Bandgap reference voltage source

A bandgap reference produces a stable dc voltage, about 1.25 V, which is almost independent of temperature, supply voltage and load. It is used in voltage regulators, ADCs, DACs and power supplies (like in the LM317).

Principle

It adds two voltages with opposite temperature coefficients so that they cancel:

• VBE of a transistor has a negative temperature coefficient (CTAT): about −2 mV/°C.
• The thermal voltage VT = kT/q has a positive temperature coefficient (PTAT): +0.086 mV/°C (VT = 25.85 mV at 300 K).

Vref = VBE + K · VT, where K is a constant set by resistors and the area ratio.

Circuit

          +VCC
        [R3]   [R3]
          |      |
          +--+   +---+       op-amp inputs taken from the
          |  |   |   |       two collectors; its output Vref
          C1 |   C2  |       drives both bases
        Q1 (n x)  Q2 (1 x)   |
          B1-----+----B2 <---+--- Vref (op-amp output)
          E1       E2
         [R1]       |
          +---------+
                |
               [R2]
                |
               GND

The op-amp (polarity chosen for negative feedback) makes the two collector voltages, and so the two collector currents, equal: I1 = I2 = I. Q1 has n times the emitter area of Q2.

Both bases are at Vref. Applying KVL: VBE1 + I R1 + 2I R2 = VBE2 + 2I R2, so

I R1 = VBE2 − VBE1 = VT ln n (the PTAT current I = VT ln n / R1)

The reference voltage is the VBE of Q2 plus the drop across R2 caused by the total current 2I:

Vref = VBE2 + 2 I R2 = VBE2 + 2 (R2/R1) VT ln n

Temperature compensation

d(Vref)/dT = dVBE/dT + K · dVT/dT = −2 mV/°C + K (0.0862 mV/°C) = 0

K = 2 (R2/R1) ln n = 2/0.0862 ≈ 23.2

Then Vref = VBE + 23.2 × 25.85 mV = 0.65 + 0.60 ≈ 1.25 V

This is about the bandgap voltage of silicon (1.12 eV, extrapolated 1.205 V at 0 K, with the final value near 1.25 V). For example, with n = 8 (ln 8 = 2.08), 2 (R2/R1)(2.08) = 23.2 gives R2/R1 ≈ 5.6.

The reference voltage does not depend on temperature to the first order, and the circuit also has good supply rejection because the currents are set by the stable Vref.