110625-0800 EDT
realoman:
A unidirectional varying DC voltage or current into a transformer primary will produce the AC component on the output. As you have said.
When you do this there is a DC bias to the transformer core flux and an unbalanced hysteresis curve. Usually a transformer operated in this manner, a single-ended audio stage, will have more air gap in the core.
A push-pull stage has a center-tapped primary and thus cancels out the two DC components. The center-tapped primary is also what was used in the car radio radio power supply.
If you apply a balanced sq-wave input to a transformer, no DC component, and make the frequency low enough relative to the equivalent shunt inductance of the primary, then the output is approximately the calculus differential of the input sq-wave as somewhat described by electric-light above. His description brought in other factors. A single RC or RL circuit can be considered an approximation to an integrator or differentiator depending upon their connection, and the circuits are given these names when used in this fashion.
(For example if you want a pulse from the leading edge of a sq-wave, then you connect a small capacitor to the sq-wave, and to a low resistance load. 100 pfd to a 1000 ohm load produces about a 0.1 microsecond pulse. Obviously an exponential decay. A diode might be needed on the output side to remove the opposite polarity pulse from the opposite sq-wave edge.)
Back to the transformer. As you raise the frequency of the sq-wave the secondary output will become more identical in shape to the primary.
The design of the vibrator type radio power supply was essentially that of a class B audio amplifier. Each side of the primary was excited only 1/2 of the time. The output voltage was approximately turns ratio times battery voltage.
Ignition coils are also transformers to raise a lower voltage to a higher one, but on the primary side the core is driven hard and high up in magnetic flux. Basically charging the magnetic field. This is a fairly slow rise because of the primary L-R time constant. On the secondary there is an induced voltage that is something less than the battery voltage times the turns ratio. Really much smaller.
But the high voltage for the spark is generated when the primary current is interrupted and the rapid collapse of the magnetic field creates a large voltage across the primary (much larger than the battery voltage), 100s to a thousand volts across the primary. This large voltage times the turns ratio produces an even larger secondary voltage, maybe 30,000 volts open circuit.
The whole operating of the circuit is more complex because of distributed capacitance and non-perfect magnetic coupling from primary to secondary. There is also the capacitance on the primary to reduce arcing of the distributor points. Typically in engines the spark plug gap would breakdown in the 10,000 V range. Highly variable based on gap, sharp edges, and pressure.
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