How does aviation radar work?

Most aviation (air traffic control) radar isn't really radar - a radio signal is sent to a transponder on the plane which broadcasts it's own position from the planes GPS, along with it's ID, route, altitude etc

You can determine height (and range) from primary radar by sweeping a narrow beam vertically and measuring the angle at which you get a strong return. Essentially it's just regular radar turned on it's side.

 

Civil aircraft don't carry primary radar, the position of other planes and the collision avoidance is managed by the transponders.
Large airliners do carry weather radar which is relatively short range and low resolution to pick up rain clouds.

Do planes still use radar altimeters or do they rely on pressure and GPS?

 

You can measure arrival times to incredible accuracy but atmospheric disturbances would be much larger than the difference in arrival time

 

Actually r, theta and phi, not x,y,z. Search and tracking are done with different modes, and sometimes by different radars. A search radar sweeps a beam which might be large. Air traffic control radars use an antenna shaped like a part of a cylinder (or paraboloid). They're wide in the rotation direction to provide a narrow beam in azimuth angle phi, but short in height to provide a fan that covers from ground to high altitudes. Phi is read off from the antenna pointing angle when a target is detected.

On fighters or air defense radar, a track mode can be activated on a detected target. By using a large antenna (large compared to a wavelength), a pencil-thin beam is formed (a beamwidth of 1 degree is typical). Thus the theta and phi angles of any target illuminated by the beam are known, and distance is found by measuring pulse round-trip travel time.

For greater accuracy needed for targeting and fire control, a "monopulse beam" is used. The antenna is split vertically into two parts whose signals are subtracted to form a narrow null at boresight. A servo points the antenna in elevation to keep the signal nulled, which can be done with extremely high accuracy. The antenna is also divided horizontally for accurate phi pointing. At the same time, all four quadrants are added in phase for the regular radar mode already mentioned.