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A DC restorer is a "clamp" circuit, and has five parts:
A low impedance signal source (not shown)
A coupling capacitor.
A switch.
A voltage supply (low impedance, could be ground).
A high impedance device (like a vacuum tube grid input) that senses/uses the output voltage (not shown).
The function of the input source is to force the input side of the capacitor to whatever voltage is being sent, even when the switch is closed.
The function of the switch is to connect the output side of the capacitor to the fixed voltage supply at the desired time.
The function of the capacitor is to charge or discharge until the voltage across the capacitor is equal to the difference between the input voltage and the desired voltage (the voltage source) when the switch is closed, then to hold that difference voltage when the switch is open (which it can do if the load is high impedance and therefore doesn't bleed off the charge.)
So..
We start with the wrong voltage across the capacitor. As the signal voltage goes up and down on the input, it goes up and down by the same amount on the output. However, the difference in voltage between the input and output is not necessarily what is desired; that is, the output has a DC offset.
At the proper time (sync or blanking), the switch is closed, forcing the output voltage to equal the desired voltage. However, the low impedance input forces the input side of the capacitor to remain the same - therefore the voltage across the capacitor is changed so that the output is correct. When the switch opens, the input voltage goes up and down with video again, and the output follows exactly, but witht the correct offset voltage.
The parts described above may be hard to see at first in some circuits, but they will be present.
Now to some nitty gritties:
The switch (diode or triode) may be gated by an external pulse, or it may be self operating (a diode). In the case of the self-operating diode, it is the sync pulse itself that causes the diode to conduct. This situation gets a little tricky - a perfect high impedance load will not work exactly right, since amplitude variations in the input may cause the negative peaks of the waveform to drift upward, and then the sync pulses will not make the diode conduct. Such self-gating clamps must have some discharge path so that the voltage on the output droops enough between sync pulses so that conduction through the diode always occurs. Gated or pulsed DC restorers avoid this problem, and can have a longer discharge time constant, so there is less droop in the waveform across one line time.
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