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Soft-start is the ability of the PWM to have a slowly-varying duty-cycle, independent of what the error amplifier wants. This is usually required at start up, to ensure currents in the components are maintained at safe levels and prevent overshoot. For example, in a buck, the output voltage is zero at start-up, so the loop would make the duty-cycle maximum, resulting in large overshoot, especially if the load current is low.
Dead-time means the difference between the maximum duty-cycle and 100%. Thus, it is a measure of the assurance you have that the switch will not be damaged, since without any dead-time the inductor does not have time to reset, possibly resulting in saturation and excessive transistor currents.
Soimetimes, dead-time refers to the time that exists between the outputs of a PWM that is designed for push-pull operation. It is important because if the two transistors are ON at the same time, large shoot-through currents will flow. Thus, dead-time is a measure of the assurance that the transistors will not be conducting at the same time.
VVV is correct. Soft start is just what it say. Limit the startup of a circuit so that it does not over shoot its limits. Those limits might be current, might be voltage. It is usaually current. When a circuit is powered up for the first time all the capacitance in the circuit needs to be energized. This creates a large inrush of energy.
As for the deadtime. Again I would agree with VVV. This is the part of the switching or active waveform that is doing no drive, or it might be waiting for a settling/recovery function to occur in a circuit.
The definitions do depend on the circuit you are applying them too.
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