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I would like to know physically what happens to the flash lamp when we use the compensation in his TTL mode (EV's +/-) I ask if the lamp will stay on for more or less time or we change the power in whatts of it ??

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A given electronic flash generally cannot modify its intensity, instead it modifies its time.

As a ballpark guide: a full power flash (power being a misnomer) is on for about 1/1000 of a second (1 msec), while a low power flash may be on the order of 1/25,000 of a second (40 µsec). High speed photography takes advantage of this by using electronic flashes at low power (actually energy) settings.

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    For small speedlights you are mostly correct. But there are plenty of different kinds of flashes that aren't speedlights. Even some mid-sized portable external battery powered "bare bulb" flashes can do it differently, and an increasing number of those and studio sized monolights are coming with TTL capabilities.
    – Michael C
    Jul 20 '20 at 9:38
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All xenon flashes work by charging up a capacitor, and then partially or fully discharging the stored energy of the capacitor through ionized xenon gas in the flash tube. (There is also a trigger needed to ionize the gas, but let's ignore that.)

Speedlights:

Speedlights work by fully charging the capacitor every time, and then using a high speed, high power switch to suddenly turn off the flow of current to the flash. As soon as the flow of current stops the xenon gas in the flash tube stops glowing almost instantly.

In manual mode, the flash lets the capacitor discharge through the flash for a fixed amount of time that is calculated to give the desired amount of light. (The change in brightness of the flash is not linear, so it isn't straightforward to figure out the amount of "burn time" required to produce a given amount of light.

TTL mode for most digital cameras uses a series of calibrated pre-flashes. The camera measures the amount of light visible in the exposure meter from those flashes, computes the amount of light needed, opens the shutter, and then fires the flash for the duration that gives the desired amount of light.

Most film cameras, on the other hand, open the shutter, begin firing the flash, and measure the total amount of light returned from the scene. When the light meter detects that the exposure is right, the system switches off ("quenches") the flash.

Studio flashes:

Some studio flashes also fully charge the capacitor and then quench the discharge electronically. Most, however, use a power setting to tell the flash how much energy to put into the flash capacitor. When the flash fires, it fully discharges the capacitor. If the capacitor was charged fully, you get a maximum power flash. If the capacitor was charged with half as much energy, you get half as much light. (I believe the relationship between capacitor energy and total light output is linear, but I'm not positive.)

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