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The word "rotodome" is a portmanteau of rotating-radome. Because of the flying saucer shape, a radar housed in a rotodome can provides coverage from the horizon to the stratosphere. The rotodome is usually unpressurized, and requires air vents to exhaust the waste heat from the radar. Rotodomes require rojos (rotary joints) for coupling RF and other signals between the radar and the cabin of the aircraft. The radar inside a rotodome can be mechanically scanned (MSA), active electronically scanned (AESA) or passive electronically scanned (PESA).
The main application of rotodomes is airborne early warning (AEW) systems. In earlier surveillance aircraft such as the E1B Tracer, a fixed radome was used to house the radar, that radome bears a strange resemblance to a certain bad hair piece. The original rotodome was placed on the E2B, which dates back to a procurement in 1956. The E2 series has flown ever since, but the latest version (E2D) is built on an entirely new airframe and its electronics are as modern as it gets. The E2D rotodome is 24 feet in diameter.
Below is a video that describes Northrop Grumman's E2D Advanced Hawkeye, a US Navy plane that is kept airborne 24/7 around aircraft carrier battle groups for "battlefield management" (i.e., serious whoop-ass). The rotodome houses antennas for long-range early-warning radar and IFF systems. Other fixed antennas on the E2D airframe provide communications.
Northrop Grumman's E2D Advanced Hawkeye
Another example of a rotodome is located on the Boeing E3 Sentry, also known as AWACS. This system is flown on Boeing 707's (which are long out of production) and first flew in 1977. The rotodome is 30 feet in diameter. In 1995, an AWACs plane crashed shortly after takeoff in Alaska, after ingesting birds into two engines, all 24 crew were killed.
Japan flies a variation of AWACs, mounted on Boeing 767s.
Below is a rare look at the radar inside a rotodome, in this case you are looking at Northrop Grumman's AN/APY-1/2 AWACS multi-mode radar inside the E-3D rotodome. This is an example of a PESA, where transmit power originates in a high-power tube, connects to the rotodome through a rojo, and is distributed to antenna elements by a feed network. Each element contains a high-power (presumably ferrite) phase shifter for beam control. We're not sure how many PESAs are inside, it could be just one, or two facing in opposite directions, but probably not three. Sometimes it is best not to guess about military stuff. Why guess, when plenty of unclassified information on AWACs is available on the Defense Technical Information Center web site, for example, this?