Short note: π-models of BJT and MOSFET

The hybrid-π model is the standard small-signal equivalent circuit for transistors in their active (BJT) or saturation (MOSFET) region. It replaces the device with linear elements and a voltage-controlled current source.

π-model of the BJT

  B o------+                +---------+------o C
           |  +             |         |
          rπ  vπ      gm vπ (↓)      ro
           |  -             |         |
  E o------+----------------+---------+------o E

• rπ = β/gm is connected between base and emitter (input resistance); vπ is the voltage across it.
• gm vπ is the collector current source, with gm = IC/VT.
• ro = VA/IC is the output resistance (Early effect).
• Parameters: gm = IC/VT; rπ = β VT/IC; re = α/gm ≈ 1/gm.

π-model of the MOSFET

  G o---                    +---------+------o D
         +                  |         |
        vgs  (open)   gm vgs (↓)     ro
         -                  |         |
  S o-----------------------+---------+------o S

• The gate is insulated, so the input resistance is infinite: the gate-source port is an open circuit (rπ = ∞) at low frequency.
• The drain current source is gm vgs, with gm = 2K (VGS - Vt) = 2 ID/(VGS - Vt) = 2√(K ID) (K in A/V², for the convention ID = K (VGS - Vt)²; with ID = ½ kn'(W/L)(VGS - Vt)², the same result is written gm = √(2 kn'(W/L) ID)). The output resistance is ro = VA/ID = 1/(λ ID).
• The body effect adds a second source, gmb vbs.

Comparison

At high frequency, the capacitances Cπ, Cμ (BJT) and Cgs, Cgd (MOSFET) are added.