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SN74LVC2G06: Best way to lower V_IH?

Part Number: SN74LVC2G06
Other Parts Discussed in Thread: SN74AUP2G06

Hello,

I have a battery-run application that requires a signal to be inverted with a circuit that draws a minimum quiescent current. The inverter output shall be open-drain (high input = low output, else Hi-Z). The inverter shall be powered by a lithium cell (Vcc = 2.7V - 4.5V), but the logic level of the input is only 1.8V. This is illustrated in the figure below for clarity:

The problem with this that I can see is that Vcc is too high to bring V_IH(min) lower than 1.8V; at least for any inverters I have looked at. 

The best I could come up with at the moment is to power an open-drain inverter such as the SN74LVC2G06, with a low-Iq LDO. Bringing Vcc down to something like 2.0V will result in a minimum V_IH level lower than 1.8V. Alternatively I could shift-up the logic, but the levels translators I have seen all have a significanly higher Iq than the LDOs I have seen.

I thought to post this here in case anyone that is more familiar with the many different TI offerings can think of an alternative solution that I missing.

The reason I mention a dual-inverter in the subject, is because I do actually have need for a second inverter. This second inverter will run on 3.3V logic, so doesn't have the same challenges as the first, but including it in the same solution is good for minimising Iq and part count as far as I can see. 

Best,

Eddie