Linearisation around an operating point

Linearisation around an operating point is the single idea that the entire course is built on: replace a device’s non-linear $i$–$v$ curve with its straight-line tangent at the Operating point, so that small signals see a linear element. Every small-signal parameter in the course — $r_d$, $g_m$, $r_o$, $r_{\pi }$ — is a slope of some non-linear characteristic evaluated at Q. Learn this pattern once and it applies to the diode, the MOSFET, and the BJT without modification.

The Taylor-expansion justification

Take any non-linear device characteristic $i=f(v)$ — the Shockley exponential for a Diode, the square law for a MOSFET. Write the total voltage as a DC bias plus a small signal: $v=V_Q+v_{sig}$. Taylor-expand $f$ about the bias point $V_Q$:

$i=f(V_Q)+{\frac{df}{dv}|}_{V_Q}{v}_{sig}+\frac{1}{2}{\frac{d^{2}f}{d{v}^2}|}_{V_Q}{v}_{sig}^2+\cdots$

The first term $f(V_Q)$ is the DC current $I_Q$ — that is the Operating point. The second term is linear in the signal $v_{sig}$, with constant coefficient ${\frac{df}{dv}|}_{V_Q}$ — that constant is the small-signal parameter (a conductance or transconductance). Everything from the third term on is non-linear distortion. The linear model is exactly the first two terms; we keep them and throw away the rest.

That throwing-away is only legitimate if the discarded terms are negligible compared with the linear term. For the diode exponential, the ratio of the quadratic term to the linear term is $\frac{1}{2}(v_{sig}/V_T)$. So the linear approximation is accurate provided

$v_{sig}\ll V_T(\approx 25\text{mV for a diode or BJT})$

For a MOSFET the square law gives a hard validity scale of $v_{gs}\ll 2V_{OV}$ (Overdrive voltage). “Small signal” is not vague — it is a quantitative bound set by the curvature of the device’s characteristic.

What linearisation produces

The slope at Q has a name and a finite value for each device:

• Diode: $r_d=V_T/I_D$ (see Diode small-signal resistance).
• MOSFET: transconductance $g_m$ and output resistance $r_o$.
• BJT: $g_m$, input resistance $r_{\pi }$, output resistance $r_o$.

Each is just a derivative of a non-linear law evaluated at the bias point. Collect them into a Small-signal model and the device becomes a linear sub-circuit, which is what makes Small-signal analysis possible. This is also why Negative feedback is so valuable: by keeping the device near a fixed operating point it keeps the signal in the region where the tangent line is a good approximation, which reduces distortion. Linearisation is the reason a fundamentally non-linear device can be a faithful linear amplifier at all.