an optimization study on cavity magnetron a thesis

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:


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 ________________


Assoc. Prof. Dr. İsmail Rafatov ________________


Assoc. Prof. Dr. Alpan Bek ________________


Assoc. Prof. Dr. Kemal Efe Eseller ________________

Electrical& Electronics Engineering Department,

Atilim University

Date: ________________

iv

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 :

v

ABSTRACT


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

vi

number of resonator and higher angular resonant frequency provide much more

power generation.

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

frequency

vii

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,

viii

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

ix

To my family

x

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.

xi

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


xii

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


2.2 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

xiii

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

xiv

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

xv

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

xvi

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

xvii

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

1

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.

2

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).

3

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.

4

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]

5

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].

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

A B C

6

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]

7

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

8

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.

9

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 (�⃗� field) and sometimes the reason is the

magnetic field (�⃗� 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 �⃗� -�⃗� fields separately, �⃗� -�⃗� fields

together or all three fields which are �⃗� field, �⃗� 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 �⃗� , then |𝐸| is equal to V/d (d is the distance between two

electrodes). Because of the field, a force (𝐹 ) is formed on an electron. This force

can be written as

𝐹 = 𝑞�⃗� (1)

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

10

𝐹 = −𝑒�⃗� (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 𝑣 , then the force on an

electron become

𝐹 = −𝑒(𝑣 × �⃗� ) (3)

This means, we can talk about the magnetic field effect if the particle is moving.

So 𝑣 ≠ 0 is the case for the �⃗� field effect. Here the force is perpendicular not only

to electron velocity, but also to �⃗� 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

𝐹 = −𝑒[�⃗� + (𝑣 × �⃗� )] (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 𝐸′ cos(𝑤𝑡). Let’s assume this field

direction is same with the �⃗� field direction. Then, Eq.4 becomes

𝐹 = −𝑒[(�⃗� + �⃗� ′ cos𝑤𝑡) + (𝑣 × �⃗� )] (5)

Here 𝐸′ is the value for AC field and 𝑤 is used for the angular frequency. So force

in Eq.5 is caused by E field, B field and AC field.

11

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

𝐹 = −𝑒�⃗� =𝑑

𝑑𝑡(𝑚𝑣 ) = 𝑚

𝑑�⃗�

𝑑𝑡 (6)

Here, 𝑣 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

−𝑒𝐸𝑥 = 𝑚𝑑𝑣𝑥

𝑑𝑡= 𝑚

𝑑2𝑥

𝑑𝑡2 (7)

−𝑒𝐸𝑦 = 𝑚𝑑𝑣𝑦

𝑑𝑡= 𝑚

𝑑2𝑦

𝑑𝑡2 (8)

−𝑒𝐸𝑧 = 𝑚𝑑𝑣𝑧

𝑑𝑡= 𝑚

𝑑2𝑧

𝑑𝑡2 (9)

Here x, y and z are components of position vector and 𝑣𝑥, 𝑣𝑦 ve 𝑣𝑧 are components

of the velocity vector.

In Fig. 12a plates are located at 𝑥 = 0 and 𝑥 = 𝑑. Potentials are equal to zero for

bottom plate and 𝑉0 for upper plate. Let’s assume that when electron enters the E

12

field at time t=0, its position is at x=y=z=0 and its initial velocities are 𝑣𝑥 = 𝑣𝑥0,

𝑣𝑦 = 𝑣𝑦0 and 𝑣𝑧 = 0. For E field, 𝐸𝑥 equals to −𝑉0 𝑑⁄ 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

𝑑𝑣𝑥

𝑑𝑡=

𝑑2𝑥

𝑑𝑡2 = −𝑒

𝑚𝐸𝑥 = −

𝑒

𝑚(−

𝑉0

𝑑) =

𝑒𝑉0

𝑚𝑑= 𝑘 (10)

𝑑𝑣𝑦

𝑑𝑡=

𝑑2𝑦

𝑑𝑡2 = 0 (11)

𝑑𝑣𝑧

𝑑𝑡=

𝑑2𝑧

𝑑𝑡2 = 0 (12)

Let’s solve these last three equations:

From equations 10, 11 and 12, we can write

𝑥 =1

2𝑘𝑡2 + 𝐴1𝑡 + 𝐵1 (13a)

𝑦 = 𝐴2𝑡 + 𝐵2 (13b)

13

𝑧 = 𝐴3𝑡 + 𝐵3 (13c)

Initially, we know that at 𝑡 = 0, 𝑥 = 𝑦 = 𝑧 = 0. Then this means that 𝐵1 = 𝐵2 =

𝐵3 = 0 and Eq.13 becomes

𝑥 =1

2𝑘𝑡2 + 𝐴1𝑡 (14a)

𝑦 = 𝐴2𝑡 (14b)

𝑧 = 𝐴3𝑡 (14c)

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

𝑑𝑥

𝑑𝑡= 𝑘𝑡 + 𝐴1 (15a)

𝑑𝑦

𝑑𝑡= 𝐴2 (15b)

𝑑𝑧

𝑑𝑡= 𝐴3 (15c)

Another condition is that at 𝑡 = 0, 𝑣𝑥 = 𝑣𝑥0, 𝑣𝑦 = 𝑣𝑦0

and 𝑣𝑧 = 0 so

𝐴1 = 𝑣𝑥0 (16a)

𝐴2 = 𝑣𝑦0 (16b)

𝐴3 = 0 (16c)

Then Eq.14 reduces to

𝑥 =1

2𝑘𝑡2 + 𝑣𝑥0

𝑡 (17a)

14

𝑦 = 𝑣𝑦0𝑡 (17b)

𝑧 = 0 (17c)

We know that 𝑘 = 𝑒𝑉0 𝑚𝑑⁄ , then Eq.17 can be written as

𝑥 = (𝑒𝑉0

2𝑚𝑑) 𝑡2 + 𝑣𝑥0

𝑡 and 𝑣𝑥0𝑡 = 𝑥 − (

𝑒𝑉0

2𝑚𝑑) 𝑡2 (18a)

𝑦 = 𝑣𝑦0𝑡 and 𝑡 =

𝑦

𝑣𝑦0

(18b)

If we use the ‘𝑡’ value in Eq.18b, Eq.18a becomes

𝑥 = (𝑒𝑉0

2𝑚𝑑) (

𝑦

𝑣𝑦0

)2

+ 𝑣𝑥0(

𝑦

𝑣𝑦0

) (19)

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

𝑣𝑥 =𝑑𝑥

𝑑𝑡= (

𝑒𝑉0

𝑚𝑑) 𝑡 + 𝑣𝑥0

(20a)

𝑣𝑦 =𝑑𝑦

𝑑𝑡= 𝑣𝑦0

(20b)

𝑣𝑧 =𝑑𝑧

𝑑𝑡= 0 (20c)

Then

𝑣 = √(𝑣𝑥2 + 𝑣𝑦

2) = √𝑣𝑦02 + [(

𝑒𝑉0

𝑚𝑑) 𝑡 + 𝑣𝑥0

]2 (21a)

𝑣 = √𝑣𝑦02 + [(

𝑒𝑉0

𝑚𝑑)2𝑡2 + 2(

𝑒𝑉0

𝑚𝑑) 𝑣𝑥0

𝑡 + 𝑣𝑥02 ] (21b)

15

If we use 𝑣𝑥0𝑡 = 𝑥 − (𝑒𝑉0/2𝑚𝑑)𝑡2 in Eq.21, we get

𝑣 = √𝑣𝑦02 + (

𝑒𝑉0

𝑚𝑑)2𝑡2 + (

2𝑒𝑉0

𝑚𝑑) 𝑥 − (

𝑒𝑉0

𝑚𝑑)2𝑡2 + 𝑣𝑥0

2 (22a)

𝑣 = √𝑣𝑥02 + 𝑣𝑦0

2 + (2𝑒𝑉0

𝑚𝑑) 𝑥 (22b)

The kinetic energy at 𝑡 = 0 is 𝐾𝐸0. When 𝑡 equals to zero and 𝑥 = 0 if we use

Eq.22b then the kinetic energy becomes

𝐾𝐸0 =1

2𝑚𝑣2 =

1

2𝑚(𝑣𝑥0

2 + 𝑣𝑦02 ) (23)

At any time ‘𝑡’, the kinetic energy is written as

𝐾𝐸𝑡 =1

2𝑚 [𝑣𝑥0

2 + 𝑣𝑦02 + (

2𝑒𝑉0

𝑚𝑑) 𝑥] (24)

The difference between Eq.23 and Eq.24 gives the gained energy in time ‘𝑡’ and

this is

∆𝐾𝐸 =1

2𝑚(

2𝑒𝑉0

𝑚𝑑) 𝑥 = (

𝑒𝑉0

𝑑) 𝑥 (25)

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

−𝑒𝑉 = −𝑒 (𝑉0

𝑑) 𝑥 = −∆𝐾𝐸 (26)

The minus sign in Eq.26 means that any decrease in potential energy is compensated

by the increase in 𝐾𝐸.

16

If initially velocities are taken as zero, this means that 𝑣𝑥𝑜= 𝑣𝑦𝑜

= 0, then

𝑣 = √2𝑒𝑉 𝑚⁄ (27)

where 𝑉 = 𝐸. 𝑥 = 𝑉0. 𝑥 𝑑⁄ 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

(𝑒 = 1.602 × 10−19 C and 𝑚 = 9.1091 × 10−31 kg) into the Eq.27, we obtain

𝑣 = 5.932 × 105√𝑉0𝑥

𝑑 𝑚/𝑠 (28)

For example, we can find the velocity at 𝑥 = 𝑑 as

𝑣 = 0.5932 × 106√𝑉0 𝑚/𝑠 (29)

It is mentioned that 𝐹 = 𝑞�⃗� 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.

17

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, 𝑣 is three

dimensional and 𝑣 = 𝑣𝑟𝑎�̂� + 𝑣∅𝑎∅̂ + 𝑣𝑧𝑎�̂�.Then, Eq.6 can be rewritten as

−𝑒

𝑚�⃗� =

𝑑

𝑑𝑡(𝑣𝑟𝑎�̂� + 𝑣∅𝑎∅̂ + 𝑣𝑍𝑎�̂�) =

𝑑

𝑑𝑡(𝑣𝑟𝑎�̂�) +

𝑑

𝑑𝑡(𝑣∅𝑎∅̂) +

𝑑

𝑑𝑡(𝑣𝑧𝑎�̂�) (30)

Here 𝑣𝑟,∅,𝑧 are velocities and 𝑣𝑟 = 𝑑𝑟 𝑑𝑡⁄ , 𝑣∅ = 𝑟𝑑∅ 𝑑𝑡⁄ and 𝑣𝑍 = 𝑑𝑧 𝑑𝑡⁄ .

Moreover, 𝑎𝑟,∅,𝑧 are the unit vectors.

Three terms in Eq.30 become

𝑑

𝑑𝑡(𝑣𝑟𝑎�̂�) = 𝑣𝑟

𝑑∅

𝑑𝑡𝑎∅̂ +

𝑑𝑣𝑟

𝑑𝑡𝑎�̂� (31)

𝑑

𝑑𝑡(𝑣∅𝑎∅̂) = −𝑣∅

𝑑∅

𝑑𝑡𝑎�̂� +

𝑑𝑣∅

𝑑𝑡𝑎∅̂ (32)

𝑑

𝑑𝑡(𝑣𝑧𝑎�̂�) =

𝑑𝑣𝑧

𝑑𝑡𝑎�̂� (33)

18

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

−𝑒

𝑚𝐸𝑟 =

𝑑𝑣𝑟

𝑑𝑡− 𝑣∅

𝑑∅

𝑑𝑡 (34)

−𝑒

𝑚𝐸∅ = 𝑣𝑟

𝑑∅

𝑑𝑡+

𝑑𝑣∅

𝑑𝑡 (35)

−𝑒

𝑚𝐸𝑧 =

𝑑𝑣𝑧

𝑑𝑡 (36)

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

𝑣 can be written as

𝑣 = 𝑣𝑟𝑎�̂� + 𝑣∅𝑎∅̂ + 𝑣𝑧𝑎�̂� =𝑑𝑟

𝑑𝑡𝑎�̂� +

𝑟𝑑∅

𝑑𝑡𝑎∅̂ +

𝑑𝑧

𝑑𝑡𝑎�̂� (37)

Here 𝑑∅ 𝑑𝑡 = 𝑤⁄ and 𝑣∅ = 𝑟𝑤, then equations 34, 35 and 36 transform to

−𝑒

𝑚𝐸𝑟 =

𝑑2𝑟

𝑑𝑡2 − 𝑟𝑤2 (38a)

−𝑒

𝑚𝐸∅ = 𝑤

𝑑𝑟

𝑑𝑡+

𝑑(𝑤𝑟)

𝑑𝑡=

1

𝑟

𝑑

𝑑𝑡(𝑟2𝑤) (38b)

−𝑒

𝑚𝐸𝑧 =

𝑑2𝑧

𝑑𝑡2 (38c)

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

𝑉 = 𝑉0ln 𝑟/𝑎

ln𝑏/𝑎 (39)

here 𝑎 is cathode radius, 𝑏 is anode radius and

19

𝐸𝑟 = −𝜕𝑉

𝜕𝑟= −𝑉0

1

𝑟 ln𝑏/𝑎 (40a)

𝐸∅ = 𝐸𝑧 = 0 (40b)

If we consider Eq.40, Eq.38 becomes

−𝑒𝑉0

𝑚𝑟 ln𝑏/𝑎=

𝑘

𝑟=

𝑑2𝑟

𝑑𝑡2 − 𝑟𝑤2 (41a)

𝑑(𝑟2𝑤)

𝑑𝑡= 0 (41b)

𝑑2𝑧

𝑑𝑡2 = 0 (41c)

Let’s assume, an electron which initially has a velocity 𝑣 = 0 enters the E-field at

𝑡 = 0 and its position is 𝑟 = 𝑎, ∅ = 0 and 𝑧 = 0. From Eq.38c, 𝑧 is zero for all ‘𝑡’

values. Besides, for Eq.38b assume that 𝑟2𝑤 = 𝑟 × 𝑟𝑤 = 𝑟 × 𝑣∅ = 𝐴. We know

that 𝑣 = 0 when 𝑡 equals to zero, then 𝐴 = 0 or 𝑤 = 0 for all ‘𝑡’ values.

Also 𝑣𝑟 = 𝑑𝑟 𝑑𝑡⁄ means that 𝑑𝑡 equals to 𝑑𝑟 𝑣𝑟⁄ . Then, Eq.38a can be written as

𝑑2𝑟

𝑑𝑡2 =𝑑𝑣𝑟

𝑑𝑡=

𝑘

𝑟 (42a)

𝑑𝑣𝑟 = (𝑘

𝑟) 𝑑𝑡 = (

𝑘

𝑟) (

𝑑𝑟

𝑣𝑟) or 𝑣𝑟𝑑𝑣𝑟 = (

𝑘

𝑟) 𝑑𝑟 (42b)

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

1

2𝑣𝑟

2 = 𝑘 ln 𝑟 + 𝐵 (43)

When we use the condition 𝑣𝑟 equals to zero at 𝑟 = 𝑎, value of 𝐵 becomes −𝑘 ln 𝑎.

Therefore,

20

1

2𝑣𝑟

2 =kln (𝑟

𝑎) (44a)

So

𝑣𝑟 = √[2𝑘 ln (𝑟

𝑎)] (44b)

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

𝑣𝑟 =𝑑𝑟

𝑑𝑡= √

2𝑒𝑉0 ln(𝑟

𝑎)

𝑚 ln(𝑏

𝑎)

(45)

When we solve the equations 34, 35 and 36, we get Eq.45. Also we mentioned

that 𝑒 = 1.602 × 10−19 C and 𝑚 = 9.1091 × 10−31 kg. Then Eq.45 becomes

𝑣𝑟 = [5.932 × 105√ln(𝑟 𝑎⁄ )

ln(𝑏 𝑎⁄ )]√𝑉0 𝑚/𝑠 (46)

If electron is at the cathode surface so if 𝑟 = 𝑎, the velocity value (𝑣) becomes zero.

However, when electron reaches to the anode surface, Eq.46 becomes

𝑣𝑟 = 5.932 × 105√𝑉0 𝑚/𝑠 (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].

21

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

𝐹 = −𝑒(𝑣 × �⃗� ) =𝑑

𝑑𝑡(𝑚𝑣 ) = 𝑚

𝑑�⃗�

𝑑𝑡 (48)

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


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

different velocities

22

The structure of this case is shown in Fig. 15a. Here velocity is again there

component and it can be written like 𝑣 = 𝑣𝑥𝑎�̂� + 𝑣𝑦𝑎�̂� + 𝑣𝑧𝑎�̂�. Likewise, the

magnetic field vector is �⃗� = 𝐵𝑥𝑎�̂� + 𝐵𝑦𝑎�̂� + 𝐵𝑧𝑎�̂�.

When we use 𝑣 and �⃗� in Eq.48 with all these three components, we obtain

𝑑𝑣𝑥

𝑑𝑡=

−𝑒

𝑚(𝑣𝑦𝐵𝑧 − 𝑣𝑧𝐵𝑦) (49)

𝑑𝑣𝑦

𝑑𝑡=

−𝑒

𝑚(𝑣𝑧𝐵𝑥 − 𝑣𝑥𝐵𝑧) (50)

𝑑𝑣𝑧

𝑑𝑡=

−𝑒

𝑚(𝑣𝑥𝐵𝑦 − 𝑣𝑦𝐵𝑥) (51)

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

𝑑2𝑥

𝑑𝑡2 =−𝑒

𝑚(𝐵𝑧

𝑑𝑦

𝑑𝑡− 𝐵𝑦

𝑑𝑧

𝑑𝑡) (52)

𝑑2𝑦

𝑑𝑡2 =−𝑒

𝑚(𝐵𝑥

𝑑𝑧

𝑑𝑡− 𝐵𝑧

𝑑𝑥

𝑑𝑡) (53)

𝑑2𝑧

𝑑𝑡2 =−𝑒

𝑚(𝐵𝑦

𝑑𝑥

𝑑𝑡− 𝐵𝑥

𝑑𝑦

𝑑𝑡) (54)

Let’s find the solution of equations from 49 to 54:

Assume that �⃗� = 𝐵0𝑎𝑧 ̂and an electron that has the velocity 𝑣 = 𝑣𝑦0𝑎�̂� initially

enters the magnetic field at the position of 𝑥 = 𝑦 = 𝑧 = 0. Also 𝑧 is zero for all

times because 𝑧 component of 𝑣 is zero and 𝑥 component of 𝑣 is zero at 𝑡 = 0. Then

Eq.52, Eq.53 and Eq.54 transform to

𝑑2𝑥

𝑑𝑡2 = −𝑒

𝑚𝐵0

𝑑𝑦

𝑑𝑡 (55a)

23

so

𝑑𝑦

𝑑𝑡= −

𝑚

𝑒𝐵0

𝑑2𝑥

𝑑𝑡2 (55b)

𝑑2𝑦

𝑑𝑡2 =𝑒

𝑚𝐵0

𝑑𝑥

𝑑𝑡 (55c)

so

𝑑𝑥

𝑑𝑡=

𝑚

𝑒𝐵0

𝑑2𝑦

𝑑𝑡2 (55d)

and

𝑑2𝑧

𝑑𝑡2 = 0 (55e)

If we use 𝑑𝑦 𝑑𝑡⁄ value in Eq.55c and 𝑑𝑥 𝑑𝑡⁄ value in Eq.55a, we get

𝑑2𝑦

𝑑𝑡2 = −𝑚

𝑒𝐵0

𝑑3𝑥

𝑑𝑡3 =𝑒𝐵0

𝑚

𝑑𝑥

𝑑𝑡 (56a)

𝑑2𝑥

𝑑𝑡2 =𝑚

𝑒𝐵0

𝑑3𝑦

𝑑𝑡3 = −𝑒𝐵0

𝑚

𝑑𝑦

𝑑𝑡 (56b)

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

𝑑2𝑣𝑥

𝑑𝑡2 = −(𝑒𝐵0

𝑚)2𝑣𝑥 and

𝑑2𝑣𝑥

𝑑𝑡2 + 𝑤02𝑣𝑥 = 0 (57a)

𝑑2𝑣𝑦

𝑑𝑡2 = −(𝑒𝐵0

𝑚)2𝑣𝑦 and

𝑑2𝑣𝑦

𝑑𝑡2 + 𝑤02𝑣𝑦 = 0 (57b)

where 𝑤0 = 𝑒𝐵0/𝑚.

The solution of Eq.57 is

𝑣𝑥 = 𝐴1 cos𝑤0𝑡 + 𝐵1 sin𝑤0𝑡 (58a)

24

𝑣𝑦 = 𝐴2 cos𝑤0𝑡 + 𝐵2 sin𝑤0𝑡 (58b)

Use conditions;

When 𝑡 = 0, 𝑣𝑥 is also zero then 𝐴1 = 0. So Eq.58a turns to

𝑣𝑥 = 𝐵1 sin𝑤0𝑡 (59a)

When 𝑡 = 0, 𝑣𝑦 equals to 𝑣𝑦0 then 𝐴2 = 𝑣𝑦0

. So Eq.58b turns to

𝑣𝑦 = 𝑣𝑦0cos𝑤0𝑡 + 𝐵2 sin𝑤0𝑡 (59b)

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

𝑑2𝑥

𝑑𝑡2 = −𝑒

𝑚𝐵0

𝑑𝑦

𝑑𝑡 so

𝑑𝑣𝑥

𝑑𝑡= −

𝑒

𝑚𝐵0𝑣𝑦 = −𝑤0𝑣𝑦 (60a)

and

𝑑2𝑦

𝑑𝑡2 =𝑒

𝑚𝐵0

𝑑𝑥

𝑑𝑡 so

𝑑𝑣𝑦

𝑑𝑡=

𝑒

𝑚𝐵0𝑣𝑥 = 𝑤0𝑣𝑥 (60b)

If we put 𝑣𝑥 and 𝑣𝑦 values into Eq.60, we find

𝑑𝑣𝑥

𝑑𝑡= −𝑤0𝑣𝑦 → 𝑤0𝐵1 cos𝑤0𝑡 = −𝑤0(𝑣𝑦0

cos𝑤0𝑡 + 𝐵2 sin𝑤0𝑡) (61a)

𝑑𝑣𝑦

𝑑𝑡= 𝑤0𝑣𝑥 → −𝑤0𝑣𝑦0 sin𝑤0𝑡 + 𝑤0𝐵2 cos𝑤0𝑡 = 𝑤0𝐵1 sin𝑤0𝑡 (61b)

Eq.61 can be used for all times. So at 𝑡 = 0, 𝐵1 is −𝑣𝑦0 and 𝐵2 is zero. Then, 𝑣𝑥

and 𝑣𝑦 become

25

𝑣𝑥 = −𝑣𝑦0sin𝑤0𝑡 (62a)

𝑣𝑦 = 𝑣𝑦0cos𝑤0𝑡 (62b)

so

𝑣 = √(𝑣𝑥2 + 𝑣𝑦

2) = 𝑣𝑦0 (62c)

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

from Eq.62

𝑥 = (𝑣𝑦0

𝑤0) cos𝑤0𝑡 + 𝐶1 (63a)

𝑦 = (𝑣𝑦0

𝑤0) sin𝑤0𝑡 + 𝐶2 (63b)

At 𝑡 = 0, 𝑥 and 𝑦 equal to zero so 𝐶1 = −𝑣𝑦0/𝑤0 and 𝐶2 = 0. Then Eq.63 turns to

𝑥 = (𝑣𝑦0

𝑤0) (cos𝑤0𝑡 − 1) (64a)

𝑦 = (𝑣𝑦0

𝑤0) sin𝑤0𝑡 (64b)

These found solutions are a circle’s parametric equations. The radius of this circle

(r) equals to 𝑣𝑦0𝑤0⁄ and also

𝑣𝑦0𝑤0⁄ = 𝑣 𝑤0⁄ = 𝑚𝑣 𝑒𝐵0⁄ (65)

where 𝑤0 =𝑒𝐵𝑜

𝑚 . The circle center is at 𝑥 = −𝑟 2⁄ and 𝑦 = 𝑟 2⁄ .

26

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

𝑣 = 𝑎𝑤0 =𝑎𝑒𝐵0

𝑚. (66)

The radius of the path of particle is

𝑟 =𝑚𝑣

𝑒𝐵𝑜. (67)

The cyclotron angular frequency caused by the circular motion is

𝑤0 =𝑣

𝑎=

𝑒𝐵0

𝑚. (68)

The period of the one turn completely is

𝑇 =2𝜋

𝑤0=

2𝜋𝑚

𝑒𝐵𝑜. (69)

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

27

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.


When we take the velocity as 𝑣 = 𝑣𝑟𝑎�̂� + 𝑣∅𝑎∅̂ + 𝑣𝑧𝑎�̂� and the magnetic flux

density as �⃗� = 𝐵𝑟𝑎�̂� + 𝐵∅𝑎∅̂ + 𝐵𝑧𝑎�̂�, we can write the components of Eq.3 as

−𝑒

𝑚(𝑣∅𝐵𝑧 − 𝑣𝑧𝐵∅) =

𝑑𝑣𝑟

𝑑𝑡− 𝑣∅

𝑑∅

𝑑𝑡 (70)

−𝑒

𝑚(𝑣𝑧𝐵𝑟 − 𝑣𝑟𝐵𝑧) =

𝑑𝑣∅

𝑑𝑡+ 𝑣𝑟

𝑑∅

𝑑𝑡 (71)

−𝑒

𝑚(𝑣𝑟𝐵∅ − 𝑣∅𝐵𝑟) =

𝑑𝑣𝑧

𝑑𝑡 (72)

It is assumed that magnetic field has only one component so �⃗� = 𝐵0𝑎�̂� as shown in

Fig. 16a and an electron with a velocity of 𝑣 = 𝑣𝑟0𝑎�̂� enters the environment of

magnetic field at 𝑟 = 𝑎 and ∅ = 𝑧 = 0. The particle does not move in the z-

direction because 𝑣𝑧 = 𝑑𝑧 𝑑𝑡⁄ = 0. Moreover, 𝑣∅ equals to 𝑟𝑤 because initially

velocity has not a component in the ∅ direction and 𝑑∅ 𝑑𝑡⁄ is 𝑤.

28

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

with different velocities

Then our equations 70, 71 and 72 become

−𝑒

𝑚𝐵0𝑤𝑟 =

𝑑2𝑟

𝑑𝑡2 − 𝑟𝑤2 (73)

𝑒

𝑚𝐵0

𝑑𝑟

𝑑𝑡=

1

𝑟

𝑑

𝑑𝑡(𝑟2𝑤) (74)

𝑑𝑣𝑧

𝑑𝑡= 0 (75)

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

29

𝑑2𝑥

𝑑𝑡2 = −𝑒

𝑚(𝐸𝑥 + 𝐵𝑧

𝑑𝑦


𝑑𝑧

𝑑𝑡) (76)

𝑑2𝑦

𝑑𝑡2 = −𝑒

𝑚(𝐸𝑦 + 𝐵𝑥

𝑑𝑧


𝑑𝑥

𝑑𝑡) (77)

𝑑2𝑧

𝑑𝑡2 = −𝑒

𝑚(𝐸𝑧 + 𝐵𝑦

𝑑𝑥


𝑑𝑦

𝑑𝑡) (78)

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

Besides, E field components are 𝐸𝑥 = −𝑉0 𝑑⁄ , 𝐸𝑦 = 0, 𝐸𝑧 = 0 and for B field �⃗� =

𝐵0𝑎�̂�, 𝐵𝑥 = 𝐵𝑦 = 0. Initially electron has a velocity of 𝑣 = 𝑣𝑦0𝑎�̂� at 𝑥 = 𝑦 = 𝑧 =

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

𝑑2𝑥

𝑑𝑡2 = −𝑒

𝑚(𝐸𝑥 + 𝐵𝑧

𝑑𝑦


𝑑𝑧

𝑑𝑡) (79)

𝑑2𝑦

𝑑𝑡2 = −𝑒

𝑚(𝐵𝑥

𝑑𝑧


𝑑𝑥

𝑑𝑡) (80)

𝑑2𝑧

𝑑𝑡2 = −𝑒

𝑚(𝐵𝑦

𝑑𝑥


𝑑𝑦

𝑑𝑡) (81)

In the cylindrical coordinate system this equations changes because of the

components and become

𝑑2𝑟

𝑑𝑡2 − 𝑟𝑤2 =𝑒

𝑚(𝐸𝑟 − 𝐵𝑧𝑤𝑟 − 𝐵∅

𝑑𝑧

𝑑𝑡) (82)

1

𝑟

𝑑

𝑑𝑡(𝑟2𝑤) = −

𝑒

𝑚(𝐸∅ + 𝐵𝑟

𝑑𝑧


𝑑𝑟

𝑑𝑡) (83)

𝑑2𝑧

𝑑𝑡2 = −𝑒

𝑚(𝐸𝑧 + 𝐵∅

𝑑𝑟

𝑑𝑡− 𝐵𝑟𝑤𝑟) (84)

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.

30

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 𝐵 = 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].

31

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

𝐹 = −𝑒[(�⃗� + �⃗� ′ cos𝑤𝑡) + (𝑣 × �⃗� )]

32


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

A parallel plate magnetron is shown in Fig. 19. Plates have 𝑉0 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 𝐵𝑧 component. Additionally, a potential that changes with time is applied. This

potential is 𝑉1 cos𝑤𝑡. 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

𝐸𝑦 = 𝐸𝑧 = 0 and �⃗� = 𝐸𝑥 = −𝑉

𝑑= (−

𝑉0

𝑑) [1 + (

𝑉1

𝑉0) cos𝑤𝑡] 𝑎�̂�, (85)

�⃗� = (−𝑉0

𝑑) [1 + 𝛼 cos𝑤𝑡]𝑎�̂� where 𝛼 = (

𝑉1

𝑉0), (86)

𝐵𝑦 = 𝐵𝑥 = 0 and �⃗� = 𝐵0𝑎�̂�. (87)

Then we obtain

𝑑2𝑥

𝑑𝑡2 = −𝑒

𝑚[−

𝑉0

𝑑(1 + 𝛼 cos𝑤𝑡) + 𝐵0

𝑑𝑦

𝑑𝑡], (88)

𝑑2𝑦

𝑑𝑡2 = −𝑒

𝑚(−𝐵0

𝑑𝑥

𝑑𝑡), (89)

33

𝑑2𝑧

𝑑𝑡2 = 0. (90)

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

If we take

𝑤0 =𝑒𝐵0

𝑚 so 𝐵0 =

𝑚𝑤0

𝑒 and

𝑒𝑉0

𝑚𝑑= 𝑘 (91)

Then Eq.88 and Eq.89 become

𝑑2𝑥

𝑑𝑡2 = 𝑘(1 + 𝛼 cos𝑤𝑡) − 𝑤0𝑑𝑦

𝑑𝑡 (92a)

𝑑2𝑦

𝑑𝑡2 = 𝑤0𝑑𝑥

𝑑𝑡 (92b)

So

𝑑𝑣𝑥

𝑑𝑡= 𝑘(1 + 𝛼 cos𝑤𝑡) − 𝑤0𝑣𝑦 (93a)

𝑑𝑣𝑦

𝑑𝑡= 𝑤0𝑣𝑥 (93b)

From Eq.93, we can obtain

𝑣𝑥 =1

𝑤0

𝑑𝑣𝑦

𝑑𝑡 (94a)

𝑣𝑦 =𝑘

𝑤0(1 + 𝛼 cos𝑤𝑡) −

1

𝑤0

𝑑𝑣𝑥

𝑑𝑡 (94b)

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

𝑑𝑣𝑥

𝑑𝑡=

1

𝑤0

𝑑2𝑣𝑦

𝑑𝑡2 = 𝑘(1 + 𝛼 cos𝑤𝑡) − 𝑤0𝑣𝑦 (95a)

34

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

𝑑𝑣𝑦

𝑑𝑡= −𝑘𝛼

𝑤

𝑤0sin𝑤𝑡 −

1

𝑤0

𝑑2𝑣𝑥

𝑑𝑡2 = 𝑤0𝑣𝑥 (95b)

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

𝑑2𝑣𝑥

𝑑𝑡2 + 𝑘𝛼𝑤 sin𝑤𝑡 + 𝑤02𝑣𝑥 = 0 (96a)

𝑑2𝑣𝑦

𝑑𝑡2 + 𝑤0𝑣𝑦 − 𝑘𝑤0(1 + 𝛼 cos𝑤𝑡) = 0 (96b)

Solution for Eq.96a is

𝑣𝑥 = 𝐴1 cos𝑤0𝑡 + 𝐵1 sin𝑤0𝑡 + 𝐶1 sin𝑤𝑡 (97)

If we use this 𝑣𝑥 in Eq.96a, we find 𝐶1 =𝛼𝑘𝑤

𝑤2−𝑤02

Then Eq.97 can be rewritten as

𝑣𝑥 = 𝐴1 cos𝑤0𝑡 + 𝐵1 sin𝑤0𝑡 +𝛼𝑘𝑤

𝑤2−𝑤02 sin𝑤𝑡 (98)

When 𝑡 equals to zero, 𝑣𝑥 value also becomes zero. So

𝐴1 = 0

𝑣𝑥 = 𝐵1 sin𝑤0𝑡 +𝛼𝑘𝑤

𝑤2−𝑤02 sin𝑤𝑡 (99)

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

𝑣𝑦 =𝑘

𝑤0

(1 + 𝛼 cos𝑤𝑡) −1

𝑤0[𝐵1 𝑤0cos𝑤0𝑡 +

𝛼𝑘𝑤2

𝑤2 − 𝑤02 cos𝑤𝑡]

35

𝑣𝑦 =𝑘

𝑤0[1 −

𝛼𝑤02

𝑤2−𝑤02 cos𝑤𝑡] − 𝐵1 cos𝑤0𝑡 (100)

When 𝑡 equals to zero, 𝑣𝑦 value also becomes zero. Then

𝐵1 =𝑘

𝑤0(1 −

𝛼𝑤02

𝑤2−𝑤02) (101)

If we use this 𝐵1 in Eq.99 and Eq.100, we get

𝑣𝑥 =𝑘

𝑤0(1 −

𝛼𝑤02

𝑤2−𝑤02) sin𝑤0𝑡 +

𝛼𝑘𝑤

𝑤2−𝑤02 sin𝑤𝑡 =

𝑑𝑥

𝑑𝑡 (102a)

𝑣𝑦 =𝑘

𝑤0[1 − (1 −

𝛼𝑤02

𝑤2−𝑤02 cos𝑤0𝑡)] −

𝛼𝑘𝑤2

𝑤2−𝑤02 cos𝑤𝑡 =

𝑑𝑦

𝑑𝑡 (102b)

Finally, after integrate the Eq.102, we find the solution for 𝑥 and 𝑦.

𝑥 =𝑘

𝑤02 [(1 −

𝛼𝑤02

𝑤2−𝑤02) cos𝑤0𝑡 −

𝛼𝑤02

𝑤2−𝑤02 cos𝑤𝑡] (103a)

𝑦 =𝑘

𝑤02 [𝑤0𝑡 − (1 −

𝛼𝑤02

𝑤2−𝑤02) sin𝑤0𝑡 −

𝑤0

𝑤

𝛼𝑤02

𝑤2−𝑤02 sin𝑤𝑡] (103b)

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, 𝛼 = 0. Then these equations

returns to

𝑣𝑥 =𝑘

𝑤0sin𝑤0𝑡, (104)

𝑣𝑦 =𝑘

𝑤0(1 − cos𝑤0𝑡), (105)

𝑥 =𝑘

𝑤02 (1 − cos𝑤0𝑡), (106)

36

𝑦 =𝑘

𝑤02 (𝑤0𝑡 − sin𝑤0𝑡). (107)

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=𝑤0 and forward translational velocity=𝑘 𝑤0⁄ . After ignoring the

translation and producing a radius of 2𝑘/𝑤02, 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 2𝑘/𝑤0. Besides, maximum velocity means

maximum kinetic energy [9].

About Fig. 20, there can be two options;

If 𝑑 ≤ 2𝑘/𝑤02, then electron fall into the anode

If 𝑑 > 2𝑘/𝑤02, then electron cannot reach the anode and turn back with in 2𝜋/𝑤0

to the cathode at a distance 2𝜋𝑘/𝑤02. 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

2𝑘/𝑤0 when it is nearest the anode. This kinetic energy is

𝐾𝐸𝑚𝑎𝑥 =1

2𝑚 (

2𝑘

𝑤0)2=

2𝑚𝑉02

𝑑2𝐵02 (108)

37

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.

𝛼𝑤02

𝑤2−𝑤02 =

1

2 and 𝑤 = 1.1𝑤0, then 𝛼 =

𝑉1

𝑉0= 0.105 (109)

Here we can see that the DC field magnitude (𝑉0) is much bigger than an AC field

magnitude (𝑉1). 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

𝑥 =𝑘

𝑤02 (1 −

1

2cos𝑤0𝑡 −

1

2cos 1.1𝑤0𝑡)

=𝑘

𝑤02(1 − cos 0.05𝑤0𝑡 cos 1.05𝑤0𝑡)

=𝑘

𝑤02 −

𝑘

𝑤02 cos 0.05𝑤0𝑡 cos 1.05𝑤0𝑡 (110)

𝑦 =𝑘

𝑤02 [(𝑤0𝑡 −

1

2sin𝑤0𝑡) −

1

1.1

1

2sin 1.1𝑤0𝑡] (111)

38

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

cathode with a maximum distance (2𝑘/𝑤02). If electron does not reach the anode, it

oscillates by decreasing the amplitudes. This continuous until it rest at 𝑥 = 𝑘/𝑤02 at

𝑤0 = 10𝜋.


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

39

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,

40

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 (𝑉0)

between anode and cathode and magnetic field (𝐵0) is in the positive z direction.

Thus, �⃗� = 𝐸𝑟𝑎�̂� and �⃗� = 𝐵𝑧𝑎�̂�. About electrons, they have cycloidal motion in the

space of magnetron.

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

𝑑2𝑟

𝑑𝑡2 − 𝑟𝑤2 =𝑒

𝑚(𝐸𝑟 − 𝐵𝑧𝑤𝑟) (112)

1

𝑟

𝑑

𝑑𝑡(𝑟2𝑤) =

𝑒

𝑚(𝐵𝑧

𝑑𝑟

𝑑𝑡) (113)

where 𝑤 = 𝑑∅ 𝑑𝑡⁄ .

Eq.113 can be written as

𝑑

𝑑𝑡(𝑟2𝑤) =

𝑒

𝑚𝑟𝐵𝑧

𝑑𝑟

𝑑𝑡=

1

2𝑤𝑐

𝑑

𝑑𝑡(𝑟2) (114)

where 𝑤𝑐 = (𝑒 𝑚⁄ )𝐵𝑧. And 𝑤𝑐 is the cyclotron angular frequency. If we integrate

the Eq.114, then we get

𝑟2𝑤 =1

2𝑤𝑐𝑟

2 + 𝑘1(𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡) (115)

41

Let’s take cathode radius as 𝑎. Then the constant 𝑘1 is −𝑤𝑐𝑎2/2 when at 𝑟 = 𝑎

and 𝑤 = 0. So another formula for 𝑤 is

𝑤 =1

2𝑤𝑐 (1 −

𝑎2

𝑟2). (116)

The kinetic energy of the electron in the magnetron is

𝐾𝐸 =1

2𝑚𝑣2 = 𝑒𝑉 (117)

From Eq.117, we can write the electron velocity 𝑣 which has two components 𝑟

and ∅,

𝑣2 =2𝑒

𝑚𝑉 = 𝑣𝑟

2 + 𝑣∅2 = (

𝑑𝑟

𝑑𝑡)2+ (𝑟

𝑑∅

𝑑𝑡)2 (118)

Let’s inner radius of anode becomes 𝑏. When electron at 𝑟 = 𝑏, 𝑉 = 𝑉0 and

𝑑𝑟 𝑑𝑡 = 0⁄ which means when the electron just grazes the inner surface of anode,

Eq.116 becomes

(𝑑∅

𝑑𝑡) =

1

2𝑤𝑐 (1 −

𝑎2

𝑏2) (119)

Eq.119 can also be written as

𝑏2 (𝑑∅

𝑑𝑡)2=

2𝑒

𝑚𝑉0 (120)

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

𝑏2 (1

2𝑤𝑐 (1 −

𝑎2

𝑏2))

2

=2𝑒

𝑚𝑉0 (121)

42

2.3 Hull Cutoff Equation for Magnetron

When we substitute 𝑤𝑐 equation in Eq.121 and do necessary arrangements, we find

following relations:

𝐵0𝑐 =(8𝑉0𝑚

𝑒)1/2

𝑏(1−𝑎2

𝑏2) (122)

𝑉0𝑐 =𝑒

8𝑚𝐵0

2𝑏2 (1 −𝑎2

𝑏2)2

(123)

For Eq.122 and Eq.123 we used 𝑤𝑐 = (𝑒 𝑚⁄ )𝐵𝑧. We call Eq.122 as the Hull cutoff

magnetic equation and result of this equation is the cutoff magnetic field (𝐵0𝑐). For

Eq.122, if 𝐵0 < 𝐵0𝑐 for a given voltage 𝑉0, 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 (𝑉0𝑐). For Eq.123, if 𝑉0 < 𝑉0𝑐,

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,

𝑚𝑣𝑡2

𝑅= 𝑒𝑣𝐵 (124)

Here 𝑅 is the path radius, 𝑣𝑡 is the tangential velocity.

The cyclotron angular frequency of the motion is

𝑤𝑐 =𝑣𝑡

𝑅=

𝑒𝐵

𝑚 (125)

One full revolution has the period which is shown below.

43

𝑇 =2𝜋

𝑤=

2𝜋𝑚

𝑒𝐵 (126)

In order to have oscillations in magnetron, construction must has an integral

multiple of 2𝜋 radians phase shift [9]. For the nth mode of the oscillation in an N

cavity magnetron, phase shift between two cavities is

∅𝑛 =2𝜋𝑚

𝑁 (127)

By adjusting the voltage of anode, it is possible to produce the oscillations.

Generally, magnetrons oscillates in 𝜋-mode. The necessary phase shift for this is

∅𝑛 = 𝜋 (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 𝜋-mode [9]

44

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

𝛽0 =2𝜋𝑛

𝑁𝐿 (129)

By using Maxwell’s equations and boundary conditions, we can obtain the solution

for ∅ component of the travelling wave electric field [9].Thus,

𝐸∅0 = 𝑗𝐸1𝑒𝑗(𝑤𝑡−𝛽0∅) (130)

The angular velocity of the travelling field is

𝑑∅

𝑑𝑡=

𝑤

𝛽0 (131)

As it understood from this relation, if the cyclotron frequency equals to the angular

frequency,

𝑤𝑐 = 𝛽0 (𝑑∅

𝑑𝑡) (132)

then field-electron interaction occurs and energy gets transferred.

2.5 Equivalent Circuit

Figure 24: Equivalent circuit for magnetrons resonator

45

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

figure are

𝑌𝑒 = the electronic admittance,

𝑉 = the RF voltage across the vane tips,

𝐶 = the capacitance at vane tips,

𝐿 = the inductance,

𝐺𝑟 = the conductance of the resonant,

𝐺𝐿 = 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

𝑄𝑢𝑛 =𝑤0𝐶

𝐺𝑟 (133)

the external quality factor is shown as

𝑄𝑒𝑥𝑡 = 𝑤0𝐶

𝐺𝐿 (134)

the loaded quality factor is shown as

𝑄𝐿 = 𝑤0𝐶

𝐺𝐿+𝐺𝑟 (135)

In these three equations, angular resonant frequency (𝑤0) is equal to 2𝜋𝑓0.

46

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

𝜂𝑐 =𝐺𝐿

𝐺𝐿+𝐺𝑟=

𝐺𝐿

𝐺𝑒𝑥𝑡=

1

(1+𝑄𝑒𝑥𝑡𝑄𝑢𝑛

) (136)

From Eq.136, we say that 𝜂𝑐 has its maximum value when 𝐺𝐿 ≫ 𝐺𝑟. 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

𝜂𝑒 =𝑃𝑔𝑒𝑛

𝑃𝑑𝑐 (137)

Here 𝑃𝑔𝑒𝑛 equals to 𝑃𝑑𝑐 − 𝑃𝑙𝑜𝑠𝑠 and it is the induced power of the RF into the anode

circuit. 𝑃𝑑𝑐 is power of dc supply and it is also 𝑉0𝐼0. 𝑃𝑙𝑜𝑠𝑠 is anode circuit’s power

lost. 𝑉0 is the anode voltage and 𝐼0 is used for the anode current. It is mentioned that

electrons generate the RF power and this equals to

𝑃𝑔𝑒𝑛 = 𝑉0𝐼0 − 𝑃𝑙𝑜𝑠𝑠 = 𝑉0𝐼0 − 𝐼0𝑚

2𝑒

𝑤02

𝛽2 +𝐸𝑚𝑎𝑥

2

𝐵𝑧2 =

1

2𝑁|𝑉|2

𝑤0𝐶

𝑄𝐿 (138)

In Eq.138, 𝑁 is resonator number, 𝑉 is the voltage in the resonator gap, 𝐸𝑚𝑎𝑥 is the

maximum value of electric field which is 𝑀1|𝑉|/𝐿, 𝛽 is the constant of phase, 𝛽𝑧 is

the magnetic flux density, 𝐿 is the distance between the vane tips and 𝑀1 is the gap

factor which is used for 𝜋-mode operation and 𝑀1 can be found by using the Eq.139.

𝑀1 = sin(𝛽𝑛𝛿 2⁄ )/(𝛽𝑛𝛿 2⁄ ) (139)

Here, for small 𝛿 values, 𝑀1 ≈ 1.

Eq.138 can be formed simply as

47

𝑃𝑔𝑒𝑛 =𝑁𝐿2

2𝑀12

𝑤0𝐶

𝑄𝐿𝐸𝑚𝑎𝑥

2 (140)

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

𝜂𝑒 =𝑃𝑔𝑒𝑛

𝑃𝑑𝑐=

(1−𝑚𝑤0

2

2𝑒𝑉0𝛽2)

(1+𝐼0𝑚𝑀1

2𝑄𝐿𝐵𝑧𝑒𝑁𝐿2𝑤0𝐶

)

(141)

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.

49

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


2𝑀12

𝑤0𝐶


2

Here,

𝑁 = resonator number,

𝐿 = the distance between the vane tips,

𝑀1 = gap factor,

𝑤0 = angular resonant frequency,

𝐶 = the capacitance at vane tips,

𝑄𝐿 = the loaded quality factor,

𝐸𝑚𝑎𝑥 = 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.

50

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

𝐸𝑟 = −𝜕𝑉

𝜕𝑟= −𝑉0

1

𝑟 ln 𝑏/𝑎

𝐸∅ = 𝐸𝑧 = 0

Here 𝑉0 is applied voltage, 𝑎 is cathode radius and 𝑏 is anode radius. Maximum

value of electric field can be observed on the cathode surface. The 𝐸𝑚𝑎𝑥 can be

written as

𝐸𝑚𝑎𝑥 = −𝑉01

𝑎 ln 𝑏/𝑎 (142)

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

𝐸 =𝜎

=𝑉

𝑑 (143)

here 𝜎 is charge density, 휀 is permittivity, 𝑑 is the distance between plates and 𝑉 is

the voltage difference between plates. We also know that charge density is

51

𝜎 =𝑐ℎ𝑎𝑟𝑔𝑒 𝑜𝑛 𝑝𝑙𝑎𝑡𝑒

𝑎𝑟𝑒𝑎 𝑜𝑓 𝑝𝑙𝑎𝑡𝑒=

𝑄

𝐴 (144)

If we use the capacitance definition, we obtain

𝐶 =𝑆𝑡𝑜𝑟𝑒𝑑 𝑐ℎ𝑎𝑟𝑔𝑒 𝑜𝑛 𝑝𝑙𝑎𝑡𝑒

𝐴𝑝𝑝𝑙𝑖𝑒𝑑 𝑣𝑜𝑙𝑡𝑎𝑔𝑒=

𝑄

𝑉=

𝑄

𝐸𝑑=

𝑄

𝜎𝑑=

𝐴

𝑑=

𝑘 0𝐴

𝑑 (145)

here 휀0 is space permittivity and equals to 8.854 × 10−12𝐹𝑎𝑟𝑎𝑑/𝑚𝑒𝑡𝑒𝑟 and 𝑘 is

relative permittivity for dielectric material between the plates and equals to 1 for

air.

3.1.3 Angular Resonant Frequency

Angular resonant frequency, 𝑤0, equals to 2𝜋𝑓0. Here 𝑓0 is the cavity resonant

frequency. We multiplies 𝑓0 with 2𝜋 because one revolution is equal to 2𝜋. 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

𝑓0 =𝑣𝑠

2𝜋√

𝐴

𝑉𝐿 (146)

52

here 𝑣𝑠 is the speed of sound. In dry air, speed of sound is approximately 340 m/s.

However, we use Eq. 147 for any cavity

𝑣𝑠𝑜𝑢𝑛𝑑 = √𝛾𝑅𝑇

𝑀 (147)

here 𝛾 is adiabatic constant and it is about the gas characteristic, 𝑅 is gas constant

(8.314 𝐽 𝑚𝑜𝑙. 𝐾⁄ ), 𝑀 is molecular mass of gas and 𝑇 is temperature. For air 𝛾 = 1.4

and 𝑀 = 28.95 𝑔/𝑚𝑜𝑙.

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

53

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

𝑤 = √𝑘

𝑚 (148)

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

𝑚 = 𝜌𝐿𝐴 (149)

here 𝐿 is the neck length and 𝐴 is the opening area of the neck. The change of

resonator volume is

𝑑𝑉 = −𝐴𝑑𝑥 (150)

here 𝑑𝑥 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

𝐾 = −𝑉𝑑𝑃

𝑑𝑉 (151)

here 𝐾 the bulk modulus (𝑁 𝑚2⁄ 𝑜𝑟 𝑃𝑎), 𝑑𝑃 is the change in applied pressure, 𝑉

is volume of the system and 𝑑𝑉 is the change in system volume. From Eq. 151, we

can write

𝑑𝑃 = 𝐾 (−𝑑𝑉

𝑉) (152)

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

54

𝑑𝑃 = 𝐾𝐴𝑑𝑥

𝑉 (153)

The net force on the air lump is

𝑑𝐹 = −𝑑𝑃𝐴 (154)

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

equations we have

𝑑𝐹 = −𝑑𝑥𝐾𝐴2

𝑉 (155)

For a spring 𝐹 = −𝑘𝑥 and the force constant is

𝑘 = −𝑑𝐹

𝑑𝑥 (156)

Then

𝑘 =𝐾𝐴2

𝑉 (157)

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

𝑤 = √𝑘

𝑚= √

𝐾𝐴2

𝑉

𝜌𝐿𝐴= √

𝐾

𝜌√

𝐴

𝑉𝐿 (158)

here √𝐾 𝜌⁄ is a form of the speed of sound. Finally, we can find cavity resonant

frequency as

𝑤 = 𝑣𝑠√𝐴

𝑉𝐿 (159)

55

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].

56

Table 1: Conductivity values of different materials [13]

Material 𝛔 [𝐒/𝐦] 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

57

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

𝑅 =𝑉

𝐼 (160)

Here 𝑉 is potential in volt, 𝐼 is current in ampere and 𝑅 is resistance. If we increase

voltage, this increases the current and 𝑉/𝐼 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 𝜌 is resistivity and specific for material. Resistance formula is

𝑅 = 𝜌𝐿

𝐴 (161)

58

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

𝑅 =𝐿

𝜎𝐴=

𝐿

𝜎𝜋𝑟2 (162)

In Eq. 162, 𝑟 is radius of resistor. Here 𝑅 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

𝛿 =1

√𝜋𝑓𝜇𝜎 (163)

where 𝜇 is permeability, 𝑓 is frequency and 𝜎 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.

𝑅𝑎𝑐 =𝐿

𝜎𝐴𝑎𝑐𝑡𝑖𝑣𝑒 (164)

here 𝐴𝑎𝑐𝑡𝑖𝑣𝑒 is the skin depth area on the conductor and equals to 2𝜋𝑟𝛿 . Then Eq.

164 becomes

59

𝑅𝑎𝑐 =𝐿

𝜎2𝜋𝑟𝛿=

𝐿

2𝑟√

𝑓𝜇

𝜋𝜎= (𝑅𝑑𝑐)

𝑟

2𝛿 (165)

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.


2𝑀12 (𝑣𝑠√

𝐴

𝑉𝐿′) (𝑘 0𝐴′

𝑑)

1

𝑄𝐿(−𝑉0

1

𝑎 ln(𝑏 𝑎⁄ ))

2 (166)

Here as mentioned before, 𝑁 is cavity number, 𝐿 is distance between vane tips, 𝑀1

is gap factor and 𝑄𝐿 is loaded quality factor. In this formula we use Eq. 149 instead

of angular resonant frequency term (𝑤0) so 𝐿′ is length of opening part of cavities

in this equation. Besides, instead of the capacitance at vane tips term (𝐶), the

capacitance form in Eq. 145 is used. Lastly, we use more detailed form of 𝐸𝑚𝑎𝑥

shown in Eq. 142.

For used magnetron as seen in Fig.31, distance between the vane tips (𝐿) is 0.3 cm,

anode radius (𝑏) is 2.9 cm, length of the opening part of the cavities (𝐿′) 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 (𝑎), gap

factor (𝑀1), the loaded quality factor (𝑄𝐿) and cavity number (𝑁) will change

according to our calculations.

60

Figure 31: Used 8 cavity magnetron for our work

There are some other fixed parameters for generated power equation. Here speed of

sound (𝑣𝑠) is taken as 340 m/s approximately, relative permittivity (𝑘) is 1 because

of air, 𝑑 same with distance between vane tips so it is 0.3 cm and applied voltage

(𝑉0) is 6000 V.

Permittivity of space (휀0) equals to 1/𝑐2𝜇0. Here 𝑐 is speed of light (2.99 ×

108 𝑚/𝑠) and 𝜇0 is permeability of free space (1.26 × 10−6 𝑊𝑏 𝐴 𝑚⁄ )[15]. Then

we can find prober permittivity of space value for power equation as

휀0 =1

𝑐2𝜇0=

1

(2.99 × 108 𝑚𝑠

)2

(1.26 × 10−6 𝑊𝑏𝐴. 𝑚

)

𝐴. 𝑠

𝑚. 𝑉

=1

8.94 × 1016 𝑚2

𝑠2 × 1.26 × 10−6 𝑉. 𝑠𝐴. 𝑚

𝐴. 𝑠

𝑚. 𝑉

휀0 = 8.88 × 10−12 𝐴.𝑠

𝑚.𝑉≈ 9 × 10−12 𝐴.𝑠

𝑚.𝑉 (167)

61

To calculate volume of the cavity (𝑉), we use 𝜋𝑟2ℎ 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

𝑉 = 𝜋 × 𝑟2 × ℎ = 𝜋 × (0.7 𝑐𝑚)2 × 5 𝑐𝑚 = 7.697 𝑐𝑚3 (168)

The area of opening part of cavity (𝐴) is calculated by multiplying height with the

distance between the vane tips. Then opening part area is

𝐴 = ℎ × 𝐿 = 5 𝑐𝑚 × 0.3 𝑐𝑚 = 1.5 𝑐𝑚2 (169)

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

𝐴′ = ℎ × 𝐿′ = 5 𝑐𝑚 × 0.6 𝑐𝑚 = 3 𝑐𝑚2 (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.

62

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

Variable Value Variable Value

𝑳 0.3 cm 𝜺𝟎 9x10-12 A.s/m.V

𝑴𝟏 1 𝑨′ 3 cm2

𝒗𝒔 340 m/s 𝒅 0.3 cm

𝑨 1.5 cm2 𝑸𝑳 10

𝑽 7.697 cm3 𝑽𝟎 6000 V

𝑳′ 0.6 cm 𝒂 1.6 cm

𝒌 1 𝒃 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

63

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


𝑵 8 𝜺𝟎 9x10-12 A.s/m.V

𝑳 0.3 cm 𝑨′ 3 cm2

𝒗𝒔 340 m/s 𝒅 0.3 cm

𝑨 1.5 cm2 𝑸𝑳 10

𝑽 7.697 cm3 𝑽𝟎 6000 V

𝑳′ 0.6 cm 𝒂 1.6 cm

𝒌 1 𝒃 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.

64

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

𝑃𝑔𝑒𝑛′ (𝑀1) = − (

𝑁𝐿2

𝑀13 ) (𝑣𝑠√

𝐴

𝑉𝐿′) (𝑘 0𝐴′

𝑑)

1

𝑄𝐿(−𝑉0

1


2 (171)

and the second derivative of Eq. 166 is

𝑃𝑔𝑒𝑛′′ (𝑀1) = (

3𝑁𝐿2

𝑀14 ) (𝑣𝑠√

𝐴

𝑉𝐿′) (𝑘 0𝐴′

𝑑)

1

𝑄𝐿(−𝑉0

1


2 (172)

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.

65

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

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

66

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


𝑵 8 𝒌 1

𝑳 0.3 cm 𝜺𝟎 9x10-12 A.s/m.V

𝑴𝟏 1 𝑨′ 3 cm2

𝒗𝒔 340 m/s 𝒅 0.3 cm

𝑨 1.5 cm2 𝑽𝟎 6000 V

𝑽 7.697 cm3 𝒂 1.6 cm

𝑳′ 0.6 cm 𝒃 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.

67

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.

𝑃𝑔𝑒𝑛′ (𝑄𝐿) = −

1

𝑄𝐿2 (

𝑁𝐿2

2𝑀12) (𝑣𝑠√

𝐴

𝑉𝐿′) (𝑘 0𝐴′

𝑑) (−𝑉0

1


2 (173)

𝑃𝑔𝑒𝑛′′ (𝑄𝐿) =

1

𝑄𝐿3 (

𝑁𝐿2

𝑀12 ) (𝑣𝑠√

𝐴

𝑉𝐿′) (𝑘 0𝐴′

𝑑) (−𝑉0

1


2 (174)

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.

68

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

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

69

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


𝑵 8 𝒌 1

𝑳 0.3 cm 𝜺𝟎 9x10-12 A.s/m.V

𝑴𝟏 1 𝑨′ 3 cm2

𝒗𝒔 340 m/s 𝒅 0.3 cm

𝑨 1.5 cm2 𝑸𝑳 10

𝑽 7.697 cm3 𝑽𝟎 6000 V

𝑳′ 0.6 cm 𝒃 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.

70

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

𝑃𝑔𝑒𝑛′ (𝑎) =

𝑁𝐿2

2𝑀12 (𝑣𝑠√

𝐴

𝑉𝐿′) (𝑘 0𝐴′

𝑑)

1

𝑄𝐿(−

2𝑉02(ln

𝑏

𝑎−1)

(ln𝑏

𝑎)

3𝑎3

) (175)

𝑃𝑔𝑒𝑛′′ (𝑎) =

𝑁𝐿2

2𝑀12 (𝑣𝑠√

𝐴

𝑉𝐿′) (𝑘 0𝐴′

𝑑)

1

𝑄𝐿(

2𝑉02(3(ln

𝑏

𝑎)

2−5 ln(

𝑏

𝑎)+3)

(ln𝑏

𝑎)

4𝑎4

) (176)

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.

71

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

72

3.2.5 Effect of Angular Resonant Frequency on Generated Power

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


2𝑀12

𝑤0𝐶


2

Firstly, we fixed some parameters. We chose cavity number 𝑁 as 8, gap factor 𝑀1

as 1 and cathode radius 𝑎 as 2 cm. Before we mentioned that distance between the

vane tips (𝐿) is 0,3 cm for our magnetron design. We showed the maximum electric

field in Eq.142 as

𝐸𝑚𝑎𝑥 = −𝑉0

1

𝑎 ln 𝑏/𝑎

Here, anode radius 𝑏 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

𝐶 𝑄𝐿⁄ term. The took loaded quality factor as 10 and the capacitance at vane tips

constant is

𝐶 =𝑘 0𝐴′

𝑑=

1×(9×10−10 𝐴.𝑠

𝑐𝑚.𝑉)×(1.3 𝑐𝑚2)

0.3 𝑐𝑚 𝐴.𝑠

𝑉= 39 × 10−10 𝐴.𝑠

𝑉 (177)

Then our equation became

𝑃𝑔𝑒𝑛 =8 × (0.3 𝑐𝑚)2

2 × 12× 𝑤0 ×

(39 × 10−10 𝐴. 𝑠𝑉

)

10

× (−6000 𝑉 ×1

(2 𝑐𝑚) × ln2.9 𝑐𝑚2 𝑐𝑚

)

2

𝑃𝑔𝑒𝑛 = 𝑤0 × 0.68 × 10−2 𝑊. 𝑠 (178)

73

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.

75

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.

76

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 × 10−3𝑊 and 37.4 × 10−3𝑊 for

77

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 × 10−2𝑊 if gap factor was 0.5 and 1.11 × 10−2𝑊

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 × 10−2𝑊. In other words, average

change was about 2.03 × 10−2𝑊 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 × 10−2𝑊. 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 × 10−3𝑊

power when the loaded quality factor was 5 and 16.6 × 10−3𝑊 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 × 10−3𝑊 and after that it became 2.1 × 10−3𝑊 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 × 10−2𝑊 and the loaded quality factor is doubled,

78

change in generated power was 2,4 × 10−2𝑊. 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 × 10−3𝑊 and 61.2 ×

10−3𝑊 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 × 10−3𝑊 between 1 cm and 1.8 cm while it was 7.63 × 10−3𝑊

approximately between 1.8 cm and 2.2 cm. When we doubled the cathode radius,

change in power became 2.1 × 10−2𝑊. 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.

79

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].

81

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