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I am giving a pulse input (from a microcontroller) to the INA/INB pin of the Gate Driver IC: https://www.ti.com/lit/ds/symlink/ucc21222-q1.pdf?ts=1708945816410&ref_url=https%253A%252F%252Fwww.ti.com%252Fproduct%252FUCC21222-Q1

Voltage level I know. What about the edge slopes that are permitted at the input of the gate driver? Also, Will the rise/fall times of the microcontroller pulses change when the gate driver is loaded? If yes, how?

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The driver is isolated, its input is a logic input, with a 200k pulldown:

enter image description here

From your MCU's point of view, it looks like any other logic gate, and current from the 200k pulldown will be tiny, so the MCU's output won't get any significant load.

If your MCU boots up with internal pullups enabled, these will probably be stronger than the 200k pulldown. So if the driver's supply comes up before the MCU has finished booting and initializing the IO pins to output low, you could have a short time when both driver inputs will register a logic high. So it could be a good idea to check what your MCU does at boot, and if necessary add an external pulldown to make sure there is no spurious triggering of the MOSFET during boot.

enter image description here

The inputs have a Schmitt trigger with a good amount hysteresis to protect against noise from switching the power device, so they don't specify any minimum rise time. But if you want to synchronize both drivers, then both signals propagation time and rise/fall times should be matched.

Basically, just connect it to the MCU's output, it will be fine.

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What about the edge slopes that are permitted at the input of the gate driver?

The device has got a programmable dead-time set by a resistor so, input rise and fall times don't need to be lightning fast but, you should ensure that both inputs are driven from the same type of output. In the data sheet it also mentions that inputs shorter than 5 ns are ignored so, as a reasonable estimation, I would say your rise time needs to be quicker than one tenth of the programmable dead-time and probably no quicker than the 5 ns from anti-glitch circuit.

Will the rise/fall times of the microcontroller pulses change when the gate driver is loaded?

Not one bit.

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