Bandgap Voltage Reference: Cancelling Temperature Drift at 1.25 V

What it is and where it hides

A bandgap voltage reference is an integrated circuit that produces a stable DC voltage — canonically near 1.25 V — that barely moves with temperature, supply voltage, or load. It is the silent workhorse of analog electronics: almost every data converter, regulator, and precision amplifier needs a fixed voltage to compare signals against, and the bandgap is the reference of choice because it can be built entirely from the bipolar transistors and resistors already available on any chip.

• ADCs and DACs use it to define full-scale; a 12-bit converter's LSB is meaningless if its reference wanders.
• LDO and switching regulators compare a divided-down output to the bandgap to set their regulated rail.
• Op-amps, comparators, and PLLs use it to bias internal currents.
• Silicon temperature sensors reuse the same PTAT term as a thermometer.

Unlike a Zener diode, which needs 6-7 V of headroom, a bandgap works from supplies as low as 1.5-2 V, which is why it became universal as chip voltages fell.

How it works: cancelling two opposing slopes

The trick is to add two voltages whose temperature slopes are equal and opposite. A forward-biased bipolar transistor's base-emitter voltage V_BE is complementary to absolute temperature (CTAT): it falls at roughly −2 mV/°C because the collector current depends exponentially on temperature through the intrinsic carrier concentration.

The counter-term comes from the difference in V_BE between two transistors run at different current densities. If transistor Q2 has N times the emitter area (or 1/N the current density) of Q1:

• ΔV_BE = (kT/q)·ln(N), where kT/q is the thermal voltage V_T (≈25.9 mV at 300 K).

Because V_T rises linearly with temperature (+0.087 mV/°C per unit), ΔV_BE is proportional to absolute temperature (PTAT) with a positive slope. Amplify ΔV_BE by a resistor ratio until its +slope exactly matches V_BE's −slope, sum them, and the temperature dependence cancels to first order. The sum sits at the extrapolated silicon bandgap voltage E_g/q ≈ 1.205 V.

The governing equation and a worked example

The output of a classic bandgap is:

V_REF = V_BE + (R2/R1)·(kT/q)·ln(N)

Symbols: V_BE = base-emitter voltage (~0.65 V, −2 mV/°C); k = Boltzmann's constant (1.381×10⁻²³ J/K); T = absolute temperature (K); q = electron charge (1.602×10⁻¹⁹ C); N = emitter-area (current-density) ratio; R2/R1 = gain resistor ratio.

Worked example. At T = 300 K, kT/q = 25.85 mV. Choose N = 8, so ln(8) = 2.079 and ΔV_BE = 53.8 mV. The PTAT slope is +0.087·2.079 = +0.181 mV/°C per unit gain. To cancel V_BE's −2 mV/°C we need a gain of 2/0.181 ≈ 11.0, i.e. R2/R1 ≈ 11. That adds 11 × 53.8 mV ≈ 0.59 V to the 0.65 V V_BE, giving V_REF ≈ 1.24 V — right at the bandgap. Residual drift from this design is typically 10-50 ppm/°C.

Designing and trimming one in practice

Real designs wrap the core in a feedback loop and then trim it. The dominant topologies are:

• Widlar (1971): the original three-transistor cell in the LM113, using a Widlar current mirror to set the PTAT current.
• Brokaw cell (1974): an op-amp forces equal currents through Q1 and Q2 (area ratio 8:1) across a shared emitter resistor; the reference is tapped at the transistor collectors. It is the most-copied topology in ICs.
• Kuijk / CMOS bandgap: an op-amp servos two branches with a diode-connected substrate PNP, the standard in CMOS.

Key practical rules: (1) trim R2/R1 at the bandgap-magic voltage — nulling drift and setting output are the same adjustment, so trimming V_REF to ~1.205 V minimizes slope. (2) Use a large N (8-24) laid out as a common-centroid array to fight mismatch. (3) Add a start-up circuit — the zero-current state is also a stable operating point. (4) Filter the PTAT resistor's noise; bandgaps are intrinsically noisy (~1-3 µV_RMS in-band before filtering).

Bandgap vs. Zener and other references

The bandgap is not always the best reference — just the most integrable. Its main rivals:

• Buried (subsurface) Zener: breakdown ~6.3 V, achieving 0.5-3 ppm/°C and far lower noise, because avalanche happens below the surface away from noisy interface traps. This is why the lowest-drift references (LM399 at ~1 ppm/°C, LTZ1000 at ~0.05 ppm/°C) are Zener-based — but they need >8 V and burn power heating themselves to a stable temperature.
• Curvature-corrected bandgaps: the raw bandgap still has a residual parabolic curvature (the V_BE(T) equation carries a T·ln(T) term). Adding a correction current squashes this to 1-5 ppm/°C, as in the LTC6655.
• XFET and floating-gate references: lower power, competitive drift, no 1.2 V floor.

The bandgap wins whenever you need a reference on the same die from a low supply — which is essentially every mixed-signal SoC.

Failure modes, trade-offs, and significance

Bandgaps have well-known limitations:

• Curvature error: first-order cancellation leaves a bowed V_REF(T) that peaks mid-range and droops at the extremes — the reason uncorrected parts spec ~20 ppm/°C over 100 °C rather than zero.
• Start-up latch: without a start-up cell the circuit can sit at 0 V forever; a classic silicon bug.
• Resistor and V_BE mismatch: a 1% area error in the N-ratio maps directly to drift; layout and dynamic element matching matter.
• Package stress / drift: mechanical stress shifts resistor values, causing long-term drift (10-100 ppm/1000 h) and hysteresis after thermal cycling.
• Noise: the PTAT gain amplifies transistor and resistor noise, so low-drift bandgaps trade bandwidth for filtering.

Significance: Widlar's insight — that the messy, temperature-sensitive V_BE hides a fundamental constant (the bandgap energy) once you subtract its PTAT part — made precise analog possible on cheap silicon. More than 50 years later, essentially every ADC, DAC, and regulator still starts with that 1.25 V.

Bandgap reference vs. other voltage-reference technologies

Reference type	Typical output	Temp drift (ppm/°C)	Notes
Bandgap (uncorrected)	1.2-1.25 V	10-50	Integratable, low supply, ~1.2 V floor
Bandgap (curvature-corrected)	1.2-2.5 V	1-5	Cancels V_BE curvature; e.g. LTC6655
Buried (subsurface) Zener	6.2-7.1 V	0.5-3	Lowest noise/drift, needs >8 V supply
Surface Zener / temp-comp	5.6-6.4 V	5-20	Simple but noisy, drifts with surface state
XFET / FGA	2.5-5 V	3-10	Low power, good long-term stability