AN OPTIMIZATION STUDY ON CAVITY MAGNETRON

A THESIS SUBMITTED TO THE GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES OF MIDDLE EAST TECHNICAL UNIVERSITY

BY

MERVE KAYAN

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN PHYSICS

JANUARY 2018

Approval of the thesis:

AN OPTIMIZATION STUDY ON CAVITY MAGNETRON

submitted by MERVE KAYAN in partial fulfillment of the requirements for the degree of Master of Science in Physics Department, Middle East Technical University by,

Prof. Dr. Gülbin Dural Ünver ______Dean, Graduate School of Natural and Applied Sciences

Prof. Dr. Altug Özpineci ______Head of Department, Physics

Assoc. Prof. Dr. Serhat Çakır ______Supervisor, Physics Department, METU

Examining Committee Members:

Prof. Dr. Enver Bulur ______Physics Department, METU

Assoc. Prof. Dr. Serhat Çakır ______Physics Department, METU

Assoc. Prof. Dr. İsmail Rafatov ______Physics Department, METU

Assoc. Prof. Dr. Alpan Bek ______Physics Department, METU

Assoc. Prof. Dr. Kemal Efe Eseller ______Electrical& Electronics Engineering Department, Atilim University

Date: ______

I hereby declare that all information in this document has been obtained and presented in accordance with academic rules and ethical conduct. I also declare that, as required by these rules and conduct, I have fully cited and referenced all material and results that are not original to this work.

Name, Last Name: MERVE KAYAN

Signature :

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ABSTRACT

AN OPTIMIZATION STUDY ON CAVITY MAGNETRON

Kayan, Merve M.S., Department of Physics Supervisor: Assoc. Prof. Dr. Serhat Çakır

January 2018, 83 pages

We studied structure of cavity magnetrons and physics behind it deeply in this thesis. The main purpose of this study is to observe parameters which affect generated power of magnetron negatively or positively. Basically, they are crossed- field devices and electrons generate RF power with the help of both electric and magnetic field. We analyzed the physics of electron motion in magnetron and came up with a power equation. Then, we studied a cylindrical hole-slot-type magnetron with specific sizes and plotted curves to visualise which parameters have an effect on power and how. It was determined as a result of analyzes that applied voltage between the anode and cathode parts, resonator number, cathode radius and angular resonant frequency are directly proportional with generated power. Contrary to this, increase in gap factor and loaded quality factor decreases the generated power. After all, for used magnetron and value range of parameters that we used, the working values which give the maximum power generation are 0.5 for gap factor, 5 for the loaded quality factor and 2.2 cm for cathode radius. Moreover, much more

v number of resonator and higher angular resonant frequency provide much more power generation.

Keywords: cavity magnetron, crossed-field devices, Helmholtz resonance frequency

vi

ÖZ

OYUKLU MAGNETRON ÜZERİNDE OPTİMİZASYON ÇALIŞMASI

Kayan, Merve Yüksek Lisans, Fizik Bölümü Tez Yöneticisi: Doç. Dr. Serhat Çakır

Ocak 2018, 83 sayfa

Bu tezde, oyuklu magnetronların yapısını ve arkasındaki fiziği derinlemesine inceledik. Bu çalışmanın asıl amacı magnetronun ürettiği gücü olumlu ya da olumsuz etkileyen değişkenleri incelemek. Temel olarak magnetronlar çapraz alanlı cihazlardır. Elektrik ve manyetik alanların yardımıyla elektronlar radyo frekanslı güç üretirler. Magnetrondaki elektronların hareketini fiziksel olarak çözümledik ve bir güç denklemine ulaştık. Sonra silindir biçiminde, belli ölçülere sahip bir magnetron tasarladık ve hangi değişkenlerin çıkış gücünü etkilediğini ve nasıl etkilediğini görselleştirmek için grafikler çizdik. Analizler sonucunda anot ve katot kısımları arasında uygulanan voltajın, çınlayıcıların sayısının, katot yarıçapının ve açısal rezonant frekansının çıkış gücüyle doğru orantılı olduğunu belirlendi. Bunun aksine açıklık faktörü ve yüklü kalite faktöründeki artış çıkış gücünü azalttı. Sonuç olarak kullandığımız magnetron ve parametrelerin değer aralıkları için en yüksek güç üretimini sağlayan çalışma değerleri açıklık faktörü için 0.5, yüklü kalite faktörü için 5 ve katot yarıçapı için 2.2 santimetredir. Ayrıca,

vii daha fazla sayıda çınlayıcı ve daha yüksek açısal resonant frekansı daha fazla güç üretimi sağlar.

Anahtar kelimeler: oyuklu magnetron, çapraz alanlı cihazlar, Helmholtz rezonans frekansı

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To my family

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ACKNOWLEDGMENTS

Above all, I owe my supervisor Assoc. Prof. Dr. Serhat Çakır a great debt of gratitude for his guidance, incredible patience, criticism, endless support, advice and continuous encouragement that enabled me to make this study. It would be impossible to finish this thesis without him so I consider myself lucky to be able to have worked under his mentorship.

I would also like to thank my unique family; my mother Belma Kayan, my father Ömer Kayan and my brother Mustafa Barış Kayan for their endless support, patience, compassionate and unconditional love. They were right beside me to support like as they did always. To have them is the best side of me.

Additional thanks to all my colleagues and friends for their continuous support. Especially, I am greatful to Mertcan Genç for his presence in my life for the last 8 years and for his endless love and support. He has always been there to make me smile and happy.

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TABLE OF CONTENT

ABSTRACT ...... v

ÖZ ...... vii

ACKNOWLEDGMENTS ...... x

TABLE OF CONTENTS ...... xi

LIST OF TABLES ...... xiv

LIST OF FIGURES ...... xv

CHAPTERS

1. INTRODUCTION ...... 1

2. BASIC PHYSICS OF MAGNETRON ...... 9

2.1 Impacts of Different Fields on Charged Particles ...... 9

2.1.1 Motion in Electric Field ...... 11

2.1.1.1 Cartesian Coordinate System ...... 11

2.1.1.2 Cylindrical Coordinate System ...... 17

2.1.2 Motion in Magnetic Field ...... 21

2.1.2.1 Cartesian Coordinate System ...... 21

2.1.2.2 Cylindrical Coordinate System ...... 27

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2.1.3 Motion in both Magnetic and Electric Field ...... 28

2.1.4 Motion in Magnetic, Electric and an AC Field ...... 31

2.1.4.1 Cartesian Coordinate System ...... 32

2.1.4.2 Cylindrical Coordinate System ...... 38

2.2 Electron Motion in Magnetron ...... 40

2.3 Hull Cutoff Equation for Magnetron ...... 42

2.4 Cyclotron Angular Frequency for an Electron ...... 42

2.5 Equivalent Circuit ...... 44

2.6 Quality Factor ...... 45

2.7 Power and Efficiency ...... 46

3. PARAMETERS WHICH AFFECT THE GENERATED POWER ...... 49

3.1 Derivations of Some Important Parameters ...... 49

3.1.1 Electric Field ...... 50

3.1.2 The Capacitance at Vane Tips ...... 50

3.1.3 Angular Resonant Frequency ...... 51

3.1.4 Electrical Conductivity ...... 55

3.2 Observations of Change in Power about Effects of Some Parameters 59

3.2.1 Effect of Cavity Number on Generated Power ...... 61

3.2.2 Effect of Gap Factor on Generated Power ...... 63

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3.2.3 Effect of Loaded Quality Factor on Generated Power ...... 66

3.2.4 Effect of Cathode Radius on Generated Power ...... 69

3.2.5 Effect of Angular Resonant Frequency on Generated Power ...... 72

4. CONCLUSIONS & DISCUSSION ...... 75

REFERENCES ...... 81

xiii

LIST OF TABLES

TABLES

Table 1 Conductivity values of different materials ...... 56

Table 2 Values of variables for cavity number-power graph ...... 62

Table 3 Values of variables for gap factor-power graph ...... 63

Table 4 Values of variables for loaded quality factor-power graph ...... 66

Table 5 Values of variables for cathode radius-power graph ...... 69

xiv

LIST OF FIGURES

FIGURES

Figure 1 Hull’s magnetron model ...... 2

Figure 2 Habann’s split-anode magnetron ...... 2

Figure 3 Multi-cavity magnetron of Hollmann ...... 3

Figure 4 Randall and Boot’s multi-cavity magnetron ...... 3

Figure 5 Basic construction of magnetron ...... 4

Figure 6 One of the resonant cavities ...... 4

Figure 7 Common cavity types ...... 5

Figure 8 Strapping alternate segments ...... 6

Figure 9 Influence of magnetic field on path of electron ...... 6

Figure 10 Microwave oven structure ...... 7

Figure 11 Radar system ...... 8

Figure 12 (a) E field between the parallel plates (b) direction of electron ...... 12

Figure 13 (a) Geometry of cylindrical diode and potentials (b) crosscut and

electric field ...... 17

Figure 14 Straight motion of electron in magnetron ...... 21

Figure 15 (a) B field between the parallel plates (b) direction of electrons with

different velocities ...... 21

Figure 16 (a) Geometry of cylindrical diode and field (b) direction of electrons

with different velocities ...... 28

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Figure 17 Electron motion in both magnetic and electric fields ...... 30

Figure 18 (a) View of cavity in the magnetron (b) equivalent parallel resonant

circuit of magnetron cavity ...... 31

Figure 19 E, B and an AC field between the parallel plates ...... 32

Figure 20 Movement of the point on the circumference of the wheel ...... 36

Figure 21 Charged particle motion in the combined field ...... 38

Figure 22 Electron paths in magnetron ...... 39

Figure 23 Force lines of an 8-cavity magnetron in π-mode ...... 43

Figure 24 Equivalent circuit for magnetrons resonator ...... 44

Figure 25 Capacitor and parallel plates with E field ...... 50

Figure 26 Cavity resonant ...... 51

Figure 27 View of simple example of cavity resonator ...... 52

Figure 28 Equivalent spring-mass system ...... 52

Figure 29 Simple circuit ...... 57

Figure 30 View of resistor ...... 57

Figure 31 Used 8 cavity magnetron for our work ...... 60

Figure 32 Cavity Number versus Generated Power Graph ...... 62

Figure 33 Gap Factor versus Generated Power Graph ...... 64

Figure 34 Gap Factor versus 1st Derivative of Power Graph ...... 65

Figure 35 Gap Factor versus 2nd Derivative of Power Graph ...... 65

Figure 36 Loaded Quality Factor versus Generated Power Graph ...... 67

Figure 37 Loaded Quality Factor versus 1st Derivative of Power Graph...... 68

Figure 38 Loaded Quality Factor versus 2nd Derivative of Power Graph ...... 68

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Figure 39 Cathode Radius versus Generated Power Graph ...... 70

Figure 40 Cathode Radius versus 1st Derivative of Power Graph ...... 71

Figure 41 Cathode Radius versus 2nd Derivative of Power Graph ...... 71

Figure 42 Angular Resonant Frequency versus Generated Power Graph ...... 73

Figure 43 Electron motion in magnetron ...... 76

xvii

xviii

CHAPTER 1

INTRODUCTION

There are two groups of microwave devices. First one is semiconductor devices which are Gunn diode, backward diode, tunnel diode, IMPATT (impact ionization avalanche transit time operation) diode, Schottky diode, varactor diode, PIN diode (p-i-n diode), transistors and integrated circuits (ICs). Second one is tube devices which are klystron, reflex klystron, traveling wave tube (TWT) and magnetron. It is more cheaply to generate and amplify high levels of microwave signals with tube devices. In this thesis, the aim is to analyze the cavity magnetron deeply.

During the last century, different types of microwave vacuum equipment have been used as an amplifier or a generator in many different areas such as: medical X-ray sources, microwave heating, communication, warfare and radar [1]. Magnetron is the most promising and popular high power microwave device because of some advantages of it. For example, it has a small size, light weight and low-cost [2]. Another positive aspect of magnetron is that it can generate high power in the range of kilowatts to megawatts. Moreover, it works with a high efficiency around 40 to 70% [3]. Magnetron is a self-excitation vacuum tube oscillator. It uses electrons with the magnetic fields and converts energy of electrons to high power radiofrequency signals [4].

The developments about the magnetron began with Heinrich Greinacher, a Swiss physicist, in 1912. He gave some basic mathematical definition about the motion of electrons in a magnetic field. In 1921, Albert Wallace Hull observed that the motion of electrons to the anode can be controlled with the influence of magnetic field.

1

Figure 1: Hull’s magnetron model [5]

Actually, he was in a competition with the opponent company and wanted to invent an amplifier that is controlled magnetically. However, he noticed the chance of radiofrequency generation and called his invention as magnetron (Fig. 1). Then in 1924, Erich Habann from Germany and Napsal August Zázek from Czechia have studied on magnetron independently. Habann used steady magnetic field as today’s magnetrons and observed oscillations in the range of 100 MHz with his split-anode magnetron (Fig. 2).

Figure 2: Habann’s split-anode magnetron [5]

Zázek has developed a magnetron that operated in the range of 1 GHz. Kinjiro Okabe from Tohoku University took a big step by developing a magnetron with the range of 5.35 GHz in 1929. Hans Erich Hollmann improved a multi-cavity magnetron and in 1938 he was granted a patent on multi-cavity magnetron in Germany (Fig. 3).

2

Figure 3: Multi-cavity magnetron of Hollmann [5]

In 1940s, engineer John Randall and Henry Boot built a multi-cavity magnetron and with this invention, England gained an advantage over Germany in the sub- marine war. These two engineers made a magnetron with more than four cavities to increase the efficiency of the radiofrequency generation (Fig. 4). In the meantime, Henry Gutton was studying about the cathodes made with barium oxide in multi- cavity magnetrons and he observed that barium oxide cathode needs lower temperature to release electron when it compared with the tungsten cathodes. In other words, this observed characteristic prolongs the magnetron life. John Randall and Henry Boot used this result in their own investigations [5].

Figure 4: Randall and Boot’s multi-cavity magnetron [5]

Physical structure of magnetron can be separated into three main parts: anode, cathode and filament and interaction space. Fig. 5 shows these parts with cavities and an output lead.

3

Figure 5: Basic construction of magnetron [6]

The anode part of magnetron is made from solid copper. As shown, it is a cylindrical block and surrounds the cathode. Each seen hole is called as a resonant cavity and they work like a parallel resonant circuit which shown in Fig. 6. The rear wall of cavity is thought as an inductive portion, like a coil with single turn and the vane tip is thought as a capacitor. The physical dimensions of the resonator determine the resonant frequency.

Figure 6: One of the resonant cavities [6]

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A single oscillated resonant cavity excites the next cavity and it oscillates too. Effected one oscillates with a phase delay, which is 180 degrees. Then, these interactions continue similarly. This continued actions form a closed slow-wave structure. Because of this feature, sometimes we use the name of “Multi-cavity Travelling Wave Magnetron” for this design. Cathode and filament are placed at the center of the magnetron and filament leads fix them in their positions with the help of leads’ rigid and large structure. Cathode has a shape like a hollow cylinder and high emission material is used for it (like barium oxide). Cathode part of magnetron provides electron that is required for energy transfer. At the center of the cathode, there is a feeding wire of the filament. If an eccentricity occurs between the cathode and anode, malfunction or an internal arcing takes place, which is an undesired event. Interaction space is the entire area between the cathode and the anode block. In this space, magnetic and electric fields affect each other and this causes a force on electrons. Around the magnetron, a magnet is mounted and this creates a magnetic field, which is parallel with the cathode axis [6].

A B C

Figure 7: Common cavity types [7]

Three common types of cavity forms are illustrated in Fig. 7. Here, A is the hole- slot-type, B is the vane-type, C is the rising-sun-type. For hole-slot and vane types, cavities are connected each other with straps as shown. However, there is not any straps in the rising sun type. About hole-slot and vane types, there should be connected alternate segments, in order that side-by-side segments have opposite poles. Therefore, they have even number of cavities. This shown in Fig. 8. For

5

rising-sun-type, large and small trapezoidal cavities are aligned respectively and this provides a stable frequency between the resonant frequencies of all cavities [7].

Figure 8: Strapping alternate segments [7]

About magnetrons, we can say that they are crossed-field devices. Electrons are released from cathode and the electric field accelerates them. After electrons increase their velocity so they gain energy, electrons direction is oblique by the magnetic field, which is perpendicular to electric field [3]. The reason of the magnetic field is the magnet placed around the magnetron. Cathode of magnetron has a negative voltage so electric field moves from the anode block to the cathode in radial direction. If there is not any magnetic field and cathode is heated, electrons move to the anode directly and uniformly as shown in Fig. 9 with the blue path.

Figure 9: Influence of magnetic field on path of electron [6]

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Electrons bends like the green path in Fig. 9 when magnetic field is weak and permanent. To have flowing plate current, electrons should reach the anode block. If we enhance the magnetic field, electrons bend sharply. Similarly, increasing the electron velocity causes an increase on the field around it and path of electrons have sharper bend. As shown in the Fig. 9 as a red path, when magnetic field reaches the critical value, electrons return to cathode without reaching the anode block. At that case, plate current drop off to a very small value. If applied magnetic field is bigger than the critical value, plate current reaches to zero. If electrons cannot reach the anode, oscillations at microwave frequencies can be produced. In other words, magnetrons work like a magnetic mirror and they trap high temperature plasma with the helping of magnetic field.

It is mentioned that magnetrons are used in countless applications. Fig. 10 shows the one of these, microwave oven. Microwave oven systems can be seperated in three parts. These are microwave source which is magnetron, waveguide feed and an oven space. Operation of microwave oven stars with the microwave generator, magnetron. Electricity comes from power outlet to the magnetron.

Figure 10: Microwave oven structure

7

Then, it transforms this energy to the high powered radio waves [8]. This magnetron works at 2.45 GHz and it produces an output power in the range of 500-1500 W. These waves reaches to the oven space with a waveguide feed and microwave cooks the foods on the rotating plate [3]. The working principle of magnetron will explain in the next chapters.

Another example is the use of magnetron in radar system. In radar systems, microwaves are generated by magnetron and the basic operating principle of magnetron is same as in previous example. Then, these waves are emitted by waveguide to an object with the intention of locating the position or the speed of the object. These signals hit the object and turn back to a receiver which is placed on the radar system. Finally, with the helping of turning signals the objects can be tracked. Fig. 11 shows this system simply [3].

Figure 11: Radar system

It will be given some information about the basic physics of magnetron in the next chapter. In Chapter 3, it is explained actual study numerically. Finally, thesis will be concluded with a conclusion & discussion part that summarizes what we learned and cap off with references.

8

CHAPTER 2

BASIC PHYSICS OF MAGNETRON

2.1 Impacts of Different Fields on Charged Particles

As we mentioned before, magnetron is a microwave tube. Operation of all tubes is directly associated with the movement of electrons. Electrons are released from the cathode because of the effect of the heat energy and these electrons’ goal is to arrive the anode. Their paths are affected by the field which is caused because of the potential difference between the electrodes. This movement of electrons sometimes changes because of the electric field (퐸⃗ field) and sometimes the reason is the magnetic field (퐵⃗ field). Since generally these tubes are used as an amplifier for weak AC signals, this AC field also affect the path of electrons. As a result, the reasons of the change on the electrons path can be 퐸⃗ -퐵⃗ fields separately, 퐸⃗ -퐵⃗ fields together or all three fields which are 퐸⃗ field, 퐵⃗ field and AC field [9]. Let’s look these three situations one by one.

Only electric field is effective. The potential difference between the electrodes is V. If electric field is 퐸⃗ , then |퐸| is equal to V/d (d is the distance between two electrodes). Because of the field, a force (퐹 ) is formed on an electron. This force can be written as

퐹 = 푞퐸⃗ (1)

Here q is the charge of the electron which is −푒. Then,

9

퐹 = −푒퐸⃗ (2)

As seen in Eq. 2, it is not important whether electron is moving or not.

If there is only magnetic field and electrons velocity is 푣 , then the force on an electron become

퐹 = −푒(푣 × 퐵⃗ ) (3)

This means, we can talk about the magnetic field effect if the particle is moving. So 푣 ≠ 0 is the case for the 퐵⃗ field effect. Here the force is perpendicular not only to electron velocity, but also to 퐵⃗ field.

If there are both electric and magnetic fields, the force on an electron can be obtained by summing Eq.2 and Eq.3. So

퐹 = −푒[퐸⃗ + (푣 × 퐵⃗ )] (4)

The Eq.4 is also called as Lorentz Force equation.

Like it is mentioned before, sometimes these tubes is used as an amplifier. In this case, we have to consider the AC field which is 퐸′ cos(푤푡). Let’s assume this field direction is same with the 퐸⃗ field direction. Then, Eq.4 becomes

퐹 = −푒[(퐸⃗ + 퐸⃗ ′ cos 푤푡) + (푣 × 퐵⃗ )] (5)

Here 퐸′ is the value for AC field and 푤 is used for the angular frequency. So force in Eq.5 is caused by E field, B field and AC field.

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This force equation can change because of the tube shape. Moreover, E or B field can be one dimensional or two or three. This also affects the force equation. However, about these fields, it is assumed that they have only one component. Besides, tube shapes cause to use the Cartesian or cylindrical systems generally [9].

2.1.1 Motion in Electric Field

If there is an electric field, the force value is also equal to the change of momentum in time. Then Eq.2 can be written as

푑 푑푣⃗ 퐹 = −푒퐸⃗ = (푚푣 ) = 푚 (6) 푑푡 푑푡

Here, 푣 is the velocity and m is the mass of the particle. Eq.6 is the general equality and this form can be used for any system.

2.1.1.1 Cartesian Coordinate System

In this case, both E and B field has three components. Then Eq.6 can be rewritten as

푑푣 푑2푥 −푒퐸 = 푚 푥 = 푚 (7) 푥 푑푡 푑푡2

푑푣 푑2푦 −푒퐸 = 푚 푦 = 푚 (8) 푦 푑푡 푑푡2

푑푣 푑2푧 −푒퐸 = 푚 푧 = 푚 (9) 푧 푑푡 푑푡2

Here x, y and z are components of position vector and 푣푥, 푣푦 ve 푣푧 are components of the velocity vector.

In Fig. 12a plates are located at 푥 = 0 and 푥 = 푑. Potentials are equal to zero for bottom plate and 푉0 for upper plate. Let’s assume that when electron enters the E

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field at time t=0, its position is at x=y=z=0 and its initial velocities are 푣푥 = 푣푥0, ⁄ 푣푦 = 푣푦0 and 푣푧 = 0. For E field, 퐸푥 equals to − 푉0 푑 and the other components are equal to zero.

Figure 12: (a) E field between the parallel plates (b) direction of electron

By using the Eq.7, Eq.8 and Eq.9 we can write

푑푣 푑2푥 푒 푒 푉 푒푉 푥 = = − 퐸 = − (− 0) = 0 = 푘 (10) 푑푡 푑푡2 푚 푥 푚 푑 푚푑

푑푣 푑2푦 푦 = = 0 (11) 푑푡 푑푡2

푑푣 푑2푧 푧 = = 0 (12) 푑푡 푑푡2

Let’s solve these last three equations:

From equations 10, 11 and 12, we can write

1 푥 = 푘푡2 + 퐴 푡 + 퐵 (13a) 2 1 1

푦 = 퐴2푡 + 퐵2 (13b)

12

푧 = 퐴3푡 + 퐵3 (13c)

Initially, we know that at 푡 = 0, 푥 = 푦 = 푧 = 0. Then this means that 퐵1 = 퐵2 =

퐵3 = 0 and Eq.13 becomes

1 푥 = 푘푡2 + 퐴 푡 (14a) 2 1

푦 = 퐴2푡 (14b)

푧 = 퐴3푡 (14c)

If we take the derivatives of x, y and z, we obtain

푑푥 = 푘푡 + 퐴 (15a) 푑푡 1

푑푦 = 퐴 (15b) 푑푡 2

푑푧 = 퐴 (15c) 푑푡 3

Another condition is that at 푡 = 0, 푣푥 = 푣푥0, 푣푦 = 푣푦0 and 푣푧 = 0 so

퐴1 = 푣푥0 (16a)

퐴2 = 푣푦0 (16b)

퐴3 = 0 (16c)

Then Eq.14 reduces to

1 푥 = 푘푡2 + 푣 푡 (17a) 2 푥0

13

푦 = 푣푦0푡 (17b)

푧 = 0 (17c)

We know that 푘 = 푒푉0⁄푚푑, then Eq.17 can be written as

푒푉 푒푉 푥 = ( 0 ) 푡2 + 푣 푡 and 푣 푡 = 푥 − ( 0 ) 푡2 (18a) 2푚푑 푥0 푥0 2푚푑

푦 푦 = 푣푦0푡 and 푡 = (18b) 푣푦0

If we use the ‘푡’ value in Eq.18b, Eq.18a becomes

2 푒푉0 푦 푦 푥 = ( ) ( ) + 푣푥0 ( ) (19) 2푚푑 푣푦0 푣푦0

Here Eq.19 is a parabola equation in the x-y plane and we know that

푑푥 푒푉 푣 = = ( 0) 푡 + 푣 (20a) 푥 푑푡 푚푑 푥0

푑푦 푣 = = 푣 (20b) 푦 푑푡 푦0

푑푧 푣 = = 0 (20c) 푧 푑푡

Then

푒푉 2 푣 = √(푣 2 + 푣 2) = √푣2 + [( 0) 푡 + 푣 ] (21a) 푥 푦 푦0 푚푑 푥0

푒푉 2 푒푉 푣 = √푣2 + [( 0) 푡2 + 2 ( 0) 푣 푡 + 푣2 ] (21b) 푦0 푚푑 푚푑 푥0 푥0

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( ) 2 If we use 푣푥0푡 = 푥 − 푒푉0/2푚푑 푡 in Eq.21, we get

푒푉 2 2푒푉 푒푉 2 푣 = √푣2 + ( 0) 푡2 + ( 0) 푥 − ( 0) 푡2 + 푣2 (22a) 푦0 푚푑 푚푑 푚푑 푥0

2푒푉 푣 = √푣2 + 푣2 + ( 0) 푥 (22b) 푥0 푦0 푚푑

The kinetic energy at 푡 = 0 is 퐾퐸0. When 푡 equals to zero and 푥 = 0 if we use Eq.22b then the kinetic energy becomes

1 1 퐾퐸 = 푚푣2 = 푚(푣2 + 푣2 ) (23) 0 2 2 푥0 푦0

At any time ‘푡’, the kinetic energy is written as

1 2푒푉 퐾퐸 = 푚 [푣2 + 푣2 + ( 0) 푥] (24) 푡 2 푥0 푦0 푚푑

The difference between Eq.23 and Eq.24 gives the gained energy in time ‘푡’ and this is

1 2푒푉 푒푉 ∆퐾퐸 = 푚 ( 0) 푥 = ( 0) 푥 (25) 2 푚푑 푑

The potential energy of electron with an ‘푥’ displacement is that

푉 −푒푉 = −푒 ( 0) 푥 = −∆퐾퐸 (26) 푑

The minus sign in Eq.26 means that any decrease in potential energy is compensated by the increase in 퐾퐸.

15

If initially velocities are taken as zero, this means that 푣푥표 = 푣푦표 = 0, then

푣 = √2푒푉⁄푚 (27)

where 푉 = 퐸. 푥 = 푉0. 푥⁄푑 and here x is again the position component in the x- direction.

Eq.27 is the solution of Eq.10, Eq.11, Eq.12 and if we substitute the constant values (푒 = 1.602 × 10−19 C and 푚 = 9.1091 × 10−31 kg) into the Eq.27, we obtain

푥 푣 = 5.932 × 105√푉 푚/푠 (28) 0 푑

For example, we can find the velocity at 푥 = 푑 as

6 푣 = 0.5932 × 10 √푉0 푚/푠 (29)

It is mentioned that 퐹 = 푞퐸⃗ so electric field is proportional to force directly. Therefore, electrons move from the cathode to the anode directly. Fig. 12b shows the movement of electron in E field between two plates.

16

2.1.1.2 Cylindrical Coordinate System

Figure 13: (a) Geometry of cylindrical diode and potentials (b) crosscut and E field

Fig. 13 shows the diode geometry in a cylindrical system. Moreover, it also shows the E field lines and moving direction of electron. In this case, 푣 is three dimensional and 푣 = 푣푟푎̂푟 + 푣∅푎̂∅ + 푣푧푎̂푧.Then, Eq.6 can be rewritten as

푒 푑 푑 푑 푑 − 퐸⃗ = (푣 푎̂ + 푣 푎̂ + 푣 푎̂) = (푣 푎̂) + (푣 푎̂) + (푣 푎̂) (30) 푚 푑푡 푟 푟 ∅ ∅ 푍 푍 푑푡 푟 푟 푑푡 ∅ ∅ 푑푡 푧 푧

Here 푣푟,∅,푧 are velocities and 푣푟 = 푑푟⁄푑푡, 푣∅ = 푟푑∅⁄푑푡 and 푣푍 = 푑푧⁄푑푡.

Moreover, 푎푟,∅,푧 are the unit vectors.

Three terms in Eq.30 become

푑 푑∅ 푑푣 (푣 푎̂) = 푣 푎̂ + 푟 푎̂ (31) 푑푡 푟 푟 푟 푑푡 ∅ 푑푡 푟

푑 푑∅ 푑푣 (푣 푎̂) = −푣 푎̂ + ∅ 푎̂ (32) 푑푡 ∅ ∅ ∅ 푑푡 푟 푑푡 ∅

푑 푑푣 (푣 푎̂) = 푧 푎̂ (33) 푑푡 푧 푧 푑푡 푧

17

By using Eq.30, Eq.31, Eq.32 and Eq.33, we can write

푒 푑푣 푑∅ − 퐸 = 푟 − 푣 (34) 푚 푟 푑푡 ∅ 푑푡

푒 푑∅ 푑푣 − 퐸 = 푣 + ∅ (35) 푚 ∅ 푟 푑푡 푑푡

푒 푑푣 − 퐸 = 푧 (36) 푚 푧 푑푡

Let’s find a solution for equations 34, 35 and 36:

푣 can be written as

푑푟 푟푑∅ 푑푧 푣 = 푣 푎̂ + 푣 푎̂ + 푣 푎̂ = 푎̂ + 푎̂ + 푎̂ (37) 푟 푟 ∅ ∅ 푧 푧 푑푡 푟 푑푡 ∅ 푑푡 푧

Here 푑∅⁄푑푡 = 푤 and 푣∅ = 푟푤, then equations 34, 35 and 36 transform to

푒 푑2푟 − 퐸 = − 푟푤2 (38a) 푚 푟 푑푡2

푒 푑푟 푑(푤푟) 1 푑 − 퐸 = 푤 + = (푟2푤) (38b) 푚 ∅ 푑푡 푑푡 푟 푑푡

푒 푑2푧 − 퐸 = (38c) 푚 푧 푑푡2

We mentioned that for this case, the motion of particle in E-field can be seen in Fig. 13 which also shows the radiuses and voltages of both cylinders. Then, the potential relation can be given as

ln 푟/푎 푉 = 푉 (39) 0 ln 푏/푎 here 푎 is cathode radius, 푏 is anode radius and

18

휕푉 1 퐸 = − = −푉 (40a) 푟 휕푟 0 푟 ln 푏/푎

퐸∅ = 퐸푧 = 0 (40b)

If we consider Eq.40, Eq.38 becomes

푒푉 푘 푑2푟 − 0 = = − 푟푤2 (41a) 푚푟 ln 푏/푎 푟 푑푡2

푑(푟2푤) = 0 (41b) 푑푡

푑2푧 = 0 (41c) 푑푡2

Let’s assume, an electron which initially has a velocity 푣 = 0 enters the E-field at 푡 = 0 and its position is 푟 = 푎, ∅ = 0 and 푧 = 0. From Eq.38c, 푧 is zero for all ‘푡’ 2 values. Besides, for Eq.38b assume that 푟 푤 = 푟 × 푟푤 = 푟 × 푣∅ = 퐴. We know that 푣 = 0 when 푡 equals to zero, then 퐴 = 0 or 푤 = 0 for all ‘푡’ values.

Also 푣푟 = 푑푟⁄푑푡 means that 푑푡 equals to 푑푟⁄푣푟. Then, Eq.38a can be written as

푑2푟 푑푣 푘 = 푟 = (42a) 푑푡2 푑푡 푟

푘 푘 푑푟 푘 푑푣푟 = ( ) 푑푡 = ( ) ( ) or 푣푟푑푣푟 = ( ) 푑푟 (42b) 푟 푟 푣푟 푟

If we integrate both sides of Eq.42b we get

1 푣2 = 푘 ln 푟 + 퐵 (43) 2 푟

When we use the condition 푣푟 equals to zero at 푟 = 푎, value of 퐵 becomes −푘 ln 푎. Therefore,

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1 푟 푣2 =kln ( ) (44a) 2 푟 푎

So

푟 푣 = √[2푘 ln ( )] (44b) 푟 푎

If we substitute 푘 value into Eq.44b, finally we get

푟 2푒푉 ln( ) 푑푟 0 푎 푣푟 = = √ 푏 (45) 푑푡 푚 ln( ) 푎

When we solve the equations 34, 35 and 36, we get Eq.45. Also we mentioned that 푒 = 1.602 × 10−19 C and 푚 = 9.1091 × 10−31 kg. Then Eq.45 becomes

ln(푟⁄푎) 푣 = [5.932 × 105√ ] √푉 푚/푠 (46) 푟 ln(푏⁄푎) 0

If electron is at the cathode surface so if 푟 = 푎, the velocity value (푣) becomes zero. However, when electron reaches to the anode surface, Eq.46 becomes

5 푣푟 = 5.932 × 10 √푉0 푚/푠 (47)

The values of velocity in Eq.29 and Eq.47 are same for a given voltage. Likely in the Cartesian case, the electron path from the cathode is direct to the anode in cylindrical system. For a magnetron, a view of impact of E field on an electron motion is shown in Fig. 14 [9].

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Figure 14: Straight motion of electron in magnetron

2.1.2 Motion in Magnetic Field Basic equation for this case is the Eq.3. Therefore, Eq.6 changes a little bit for this case and it becomes

푑 푑푣⃗ 퐹 = −푒(푣 × 퐵⃗ ) = (푚푣 ) = 푚 (48) 푑푡 푑푡

Let’s find the solution for Eq.48 again for two coordinate systems.

2.1.2.1 Cartesian Coordinate System

Figure 15: (a) B field between the parallel plates (b) direction of electrons with different velocities

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The structure of this case is shown in Fig. 15a. Here velocity is again there component and it can be written like 푣 = 푣푥푎̂푥 + 푣푦푎̂푦 + 푣푧푎̂푧. Likewise, the magnetic field vector is 퐵⃗ = 퐵푥푎̂푥 + 퐵푦푎̂푦 + 퐵푧푎̂푧.

When we use 푣 and 퐵⃗ in Eq.48 with all these three components, we obtain

푑푣 −푒 푥 = (푣 퐵 − 푣 퐵 ) (49) 푑푡 푚 푦 푧 푧 푦

푑푣 −푒 푦 = (푣 퐵 − 푣 퐵 ) (50) 푑푡 푚 푧 푥 푥 푧

푑푣 −푒 푧 = (푣 퐵 − 푣 퐵 ) (51) 푑푡 푚 푥 푦 푦 푥

Because the first derivative of the position vector is the velocity vector, we can change the form of Eq.49, Eq.50 and Eq.51 and they become

푑2푥 −푒 푑푦 푑푧 = (퐵 − 퐵 ) (52) 푑푡2 푚 푧 푑푡 푦 푑푡

푑2푦 −푒 푑푧 푑푥 = (퐵 − 퐵 ) (53) 푑푡2 푚 푥 푑푡 푧 푑푡

푑2푧 −푒 푑푥 푑푦 = (퐵 − 퐵 ) (54) 푑푡2 푚 푦 푑푡 푥 푑푡

Let’s find the solution of equations from 49 to 54: ⃗ Assume that 퐵 = 퐵0푎̂푧 and an electron that has the velocity 푣 = 푣푦0푎̂푦 initially enters the magnetic field at the position of 푥 = 푦 = 푧 = 0. Also 푧 is zero for all times because 푧 component of 푣 is zero and 푥 component of 푣 is zero at 푡 = 0. Then Eq.52, Eq.53 and Eq.54 transform to

푑2푥 푒 푑푦 = − 퐵 (55a) 푑푡2 푚 0 푑푡

22 so

푑푦 푚 푑2푥 = − 2 (55b) 푑푡 푒퐵0 푑푡

푑2푦 푒 푑푥 = 퐵 (55c) 푑푡2 푚 0 푑푡 so

푑푥 푚 푑2푦 = 2 (55d) 푑푡 푒퐵0 푑푡 and

푑2푧 = 0 (55e) 푑푡2

If we use 푑푦⁄푑푡 value in Eq.55c and 푑푥⁄푑푡 value in Eq.55a, we get

2 3 푑 푦 푚 푑 푥 푒퐵0 푑푥 2 = − 3 = (56a) 푑푡 푒퐵0 푑푡 푚 푑푡

2 3 푑 푥 푚 푑 푦 푒퐵0 푑푦 2 = 3 = − (56b) 푑푡 푒퐵0 푑푡 푚 푑푡

We can change the form of Eq.56 and it turns to

푑2푣 푒퐵 2 푑2푣 푥 = − ( 0) 푣 and 푥 + 푤2푣 = 0 (57a) 푑푡2 푚 푥 푑푡2 0 푥

푑2푣 푒퐵 2 푑2푣 푦 = − ( 0) 푣 and 푦 + 푤2푣 = 0 (57b) 푑푡2 푚 푦 푑푡2 0 푦

where 푤0 = 푒퐵0/푚. The solution of Eq.57 is

푣푥 = 퐴1 cos 푤0푡 + 퐵1 sin 푤0푡 (58a)

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푣푦 = 퐴2 cos 푤0푡 + 퐵2 sin 푤0푡 (58b)

Use conditions;

When 푡 = 0, 푣푥 is also zero then 퐴1 = 0. So Eq.58a turns to

푣푥 = 퐵1 sin 푤0푡 (59a)

When 푡 = 0, 푣푦 equals to 푣푦0 then 퐴2 = 푣푦0. So Eq.58b turns to

푣푦 = 푣푦0 cos 푤0푡 + 퐵2 sin 푤0푡 (59b)

With all these, Eq.55a and Eq.55c can be written as

푑2푥 푒 푑푦 푑푣 푒 = − 퐵 so 푥 = − 퐵 푣 = −푤 푣 (60a) 푑푡2 푚 0 푑푡 푑푡 푚 0 푦 0 푦 and

푑2푦 푒 푑푥 푑푣 푒 = 퐵 so 푦 = 퐵 푣 = 푤 푣 (60b) 푑푡2 푚 0 푑푡 푑푡 푚 0 푥 0 푥

If we put 푣푥 and 푣푦 values into Eq.60, we find

푑푣 푥 = −푤 푣 → 푤 퐵 cos 푤 푡 = −푤 (푣 cos 푤 푡 + 퐵 sin 푤 푡) (61a) 푑푡 0 푦 0 1 0 0 푦0 0 2 0

푑푣 푦 = 푤 푣 → −푤 푣 sin 푤 푡 + 푤 퐵 cos 푤 푡 = 푤 퐵 sin 푤 푡 (61b) 푑푡 0 푥 0 푦0 0 0 2 0 0 1 0

Eq.61 can be used for all times. So at 푡 = 0, 퐵1 is −푣푦0 and 퐵2 is zero. Then, 푣푥 and 푣푦 become

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푣푥 = −푣푦0 sin 푤0푡 (62a)

푣푦 = 푣푦0 cos 푤0푡 (62b) so

2 2 푣 = √(푣푥 + 푣푦 ) = 푣푦0 (62c)

We said that the first derivative of the position vector is the velocity vector. Then, from Eq.62

푣푦0 푥 = ( ) cos 푤0푡 + 퐶1 (63a) 푤0

푣푦0 푦 = ( ) sin 푤0푡 + 퐶2 (63b) 푤0

At 푡 = 0, 푥 and 푦 equal to zero so 퐶1 = −푣푦0/푤0 and 퐶2 = 0. Then Eq.63 turns to

푣푦0 푥 = ( ) (cos 푤0푡 − 1) (64a) 푤0

푣푦0 푦 = ( ) sin 푤0푡 (64b) 푤0

These found solutions are a circle’s parametric equations. The radius of this circle ⁄ (r) equals to 푣푦0 푤0 and also

⁄ ⁄ ⁄ 푣푦0 푤0 = 푣 푤0 = 푚푣 푒퐵0 (65)

푒퐵 where 푤 = 표 . The circle center is at 푥 = − 푟⁄2 and 푦 = 푟⁄2. 0 푚

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When there is a constant magnetic field, let’s suppose that the energy of the particle does not change. The linear velocity is related with the angular velocity which can be found from equations 49, 50 and 51. Particle’s linear velocity is

푎푒퐵 푣 = 푎푤 = 0. (66) 0 푚 The radius of the path of particle is

푚푣 푟 = . (67) 푒퐵표

The cyclotron angular frequency caused by the circular motion is

푣 푒퐵 푤 = = 0. (68) 0 푎 푚

The period of the one turn completely is

2휋 2휋푚 푇 = = . (69) 푤0 푒퐵표

From last four relations, it is obtained that; Magnetic field uses force on the electron and this force is perpendicular to the motion of electron continuously. Thus, there is no work is done and electron velocity does not change.

The magnetic field causes a circular path of electron. In other words, force direction of the motion changes. However, force magnitude remains constant.

The velocity of the electron directly affects the radius of the circular motion of the particle. However, radius or velocity have no effect on the period or angular

26

velocity. In other words, if velocity of electron increases, then the radius of circular path is also increases [9].

Fig. 15b illustrates that if velocity of an electron is low enough, it may return to the cathode after releasing. However, it reaches to anode if electron has an efficiently high velocity.

2.1.2.2 Cylindrical Coordinate System

When we take the velocity as 푣 = 푣푟푎̂푟 + 푣∅푎̂∅ + 푣푧푎̂푧 and the magnetic flux density as 퐵⃗ = 퐵푟푎̂푟 + 퐵∅푎̂∅ + 퐵푧푎̂푧, we can write the components of Eq.3 as

−푒 푑푣 푑∅ (푣 퐵 − 푣 퐵 ) = 푟 − 푣 (70) 푚 ∅ 푧 푧 ∅ 푑푡 ∅ 푑푡

−푒 푑푣 푑∅ (푣 퐵 − 푣 퐵 ) = ∅ + 푣 (71) 푚 푧 푟 푟 푧 푑푡 푟 푑푡

−푒 푑푣 (푣 퐵 − 푣 퐵 ) = 푧 (72) 푚 푟 ∅ ∅ 푟 푑푡

It is assumed that magnetic field has only one component so 퐵⃗ = 퐵0푎̂푧 as shown in

Fig. 16a and an electron with a velocity of 푣 = 푣푟0푎̂푟 enters the environment of magnetic field at 푟 = 푎 and ∅ = 푧 = 0. The particle does not move in the z- direction because 푣푧 = 푑푧⁄푑푡 = 0. Moreover, 푣∅ equals to 푟푤 because initially velocity has not a component in the ∅ direction and 푑∅⁄푑푡 is 푤.

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Figure 16: (a) Geometry of cylindrical diode and field (b) direction of electrons with different velocities

Then our equations 70, 71 and 72 become

−푒 푑2푟 퐵 푤푟 = − 푟푤2 (73) 푚 0 푑푡2

푒 푑푟 1 푑 퐵 = (푟2푤) (74) 푚 0 푑푡 푟 푑푡

푑푣 푧 = 0 (75) 푑푡

The solution of these equations can be found as in the part of Cartesian coordinate system and we get the same results as found in Section 2.1.2.1. Again electron has a circular motion as shown in Fig. 16b. Here the velocity and the radius of circular path are vary. As seen, electrons have a lower velocities returns the cathode but faster electrons reaches to the anode.

2.1.3 Motion in both Magnetic and Electric Field

To explain this case, we start from the Eq.4. The combination of Eq.6 and Eq.48 can be written in the rectangular coordinate system as

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푑2푥 푒 푑푦 푑푧 = − (퐸 + 퐵 − 퐵 ) (76) 푑푡2 푚 푥 푧 푑푡 푦 푑푡

푑2푦 푒 푑푧 푑푥 = − (퐸 + 퐵 − 퐵 ) (77) 푑푡2 푚 푦 푥 푑푡 푧 푑푡

푑2푧 푒 푑푥 푑푦 = − (퐸 + 퐵 − 퐵 ) (78) 푑푡2 푚 푧 푦 푑푡 푥 푑푡

Let’s assume that in Fig. 14 there is an electric field and a magnetic field together.

Besides, E field components are 퐸푥 = − 푉0⁄푑, 퐸푦 = 0, 퐸푧 = 0 and for B field 퐵⃗ =

퐵0푎̂푧, 퐵푥 = 퐵푦 = 0. Initially electron has a velocity of 푣 = 푣푦0푎̂푦 at 푥 = 푦 = 푧 = 0. As it was explained before velocity value in the z-direction is zero. Then, the equations 76, 77 and 78 change their forms and become

푑2푥 푒 푑푦 푑푧 = − (퐸 + 퐵 − 퐵 ) (79) 푑푡2 푚 푥 푧 푑푡 푦 푑푡

푑2푦 푒 푑푧 푑푥 = − (퐵 − 퐵 ) (80) 푑푡2 푚 푥 푑푡 푧 푑푡

푑2푧 푒 푑푥 푑푦 = − (퐵 − 퐵 ) (81) 푑푡2 푚 푦 푑푡 푥 푑푡

In the cylindrical coordinate system this equations changes because of the components and become

푑2푟 푒 푑푧 − 푟푤2 = (퐸 − 퐵 푤푟 − 퐵 ) (82) 푑푡2 푚 푟 푧 ∅ 푑푡

1 푑 푒 푑푧 푑푟 (푟2푤) = − (퐸 + 퐵 − 퐵 ) (83) 푟 푑푡 푚 ∅ 푟 푑푡 푧 푑푡

푑2푧 푒 푑푟 = − (퐸 + 퐵 − 퐵 푤푟) (84) 푑푡2 푚 푧 ∅ 푑푡 푟

Equations from 79 to 84 can be obtained by the similar steps in the previous sections. These equations explain the behavior of the electron in combined electric and magnetic field.

29

In the resulting solutions of the motion in combined E, B and an AC field, if AC field is removed, then we get same results with equations from 79 to 81.

The reason of circular path is magnetic field and linear path due to the electric field. In Fig. 17, seen curvature of the path is an effect of amplitudes of both magnetic an electric fields. This figure also shows different paths. Here, if 퐵 = 0, then electron

Figure 17: Electron motion in both magnetic and electric fields [9]

motion is straight like as path x. When B field is increased a little bit, B field exerts a force on electron and bends its path to the left (path y). So if increase in B field reaches the sufficient value, then path becomes sharper, electron just graze the anode and returns to the cathode like path z. For path z, required B field is called as cutoff field. So with a cutoff field, anode current becomes zero. If B continues to increase after this critical value, electron returns cathode even sooner (path w) [9].

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Figure 18: (a) View of cavity in the magnetron (b) equivalent parallel resonant circuit of magnetron cavity [9]

Magnetron cathode produces electrons and they go to the anode with curved paths. Then in cavities, oscillating B and E fields are formed. The gathering of the electrons at the ends of the cavities causes capacitance. Flowing current around the cavities also causes inductance. Therefore, each one of the cavities works like a parallel resonant circuit. This is shown in Fig. 18 [9].

2.1.4 Motion in Magnetic, Electric and an AC Field

For this case, the starting point is Eq.5 which is

퐹 = −푒[(퐸⃗ + 퐸⃗ ′ cos 푤푡) + (푣 × 퐵⃗ )]

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2.1.4.1 Cartesian Coordinate System

Figure 19: E, B and an AC field between the parallel plates

A parallel plate magnetron is shown in Fig. 19. Plates have 푉0 and 0 voltages and there is an E field in the x-direction. Besides the E field, there is a B field which has only 퐵푧 component. Additionally, a potential that changes with time is applied. This potential is 푉1 cos 푤푡. In order that all time-varying electric field is related to time- varying magnetic field or quite the opposite, any such related fields are not taken consideration. Therefore, for this case, electric and magnetic fields do not satisfy Maxwell’s equations [9]. Then, by using all these and Fig. 19, we can write

푉 푉0 푉1 퐸푦 = 퐸푧 = 0 and 퐸⃗ = 퐸푥 = − = (− ) [1 + ( ) cos 푤푡] 푎̂푥, (85) 푑 푑 푉0

푉0 푉1 퐸⃗ = (− ) [1 + 훼 cos 푤푡]푎̂푥 where 훼 = ( ), (86) 푑 푉0

퐵푦 = 퐵푥 = 0 and 퐵⃗ = 퐵0푎̂푧. (87)

Then we obtain

푑2푥 푒 푉 푑푦 = − [− 0 (1 + 훼 cos 푤푡) + 퐵 ], (88) 푑푡2 푚 푑 0 푑푡

푑2푦 푒 푑푥 = − (−퐵 ), (89) 푑푡2 푚 0 푑푡

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푑2푧 = 0. (90) 푑푡2

Let’s find the solutions of Eq.88, Eq.89 and Eq.90:

If we take

푒퐵 푚푤 푒푉 푤 = 0 so 퐵 = 0 and 0 = 푘 (91) 0 푚 0 푒 푚푑

Then Eq.88 and Eq.89 become

푑2푥 푑푦 = 푘(1 + 훼 cos 푤푡) − 푤 (92a) 푑푡2 0 푑푡

푑2푦 푑푥 = 푤 (92b) 푑푡2 0 푑푡

So 푑푣 푥 = 푘(1 + 훼 cos 푤푡) − 푤 푣 (93a) 푑푡 0 푦

푑푣 푦 = 푤 푣 (93b) 푑푡 0 푥

From Eq.93, we can obtain

1 푑푣푦 푣푥 = (94a) 푤0 푑푡

푘 1 푑푣푥 푣푦 = (1 + 훼 cos 푤푡) − (94b) 푤0 푤0 푑푡

When we differentiate Eq.94a and then use Eq.93a, it gives

2 푑푣푥 1 푑 푣푦 = 2 = 푘(1 + 훼 cos 푤푡) − 푤0푣푦 (95a) 푑푡 푤0 푑푡

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Similarly differentiate Eq.94b and the use Eq.93b, this gives

2 푑푣푦 푤 1 푑 푣푥 = −푘훼 sin 푤푡 − 2 = 푤0푣푥 (95b) 푑푡 푤0 푤0 푑푡

By using Eq.95a and Eq.95b, we can find

푑2푣 푥 + 푘훼푤 sin 푤푡 + 푤2푣 = 0 (96a) 푑푡2 0 푥

푑2푣 푦 + 푤 푣 − 푘푤 (1 + 훼 cos 푤푡) = 0 (96b) 푑푡2 0 푦 0

Solution for Eq.96a is

푣푥 = 퐴1 cos 푤0푡 + 퐵1 sin 푤0푡 + 퐶1 sin 푤푡 (97)

훼푘푤 If we use this 푣푥 in Eq.96a, we find 퐶1 = 2 2 푤 −푤0 Then Eq.97 can be rewritten as

훼푘푤 푣푥 = 퐴1 cos 푤0푡 + 퐵1 sin 푤0푡 + 2 2 sin 푤푡 (98) 푤 −푤0

When 푡 equals to zero, 푣푥 value also becomes zero. So

퐴1 = 0

훼푘푤 푣푥 = 퐵1 sin 푤0푡 + 2 2 sin 푤푡 (99) 푤 −푤0

If Eq.99 is used in Eq.94b, 푣푦 can be written as following

푘 1 훼푘푤2 ( ) 푣푦 = 1 + 훼 cos 푤푡 − [퐵1 푤0cos 푤0푡 + 2 2 cos 푤푡] 푤0 푤0 푤 − 푤0

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2 푘 훼푤0 푣푦 = [1 − 2 2 cos 푤푡] − 퐵1 cos 푤0푡 (100) 푤0 푤 −푤0

When 푡 equals to zero, 푣푦 value also becomes zero. Then

2 푘 훼푤0 퐵1 = (1 − 2 2) (101) 푤0 푤 −푤0

If we use this 퐵1 in Eq.99 and Eq.100, we get

2 푘 훼푤0 훼푘푤 푑푥 푣푥 = (1 − 2 2) sin 푤0푡 + 2 2 sin 푤푡 = (102a) 푤0 푤 −푤0 푤 −푤0 푑푡

2 2 푘 훼푤0 훼푘푤 푑푦 푣푦 = [1 − (1 − 2 2 cos 푤0푡)] − 2 2 cos 푤푡 = (102b) 푤0 푤 −푤0 푤 −푤0 푑푡

Finally, after integrate the Eq.102, we find the solution for 푥 and 푦.

2 2 푘 훼푤0 훼푤0 푥 = 2 [(1 − 2 2) cos 푤0푡 − 2 2 cos 푤푡] (103a) 푤0 푤 −푤0 푤 −푤0

2 2 푘 훼푤0 푤0 훼푤0 푦 = 2 [푤0푡 − (1 − 2 2) sin 푤0푡 − 2 2 sin 푤푡] (103b) 푤0 푤 −푤0 푤 푤 −푤0 To make a comment about Eq.102 and Eq.103, we have to change their forms. Therefore, there are two cases.

Lack of AC field: If an AC field does not applied, 훼 = 0. Then these equations returns to 푘 푣푥 = sin 푤0푡, (104) 푤0

푘 푣푦 = (1 − cos 푤0푡), (105) 푤0

푘 푥 = 2 (1 − cos 푤0푡), (106) 푤0

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푘 푦 = 2 (푤0푡 − sin 푤0푡). (107) 푤0

Eq.106 and Eq.107 are cycloid’s parametric equations. A point on the rolling wheel traces this curve. For this wheel motion, there are two velocity components; angular velocity=푤0 and forward translational velocity=푘⁄푤0. After ignoring the 2 translation and producing a radius of 2푘/푤0 , the wheel radius can be obtained from the linear velocity of a point on the circumference. The maximum velocity is reached nearest to the anode and this is 2푘/푤0. Besides, maximum velocity means maximum kinetic energy [9].

About Fig. 20, there can be two options;

2 If 푑 ≤ 2푘/푤0 , then electron fall into the anode 2 If 푑 > 2푘/푤0 , then electron cannot reach the anode and turn back with in 2휋/푤0 2 to the cathode at a distance 2휋푘/푤0 . This distance is also twice the wheel radius.

Figure 20: Movement of the point on the circumference of the wheel [9]

It was mentioned that particle has maximum kinetic energy with a velocity of

2푘/푤0 when it is nearest the anode. This kinetic energy is

2 2 1 2푘 2푚푉0 퐾퐸푚푎푥 = 푚 ( ) = 2 2 (108) 2 푤0 푑 퐵0

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This energy returns to electric field completely before the electrons turn back to the cathode. Like it was said before this case is similar to case in Section 2.1.3 because there are only E and B fields.

AC field present: Let’s look what happens when AC field is applied.

2 훼푤0 1 푉1 2 2 = and 푤 = 1.1푤0, then 훼 = = 0.105 (109) 푤 −푤0 2 푉0

Here we can see that the DC field magnitude (푉0) is much bigger than an AC field magnitude (푉1). This situation is observed when a small ac signal is to be amplified at the cost of a large applied dc source [9].

When above relations are considered, Eq.103 becomes

푘 1 1 푥 = 2 (1 − cos 푤0푡 − cos 1.1푤0푡) 푤0 2 2

푘 = 2 (1 − cos 0.05푤0푡 cos 1.05푤0푡) 푤0

푘 푘 = 2 − 2 cos 0.05푤0푡 cos 1.05푤0푡 (110) 푤0 푤0

푘 1 1 1 푦 = 2 [(푤0푡 − sin 푤0푡) − sin 1.1푤0푡] (111) 푤0 2 1.1 2

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Figure 21: Charged particle motion in the combined field [9]

Whit Eq.110 and Eq.111, the behavior of a charged particle in the combined field is explained. Fig. 21 shows this motion. The motion of electron starts from the 2 cathode with a maximum distance (2푘/푤0 ). If electron does not reach the anode, it 2 oscillates by decreasing the amplitudes. This continuous until it rest at 푥 = 푘/푤0 at

푤0 = 10휋.

2.1.4.2 Cylindrical Coordinate System

In the previous section, I showed the electron path in the presence of E and B field. In this section, I will explain the travelling path of electron in magnetron when there are E field, B field and AC field. The radio frequency field (RF) is an alternating current so this current generates an electromagnetic field which is called as a RF field [10].There is a RF field inside all cavities and RF field changes the path of electrons. With the help of shown paths in Fig. 22, we will understand the behavior of electrons in magnetron.

Electron a: Tangential component of electric field arise from the RF field in the magnetron. This tangential component prevents the tangential velocity of electron when electron ‘a’ came to point 1. Hence, electron ‘a’ is geared down and transmits its energy to the RF field. The magnetic field force on electron is decreased because of this slowdown. Consequently, electron moves closer to the anode. Then, electron ‘a’ comes to the point 2, field polarity becomes reversed and electron ‘a’ is geared

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down again and gives energy to the RF field. As a result, over again B field force effect on electron ‘a’ decreases. In other words, every time E field polarity reverses when electrons come at a proper position for interaction. In this way, electrons spend a lot of time in interaction space and turn around the cathode many times before they reach the anode.

Figure 22: (a) Electron paths in magnetron [9]

Electron b: Due to the location of electron ‘b’, RF field accelerates it so it gets energy from RF field. Thus the magnetic force on it increases. As compared with electron ’a’, ‘b’ spends much less time in interaction space. It turns back to the cathode sooner than the electron return in absence of RF field. Given energy to the RF field must be much more than the receiving energy. There are many electron like ‘a’ and ‘b’. However, electron ‘b’ spends less time in the RF field when it is compared with the electron ‘a’. Thus, ‘b’ takes energy from RF field but ‘a’ gives much more of extracted energy to the RF field. Moreover, ‘a’ give energy again and again while ‘b’ takes energy once or twice. This differences between electron ‘a’ and ‘b’ provide sustained oscillations.

Electron c: This electron also makes energy contribution to the RF field like as electron ‘a’. However, tangential component of electron ‘c’ is not much powerful by comparison with electron ’a’ so it cannot give much energy like ‘a’. However,

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it runs across with the radial RF field component and it affects acceleration radially. Magnetic field exerts force on electron ‘c’ strongly at this junction point and electron ‘c’ returns to the cathode. For electron ‘d’, similarly magnetic field also slows down it tangentially. Therefore, electron ‘d’ is grabbed by the favored electrons which are in equilibrium position [9].

2.2 Electron Motion in Magnetron

In conventional (cylindrical) magnetron, there is a radially applied voltage (푉0) between anode and cathode and magnetic field (퐵0) is in the positive z direction.

Thus, 퐸⃗ = 퐸푟푎̂푟 and 퐵⃗ = 퐵푧푎̂푧. About electrons, they have cycloidal motion in the space of magnetron.

For given E and B field, Eq.82 and Eq.83 reduce to

푑2푟 푒 − 푟푤2 = (퐸 − 퐵 푤푟) (112) 푑푡2 푚 푟 푧

1 푑 푒 푑푟 (푟2푤) = (퐵 ) (113) 푟 푑푡 푚 푧 푑푡 where 푤 = 푑∅⁄푑푡. Eq.113 can be written as

푑 푒 푑푟 1 푑 (푟2푤) = 푟퐵 = 푤 (푟2) (114) 푑푡 푚 푧 푑푡 2 푐 푑푡

where 푤푐 = (푒⁄푚)퐵푧. And 푤푐 is the cyclotron angular frequency. If we integrate the Eq.114, then we get

1 푟2푤 = 푤 푟2 + 푘 (푐표푛푠푡푎푛푡) (115) 2 푐 1

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2 Let’s take cathode radius as 푎. Then the constant 푘1 is −푤푐푎 /2 when at 푟 = 푎 and 푤 = 0. So another formula for 푤 is

1 푎2 푤 = 푤 (1 − ). (116) 2 푐 푟2

The kinetic energy of the electron in the magnetron is

1 퐾퐸 = 푚푣2 = 푒푉 (117) 2

From Eq.117, we can write the electron velocity 푣 which has two components 푟 and ∅,

2푒 푑푟 2 푑∅ 2 푣2 = 푉 = 푣2 + 푣2 = ( ) + (푟 ) (118) 푚 푟 ∅ 푑푡 푑푡

Let’s inner radius of anode becomes 푏. When electron at 푟 = 푏, 푉 = 푉0 and 푑푟⁄푑푡 = 0 which means when the electron just grazes the inner surface of anode, Eq.116 becomes

푑∅ 1 푎2 ( ) = 푤 (1 − ) (119) 푑푡 2 푐 푏2

Eq.119 can also be written as

푑∅ 2 2푒 푏2 ( ) = 푉 (120) 푑푡 푚 0

When we substitute Eq.119 in Eq.120, we get

2 1 푎2 2푒 푏2 ( 푤 (1 − )) = 푉 (121) 2 푐 푏2 푚 0

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2.3 Hull Cutoff Equation for Magnetron

When we substitute 푤푐 equation in Eq.121 and do necessary arrangements, we find following relations:

8푉 푚 1/2 ( 0 ) 푒 퐵0푐 = 푎2 (122) 푏(1− ) 푏2

2 푒 푎2 푉 = 퐵2푏2 (1 − ) (123) 0푐 8푚 0 푏2

For Eq.122 and Eq.123 we used 푤푐 = (푒⁄푚)퐵푧. We call Eq.122 as the Hull cutoff magnetic equation and result of this equation is the cutoff magnetic field (퐵0푐). For

Eq.122, if 퐵0 < 퐵0푐 for a given voltage 푉0, then electron cannot arrive the anode. Like for like-bases, name of the Eq.123 is the Hull cutoff voltage equation and if we solve this equation we get the cutoff voltage (푉0푐). For Eq.123, if 푉0 < 푉0푐, again electron cannot reach the anode [9].

2.4 Cyclotron Angular Frequency for an Electron

As mentioned earlier, the magnetic field and the cycloidal path of electron are orthogonal to each other. The centrifugal force on electron equals to the pulling force. So,

푚푣 2 푡 = 푒푣퐵 (124) 푅

Here 푅 is the path radius, 푣푡 is the tangential velocity. The cyclotron angular frequency of the motion is

푣 푒퐵 푤 = 푡 = (125) 푐 푅 푚

One full revolution has the period which is shown below.

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2휋 2휋푚 푇 = = (126) 푤 푒퐵

In order to have oscillations in magnetron, construction must has an integral multiple of 2휋 radians phase shift [9]. For the nth mode of the oscillation in an N cavity magnetron, phase shift between two cavities is

2휋푚 ∅ = (127) 푛 푁

By adjusting the voltage of anode, it is possible to produce the oscillations. Generally, magnetrons oscillates in 휋-mode. The necessary phase shift for this is

∅푛 = 휋 (128) Fig. 23 illustrates the force lines of an eight-cavity magnetron in π-mode. Here, successive descent and ascent of field in cavities can be thought as a travelling wave. When field decelerates the electrons and each passing of electrons near the cavities is occur, electrons give energy to the travelling wave [9].

Figure 23: Force lines of an 8-cavity magnetron in 휋-mode [9]

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If the distance between the cavities is 퐿, then the phase constant is

2휋푛 훽 = (129) 0 푁퐿

By using Maxwell’s equations and boundary conditions, we can obtain the solution for ∅ component of the travelling wave electric field [9].Thus,

푗(푤푡−훽0∅) 퐸∅0 = 푗퐸1푒 (130)

The angular velocity of the travelling field is

푑∅ 푤 = (131) 푑푡 훽0

As it understood from this relation, if the cyclotron frequency equals to the angular frequency, 푑∅ 푤 = 훽 ( ) (132) 푐 0 푑푡 then field-electron interaction occurs and energy gets transferred.

2.5 Equivalent Circuit

Figure 24: Equivalent circuit for magnetrons resonator

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Fig. 24 shows an equivalent circuit for magnetrons resonator. The values in the figure are

푌푒 = the electronic admittance, 푉 = the RF voltage across the vane tips, 퐶 = the capacitance at vane tips, 퐿 = the inductance,

퐺푟 = the conductance of the resonant,

퐺퐿 = the load conductance per resonator. Each one of the resonators contains a similar resonant circuit like as in the Fig. 24 [9].

2.6 Quality Factor

For a resonant circuit, the uncharged quality factor is shown as

푤0퐶 푄푢푛 = (133) 퐺푟 the external quality factor is shown as

푤0퐶 푄푒푥푡 = (134) 퐺퐿 the loaded quality factor is shown as

푤0퐶 푄퐿 = (135) 퐺퐿+퐺푟

In these three equations, angular resonant frequency (푤0) is equal to 2휋푓0.

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2.7 Power and Efficiency

In magnetrons, there are two values of efficiency term. First one is the circuit efficiency and this can be shown as

퐺퐿 퐺퐿 1 휂푐 = = = 푄 (136) 퐺퐿+퐺푟 퐺푒푥푡 (1+ 푒푥푡) 푄푢푛

From Eq.136, we say that 휂푐 has its maximum value when 퐺퐿 ≫ 퐺푟. This means magnetron has heavy loading. For some cases, this does not desire because this cause a sensitive tube in loading. The second value of efficiency term is electronic efficiency which is

푃푔푒푛 휂푒 = (137) 푃푑푐

Here 푃푔푒푛 equals to 푃푑푐 − 푃푙표푠푠 and it is the induced power of the RF into the anode circuit. 푃푑푐 is power of dc supply and it is also 푉0퐼0. 푃푙표푠푠 is anode circuit’s power lost. 푉0 is the anode voltage and 퐼0 is used for the anode current. It is mentioned that electrons generate the RF power and this equals to

2 2 푚 푤0 퐸푚푎푥 1 2 푤0퐶 푃푔푒푛 = 푉0퐼0 − 푃푙표푠푠 = 푉0퐼0 − 퐼0 2 + 2 = 푁|푉| (138) 2푒 훽 퐵푧 2 푄퐿

In Eq.138, 푁 is resonator number, 푉 is the voltage in the resonator gap, 퐸푚푎푥 is the maximum value of electric field which is 푀1|푉|/퐿, 훽 is the constant of phase, 훽푧 is the magnetic flux density, 퐿 is the distance between the vane tips and 푀1 is the gap factor which is used for 휋-mode operation and 푀1 can be found by using the Eq.139.

푀1 = sin(훽푛훿⁄2)/(훽푛훿⁄2) (139)

Here, for small 훿 values, 푀1 ≈ 1. Eq.138 can be formed simply as

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2 푁퐿 푤0퐶 2 푃푔푒푛 = 2 퐸푚푎푥 (140) 2푀1 푄퐿

Then by using Eq.140, we can write electronic efficiency equation as

2 푚푤0 (1− 2) 푃푔푒푛 2푒푉0훽 휂푒 = = 2 (141) 푃푑푐 퐼0푚푀1푄퐿 (1+ 2 ) 퐵푧푒푁퐿 푤0퐶

In this chapter we explained the physics behind of magnetron. Finally we achieved the general power equation (Eq.140) and explained the parameters in this equation. In next chapter, we will come to actual work and show the derivations of some of these parameters and rewrite the generated power equation more detailed form. In Chapter 3, our aim is to analyze the effects of some critical parameters on power generation in magnetron.

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CHAPTER 3

PARAMETERS WHICH AFFECT THE GENERATED POWER

We talked about the generated power in Chapter 2. We showed that electrons generate the RF power in Eq.140 which equals to

2 푁퐿 푤0퐶 2 푃푔푒푛 = 2 퐸푚푎푥 2푀1 푄퐿 Here,

푁 = resonator number, 퐿 = the distance between the vane tips,

푀1 = gap factor,

푤0 = angular resonant frequency, 퐶 = the capacitance at vane tips,

푄퐿 = the loaded quality factor,

퐸푚푎푥 = the maximum value of electric field.

In this chapter we will analyze that how some of these parameters change the generated power.

3.1 Derivations of Some Important Parameters

In power equation, some parameters must be written in different forms to observe exact effects on power. Therefore, firstly we should get the bottom of these parameters.

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3.1.1 Electric Field

We mentioned that negatively charged cathode and positively charged anode block cause an electric field between each other. This radial field was derived in Eq.40 which is 휕푉 1 퐸 = − = −푉 푟 휕푟 0 푟 ln 푏/푎

퐸∅ = 퐸푧 = 0

Here 푉0 is applied voltage, 푎 is cathode radius and 푏 is anode radius. Maximum value of electric field can be observed on the cathode surface. The 퐸푚푎푥 can be written as 1 퐸 = −푉 (142) 푚푎푥 0 푎 ln 푏/푎

3.1.2 The Capacitance at Vane Tips

Figure 25: Capacitor and parallel plates with E field

In magnetron, we can think vane tips as parallel plates and start with finding the capacitance of parallel plates like in Fig. 25. Electric field between the plates are

휎 푉 퐸 = = (143) 휀 푑 here 휎 is charge density, 휀 is permittivity, 푑 is the distance between plates and 푉 is the voltage difference between plates. We also know that charge density is

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푐ℎ푎푟푔푒 표푛 푝푙푎푡푒 푄 휎 = = (144) 푎푟푒푎 표푓 푝푙푎푡푒 퐴

If we use the capacitance definition, we obtain

푆푡표푟푒푑 푐ℎ푎푟푔푒 표푛 푝푙푎푡푒 푄 푄 푄휀 퐴휀 푘휀 퐴 퐶 ======0 (145) 퐴푝푝푙𝑖푒푑 푣표푙푡푎푔푒 푉 퐸푑 휎푑 푑 푑

−12 here 휀0 is space permittivity and equals to 8.854 × 10 퐹푎푟푎푑/푚푒푡푒푟 and 푘 is relative permittivity for dielectric material between the plates and equals to 1 for air.

3.1.3 Angular Resonant Frequency

Angular resonant frequency, 푤0, equals to 2휋푓0. Here 푓0 is the cavity resonant frequency. We multiplies 푓0 with 2휋 because one revolution is equal to 2휋. Fig.26 shows simple cavity resonant system and we use Eq. 146 to find the value of the cavity resonant frequency which is

Figure 26: Cavity resonant

푣 퐴 푓 = 푠 √ (146) 0 2휋 푉퐿

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here 푣푠 is the speed of sound. In dry air, speed of sound is approximately 340 m/s. However, we use Eq. 147 for any cavity

훾푅푇 푣 = √ (147) 푠표푢푛푑 푀 here 훾 is adiabatic constant and it is about the gas characteristic, 푅 is gas constant (8.314 퐽⁄푚표푙. 퐾), 푀 is molecular mass of gas and 푇 is temperature. For air 훾 = 1.4 and 푀 = 28.95 푔/푚표푙.

Eq. 146 is Helmholtz resonance frequency formula. To derive this formula, let’s look at Fig.27

Figure 27: View of simple example of cavity resonator [11]

In this system, springiness of air is the reason of vibration. Here, there is an air lump at the neck. The air force pushes the air lump down so it compresses the inside air. Then pressure drives out the lump. In other words, inside air volume works like a mass on a spring system in Fig.28 and mass m of air in the neck oscillates in and out.

Figure 28: Equivalent spring-mass system

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For spring-mass system, angular frequency formula with spring constant, k, is

푘 푤 = √ (148) 푚

Air lump mass can be calculated with the density of the air (휌) and it is found as

푚 = 휌퐿퐴 (149) here 퐿 is the neck length and 퐴 is the opening area of the neck. The change of resonator volume is

푑푉 = −퐴푑푥 (150) here 푑푥 is the air lump displacement and the volume decrease causes minus sign.

The bulk modulus is the other parameter for derivation of Eq. 146. The bulk modulus is the ability of a material to resist deformation in terms of volume change, when subject to compression under pressure. The relation is

푑푃 퐾 = −푉 (151) 푑푉 here 퐾 the bulk modulus (푁⁄푚2 표푟 푃푎), 푑푃 is the change in applied pressure, 푉 is volume of the system and 푑푉 is the change in system volume. From Eq. 151, we can write

푑푉 푑푃 = 퐾 (− ) (152) 푉

If we insert Eq. 150 into Eq. 152 we obtain

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퐴푑푥 푑푃 = 퐾 (153) 푉

The net force on the air lump is

푑퐹 = −푑푃퐴 (154)

Because of the acting outward we used minus sign in Eq. 154. From last two equations we have

퐾퐴2 푑퐹 = −푑푥 (155) 푉

For a spring 퐹 = −푘푥 and the force constant is

푑퐹 푘 = − (156) 푑푥 Then

퐾퐴2 푘 = (157) 푉

With the help of mass and force constants, we can obtain frequency as

퐾퐴2 푘 퐾 퐴 푤 = √ = √ 푉 = √ √ (158) 푚 휌퐿퐴 휌 푉퐿

here √퐾⁄휌 is a form of the speed of sound. Finally, we can find cavity resonant frequency as

퐴 푤 = 푣 √ (159) 푠 푉퐿

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Frequency formula shows that, smaller opening gives lower frequency since air can rush in and out slower. Besides, smaller volume gives higher frequency because less air must move out to relieve a given pressure excess. Lastly, shorter neck gives higher frequency by reason of there is less resistance to air moving in and out [12].

3.1.4 Electrical Conductivity

Conductivity is about current flow through a material. In more detail, for a given electric field in a material, a lower conductivity material will produce less current flow than a high conductivity material.

Loss in power and conductivity are proportional. We use ‘lossless’ word for a zero conductivity material which are air, vacuum etc. If conductivity is bigger than zero, ‘loosy’ word can use for these materials that are salt water, silicon etc. Finally, some materials such as metals, copper, silver, etc. are named as ‘conductors’. Conductors have far greater conductivity which is approximately infinite. Table 1 contains conductivity value and classification of some materials [13].

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Table 1: Conductivity values of different materials [13]

Material 훔 [퐒/퐦] Classification

Silver 6.3 x 107 Conductor

Copper 6.0 x 107 Conductor

Aluminum 3.5 x 107 Conductor

Tungsten 1.8 x 107 Conductor

Nickel 1.4 x 107 Conductor

Iron 1.0 x 107 Conductor

Mercury 1.0 x 106 Conductor

Carbon 2.0 x 103 Lossy

Sea Water 4.8 Lossy

Germanium 2.17 Lossy

Silicon 1.6 x 10-3 Lossy

Glass ~10-12 Lossless

Rubber ~10-14 Lossless

Air ~10-15 Lossless

Teflon ~10-24 Lossless

Vacuum 0 Lossless

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To derive conductivity we should know that conductivity is the inverse of resistivity of material. Well then, what is resistivity exactly? The answer starts with Fig.29.

Figure 29: Simple circuit

푉 푅 = (160) 퐼

Here 푉 is potential in volt, 퐼 is current in ampere and 푅 is resistance. If we increase voltage, this increases the current and 푉/퐼 ratio stays same so increase in voltage never changes the resistance. In other words, resistance has a constant value and it changes only if we changes resistor material, makeup, size or dimensions. To understand better, the bigger view of resistor is shown in Fig.30.

Figure 30: View of resistor

Here 휌 is resistivity and specific for material. Resistance formula is

퐿 푅 = 휌 (161) 퐴

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Resistivity gives an idea of how much something naturally resists current and conductivity tells how much something naturally allows current. I mentioned that inverse of resistivity gives conductivity. Then the other form of resistance is

퐿 퐿 푅 = = (162) 휎퐴 휎휋푟2

In Eq. 162, 푟 is radius of resistor. Here 푅 is DC resistance for a conductor. At DC, charge carriers are equally separeted through the whole cross section area of resistor.

With the increase in frequency, the magnetic field at the inductor center increases and this causes an increase on the reactance near the center of resistor. Therefore, charges in resistor moves to edges from the center. Thus, the current density decreases at the center while it increases at the edges. This situation is explained as ‘skin effect’. Besides, ‘skin depth’ is the depth into conductor where the current density falls to 37% of its surface value. The skin depth formula is

1 훿 = (163) √휋푓휇휎 where 휇 is permeability, 푓 is frequency and 휎 is conductivity of the material.

Resistance and frequency are proportional to each other and skin depth dependent resistance is named as an AC resistance. Eq. 164 shows a formula for an AC resistance approximately.

퐿 푅푎푐 = (164) 휎퐴푎푐푡푖푣푒

here 퐴푎푐푡𝑖푣푒 is the skin depth area on the conductor and equals to 2휋푟훿 . Then Eq. 164 becomes

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퐿 퐿 푓휇 푟 푅 = = √ = (푅 ) (165) 푎푐 휎2휋푟훿 2푟 휋휎 푑푐 2훿

As seen in Eq. 165, AC resistance proportionally changes with the square root of frequency [14].

3.2 Observations of Change in Power about Effects of Some Parameters

We did our observations by taking generated power formula in Eq. 166 and a magnetron like as in Fig.31 into consideration.

2 ′ 2 푁퐿 퐴 푘휀0퐴 1 1 푃푔푒푛 = 2 (푣푠√ ′) ( ) (−푉0 ) (166) 2푀1 푉퐿 푑 푄퐿 푎 ln(푏⁄푎)

Here as mentioned before, 푁 is cavity number, 퐿 is distance between vane tips, 푀1 is gap factor and 푄퐿 is loaded quality factor. In this formula we use Eq. 149 instead ′ of angular resonant frequency term (푤0) so 퐿 is length of opening part of cavities in this equation. Besides, instead of the capacitance at vane tips term (퐶), the capacitance form in Eq. 145 is used. Lastly, we use more detailed form of 퐸푚푎푥 shown in Eq. 142.

For used magnetron as seen in Fig.31, distance between the vane tips (퐿) is 0.3 cm, anode radius (푏) is 2.9 cm, length of the opening part of the cavities (퐿′) is 0.6 cm, radius of cavities is 0,7 cm and the distance between magnetron center and the outermost point of cavity surface from the center is 4.9 cm. Cathode radius (푎), gap factor (푀1), the loaded quality factor (푄퐿) and cavity number (푁) will change according to our calculations.

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Figure 31: Used 8 cavity magnetron for our work

There are some other fixed parameters for generated power equation. Here speed of sound (푣푠) is taken as 340 m/s approximately, relative permittivity (푘) is 1 because of air, 푑 same with distance between vane tips so it is 0.3 cm and applied voltage

(푉0) is 6000 V.

2 Permittivity of space (휀0) equals to 1/푐 휇0. Here 푐 is speed of light (2.99 × 8 −6 10 푚/푠) and 휇0 is permeability of free space (1.26 × 10 푊푏⁄퐴 푚)[15]. Then we can find prober permittivity of space value for power equation as

1 1 퐴. 푠 휀 = = 0 푐2휇 푚 2 푊푏 푚. 푉 0 (2.99 × 108 ) (1.26 × 10−6 ) 푠 퐴. 푚 1 퐴. 푠 = 푚2 푉. 푠 푚. 푉 8.94 × 1016 × 1.26 × 10−6 푠2 퐴. 푚 퐴.푠 퐴.푠 휀 = 8.88 × 10−12 ≈ 9 × 10−12 (167) 0 푚.푉 푚.푉

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To calculate volume of the cavity (푉), we use 휋푟2ℎ formula. Here ℎ is height of magnetron. In Fig.31, one dimensional view of magnetron is seen. However, it also has a height and we take it as 5 cm. Then cavity volume is

푉 = 휋 × 푟2 × ℎ = 휋 × (0.7 푐푚)2 × 5 푐푚 = 7.697 푐푚3 (168)

The area of opening part of cavity (퐴) is calculated by multiplying height with the distance between the vane tips. Then opening part area is

퐴 = ℎ × 퐿 = 5 푐푚 × 0.3 푐푚 = 1.5 푐푚2 (169)

We used 퐴′ term for plate area in Eq. 145 and it equals

퐴′ = ℎ × 퐿′ = 5 푐푚 × 0.6 푐푚 = 3 푐푚2 (170)

3.2.1 Effect of Cavity Number on Generated Power

In this section, we observed that how generated power changes with respect to cavity number. We worked with Eq. 166 and we changed the cavity number from 4 to 12. Moreover, increment of resonant number was 2 because we should have even number of cavities in order that side-by-side segments have opposite poles. This was also shown in Fig.8. For a true observation, we kept fixed the other variables in the formula. The values of parameters which we used are shown in Table 2.

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Table 2: Values of variables for cavity number-power graph

Variable Value Variable Value

-12 푳 0.3 cm 휺ퟎ 9x10 A.s/m.V ′ 2 푴ퟏ 1 푨 3 cm

풗풔 340 m/s 풅 0.3 cm 2 푨 1.5 cm 푸푳 10 3 푽 7.697 cm 푽ퟎ 6000 V 푳′ 0.6 cm 풂 1.6 cm 풌 1 풃 2.9 cm

In graph, we used the data in Table 2 and created the cavity number versus generated power graph. This is shown in Fig.32 and it is understood that resonator number and generated power are directly proportional.

Figure 32: Cavity Number versus Generated Power Graph

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3.2.2 Effect of Gap Factor on Generated Power

After resonator number, we analyzed the effects of gap factor on generated power. We again used Eq. 166 and we took the gap factor values from 0.5 to 1.5. Moreover, increment of gap factor values was 0.01 in order to obtain more smoothly graph. For an accurate observation, we kept some variables fixed. The used variables can be seen in Table 3.

Table 3: Values of variables for gap factor-power graph

Variable Value Variable Value

-12 푵 8 휺ퟎ 9x10 A.s/m.V 푳 0.3 cm 푨′ 3 cm2

풗풔 340 m/s 풅 0.3 cm

2 푨 1.5 cm 푸푳 10

3 푽 7.697 cm 푽ퟎ 6000 V 푳′ 0.6 cm 풂 1.6 cm

풌 1 풃 2.9 cm

The curve of gap factor versus generated power is shown in Fig.33. As seen, when we increase the gap factor value, generated power decreases exponentially.

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Figure 33: Gap Factor versus Generated Power Graph

After that, we took the first and second derivatives of power with respect to gap factor. The first derivative of Eq. 166 is

2 ′ 2 ′ 푁퐿 퐴 푘휀0퐴 1 1 푃푔푒푛(푀1) = − ( 3 ) (푣푠√ ′) ( ) (−푉0 ) (171) 푀1 푉퐿 푑 푄퐿 푎 ln(푏⁄푎) and the second derivative of Eq. 166 is

2 ′ 2 ′′ 3푁퐿 퐴 푘휀0퐴 1 1 푃푔푒푛(푀1) = ( 4 ) (푣푠√ ′) ( ) (−푉0 ) (172) 푀1 푉퐿 푑 푄퐿 푎 ln(푏⁄푎)

From Eq. 171 and Eq. 172 we formed Fig.34 and Fig.35. In graph of first derivative, we saw that maximum changes are observed from 0.5 to 0.8. Moreover, it reached the saturating point at 1.5. About Fig.35, our aim was to see whether there is a maximum and a minimum points or not. If there was a peak in this graph, we would choose peak value as an optimized gap factor value but there was not.

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Figure 34: Gap Factor versus 1st Derivative of Power Graph

Figure 35: Gap Factor versus 2nd Derivative of Power Graph

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3.2.3 Effect of Loaded Quality Factor on Generated Power

Thirdly, we observed the changes in generated power when loaded quality factor value is changed. We used the loaded quality factor values between 5 and 15 and again we increased the value by 0.01 because of the same reasons before. The fixed values for Eq. 166 are listed in Table 4.

Table 4: Values of variables for loaded quality factor-power graph

Variable Value Variable Value 푵 8 풌 1 -12 푳 0.3 cm 휺ퟎ 9x10 A.s/m.V ′ 2 푴ퟏ 1 푨 3 cm

풗풔 340 m/s 풅 0.3 cm 2 푨 1.5 cm 푽ퟎ 6000 V 푽 7.697 cm3 풂 1.6 cm 푳′ 0.6 cm 풃 2.9 cm

We got the Fig.36 which shows the power changes according to loaded quality factor. It is seen that if we increase the loaded quality factor value, the generated power decreases exponentially.

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Figure 36: Loaded Quality Factor versus Generated Power Graph

Then we looked the derivatives of power with respect to loaded quality factor. After all, we obtained Eq. 173 for first derivative and Eq. 174 for second derivative.

2 ′ 2 ′ 1 푁퐿 퐴 푘휀0퐴 1 푃푔푒푛(푄퐿) = − 2 ( 2) (푣푠√ ′) ( ) (−푉0 ) (173) 푄퐿 2푀1 푉퐿 푑 푎 ln(푏⁄푎)

2 ′ 2 ′′ 1 푁퐿 퐴 푘휀0퐴 1 푃푔푒푛(푄퐿) = 3 ( 2 ) (푣푠√ ′) ( ) (−푉0 ) (174) 푄퐿 푀1 푉퐿 푑 푎 ln(푏⁄푎)

Thereafter, with the help of last two equations we obtain Fig. 37 and Fig. 38 for derivatives of generated power. In the graph of first derivative, we observed the maximum change gap as between 5 and 8. Moreover, from Fig.37 it is seen that 15 is saturating point for the loaded quality factor. Moreover, the graph of second derivative again did not give us a peak so there is not exact loaded quality factor value.

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Figure 37: Loaded Quality Factor versus 1st Derivative of Power Graph

Figure 38: Loaded Quality Factor versus 2nd Derivative of Power Graph

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3.2.4 Effect of Cathode Radius on Generated Power

After loaded quality factor, we looked for the effects of cathode radius. We changed the cathode radius value from 1 cm to 2.2 cm in Eq. 166. Again to have a smooth graphic, we used increments as 0.01 cm. The unchanged values for this part are shown in Table 5.

Table 5: Values of variables for cathode radius-power graph

Variable Value Variable Value 푵 8 풌 1 -12 푳 0.3 cm 휺ퟎ 9x10 A.s/m.V ′ 2 푴ퟏ 1 푨 3 cm

풗풔 340 m/s 풅 0.3 cm 2 푨 1.5 cm 푸푳 10 3 푽 7.697 cm 푽ퟎ 6000 V 푳′ 0.6 cm 풃 2.9 cm

With the help of the values in Table 5, we obtained Fig. 39 which is a graphic of cathode radius versus generated power. Here, we see an exponential increase in power so if we use bigger cathode, we get larger power.

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Figure 39: Cathode Radius versus Generated Power Graph

Then we took the derivative of generated power with respect to cathode radius. The first derivative is

2 푏 푁퐿2 퐴 푘휀 퐴′ 1 2푉0 (ln −1) 푃′ (푎) = (푣 √ ) ( 0 ) (− 푎 ) (175) 푔푒푛 2 푠 ′ 푏 3 2푀1 푉퐿 푑 푄퐿 (ln ) 푎3 푎

2 2 푏 푏 푁퐿2 퐴 푘휀 퐴′ 1 2푉0 (3(ln ) −5 ln( )+3) 푃′′ (푎) = (푣 √ ) ( 0 ) ( 푎 푎 ) (176) 푔푒푛 2 푠 ′ 푏 4 2푀1 푉퐿 푑 푄퐿 (ln ) 푎4 푎

After that, we formed the graph of cathode radius versus the first derivative of generated power as in Fig. 40 and the second derivative of generated power as in Fig. 41.

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Figure 40: Cathode Radius versus 1st Derivative of Power Graph

In the graph of first derivative, maximum change occurred between 1.8 cm and 2.2 cm and saturation point did not observed. Moreover, There was not any peak point in the graph of second derivative of power with respect to cathode radius like as previous second derivative graphs.

Figure 41: Cathode Radius versus 2nd Derivative of Power Graph

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3.2.5 Effect of Angular Resonant Frequency on Generated Power

In this section again we worked with Eq.140 which is

2 푁퐿 푤0퐶 2 푃푔푒푛 = 2 퐸푚푎푥 2푀1 푄퐿

Firstly, we fixed some parameters. We chose cavity number 푁 as 8, gap factor 푀1 as 1 and cathode radius 푎 as 2 cm. Before we mentioned that distance between the vane tips (퐿) is 0,3 cm for our magnetron design. We showed the maximum electric field in Eq.142 as

1 퐸 = −푉 푚푎푥 0 푎 ln 푏/푎

Here, anode radius 푏 was 2.9 cm and voltage value was 6000 volt. We just want to observe how the generated power changes with frequency so we can also fixed

퐶⁄푄퐿 term. The took loaded quality factor as 10 and the capacitance at vane tips constant is

−10 퐴.푠 2 푘휀 퐴′ 1×(9×10 )×(1.3 푐푚 ) 퐴.푠 퐴.푠 퐶 = 0 = 푐푚.푉 = 39 × 10−10 (177) 푑 0.3 푐푚 푉 푉

Then our equation became

−10 퐴. 푠 8 × (0.3 푐푚)2 (39 × 10 ) 푃 = × 푤 × 푉 푔푒푛 2 × 12 0 10 2 1 × (−6000 푉 × ) 2.9 푐푚 (2 푐푚) × ln 2 푐푚

−2 푃푔푒푛 = 푤0 × 0.68 × 10 푊. 푠 (178)

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Finally, we formed angular resonant frequency versus generated power figure (Fig. 42).

Figure 42: Angular Resonant Frequency versus Generated Power Graph

As seen, angular resonant frequency and generated power are directly proportional to each other.

In this chapter, we observed that applied voltage, resonator number, cathode radius and angular resonant frequency are directly proportional with generated power. On the other hand, increase in gap factor or loaded quality factor decreases the power generation.

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CHAPTER 4

CONCLUSION & DISCUSSION

Medical X-ray sources, radiotherapy, microwave heating, industrial heating, communication, warfare, cargo scanning and radar are some of the areas that used magnetron. In short, the usage area of magnetron is very wide. This causes that producing magnetrons with different types, geometries, sizes and power is necessary. The studies on magnetron started in 1912 and since then, it has been being improved a lot [5]. With the effect of changing and proceeding technology, studies on magnetron will continue to increase.

In this study, we have mentioned construction, working principle and0 critical parameters of magnetrons. Magnetrons are crossed-field devices and generate microwaves. The frequency range of microwaves is about between 300 MHz and 300 GHz. Its small size, light weight, low-cost and working with high efficiency make magnetrons most promising and popular high power microwave device. Simply, in the middle of it there is a cathode part, then there is an interaction space roundly and it ends up with an anode part which surrounds the interaction space (Fig. 5). All system starts at cathode. Electrons are spread from cathode and they are kept in the space with the help of electric, magnetic and AC fields. Electrons move directly from cathode to anode with the help of electric field which is in radial direction. Then, magnetic field which is perpendicular to electric field bends them. After all, we want to have very long electron path so electrons should stay in space very long time in order to have better and more efficient microwave generation. Therefore, electrons should have cycloidal motion in the interaction space of cylindrical magnetron and this motion is shown in Figure 43.

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Figure 43: Electron motion in magnetron [16]

The magnetron geometry and used materials are basic factors which effect efficiency. First, augmenting the cavity number is the one way of increasing the efficiency of the radiofrequency generation. Secondly, used cathode material can release electron in low temperature in order to increase lifetime of cathode so lifetime of magnetron. For example, barium oxide is better than tungsten because of the working ability in low temperatures and its high emission feature.

To understand the magnetron principle, firstly we got the bottom of the electron motion at the influence of different field. We explained physics behind of magnetron theoretically. Then, we continued to work about generated power of magnetron and checked theoretical facts numerically.

We started creating a magnetron with some specific dimensions (Fig. 31). We use this magnetron in our research. First parameter is cavity number. There should be connected alternate segments, in order that side-by-side segments have opposite poles. Therefore, they have even number of cavities. We observed that cavity number and generated power are directly proportional (Fig. 32). That means if we increase cavity number, generated power will also increase. Therefore, there is not a best working space so if we augment the cavities then generated power continues to increase. For our magnetron, when we use cavity number as 4, generated power becomes 12.5 × 10−3푊 and 37.4 × 10−3푊 for

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12-cavity magnetron. Second one is the gap factor. Gap factor is a unitless parameter used for a specific operation mode which causes the greatest output power. We changed the gap factor value and drew a curve (Fig. 33). Here, power and gap factor were inversely proportional. In other words, power decreases exponentially when we increase the gap factor value. For used magnetron, generated power was 9.99 × 10−2푊 if gap factor was 0.5 and 1.11 × 10−2푊 for gap factor value of 1.5. Then, we took the derivative of power with respect to gap factor in order to find the fastest change (Fig. 34). In graph of first derivative, we saw that maximum changes are observed from 0.5 to 0.8. When gap factor equaled to 0.8, power was 3.9 × 10−2푊. In other words, average change was about 2.03 × 10−2푊 for each 0.1 increase in gap factor until the value of 0.8. However, after 0.8, for each 0.1 increase, the average change became 0.4 × 10−2푊. Moreover, it reached the saturating point at 1.5 so after this value increase in gap factor do not change power effectively (Fig. 34). Lastly, we looked the second derivative of power with respect to gap factor. Our aim was to see whether there is a maximum and a minimum points or not. If there was a peak in this graph, we would choose peak value as an optimized gap factor value but there was not (Fig. 35). The loaded quality factor is the third parameter. When we increase the loaded quality factor value, we observed an exponential decrease in generated power (Fig. 36). We got 49.9 × 10−3푊 power when the loaded quality factor was 5 and 16.6 × 10−3푊 at 15. Then we looked first derivative graph as before and determined the maximum change gap as between 5 and 8 (Fig. 37). Between 5 and 8, the average change was about 6.2 × 10−3푊 and after that it became 2.1 × 10−3푊 approximately. In other words, after the value of 8, the power change is not too much and after the value of 15, it does not change. Therefore, 15 is saturating point for the loaded quality factor (Fig. 37). Moreover, second derivative again did not give us a peak so there is not exact loaded quality factor value (Fig. 38). If we make a comparison between gap factor graphs and loaded quality factor graphs, change in gap factor affects the power more. For example, when we doubled gap factor, generated power decrease became 7.7 × 10−2푊 and the loaded quality factor is doubled,

77 change in generated power was 2,4 × 10−2푊. This also seen in theoretical power equation (Eq. 166). As seen in equation, gap factor used in square form. Fourth parameter is cathode radius. This time figure gave as an exponential increase on generated power when we used larger cathode (Fig. 39). When cathode radius was 1 cm, power generation became 19,9 × 10−3푊 and 61.2 × 10−3푊 if it was 2.2 cm. In the graph of first derivative, maximum change occurred between 1.8 cm and 2.2 cm (Fig. 40). The average change was about 1.35 × 10−3푊 between 1 cm and 1.8 cm while it was 7.63 × 10−3푊 approximately between 1.8 cm and 2.2 cm. When we doubled the cathode radius, change in power became 2.1 × 10−2푊. Therefore, by looking our magnetron, we can say that cathode radius changes are less effective on power when it is compared with gap factor and it has almost same effect on power generation with loaded quality factor. Moreover, in derivative graph, saturation point did not observed (Fig. 40). There was not any peak point in the graph of second derivative of power with respect to cathode radius like as previous second derivative graphs (Fig. 41). Angular resonant frequency is the last parameter that we examine. We used constant values for variables in power equation except frequency and we came up to a linear equation (Eq. 178). Then, we obtained a power versus frequency figure which shows they are directly proportional to each other (Fig. 42). Therefore, there is not any working space. If we continue to increase frequency, generated power is also increases.

Thus far, we told about what we did but we also have some plans in the matter of what we do next and there are some aims and matters that can be clarified. Firstly, we can work with 2D and then 3D simulations of magnetron that have different shapes or sizes and learn the working principle and effects of different parameters visually so more clearly. Before we said that barium oxide cathode is better than tungsten one. Then secondly, material properties can be studied more in detail. Therefore, we can do numerical or visual observations about which material is more useful or how much effects they have exactly.

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In conventional magnetrons, applied voltage is about hundred volts to kilovolts and power levels are about watts to kilowatts. On the other side, for relativistic magnetrons these becomes several hundred kilovolts to megavolts and generates a power at gigawatts [17]. Therefore, we can use the description of cavity magnetron’s extrapolation for relativistic magnetrons. Variously from conventional magnetrons, efficiency for relativistic magnetrons is about 10- 40%. Mode control which is improved and better intellection of phase-locking can be solutions of this problem. Moreover, higher voltage operation with a magnetron and multiple magnetrons’ phase locked operation can cause an increase in power. In the future we will have more powerful and compact ones [18].

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