Dynamic resistance of diode, small signal

The dynamic (incremental or ac) resistance rd of a forward-biased diode is the reciprocal of the slope of its i–v curve at the operating point: rd = dv/di at the Q-point. It is the resistance the diode shows to a small ac signal riding on a dc bias.

Derivation

Shockley equation for forward bias (v ≫ ηV_T):

iD = IS [ exp(vD / (η VT)) - 1 ]  ≈  IS exp(vD / (η VT))

Here IS is the saturation current, η the emission coefficient (1 to 2) and VT = kT/q ≈ 25 mV at room temperature.

Let the diode be biased at (VD, ID) and let a small signal vd be added:

vD = VD + vd
iD = IS exp[(VD + vd)/(η VT)]
   = IS exp(VD/(η VT)) · exp(vd/(η VT))
   = ID · exp(vd/(η VT))

For a small signal (vd ≪ η VT) use exp(x) ≈ 1 + x:

iD = ID (1 + vd/(η VT)) = ID + (ID/(η VT)) vd

So the total current is the dc part ID plus an ac part id = (ID/(η VT)) · vd. Therefore

id / vd = ID / (η VT)

        rd = vd / id = η VT / ID

The same result comes from differentiating: 1/rd = diD/dvD = iD/(η VT) = ID/(η VT) at the Q-point.

Conditions and meaning

• Valid for vd ≲ 10 mV (η = 1) so the exponential can be linearised.
• At room temperature with η = 1: rd = 25 mV / ID. For ID = 1 mA, rd = 25 Ω; for ID = 0.1 mA, rd = 250 Ω.
• rd depends on the dc bias current, so the diode acts as a current-controlled resistor. The small-signal model is simply the resistor rd.