With decreasing frequency, the penetration depth of the alternating current into metal increases (skin effect). This reduces the ohmic losses in the resonator and less heat is generated

Efficiency regarding what exactly? - Conversion from DC to microwaves? Power coupling into the plasma?...

Thanks. I meant conversion from DC to microwaves.

With decreasing frequency, the penetration depth of the alternating current into metal increases (skin effect). This reduces the ohmic losses in the resonator and less heat is generated

Dear Herbert Weinder,

Thank you for a short and clear explanation. That is exactly what I needed.

It is also to do with the speed the electrons need to have, to synchronise with the microwave fields on the resonant structure. At high frequencies, for the same size structure, the electrons have to travel faster and so hit the anode harder, wasting more energy, and giving less efficiency.

Dear Malcolm White,

That is very valuable information. Thank you a lot! Things are getting more a more clear to me.

Total efficiency is the sum of electronic efficiency and circuit efficiency.

1) Suppose a magnetron has 80-90% efficiency at 915 MHz and you want to achieve same in X band. This doesn't happen. You get generally less than 60%. If we determine dimensions of the magnetron for 80% efficiency at 9 GHz, we will get very small physical dimensions, that are not practically realizable. Also, if you increase power level with frequency, electronic efficiency decreases.

2) Circuit efficiency decreases with the increase in the frequency due to the losses. Skin depth and ohmic loss can be one of the reason for this which is already pointed out.

It isn't mainly ohmic losses. Most of the losses are in the energy the electrons have when they hit the anode. The ohmic losses are very small compared to that. Even at 10 GHz the cavities have a Q of over 100, so the copper losses are a lot less than the heat from the electrons hitting the anode.

If there are 12 cavities, the electrons have to travel once round the diameter of the anode every six microwave cycles. This velocity give an energy, which then translates into electron volts, which you can't help losing when the electrons hit the anode. For a 1 kV magnetron it might be 300 ev - so not better than 70% efficient.

There are additional losses because some of the input power goes to keeping the cathode hot, (electrons in the wrong position for synchronisation are driven into the cathode) so that it emits the required current (except for cold-cathode magnetrons), but this is not as much as is lost at the anode.

Read Collins from the MIT series. It says most of what there is to say about magnetrons, though there have been lots of developments since. https://www.jlab.org/ir/MITSeries/V6.PDF. I was a magnetron design engineer at the start of my career and have read all of it and worked through most of the equations.

Thank you, Malcolm White for adding insights.

See in attached file

Dear Sandeep Kumar Vyas,

That is great! Everything I needed very short and very clear.

Best regards

Mateusz Wnukowski

Dear Malcolm White and Mohit Kumar Joshi,

Thank you for your detailed explanation. They helped a lot.

Best regards

Mateusz Wnukowski

This is a very interesting question, indeed. I, personally, do not have any definitive answer to this question, as of today. Perhaps, it is somehow connected with a frequency difference. And of course, it does not have anything common with the Ohmic losses or skin effect ...

I think the answer to this question might be found, and should be found, by somebody who will be able to computer simulate both 915 MHz and 2.45 GHz magnetrons operation and see all the differences. Computer PIC simulations will give the deep inside view on how these magnetrons operate and what is the electronic efficiency of these magnetrons.

And, by the way, kudos to Mateusz for his very, very interesting question!

In my opionion, higher frequency magnetron need higher electron drift velocity to generate microave. And higher drift velocity makes electron spokes travels faster toward the anode blocks, which makes a shorter beam wave interacting time. Thus a lower power conversion effeciency will be obtained.

There are two different electron drifts that determine a magnetron operation. Velocity of azimuthal electron drift in crossed “External” radial electric field and “External” axial magnetic field needs to be synchronized with azimuthal phase velocity of the “Induced” electromagnetic wave. In this sense, of course, “higher frequency magnetron needs higher electron drift velocity to generate microwaves”. Velocity of radial electron drift, inside microwave spokes from cathode to anode, is determined by crossed “Induced” azimuthal electric field and “External” axial magnetic field. So, the radial electron drift from electron hub boundary “toward the anode blocks” does not depend, directly, on frequency of magnetron operation, but depends on the power generated by a magnetron, which is determined by the “Induced” oscillating electric field and its azimuthal component particularly.

Velocity of the radial electron drift should be high enough to allow electrons to be “in-phase”, which means to be within the same electron spoke, on their journey from cathode to anode to provide maximum electronic efficiency of a magnetron operation. Generally speaking, a 915 MHz magnetron provides much more power than a 2.45 GHz magnetron does. The later means that a radial electron drift velocity in a 2.45 GHz magnetron might be not so high to allow electrons to travel all their way from the electron hub boundary “toward the anode blocks”. Just because of this, electrons in a 2.45 GHz magnetron go, eventually, from “in-phase” to “out-phase” somewhere between the cathode and the anode. Contrary to this, a radial electron drift velocity in a 915 MHz magnetron might be just enough to allow electrons to travel all their way from the electron hub boundary “toward the anode blocks” and provide, by this way, the maximum electronic efficiency of the magnetron operation.

So, the answer to the question of why “915 MHz systems have higher efficiency (ca. 85%) than the 2.45 GHz (ca. 65%)” might be as derived above – This is because power generated by 915 MHz magnetron is much higher than power generated by 2.45 GHz magnetron, which allows electrons within microwave spokes to be “in-phase” all the time up to a moment when they strike an anode, even if an “External” magnetic field in magnetrons is different.

Andrey

You have missed the important point that the electrons have to get from one magnetron cavity to the next in half an RF cycle in order to remain in phase (alternate cavities are half a cycle out of phase). This determines the "drift" velocity, which largely determines the kinetic energy wasted when the electrons hit the anode. The "drift" velocity is usually higher than the radial velocity, and increases proportional to frequency unless the cavity period is also scaled inversely with frequency, (which can give problems with power dissipation because the metal bits get very thin).

Absolutely agree! I would like just mention that there is an "azimuthal drift" velocity and a "radial drift" velocity. While the azimuthal drift velocity is determined by (or, it would be better to say, depends on) a frequency of magnetron operation, the radial drift velocity is determined by an oscillating electric field intensity within magnetron resonators or by a microwave power extracted from the magnetrons resonators.

Thanks Andrey. The radial drift is what actually gives the electrons energy - as they "fall" outwards they gain azimuthal speed (energy), most of which is then transferred to the microwave field so that the speed actually stays close to the synchronisation speed. The trick is to make the kinetic energy required at the anode radius a small proportion of the total electron volts due to the radial "drop". Increasing the voltage will do that, but requires increased magnetic fields or a bigger anode (longer period or more cavities, with complexity or stability problems I think). It's been along time since I did this stuff! I may have some of the trade-offs wrong.

While many of the previous answers are very good, they address the physics of why there is a difference in efficiencies between 915 MHz and 2450 MHz magnetrons. However, I think the real answer to the question is more a matter of choice made by the magnetron design engineer. Both 915 MHz and 2450 MHz magnetrons can have the same efficiency if the designer so desired, and in fact this is the case for the two types having the same output power rating (i.e. ~70% for tubes rated for 5 kW). There is a compromise between efficiency and operational stability. That is, at higher efficiency the magnetron requires greater isolation from reflection in order to maintain output power and frequency stability. Microwave oven magnetrons are designed with lower efficiency because they are not isolated at all from reflections. Otherwise, under light load conditions they would behave erratically. The same applies for many industrial applications of 2450 MHz magnetrons. In contrast, high power (up to 100 kW) 915 MHz magnetrons require greater efficiency due to physical limitations in heat dissipation. The consequence is they require good isolation for operational stability.

So, the question is – Why does a magnetron require greater isolation from reflection when it works with higher efficiency? What is the difference in a fundamental physics of a magnetron operation in these two operational conditions – magnetron operation with higher efficiency vs. magnetron operation with low efficiency?

To get high efficiency in a magnetron you need the electron volts tied up in the electron kinetic energy when they hit the anode to be small compared to the electron volts they gain as they drop the several hundred or thousand volts from the cathode to the anode. The difference goes into the microwave power, except for a bit which makes the cathode hot. The electron volts in the kinetic energy is fixed by the internal diameter of the anode and the number of cavities - the electrons have to be travelling at the right speed so that they go past one cavity every half-cycle of the microwave frequency. That means the efficiency depends on how much bigger than this the drive voltage can be. As the drive voltage gets bigger it becomes harder for the fields in the anode cavities to control the electrons, and it only works for a narrow range of conditions. If the magnetron is working into a poor load, then the fields in the anode cavities are different, because of the reflected power, and the magnetron doesn't run so well. This is plotted on a Rieke diagram, where the power, efficiency and operating frequency are plotted as contours on a Smith chart to show how all three vary with the magnitude and phase of the reflected power.

What is the "drive voltage" definition - the cathode-anode voltage drop?

Yes. It looks like a climb, but because electrons are negative they drop.

1. Yes, I agree that to get high efficiency in a magnetron, the electron kinetic energy near the anode should be as low as possible. Which means that all energy acquired by an electron from the cathode-anode voltage drop, while it is “accelerated” on its way from the cathode to the anode, is converted into the microwave energy.

2. No, I do not agree that the electron kinetic energy near the anode is fixed by geometry of the slow-wave structure of the magnetron. There should be a synchronization between (i) phase velocity of electromagnetic wave “tied up” in the magnetron operating mode (usually pi-mode), and (ii) drift velocity of electrons in crossed electric and magnetic fields. The drift velocity should be little bit higher than the phase velocity to allow electron kinetic energy to be transferred into the electromagnetic energy. It is like electrons are pushing electromagnetic wave and, by such a way, transfer their kinetic energy into the electromagnetic energy. Geometry of the slow-wave structure of magnetron (anode diameter and number of resonators) defines magnetron operating mode and, by such a way, phase velocity of electromagnetic wave. The drift velocity of electrons is defined by crossed electric and magnetic fields in the magnetron operation space, between the anode and the cathode of the magnetron.

3. Yes, I agree that efficiency depends on the “drive voltage”, which is cathode-anode voltage drop. Assuming that the “right voltage” is a synchronous voltage, at which phase velocity equals to drift velocity, the “drive voltage” should be bigger than the “right voltage” … just a little bit. Magnetron does not work if the “drive voltage” is lower than the “right voltage”. However, as the “drive voltage” gets bigger and bigger than the “right voltage”, especially in strapped magnetrons, synchronization gets worse and worse and efficiency gets lower and lower. But we need to remember that “drive voltage” itself is not the only force that defines the drift velocity of electrons. It is a combination of crossed electric and magnetic fields that defines drift velocity of electrons.

4. I’m not absolutely sure how the efficiency depends on the load/reflected power. In strapped low-power magnetrons, for example, microwave power is extracted from one cavity or from one vane only, and, by such a way, the reflected power affects oscillating electric field in one cavity or oscillating voltage at one vane only. This is the question for me. Perhaps, this difference in oscillating electric fields in cavities (oscillating voltages at vanes) indeed results in the magnetron efficiency decrease.

Andrey D. Andreev

2. All the magnetrons that I came across operated in the pi mode - which meant that the resonant mode that was excited had alternate cavities 180 degrees out of phase - pi radians. A lot of work on cavity design was to arrange that this mode had the lowest phase velocity around the inner circumference of the anode, so that it is the mode that captures the electron beam as the voltage rises when it is turned on, and so that it is hard for it to jump into other modes when it is on. The ring-strap arrangement adds capacitance to the pi mode and lowers its frequency (more than it does for other modes) so it moves down in the dispersion diagram (omega beta diagram) so its phase velocity is lower so the synchronous velocity of the electrons is lower than for other modes. There may be some other modes with lower synchronous velocity but these couple weakly.

The rising-sun cavity arrangement has the same aim, of lowering the frequency of the pi mode, or increasing the frequency of the other modes.

The synchronous velocity is when the electron spoke (the electron beam sorts itself into spokes like a wheel. If this doesn't happen then the magnetron has not "captured" the beam) travels from one cavity to the next at the phase velocity. Usually this means in one half cycle. It could be 3 half cycles, or 5 or more, but I have never come across anything but one half cycle. This means that the synchronous velocity is the distance between the vane centres in one half cycle. The electrons in the spoke travel at or close to the synchronous velocity. That is what it has to be, and this energy gets wasted. That is the energy I described in my previous post.

The way that it works is like when a power boat is not quite going fast enough to plane. Its speed is determined by the maximum speed of the wave it generates, and all the energy of the motor goes into making that wave bigger. The dynamics of the situation mean the boat doesn't go faster or slower than that wave, but at the same speed, except for very small changes - if the motor power is increased the boat moves slightly higher up the wave and a bit forward and the wave gets bigger, if the motor power is reduced the boat moves slightly lower down the wave and a bit further back, and the wave gets smaller.

When the magnetron is operating stably, the electron spokes or bunches travel at the same speed as the electromagnetic field pattern, which is the phase velocity, which for the pi mode is one cavity period in one half microwave cycle. The individual electrons in the spoke or bunch travel very close to this speed, and still have that speed when they hit the anode. The way that the crossed field works is that because of the magnetic field the electrons cannot move out to a larger radius without losing speed, but when they lose speed and move out to a larger radius they immediately gain the speed back again because they have fallen down the voltage hill. This is a feedback mechanism that keeps the average speed of the bunch or spoke constant, and so the kinetic energy content of the electrons in the spoke constant.

4. The cavities of a magnetron are strongly coupled together, and typically store 100 or more times as much energy as is coupled out each cycle (see the definition of Q), so the energy is fairly evenly distributed among the cavities. The coupling does affect the efficiency, because there are two loss mechanisms in the cavities. There is resistive loss in the cavities, and there is loss to the output. These are described by Qo (internal losses) and Qe (external losses), and also the loaded Q is used, QL=1/(1/Qo+1/Qe) (total losses). If Qo and Qe are the same then the same amount of power is lost in the cavities as goes from the output, and the magnetron will be less than 50% efficient. Usually Qo is much bigger than Qe so not much power is lost in the cavities (they are polished copper) so this doesn't affect the efficiency much.

The efficiency depends on the load in two ways. Firstly, if the load isn't matched, then any reflected power is likely to be wasted. Secondly, if the magnetron sees a reflection it affects its operation and the frequency and power (and efficiency) can change. Look up Rieke diagrams which I think I flagged up in the previous post. You don't seem to have followed that up.

In a short transmission line it would be possible, but unwise, to have the magnetron connected to a load with a poor match so that the output became part of the magnetron resonant circuit (this is how a mismatch changes the frequency). However this would probably be a bit touchy.

The group I was with at MOV/EEV built magnetrons with 2 outputs and used a moveable short in one of the outputs to tune the frequency of the magnetron. The power and efficiency usually changed too, but by an acceptable amount.

From "Principles and Practice of RADAR," R.S.H. Boulding, Seventh Edition, 1963, p.94. "It will be noted that efficiency is increased by increasing Vt [threshold voltage], which is only another way of saying by increasing the magnetic field, since the two are related."

And since 915 MHz magnetron has higher threshold voltage, higher operating magnetic field and anode voltage than 2.45 GHz magnetron does, 915 MHz magnetron has higher efficiency than 2.45 magnetron.