Beta Current Gain
Beta (β) is the common-emitter current gain of a bipolar transistor: the ratio of collector current to base current, β = IC / IB. A transistor with a beta of 100 that is passing 10 microamperes into its base can control about 1 milliampere in its collector, which is why a bipolar transistor is often described as a current-controlled current source — a small current at the base steering a much larger one through the collector. It is the single number most people mean when they say how much a transistor "amplifies", and it is the figure that datasheets print in the largest type.
Beta is derived from alpha and inherits all of its dependence on the physics of the base. The two gains are linked by β = α / (1 − α), or equivalently α = β / (β + 1). Because alpha is so close to 1, a tiny change in the fraction of carriers lost in the base produces a large change in beta: the difference between α = 0.99 and α = 0.995 is the difference between a beta of 99 and a beta of 199. Beta is therefore an exquisitely sensitive indicator of base quality, and it is also the reason that identical-looking transistors from the same wafer can have noticeably different gains.
Beta is not a constant, and treating it as one is the classic beginner's error. It varies with collector current, rising from a low value at microampere currents, peaking in the middle of the device's range, and falling again at high current as high-level injection sets in. It rises with temperature — roughly by a fraction of a percent per degree Celsius — which feeds the thermal-runaway danger in poorly biased stages. It also depends on collector-emitter voltage through the Early effect, and it varies widely from device to device: a given type number may be specified with a spread of two-to-one or three-to-one between the weakest and strongest acceptable parts. For that reason a design that works only for a beta of exactly 100 is a bad design, and good circuits are arranged so that their behaviour depends on resistor ratios rather than on beta.
The two printed symbols, hFE and hfe, mean different things. Capital letters in the subscript denote the DC (large-signal) value, hFE, which is just IC divided by IB at a particular operating point. Lowercase hfe is the small-signal value — the slope of the collector-current-versus-base-current curve — and at moderate frequencies the two are usually within a modest factor of each other but are not identical. The general-purpose 2N3904, for example, is specified with a DC gain of about 100 to 300 at a collector current of 10 milliamperes, a range that illustrates how loosely the figure is controlled. Many small-signal parts are also sold in gain groups: the BC547 family has A, B and C suffixes for increasing ranges of beta, so that designers can buy a tighter band than the generic type allows.
In switching service the useful number is not the datasheet beta but a deliberately smaller forced beta. A transistor driving a lamp or relay is meant to be saturated, with the collector-emitter voltage down to a few tenths of a volt, and this requires more base current than IC / β would suggest. The common rule is to design for a forced beta of about 10, supplying ten times less base current than the collector needs at the unsaturated gain. That margin absorbs the minimum-beta part, high temperature and ageing; if the designer instead supplies base current from the typical beta, a weak transistor will sit half-saturated, run hot and waste power. Saturation also slows turn-off, because the extra stored charge in the base must be removed before the collector current falls.
The main misconception is that more beta is always better. High-beta transistors often have lower breakdown voltage, more temperature sensitivity and, at the extreme, super-beta devices with very thin bases that are fragile. In amplifiers the stage gain is set largely by resistor values and the transconductance of the device rather than by beta, and a good bias network makes the result almost independent of it. Modern relevance is easy to find: beta still determines the base current a microcontroller pin must source to drive a bipolar switch, the input current of an op-amp's bipolar input stage, and the accuracy of a current mirror, where finite beta introduces a small gain error.
Read beta as the transistor's leverage: not a fixed constant but a context-dependent multiplier that exists only because the base loses so little current, that wanders with current, temperature and the individual part, and that careful designers treat as a loosely known range to be designed around rather than a number to be relied upon.