Nothing happens when you turn on the power until you twist one of the shafts; then the other turns in precise synchronism with the first.
And the control twist was often provided by a mechanical analog computer. The fire control computers on these warships were works of art.
[F]or some reason the resolver and the computer were given 800 Hz AC signals from separate oscillators. They were locked in frequency but not in phase, so when they powered up they settled down into one of several (three?) relationships at random.
The radar had three modes: SLEW, AUTO, and LGC. SLEW means manual control. The radar antenna is stationary unless the crew manually operates the shaft and trunnion driver switches. You would use this to roughly point the antenna in the general direction of the CSM. AUTO mode is useful when the antenna is pointed close enough to the CSM that its own electronics can maintain a track. If the CSM drifts off center, the controller electronics can send shaft and trunnion drive commands.
But mostly you wanted LGC mode, in which case the radar was sending range and angle data to the computer via counter interrupts. Also, the LGC could command the radar antenna by sending shaft and trunnion commands. Due to mission ops overkill, flight controllers generally knew what approximate shaft and trunnion angles would apply to all the steerable antennas at all times, given the spacecraft orientation, and it was generally easiest for the crew to aim the antenna by switching out of SLEW or AUTO mode into LGC mode and punching in the shaft and trunnion angles read up to them.
But no one entirely trusted the computer, so there were methods of flying both spacecraft that didn't rely on the computer. And in the case of having to fly a rendezvous under AGS control or (heavens forbid) manual control, there had to be a way of interpreting radar data that assumed the LGC was entirely dead -- including its 800 Hz reference. Thus in SLEW and AUTO modes, AGS got the range and angle data crammed into it by the separate power supply. And the cockpit displays were also driven by the same reference signal. In theory an ascent and rendezvous could be flown by reading off the raw shaft and trunnion angle displays and the range strip. But I'd bet only Aldrin would have been able to do it.
Why in heaven's name they didn't phase-lock the LGC and radar power supplies to start with, I'll never know. But I think you are probably on the right track when you note that slightly out-of-phase signals were not considered especially problematic, especially among analog control system designers who were just cutting their teeth on digital controls.
Incidentally I ran into a similar problem in 2002. A contractor putting together an assembly for us "fixed" a design error by tying two tachometer wires to the same contact pin, from different rotating components. The "fix" was discovered late enough in the process that I and my best software engineer had to figure out a way to decode the convolved tachometer pulses on the wire. And yes, as you insinuate, the signals can combine such that there are twice as many pulses as expected, the correct number of pulses (only much longer), or the agonizingly ambiguous case where the falling edge of the outgoing signal combines with the rising edge of the incoming signal. Luckily these were square wave pulses and we got by until we could justify the expense of reworking the assembly. I have one of these assemblies on my Shelf of Shame in the office.
