Current Gain
Current gain is the ratio of a transistor's output current to the input current that controls it. For a bipolar junction transistor it is usually the common-emitter figure called beta, or hFE, defined as the collector current divided by the base current (β = IC / IB), and typically it lies from a few tens to a few hundred. A beta of 100 means that each microamp of base current is accompanied by about 100 microamps of collector current, so the transistor is a current-controlled current source with a multiplication factor.
Alpha and beta describe the same physics from two sides, and they are linked by a simple formula. Alpha is the collector current divided by the emitter current (α = IC / IE), always slightly below 1 because a few carriers recombine in the base; values of 0.98 to 0.998 are common. Since the emitter current is the sum of the base and collector currents, beta follows from alpha as β = α / (1 − α). An alpha of 0.99 therefore gives a beta of 99, and an alpha of 0.995 gives 199. This is why a tiny improvement in how many carriers survive the base produces a large change in beta, and why a thin, lightly doped base, which the base entry covers, pays off so strongly. The early point-contact transistor of 1947 could actually have an alpha above 1, because of how its contacts behaved, a feature that later junction devices did not share.
Beta is not a constant of the part, and this is the central pitfall of the idea. It rises with temperature, peaks at some intermediate collector current and falls at both very low and very high currents, and it changes with collector voltage through the Early effect. Two transistors with the same part number, from the same reel, can differ by a factor of two or more. Datasheets reflect this by giving a range, such as 100 to 300 for a small-signal device, and often only a minimum at a stated current. A circuit that needs a particular beta to work is therefore fragile: it may run on the bench and fail on a hot day or with a replacement part.
Current gain should not be confused with transconductance, which is the more fundamental quantity. A bipolar transistor is really controlled by the base-emitter voltage, and the collector current follows an exponential law in that voltage. The ratio of the change in collector current to the change in base-emitter voltage is the transconductance, gm = IC / VT, where VT is the thermal voltage, about 26 millivolts at room temperature. At a collector current of 1 milliamp that gives about 38 milliamps per volt, a number set by physics and independent of the particular transistor. Beta then only tells how much base current this collector current costs. This is also why the small-signal voltage gain of a common-emitter stage is set mostly by the collector resistor and the bias current, and why field-effect transistors, which draw almost no gate current, are described with transconductance alone.
Designers deal with the variability by making beta matter as little as possible. Negative feedback, such as an emitter resistor or a voltage-divider bias network that is stiff compared with the base current, makes the operating point follow the resistors rather than the transistor. In switching service, engineers deliberately force a low gain, supplying about one tenth of the collector current as base current (a forced beta of ten) so that the transistor saturates reliably whatever its actual beta. The Darlington pair multiplies two betas for very high gain at the cost of speed and a higher saturation voltage, and current mirrors and differential pairs rely on two transistors on one chip matching each other rather than on any absolute value. The 2N2222 and BC547 families are quoted with broad gain ranges for precisely this reason.
Gain figures also show up well beyond the textbook bipolar transistor. The idea of ratio of output to input current appears as the current transfer ratio of an optocoupler, the gain of a photomultiplier or avalanche photodiode, and the current gain cut-off frequency, fT, which marks the frequency at which a transistor's current gain falls to one and which measures how fast a process is. In modern chips, MOS transistors have effectively infinite DC current gain because the gate draws no steady current, so beta is mostly a concern of analogue, power and radio-frequency bipolar design, including silicon-germanium transistors in phones and radar, where careful engineering raises gain and speed together.
Read current gain as a convenient but loose multiplication factor: a rough rule that a bipolar transistor turns a small base current into a larger collector current, whose exact value drifts with temperature, bias and the individual part, so good circuits are built to work whatever it turns out to be.