Transconductance gm of BJT

Transconductance (gm) is the ratio of the change in output (collector) current to the small change in input voltage (VBE) that causes it, at a fixed VCE. It measures how effectively the input voltage controls the output current: gm = ∂iC/∂vBE (units: A/V or S, mA/V).

Derivation for BJT

In the active region the collector current is

iC = IS exp(vBE / VT)

Let vBE = VBE + vbe, where VBE is the dc bias and vbe the small signal:

iC = IS exp(VBE/VT) · exp(vbe/VT) = IC exp(vbe/VT)

For vbe ≪ VT (about < 10 mV), exp(x) ≈ 1 + x:

iC = IC (1 + vbe/VT) = IC + (IC/VT) vbe

The signal part is ic = (IC/VT) vbe, so

gm = ic/vbe = ∂iC/∂vBE |at Q = IC / VT

Or by differentiation: gm = d/dvBE [IS exp(vBE/VT)] = iC/VT = IC/VT.

Remarks

• At room temperature VT = 25 mV, so gm = 40 IC; for IC = 1 mA, gm = 40 mA/V.
• gm is proportional to the dc collector current only (independent of the transistor's size).
• Related quantities: rπ = β/gm and re = α/gm ≈ 1/gm.
• Voltage gain of CE stage: Av = -gm RC.