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I can't understand how MOSFET LDO works and when it would be low drop.

Below, there is a simple schematic with N channel mosfet and TL431. Providing gate voltage, D-S will open, but minimum Gate-Source voltage for eg. IRF540 is 4V (for 1A drain current). Load is connected between source and GND, so minimum Gate-GND voltage is Vout+Vgs=Vout+4V, so whole Vin is minimum Vout+4V.

Do I understand it correctly?

Circuit schematic

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    \$\begingroup\$ Using a regular NMOS for the series regulation probably is mutually exclusive with it being a LDO. Typical series regulation parts are PFET/PNP. This still could work as a linear regulator, but many people conflate/confuse the two terms. \$\endgroup\$ – W5VO May 19 '18 at 15:28
  • \$\begingroup\$ @W5VO You are aware that an LDO regulator is a linear regulator? \$\endgroup\$ – marcelm May 19 '18 at 17:25
  • \$\begingroup\$ @marcelm And that's the exact problem that I'm talking about - so many people are used to the statement "LDO == Linear Regulator" that they assume the commutative property "Linear Regulator == LDO". Next time find a way of saying that without patronizing people. \$\endgroup\$ – W5VO May 19 '18 at 21:42
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Your circuit works as an LDO regulator but only because there is a 5 volt power source that can lift the gate 5 volts higher than the input supply voltage. This means that if you only need 4 volts gate-source there will be virtually a zero voltage drop-out performance.

All linear LDO regulators that I know of don't have this useful but impractical voltage source hence they use PMOSFET transistors.

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    \$\begingroup\$ Actually there are a fair number of LDOs with a bias input intended to be used with a low-current higher voltage rail to get lower dropout, so sometimes when the extra voltage source exists anyway it can help get very low dropout on another rail: ti.com/lit/ds/symlink/tps7a10.pdf \$\endgroup\$ – John D May 19 '18 at 16:11
  • \$\begingroup\$ @JohnD Nice find so now I can say that I know of no regular LDO regulators and only 1 irregular LDO regulator (that requires a low current bias voltage 1.4 volts above Vin). \$\endgroup\$ – Andy aka May 19 '18 at 16:23
  • \$\begingroup\$ So only solution is to change N MOSFET to NPN BJT? Am I able to reduce dropout using P channel and TL431 in similar configuration? \$\endgroup\$ – kamillabaarnowak May 19 '18 at 16:53
  • \$\begingroup\$ @kamillabaarnowak - no with a PMOSFET you will need an extra inversion of the signal and you get an accompanying level of instability that you have to fight against / correct in the design. Not a simple step. Anyway that question, if you are interested in a fuller answer would best be served by googling how they work and if you have issues ask a new question here. \$\endgroup\$ – Andy aka May 19 '18 at 19:33

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