Small-signal analysis

Small-signal analysis is the universal technique for handling a non-linear device that carries a DC bias and a small time-varying signal at the same time. The idea: find the DC operating point with the full non-linear model, then linearise the device about that point and treat the small signal with ordinary linear circuit analysis. Every amplifier in this course — diode, MOSFET, BJT — is analysed this way, with the same three steps and only the device’s small-signal element changing.

Why split DC and AC at all

Devices like the diode and transistor are non-linear: doubling the input does not double the output. You cannot just “solve the circuit” because superposition does not hold for non-linear elements. The escape is to recognise that the signal is small. Around a fixed bias point the device’s $i$–$v$ curve is approximately a straight line (see Linearisation around an operating point), and a straight line is a linear element. So you do the non-linear work once, at DC, to find where on the curve you are sitting; then the small wiggle around that point sees only the local tangent, which is linear, and linear circuits obey superposition.

The three-step procedure

Step 1 — DC analysis. Set all AC (signal) sources to zero. Find the Operating point of every device using its large-signal model — the Constant-voltage-drop model or Exponential diode model for a diode, the square-law model for a MOSFET. This gives the DC currents and voltages ($I_D$, $V_{GS}$, $I_C$, …) the device sits at with no signal.

Step 2 — AC small-signal analysis. Set all DC sources to zero: replace DC voltage sources with short circuits and DC current sources with open circuits. Replace each device with its Small-signal model (for a diode, the resistance $r_d=V_T/I_D$ from Step 1). Solve the resulting linear circuit for the small-signal output.

Setting DC voltage sources to shorts and DC current sources to opens is just the Superposition principle applied to the linearised circuit: a fixed source contributes nothing to the change in a quantity, and a voltage source that does not change is, to a varying signal, indistinguishable from a short (it has zero AC voltage across it); a current source that does not change looks like an open (zero AC current through it).

Step 3 — Combine. The total voltage or current at any node is the DC value from Step 1 plus the AC value from Step 2. The DC sets where the device operates; the AC is the signal swinging about that point.

Step 1 DC (operating point); Step 2 AC (device → small-signal model); Step 3 total = DC + AC.

The same procedure everywhere

The power of this method is that it does not change from device to device. Only the small-signal element swapped in at Step 2 differs: $r_d=V_T/I_D$ for a diode (see Diode small-signal resistance), the $g_m$, $r_o$ pair for a MOSFET, the $g_m$, $r_{\pi }$, $r_o$ set for a BJT. Once you can do it for a diode you can do it for any amplifier — that is why the diode example is taught first. See Linearisation around an operating point for why the tangent approximation is legitimate and what limits its validity.