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Could I get a MOSFET to behave like a BJT (with respect to the large signal and hybrid-pi model) if I just tie a resistor from gate to source?

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No.

  1. A BJT has a very definite relationship between VBE and currents. MOSFETs have lower gm (dI_drain/dVGS) which is not as precise as BJTs.
  2. High voltage MOSFETs have different characteristics as VDS exceeds about 5-10 V (LDMOS pinch off); no such breakpoint exists in NPNs.
  3. BJTs have non-zero VCE as IC goes to 0; MOSFETs behave like resistors.
  4. The current that flows in your R_GS is linear with VGS; a BJT is exponential. There is no definite relationship (like beta) between resistor current and drain current.
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