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TPS74901KTWR LDO Linear Regulator's Power Efficiency

Other Parts Discussed in Thread: TPS74901, PTH04T240W

I have 12V and 5V supplies only coming from my VME chassis that I can use to feed the TPS74901. I need to generate 1V output for my FPGA and I have to use the TPS74901 because I already have plenty of this part available from our inventory. The problem is the power efficiency becomes an issue at 20% if I use 5V supply and becomes even worse if I used 12V. I'm thinking of using a zener that is rated for 1.2V or less but these types of zener don't seem to exist. I thought about using a simple voltage divider circuit but using resistor to dump voltage and current doesn't appear to be a viable solution due to the excessive heat dissipation.

Does anybody have a better solution to improve the power efficiency of the TPS74901 using the supply inputs mentioned previously?

Thanks,

Jojo