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Whitepaper | CST AG 2013

A Multiphysics Approach to Magnetron and Microwave Oven DesignThe magnetrons used in microwave ovens operate on the same frequency band as Wi-Fi equipment, and the radiation they release can interfere with the operation of wireless networks. This paper pre-sents a multiphysics simulation of a magnetron using CST STUDIO SUITE®, with the aim of testing the electrical, magnetic, thermal and mechanical characteristics of a low-interference magnetron design. The simulation results are then compared to measurements made experimentally, and the two sets of results are shown to be in good agreement.

Magnetrons are widely used as RF power sources because they offer high energy conversion efficiency (around 75%) at a low cost. The magnetron was invented during World War II, when its small size, high power and short wavelength made it ideal for use in radar, but with mass production and the development of automatic manufacturing techniques, magnetrons made the move into the home as the radiation source in microwave ovens.

The problem with the microwave oven’s widespread adoption is that magnetrons generate significant amounts of RF noise; the ones used in ovens radiate at around 2.45 GHz. Historically this frequency band was reserved for noisy industrial applica-tions, and little consideration was given to the risk of elec-tromagnetic interference (EMI), but the rise of short-range wireless communication systems such as Wi-Fi and Bluetooth operating on the same frequencies means the microwave ov-en is now a major potential source of interference to computer networks and mobile communications.

The risk of interference has forced researchers and engineers to look for ways to improve the EMI profile of the ovens. There have already been a number of innovations made in magne-tron design meant to reduce interference. For instance, mag-netrons can be designed with inhomogeneous magnetic fields to inhibit the production of stray modes, and the cathode heat-ers can be designed to switch off when not needed, so currents in the filament do not interact with the fields and change the resonant frequency of the magnetron. These approaches both result in a cleaner EM spectrum.

However, although these approaches are good at reducing the sideband noise caused by unwanted resonances in the magne-tron, they can’t do much about the radiation that the magne-tron produces at its intended mode. Since we can’t get rid of this noise, the only way to stop it interfering with Wi-Fi is to change the magnetron’s resonant frequency.

Although magnetrons are restricted to the same frequency band as Wi-Fi, the way the Wi-Fi spectrum is divided up into

channels means that there is still room to reduce the risk of interference, by shifting the magnetron’s resonant frequency away from the middle of the spectrum towards the outer edges, as shown in Figure 1. This opens up more channels and increases the chances that the network equipment can find a free channel to communicate on.

Figure 1 Channels in the 2.4 GHz Wi-Fi spectrum. Shifting the microwave oven to

a higher operating frequency reduces the number of channels it can interfere with

However, changing the resonant frequency of a magnetron means changing its electrical properties significantly, and in testing it’s important to be aware of how these changes can af-fect its magnetic, mechanical, thermal and farfield properties.

There are a number of components that an oven magnetron needs to generate microwaves, illustrated in Figure 2:

ó A heated filament cathode, which acts as the electron source ó An anode with multiple resonant cavities, separated

by metal vanes. The walls of these cavities form simple LC circuits

ó Metal rings, known as straps, which connect alternate vanes and ensure they remain in phase, suppressing unwanted modes

ó A pair of permanent ceramic ring magnets – one above the anode and one below – to steer the electrons around the cathode

ó Iron pole pieces between the magnets and the anode to focus the magnetic field

ó An antenna through which the microwaves are transmitted

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As well as these, simulations also have to take into account the structural elements of the magnetron. The magnetron is held together by a yoke – a metal frame around the magnetron – with a heat sink between the anode and the yoke. All of these parts will have to be adjusted or tested during these simulations.

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Figure 2 The structure of a typical magnetron in cross-section

1. Top permanent ceramic ring magnet2. Top pole piece3. Bottom pole piece4. Bottom permanent ceramic ring magnet5. Antenna6. 10 vane resonant cavity7. Yoke8. Heat sink fi ns

Antenna cap, heated fi lament cathode, fi lament insulator and harmonic choke are not shown

Their properties can all be tested in the lab with prototypes. However, producing a prototype is time consuming and ex-pensive, and fi nding and fi xing a fault means redesigning and rebuilding the prototype. Modeling offers a way of shortening the design cycle by reducing the number of prototyping stages needed. The properties of the magnetron can be simulated on the computer and faults identifi ed before the fi rst prototype is built.

Finding the cold resonant frequency with the eigenmode solver

The fi rst step of the process is to change the magnetron’s re-sonant frequency. Magnetrons generate microwaves from the oscillations of fi elds in the anode’s cavities, driven by the rota-tion of a wheel of electrons. Inside the magnetron, electrons tend to spiral out from the cathode to the anode. However, because the vanes of the anode form resonant circuits, the po-tential distribution over the anode is not constant.

Like most magnetrons, ours operates at the π-mode, so there is a 180° phase shift between pairs of adjacent vanes – in other words, because alternate vanes are strapped together, each one has the opposite polarity to its neighbors. The moving fi eld in the magnetron at π-mode accelerates electrons in some parts of the chamber and decelerates those in other parts. Since the anode has 10 vanes, this produces an electron beam with fi ve spokes. As the LC circuits oscillate, the potentials swap sides and the electron beam is rotated – this rotation in turn pumps more energy into the circuits and sustains the oscillations.

Changing the resonant frequency is a matter of changing the behavior of these LC circuits by altering the geometry of the anode. In this case, we adjusted its capacitance by changing the gap between the anode and its straps. Parameterizing this gap is a simple way of fi nding the geometry that will give us the resonant frequency we need. The eigenmode solver of CST MICROWAVE STUDIO® (CST MWS®) provides fast calcula-tions of the magnetron’s cold resonant frequency; for the mod-el used in this example, which had 94,329 tetrahedral mesh cells, the eigenmode solver took 6 minutes and 6 seconds to calculate its resonant frequency on workstation with dual 2.4 GHz Intel Xeon E5645 processors and 128 GB RAM. With the ei-genmode solver, we can quickly run an optimizer or perform a parameter sweep over a range of different values for the strap gap to fi nd a magnetron with the right resonant frequency.

Figure 3 The eigenmode of the magnetron

However, the cold resonant frequency is only an approxima-tion of the resonance when the magnetron is actually in use. When the electron beam inside the magnetron interacts with the electromagnetic fi elds, it disturbs them, and this changes the resonances of the cavities. For an accurate simulation of the magnetron when it’s in use, we need to be able to model the electron beam itself.

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Modeling the fields in CST EM STUDIO

Before we inject the particles, we need to set up the fi elds that will steer them. To study the fi elds inside the magnetron, we use the electrostatic fi eld solver and the magnetostatic fi eld solver, available in CST EM STUDIO® (CST EMS).

The permanent magnets ensure that the particles orbit correct-ly around the central cathode, so it is important to make sure the magnetic fi elds behave as expected inside the magnetron. These fi elds are not trivial: to make sure the magnetic fi elds are focused in the interaction region between the anode and cathode, there are two hollow truncated cones of iron between the magnets and the electrodes. These are the pole pieces, and they act as ferromagnets in the presence of the permanent magnets. These focus the fi eld, so that it is strong and fairly homogeneous in the area between the anode and cathode.

Figure 4 The magnetic fi eld within the magnetron. The magnets, pole pieces, yoke

and interaction region together form a magnetic circuit

Figure 5 The magnitude of the magnetic fi eld, measured along an off-axis vertical line

The magnetostatic fi eld solver is capable of taking the non-linear properties of the materials into account. This makes it ideal for simulating the iron pole pieces, whose ferromag-netism varies strongly with the fi eld around it.

Figure 4 shows a vector plot of the magnetic fi eld inside the magnetron, and Figure 5 shows the magnitude of the fi eld along a line parallel to the axis (r = 2 mm). Only the magnets, pole pieces and yoke have been included in the simulation – the electrodes and the casing could have been included, but they have little effect on the magnetic fi eld and would have extended the simulation time.

The B-fi eld maximum from the simulation agrees well with the design value of 0.19 T. The largest peak corresponds to the fi eld in the interaction region, while the smaller peaks are the fi elds within the inner radius of the magnets. As shown in the 3D model, the fi elds are localized almost entirely to the area between the pole pieces. The magnetic current returns through the yoke, and there is virtually no B-fi eld outside it – the yoke makes a good shield.

While the magnetic fi eld gives the particles angular motion, the electric fi eld gives them radial motion. Together, the mag-netic and electric fi elds defi ne the drift velocity of the particles. The drift velocity must be synchronous to the phase velocity of the operating mode, so the electrons orbit the cathode at the right speed to generate resonances in the cavities.

Figure 6 shows a scalar isoline plot of the electric potential dis-tribution between anode and cathode. The fi eld varies evenly in the interaction region.

Figure 6 The electric fi eld inside the magnetron before electron emission

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This fi eld is purely electrostatic, induced by the potential ap-plied to the electrodes. The simulation does not take into ac-count the variations in the fi eld induced by oscillations in the LC circuits, which cause the electron beam to develop spokes. A full dynamic simulation will need to be able to simulate the time-dependent electric fi eld that generates the microwaves, and that is where the particle simulation comes in.

Simulating the electron beam with the PIC solver

To study the electron beam, we use CST PARTICLE STUDIO® (CST PS). CST PS, CST EMS and CST MWS are all part of CST STUDIO SUITE. They share an integrated design environ-ment and so the process of importing models and fi elds from one to another is simple.

Since a microwave oven magnetron is not a relativistic device, we can simulate the behavior of the electrons best using the Child-Langmuir space charge model, with a space charge lim-ited current of 20 A. The cathode is designated as a particle source, spraying electrons throughout the magnetron cham-ber. The particle-in-cell solver (PIC) available in CST PS then lets us model this electron cloud as it forms a coherent beam. An example of a particle beam in the π-mode is shown in Figure 7.

Figure 7 The spoked particle beam. Vanes alternately attract and repel electrons

Moving charges induce electric and magnetic fi elds, and these fi elds affect the paths of the moving charges. The PIC solver takes into account the coupling between the electrons and the fi eld, simulating the motion of the particles and the fi eld around them in the time domain. Because it can model par-ticles over their whole path, the PIC solver gives us the fi elds over an extended period of time, rather than just at one in-stant. With it we can examine the beam loading effect – the

interaction between the particle beam and the fi elds within the magnetron which alters its resonant frequency.

The PIC solver can take advantage of GPU computing to speed up the simulation process substantially. When the particle beam inside the magnetron was simulated, with a 2,567,708 mesh cell model and an average of 1.22 million electrons in the beam, it took 6 hours and 15 minutes to run on a computer with dual 2.4 GHz Intel Xeon E5620 processors, 48 GB RAM and a Tesla C2050 GPU. With GPU computing disabled, the same computer took 32 hours and 48 minutes to carry out the simu-lation, more than 5 times longer. This speed-up scales almost linearly with the number of particles – the more particles in the simulation, the more useful GPU computing is.

From the signals generated by the PIC Solver, we can calculate a number of the magnetron’s properties. Most importantly, it gives us the true resonant frequency of the magnetron. The ends of the magnetron can be designated as waveguide ports, and the solver will calculate the modes at these ports, along with the line impedance between them. The time signal at the output port corresponds to the power of the magnetron. We can also turn the time signal into a spectrum in the frequency domain by using a Fourier transform.

Current and voltage monitors can also be defi ned with the PIC Solver. These can give us the power drawn by the cathode which, when combined with the output power of the magne-tron, lets us calculate its effi ciency. The port and monitor defi -nitions for this model are shown in Figure 8 – the red rectan-gles are the waveguide ports and the lines marked with circles are the monitors.

Figure 8 The set-up of the ports and monitors

As shown in Figure 9, for the fi rst 40 nanoseconds of opera-tion, the magnetron produces little power. This is because the particle beam has not formed spokes, so the oscillations are not yet coherent. After this time, its output grows rapidly, and once it is resonating it produces a clean, constant signal. This signal effectively represents the square root of the output

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power – in other words, the rms output power is half the square of the peak signal. The calculated output power of the magnetron is 1095.12 W (half of 46.82), close to its rated value of 1000 W, and taking the Fourier transform of the time sig-nals, as shown in Figure 10, gives us the hot resonant frequen-cy, 2.471 GHz. For comparison, this particular magnetron had a cold resonant frequency of 2.49 GHz – the 20 MHz difference is due to this beam loading effect.

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Figure 9 The output of the magnetron over time. The magnetron takes around 50

nanoseconds to start resonating

Figure 10 The frequency spectrum of the magnetron in linear scaling

Thermal and mechanical analysis

Once the magnetron itself has been tested, the next step of the design process is to test the heat sinks. The inductive and capacitive components of the magnetron give it a signifi cant impedance; in a typical magnetron, 30% to 50% of the input energy is lost as heat. Magnetrons are quite sensitive to heat-ing, because of the ceramic magnets used to steer the beam. If these magnets are heated past their Curie temperature, they

will lose their magnetism, and the magnetron will no longer be able to generate the wheel of electrons needed to maintain stable oscillations. To prevent this happening, the heat needs to be carried away effi ciently.

The analysis of the heat sink was carried out in CST MPHYSICS® STUDIO (CST MPS), a combined thermal and me-chanical analysis tool integrated into CST STUDIO SUITE. MPS can use results from simulations in other tools such as MWS, PS and EMS to study the thermal effects of fi elds, currents and particle collisions on the model.

The model was heated by a volumetric heat source of 585 W representing, as a test of the worst-case scenario, the heat loss generated by a magnetron with an effi ciency of 42%. Cooling is provided by a fan blowing 1.0 m3/min of air through the structure. Figure 11 shows the temperature distribution for the magnetron in use, as calculated by the thermal solver. The Curie temperature of the ceramic magnets is around 450 °C, so it is vital that even in this worst case, they stay well below that limit. The analysis shows that the highest temperature in the magnetron in this simulation was just 352 °C, and the magnets were cooler still at around 200 °C to 250 °C.

Figure 11 The temperature distribution across the magnetron

CST DESIGN STUDIO™ lets multiple simulation tasks be chained together, with the results from one simulation being sent to the next. This means the temperature distribution can be easily imported into a mechanical simulation, so that the thermal expansion of the magnetron parts can be calculated by the structural mechanics solver. In this case, the assumption is made that the ends of the magnetron are fi xed. This caus-es the heat sink and yoke to bulge outwards. The distortion shown in Figure 12 is exaggerated to allow easier visualization.

Investigating shielding and interference effects in CST MWS

So far, this article has examined the magnetron on its own, without any load and with nothing but empty space sur-rounding it. This is fi ne for calculating the properties of the magnetron itself, but of course, in the real world this is not all that a microwave oven designer needs to take into account. A

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full simulation would not be complete without taking into ac-count the casing of the oven and the food being cooked inside it, both of which can signifi cantly alter the fi elds generated by the magnetron.

Figure 12 The magnetron after thermal expansion (exaggerated)

In an ideal oven, which cooked food perfectly evenly, the elec-tric fi eld would be homogeneous. However, the cavity of the oven refl ects and traps the waves, producing a fi eld with a complex pattern of constructive and destructive interference. These are what cause the familiar hot spots and dead spots seen when using a microwave oven, where the food is either overheated or barely heated at all.

To test how even the fi eld is inside the microwave, a test load consisting of a cylinder of water is placed in the middle of the microwave oven, and the cooking process simulated using the time domain solver. With GPU acceleration, this simulation took around 1 hour 15 minutes. The fi eld was then visualized as a scalar carpet plot on a plane cutting horizontally through the oven, as shown in Figure 13. The shortening of the wavelength within the water can clearly be seen, as can the formation of peaks and troughs where the fi eld is stronger or weaker. Without something to even out the heating, like a turntable, these would lead to hot spots and dead spots.

Figure 13 The electric fi eld within and around the oven

This image also shows that the fi eld outside the oven appears to be negligible. The goal of this multiphysics analysis was to test a low-interference magnetron, so it is also very important to study the emissions from the oven. The radiation produced by microwave ovens can also damage the eyes and possibly poses other health risks, so we want to be sure that users are not being exposed to the microwave energy. Ideally, the oven would leak no radiation at all – all the energy would remain within the oven cavity. In general, a simple metal box provides a good shield for microwave radiation. However, to actually cook food in the oven, there has to be a door in the front. The seam between the door and the case, and in some cases the panel of the door itself, can allow radiation to leak out.

CST MWS includes tools for adding compact models of details like vents, slots, seams and panels available in CST STUDIO SUITE. These compact models have the same electrical properties as a full model, but they have a simpler geometry than the full scale model and are far easier for the solvers to simulate. Using compact models speeds up the pro-cess of designing and simulating the oven casing.

power (f=2.45) 1sqr(1000),0 (peak)

Cutplane normal: 0, 1, 0Cutplane position: -185Component: Abs2D Maximum: 84,912D Minimum: 0,3331Frequency: 2.45

Figure 14 The fi eld distribution 5 cm in front of the door, with the maximum power

highlighted

Once we have a case, we need to test whether it controls the electric fi eld correctly. The transient solver available with CST MWS offers tools for visualizing both the local fi eld dis-tribution and the farfi eld of the oven, for a complete illustra-tion of its electric fi elds. Figure 14 shows the nearfi eld at a dis-tance of 5 cm in front of the door, along with a calculation the

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maximum power radiated through the door. The normal limit set for radiation leakage is 5 mW/cm2, which is equivalent to 50 W/m2. As shown, across most of the area, the door barely leaks at all. However, there are two points where the power density is much greater. In the bottom spot, this power signifi cantly ex-ceeds the limit, with a maximum power density of 84.91 W/m2.Before this oven could be manufactured, the design of the door and the seal would have to be redesigned. One advantage of simulation is that this fault was discovered without having to build and test a full prototype.

Distance/m

Figure 15 Farfi eld monitors at 5 cm and 1 m

As well as the nearfi eld effectiveness, the designer may also want to examine the effectiveness of the shielding of the mi-crowave oven as a whole, using a farfi eld analysis. As an ad-ditional demonstration of the fi eld simulation capabilities of CST MWS, two farfi eld monitors are added to the model, as shown in Figure 15. These calculate the farfi eld at that particu-lar point when the simulation is run. With a farfi eld monitor,

the farfi eld can be probed at a distance from the oven without having to extend the simulation boundaries all the way to the monitor. This means that the solver does not have waste time calculating fi elds over wide areas of empty space, and this can speed up the simulation dramatically.

Figure 16 shows the 3D farfi eld plot of the microwave oven’s emissions at its resonant frequency at a distance of 1 meter. This replicates a standard laboratory test and allows a quick visualization of the fi elds around the oven.

Figure 16 The farfi eld at 1 m from the oven

CST Module Part Quantity Simulated value Measured value

CST MWS

Resonator Q_unloaded 1649.3 1631.0

Resonator Frequency [GHz] 2.49 2.520

MWO Frequency [GHz] 2.45 2.45

CST EMS Magnet B-fi eld [Tesla] 0.189 0.19

CST PS

Resonator Spokes [EA] 5 5

Resonator Frequency [GHz] 2.471 2.481

Resonator Power [Watt] 1095.12 1000

CST MPS

Anode 335 320

Heat sink Temp. [°C] 270 221

Yoke 171 85

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Comparing the simulation to a prototype

Simulation can be a great way to reduce development costs, but only if the simulation is able to replicate laboratory measurements. Fast, high accuracy simulations can make the design process quicker and cheaper, and decrease the time it takes to bring a product to market.

To demonstrate the accuracy of the simulations, a prototype oven was built according to the simulated design and tested in the laboratory. The magnetron’s measured characteristics are shown in the table on page 7.

The simulation results agree closely with the measurements. The simulation was able to accurately predict the power and

frequency of the output microwaves, and correctly simulated

the beam formation. It was also correct that the temperature of

the magnetron remained below the Curie point of the magnets, which is important to ensure the reliable operation of the oven.Some quantities differed between the simulation and the ex-periment. In particular, the temperatures of the yoke and heat sink were significantly lower in the actual magnetron. This ap-pears to be due to a change in the heat sink design between modeling and prototyping – the heat sink tested in the labora-tory had double-bent fins, and the contacts between the fins and the yoke were different. The high Q-factor and low reso-nant frequencies might be due to overestimating the conduc-tivity of the resonant circuits.

Summary

The process of designing and testing a new magnetron is complicated, drawing on a number of different disciplines of physics. In this one workflow, we used an eigenmode solver to investigate the resonator, a particle-in-cell solver to model the beam effects, an electrostatic solver to examine the elec-trodes, a magnetostatic solver to investigate the magnets, thermal and mechanical solvers to check the heat sink and a transient solver to visualize the fields around the magnetron. With acceleration methods such as parallelization and GPU computing, simulations of complex properties can be carried out in hours or minutes, and the results of these simulations visualized clearly.

CST STUDIO SUITE brings the many different solvers needed for such a workflow together into one integrated design envi-ronment, allowing a true multiphysics analysis of the magne-tron that can replicate the measurement process and reduce the need for prototyping and experimentation in the design process.

AuthorsDr. Monika Balk, CST AG – Support and EngineeringSeung-Won Baek, CST of America, Inc. – Support

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CHANGING THE STANDARDS