whereas the components 3f, 5f, 6f, etc., and f/2, f/3. f/4, etc., the transfered amount of energy is negligible. Two Cavity Klystron

The two-cavity klystron is a widely used microwave amplifier operated by the principles of velocity and current modulatin. It consists of an elctron gun, focussing and accelerating grids, two identical cavities separated by a distance and at the far end a gounded collector plaate. The elctron gun emits electrons from the surace of its cathode and then they are focussed into a bearm. Using a dc accelerating positive voltage the beam is accelerated to high velocities.

Characteristics of two cavity klystron

Efficiency: about 40% Power output in CW mode: upto 500 kW at 10 GHz Power output in Pusled mode: upto 30 MW at 10 GHz Power gain: about 30 dB

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Unit-V Microwave Engineering

Fig 5.1 Scematic diagram of two cavity klystron

Components of two cavity klystron

1. Cathode: Source of electrons 2. Anode: for formation of electron beam 3. Bucher Cavity: A reentrat type resonator cavity which is kept at a +ve voltage

of Vo w.r.t. cathode to effect acceleration of electrons. RF input voltage of V1 Sin ωt is applied to buncher cavity. 4. Catcher cavity: Similar to buncher cavity. The amplified output signal

V2 Sin ωt is obtained from this cavity. 5. Collector: The elctrons after transfer of enerty to catcher cavity are collected by the collector.

Let us define varous parameters used in the description and operation of two cavity klystron.

Vo = DC voltage between cathode and buncher cavity.

V1 = Amplitude of input RF signal, V1<< Vo ω = 2πf = Input signal angular frequency. It is also equal to resonant frequency of both the cavities. vo = Uniform velocity of electrons between cathode and buncher cavity. to = Time at which electrons enter the buncher cavity t1 = Time at which electrons leave the buncher cavity

 = Transit time of electrons in the buncher cavity = t1 - to

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Unit-V Microwave Engineering

θg = Angle /Phase vaiation of input signal during the tansit time = ωŤ β = Beam Coupling Coefficient of the buncher / Catcher Cavity

When the electrons enter the buncher cavity with uniform velocity „ v0 ‟ interact with the field due to input RF signal V1sin ωt. The time varying field in the cavity cause the electrons to accelerate or decelerate and there by electrons undergo velocity modulation.

Let v(t1) = Velocity of electrons at t= t1 at the output of buncher cavity This is a time varying quantity Refer fig-5.3 for velocity modulation of electrons Let d = cavity with of buncher / catcher cavity L = spacing between buncher and catcher cavities * „L‟ is the design parameter for optimum performance of the klystron amplifier

t2 = Time at which electrons enter the catcher cavity

t3= Time at which electrons leave the catcher cavity d    t1  t0 (5.2) vo

Because V1<

Evaluation of v(t1) v(t1) is the instantaneous velocity of electrons which is a time varying quantity and primarily depends upon the average voltage in the gap during the time „ ‟ i.e. during the time period (transit) the electrons are influenced by the field.

Vavg = Average voltage in the gap during time „ ‟

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Unit-V Microwave Engineering

1 t1 V  V sint dt avg   1 t0

V t = 1 cost1  t0

V1 V1 Vavg  cost1  cost0  Vavg  costo  cost1   =  Where  = t1- t0 d t1 = t0 +  = t0 + vo V   d  1   Vavg  cost0  cost0      v0  d Where θg =   = (5.3) v0 V V  1 cost  cos(t  ) avg  0 0 g

 g Let A = ωt0 + 2  and B = g 2 Since cos (A-B) – cos (A+B) = 2 sin A sin B

sin(d / 2v )  d  0   Vavg  V1   sint0    d / 2v0   2v0  sin / 2    g  g  = V1 sint0   g / 2  2  βi = beam coupling coefficient of input (buncher) cavity by definition

sing / 2 β = i  / 2 g (5.4)     g  Vavg  V1i sint0    2  2eV As we have seen earlier v  0 0 m 2e    |||ly (t ) =  g  v 1  . V1i sint0   V0 m  2  2e   V    i 1  g   V0 1 sint0   m  V0  2 

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Unit-V Microwave Engineering

iV1 Since V1<

x Using binomial expansion 1 x 1 for x 1 2

  iV1   g  v(t1 )  v0 1 sint0   2V  2   0  

Since  = t1-t0

θg =  =  t1 -  t0

 t0 =  t1+  t0

 t0+ θg /2 =  t1 - θg+ θg / 2 =  t1- θg /2

  iV1   g  v(t1 )  v0 1 sinwt1   2V  2   0   (5.5)

 iV1   g  v(t1)  v0 1 sinwt0   2V  2   0   (5.6)

Numerical : The parameters of a 2 cavity klystron are

v0 = 1000 V, I0 = 25 mA(beam current)

d = 1mm, f = 3 GHz

Find out βi beam coupling coefficient

6 6 Solution v0  0.59310 V0  0.59310  1000

= 1.88 x 107 m/s

d 2 3109 103    1.002 rad g v 1.88107 0 sin 2 sin0.5   g   0.958 i  2 0.5 g Bunching Process of Electrons

All the electrons in the beam will drift with a uniform velocity of “ v0 ”at t =t0 i.e. at time of entry into the buncher cavity. For t2 > t > t0 i.e. in the cavity gap the velocity of electrons vary with time depending upon the instantaneous field V1sinwt

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Fig 5.2: Bunching process in 2-cavity klystron Consider three arbitrary electrons a, b and c passing thro the gap when the field is – ve max, zero and +ve max respectively at time instances ta, tb and tc.

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Fig 5.3 : Appligate diagram

Velocity of electron „b‟ = vb = v0 (field zero)

Velocity of electron „a‟ = va < v0 = vmin (-ve field)

Velocity of electron „c‟ = vc > v0 = v0 = vmax (+ve field) Let us consider that these three electrons draft with different velocities and meet (bunch) together at t = td at a length ∆L from buncher cavity.

∆L= vmin (td-ta) (5.7)

∆L= (td-tb) (5.8)

∆L= vmax (td-tc) (5.9)

tc-tb = tb-ta= π / 2w (1/4 of time period) (5.10)

∆L= (td - ta) = (td - tb +π/2ω) (5.7A)

∆L= (td - tc) = (td - tb –π/2ω) (5.8A)

 iV1  g  We have v(t1)  v0 1 sint0   2V  2   0  

From above equation

  V  v  v(t )  v 1 i 1  max 1 0  2V   0  (5.11)

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  V  v  v(t )  v 1 i 1  min 1 0  2V   0  (5.12)

Substituting equation 12, 11 in equation 7A and 8A

  v0iV1 v0iV1   ∆L = v0 (td tb)  v0  (td  tb )   2 2V 2V 2  0 0  (5.13)

  v0iV1 v0iV1   ∆L = v0 (td tb)   v0  (td  tb )   2 2V 2V 2  0 0  (5.14)

Subtracting Eqn 5.14 from Eqn 5.13   V  V  v  v i 1 (t t )  v i 1  0 0 2 0 2V d b 0 2V 2 0 0 v  V   V  0 i 1 (t t )  v 1 v i 1 2V d b 0 2 0 2V 2 0 0

v0iV1   iV1  v0 (td tb)  v0 1   2V 2 2V 2 0  0   V sin ce i 1 1 2V0

V t t  0 d b w V i 1 (5.15) From Equation 5.8 and 5.15 V L  v 0 0 w V i 1 (5.16)

∆L is theoretical value of distance from buncher cavity at which bunching of electrons takes place. Refer Fig -5.1

Equation 5.16 gives the design parameter for spacing between buncher and catcher cavities.

However equation – 5.16 is only an approximation, because mutual repulsive force between the electrons in the highly densed beam are not taken into consideration. It

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Unit-V Microwave Engineering will be seen later that optimum spacing between the two cavities “L” optimum is given by (for maximum degree of bunching)

3.682v0V0 Loptimum  iV1

Which is closer to equation – 5.16 (For derivation refer equation – 5.28) Let T= Transit time for on electron travel distance „L‟ (function of „t‟) L = spacing between two cavities

Let T0 = Transit time for electron when the field in buncher cavity is i.e. v (t1) = v0

L T  0 v 0 (5.18) L T  v(t ) 1

Substituting for v(t1) from equation 5

L T   iV1  g  v0 1 sint1    2V0  2 

1 L  iV1  g  T  1 sint1   v 2V  2  0  0  

-1 Using binomial expansion (1+x) = 1-x for x<<1 and V1<

 iV1  g  T  T0 1 sint1   2V  2   0   (5.19) Multiplying above equation by „ω‟

 iV1  g  T  T0 1 sint1   V  2   0   (5.20)

Let θ0 = Angular variation in the signal during time „T0‟

θ0 = ωT0

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T  V    0 i 1  g  T  T0  sint1   V0  2   T   X sint  g 0 1 2

iV1 where X  0 2V0 (5.21)

X is called Bunching parameter of Klystron

The second design criterion is that the maximum energy will be transferred by the electrons to the catcher cavity when the bunch enters the cavity while the field is at negative peak. Assuming the buncher and catcher cavities are at same phase the above condition can be expressed mathematically

θ0 = ωT0 = 2πn – π/2 = 2πN (5.21A)

Where n is an integer and N is number cycles the angle has undergone chages during the transit time T0

Expression for output current

Let us try to establish the relation between I0 = dc current passing through buncher vacity and „i2‟ ac current in the catcher cavity.

Making use of law of conservation

Let charge „dQ0‟ pass through the buncher gap at a time interval „dt0‟ and we will assume the same amount of charge passes through the catcher gap later in time interval „dt2‟

dQ0 = I0 dt0

dQ  I dt  i dt 0 0 0 2 2 (5.22)

We have earlier defined t0, t1, t2 such that

 =t1 – t0 t1 = t0+ Ť

T= t2-t1, T= t2-(t0+ Ť)

From equation 19

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 iV1  g  T  t2  t1  T0 1 sint1   2V  2   0  

  iV1   g  t2  t0   T0 1 sint1   2V  2   0  

Multiplying by „ω‟

  iV1   g  t2  t0    T0 1 sint1   2V  2   0  

  V    t  t    0 i 1 sint  g  2 0 g 0 2V  0 2  0     V we have X  0 i 1 V 0

 g  t2  t0 g 0  X sint0    2 

Differentially above equation w.r.t. „t0‟

θg, θ0 are constants w.r.t. „t‟

dt     2   X cost  g and dt  0 2  0  

  g  dt2  dt0 1 X cost0    2     (5.23)

From equation 22 and 23

  g  I0dt0  i2 dt2  i2 dt0 1 X cost0     2  I i (t )  0 2 0    1 X cost  g   0 2    (5.24) 2 1 i2, the beam current at catcher cavity is a periodic waveform of  ,  f period about dc current I0

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 i2  I0  2I00 J n ()cos n(t2  T0 ) n1 (5.25)

Where n = integer

Derivations of above equation is out of preview of the syllabus

th st Jn(x) = n order bessel function of 1 kind

We are interested in the fundamental component i.e. n=1, ac beam current

Neglecting dc current and higher order of ac current i.e. n>2

If = fundamental component of ac current from equation 25 in catcher cavity whose beam complex coefficient = β0

I  2I J (X ) cos w (t  T ) f 0 1 0 2 0 (5.26)

Let Ifmax= magnitude of If in catcher cavity I  2 I J (X ) 2 0 0 1 (5.27)

Fig 5.4: Bessel function Jn (nX)

J1(X) is maximum at X = 1.841 i.e. J1(1.841) = 0.582 from Bessel function

Where X = Bunching parameter of Klystron as defined in equation 21

 V  V L X  i 1   i 1 w 2V 0 2V v 0 0 0

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L → Loptimum as X → 1.841 21.841V  3.682 V Loptimum  0 0  0 0  V   V i 1 i 1 (5.28)

The same equation is given theoretically as equation 5.17 earlier

The Output power and beam loading

Let I2 is max value of current in the catcher cavity with β=β0

From equation 5.27

I2 = 2β0I0J1(X) (5.29)

Power Output and efficiency of Klystron

The equivalent output circuit of Klystron is

Fig 5.5 : Output equivalent circuit

Where Rsho = Wall resistance of catcher cavity

RB = beam loading resistance

RL = External load resistance

Rsh = Effective shunt resistance = Rsho || RB || RL i f max I2 2I00 J1(X ) irms    2 2 2

2 2 2 2 Poutput  i rms Rsh  2I  J (X )R   I J (X )V 0 0 1 sh 0 0 1 2 (5.29A)

max imum output voltage  V  i Rsh  2 I J (X )R 2 f rms 0 0 1 sh (5.30) P 2I 2 2 J 2 (X )R Efficiency   0  0 0 1 sh p V I dc 0 0

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Unit-V Microwave Engineering

 I J (X )V V   0 0 1 2   J (X ) 2 V I 0 1 V 0 0 0 (5.31)

Maximum theoretical efficiency of Klystron is

β0 =1, J1(X)=0.582, V2=V0

ηmax = 58.2 % theoretical

Practically η ≈ 40%

Condition for maximum transfer of energy to catcher cavity

From equation 5.21A we have θ0 = ωT0 = 2πn – π/2 = 2πN and from equation 5.21

 V  V X  i 1 o  i 1 2N 2V 2V 0 0

2 X 1.841 V1 1.841 ( )max= = (5.32) V0 2N N

Mutual Conductance ‘Gm’

퐼2 2 퐼푂 훽𝑖 퐽1 푋 Gm = = (5.33) 푉1 푉1 From equation 5.21 we have

훽 푉 훽 푉 휔 퐿 X = 𝑖 1 휃표 = 𝑖 1 2푉표 2푉표 푣표

2 푣푂 푉표 푋 V1= (5.34) 휔훽 𝑖 퐿

Substituting V1 in equation 5.33

2 퐼푂 훽표 퐽1 푋 Gm = ω 훽𝑖 L 2푣표 푉표 푋

Let βi = βo = β

2 훽 휔 퐿 퐽1 푋 퐼푂 Gm = (5.35) 푣표 푋 푉표

Let Ro be the dc beam resistance of buncher cavity.

Let Go be the dc beam conductance of buncher cavity.

퐼푂 푉푂 Go = and Ro = 푉표 퐼푂

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2 훽 휔 퐿 퐽1 푋 Gm = Go (5.36) 푣표 푋

Gm 훽 2휔 퐿 퐽 = 1 푋 (5.37) Go 푣표 푋

Gm The maximum value of is obtained by J1(X) = 0.582 for X – 1.841 and β = 1 Go

Gm 0.316휔 퐿 max= (5.38) Go 푣표

Voltage gain Av:

푉2 Av = substituting for V2 from equation 5.30 푉1

푉2 2 훽표 퐼표 퐽1 푋 푅푠ℎ Av = = 푉1 푉1

substituting for V1from equation 5.34, and taking βi = βo = β

훽표 퐼표 퐽1 푋 푅푠ℎ Av = ω 훽𝑖 L (5.39) 푣표 푉표 푋

ω훽𝑖L Since = 휃표 푣표

2 훽 휃표 퐽1 푋 퐼푂 Av = 푅푠ℎ (5.40) 푋 푉표

Substituting for Gm from equation 5.35

Av = Gm 푅푠ℎ (5.41)

Multi Cavity Klystron

 Typical gain of 2-cavity klystron is 30 dB. This gain is not adequate in many applications  In order to achieve higher gain, several two cavity resonant tubes are connected in cascade in which output of each of the tubes is fed as input to the following tube.

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Unit-V Microwave Engineering

 The intermediate cavities are place at a distance so that the bunching parameter X = 1.841 with respect to the previous cavity.  The intermediate cavity acts as a buncher with the passing electron beam inducing a more enhanced RF voltage than the previous cavity, which in turn sets up an increased velocity modulation.  Typical gain achievable by a multi cavity klystron is of the order of 50 dB with bandwidth of about 80 M Hz.  A multi cavity klystron amplifier produces high gain and narrow bandwidth if all the cavities are tuned to the same frequency.  When each of the cavities are tuned to slightly different frequencies (Staggered tuning), the bandwidth will appreciably increase but at the cost of the gain.

Schematic diagram of a 4 –cavity klystron

Beam current density in klystrons and plasma frequency

While carrying out the mathematical analysis of 2-cavity klystron earlier, the space charge effect (mutual repulsive forces between the electrons) was neglected. This is acceptable for a low power amplifier with small density of electrons in the beam.

However, when high power klystron tubes are analysed, the electron density in the beam is large and forces of mutual repulsion of electrons cannot be neglected.

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Unit-V Microwave Engineering

When the electrons perturbate (oscillate) in the electron beam, the electron density consists of a dc part t RF puturbation caused by the electron bunches. The space change forces within electron bunch vary with shape and size of an electron beam.

Mathematically, the charge density and velocities of perturbations are given bye

Charge density = ρ = - B cos (βez) cos (wqt+ θ) (5.42)

Velocity perturbation = v = - C sin (βez) sin (wqt+ θ) (5.43)

Where

B = constant of charge perturbation C= constant of velocity perturbation

βe = ω / v0 = is the dc – phase constant of electron beam

ωp = Plasma frequency e   0 p (5.44)  0 m

ωq = R. ωp is the reduced plasma frequency θ = phase angle of oscillation

ρ0 = dc electron charge density R = Reduction factor

v0= dc electron velocity

ρ = Instantaneous RF charge density 2- cavity klystron as on oscillator

 An amplifier can be used as on oscillator with appropriate feedback

 The 2-cavity klystron oscillator is obtained by simply providing a feedback loop between the simply providing a feedback loop between the input and output cavities of the klystron.

 The condition for sustained oscillations is       2n rad 2 (5.45) Where θ = Total phase shift in the resonators and the feedback cable     phase angle between buncher and catcher voltages 2 n = an integer

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Unit-V Microwave Engineering

If the two resonators oscillate in phase i.e. θ = 0 then   2n   for max power output 2

 If the resonators are de-funed, the oscillations can be obtained over a wide range of frequencies

 The frequency stability of oscillator is obtained by (a) controlling the temperature of resonation (b) Use of regulated power supplies

Solved problems on 2-cavity Klystron eg1.The parameters of a two cavity Klystron are: (May 2009) Input power = 10 mW Voltage gain = 20 dB

Rsh(Input cavity) = 25 kΩ

Rsh(Output cavity) = 35 kΩ Load resistance = 40 kΩ Calculate (a) Input voltage (b) Output voltage (c) Power output

Solution

From input ac equivalent circuit

2 푉1 Pac in = 푅푠ℎ𝑖푛

V1 = 푃푎푐𝑖푛 ∗ 푅푠ℎ𝑖푛 = 250 = 15.81 V

푉2 Av = 20 log = 20 dB 푉1

푉2 푉2 log = 1, = 10, V2 = 158.1 V 푉1 푉1

푉2 Power output = 2 = 1.339 W 푅퐿 ||푅푠ℎ표푢푡 eg2: A 2-cavity Klytron amplifier has following parameters (May 2007)

Vo = 1200 V Io = 25 mA Ro = 30 kΩ f = 10 GHz

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Unit-V Microwave Engineering

d = 1 mm L = 4 cm Rsh = 30 kΩ Calculate (a) Input voltage for max output (b) Voltage gain (c) Klystron efficiency

Solution 3.682v V L  0 0 optimum  V i 1

3.682 푣표 푉표 ∴ V1 = 휔훽𝑖 퐿 v  0.593106 V  0.593106  1200 0 0 = 0.2054 X 108 m/s  d 2 10109 103     3.058rad g v 2.054107 0 sin 2 sin1.529   g   0.653 i  2 1.529 g 3.682 푣표 푉표 V1 = X = 55.3 V 휔훽𝑖 퐿

V2  20 I0 J1 (X )Rsh

= 2 X 0.653 X 25 X10-3 X 0.582 X 30 X 103 = 570.069 V

푉2 Av = 20 log = 20.26 dB 푉1 V    J (X ) 2  0 1 V 0

0.653 푋 0.582 푋 570.069 = 0.1805 or 18.05% 1200

eg 3: A 2-cavity klystron amplifier is tuned at 3 G Hz. The drift space length is 2 cm and beam current is 25 mA. The Catcher voltage is 0.3 times the beam voltage and β = 1. Calculate

(a) Power output and efficiency for N = 5.25 (b) Beam voltage, input voltage and output current for N=5.25 Solution

휔 퐿 Θo = = 2πN 푣표

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휔 퐿 v  0.593106 V = 0 0 2πN

휔 퐿 2 V0 = [ ] = 371.4 V πN X 0.593 X 106

V2 = 0.3 V0 = 111.4 V

From Eqn 5.32 we have

V1 1.841 ( )max = = 0.11 V0 N

V1 max = 0.11 x 371.4 = 41.46 V

From equation 5.29 Poutput   I J (X )V 0 0 1 2 = 25 X 10-3 X 0.582 X 111.4 = 1.62 W

Pout 1.62퐿    = 0.174 or 17.4% 371.4 X 25 X 10−3 V0 I 0 eg4: A 2-cavity klystron operates at 5 GHz with a dc voltage of 10 kV and a 2 mm cavity gap. For a given RF voltage the magnitude of gap voltage is 100 volts Calculate

(a) Transit angle (b) Velocity of electron leaving the gap

6 8 v0  0.59310 10000 = 0.593 X 10 m/s

d    33.7 X 10-12 s vo

 d 2 5109  2x 103    1.058 rad g v v 0 0 sin 2   g  0.954 i  2 g

From equation 5.11 and 5.12 we have

  V  0.954x100  i 1    8 vmax  v0 1   0.593 X 108 1  = 0.5986 x 10 m/s  2V0   2x10000 

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Unit-V Microwave Engineering

  V   0.954x100  8 v  v(t )  v 1 i 1   0.593 X 108 1  =0.59 x 10 m/s min 1 0  2V  2x10000 lllly  0    eg 5: : A four cavity klystron amplifier has the following parameters

V0 = 18 kV , I0 = 2.25 A , d = 1 cm

f = 10GHz, V1 = 10V (rms) , βi = β0 = 1 -8 3 ρ0 = dc electron beam density = 10 C/ m Determine

1. dc electron velocity ‘υ0’ 2. dc electron phase constant ‘βe’ 3. plasma frequency ‘ωp’ 4. reduced plasma frequency ‘ωq’ for R = 0.5 5. Reduced plasma phase constant ‘βq’ 6. Transit time across input gap

Solution

(a) The dc electron velocity is 6 8 0  0.59310 18,000  0.79610 m/ s

(b) The dc electron phase constant is  2 10109     7.89102 rad / m e  0.796108 0

(c) The plasma frequency 11 8 1/ 2 e0 1.759 10 10  7  p    12   1.4110 rad / s o m  8.85410  (d) The reduced plasma frequency 7 7 ωq = R. ωp = 0.5 x 1.41 x 10 = 0.705 x 10 rad / s

(e) The reduced plasma phase constant

 0.705107   q   0.088 rad / m q  0.796108 0

(f) Transit time across the gap

d 102    8  0.1256 ns 0 0.79610

Eg 6: A 2-cavity klystron operates at 10 GHz with Department of ECE, CMR College of Engineering & Technology Page 23

Unit-V Microwave Engineering

V0 = 10 kV , I0 = 3.6 mA , L= 2 cm

Gshout = 20 μ mhos, β = o.92, Rshin = 80 k ohms Findout (a) Max voltage gain (b) Max Power gain

Solution

6 8 0  0.59310 10,000  0.59310 m/ s

 L 2 10109  2x 102 o    21.187 rad v v 0 0

Rshout = 1/ Gshout = 50 k Ohms

For Max Av, J1(X) = 0.582 and x = 1.841, From equation 5.40

2 훽 휃표 퐽1 푋 퐼푂 .92 푥 .92 푥 21.187 푥 .582 푥 3.6 푥 50 Av = 푅푠ℎ = = 1.020 푋 푉표 18410

2 ퟏ.ퟎퟐ 풙 ퟏ.ퟎퟐ 풙 ퟖퟎ Power gain = Av Rshin /Rshout = = 1.664 ퟓퟎ

Eg 7 : 2-cavity klystron operates at 4 GHz with V0 = 1 kV , I0 = 22 mA , d= 1

mm, Rshout = 20 μ mhos, L = 3 cm . If the dc beam conductance and catcher cavity total equivalent conductance are 0.25 x 10-4 mhos and 0.3x 10-4 mhos respectively

Findout (a) Beam coupling coefficient (b) dc transit angle in the drift space

© Input cavity voltage V1 for max V2 (d) Voltge gain and efficiency (neglecting beam loading)

Solution

6 8 0  0.59310 1000  0.18810 m/ s

 d 2  4109 1x 10  3    1.336 rad g v v 0 0

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 L 2  4109 3x 102 o    40.1 rad v v 0 0

3.682 푣표 푉표 V1 = 푋 = 99.05 V considering X= 1.841 휔훽𝑖 퐿

From equation 5.41

2 2 훽 휃표 퐽1 푋 퐼푂 훽 휃표 퐽1 푋 퐼푂 Av = 푅 = = 7.988 푋 푉 푠ℎ 푋 퐺 푉 표 푠ℎ 표

V2 = Av V1 = 7.988 x 99.05 = 791.2 V

From equation 5.31

V2   0 J1(X ) V0 Considering J1(X) = 0.582

= 0.4268

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Unit-V Microwave Engineering

REFLEX KLYSTRON

Reflex klystron is a single cavity law power microwave oscillator. The characteristics of Reflex Klystron are

Power output: 10- 500mw Frequency range: 1 to 25 GHz Efficiency: 10-20% Applications

1. Widely used in the as a source for microwave experiments

2. Local oscillator in microwave receivers

The theory of the 2-cavity klystron can be applied to the analysis of Reflex klystron with slight modifications

Fig 5.6: Schematic digram of Reflex Klystron

The components of Reflex Klystron are

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1. Cathode

2. Anode grid

3. Cavity resonator at potential of +v0 w.r.t cathode

4. Repeller at potential of - vr w.r.t. cathode

Formation of electron beam with uniform velocity v0 up to civility resonator is similar to that of 2-cavity klystron

6 v0  0.59310 V0 m/ s

Due to dc voltage in the cavity circuit, RF noise is generated in the cavity. This em noise field in the cavity get pronounced at cavity resonant frequency and acts as a

small signal microwave voltage source of V1 sin wt.

The electron beam with uniform velocity v0 when enters the cavity undergoes velocity modulation as in the case of 2-cavity klystron.

Let t0 = time at which electron enters the cavity gap

t1 = time at which electron leave the cavity gap d = cavity gap Z = Axis as shown in schematic diagram Z = 0 at the input gap of cavity Z = d at the output gap of cavity Z = L at the reseller From equation 5 of 2 cavity klystron

 iV1   g   (t1 )  0 1 sinwt1   2V  2   0  

Some electrons are accelerated by the accelerating field (during +ve cycle of RF field) and enter the repeller space with greater velocity compared to the electrons with unchanged velocity, some electrons are decelerated by the decelerating field (during –ve cycle of RF field) and enter repeller space with less velocity All the electrons entering repeller space are retarded by the repeller which is at a –ve

potential of -vr. All the electrons are turned back and again enter the cavity in a

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bunched manner. The bunch re enter the cavity and when field in the cavity is a retarding field bunches convey kinetic energy to the cavity. The cavity converts this kinetic energy into electron magnetic energy at the resonant frequency resulting in

the sustained oscillations and therefore the output of the cavity is V1 sin ωt

Fig 5.7: Applegate diagram of Reflex Klystron

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Let „b‟ be the reference electron at t = t2 for our analysis. Electron „b‟ is passing through the cavity gap while the field is zero (-ve shape) when the electrons a,b,c… leave the cavity i.e. at z = d, the velocity is given by equation

 iV1   g   (t1 )  0 1 sinwt1   2V  2   0   (5.46)

These electrons are subjected to retarding field due to repeller voltage during the drift space from z = d to z = L. the retarding field in the drift space is given by

V V V sin wt E  r 0 1 L (5.47)

The force equation for an electron in the repeller region is given by

2 d z Vr V0 V1 sin wt m 2  eE  e dt L

Since V1<

2 d z Vr V0  m 2  e dt L (5.48)

Integrating the above equation

Where

K1 = integration constant

t0 = Time at the electron enters the gap

t1 = Time at the electron leave the gap

t2 = Time at the electron re-enters the gap due to retarding field

at t = t1, z = d, v(t1) = dz / dt

K1 = dz/ dt = v(t1) Integrating the above equation once again

 eV V  t t z  0 r (t  t )dt  v(t )dt mL  1  1 t1 t1

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V0 Vr  2 z  e (t t1)  v(t1)(t t1)  K2 2mL

At t= t1, z= d

K2 = d

V0 Vr  2 z  e (t  t1 )  v(t1 )(t  t1 )  d 2mL (5.49)

At t = t2 electrons returns of cavity after retardation at t = t2, z=d substituting this in above equation.  eV V  d  0 r (t  t )2  v(t )(t t )  d 2mL 2 1 1 2 1  eV V  o  0 r (t t )2  v(t )(t  t ) 2mL 2 1 1 2 1

Let T be rannd trip transit time = t2-t1

 eV0 Vr  2  O  (t2  t1) (t2  t1)  v(t1)  2mL 

e V0 Vr  (t2  t1 )  v(t1 ) 2mL (5.50) 2mL T' t2  t1  v(t1 ) eV0 Vr 

 iVi  g   T0 '1 sin wt1    2V0 2  2mL where T ' v 0 eV V  0 0 r (5.51)

TO‟Is the round trip transit time of electron „b‟ which is learning the cavity at velocity v(t1) = v0

T‟0 is a function of Vr

ω(t2-t1) = ωT

 V    i i  g  ' T' T'0 T'0 sinwt1   2V0  2 

Let ωT0 = θ0, From equation 5.21 we have Department of ECE, CMR College of Engineering & Technology Page 31 iVi X ' 0 ' 2V0 Unit-V Microwave Engineering

   ' ' X 'sinwt  g ' 0  1 2    (5.52)

Where X‟ is bunching parameter of Reflex Klystron

Power output and efficiency of Reflex Klystron

In case of 2-cavity klystron, we had seen that the maximum transfer of kinetic energy to the cavity takes place when the electron bunch enters when the field is –ve peak

Similarly in the case of reflex Klystron, the bunch must enter cavity when the field is +ve peak. (This is because the direction of electron bunch entering into the cavity is 1800 opposite to that of 2-cavity Klystron)

Considering the above condition we can see from the applegate diagram that the round trip transit time of reference electron is

 1   (t2  t1 )  T0  n  2  2n   2N  4  2 (5.53)

When n = 1,2,3……………….

Let N= n-1/4 is called the mode number

Therefore θ0 = 2πN

Applying the same analogy of 2-cavity klystron and using equation 25, the current in the cavity can be expressed as  ' i2  I 0  2I 0 J'n (n )cos (t2 0  g  (5.54) n1 The fundamental component of current in the cavity if at n =1 is i   i  2I  J (1 )cos(t  ) f i 2 n1 0 i 1 2 0 (5.55)  g 0

Maximum magnitude of fundamental component current in the cavity I2 1 I 2  2I 0 i J1 ( ) (5.56)

V2 = output voltage of the cavity = V1 (except for the phase difference)

1 V1 V2  2I0i J1( )Rsh (5.57)

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2  2I  J (1)  2  0 i 1  output power  Paz  irms.Rsh    Rsh (5.58)  2 

2 2 2 1 Pac  2I 0 i J1 ( )Rsh (5.59)

1 Pac  V1.I 0 i J1 ( ) (5.60)

We have earlier seen that

1 iV1 iV1    X  0  2n   2V0 2V0  2 

V 2X 1 1  (5.61) V0    i 2n    2 

P V I  J (1 ) ac  Power Efficiency    1 0 i 1 (5.62) Pdc V0 I 0

V  J (1 ) 2X 1 J (1 )  1 . i 1  1 V I    0 0 2n    2 

2X 1 J (1 )   1    2n    2  (5.63)

1 1 1 The product X1J1(X ) is maximum at X = 2.408, J1(X ) = 0.52

1 1 1 X J1(X )max=1.25 at X =2.408

21.25   max    2n    2 

At n = 2 (n=1 too short a value)

ηmax = 0.227 or 22.7 (5.64)

From equation 5. 50 we have

2mLv0 2e T'0  where v0  V e(V0 Vr ) m 0

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From equation 5.53, ωT‟0 = 2πn-π/2

2mLv 2mL 2e V  T  0  . 0  2n  0 e(V V ) e m V V 2 0 r 0 r (5.65)

2    2n   V 2 e 0    . (V V ) 2 8 2 L2 m 0 r (5.66)

The above equation gives relationship between V0, Vr and „n‟ for given V0, n= f(Vr)

2X 1V From equation 5. 60 we have V  0 1    i 2n    2 

1 From equation 5.58, Pac = V1I0βiJ1(X )

2X 1V I  J (1) 2V I X 1J (1) P  0 0 i 1  0 0 1 (5.67) ac     i 2n   2n   2  2

Substituting for 2πn-π/2 from equation 5.65

2V I X 1J (1) m eV V   0 0 1 . 0 r 2mL 2e V0

1 1 V0I0 X J1( )V0 Vr  e power output  Pac  (5.68) L 2mV0

Equation 5.68 gives relationship between relationship between

1. Vr and Pac 2. ω and Pac

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Fig 5.8: Power output and frequency characteristics of reflex klystron

Power output frequency as a function of Vr for various modes

The above fig gives the characteristic of Reflex Klystron

Electronic Admittance of Reflex Klystron

We have seen earlier that the fundamental component of current in the cavity is

1 I f  2I0 i J1 ( )cos(t2 0 )

The above equation in the phasor form is

I  2I  J (1 )e j0 f 0 i 1 (5.69)

There is a phase difference of π/2 between the voltages V1at t= t0 and V2 at t= t2

-ejπ/2 V2=V1 (5.70)

The electronic admittance Ye is defined as the ratio of if to V2

Ye = if / V2 (5.71)

From equation 5.69, 5.70 and 5.71

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1    j 0 . I 0 2 J1 ( )  2  Ye  i '0 . 1 e V0 X

I0 where Y0  =dc beam admittance V0

2 1    j 0 . i '0 J1 ( ) 2 Y  Y . e   (5.72) e 0 X 1

Ye is a complex quantity and can be written as Ye = Ge + j Be

It can be seen that the amplitude of electronic admittance is proportional to dc beam admittance and also transit angle or N

The equivalent circuit of Reflex Klystron is given by

Fig 5.9: Equivalent circuit of reflex klystron

Where Gc = copper loss conductance

Gb = Beam loading conductance

GL = Load conductance

L and C are energy storage elements of the cavity

G = Conductance of the cavity including external load =

G = Gc + Gb + GL = 1 / Rsh (5.73)

Equation 5.69 can also represented as

Ye = Ge + jBe (5.74)

Where Ge = Electronic conductance

Be = Electronic Sustenance

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The necessary condition for oscillations is that the magnitude of the real part of the admittance „Ge‟ should not be less than total conductivity of the cavity „G‟

|-Ge| ≥ G (5.75)

This condition is pictorially represented in the figure below

Fig 5.10: Electronic Admittance spiral of reflex klystron

Electronic and mechanical tuning of Reflex Klystron

For using the device as varying frequency oscillator

Electronic tuning is possible by adjustment of repeller voltage „Vr‟. the tuning range is about ± 8 MHz in X – band. For higher bands tuning to an extent of ±80 MHz is practicable.

From equation 6.66, we have

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2 2 8mL V0 2 (V0 Vr )  2     2n   .e  2  (5.76)

Differentiating Vr w.r.t ω

2 dVr 16mL V0 2(Vr V0 )  2  d    2n   .e  2 

2 dVr 8mL V0 1 .  2 d    Vr V0 e2n    2 

Substituting for Vr V0 from equation 5.76 above

   2 2n   dVr 8mL V0  2  e  2 d    2L 2mV0 e2n    2  dV 4L V m L 8mV r  . 0  0 d  2e  e 2n  2n  2 2   2f dV 2L 8mV r  0 df  e 2n  2 (5.77)

dVr df  2L 8mV0  e 2n  2 (5.78)

Equation 5.78 gives the relation between variation Vr and the resulting variation in the frequency.

Mechanical Tuning of Reflex Klystron

The resonant frequency of the cavity can be adjusted using following two methods

1. The frequency of resonance is mechanically adjusted by adjustable screws using the method called post.

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2. The walls of the cavity are moved slightly in and out by means of a adjustable screw which inturn tightens or loosens small bellows. This will result in variation of dimensions of the cavity and then the resonantly frequency

Solved problems on Reflex Klystron

Eg1: The parameter gives for a reflex Klystron are

Vr = 2kV, V0 = 500V , L = 2cm, n=1, f = 2 GHz

Find at the variation in frequency df for dVr factor = 0.02 or 2% from equation 5.78

0.02 2000 df  2  2102 8 91031 500  1.61019 2  2 df  10MHz

Eg 2. A Reflex klystron operates at the peak mode of n=2

And V0 = 300V

I0 = 20mA

V1 = 40V

Calculate (a) input power in watts ‘Pdc’

(b) output power in watts ‘Pac’

(c) Efficiency n

Solution

(a) Input power in watts -3 Pdc = V0I0 = 300 x 20 x 10 = 6 W

(b) Output power in watts, using equation 5.67

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2V I X 1 J (1 ) P  0 0 1 ac  2n  2 1 1 Assu min g X J1 ( )  1.25 max 2 300 20103 1.25 P   1.36walts ac  4  2 (c)Efficiency „n‟ P 1.36   ac   0.2267 or 22.67% dc 6.0

Eg 3. Given parameters of a reflex klystron

V0 = 400Vm Rsh = 20 kΩ, f = 9GHz L = 1mm, n=2

find (a) Repeller Voltage Vr (b) Efficiency ‘η’

Solution : from equation 5.76 2L 2V m V  V  . 0 r 0  2n  e 2 2 9109 103 29103  400 V  400  . r  19 4  1.610 2 36 106  20 2 0.5691011 V  400  . r 7 2 720106  0.11381010 Vr  400  7 2

Vr  1093.9V

(c) Efficiency η From equation – 5.63 2X 1J (1)   1  2n  2 Assu min gX 1J (X )   1.25 max 21.25    0.227 or 22.7 3.5

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Unit-VI Microwave Engineering

UNIT-VI

HELIX TWT AND M-TYPE TUBES

Helix Travelling Wave Tubes

In the previous unit klystrons and reflex klystrons were analysed in some detail. When it comes to study of TWTs it is appropriate to compare the basic operating principles of both TWT and the klystron. In the case of TWT, the microwave circuit is non-resonant and the wave propagates with the same speed as the electrons in the beam. The initial effect on the beam is a small amount of velocity modulation caused by the weak electric fields associated with the traveling wave. Just as in the klystron, this velocity modulation later translates to current modulation, which then induces an RF current in the circuit, causing amplification. However, there are some major differences between the TWT and the klystrons.

1. The interaction of electron beam and RF field in the TWT is continuous over the entire length of the circuit, but the interaction in the klystron occurs only at the gaps of a few resonant cavities. 2. The wave in the TWT is a propagating wave; the wave in the klystron is not. 3. In the couple cavity TWT there is a coupling effect between the cavities, whereas each cavity in the klystron operates independently.

A traveling-wave tube (TWT) is a Microwave Amplifier with following characteristics:

1. Low Power Amplifier: up to 10 kW 2. Frequency Range: 3 G Hz – 50 G Hz 3. Wide Band width: about 800 MHz 4. Power gain: upto 60 dB 5. Efficiency: 20 – 40 %

Comparison between TWT and Klystron Amplifier

S.No. Klystron Amplifier TWT Amplifier 1 Linear beam „O‟ Type Linear beam „O‟ Type 2 Uses input and output resonant Uses non-resonant wave circuit cavities 3 Narrow band amplifier Wide band amplifier BW 800 MHz 4 Interaction between electrons Longer interaction and the field is very short 5 Non propagating wave Propagating wave

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The TWT operates on the principle of slow wave. It is a not resonant „O‟-Type microwave device. Its operation is base on the interaction between the waves in the travelling wave structure and the electronic beam. The main elements of the TWT Amplifier are:

(1) Electron gun; (2) RF input (3) Magnets (4) Attenuator (5) Helix coil (6) RF output (7) Vacuum tube (8) Collector

Fig 6.1: Cutaway view of a TWT.

Description

The device is an elongated vacuum tube with an electron gun (a heated cathode that emits electrons) at one end. A magnetic containment field around the tube focuses the electrons into a beam, which then passes down the middle of a wire helix that stretches from the RF input to the RF output, the electron beam finally striking a collector at the other end. The applied RF signal propagates around the turns of the helix and produces and electric field at the center of the helix, with direction of propagation along helix axis.

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Fig 6.2: Helix Traveling Wave Tube (a) Schematic Diagram (b) Simplified Circuit

Fig 6.3: Helical Slow Wave Structure (a) Helical Coil (b) One turn of helix Department of ECE, CMR College of Engineering & Technology Page 3

Unit-VI Microwave Engineering

This is termed as O-type traveling wave tube. The slow-wave structure is either the helical type or folded back type. The applied signal propagates around the turns of the helix and produces an electric field at the center of the helix, directed along the helix axis. The helix acts as a delay line, in which the RF signal travels at near the same speed along the tube as the electron beam. The axial electric field progresses with a velocity that is very close to the velocity of light multiplied by the ratio of helix pitch to helix circumference. When electrons enter the helix tube, an interaction takes place between the moving axial field and the moving electrons. On the average, the electrons transfer energy to the wave on the helix. This interaction causes the signal wave on the helix to become larger. The electrons entering the helix at zero fields are not affected by the signal wave; those electrons entering the helix at the accelerating field are accelerated, and those at the retarding field are decelerated. As the electrons travel further along the helix, they bunch at the collector end. The bunching shifts the phase by π/2. Each electron in the bunch encounters a stronger retarding field. Then the microwave energy of the electrons is delivered by the electrons bunch to the wave on helix. The amplification of the signal wave is accomplished.

An attenuator placed on the helix, usually between the input and output helices, prevents reflected wave from traveling back to the cathode and there by suppresses the oscillations if any.

Higher powered TWT‟s usually contain beryllium oxide ceramic as both a helix support rod and in some cases, as an electron collector for the TWT because of its special electrical, mechanical, and thermal properties.

Slow wave structures

Slow-wave structures are special circuits that are used in microwave tubes to reduce the wave velocity in a certain directions so that the electron beam and the signal wave can interact. The phase velocity of a wave in ordinary waveguides is greater that the velocity of light in a vacuum. In the operation of traveling wave and magnetron type devices, the electron beam must keep in step with the microwave signal. Since the electron beam can be accelerated only to velocities that are about a fraction of the velocity of light, a slow wave structure must be incorporated in the microwave devices so that the phase velocity of the microwave signal can keep pace with that of the electron beam for effective interactions. Several types of slow-wave structures are shown in the figure given below.

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Unit-VI Microwave Engineering

Fig 6.4: Slow wave structures (a) Helical line (b) Folded-back line (c) Zigzag line (d) Inter-digital line (e) Corrugated waveguide.

Traveling-wave tube amplifier vp = phase velocity of the em wave along the axial direction = ω / β p = helix pitch d = diameter of the pitch ψ = pitch angle β = phase constant along the axis. ω = angular frequency of the wave. vg = Group velocity of the em wave.

βo = phase constant of dc electron beam along the axis.

βe = phase constant of the velocity modulated electron beam along the axis. 1/2 vo = velocity of dc electron beam = [2eVo/m] l = length of the slow wave structure. 푙 N = 휆푒

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Unit-VI Microwave Engineering

Applications

1. TWTAs are commonly used as amplifiers in satellite transponders, where the input signal is very weak and the output needs to be high power. 2. TWT is used as transmitter amplifier particularly in airborne and shop borne fire- control radar systems, Satellites, and in electronic warfare and self-protection systems. In these types of applications, a control grid is typically introduced between the TWT's electron gun and slow-wave structure to allow pulsed operation. The circuit that drives the control grid is usually referred to as a grid modulator.

Another major use of TWTAs is for the electromagnetic compatibility (EMC) testing industry for immunity testing of electronic devices.

Mathematical Analysis

The phase velocity „Vp „of em wave in the axial direction is given by

풑 Vp = 푐 = c.sin ψ (6.1) 풑ퟐ+(흅풅)ퟐ

If the helix is filled with dielectric material, then

풑 Vp = 푐 (6.2) 흁흐 풑ퟐ+(흅풅)ퟐ

For small pitch angle, p << d, the eqn 6.1 becomes 풑 Vp = 푐 (6.3) 흅풅

If λa is the wavelength along the axial direction

ퟐ흅훌퐚 퐟 훌퐚 훚 Vp = λa.f = = 2휋푓 = (6.4) ퟐ흅 ퟐ흅 휷

The group velocity Vg is given by

훛훚 Vg = (6.5) 흏휷

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Unit-VI Microwave Engineering

Fig 6. 5: Brillouin Digram of Helical structure

Fig 6.5 shows the ω-β or Brillouin diagrm for a helical slow wave structure. The diagram is very useful in designing a helix slow wave structure. Once β is found, vp can be computed from eqn 6.4 for a given dimension of the helix. The group velocity of the wave is merely the slope of the curve given by eqn 6.5.

The motion of electrons in the helix type TWT can be quatitatively analyzed interms of the axial electric field. If the travelling wave is propagating in the z-direction.

Ez = z-component of electric field = E1 sin(ωt –βpz) (6.6)

Where βp = axial phase constant of em wave

훚 βp = 푽퐩

The equation of motion of electron is given by

퐝퐯 m = - e E1 sin(ωt –βpz) (6.7) 퐝퐭

The electron velocity is given by dc + ac components

v = v0 + ve cos (ωet +θe) (6.8)

Where

v0 = dc elctron velocity

ve = magnitude of velocity fluctuation

ωe = Angular frequency of velocity fluctuations

θe = Phase angle of fluctuations

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Unit-VI Microwave Engineering

퐝퐯 = - ve ωe sin(ωet +θe) (6.9) 퐝퐭

Substituting eqn 6.9 in eqn 6.7

m ve ωe sin(ωet +θe) = e E1 sin(ωt –βpz) (6.10)

z = distance travelled by the electrons = vo (t - t0) (6.11)

Substituting eqn 6.11 in eqn 6.10

m ve ωe sin(ωet +θe) = e E1 sin{ωt –βp vo (t - t0)} (6.12)

m ve ωe sin(ωet +θe) = e E1 sin{(ω –βpvo)t - βpvo t0)} (6.13)

Comparing LHS and RHS terms of above equation we can write

퐞퐄ퟏ ve = (6.14) 풎흎풆

ωe = βp(vp - vo) (6.15)

θe = βpvo t0 (6.16)

It can be seen from equation 6.14 that the magnitude of velocity fluctuation of the electron beam is directly proportional to the magnitude of the axial electric field.

Convection current

The convection current „i‟ induced in the electron beam by axial electric field is given by (Derivation not in the purview of the syllabus)

βe 퐈퐨퐄ퟏ i = 푗 2 (6.17) ퟐ 푽퐨(풋βe−γ )

Where Io = dc current

푉o = dc voltage

훚 2eV0 Where βe = , v0  풗퐨 m

and γ= αe +j βe propagation constant of axial wave

Axial Field

Equivalent circuit of slow wave structure is given by

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UNIT IV

Contents:  Cross-field Tubes  Introduction  Cross field effects  Magnetrons-different types, cylindrical travelling wave magnetron-Hull cutoff and Hartree conditions, modes of resonance and PI- mode operation, separation of PI-mode, O/P characteristics.  Microwave Semiconductor Devices:  Introduction to Microwave semiconductor devices, classification, applications  Transfer Electronic Devices, Gunn diode - principles, RWH theory, Characteristics, Basic modes of operation - Gunn oscillation modes, LSA Mode, Varactor diode, Parametric amplifier, Introduction to Avalanche Transit time devices (brief treatment only), Illustrative Problems

Introduction: Magnetron is a grouping of a simple diode vacuum tube together with built in cavity resonators and an exceptionally powerful magnet. There are three types of magnetrons:

. Negative resistance type . Cyclotron frequency type . Travelling wave or Cavity type Negative resistance magnetrons make use of negative resistance between two anode segments. Cyclotron frequency magnetrons depends upon synchronism amid an alternating component of electric field and periodic oscillation of electrons in a direction parallel to this field. Cavity type magnetrons depends upon the interface of electrons with a rotating electromagnetic field of constant angular velocity. CONSTRUCTION

A magnetron consist of a cathode which is used to release the electrons and number of anode cavities and a permanent magnet is placed on the flipside of cathode and the space between the anode cavity and the cathode is called interacting space. The electrons which are emitted from the cathode moves in diverse path in the interacting space depending upon strength of electric and magnetic fields applied to the magnetron.

Types of Magnetrons

There are three main types of Magnetrons.

Negative Resistance Type

 The negative resistance between two anode segments, is used.  They have low efficiency.  They are used at low frequencies (< 500 MHz).

Cyclotron Frequency Magnetrons

 The synchronism between the electric component and oscillating electrons is considered.  Useful for frequencies higher than 100MHz.

Travelling Wave or Cavity Type

 The interaction between electrons and rotating EM field is taken into account.  High peak power oscillations are provided.  Useful in radar applications.

Cavity Magnetron

The Magnetron is called as Cavity Magnetron because the anode is made into resonant cavities and a permanent magnet is used to produce a strong magnetic field, where the action of both of these make the device work.

Construction of Cavity Magnetron

A thick cylindrical cathode is present at the center and a cylindrical block of copper, is fixed axially, which acts as an anode. This anode block is made of a number of slots that acts as resonant anode cavities.

The space present between the anode and cathode is called as Interaction space. The electric field is present radially while the magnetic field is present axially in the cavity magnetron. This magnetic field is produced by a permanent magnet, which is placed such that the magnetic lines are parallel to cathode and perpendicular to the electric field present between the anode and the cathode.

The following figures show the constructional details of a cavity magnetron and the magnetic lines of flux present, axially.

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This Cavity Magnetron has 8 cavities tightly coupled to each other. An N-cavity magnetron has N modes of operations. These operations depend upon the frequency and the phase of oscillations. The total phase shift around the ring of this cavity resonators should be 2nπ where n is an integer.

If ϕv represents the relative phase change of the AC electric field across adjacent cavities, then

ϕv=2πn/N

Where n=0,±1,±2,±(N/2−1),±N/2

Which means that N/2 mode of resonance can exist if N is an even number.

If,

n=N/2thenϕv=π

This mode of resonance is called as π−mode.

n=0thenϕv=0

This is called as the Zero mode, because there will be no RF electric field between the anode and the cathode. This is also called as Fringing Field and this mode is not used in magnetrons.

Operation of Cavity Magnetron

When the Cavity Klystron is under operation, we have different cases to consider. Let us go through them in detail.

Case 1

If the magnetic field is absent, i.e. B = 0, then the behavior of electrons can be observed in the following figure. Considering an example, where electron a directly goes to anode under radial electric force.

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Case 2

If there is an increase in the magnetic field, a lateral force acts on the electrons. This can be observed in the following figure, considering electron b which takes a curved path, while both forces are acting on it.

Radius of this path is calculated as

R=mv/eB

It varies proportionally with the velocity of the electron and it is inversely proportional to the magnetic field strength.

Case 3

If the magnetic field B is further increased, the electron follows a path such as the electron c, just grazing

the anode surface and making the anode current zero. This is called as "Critical magnetic field" (Bc), which is the cut-off magnetic field. Refer the following figure for better understanding.

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Case 4

If the magnetic field is made greater than the critical field,

B>Bc

Then the electrons follow a path as electron d, where the electron jumps back to the cathode, without going to the anode. This causes "back heating" of the cathode. Refer the following figure.

This is achieved by cutting off the electric supply once the oscillation begins. If this is continued, the emitting efficiency of the cathode gets affected.

Operation of Cavity Magnetron with Active RF Field

We have discussed so far the operation of cavity magnetron where the RF field is absent in the cavities of the magnetron (static case). Let us now discuss its operation when we have an active RF field.

https://www.rgpvonline.com/guide.html Page no: 12 As in TWT, let us assume that initial RF oscillations are present, due to some noise transient. The oscillations are sustained by the operation of the device. There are three kinds of electrons emitted in this process, whose actions are understood as electrons a, b and c, in three different cases.

Case 1

When oscillations are present, an electron a, slows down transferring energy to oscillate. Such electrons that transfer their energy to the oscillations are called as favored electrons. These electrons are responsible for bunching effect.

Case 2

In this case, another electron, say b, takes energy from the oscillations and increases its velocity. As and when this is done,

 It bends more sharply.  It spends little time in interaction space.  It returns to the cathode.

These electrons are called as unfavored electrons. They don't participate in the bunching effect. Also, these electrons are harmful as they cause "back heating".

Case 3

In this case, electron c, which is emitted a little later, moves faster. It tries to catch up with electron a. The next emitted electron d, tries to step with a. As a result, the favored electrons a, c and d form electron bunches or electron clouds. It called as "Phase focusing effect".

This whole process is understood better by taking a look at the following figure.

Phase Focusing Effect

https://www.rgpvonline.com/guide.html Page no: 13 Figure A shows the electron movements in different cases while figure B shows the electron clouds formed. These electron clouds occur while the device is in operation. The charges present on the internal surface of these anode segments, follow the oscillations in the cavities. This creates an electric field rotating clockwise, which can be actually seen while performing a practical experiment.

While the electric field is rotating, the magnetic flux lines are formed in parallel to the cathode, under whose combined effect, the electron bunches are formed with four spokes, directed in regular intervals, to the nearest positive anode segment, in spiral trajectories.

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1. INTRODUCTION

The application of two-terminal semiconductor devices at microwave frequencies has been increased usage during the past decades. The CW, average, and peak power outputs of these devices at higher microwave frequencies are much larger than those obtainable with the best power transistor.

The common characteristic of all active two-terminal solid-state devices is their negative resistance. The real part of their impedance is negative over a range of frequencies.

In a positive resistance the current through the resistance and the voltage across it are in phase. The voltage drop across a positive resistance is positive and a power of (I2 R) is dissipated in the resistance.

In a negative resistance, however, the current and voltage are out of phase by 180°. The voltage drop across a negative resistance is negative, and a power of (-I2R) is generated by the power supply associated with the negative resistance.

In other words, positive resistances absorb power (passive devices), whereas negative resistances generate power (active devices).

The differences between microwave transistors and transferred electron devices (TEDs) are fundamental. Transistors operate with either junctions or gates, but TEDs are bulk devices having no junctions or gates. The majority of transistors are fabricated from elemental semiconductors, such as silicon or germanium, whereas TEDs are fabricated from compound semiconductors, such as gallium arsenide (GaAs), indium phosphide (InP), or cadmium telluride (CdTe).

TEDS operate with ‘hot’ electrons whose energy is much greater than thermal energy.

2.GUNN DIODES-GaAs DIODE

Gunn Diode is a one kind of transferred electronic device and exhibits negative resistance characteristic.

Gunn-effect diodes are named after J. B. Gunn, who in 1963 discovered periodic fluctuations of current passing through then-type gallium arsenide (GaAs) specimen when the applied voltage exceeded a certain critical value.

These are bulk devices in the sense that microwave amplification and oscillation are derived from the bulk negative-resistance property of uniform semiconductors rather than from the junction negative-resistance property between two different semiconductors, as in the tunnel diode.

3. GUNN EFFECT

A schematic diagram of a uniform n-type GaAs diode with ohmic contacts at the end surfaces is shown in Fig.1.

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J. B. Gunn observed the Gunn effect in the n-type GaAs bulk diode in 1963.

Above some critical voltage, corresponding to an electric field of 2000-4000 volts/cm, the current in every specimen became a fluctuating function of time. In the GaAs specimens, this fluctuation took the form of a periodic oscillation superimposed upon the pulse current. The frequency of oscillation was determined mainly by the specimen, and not by the external circuit. The period of oscillation was usually inversely proportional to the specimen length and closely equal to the transit time of electrons between the electrodes, calculated from their estimated velocity of slightly over 107 cm/s.

Figure 1. Schematic diagram for ntype GaAs diode.

From Gunn's observation the carrier drift velocity is linearly increased from zero to a maximum when the electric field is varied from zero to a threshold value. When the electric field is beyond the threshold value of 3000 V/cm for the n-type GaAs, the drift velocity is decreased and the diode exhibits negative resistance.

Figure 2. Drift velocity of electrons in n-type GaAs versus electric field.

4. RIDLEY-WATKINS-HILSUM (RWH) THEORY

The fundamental concept of the Ridley-Watkins-Hilsum (RWH) theory is the differential negative resistance developed in a bulk solid-state III-V compound when either a voltage (or electric field) or a current is applied to the terminals of the sample. There are two modes of

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UNIT II: SEMICONDUCTOR MICROWAVE DEVICES AND APPLICATIONS negative-resistance devices: voltage-controlled and current-controlled modes as shown in Fig.3.a and Fig.3.b

Figure 3. Diagram of negative resistance

In the voltage-controlled mode the current density can be multivalued, whereas in the current- controlled mode the voltage can be multivalued. The major effect of the appearance of a differential negative-resistance region in the current density-field curve is to render the sample electrically unstable. As a result, the initially homogeneous sample becomes electrically heterogeneous in an attempt to reach stability. In the voltage-controlled negative- resistance mode high-field domains are formed, separating two low-field regions. The interfaces separating low and high-field domains lie along equipotential; thus they are in planes perpendicular to the current direction as shown in Fig. 4(a). In the current-controlled negative-resistance mode splitting the sample results in high-current filaments running along the field direction as shown in Fig. 4(b).

Figure 4.Diagrams of high field domain and high current filament.

Expressed mathematically, the negative resistance of the sample at a particular region is

= negative resistance ..(1) = If an electric field E0 (or voltage V0) is applied to the sample, for example, the current density lo is generated. As the applied field (or voltage) is increased to E2 (or V2), the current density is decreased to J2. When the field (or voltage) is decreased to E1 (or V1), the current density is increased to J1 . These phenomena of the voltage-controlled negative resistance are shown in Fig. 5(a). Similarly, for the current-controlled mode, the negative- resistance profile is as shown in Fig. 5(b).

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Figure 5.Multiple values of current density for negative resistance.

5. TWO-VALLEY MODEL THEORY

According to the energy band theory of then-type GaAs, a high-mobility lower valley is

Valley Effective Mass Mobility Separation

Me μ ∆E

separated by an energy of 0.36 eV from a low-mobility upper valley as shown in Fig. 6.

Figure 6.Two-valley model of electron energy versus wave number for n-type GaAs.

TABLE 1. DATA FOR TWO VALLEYS IN GaAs

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Lower Mel = 0.068 μl = 8000 cm2/v-sec ∆E = 0.36 eV

Upper Meu= 1.2 μu = 180 cm2/v-sec ∆E = 0.36 eV

Electron densities in the lower and upper valleys remain the same under an equilibrium condition. When the applied electric field is lower than the electric field of the lower valley (E

Figure 7 Transfer of electron densities.

When the applied electric field is higher than that of the lower valley and lower than that of the upper valley (Ee

..(2) σ = en μ + nμ Where,e = the electron charge

µ = electron mobility n = ne + nu is the electron density

When a sufficiently high field Eis applied to the specimen, electrons are accelerated and their effective temperature rises above the lattice temperature. Furthermore, the lattice temperature also increases. Thus electron density n and mobility µ are both functions of electric field E. Differentiation of Eq. (7-2-2) with respect to E yields

..(3) = e μ + μ + e n + n

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If the total electron density is given by n = and it is assumed that and are n + n μ μ proportional to Ep, where p is a constant, then

..(4) n + n = = 0

..(5) = −

..(6) and ∝ = pE = p ∝ p = μ Substitution of Equation (4) to (6) into Eq. (3) results in

…(7) = e μ − μ + e n μ + nμ

Then differentiation of Ohm's law J = σE with respect to E yields

..(8) = σ + E Equation (8) can be rewritten

/ ..(9) = 1 + / Clearly, for negative resistance, the current density J must decrease with increasing field E or the ratio of dJ/dE must be negative. Such would be the case only if the right-hand term of Eq. (9) is less than zero. In other words, the condition for negative resistance is

/ ..(10) − / > 1 Substitution of Equation (2) and (7) with f = nu/nl results in [2]

– − − p > 1 ..(11) 6. APPLICATION OF GUNN DIODE

In Radar Transmitters (police Radar, CW Doppler radar).

Pulsed Gunn diode oscillators used in transponders, for air traffic control and in industry telemetry system.

Fast combinational and sequential logic amplifier.

As pump source in preamplifier.

In microwave receiver as low and medium power oscillator.

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Domain Formation:

Differential resistance occur when an electric field of a certain range is applied to a multivalley semiconductor, such as then-type GaAs. due to that decrease in drift velocity with increasing electric field. Its leads to formation of a high-field domain for microwave generation andamplification.

In the n-type GaAs diode the majority carriers are electrons. When a small voltage is applied to the diode, the electric field and conduction current density are uniform diode.

σV J = σE = U = ρϑ U L Where

J = conduction current density = conductivity

Ex = electric field in the x direction

L = length of the diode

V = applied voltage

p = charge density

v = drift velocity

U = unit vector

The current is carried by free electrons that are drifting through a background of fixed positive charge.

When the applied voltage is above the threshold value, which measuredabout 3000 V/cm times the thickness of the GaAs diode, a high-field domain is formed near the cathode that reduces the electric field.

V = − E dx

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The high field domain then drifts with the carrier stream across the electrodes and disappears at the anode contact. When the electric field increases, the electron drift velocity decreases and the GaAs exhibits negative resistance.

As shown fig 1(b) below there exists an excess (or accumulation) of negative charge that could be caused by a random noise fluctuation or possibly by a permanent nonuniformity in doping in the n-type GaAs diode.

An electric field is then created by the accumulated charges as shown in Fig 1(d). The field to the left of point A is lower than that to the right. If the diode is biased at point EA on the J- E curve, implies that the carriers (or current) flowing into point A are greater than those flowing out of point A,therefore increasing the excess negative space charge at A. when the electric field to the left of point A is lower than it was before, the field to the right is then greater than the original one, resulting in an even greater space-charge accumulation. process continues until the low and high fields both reach outside the differential negative- resistance region Fig1(a).

This process, depends on condition that the number of electrons inside the crystal is large enough to allow the necessary amount of space charge to be built up during the transit time of the space-charge layer.

The electric field in side the dipole domain would be greater than the fields on either side of the dipole in Fig 2.(c). Because of the negative differential resistance, the current in the low- field side would be greater than that in the high-field side.

Then the dipole field reaches a stable condition and moves through the specimen toward the anode. When the high-field domain disappears at the anode, a new dipole field starts forming at the cathode and the process is repeated.

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Figure 1: formation of an electron accumulation layer in GaAs.

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Figure2: formation of an electron dipole layer in GaAs.

Properties of High field domain

Will start to form whenever the electric field in a region of the sample increases above the threshold E.When the electric field increases, the electron drift velocity decreases and the GaAs diode exhibits negative resistance.

If additional voltage is applied, the domain will increase in size and absorb more voltage than was added and the current will decrease.

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UNIT II: SEMICONDUCTOR MICROWAVE DEVICES AND APPLICATIONS domain will not disappear before reaching the anode unless the voltage is dropped appreciably below threshold.

New domain formation can be prevented by decreasing the voltage slightly below threshold.

Domain will modulate the current through a device as the domain passes through regions of different doping and cross-sectional area,or domain may disappear. Effective doping may varied in region.

The domain length is inversely proportional to the doping. devices with the same product of dopingmultiplied by length will behave similarly in terms of frequency multiplied by length.

Domain can be detected by a capacitive contact.Presence of a domain anywhere in a devicecan be detected by a decreased current.

Modes of operation of Gunn diode

Figure 3: Modes of operation for Gunn diodes

Gunn first announced his observation of microwave oscillation in the n-type GaAs and n-type InP diodes in 1963, various modes of operation have been developed, depending on the material parameters and operating conditions.

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Formation of strong space-charge instability depends on the conditions that enough charge is available in the crystal. four basic modes of operation of uniformly doped bulk diodes with low-resistance contact are as follows

Transit Time Domain mode

Delayed Domain mode

Quenched Domain mode limited space charge accumulation mode

1.Transit TimeDomain mode: in the region where the product of frequency multiplied by length is about 107 cm/s and the product of doping multiplied by length is greater than l012/cm2.the device is unstable because of the cyclic formation of either the accumulation layer or the high-field domain. f = Vd/L in these mode is slightly sensitive to the applied voltages since the drift velocity Vd depends on the bias voltages.

Vd= f .L = 107 cm/s when Vd = VS, thenhigh field domain is stable.

Bias voltage is normally maintain little higher Eth .

At this instant Oscillation period = Transit Time (τo = τt).

Operating ‘f’ depends on ‘Vd’ hence on bias voltage > Eth.

It is a low power,low efficiency mode and requires that operating frequency laser then 30GHz.

These limit on frequency is due to that device length .

2. Delayed Domain mode:This mode is defined in the region where the product of frequency times length is about 107 cm/s and the product of doping times length is between 1011 and l012/cm2.

When transit time is chosen that domain is collected E

Oscillation period is greater than transit time τo < τt

This dealy inhabited mode has an 20 % Efficiency.

Operatingfrequency can be less than or equal to Gunn Mode frequency

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3.Quenched Domain mode: This mode is defined in the region where the product of frequency times length is above 107 cm/s and the quotient of doping divided by frequency is between 2 x 104and 2 x 105.

It is bias field drops below sustaining field Esduring the negative half cycle domain collapses before it reaches the anode.i.e The domain disappear somewhere in the sample itself.

Operating frequency will be higher than Gunn Mode and delayed mode, certainly this depend on the external circuit.

When bias field swings back above threshold value Vth, new domain formed and process repeats,hence in that mode domain is quenched before it reches the anode.

Frequency of resonant circuit then the transit time frequency is 13%.

4.Limited Space Charge Accumulation mode: This mode occurs only when there is either Gunn or LSA oscillation, and it is usually at the region where the product of frequency times length is too small to appear in the figure. When a bulk diode is biased to threshold, the average current suddenly drops as Gunn oscillation begins.

It gives high power upled high efficiency

The domain is not allowed to form

RF voltage and frequency are so chosen that they do not have sufficient time to form domain above threshold.

IN LSA mode high power and high η20%,16 to 23% compare to 5% for gunn mode The field No peak value permits high operating voltage.

Operating frequency is 0.5-50 times more than Gunn Mode.

It can be used up to 100 GHz and is dependent on external resonating circuits.

High Power and High Efficiency.

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Parametric Amplifier

Parametric amplifier first propounded by Lord Reyleigh in 1880’s for mechanical system and by R.Harteley in 1930’s for electrical applications. A parametric amplifier is one that uses non linear reactance or a time varying reactance for its application. In fact parametric devices basically depend on the possibility of increasing the energy of the signal at one frequency by supplying energy at some other frequency.

Consider the simple tank circuit in which we can separate the plates of capacitor used mechanically. Amplification is obtained by pulling capacitor plates apart when ‘V’ and ‘Q’ are at maximum value.

Q ε ε A V = & C = 0 r C d

To obtain amplification capacitor plates are pulled apart when the charge and the voltage are at their maximum. Because of electric field between the plates,it requires an expenditure of energy to pulled the plates apart. This mechanical energy appears as additional electric energy stored in the capacitor.The voltage and charge continue the oscillation towards zero. At zero voltage the capacitance plates are brought back to their original separation and this requires no expenditure of energy as the electric field is also zero now.The voltage and charge now swing to their wave maximum at which plates are pulled apart once again and the process can be continued at each maximum and minimum voltage and hence signal builts up.

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UNIT II: SEMICONDUCTOR MICROWAVE DEVICES AND APPLICATIONS

Fig.1.1 Tank circuit

Varactor diode is the most widely used active element in parametric amplifier. Parametric amplifier is low noise amplifier because no resistance is involved in the amplification process. There will be no thermal noise as the active device involved is reactive. Amplification is obtained if the reactance is varied at some frequency higher than the frequency of the signal being amplified.

1.2 Amplification Mechanism of a Parametric Amplifier :

In a parametric amplifier the pump generator acts as a local oscillator and varactor diode C(t) as the mixer as shown in fig 1.2.

Fig 1.2 Equivalent circuit of parametric amplifier

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UNIT II: SEMICONDUCTOR MICROWAVE DEVICES AND APPLICATIONS

The signal frequency f s and pump frequency f c are mixed in a non linear capacitor C(t) to generate voltages at fundamental frequencies f p and f s as well as the sum and the difference frequencies mf p ± nf s across C(t). An output voltage at f o is obtained across load resistance RL. The output circuit which does not require external excitation is called Idler circuit.The output frequency f o in the Idler circuit is expressed as the sum and the difference frequency of fs and f p.

If f o> f s the device called up converter.

If f o < f p the device called down converter.

1.3 Parametric Up Converter(PUC)

In a PUC the output equal to sum of f s and f p(f o=f s +f p)and there is no power flow in the parametric amplifier at frequency other than the signal,pump output frequencies.

For PUC,we define three parameters the power gain, noise figure and bandwidth.

1.Power Gain: The maximum power gain for PUC is given by Maximum Power gain f x = s 2 fo 1( + 1+ x) where f x = s (γQ)2

fo and f o=f s +f p 1 Q = 2πfCR d where R d= series resistance of pn junction

= figure of merit for non linear capacitance

As R d tends to zero, Q tends to ∞ and gain degradation factor is unity. 2.Noise Figure: The noise figure(F) ,for PUC is given by

2T  1 1  F =1+ d  +  T γQ (γQ)2 o  

Where T d= diode temperature in K

To= ambient temprature(300 K)

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UNIT II: SEMICONDUCTOR MICROWAVE DEVICES AND APPLICATIONS

In typical microwave diode =10,f o/f s=300 K, the minimum noise figure is 0.90 dBwhich is far less as compared to 3 to 4 dB of TWT.

3. Bandwidth: The bandwidth of PUC is given by

γ o fi BW = 2 fs ×Gain For a typical microwave diode f o/fs=10 and = 0.2 and BW=1.26.

1.4 Parametric Down Converter(PDC):

For parametric amplifier to act as a PDC,f s=f p+f 0. This means that input power must feed into the Idler circuit and the output power must move out from the signal circuit.The gain of PDC is given by

Gain(actually loss) f x = s 2 fo 1( + 1+ x)

Expression for Noise figure and BW of PDC is similar to PUC

Negative resistance parametric amplifier

If the significant portion of power flow flows only at f s ,f p and f i , regenrative condition with possibility of oscillation at both signal and Idler frequency will occur.When the mode operates below the oscillation threshold the device behave as bilateral negative resistance para amp.

Degenerate Parametric amplifier :

The degenerate parametric amplifier or oscillator is defined as a negative resistance amplifier with the signal frequency(fs)equals to idler frequency(fi).Since the idler frequency(fi)is the difference between the pump frequency(fp)and the signal(fs),the signal frequency is just half of pump frequency(fp).The bandwidth and gain are same as upconverter.It is a simple device uses a relatively low pump frequency(fp)and low noise figure.

Non-degenerate parametric amplifier :

In this, the signal and idler frequencies are not same but are clearly seperated.The pump frequency need not be multiple of signal frequency .

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UNIT II: SEMICONDUCTOR MICROWAVE DEVICES AND APPLICATIONS

There are three parameters of PDC power gain,bandwidth and noise figure.

1.Power Gain: The output power is taken from the resistance R i at a frequency f i and converter gain from f s to f i is given by

Gain 4 f Rg R α = i i f R R 1( −α)2 s TS Ti Where fi = f P - fS

Rg = Signal generator output resistance

Ri = Idler generator output resistance

Rts = Total series resistance at fS

Rti = Total series resistance at fi

α = R / R ts

2 R = Equivalent Noise resistance γ R = 2 ω ωig C RT i

2. Noise Figure: It is same as PUC.

3.Bandwidth: It is given by γ f BW = o i 2 fs ×Gain

If gain = 20dB, fi = 4 fS and γ =0.3 then BW = 0.03

Advantages and limitations of parametric amp:

1)Noise figure: Typical value 1-2 dB.

2) Frequency:40-200 GHz.

3) Bandwidth: Generally small due to presence of tuned circuits.

4) Gain: typical value 20-80 dB.

Applications of parametric amp:

1)Radio telescopes.

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UNIT II: SEMICONDUCTOR MICROWAVE DEVICES AND APPLICATIONS

2)Space probe tracking and communications.

3)Tropospheric scatter receivers.

4)Long range RADAR.

5)Satellite ground stations.

6)Radio astronomy etc.

Questions asked in university exam:

1. Explain the principle of operation of parametric amplifier with the help of neat diagram. Also explain degenerate and non-degenerate modes of operation.(W-17)

2. Show that negative resistance device when use in system act as amplifier.(W-18)

3. Explain operation of parametric amplifier as up converter and down converter.Why does it offer low noise amplification? (S-16, W-16, W-15)

IMPATT:-

• IMPATT diode stands for Impact Avalanche and Transit Time Diode. • A theoretical Read diode made of an n+ -p-i-p+ or p+ -n-i-n+ structure has been analyzed. • Its basic physical mechanism is the interaction of the impact ionization avalanche and the transit time of charge carriers. Hence the Read-type diodes are called IMPATT diodes. • These diodes exhibit a differential negative resistance by two effects: 1. The impact ionization avalanche effect, which causes the carrier current Io(t) and the ac voltage to be out of phase by 90° 2. The transit-time effect, which further delays the external current I(t) relative to the ac voltage by 90° . • IMPATT diodes consist of a high doping avalanching region followed by a drift region where the field is low enough that the carriers can traverse through it without avalanching. • The Read diode is the basic type in the IMPATT diode family. The others are the one- sided abrupt p-n junction, the linearly graded p-n junction (or double-drift region), and the p-i-n diode . • The principle of operation of these devices, however, is essentially similar to the mechanism described for the Read diode.

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UNIT II: SEMICONDUCTOR MICROWAVE DEVICES AND APPLICATIONS

Fig: Three typical silicon IMPATT diodes.

Negative Resistance: Small-signal analysis of a Read diode results in the following expression for the real part of the diode terminal impedance. R = Rs + ∈ Where Rs = passive resistance of the inactive region.

= carrier drift velocity L = length of the drift space-charge region

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UNIT II: SEMICONDUCTOR MICROWAVE DEVICES AND APPLICATIONS

A = diode cross section

= semiconductor dielectric permittivity ∈

Moreover, ( ) is the transit angle, given by

= = and w, is the avalanche resonant frequency, defined by

′ =

Where , ∝ = derivative of the ionization coefficient with respect to the electric field. • The variation of the negative resistance with the transit angle when is > plotted.

• The peak value of the negative resistance occurs near = . For transit angles larger than and approaching 3 /2, the negative resistance of the diode decreases rapidly.

• The Read-type IMPATT diodes work well only in a frequency range around the transit angle.

i.e.

f = =

Fig: Negative resistance versus transit angle.

Power Output and Efficiency:

The maximum voltage that can be applied across the diode is given by

= L where L is the depletion length and Em is the maximum electric field.

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UNIT II: SEMICONDUCTOR MICROWAVE DEVICES AND APPLICATIONS

The maximum current is given by

= A = A

= ∈ A

= ∈

Therefore the upper limit of the power input is given by

= I

----(1) = ∈ The capacitance across the space-charge region is defined as

C = ∈ ----(2) ∈ = and C =

∈=

Power, =

=

= = 1

= The efficiency of the IMPATT diodes is given by

=

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UNIT II: SEMICONDUCTOR MICROWAVE DEVICES AND APPLICATIONS

= ( )( ) TRAPATT DIODE :-

• The abbreviation TRAPATT stands for trapped plasma avalanche triggered transit mode, a mode first reported by Prager . It is a high-efficiency microwave generator capable of operating from several hundred megahertz to several GHz . • The basic operation of the oscillator is a semiconductor p-n junction diode reverse biased to current densities well in excess of those encountered in normal avalanche operation. High-peak-power diodes are typically silicon n+ -p-p+ (or p+ -n-n+) structures with then-type depletion region width varying from 2.5 to 12.5 µ,m. • The doping of the depletion region is generally such that the diodes are well "punched through" at breakdown; that is, the de electric field in the depletion region just prior to breakdown is well above the saturated drift-velocity level. • The device's p+ region is kept as thin as possible at 2.5 to 7.5 µ,m. The TRAPATT diode's diameter ranges from as small as 50 µ,m for CW operation to 750 µ,mat lower frequency for high peak-power devices.

OPERATION

Fig: Voltage and Current Waveforms for TRAPATT Diode.

• A typical voltage waveform for the TRAPATT mode of an avalanche p+ -n-n+ diode operating with an assumed square wave current drive is shown in Fig. • These analyses have shown that a high-field avalanche zone propagates through the diode and fills the depletion layer with a dense plasma of electrons and holes that become trapped in the low-field region behind the zone.

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UNIT II: SEMICONDUCTOR MICROWAVE DEVICES AND APPLICATIONS

• At point A the electric field is uniform throughout the sample and its magnitude is large but less than the value required for avalanche breakdown. The current density is expressed by = where is the semiconductor dielectric permittivity of the diode. ∈ • At the instant of time at point A, the diode current is turned on. Since the only charge carriers present are those caused by the thermal generation, the diode initially charges up like a linear capacitor, driving the magnitude of the electric field above the breakdown voltage. • When a sufficient number of carriers is generated, the particle current exceeds the external current and the electric field is depressed throughout the depletion region, causing the voltage to decrease. This portion of the cycle is shown by the curve from point B to point C. • During this time interval the electric field is sufficiently large for the avalanche to continue, and a dense plasma of electrons and holes is created. As some of the electrons and holes drift out of the ends of the depletion layer, the field is further depressed and "traps" the remaining plasma. The voltage decreases to point D. • A long time is required to remove the plasma because the total plasma charge is large compared to the charge per unit time in the external current. • At point E the plasma is removed, but a residual charge of electrons remains in one end of the depletion layer and a residual charge of holes in the other end. As the residual charge is removed, the voltage increases from point E to point F. • At point F all the charge that was generated internally has been removed. This charge must be greater than or equal to that supplied by the external current; otherwise the voltage will exceed that at point A. • From point F to point G the diode charges up again like a fixed capacitor. At point G the diode current goes to zero for half a period and the voltage remains constant at VA until the current comes back on and the cycle repeats. The electric field can be expressed as , = − + ∈ ∈

where is the doping concentration of then region and x is the distance. Thus the value of t at which the electric field reaches at a given distance x into the depletion region is obtained by setting , yielding , =

= Differentiation of above equation w.r.to time t results in

≡ =

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UNIT II: SEMICONDUCTOR MICROWAVE DEVICES AND APPLICATIONS where is the avalanche-zone velocity. • Thus the avalanche zone (or avalanche shock front) will quickly sweep across most of the diode, leaving the diode filled by a highly conductng plasma of holes and electrons whose space charge depresses the voltage to low values. Because of the dependence of the drift velocity on the field, the electrons and holes will drift at velocities determined by the low-field mobilities , and the transit time of the carriers can become much longer than =

where is the saturated carrier drift velocity. • Thus the TRAPATT mode can operate at comparatively low frequencies, since the discharge time of the plasma-that is, the rate Q/ I of its charge to its current can be considerably greater than the nominal transit time Ts of the diode at high field. • Therefore the TRAPATT mode is still a transit-time mode in the real sense that the time delay of carriers in transit (that is, the time between injection and collection) is utilized to obtain a current phase shift favourable for oscillation

POWER OUTPUT AND EFFICIENCY

• RF power is delivered by the diode to an external load when the diode is placed in a proper circuit with a load. The main function of this circuit is to match the diode effective negative resistance to the load at the output frequency while reactively terminating (trapping) frequencies above the oscillation frequency in order to ensure TRAPATT operation. • To date, the highest pulse power of 1.2 kW has been obtained at 1.1GHz (five diodes in series) , and the highest efficiency of 75% has been achieved at 0.6 GHz .

AVALANCHE TRANSIT TIME DEVICES

IMPATT TRAPPAT

Operating Frequency 0.5-100 GHZ 1-10 GHZ

Noise Figure 30 dB 60 dB

Not used as an amplifier

Power output 1W (CW), 400 W (Pulsed) Several 100 W (Pulsed)

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UNIT-V MICROWAVE MEASUREMENTS

Contents:  Description of Microwave Bench – Different Blocks and their Features, Precautions;  Waveguide Attenuators – Resistive Card, Rotary Vane types;  Waveguide Phase Shifters – Dielectric, Rotary Vane types.  Network Analyzer, Power Meter, Spectrum Analyzer  Microwave Power Measurement – Bolometer Method  Measurement of Attenuation, Frequency, VSWR, Cavity Q.  Impedance Measurements.

Introduction:

In low frequency circuits parameters such as voltage, current, etc can be measured and from these impedance, power factor and phase angle can be calculated. At microwave frequencies it is more convenient to measure power instead of V and I. Properties of devices and circuits at microwave frequencies i.e characterized by S- parameters, power, frequency and VSWR and noise figure. Direct microwave measuring devices are vector network analyzers, spectrum analyzers and power meters. Due to their complications and high cost, microwave measurements in lab are often carried out using 1 kHz square wave modulating signal which modulates the microwave test signal. Measurements of microwave power, frequency, VSWR and losses in the network have certain procedures. The procedures must be followed in order to obtain accurate results.

Important measurement devices

 The tunable detectors are used to demodulate the signal and couple the required output to high frequency scope analyzer. The low frequency demodulated output is detected using non reciprocal detector diode mounted in the microwave transmission line.  Slotted section with line carriage is a microwave sectioned coaxial line connecting a coaxial E-field probe which penetrates inside a rectangular waveguide slotted section. The longitudinal slot is cut along the center of the waveguide broad walls. The probe is made to move along the slotted wall which samples the electric field proportional to probe voltage.  Main purpose of slotted section with line carriage is

1. For determination of location of voltage standing wave maxima and minima along the line. 2. Measure the VSWR and standing wave pattern. 3. Wavelength. 4. Impedance. 5. Reflection co-efficient. 6. Return loss measurement.  VSWR meter is a highly sensitive, high gain, low noise voltage amplifier tuned normally at fixed frequency of 1KHZ of which microwave signals modulated. This meter indicates calibrated VSWR reading for any loads.

Power Measurement

 Power is defined as the quantity of energy dissipated or stored per unit time.

 Microwave power is divided into three categories – low power (less than 10mW), medium power (from 10mW to 10W) and high power (greater than 10w).

 The general measurement technique for average power is to attach a properly calibrated sensor to the transmission line port at which the unknown power is to be measured.

 The output from the sensor is connected to an appropriate power meter. The RF power to the sensor is turned off and the power meter zeroed. This operation is often referred to as “zero setting” or “zeroing.” Power is then turned on.  The sensor, reacting to the new input level, sends a signal to the power meter and the new meter reading is observed.

 There are three popular devices for sensing and measuring average power at RF and microwave frequencies. Each of the methods uses a different kind of device to convert the RF power to a measurable DC or low frequency signal. The devices are the diode detector, the bolometer and the thermocouple.

Diode Detector

The low-barrier Schottky (LBS) diode technology which made it possible to construct diodes with metal-semiconductor junctions for microwave frequencies that was very rugged and consistent from diode to diode. These diodes, introduced as power sensors in 1974, were able to detect and measure power as low as −70 dBm (100 pW) at frequencies up to 18 GHz.  Bolometer Sensor: Bolometers are power sensors that operate by changing resistance due to a change in temperature. The change in temperature results from converting RF or microwave energy into heat within the bolometric element.There are two principle types of bolometers ,barretters and thermistors. A barretter is a thin wire that has a positive temperature coefficient of resistance. Thermistors are semiconductors with a negative temperature coefficient. Thermistor elements are mounted in either coaxial or waveguide structures so they are compatible with common transmission line systems used at microwave and RF frequencies. Power meters are constructed from balanced bridge circuits. The principal parts of the power meter are two self-balancing bridges, the meter-logic section, and the auto-zero circuit. The RF Bridge, which contains the detecting thermistor, is kept in balance by automatically varying the DC voltage Vrf, which drives that bridge. The compensating bridge, which contains the compensating thermistor, is kept in balance by automatically varying the DC voltage Vc, which drives that bridge. The power meter is initially zero-set (by pushing the zero-set button) with no applied RF power by making Vc equal to Vrfo (Vrfo means Vrf with zero RF power). After zero-setting, if ambient temperature variations change thermistor resistance, both bridge circuits respond by applying the same new voltage to maintain balance.

 Thermocouple Sensors

Thermocouple sensors have been the detection technology of choice for sensing RF and microwave power since their introduction in 1974. The two main reasons for this evolution are:

1) they exhibit higher sensitivity than previous thermistor technology, and

2) they feature inherent square-law detection characteristic (input RF power is proportional to DC voltage out). Since thermocouples are heat-based sensors, they are true “averaging detectors.”Thermocouples are based on the fact that dissimilar metals generate a voltage due to temperature differences at a hot and a cold junction of the two metals. The power sensor contains two identical thermocouples on one chip, electrically connected as in Figure.

The thermocouples are connected in series as far as the DC voltmeter is concerned. For the RF input frequencies, the two thermocouples are in parallel, being driven through coupling capacitor Cc. Half the RF current flows through each thermocouple. Each thin-film resistor and the silicon in series with it have a total resistance of 100 Ω. The two thermocouples in parallel form a 50 Ω termination to the RF transmission line. The lower node of the left thermocouple is directly connected to ground and the lower node of the right thermocouple is at RF ground through bypass capacitor Cb. The DC voltages generated by the separate thermocouples add in series to form a higher DC output voltage. The principal advantage, however, of the two thermocouple scheme is that both leads to the voltmeter are at RF ground; there is no need for an RF choke in the upper lead. If a choke were needed it would limit the frequency range of the sensor. For a square wave modulated signal the peak power can be calculated from the average power measured as P av.

FrequencyMeasurement Microwave frequency can be measured by a number of different mechanical and electronic techniques. The Mechanical techniques are, Slotted Line Method (Indirect Method) The standing waves setup in a transmission line or a waveguide that produce minima for every half wavelength.

These minima are detected and the distance between them is measured.

where λ0 – free space wavelength λc - cut off wavelength λg – guide wavelength dmin can be measured by the slotted line probe carriage.

Resonant Cavity Method (Direct Method) The most commonly used type of microwave frequency meter is wave meters. It consists of a cylindrical or coaxial resonant cavity. The size of the cavity can be altered by adjustable plunger. The cavity is designed in such a way that for a given position of the plunger, the cavity is resonant only at one frequency in the specified range.

The cavity is coupled to the waveguide through an iris in the narrow wall of the waveguide. If the frequency of the wave passing through the waveguide is different from the resonance frequency of the cavity, the transmission is not affected. If these two frequencies coincide then the wave passing through the waveguide is attenuated due to power loss. It will be indicated as a dip in the meter. Electronic Technique Counter

An accurate measurement of microwave frequency can be measured here. The input signal is divided into two equal signals by a resistive power divider. These two parts of the signal are fed to 2 mixers. The mixer 1 is used in the input PLL (Phase Locked Loop) and the mixer 2 is used to determine the harmonic number. The frequency f1 of the input PLL is also fed to the direct counter circuits.The input PLL consists of a voltage controlled oscillator (VCO), mixer, an IF amplifier, a phase detector and a gain control block. The VCO searches over its range until an IF signal equal to 20MHz is found. Phase lock occurs when the phase detector output sets the VCO frequency f1 such that obtained. Counting the number of zero crossing for the period of f0, determines the harmonic number n of the phase lock loop. The input frequency is then calculated by presetting into IFref counter, measuring f1 and extending gate time according to number n.

VSWR Measurements: In a microwave network, if load impedance and line impedance are not matched, signal fed from the source is reflected again towards source causing standing wave pattern in the network. Voltage Standing Wave Ratio is a measure used for finding the magnitude of ration of reflected signals maximum and minimum amplitudes. V max S  (1)

Low VSWR Measurements: (S<20)

1. Microwave Source is energized with 1 KHz square wave signal as carrier.

2. Tunable passive components are so adjusted to get reading across the VSWR meter in 30 dB scale.

3. Detector (Tunable probe detector) is adjusted to get maximum power across the VSWR meter.

4. Slotted line carriage is moved from the load towards source to find the standing wave minimum position.

5. By adjusting the gain control knob of VSWR meter and attenuator the reading across the VSWR meter is made as 1 or 0 dB known as normalization.

6. Again the slotted carriage is moved towards source to find the next minimum position. The reading shown at this point in the VSWR meter is the ratio of magnitude of reflected signals minimum and maximum voltages .

VSWR =VMAX/VMIN

7. VSWR meter has three different scales with different ranges as specified below.

a) NORMAL SWR Scale 1

b) NORMAL SWR Scale 2 c) EXPANDED SWR Scale 3

8. If the device under test (DUT) is having the range of VSWR 1 – 4, reading is

taken from the first scale from the top (NORMAL SWR Scale 1 – 1 – 4).

9. If the device under test (DUT) is having the range of VSWR 3.2 – 10, reading is taken from the second scale from the top (NORMAL SWR Scale 2 (3.2 – 10).

10. If the device under test (DUT) is having the range of VSWR 1 – 1.33, reading is taken from the third scale from the top (EXPANDED SWR Scale 3 (1 – 1.33).

11. If the device under test (DUT) is having the range of VSWR 10 – 40, a 20 dB range is selected in the VSWR meter and reading is taken from the first scale from the top (NORMAL SWR Scale 1 – 1 – 4) which is then multiplied by 10 for getting the actual reading. Possible Errors in Measurements:

1. Detector may not work square law region for both Vmax. and Vmin.

2. Depth of the probe in the slotted line carriage is made as minimum. If not, it may cause reflections in addition to the load reflections.

3. For the device having low VSWR, connector used for measurement must have proper matching with line impedance.

4. If the geometrical shape of the slotted line is not proper, Vmax. (or) Vmin. Value will not constant across the slotted line.

5. If the microwave signal is not properly modulated by a 1 KHz square wave, then signal becomes frequency modulated thereby it causes error in the Vmin. value measured. The value becomes lower than the actual.

6. Residual VSWR of slotted line carriage may cause error in the measurements.

High VSWR Measurements - Double Minima Method - (S>20)

Measurement of high VSWR needs separate procedure because the detector may not be tuned to work in square law region. An alternate method known as double minimum method is used for finding high VSWR with the same experimental set up as shown above. 1. Microwave Source is energized with 1 KHz square wave signal as carrier.

2. Tunable passive components are so adjusted to get reading across the VSWR meter in 30 dB scale.

3. Detector (Tunable probe detector) is adjusted to get maximum power across the VSWR meter.

4. Slotted line carriage is moved from the load towards source to find the standing wave minimum position. Let it be d1.

5. Slotted line carriage is moved further to find the next immediate minimum position. Let it be d2. Now g = 2 (d1 - d2)

6. By adjusting the gain control knob of VSWR meter and attenuator the reading across the VSWR meter is made as 3 dB at this minimum position.

7. By taking this point as reference, slotted line carriage is moved on either side. The points at which the VSWR meter shows 0 dB reading on both sides are noted as x1 and x2. VSWR Measurements by Return Loss (Reflectometer) Method: To overcome the difficulties faced in slotted line carriage for measuring VSWR, reflectometer can be used. Reflectometer is a device having two directional couplers combined together with ideal coupling factor and directivity. It is a four-port device. i. Microwave Source is energized with 1 KHz square wave signal as carrier.

ii. Tunable passive components are so adjusted to get reading across the VSWR meter in 30 dB scale.

iii. Detector (Tunable probe detector) is adjusted to get maximum power across the VSWR meter.

iv. Port 2 is with a movable short and is adjusted for getting the output across the detector to unity in VSWR meter. Port 3 is matched terminated. v. VSWR meter and matched load at port4 and port 3 are interchanged. The output of the port3 is noted which should be ideally equal to the output from port 4. vi. Without disturbing the VSWR meter adjustment, the unknown load is connected at port 2 by replacing the short and the output at port3 is noted to obtain . This method is well suited for loads having low VSWR. The major sources of errors are

1. Unstability of the signal source causes a change of signal power level during measurement of input and reflected signals.

2. Non-ideal directional couplers and detectors are also sources of error. Attenuation and Insertion loss

When a device or network is inserted in the transmission line, part of the input signal power is reflected from the input terminal and the remaining part which actually enters the network is attenuated due to the non-zero loss of the network. Insertion loss = Reflection loss + Attenuation loss

 Insertion loss is defined as difference in power arriving at the terminating load with or without the network in circuit. Insertion loss (dB) =10 log (P0/Pi)

 Attenuation is defined as a power ratio between 2 points along a transmission path. It is measured under matched conditions. Commonly 2 methods are used to measure the attenuation.

Power ratio Method

Measure of power delivered to the load

1. First without network as P1. 2. Then with network whose attenuation is to be measured as P2.

The ratio of second to the first power gives attenuation.

Attenuation = Power( without N/W) Power( with N/W

RF substitution Method

If the input power is low and the device (network) has a high attenuation, then this method is used. This is a simple method in which the output power from the unknown device or network is measured. This network is then replaced by a calibrated attenuator which is adjusted until output power is the same. Under this condition the attenuation of the two is same and can read off from the attenuator.

NETWORK ANALYZER

A network analyzer is an instrument that measures the network parameters of electrical networks. Today, network analyzers commonly measure s– parameters because reflection and transmission of electrical networks are easy to measure at high frequencies, but there are other network parameter sets such as y- parameters, z-parameters, and h- parameters. Network analyzers are often used to characterize two-port networks such as amplifiers and filters, but they can be used on networks

with an arbitrary number of ports. Network analyzers are used mostly at high frequencies; operating frequencies can range from 5 Hz to 1.05 THz. Special types of network analyzers can also cover lower frequency ranges down to 1 Hz. These network analyzers can be used for example for the stability analysis of open loops or for the measurement of audio and ultrasonic components.

The two basic types of network analyzers are

scalar network analyzer (SNA)—measures amplitude properties only vector network analyzer (VNA)—measures both amplitude and phase properties A VNA is a form of RF network analyzer widely used for RF design applications. A VNA may also be called a gain-phase meter or an automatic network analyzer.An SNA is functionally identical to a spectrum analyzer in combination with a tracking generator. Another category of network analyzer is the microwave transition analyzer (MTA) or large signal network analyzer (LSNA), which measure both amplitude and phase of the fundamental and harmonics. The MTA was commercialized before the LSNA, but was lacking some of the user- friendly calibration features now available with the LSNA.