How a transistor actually amplifies

The phrase small-signal gain is one of those engineering expressions that manages to sound both technical and unhelpful at the same time. A transistor amplifies. Fine. How?

The answer, when you work through it, is not that a transistor makes electrons stronger. Nothing in the device multiplies anything in a physical sense. It controls how hard a separate, already-energized current is allowed to flow, and then a nearby resistor turns that variable current back into a variable voltage, and the variable voltage is bigger than the one you started with. That is it. Everything else is details, and the details are where it gets fun.

We'll build this up one scene at a time using an NPN bipolar junction transistor, the sort of thing that still comes in a TO-92 three-legged black lump if you buy a bag from Mouser.

What is a BJT?

A bipolar junction transistor is a three-terminal semiconductor device: a sandwich of doped silicon with a wire attached to each of its three layers. The NPN kind we'll be using stacks n-type silicon, p-type silicon, and n-type silicon, in that order. The three terminals attached to those layers are called, from outside to middle to outside, the collector, the base, and the emitter.

"N-type" and "p-type" are just two versions of silicon that have been doped with impurities to give them either a surplus of free electrons (n-type) or a surplus of the absence of free electrons, which solid-state physicists call holes and treat like positive charge carriers that move around independently. The bipolar in the name is an admission that both kinds of carriers end up doing useful work inside the device. Most semiconductor designs try to avoid that. The BJT embraces it.

The practical consequence is simple: a small current flowing into the base lets a much larger current flow from the collector to the emitter. That is the whole trick, and once you accept that it happens, the amplifier parts follow.

A transistor is a valve

Start with the classical way a ham thinks about a BJT. A small current flowing into the base allows a much larger current to flow down through the collector. The ratio between the two is called β, and for most small-signal transistors it sits somewhere between 100 and 300. It is also the one number on the datasheet that is most likely to be wrong about your specific part.

That's it. Move the base current up, the collector current moves up in lockstep. You have not yet amplified a voltage, because nothing in the picture is a voltage source. All you've done is built a current-controlled current source. Interesting, but not a radio.

The load line: where current meets a resistor

To get a voltage out, stick a resistor in series with the collector. The supply provides V_CC. The transistor pulls some collector current. The voltage dropped across the resistor is whatever Ohm's law says it has to be, and V_CE is whatever's left over.

The family of curves shows how I_C depends on V_CE for various base currents. They're nearly flat because an ideal BJT is a current source, not a resistor. The orange line is the load line, the set of all (V_CE, I_C) points compatible with your V_CC and R_C. Wherever the load line meets the base-current curve you chose, that is the operating point (or, more ceremoniously, the Q-point). Turn the base current up, Q-point slides up the load line toward the top-left. Turn it up too far and you hit the saturation wall, where V_CE flattens against the supply rail minus a few tenths of a volt and the transistor gives up on being a current source. Turn it down to zero and you're in cutoff, V_CE sitting at the full V_CC because no current is flowing.

The useful region is the one in the middle. The entire game of building an amplifier is making sure you stay there on purpose.

What's happening as V_BE changes

The base current is actually controlled by the voltage across the base-emitter junction, V_BE. That junction is a forward-biased diode, and diodes respond exponentially to voltage: small changes near the knee produce large changes in current, until eventually you drop the entire supply across the collector resistor and can't go any further.

Drag V_in (which is V_BE here) and watch V_out. The three regions of the earlier scene show up as three regions of the curve: a flat portion at the top where nothing happens, a steep diagonal in the middle where a few millivolts of input produce volts of output, and a flat portion at the bottom where you're welded to the rail. The slope of the steep part is the voltage gain. It is not an especially well-behaved function of V_in, but it's enormous, and that's enough.

Notice that the amplifier is upside down. V_in going up makes V_out go down. This is the standard "common-emitter" amplifier, and its inversion is a consequence of the fact that more base drive means more collector current means more voltage dropped across R_C means less voltage left on the collector. The inversion does not bother anyone except people reading their first schematic.

Amplification, live

Pick a bias near the middle of the steep part. Add a small AC signal on top of the bias. The signal walks V_BE back and forth by a few millivolts. That walks I_C by a few milliamps (multiplied by the exponential). That walks V_CE by a few volts (multiplied by R_C). The result is a bigger, flipped version of the input.

Crank the input amplitude high enough and the output swing starts slamming into the rails. The peaks clip. A real amplifier design is mostly the art of picking a bias point such that the signal you care about fits cleanly between V_CC and V_CEsat with some margin. Hi-fi engineers and ham radio operators fight about exactly how much margin. The transistor does not have strong opinions on this either way.

Where the number comes from

All of the above is captured in two small formulas that turn the exponential behavior into linear behavior near a chosen operating point.

The rate at which collector current changes per volt of base drive is called transconductance:

g_m = I_C / V_T

where V_T is the thermal voltage, about 26 mV at room temperature. That is a number that comes directly from the physics of a PN junction and has been the same temperature coefficient since semiconductor devices were invented.

The voltage gain of this common-emitter stage is then:

A_v = -g_m * R_C

A BJT biased at 2 mA with a 1 kΩ collector resistor has a transconductance of about 77 mA/V and a voltage gain of about 77. Run it at 5 mA and the gain climbs to around 190. Run it at 100 μA and you have a gain of about 4 and should probably consider a different amplifier topology. None of this depends on β, by the way, which is why a well-designed amplifier doesn't care very much that β varies from part to part.

Everything else, and there is a lot of else, is decoration on this core. Bypass capacitors, emitter degeneration, cascode stages, differential pairs, current mirrors, Darlingtons, bootstraps. They all exist to make the basic idea above more linear, more bandwidth-y, more stable with temperature, or less dependent on a specific transistor's specific quirks. But underneath the decoration there's still a valve controlling a current, and a resistor turning that current into a voltage, and the voltage is bigger than the one you started with. The rest is bookkeeping.