Calculating MOSFET gain

[billings]

I'm futzing around with the peppermill circuit on the breadboard, and wanted to futz with the gain on the MOSFET boosting stage.  I saw the 0.5 (Rd / Rs) formula used in the fetzer valve article for JFETs, but given that the MOSFET booster is at unity gain with 10k at both source and drain it would seem that formula doesn't apply.  Did a search on the forums here and didn't find anything.  My text only seems to mention the math for grounded emitter depletion-mode devices, which doesn't really help me out much.  Anyone know the right math here?

[R.G.]

A MOSFET has a transconductance.

That is, for a small differential voltage on its gate-to-source, it causes a certain current to flow. I happen to know that power MOSFETs usually have a transconductance of around one ampere per volt, or about one Siemen. The Siemen is a unit of 1/(1 ohm), or the inverse of resistance, which is volts per ampere. A common BS170 has a transconductance of about 320mS or 0.32A/V.

The MOSFET transconductance causes current to flow based on the gate-source differential. If you put a resistor between the source and ground, that resistor causes a negative feedback voltage that subtracts from any voltage you put on the gate - just like an emitter resistor does for a BJT in an emitter follower. The long and short of it is that the "gain" of a MOSFET from gate to source is just slightly under one.

However, the source current flows in the drain circuit, too. In fact, EXACTLY the same current flows in the drain as in the source - that's because the gate is insulated from them by twenty volts worth of glass. If you put a resistor in the drain, the drain voltage will vary by the source current times the drain resistor. And the gain must and always will be -Rd/Rs.

What if you bypass part of the source resistor? Then the gain will be -Rd/Rsu where Rsu is the unbypassed part.

What if you bypass ALL of the source resistor? Then the gain will be -GmRd, where Gm is the transconductance, which varies from device to device.

[puretube]

The SIEMENS is named after the guy
which the street adjacent to my backyard is named after.

{Besides having invented (a.o.) the pointer telegraph
and the dynamo,
the company he established provides over one quarter
of the inhabitants of OHM`s birthplace
(which now is the home of Siemens-HQ) with jobs,
which make this place the 3rd-richest german city
concerning purchasing power...}

The unit "Siemens" is used the same in singular and in plural.

[old tube-purists remember it also being named "mho"
(`Ohm` spelled backwards...) in the not-so-long-ago past].


btw.: in the native language of the guy my old school was named after,
the unit that was named after the law he discovered
is being used as "Ohm" in singular as well as in plural...

[R.G.]

It would really be better if all us native speakers of English could actually use the language correctly... 

[Rob Strand]

> . If you put a resistor in the drain, the drain voltage will vary by the source current times the drain resistor. And the gain must and always will be -Rd/Rs.

That's not quite right the negative feedback on Rs isn't perfect the Gm part still has effect when the source resistor is present.  It's the same sort of thing with a BJT where you have to include 're' when the emitter is bypassed or unypassed.  So you end-of with the general expression,

  Gain = - Rd / (Rs_unbypassed_part + rs)

where,

   rs = 1/gm.

So when Rs_unbypassed part = 0 you get  Gain = -Rd / rs = - Rd * (1/rs) = Rd * gm ie. the normal fully bypassed source gain equation you gave.

Note gm (or yfs) isn't the gm in the datasheet it needs to be recalculated at the MOSFET is operating current..  The quick way to do this is,

  gm_actual = sqrt( Id_actual / Id_datasheet) * gm_datasheet

where,

  gm_ actual is the gm of the MOSFET in the circuit
  Id_actual is the drain (or source) current in the circuit
  Id_datasheet is the Id that gm_datasheet is specified at in the datasheet
  gm_datasheet is the gm specified in the datasheet.

A final caveat is most MOSFET data sheets are specified at high operting currents (1A +) whereas effects use less than 10mA.  The parameters in the data sheet don't tend to work that well since the behaviour of the MOSFET changes at low currents.

All the above equations actually work for JFETS as well as MOSFETs.

[Rob Strand]

FYI:  This article shows the gain for JFETs.  It's the best one I could find in the time, sorry no MOSFETs.   The equation given is the same as the one I gave but it's writtent slightly differently - replace the gm's with 1/rs to get my equation.

http://www.interfet.com/pdf/App_Notes.pdf