From switch to amplifier
If you arrived here from the digital tracks, you already know the MOSFET as a switch. Drive the gate high and it turns fully on (a closed switch, a hard 0 dragged to ground); drive it low and it turns fully off (an open switch). The whole digital world is built on that clean two-state picture: 1 or 0, on or off, nothing in between. It is a beautiful abstraction, and it is exactly why a billion-gate chip can be reasoned about at all.
Analog throws that abstraction away. The trick is to stop slamming the gate to the rails and instead hold it partway — to bias the transistor at some steady in-between voltage and keep it there. In that half-open state the device is no longer a switch with two settings; it becomes a knob. Nudge the gate voltage up a hair and the current through the device rises a little. Nudge it down and the current falls. The output current *tracks* the input voltage continuously, smoothly, with no 1-or-0 cliff anywhere. That continuous tracking is the seed of every amplifier you will ever design.
Why would a tiny gate wiggle be worth anything? Because that wiggle can be your signal — a faint voltage from a microphone, an antenna, a sensor — and if a small input wiggle produces a *bigger* output wiggle, you have gain. You have made the signal stronger without adding any new information to it. That is amplification, and it is the job this device was secretly built to do the moment you stopped treating it as a switch.
The MOSFET in saturation
To get that knob-like behavior, you have to bias the transistor in a specific operating region called saturation (confusingly, also called the *active* region — same thing). A MOSFET has three regions: cut-off (gate below threshold, the device is off — the digital '0' switch), triode (also called linear or ohmic, where it behaves like a voltage-controlled resistor — the digital 'on' switch), and saturation, the sweet spot in between where it acts like a current source. Analog amplifiers live almost entirely in saturation.
The condition for saturation is simple to state. Define the overdrive voltage V_ov = V_GS − V_th — how far the gate-source voltage sits *above* the threshold V_th. The device is in saturation whenever the drain-source voltage is at least that large: V_DS ≥ V_ov. Below that you slide into triode; you generally keep enough V_DS headroom to stay safely in saturation. V_ov is the single most important design choice you make for a transistor — it sets how 'turned on' the device is, and (as you will see next) it sets the gain too. Typical analog overdrives are small, often 100–200 mV.
.dc VGS 0 1.2 0.01 * Sweep gate voltage, plot drain current Id(VGS): * below V_th -> Id ~ 0 (cut-off) * above V_th -> Id ~ k*(Vgs-Vth)^2 (saturation, square-law) VDS d 0 0.9 ; hold drain high so device stays in saturation VGS g 0 0.0 ; this is the source we sweep
In saturation the drain current follows the classic square-law: I_D ≈ ½ · µCox · (W/L) · V_ov². Read it slowly. The current does not depend on V_DS very much at all — push more voltage across the device and the current barely budges. That is the defining behavior of a current source: it sets a current and holds it roughly constant regardless of the voltage across it. So the headline of this whole section is one sentence worth memorizing: a MOSFET in saturation is a voltage-controlled current source. The gate voltage commands a current, and the device delivers it.
Transconductance: the key number
If a transistor in saturation is a voltage-controlled current source, the obvious next question is: how strong is that control? Wiggle the gate by one millivolt — how much does the drain current move? That sensitivity, current-out per volt-in, is the single number analog designers reach for first. It is called transconductance, written gm, and it is measured in siemens (amps per volt). Its formal definition is just the slope of the current-versus-gate-voltage curve at your bias point: gm = dI_D / dV_GS.
Intuition first, then the formula. Picture the square-law curve from the last section. At your chosen bias point, gm is simply how steep that curve is right there — a steep slope means a small gate wiggle throws a big current swing (lots of control, lots of potential gain); a shallow slope means the gate barely matters. Bias higher up the curve, where it is steeper, and you get more gm. That is the whole picture: gm is the steepness of the knob.
gm = dId/dVgs ; definition: slope of the Id-Vgs curve gm = 2 * Id / Vov ; the everyday design form (in saturation) ; intuition: more bias current Id, OR smaller overdrive Vov -> more gm
Differentiate the square-law and you get the form you will use every day: gm = 2·I_D / V_ov. This little equation is dense with design guidance. Want more gm? Either push more bias current I_D through the device, or shrink the overdrive V_ov (bias closer to threshold). Both make the knob more sensitive — but neither is free. More current burns more power; a smaller overdrive eats into your voltage headroom and slows the device down. Almost every analog trade-off you will meet later is, at bottom, an argument about where to set gm.
The small-signal trick
There is a problem we have been quietly stepping around. The square-law is curved — it is nonlinear. If you feed a clean sine wave into a curved transfer characteristic, the output comes out distorted, with the top and bottom of the wave stretched differently. How can a bent, nonlinear device ever make a faithful amplifier? The answer is one of the most important ideas in all of analog electronics: the small-signal trick, also called the small-signal model.
The trick is to split every voltage and current into two parts: a big steady DC value (the bias, or [[bias-point|operating point]]) plus a tiny wiggle on top (the signal). You bias the transistor at a fixed point on that curved characteristic, then assume your signal is small enough that it only ever explores a tiny stretch of curve right around that point. And here is the magic: *any* smooth curve, looked at closely enough over a small enough span, looks like a straight line. Zoom far enough into a circle and it looks flat; zoom into the square-law at your bias point and it looks linear. Over that little span, the curvy device behaves like a clean linear gain block.
Once you have made that split, you do something that feels almost like cheating: you throw the DC away and analyze only the wiggles. The bias just sets the stage; the signal is what you amplify. In the small-signal world the transistor collapses to a wonderfully simple statement — the small drain-current wiggle equals gm times the small gate-voltage wiggle: i_d = gm · v_gs. A current source controlled by a voltage, perfectly linear, with gm as the dial. That one linear relationship is what makes the curvy, temperature-dependent, messy real device tractable on paper.
Voltage gain
We finally have everything we need to build a real amplifier. The transistor turns a gate-voltage wiggle into a drain-current wiggle (that is gm). But our input was a voltage and we usually want our output to be a voltage too — so we need to turn that current wiggle *back* into a voltage. The oldest trick in the book: run the current through a resistor. A current wiggle i_d flowing through a load resistor R_D produces a voltage wiggle i_d · R_D across it, by Ohm's law. This single-transistor-plus-load arrangement is the common-source amplifier, the workhorse gain stage of analog design.
Now chain the two steps. The input voltage wiggle v_gs creates a current wiggle gm·v_gs; that current flows through R_D and creates an output voltage wiggle of gm·v_gs·R_D. Divide output by input and the v_gs cancels, leaving the voltage gain: A_v = −gm·R_D. The size of the gain is gm times the load resistance — both levers you control. The minus sign says the output is inverted: when the input rises, more current flows, which pulls the output node *down*. Push the gate up, the drain falls. That inversion is a feature, not a bug; you will use it constantly.
* Common-source gain, small-signal: Av = -gm * RD * Example: gm = 1 mS (0.001 S), RD = 10 kohm * Av = -(0.001)*(10000) = -10 -> a 1 mV input wiggle -> 10 mV output .ac dec 20 1 1G ; sweep frequency, plot |Av| vs f to see the gain .op ; first solve the bias point so gm is well-defined
Output resistance & intrinsic gain
To push gain up, A_v = −gm·R_D begs you to make R_D enormous. But there is a hidden ceiling, and it lives inside the transistor itself. Remember we said the saturation current is *almost* independent of V_DS? 'Almost' is the operative word. In reality the current creeps up slightly as V_DS rises — the device is not a perfect current source. That gentle slope means the transistor has its own finite output resistance, written ro, sitting in parallel with whatever load you attach. It is the resistance the transistor's drain presents all by itself.
You can estimate ro as ro ≈ V_A / I_D, where V_A is the Early voltage, a device parameter capturing how flat that saturation curve is (a bigger V_A means a flatter, more ideal current source and thus a larger ro). The consequence is sobering: no matter how huge a load resistor you bolt on, the effective load can never exceed ro, because ro is always there in parallel. The best you can ever do with a *single* transistor is to let ro itself be the load — and that gives the maximum gain one device can deliver.
ro = Va / Id ; transistor's own output resistance Av_max = -gm * ro ; intrinsic gain: best a single device can do Av_max = -(2*Id/Vov)*(Va/Id) = -2*Va/Vov ; the Id cancels! ; note: the gain ceiling depends on Va and Vov, NOT on the bias current
That maximum is called the intrinsic gain, and it equals gm · ro — the product of the two numbers this whole guide has been building toward. It is the hard ceiling on the voltage gain a single transistor can provide. Substitute the design forms and something striking falls out: gm·ro = (2I_D/V_ov)·(V_A/I_D) = 2·V_A / V_ov. The bias current cancels completely. You cannot buy more intrinsic gain by spending more current — the ceiling is set only by the Early voltage (a property of the process and device length) and your overdrive. In modern short-channel processes intrinsic gain is often modest, perhaps 10× to a few tens, which is exactly why real designs stack and combine transistors to climb past it.