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Commercial magnetron from a microwave oven
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The magnetron is the tube was what enabled airborne radar at microwave frequencies during WWII. Invented by Microwave Hall of Famer Albert Wallace Hull, fellow Hall-of-Famer Percy Spencer was later figured out how to productize the expensive and labor intensive maching process that the British were using to manufacture C-band maggies in the late 1930s. Marvin Bock was in charge of commercializing the Radarrange in the late 1940s.
The beauty of the magnetron in WWII was that it provided high power (hundreds of watts) at extremely high frequency (C-band!) which allowed radar systems to use a parabolic reflector for an antenna; this reflector was small enough to be housed inside the nose of an airplane, behind an aerodynamic radome, rather than a drag-inducing dipole array mounted on the exterior of the plane. Additionally, the magnetron's high frequency afforded the radar operator a much sharper image of the target than that offered by the dipole array. To the end of the war the Germans had to use dipole arrays on their airplanes because their radars had an upper frequency range of about 200 MHz.
A magnetron can either be a source of microwaves (an oscillator) or an amplifier.
The word "magnetron" is a portmanteau, combining "magnet" and "electron".
Are you as smart as a fifth grader?
The information below was originally written for a fifth grade class in Eugene Oregon that asked the question, "what's that big magnet inside the microwave oven for?" The Unknown Editor was obliged to "cook up" an answer, but perhaps delivered more than they wanted!
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Your question is a good one! I must confess I have never had reason to understand the details of a magnetron, but I will attempt to give you an explanation that might help.
Question: what does a magnetron have in common with a pokemon? They are both examples of portmanteau, where two words are combined into one new word.
Engineers use portmanteaux all the time, although most of them don't even know what the word means!
Weedle, #13 Pokemon
First of all, it is hard to believe that people figured all this stuff out a long time ago. Some very clever thought went into the magnetron designs in the late 1930s, by mathematicians, then scientists, then engineers. Raytheon was involved in productizing the device, the British invented it but the way they manufactured it was labor intensive. Percy Spencer came up with a way to replace expensive machining with a stack of stampings which was much, much cheaper. Today a secret invention that helped win WWII (enabling airborne radar) is manufactured in China to heat your lunch! But I digress...
Now then, vacuum electronics used to be the king of all electrical devices like radios and televisions, before the "transistor age" which started in the 1950s. Tubes, like transistors, can perform many functions, like amplifiers, switches, TV screens, even computers (like the ENIAC which consumed enough electricity to light up Eugene Oregon). Electronics was much cruder way back in the day!
The way a vacuum tube works is with sufficient temperature and very high voltage (electric field), electrons can be boiled off of one metal and travel to another one, through the vacuum, as opposed to through a wire. The reason that TVs and radios used to have to warm up, was because the heaters in the tubes had to get hot enough to boil the electrons. This need for heat is the opposite of transistors, where heat is considered the biggest enemy of reliability.
Electricity and magnetism are very interrelated. The easiest examples to think about are motors and generators. Although they don't all use permanent magnets, they all exploit the interaction of electrons to a magnetic field.
A funny thing about that interaction... when an electron is moving in one direction (say East) if it encounters a magnetic field that crosses its path (North-South) it is deflected upwards, not sideways! This is kind of like (but unrelated) to a gyroscope, when you try to rotate it in one direction it fights back at a 90 degree angle to the force you are applying.
So now we get to the magnetron...
In a "maggie" the conductor at the center is heated. Then a huge DC voltage (equivalent to a few thousand AA batteries connected in series!) is put between the center conductor and the outer conductor. The voltage is enough to really hurt you or kill you, so don't mess around with an oven that is partially disassembled! The voltage is stepped up from 120 volts that the power company provides to your wall outlets, then converted from alternating current (AC) to direct current (DC). Ask your teacher to explain AC and DC... the result is that many, many electrons flow through the vacuum from the center to the outer conductor, concentrically. At this point, we don't have any conversion of "house current" to microwave current, microwaves are a form of alternating current but at 40,000,000 times the frequency that the power company sent to your house!
This little Maggie is a fourth grader and hasn't learned to sharpen her giant pencil. She's a pretty good swimmer though!
The giant magnet in a microwave that you referred to is positioned to put an extreme magnetic field up and down through the magnetron (north-south in magnet lingo), while the electrons are moving from center to outside (concentrically). The effect is that the magnet defects the electrons sideways. By careful design, the magnet can spin the electrons around the gap in the magnetron, when the force the magnet exerts is equal to the centrifugal force of the spinning electrons. So what you have is a "gas" of electrons, spinning like a tornado around and around inside the magnetron! A beautiful thing, but not yet a source of microwave energy.
Schematic borrowed from Wikipedia, electron path in red
Now picture when you are driving in a car on the highway, and someone opens one of the rear windows... and all you can hear is that wump wump wump noise and it drives you crazy! This is because the car has a resonance at a very low frequency. A flute also converts wind to sound, but at a much higher pitch because the flute's resonant cavity is so much smaller than the inside of a car. Both examples convert one form of energy (wind) to another (sound). This is exactly what happens in a magnetron! Those little chambers in the structure resonate at a certain frequency when the electron cloud blows past them. Thus one form of energy (electricity from the wall outlet, which is stepped up to very high voltage inside the microwave oven) is converted to another (microwaves). The energy is simply picked off by sticking a wire or antenna (shown in brown) into one of the magnetron's cavities, and the energy travels along the wire and through a waveguide to a second antenna that sends the energy to your food. A waveguide is merely a hollow metal pipe that the wave energy can travel through with low loss, like when you speak though a pipe and a friend listens on the other end. Indeed, there are many analogies between microwaves and sound waves, they are actually of quite similar dimension (wavelength), the real difference is that microwaves travel at 1,000,000,000 feet per second while sound travels at "only" 1000 feet per second!
Maybe I've given you too much to think about all at once, let's just simplify it. The magnet is used to spin the electrons in a circle, and the cavities are there to steal energy from the spinning cloud and generate 2,400,000,000 cycles per second of radio waves, at a power level high enough to cook your lunch. Note that the magnet doesn't supply a lick of energy to the system (the power company and Mom's checkbook get credit for that), the magnet just happens to guide the electrons and trick them into converting their energy to something we can use (a warm and tasty snack, just be sure to "run it off" at lunchtime!)