RF cavities and related devices

[artis]

Notice: I was told to make my own thread about the subject so here it is., please move it to a better subforum if there is a better place for it.At first I would like to ask three questions.
1) A torus shaped resonant cavity, like the ones found in particle accelerators and klystrons, is it a high frequency LC circuit, the parallel plate structure with the hole in the middle for beam passing is acting like the capacitor while the torus shaped outer part is a very small one loop inductor?
Is it then true that both in a klystron and also in a particle accelerator the cavity interacts with the beam of charged particles by the E field which is between the cavities plate like structure and the toroidal shape B field in the cavity torus is just a side effect of the charge running back and forth between the cavity plates connected by the torus bell or however it is called.?
2) Is my understanding of the klystron correct, both the input and output cavities work the same way only the input cavity is supplied by a RF signal of low amplitude (weak) which then creates a low strength RF oscillation in the input cavity, the electron gun emitted electrons pass by the cavities center hole where they are disturbed by the RF E field that forms between the cavities center plates and so in one half period of the signal passing electrons get accelerated while in the other half period electrons get de-accelerated or pushed back against, further down the "drift tube" this creates regions along the beam path of higher and lower electron concentrations with corresponding higher and lower E field along the beam path, as these regions pass the second or multiple second cavities they induce a RF E field between the cavity plates which then oscillates the cavity. The extra gained energy (wave amplitude) is due to the electron beam gaining energy as it gets accelerated towards the anode.
3) It seems that at RF and microwave frequencies the wavelength is small enough so that along a waveguide the EM wave can reach multiple positive and negative peaks which induce opposing E and B field lines along the wavepath inside the waveguide, is it possible to have a RF or microwave solenoid or something remotely similar where for example the B field lines all point in the same direction through a surface of some given area like it normally is with a low frequency or DC electromagnet?

I read about transverse electric and magnetic modes where one field is transverse to the wave propagation direction while the other is parallel to the direction of propagation,is it possible to have a situation where one of the fields either E or B is parallel to the path of the waveguide and at peak points would resemble the field lines coming out of a solenoid magnet?Thank you very much.

 

[sophiecentaur]

Where have you been getting your information from`? You seem to be mixing ideas about the cavity and a waveguide. Why? There are loads of available hits from a search on Klystron theory. You don't even need a waveguide to handle the signals in and out of a klystron. The coupling in and out a a cavity will be achieved differently for waveguide and co-ax.
1. You can say there is a 'Equivalent LC" circuit but that's all. A cavity is doesn't consist of two lumped components, the equivalent L and C are different in different places in the cavity. There is a maximum of alternating Voltage across the gap and a minimum of Current at or near resonance and the RF power is introduced at a place in the cavity that's well matched to the 'load' that is provided by the electron beam going through it. The cavity and coupling may be arranged to be narrow band or wide band - for a TV channel, the Q may be only around 50 or so. For a narrow band signal, the Q could be ten times that.
2. A cavity will interact with an electron beam, according to the varying E field across the gap. Power will flow one way or the other, depending on the details of the phases. If you read about klystrons, you will come across the expressions Velocity Modulation (at the input) and Density Modulation (the electron bunching) which couples high RF power into the output.

The extra gained energy (wave amplitude) is due to the electron beam gaining energy as it gets accelerated towards the anode.

This doesn't read right. The electron beam gains all its energy in the 'gun' which is upstream of the cavities. Have you not seen a diagram of a two cavity klystron?
3. Just what is all this about? How des it relate to klystron operation? "Solenoid"?

Read more and hesitate before coming to your own conclusions. This is all very well established stuff that's taught at many levels. Search for something that genuinely suits your level.

 

[artis]

So most klystrons use coaxial cable both for input and output coupling?

I meant to say that the electron beam gains its energy as it passes from cathode towards anode? as the PD across them is what accelerates the electrons correct? When I hear the word "gun" I understand that as only the filament that does thermal emission and the acceleration is done by the E field across cathode-anode? I see in diagrams in google that the anode in a klystron is past the output cavities at the very end of the klystron.

I think I understand the velocity and density modulation, because as the emitted electrons accelerate towards the anode the input cavity accelerates some of them even more while brakes the others depending on the E field polarity on each half period of the input RF wave, so further down the tube this creates bunches of electrons more closely together and spaces between them with very little electrons that is why its called density at the output?Narrow band and wide band , does that refers to the frequency spectrum the klystron is designed to operate between? Does that imply TV needs a wider spectrum while some specific application like probably particle accelerators can get away with very narrow frequency range?My third question was not directly related to klystrons, that is why I made the title of the thread "RF cavities and related devices".
With respect to the question I wanted to know is it possible to produce a B field that would run perpendicular to a surface, like a sheet of metal for such high frequencies as RF or microwaves? I understand there is an E field inside a klystron cavity which runs parallel to the beam path, I was thinking of something similar just the B field instead of E. In other words say I need to have a B field running through a conducting copper sheet, would I be able to produce such a field at those frequencies and do as I said? I hope you understand what I am asking. :)
Thank you.

 

[sophiecentaur]

The UHF klystrons I was involved with (up to 800MHz or so) had coax feeds.

If you look at the diagrams of klystrons you will see an Anode upstream of the cavities and a Collector downstream. The collector has to absorb all the surplus Kinetic Energy of the beam. There would be no point in giving it an extra positive charge.

You can read better descriptions of the bunching phenomenon that I have time to describe. You seem to have got the gist of it, though.

Bandwidth and gain tend to be conflicting requirements. A narrow band klystron will have more gain so it needs a lower power drive amplifier.

A B field doesn't add to the Kinetic Energy of a charged particle; it merely deflects it. the frequency of a B field can be anything you want (or at least anything you can generate). I haven't understood where you are going with the B field thing. The fields in a cavity tend to be limited by the resistance of the walls and, of course, the load.

 

[tech99]

My third question was not directly related to klystrons, that is why I made the title of the thread "RF cavities and related devices".
With respect to the question I wanted to know is it possible to produce a B field that would run perpendicular to a surface, like a sheet of metal for such high frequencies as RF or microwaves? I understand there is an E field inside a klystron cavity which runs parallel to the beam path, I was thinking of something similar just the B field instead of E. In other words say I need to have a B field running through a conducting copper sheet, would I be able to produce such a field at those frequencies and do as I said? I hope you understand what I am asking. :)
Thank you.

Have a look at the field distribution for the TE01 mode in a circular waveguide. But it doesn't help us make a klystron.

 

[Klystron]

Suggest studying other amplitron tubes to broaden understanding. Traveling-wave tubes (TWTs) and even standing-wave tubes represent the concepts of RF amplification through drift tubes perhaps with more clarity than klystron construction *. Density modulation contains subtle distinctions from more typical amplitude (AM), frequency (FM), and phase modulations (PM).

I found deeper understanding of electronics came with studying electron cycloid rotation in magnetrons and related devices. The true heart of most pulsed RF transmitters IMO lies in the design of the pulse forming network and synchronization circuits; signal and timing.

The previous thread mentioned by the OP referenced resonant cavities (including superconducting cavities) fed by RF amplifiers used within collider experiments. Even deeper understanding of electronics IMO comes with studying receivers such as the venerable "Dicke Fix" receiver designed by Robert Dicke https://en.wikipedia.org/wiki/Robert_H._Dicke while at MIT.

* Point being most commercial tubes are designed for specific applications with power, beam density, frequency, etc., requirements.

 

[artis]

Why I ask about the B field is because, while reading about this I imagined an interesting idea. Take a disc and spin it in a homogeneous B field , what you get? current generated. the same thing can be done linearly by moving a conducting metal sheet in a B field and having sliding contacts at the sides. In this way one can apply a AC field and the result would be the same AC but with increased amplitude or current. I was wondering can one amplify an RF or microwave signal in such a manner?
Lorentz force directly affects electrons in a moving conductor and the force the electrons feel changes with changes in the B field, so technically this works like an amplifier. That is why I asked.
At such high frequency is it possible to even construct a device (waveguide or something) that would allow for B field lines all going in single direction through a given surface area?

PS. °Can klystron reach into the microwave frequency spectrum and have a magnetron as its input signal source?

 

[tech99]

Klystrons are normally used for microwave frequencies I believe, and also down to TV Band IV.
Regarding magnetrons and klystrons, as the latter can be used to amplify a stable source, I am surprised that the military choose them for power over frequency stability. On the other hand, a magnetron is a self excited oscillator, so can be pulled by the load and will be sensitive to operating voltages and temperature.They are also known to have spurious output frequencies.

 

[sophiecentaur]

PS. °Can klystron reach into the microwave frequency spectrum and have a magnetron as its input signal source?

What have you been reading about klystrons? All the chatty sources talk about the frequencies at which they operate.
What frequencies are mentioned in the source at the top about accelerators?

 

[artis]

My apologies, I missed the intro part in wiki article which clearly says they operate from RF up to microwave.

So if the frequency drifts wildly in a magnetron based on what folks have said here then why do they use it as a frequency source in applications like radars? Why not use solid-state semiconductor analogs like the gunn diode or similar? I suppose a microwave oven is perfectly fine with an unstable frequency but a radar might not be?I looked at the diagrams again and now I see the anode is located not far away from the cathode in a klystron, wiki article shows before the input cavity, so then I have a question, if the whole accelerating E field is before any of the cavities and drift tube between them then how come the electron beam stays focused along its path and there is no divergence of the beam, some large klystrons seem to stand taller than a basketball player, so what keeps the beam focused up until the end output cavities?

thanks.

 

[sophiecentaur]

if the frequency drifts wildly in a magnetron based on what folks have said here then why do they use it

When a magnetron is chosen, the spec will be appropriate for the application - as in all Engineering choices.

Why not use solid-state semiconductor analogs like the gunn diode or similar?

Power? Look at the specs; it's the numbers that count in Engineering.
Again, what are you actually reading (not skimming) about this topic?

so what keeps the beam focused up until the end output cavities?

The details of klystrons appear all over the place. You need to look at diagrams and read the labels. Google Klystron Images and take your pick.

 

[artis]

I have looked at the diagrams and only in a few I see some magnets shown alongside the drift tube while in others there is nothing shown alongside the drift tube, can you please answer my question as to what really confines the electrons from spreading all over the place as they get past anode and into the drifttube?

thanks.

 

[sophiecentaur]

in a few I see some magnets shown alongside the drift tube while in others there is nothing shown alongside the drift tube

I can't think of any particular reason why they could be omitted - except it's fairly obvious that something is needed to focus the beam. Components are often missing from diagrams when they have a secondary or support function. I do remember using a couple of Travelling Wave Tubes which used a sort of jacket round the tube and that was a permanent magnet - the diameter of the internal space, where the tube sat had variable spacing. There may have been some extra beam focus control with coils but I cannot remember. Have you done your reading around and found out how the beam is focussed?
Relative massive ions / protons would probably diverge less than low mass electrons (?).

 

[artis]

that was what I thought of too that there must be some means of keeping the beam together and it cannot be done by electric field as then it would form a capacitor which when charged would then have no force on the electrons at all, so B field is used, I see.

Also I read a Stanford paper that describes large power klystrons that have an output RF power over 1000kW or some even 50+ MW, now given an average klystrons is about 50 to 70% efficient at those frequencies and power levels then I assume a 50MW klystrons " power grid consumption" would be something along 70MW? Is this the reason large particle accelerators need something like a medium sized power plant to supply their electricity needs?Also if magnets are used for the beam focusing and the electrons in the drift tube are past anode, sin't there some de-acceleration happening and synchrotron radiation emitted ?

 

[Klystron]

I have looked at the diagrams and only in a few I see some magnets shown alongside the drift tube while in others there is nothing shown alongside the drift tube, can you please answer my question as to what really confines the electrons from spreading all over the place as they get past anode and into the drifttube?

thanks.

AFAIK klystrons operate like other vacuum-tube technology of that era. Suggest you visit manufacturer websites such as Varian Associates and Raytheon to gather specific data. Stanford University and SRI International have or had klystron exhibits; at the birthplace, so to speak.

 

[sophiecentaur]

Could it have been the anode of an old TWT?

Do you mean Anode or Collector? The collector is the big beefy thing thing that gets very hot - dumping all the remaining Power of the beam.

 

[Klystron]

My honest response at this late date must be "D* if I know!". I find this thread unsettling. Shouldn't we be discussing resonance as theory?

The part probably came from an obsolete "reflex" device; though some airmen thought it was a rocket nozzle (wrong material and shape).

 

[sophiecentaur]

It was probably part of an old air conditioner - just to add confusion.

 

[artis]

Since this thread tile says "RF cavities and related devices" I would like to ask something about RF waves.
I now know that in an accelerator or klystron RF cavity for that matter there is an E field between the parallel planes of the cavity closer to the center, are there are devices or cavities where there would be homogeneous B field lines all pointing in the same direction just like with the E field?

I tried google searching "transverse electric" or transverse magnetic but had no luck, even though one field is transverse the other field should be then in the direction of the waveguide but the field lines seem not to be homogeneous across all the length of the guide instead forming little loops, is that because of the short wavelength of the RF wave?

 

[sophiecentaur]

B field lines all pointing in the same direction

You can have a region in which the B field lines are all in the same direction and 'uniformly spaced' - i.e. the Field is the same. But that's only by using a magnetic substance and you cannot achieve this over more than a small region in a cavity. The H field lines always form endless loops. The E and H fields in a Co-ax cable are transverse to the axis, the E lines are straight and radial and the H lines are circles. It's similar for a klystron cavity where the H lines go around the O of the cavity and the E fields are across the gap.
There are countless sources of images for the fields in waveguides and co-ax cable. The fields in co-ax are TEM, both transverse to the axis which is the direction of power flow. I can't understand how your googling, apparently yielded no results. Perhaps if you include the term "MODE" ??

 

[artis]

Well what I meant is that I hoped to google a cavity or waveguide where the B field lines are homogeneous and all in the same direction in a given space but now after your answer I realize what I already thought that for such high frequencies as RF and microwaves the wavelength is so small that forming ordinary coil electromagnets where the b field loops around the whole magnet and at any given space field lines are all in the same direction and mostly uniform is impossible.

Speaking about what you mentioned about the co ax and similar waveguides then yes there are images and I can imagine myself the E and B fields being perpendicular to one another and them forming these regions.

So let me get this as close as I can, in a waveguide looking at the B field for example the field lines loop in one direction and after a small distance they loop in the other direction as the field direction and peak goes back and forth over small distances which are described by the wavelength of the field. The same happen with the E field just that the lines are perpendicular to the B field ?

Sorry my writing is sloppy but I think I mostly now get this.
So due to the nature and physics involved I then conclude that the RF cavity can only be made with the E field lines being all parallel and in the same direction in the middle while the B field loops the outer torus but can't be made the other way around?

 

[artis]

So let me ask one other thing, if I had an electromagnet with a metal core or maybe an air core if RF frequencies are involved, is it possible to make such a short low inductance few turn winding and power it from an RF source and then get uniform field lines all in the same direction out of the axial ends of the electromagnet much like from an ordinary metal core wire coil electromagnet?

 

[gneill]

So let me ask one other thing, if I had an electromagnet with a metal core or maybe an air core if RF frequencies are involved, is it possible to make such a short low inductance few turn winding and power it from an RF source and then get uniform field lines all in the same direction out of the axial ends of the electromagnet much like from an ordinary metal core wire coil electromagnet?

If you could it would likely take the form of a Helmholtz coil, essentially two separate coils facing each other and with certain relationships in the geometry of the setup.

https://en.wikipedia.org/wiki/Helmholtz_coil

Producing RF frequency magnetic fields of a practical strength is very energy intensive. In order to get a reasonable magnetic field you need to drive a LOT of current at a very high frequency through a quite low impedance coil.

 

[artis]

When you said "if you could" did you mean it is not possible?

Well the way I understand it is that since the frequency is so high in any normal coil there would be a very high back EMF which would block most current from passing through so only a coil with low impedance made of one or few turns would let reasonable amount of current pass in each cycle ?
The typical RF cavity is a single turn coil?

So assuming I make two Helmholtz coils each having one turn and space them some distance apart in the middle of the coils the B field would be quite homogeneous and field lines would point in single direction like in a normal lower frequency setup?

 

[sophiecentaur]

The typical RF cavity is a single turn coil?

A "coil" is often referred to a lumped component because it is small compared with the wavelength of the signal. A waveguide works because of its dimensions and it will not propagate a signal with a frequency lower than than its cutoff frequency.
I really do get the impression that you are rummaging about in your box of ideas to find an example of where you can find a small region inside a component over which the magnetic field is uniform. But to what end? Take a small enough region, anywhere and it will satisfy that condition. Because of the inherent difference in the divergence of E and H fields, we wouldn't be looking for this sort of thing.
What application did you have in mind for this region of uniform RF H field?

 

[artis]

I don't have a specific application in mind I was just curious when I was googling and only found homogeneous E field inside cavities but couldn't find a B field analog.
Now I start to realize that it is complicated if even possible due to the very small wavelengths.
I still wonder whether two Helmholtz like few or single turn coils would do the trick, sure the coil diameter would surpass the size of the wavelength.

Speaking about waveguides, is the reason why they work with specific frequencies to do with the fact that as the E and B fields travel being perpendicular to each other with the right size and frequency guide the peaks of the waves are (at or near?) the metallic walls of the guide and so get reflected but say for the same size guide a smaller wavelength( higher frequency) waves peaks would not reach the walls and so the wave would travel similarly as in air and so would loose its energy much faster?

 

[Klystron]

I don't have a specific application in mind I was just curious when I was googling and only found homogeneous E field inside cavities but couldn't find a B field analog.
Now I start to realize that it is complicated if even possible due to the very small wavelengths.
I still wonder whether two Helmholtz like few or single turn coils would do the trick, sure the coil diameter would surpass the size of the wavelength.

Speaking about waveguides, is the reason why they work with specific frequencies to do with the fact that as the E and B fields travel being perpendicular to each other with the right size and frequency guide the peaks of the waves are (at or near?) the metallic walls of the guide and so get reflected but say for the same size guide a smaller wavelength( higher frequency) waves peaks would not reach the walls and so the wave would travel similarly as in air and so would loose its energy much faster?

The data, drawings, and information in this article may help your understanding https://en.wikipedia.org/wiki/Cutoff_frequency#Waveguides

Cutoff frequencies (or wavelengths), bandpass and bandwidth measurements help my understanding of RF energy propagation in waveguide. Much has changed teaching electronics with improved understanding of emf since I worked on radar networks, so my old methods are suspect yet I find visualizing combined orthogonal electric and magnetic fields expedient.

[Quality factor arose in another thread. If you have time to read this, it will help understand coils and resonance https://en.wikipedia.org/wiki/Q_factor ]

 

[artis]

Well the wiki article has some complicated math in it and I can't think in terms of numbers I can only in terms of visuals and 3d models.
I was hoping to get a confirmation or denial of my question with regards to why certain size waveguides work for certain range of frequencies and my assumption was that it is because if the frequency is too low the wavelength is too large and it sort of runs into the wall of the waveguide while too high frequency the wave can't reach the wall or something along those lines?
Surely the metallic wall of a waveguide and the wave reflection has something to do with this.One more question, I understand that RF cavities come in different sizes for different frequencies and power levels but what would be the average cavity capacitance for a medium power klystron or accelerator cavity? I feel it is very small given how far apart the plates are and their surface area

 

[Klystron]

Well the wiki article has some complicated math in it and I can't think in terms of numbers I can only in terms of visuals and 3d models.
[snip]

Agree that wiki math has gotten intense, but 'math is the language of physics' and forum rules require it. Can you refresh knowledge of algebra and trig and natural logarithms? Do you understand capacitance and induction in basic electronic circuits? Please state your goals and an E.E. mentor can help you learn.

I'll look for non-math learning materials like the Army uses for operators but they are superficial and lack basis.

I understand waveguide dimensions from calculating multiples and fractions of the reference wavelength Lambda. Wavelength is proportional to "c" (speed of light in vacuum) divided by frequency in hertz. Your choice of units for c determines your Lambda units.

 

[anorlunda]

One more question, I understand that RF cavities come in different sizes for different frequencies and power levels but what would be the average cavity capacitance for a medium power klystron or accelerator cavity? I feel it is very small given how far apart the plates are and their surface area

I can understand that the math may be hard to follow. But do you understand how we can't answer a question that contains the words average, medium, small, far? If you can't speak in math, at least speak in numbers.

 

[sophiecentaur]

I have a real problem when people say they want Science without Maths. Most people accept that Maths is part of their lives in general and wouldn't 'demand', for instance, to be given an explanation about the state of their credit card account which doesn't involve some Maths. They may, of course, just accept the bottom line of their bill every month (as many of us do) and that's all they can really expect from a non-mathematical approach to EM waves in cavities. Without Maths, you just have to accept some things and not expect to be able to predict or expand on things. No problem - just don't get cross when the explanations are unapproachable.

 

[artis]

I do understand basics , capacitance and inductance I understand those I also can imagine how the E field charges up the cavity plates and how then they discharge through the torus and then charge up the capacitor with opposite polarity and how the frequency of this LC circuit is determined by the inductance of the torus and capacitance of the cavity plates. I think I also understand how waveguides work in basic but I can't calculate these complicated formulas.

This being said I don't believe there isn't any simple explanation for why a certain size waveguide only works for certain frequency range waves. I can imagine optical fiber which I assume does the same function for waves only in different spectrum of the EM scale. The fiber as I understand it is a long long cable which acts as a long thin mirror and light is reflected from a mirror surface so the mirror like fiber helps minimize losses and helps light propagate, although optical mirrors cannot reflect non visible spectrum EM radiation like infrared near as good as visible so here I can understand why the fiber cable can act as a waveguide for certain frequencies only.

I feel that for RF and microwave waveguides since they are metallic the reason why they are good for certain frequencies is more about their size and the wavelength of the wave itself than material properties like in a fiber cable.
What I imagined is this. For example a 400Mhz wave has a fixed wavelength and the waveguides objective is to help that wave propagate, just like an optical fiber reflects light pulses a metallic wall reflects RF wave. I read that for the simplest situation the rectangular waveguide has to have a width that matches that of the wave. The example given was a transverse electric TE10 mode where the E field is transverse to the length of the waveguide. Only I did not understand whether the wavelength is one full period of the wave or half period because the pictures were showing a half wave.
Anyway this is probably as far as I will be able to understand this.

 

[artis]

By the way I couldn't find in google what is the average capacitance of an RF cavity?

 

[tech99]

Notice: I was told to make my own thread about the subject so here it is., please move it to a better subforum if there is a better place for it.At first I would like to ask three questions.
1) A torus shaped resonant cavity, like the ones found in particle accelerators and klystrons, is it a high frequency LC circuit, the parallel plate structure with the hole in the middle for beam passing is acting like the capacitor while the torus shaped outer part is a very small one loop inductor?
Is it then true that both in a klystron and also in a particle accelerator the cavity interacts with the beam of charged particles by the E field which is between the cavities plate like structure and the toroidal shape B field in the cavity torus is just a side effect of the charge running back and forth between the cavity plates connected by the torus bell or however it is called.?
2) Is my understanding of the klystron correct, both the input and output cavities work the same way only the input cavity is supplied by a RF signal of low amplitude (weak) which then creates a low strength RF oscillation in the input cavity, the electron gun emitted electrons pass by the cavities center hole where they are disturbed by the RF E field that forms between the cavities center plates and so in one half period of the signal passing electrons get accelerated while in the other half period electrons get de-accelerated or pushed back against, further down the "drift tube" this creates regions along the beam path of higher and lower electron concentrations with corresponding higher and lower E field along the beam path, as these regions pass the second or multiple second cavities they induce a RF E field between the cavity plates which then oscillates the cavity. The extra gained energy (wave amplitude) is due to the electron beam gaining energy as it gets accelerated towards the anode.
3) It seems that at RF and microwave frequencies the wavelength is small enough so that along a waveguide the EM wave can reach multiple positive and negative peaks which induce opposing E and B field lines along the wavepath inside the waveguide, is it possible to have a RF or microwave solenoid or something remotely similar where for example the B field lines all point in the same direction through a surface of some given area like it normally is with a low frequency or DC electromagnet?

I read about transverse electric and magnetic modes where one field is transverse to the wave propagation direction while the other is parallel to the direction of propagation,is it possible to have a situation where one of the fields either E or B is parallel to the path of the waveguide and at peak points would resemble the field lines coming out of a solenoid magnet?Thank you very much.

The thread has been going for ages, and is a bit hard to follow, so I would like to comment on your original question.
1) Correct. Similar action is seen in the cavities of a magnetron, which are big holes drilled in a copper block with a slit facing the electrons. When they made the first cavity magnetron, Randall and Boot based the design of the resonator on the original wire loop used by Hertz. The wire is nominally half a wavelength long but the capacitance between the balls at the end make it less than this - the same with the magnetron and klystron resonators. The hole in the centre is itself too small to accommodate a wave.
2) Correct, with the addition from Sophie Centaur that the additional energy comes from the high tension (B+) supply.
3) If you take a metal pipe, it is possible to fit a wave into it provided an internal dimension is greater than half a wavelength. You can see this if you try to send a wave between two metal plates which are spaced too closely - there is no transmission if closer than half a wavelength.
The modes in a waveguide are just the field patterns arising from a propagating wave. Whereas in free space the E and B fields are at right angles to the direction of propagation, in a waveguide there is also a component along that direction. There is a good practical description in "Antennas" by J D Kraus.

 

[sophiecentaur]

I do understand basics , capacitance and inductance I understand those I also can imagine how the E field charges up the cavity plates and how then they discharge through the torus and then charge up the capacitor with opposite polarity and how the frequency of this LC circuit is determined by the inductance of the torus and capacitance of the cavity plates. I think I also understand how waveguides work in basic but I can't calculate these complicated formulas.

This demonstrates your problem. I tried earlier to point out that a cavity is NOT just two lumped components but you are not taking that on board. Perhaps a mechanical equivalent could help here. Imaging you have a wave tank with water sloshing from one to end. You can excite it at its resonant frequency, with nodes each end and an antinode in the middle. The same frequency can be obtained with a single mass and a massless spring. You are implying that the tank can be described as just two components but where is the spring and where is the mass in the water tank? You ask what is the Capacitance of the cavity and what is the inductance and that is not answerable.
Also, waveguides cannot be treated as simple LC structures. Depending on where you 'drive' them from (where you introduce the power with a loop or a slot etc.), a waveguide can appear to have a Low Impedance or a High Impedance.
You say you don't understand complicated formulae so you just have to take what I say as basically true. It's a door to which you have no access until you are prepared for a bit of Maths. There is no simple arm waving model for cavities and waveguides. I can suggest that you do a Google Images search for fields and waves in waveguides and hunt around the numerous pictures you will find.
One important thing about waveguides (@tech99 mentions it, above) is that the only work when they are big enough to support a progressive wave. It's a very sophisticated concept and is only a limit for transverse waves; a wave tank can propagate longitudinal waves of any wavelength and the two waves are very different in this respect.