Current Gain (Beta)

Part of The Transistor

Understanding, measuring, and designing around the transistor’s current amplification factor.

Why This Matters

Beta (β or hFE) — the transistor’s current gain — is the single parameter most designers encounter first and most often. It quantifies how much the transistor amplifies: a base current of 10 µA produces a collector current of 10 µA × hFE. For hFE = 100, that is 1 mA. A transistor’s usefulness as an amplifier and its behavior as a switch both depend heavily on this parameter.

The practical challenge: hFE varies enormously. Between identical transistors from the same production batch, gain can span 3:1. Between the same transistor at different temperatures, gain changes by 30-50%. Between the same transistor at different collector currents, gain rises and falls with a characteristic hump. These variations are not defects — they are intrinsic to the physics. Designing circuits that work correctly despite them is the engineer’s task.

For a rebuilding civilization fabricating transistors, understanding hFE helps set fabrication targets (what base width, what lifetime, achieves the gain needed?), design bias networks that tolerate gain variation, and sort devices into appropriate bins for different applications.

Physical Origin of hFE

hFE = IC / IB. The collector current is the injected minority carrier current that successfully crosses the base. The base current is the recombination current — carriers lost in transit plus the input current needed to maintain the base charge.

Three factors determine hFE:

Base transport factor (αT): Fraction of injected carriers that cross the base without recombining. For base width WB and minority carrier diffusion length L: αT ≈ 1 - (WB/L)² / 2 (for WB << L)

To maximize αT: minimize WB and maximize L (by increasing minority carrier lifetime through purity).

Emitter injection efficiency (γ): Fraction of emitter current due to minority carriers injected into the base (vs. back-injection of majority carriers from base into emitter). γ = IE_minority / IE_total. Maximized by heavy emitter doping relative to base doping.

Recombination in depletion zone: Some carriers recombine within the emitter-base depletion zone. This component, characterized by ideality factor n=2, reduces effective gain at low currents.

Total hFE: hFE ≈ (αT × γ) / (1 - αT × γ)

For αT = 0.98, γ = 0.99: αF = 0.9702, hFE = 32. For αT = 0.99, γ = 0.998: αF = 0.988, hFE = 82. The gain rises rapidly as transport factor and injection efficiency approach 1.

hFE vs. Collector Current

hFE is not constant across all collector currents. It has a characteristic hump:

Low current region: hFE is low because depletion zone recombination current (n=2 component) dominates base current. As IC increases, the n=1 diffusion component grows faster and gain rises.

Peak region: At intermediate currents (often 1-10 mA for small signal transistors), hFE reaches its maximum. This is the specified hFE in datasheets unless otherwise noted.

High current region: hFE falls because of high injection effects (minority carrier concentration approaches majority), Kirk effect (base region widens at high current densities — “base pushout”), series resistance voltage drops, and other second-order effects.

Implication for circuit design: Bias the transistor at IC in the middle of its high-gain region. For switching applications, the saturation transistor operates at high current — the gain is lower than the peak, meaning more base drive is needed than a naive hFE estimate suggests.

Measurement: Measure hFE at several collector currents by varying the collector load resistance: IC = 0.01 mA, 0.1 mA, 1 mA, 10 mA (if device can handle it). Plot hFE vs. IC. The peak and the high-current rolloff are visible. Design around the plateau region.

hFE vs. Temperature

hFE increases with temperature for silicon BJTs, typically 0.5-1% per °C. A transistor with hFE = 100 at 25°C has hFE ≈ 150 at 75°C. For germanium, the relationship is similar but temperature is more limited.

The reason: minority carrier lifetime and mobility both change with temperature. Minority carrier lifetime increases with temperature (reduced capture cross-section at traps). Higher lifetime means higher transport factor αT, which means higher hFE.

The temperature coefficient of hFE has circuit implications:

• CE amplifier gain changes with temperature (about +0.5%/°C, or +50% over 100°C range)
• Transistor switches need base overdrive at low temperature where hFE is lowest
• Current mirrors using hFE ratios require temperature-matched devices

Compensation strategy: Use circuit designs whose behavior depends on resistance ratios rather than absolute hFE values. Voltage divider bias is one example — the collector current is set by the emitter resistor voltage, not by hFE directly. Differential pairs and current mirrors track temperature because both transistors experience the same temperature changes.

hFE vs. Voltage (Early Effect)

In the active region, hFE and IC are not perfectly independent of VCE. As VCE increases, the collector-base depletion zone widens, reducing the effective base width WB. Narrower base means fewer recombinations, higher αT, and higher hFE. This base-width modulation with voltage is the Early effect.

Characterized by VA (Early voltage, typically 50-200V for bipolar transistors): IC = IS × e^(VBE/VT) × (1 + VCE/VA)

The slope of the IC vs VCE curves in the active region has slope = IC/VA (not perfectly horizontal). In a common-emitter circuit, this finite slope represents an output resistance ro = VA/IC.

Practical consequence: For circuits requiring high DC gain or precise current mirroring, choose transistors with high VA. For most amplifiers, VA is high enough to ignore.

Measuring hFE Accurately

DC measurement (simple): Force IB = 10 µA (5V through 430 kΩ). Measure IC (voltage across 1 kΩ collector resistor / 1000). hFE = IC/IB. Accurate to ±5% with good voltmeter.

h-parameter bridge: Use a Wheatstone bridge with one arm containing the transistor, balanced against known impedances. More accurate than direct measurement, but requires more equipment.

curve tracer method: From the IC vs. VCE family of curves, hFE = ΔIC / ΔIB at the operating point. Read from the curve family.

AC measurement (hfe): At a specific frequency, measure small-signal current gain. hfe ≠ hFE in general — hfe decreases with frequency above the beta cutoff frequency (fβ ≈ fT/hFE). Relevant for RF circuit design.

Designing Around hFE Variation

The guiding principle: make circuit performance depend on resistor ratios, not transistor gain.

Voltage divider bias: Operating point determined by VB (set by resistors) and RE (feedback resistor). IE = (VB - VBE) / RE. IC ≈ IE. Variation in hFE from 50 to 300 changes IC by less than 10% if the bias is well designed.

Emitter degeneration: Adding an unbypassed RE makes the stage gain Av = -RC/(RE + re) ≈ -RC/RE for RE >> re. This gain is determined by resistors only — insensitive to transistor gain. At the cost of signal gain magnitude.

Feedback: Collector feedback, emitter feedback, or operational-amplifier-style negative feedback all make circuit performance less sensitive to transistor parameters. Deep negative feedback is the ultimate: loop gain >> hFE variation means the closed-loop gain is almost entirely determined by feedback resistors.

Sorting: For matched-pair applications (differential amplifiers, current mirrors) where transistor parameters must match each other rather than match a specific absolute value, sort devices by hFE into narrow bins. Matching within ±10% is achievable with careful measurement.

Keep records of hFE values measured for every device in your inventory. A searchable logbook enables selecting matched devices when needed without measuring every candidate again.