Hybrid pi and T model, rπ and re

Hybrid-π model

The hybrid-π model is used for small signals in the active region. It has rπ between the base and the emitter, a voltage-controlled current source gm vπ between the collector and the emitter, and (optionally) ro in parallel with the source.

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

gm = IC/VT , rπ = vπ/ib = β/gm , ro = VA/IC.

T-model

The T-model is used when the base is the common terminal or when there is a resistor in the emitter. It has re between the emitter and the base, and a current source α ie between the collector and the base.

    C o-------( )  α ie
              |
    B o-------+
              |
             re
              |
    E o-------+

re = VT/IE = α/gm.

Relation between rπ and re

rπ is the resistance seen at the base: rπ = vbe/ib. re is the resistance seen at the emitter: re = vbe/ie. The same vbe appears in both, and ie = (β+1) ib. Hence

rπ = vbe/ib = vbe/[ie/(β+1)] = (β+1)(vbe/ie)

        rπ = (β+1) re

Also, rπ = β/gm and re = VT/IE = α/gm, so rπ/re = β/α = β + 1. For IC = 1 mA, β = 100: gm = 40 mA/V, rπ = 2.5 kΩ, re ≈ 24.8 Ω (rπ = 101 × 24.8 Ω).