Download A Multiphysics Approach to Magnetron and Microwave Oven Design

2013
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A Multiphysics Approach to
Magnetron and Microwave
Oven Design
The 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 presents 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 applications, and little consideration was given to the risk of electromagnetic 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 oven 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 magnetron design meant to reduce interference. For instance, magnetrons can be designed with inhomogeneous magnetic fields
to inhibit the production of stray modes, and the cathode heaters 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 magnetron, they can’t do much about the radiation that the magnetron 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 affect 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:
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heated filament cathode, which acts as the electron source
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An anode with multiple resonant cavities, separated
by metal vanes. The walls of these cavities form simple
LC circuits
Metal rings, known as straps, which connect alternate
vanes and ensure they remain in phase, suppressing
unwanted modes
A pair of permanent ceramic ring magnets – one above
the anode and one below – to steer the electrons around
the cathode
Iron pole pieces between the magnets and the anode to
focus the magnetic field
An antenna through which the microwaves are transmitted
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Magnetron
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
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Top permanent ceramic ring magnet
Top pole piece
Bottom pole piece
Bottom permanent ceramic ring magnet
Antenna
10 vane resonant cavity
Yoke
Heat sink fins
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 field 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 five
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 finding the geometry that will give
us the resonant frequency we need. The eigenmode solver of
CST MICROWAVE STUDIO® (CST MWS®) provides fast calculations of the magnetron’s cold resonant frequency; for the model 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 eigenmode solver, we can quickly run an optimizer or perform a
parameter sweep over a range of different values for the strap
gap to find a magnetron with the right resonant frequency.
Antenna cap, heated filament cathode, filament 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 expensive, and finding and fixing 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 identified before the first prototype
is built.
Finding the cold resonant frequency with the
eigenmode solver
The first step of the process is to change the magnetron’s resonant frequency. Magnetrons generate microwaves from the
oscillations of fields in the anode’s cavities, driven by the rotation 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 potential distribution over the anode is not constant.
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Figure 3 The eigenmode of the magnetron
However, the cold resonant frequency is only an approximation of the resonance when the magnetron is actually in use.
When the electron beam inside the magnetron interacts with
the electromagnetic fields, 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 fields that
will steer them. To study the fields inside the magnetron, we
use the electrostatic field solver and the magnetostatic field
solver, available in CST EM STUDIO® (CST EMS).
The permanent magnets ensure that the particles orbit correctly around the central cathode, so it is important to make sure
the magnetic fields behave as expected inside the magnetron.
These fields are not trivial: to make sure the magnetic fields
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 field, so that it is strong and fairly
homogeneous in the area between the anode and cathode.
Magnetron
The magnetostatic field solver is capable of taking the nonlinear properties of the materials into account. This makes
it ideal for simulating the iron pole pieces, whose ferromagnetism varies strongly with the field around it.
Figure 4 shows a vector plot of the magnetic field inside the
magnetron, and Figure 5 shows the magnitude of the field
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 field and would have
extended the simulation time.
The B-field maximum from the simulation agrees well with
the design value of 0.19 T. The largest peak corresponds to
the field in the interaction region, while the smaller peaks are
the fields within the inner radius of the magnets. As shown
in the 3D model, the fields are localized almost entirely to the
area between the pole pieces. The magnetic current returns
through the yoke, and there is virtually no B-field outside it –
the yoke makes a good shield.
While the magnetic field gives the particles angular motion,
the electric field gives them radial motion. Together, the magnetic and electric fields define 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 distribution between anode and cathode. The field varies evenly
in the interaction region.
Figure 4 The magnetic field within the magnetron. The magnets, pole pieces, yoke
and interaction region together form a magnetic circuit
Figure 5 The magnitude of the magnetic field, measured along an off-axis vertical line
Figure 6 The electric field inside the magnetron before electron emission
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This field is purely electrostatic, induced by the potential applied to the electrodes. The simulation does not take into account the variations in the field 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 field 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 environment and so the process of importing models and fields 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 limited current of 20 A. The cathode is designated as a particle
source, spraying electrons throughout the magnetron chamber. 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.
Magnetron
interaction between the particle beam and the fields 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 simulation, 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 defined with the
PIC Solver. These can give us the power drawn by the cathode
which, when combined with the output power of the magnetron, lets us calculate its efficiency. The port and monitor definitions for this model are shown in Figure 8 – the red rectangles are the waveguide ports and the lines marked with circles
are the monitors.
Figure 7 The spoked particle beam. Vanes alternately attract and repel electrons
Moving charges induce electric and magnetic fields, and these
fields affect the paths of the moving charges. The PIC solver
takes into account the coupling between the electrons and
the field, simulating the motion of the particles and the field
around them in the time domain. Because it can model particles over their whole path, the PIC solver gives us the fields
over an extended period of time, rather than just at one instant. With it we can examine the beam loading effect – the
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Figure 8 The set-up of the ports and monitors
As shown in Figure 9, for the first 40 nanoseconds of operation, 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 signals, as shown in Figure 10, gives us the hot resonant frequency, 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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Magnetron
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 efficiently.
The analysis of the heat sink was carried out in CST
MPHYSICS® STUDIO (CST MPS), a combined thermal and mechanical 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 fields, 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 efficiency 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 9 The output of the magnetron over time. The magnetron takes around 50
nanoseconds to start resonating
Figure 11 The temperature distribution across the magnetron
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 significant
impedance; in a typical magnetron, 30% to 50% of the input
energy is lost as heat. Magnetrons are quite sensitive to heating, because of the ceramic magnets used to steer the beam. If
these magnets are heated past their Curie temperature, they
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 fixed. This causes 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 surrounding it. This is fine 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 account the casing of the oven and the food being cooked inside
it, both of which can significantly alter the fields generated by
the magnetron.
Figure 12 The magnetron after thermal expansion (exaggerated)
In an ideal oven, which cooked food perfectly evenly, the electric field would be homogeneous. However, the cavity of the
oven reflects and traps the waves, producing a field 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 field 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 field 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 field is stronger or weaker.
Without something to even out the heating, like a turntable,
these would lead to hot spots and dead spots.
Magnetron
This image also shows that the field 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 process of designing and simulating the oven casing.
power (f=2.45) 1sqr(1000),0 (peak)
Cutplane normal: 0, 1, 0
Cutplane position: -185
Component: Abs
2D Maximum: 84,91
2D Minimum: 0,3331
Frequency: 2.45
Figure 14 The field distribution 5 cm in front of the door, with the maximum power
highlighted
Figure 13 The electric field within and around the oven
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Once we have a case, we need to test whether it controls
the electric field correctly. The transient solver available with
CST MWS offers tools for visualizing both the local field distribution and the farfield of the oven, for a complete illustration of its electric fields. Figure 14 shows the nearfield at a distance of 5 cm in front of the door, along with a calculation the
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Magnetron
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 significantly exceeds 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.
the farfield 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 fields over wide areas of empty space, and this can
speed up the simulation dramatically.
Figure 16 shows the 3D farfield 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 fields around the oven.
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ce/
an
ist
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Figure 15 Farfield monitors at 5 cm and 1 m
As well as the nearfield effectiveness, the designer may also
want to examine the effectiveness of the shielding of the microwave oven as a whole, using a farfield analysis. As an additional demonstration of the field simulation capabilities of
CST MWS, two farfield monitors are added to the model, as
shown in Figure 15. These calculate the farfield at that particular point when the simulation is run. With a farfield monitor,
CST Module
CST MWS
CST EMS
CST PS
Part
Quantity
Simulated value
Measured value
Resonator
Q_unloaded
1649.3
1631.0
Resonator
Frequency [GHz]
2.49
2.520
MWO
Frequency [GHz]
2.45
2.45
Magnet
B-field [Tesla]
0.189
0.19
Resonator
Spokes [EA]
5
5
Resonator
Frequency [GHz]
2.471
2.481
Resonator
Power [Watt]
1095.12
1000
335
320
270
221
171
85
Anode
CST MPS
Figure 16 The farfield at 1 m from the oven
Heat sink
Yoke
Temp. [°C]
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Magnetron
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 experiment. In particular, the temperatures of the yoke and heat
sink were significantly lower in the actual magnetron. This appears to be due to a change in the heat sink design between
modeling and prototyping – the heat sink tested in the laboratory had double-bent fins, and the contacts between the fins
and the yoke were different. The high Q-factor and low resonant frequencies might be due to overestimating the conductivity 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 electrodes, 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 environment, allowing a true multiphysics analysis of the magnetron that can replicate the measurement process and reduce the
need for prototyping and experimentation in the design process.
Authors
Dr. Monika Balk, CST AG – Support and Engineering
Seung-Won Baek, CST of America, Inc. – Support
CST AG
Bad Nauheimer Str. 19
64289 Darmstadt, Germany
[email protected]
http://www.cst.com
CHANGI NG TH E STAN DARDS
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