" A magnetic doughnut "
This is the ferrite ring, correct?
Well yes. However in hindsight, I've never actually checked to see if it was indeed magnetic. That was silly of me, but we are talking about the same thing
" But, they also respond to DC if it's a transient spike. That's why, when the circuit is first powered up, the voltage increases. "
Oh, I did actually not know that...
What's required is that the electricity "cuts" across the coil, inducing voltage in the secondary. With AC, the flow goes positive, returns to neutral, then goes negative. Each cycle fulfills the requirement. DC goes from zero to full positive, and in doing so it does "cut" across the field, resulting in a single spike.
"So, when a device with caps in the primary circuit is turned, the >> current << flow is very high until the capacitors have fully charged. "
This is because the capacitor limits the voltage until it's discharged, right?
You're not even close on this one.
Capacitors store electricity quantitatively! That is to say they store "amperes". But it is stored at the prevailing "pressure", which is voltage. A much better explanation is that they stabilize the voltage of the circuit
The classic use of large electrolytic capacitors, is in the final output stage of all audio power amplifiers. Normally, the output stage of an amplifier is directly connected to the B+ supply, which comes directly off the PSU. The caps perform 2 functions. They reduce "AC ripple" in the DC side of the PSU, (The "rectifiers" in the PSU convert AC input to DC, but it is actually still in the form of a half sine wave, or can also be with the negative AC pulse inverted to positive. In this case both the "DC" pulses are half AC sine waves). And returning to topic, the caps second function is to prevent amplifier loading from dropping the B+ voltage. As the voltage drops, caused by perhaps, somebody blasting a power chord on the guitar, the capacitor discharges to maintain voltage equilibrium in the circuit, When the peak passes the caps recharge from the PSU.
When the amp is first turned, the caps are empty, and the current inrush is formidable. Since the output transistors are wired directly to the PSU, the excess startup current would fry them. Hence, the "speaker protection relay", which separates them from the circuit until it is at operating equilibrium. The current inrush is so massive in this type circuit, it technically is almost a short. At the very least, the output transistors would perceive it to be a short.
In its most basic form a capacitor is two plates in close proximity. Because opposite charges attract, (just like opposite magnetic poles). the negative potential "holds" the positive charge on the positive plate. The size of the plates and their proximity, define the quantity of electricity that can be stored.A capacitor will pass AC, because the charge potential reverses, the positive plate becomes the negative plate, over and over. Given this template, it should be easy to figure out why the capacitor will block DC and pass AC in the same circuit.
Now, about Lenz's law: "An induced current is always in such a direction as to oppose the motion or change causing it." Quote of Wikipedia really, it's a bit tricky to explain it otherwise.
But one famous effect of this law is that phenomenon when a circuit is broken, the voltage spikes tremendously because it's trying to stay "circuited". So for an example, a Tesla coil needs a voltage safety gap, otherwise Lenz's law would make sure to burn out the entire transformer when the device is shut off because the voltage spikes.
This is because the field collapses and "cuts" across the inductor. The DC drop is acting as an AC swing toward negative. (At least that explanation is close enough to satisfy me).