The RDC system of the U.S. used "Synchroes" which were a kind of amplified, multi-stage, three-axis transformer that could boost control signals output by a weapon director or calculator directly until they were powerful enough to act as control circuits in the moving mounts/turrets, all without any loss in accuracy and no "man in the loop." THEY NEED AC ELECTRICAL POWER TO FUNCTION!!!!! The introduction of AC power and RDC were simultaneous in the newer U.S. Navy ships introduced during the 1930's, with older ships only gradually being retro-fitted with AC until WWII, when the work became much more important.
The British did not use AC power until they began adopting U.S. AA directors to some of their ships during WWII. They previously employed a hybrid automatic stepped coarse control using DC switches and a man in the final fine control stage, which is rather poor design in AA gunnery against rapidly maneuvering targets, as you can imagine. The VANGUARD was the first ship with extensive AC power from scratch, but even here they did not have as good an RDC system as the U.S. Navy used.
Japanese anti-surface-ship fire control was pretty good, but it was based on shooting at large targets and getting hits on or just short of the target by Type 91 diving APC projectiles. They tightened up their spread between individual splashes as much as possible to "clobber" a target if they got the range, but this meant that they would miss the target altogether if they got the range wrong. In theory, a tight spread may be fine, but under actual battle conditions, the errors are such that it is better to allow a wider spread of splashes to take up the slack in fire control errors, meaning that there is an optimum compromise between gun accuracy and fire control accuracy. The U.S. recognized this and purposely "detuned" their fire control/gun alignment systems to give this maximum overlap at what they considered the optimum expected range. With radar, the fire control range became better, so less slack was needed and tighter salvoes could be used, which greatly helped against aircraft, especially with VT fuzes to "fill in the cracks" in the fire control solution.
Radar beats optical fire control under most cases because visibility is rarely optimum, especially at long range, and because it rapidly gives a series of ranges that can much more rapidly be used to smooth out the actual motion of the target from the artifacts created by various errors. Also, stereoscopic range finders require special training and fewer people are available to use them because of this, making it more difficult on those few when constant operation is needed.
To my knowledge, all U.S. Navy "Range-keepers" (the original name of the analog ballistic computers used in director-control systems in the U.S. Navy; the first, the Mark 1, being made by Sperry-Gyroscope, now Unisys, before WWI) used differentiation and integration (calculus) to determine the current motion of the target so that it could be extrapolated to give a future aim point for gun laying. The future motion is always assumed to be a straight line (even with sophisticated computers today, trying to guess at a curved path is usually a waste of time), with any change in target trajectory (direction or speed) causing a short delay until the "fire control solution" again settles. Constant target maneuvering is why guided missiles with homing warheads (active or semi-active) are needed for any kind of long range in AA weapons; controlling the weapon from the ship causes unacceptable delays when the weapon is near the target and even tiny errors can mean a miss.
The more computers, up until the number of directors/range-finders is reached, the more separate targets can be engaged simultaneously, but the fewer weapons are being fired at each. Against aircraft, which are thin-skinned, using high-rate-of-fire machine guns and automatic cannon, even a single weapon is potentially sufficient, but this is not true when the rate of fire of the weapon is too low, which requires salvo fire from several weapons. For this reason, it is important to carefully design a weapon system to minimize the number of expensive, delicate computers to optimize the system against the most targets that can be fired against with some reasonable chance of success with the weapons suite being controlled by all of the fire control systems--just adding more computers does not mean that more targets are capable of being hit, no matter how many directors you have! Redundant systems for back-up must be included in this study, since things break.
Fire Control on the Prinz Eugen.
U.S. evaluations of the PRINZ EUGEN were mixed. The U.S. sailors considered it a yacht (!) compared to U.S. heavy cruisers and the powerplant was so hard to keep running that some of the original German crew had to be hired on to do so when steaming it to Bikini Atoll.
It had twice as many independent computer-controlled directors so that it could simultaneously engage twice as many targets -- with less weapons per target to be sure--than U.S. warships could unless they went to local control, though the German Navy fire-control optics and computer system was not considered superior to the U.S. Navy system as to getting the range and hitting the target. U.S. warships had much superior radar control, also, by the end of WWII.
Both the U.S. and German Navy had adopted AC electric power in their post-1930 warships (during WWII all U.S. warships were retrofitted from DC to AC, to my knowledge) and, as a result, could use direct, amplified, fully-automatic control of their weapons from the gun director via the ballistic computer, without any human-in-the-loop "follow-the-pointer" manual aiming needed (except as a casualty backup).
The method for doing this differed somewhat, however, between the U.S. and German fire-control systems. The U.S. Navy had developed vacuum-tube power amplifiers connected into three-pole "synchroes" that acted much like locked-together compass needles, one "master" in the director and, after being "suped up" in the amplifier, a second "slave" directly connected to the mount controls and to any repeater indicators in other parts of the ship. The slaves in the mount had enough strength to adjust the turret's electric and hydraulic motor controls by themselves without a man having to do so. To increase accuracy, a 1-to-1 (360 degrees per turn of the mount for train) or 2-to-1 (180 degrees per turn for elevation) synchro for "coarse" positioning and a second 10-to-1-ratio (36 degrees per turn) synchro for "fine" positioning were usually tied in-line, one pair for elevation and one pair for train, with the fine-tuning of the final aim point being made by the "fine" synchroes after the 1- or 2-speed "coarse" synchroes had moved the mount to within a couple of degrees of the target in both train and elevation (special cross-connected circuits handled this selection originally, but, later, computer logic did this selection of which synchro to use when for optimum aiming). By using three poles in the pick-off from the director's position "needle" (one synchro for train and one synchro for elevation), an exact (within the tolerance of the rotating gears, which is why the two, coarse and fine, synchroes were used) unambiguous needle position could be obtained that was all amplified simultaneously by the same vacuum-tube amplifier circuit, so even if the amplifier was slightly out of adjustment, all this did was increase or reduce total power, not change the angle data. Only AC systems could act in this transformer-like manner, not DC systems, which needed stepped power circuits and, almost always, a man for final adjustment of aim.
The German system used "magnetic amplifiers" which were also a kind
of AC electric transformer, but did not use any vacuum tubes to amplify
the leverage (voltage) of the control circuits--they acted more like the
nested coils in the ignition system of automobiles. They seem to
have been quite satisfactory and U.S. ordnance designers studied them carefully
during and after WWII. In some functions, the German-developed magnetic
amplifiers turned out to be best and the U.S. Navy adopted them after WWII,
but synchroes seem to have been best for raw power output for controlling
weapon train and elevation machinery and are still used in many cases by
many navies, though digital computer controls are gradually replacing them