JOVANA
Library Glossary Getting Started Three Levels Fields How it works Mission
Join the mission
All guides

The MOSFET & Small-Signal Thinking

There are more [[mosfet|MOSFETs]] on Earth than there are grains of sand or stars in the observable universe — by a wide margin. Every phone, every laptop, every car holds billions of them. In rung 3 you met the BJT, controlled by a *current*. The MOSFET is controlled by a *voltage* on a gate it barely touches, which is exactly why we could pack billions onto a fingernail. Here you'll learn how it works, and then learn the single most important trick in all of analog design: how to make a tiny wiggle on the gate come out as a *big* wiggle on the drain.

A faucet you turn with a charged plate

Picture a garden hose pinched shut. Water (current) wants to flow from one end to the other, but the pinch blocks it. Now imagine you could *un-pinch* the hose not by touching the water, but by holding a charged plate near it — the closer the plate's charge, the wider the channel opens. That is, almost exactly, a MOSFET. The name spells out the sandwich: Metal–Oxide–Semiconductor Field-Effect Transistor. A metal (or polysilicon) *gate* sits on top of a wafer-thin insulating layer of oxide — usually silicon dioxide, the same stuff as glass and sand — which sits on the silicon body. The gate never touches the silicon underneath. It rules by field effect: the electric field from charge on the gate reaches *through* the insulator and summons a conducting channel below.

An n-channel MOSFET (NMOS) has three terminals you'll meet over and over: the gate (G), the drain (D), and the source (S). (There's a fourth, the body or bulk, usually tied to the source — park it for now.) Current flows between drain and source *through* the channel; the gate just decides how wide that channel is. Because the gate sits behind an insulator, almost no current flows *into* it. That single fact — a gate you steer with voltage instead of current — is why the MOSFET conquered the planet.

Threshold, and the square law

Nothing happens until you reach a tipping point. Raise the gate-to-source voltage V_GS from zero and, for a while, the channel stays empty: the drain and source are isolated, the transistor is OFF. Cross a magic number called the threshold voltage V_T (often around 0.4–0.7 V in modern parts) and a thin sheet of electrons — the inversion layer — suddenly appears under the oxide, bridging drain to source. The transistor turns ON. How far you push *above* threshold is what matters; engineers call that headroom the overdrive voltage, V_OV = V_GS − V_T.

Once on, the MOSFET lives in one of two regimes depending on the drain voltage. With a small V_DS the channel behaves like a voltage-controlled resistor — the triode (or linear) region, the home of switches. But raise V_DS enough and the channel near the drain *pinches off*; the current stops caring about V_DS and flattens into a near-constant value. This is the saturation region, and it is where amplifiers live. In saturation the textbook model is the famous square law:

Saturation (long-channel, ideal):

    I_D = (1/2) * k_n * (W/L) * (V_GS - V_T)^2

    k_n = mu_n * C_ox     (process transconductance, A/V^2)
    W/L = channel width / length  (the designer's main knob)
    V_OV = V_GS - V_T     (overdrive)

Worked example  (k_n*W/L = 1 mA/V^2,  V_T = 0.5 V):

    V_GS = 1.0 V  ->  V_OV = 0.5 V  ->  I_D = 0.5*(1m)*(0.5)^2 = 125 uA
    V_GS = 1.5 V  ->  V_OV = 1.0 V  ->  I_D = 0.5*(1m)*(1.0)^2 = 500 uA

    Double the overdrive -> 4x the current.  That's the "square".
The square law in saturation: drain current grows with the **square** of overdrive. Width/length (W/L) is the geometry a chip designer chooses; everything else is set by the process.

First you bias, then you wiggle

Here's the conceptual leap that turns a switch into an amplifier. A microphone, a guitar pickup, an antenna — these produce *tiny* AC signals riding around zero. If you fed such a signal straight into a MOSFET's gate, it would spend half its time below threshold (transistor OFF, output dead) and produce a mangled, clipped mess. The fix is to first establish a steady DC operating point — a [[bias-point|bias point]] — that parks the transistor comfortably in saturation. *Then* you let the real signal ride on top as a small perturbation.

This is the heart of transistor biasing, and it embodies a deep idea: superposition of a big DC reality and a small AC story. Every gate voltage, every drain current splits into two parts — a capital-letter DC value plus a lowercase wiggle:

    v_GS(t)  =  V_GS   +   v_gs(t)
               ^^^^^      ^^^^^^^
               DC bias    small AC signal  (|v_gs| << V_OV)

    i_D(t)   =  I_D    +   i_d(t)

    total  =  big steady part  +  tiny moving part
Notation convention you'll see everywhere: UPPER-case subscript & symbol = DC bias; lower-case = the small AC signal. Total instantaneous value mixes both.
  1. Set the DC bias. Choose a V_GS (via a resistor network or current mirror) so that I_D and V_DS land squarely in saturation, with room to swing up and down. This is the operating point.
  2. Add the signal. Couple the real AC input onto the gate (often through a capacitor that blocks DC). Now v_GS gently rocks above and below the bias point.
  3. Analyse small-signal only. Mentally subtract the DC. Treat the wiggle as if the circuit were linear around the bias point — this is the small-signal model.

Transconductance: the size of the lever

If small-signal thinking is the method, [[transconductance|transconductance]] is the prize. It answers one question: when the gate voltage wiggles by a tiny amount, how much does the drain current wiggle in response? It's the slope of the I_D-versus-V_GS curve *right at your bias point*, and it gets the symbol g_m:

    g_m  =  d(I_D)/d(V_GS)   evaluated at the bias point

  Differentiate the square law,  I_D = (1/2) k_n (W/L) (V_GS - V_T)^2 :

    g_m  =  k_n (W/L) (V_GS - V_T)   =   k_n (W/L) * V_OV

  Two extremely useful equivalent forms:

    g_m  =  2 * I_D / V_OV          (current per volt of overdrive)
    g_m  =  sqrt( 2 * k_n (W/L) * I_D )

  Units: amperes per volt = siemens (S).  Often quoted in mS or mA/V.
Transconductance g_m is the local slope of the transfer curve. The form g_m = 2·I_D/V_OV is worth memorising: it ties the lever directly to your bias current.

Why care? Because g_m is exactly the gain mechanism. Tie a resistor R_D from the drain up to the supply. The signal current i_d = g_m · v_gs flows through it, and by Ohm's law it carves out a voltage v_d = −i_d · R_D = −g_m · R_D · v_gs across it. So the small-signal voltage gain of this stage is a startlingly clean expression:

    A_v  =  v_d / v_gs  =  - g_m * R_D

  Numbers, from our bias:  I_D = 500 uA, V_OV = 1.0 V
      g_m = 2*I_D/V_OV = 2*(500u)/1.0 = 1 mA/V = 1 mS
      with R_D = 10 kOhm:
      A_v = -(1mS)*(10k) = -10

  A 5 mV wiggle on the gate  ->  a 50 mV wiggle on the drain.
  The minus sign = inversion: gate up, drain down.
The common-source stage: gain is just −g_m·R_D. The wiggle on the gate comes out roughly ten times larger (and upside-down) at the drain.

The small-signal model in one picture

Once you've extracted g_m at the bias point, you can throw away the messy nonlinear transistor and replace it — *for signals only* — with a tiny linear cartoon. The gate is an open circuit (it draws no current). The drain is a current source whose strength is g_m·v_gs, controlled by the gate-source voltage. One more refinement: a real channel doesn't hold current *perfectly* constant in saturation — it drifts up slightly as V_DS rises (channel-length modulation), which we capture as a finite output resistance r_o in parallel. Here is the whole model:

   Small-signal model of a MOSFET (in saturation):

     gate o-------+                    +-------o drain
     (G)          |                    |       (D)
                  |          ^         _|_
     v_gs   [open gate]   ( g_m*v_gs )  |  r_o   (output resistance,
                  |        controlled   |        from channel-length
                  |        current src  |        modulation)
   source o-------+--------------+------+-------o source
     (S)                        common

   Gate draws ZERO current.  Drain current = g_m * v_gs.
   Full stage gain with load R_D:   A_v = - g_m * (r_o || R_D)
The hybrid-pi small-signal model. Strip away DC, keep only wiggles: one open gate, one controlled current source g_m·v_gs, one output resistance r_o.

This little three-element sketch is one of the most powerful tools in electronics. It lets you analyse a wildly nonlinear device with nothing more than linear circuit laws — KCL, KVL, Ohm's law — the same tools you mastered in earlier rungs. The BJT gets its own version of this trick too, formalised long ago as the hybrid (h) parameters; the MOSFET's g_m-and-r_o model is the same philosophy, born of an insulated gate. Master this split — bias in big letters, signals in small ones — and the entire field of analog design opens up.