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What exactly will happen if I use 1.2K ohms at LIN master node pull-up? Why does the LIN protocol spec recommend to use 1K typically?

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What happens is that then rising edges are slightly slower. It might not be a serious issue, but it does make difference depending on this being one-off hobby project or official product that goes through compliance testing which might fail.

Sometimes specifications define something because otherwise there would be too varying implementations that are not compatible.

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