How a Brokaw Bandgap Voltage Reference Achieves Stability

A Brokaw bandgap voltage reference achieves exceptional stability in electronic circuits by combining a negative temperature coefficient and a positive temperature coefficient so that their thermal drifts cancel each other out. Invented by Paul Brokaw in 1974, this fundamental analog circuit utilizes two bipolar junction transistors operating at different current densities to generate a predictable voltage output of approximately 1.25 volts. This article explains the mechanics of these opposing temperature coefficients, how the Brokaw cell merges them using an operational amplifier and resistor network, and why this balance results in a reliable, temperature-independent voltage reference.

To understand how the circuit works, you must first look at the negative temperature coefficient, often referred to as Complementary to Absolute Temperature (CTAT). The base-emitter voltage (\(V_{BE}\)) of a bipolar junction transistor naturally decreases as the temperature rises. At room temperature, this drop occurs at a very predictable rate of roughly -2 millivolts per degree Celsius. If a circuit relied solely on a single transistor for its reference voltage, the output would drop significantly as the device warmed up, making it unusable for precision electronics.

To counteract this downward drift, the Brokaw cell generates a positive temperature coefficient, known as Proportional to Absolute Temperature (PTAT). The circuit uses two identical transistors, Q1 and Q2, which are forced to operate at different current densities. Usually, this is done by making the emitter area of Q2 larger than Q1 by a specific ratio, while pushing the exact same amount of collector current through both. Because of this size difference, a voltage difference (\(\Delta V_{BE}\)) develops between the base-emitter voltages of the two transistors. According to semiconductor physics, this voltage difference is directly proportional to the absolute temperature, meaning it increases at a steady, predictable rate as the temperature rises.

The magic of the Brokaw architecture lies in how it merges the CTAT and PTAT voltages within a single feedback loop. The bases of both Q1 and Q2 are tied together, and an operational amplifier is used to sense the currents in their collectors, adjusting the base drive to keep the currents equal. A small resistor (R1) is placed in series with the emitter of the larger transistor (Q2) to develop the \(\Delta V_{BE}\) voltage. Both emitter paths then join together and flow through a second, shared resistor (R2) to the ground.

Because the current flowing through R1 is determined by \(\Delta V_{BE}\), that current is completely PTAT. This PTAT current, along with the identical current from Q1, flows through the shared resistor R2, creating a larger, amplified PTAT voltage drop across R2.

The final reference voltage is taken at the shared base of the transistors. Mathematically, the voltage at this node is the sum of the base-emitter voltage of Q1 (which is CTAT) and the amplified voltage drop across the resistor network (which is PTAT). By carefully selecting the ratio of the resistors R1 and R2, a circuit designer can scale the positive temperature coefficient of the resistor network so that it equals the exact magnitude of the negative temperature coefficient of the transistor.

When the positive slope of the PTAT voltage (+2 mV/°C) is added to the negative slope of the CTAT voltage (-2 mV/°C), the net temperature coefficient becomes zero. This summation reliably hits a "magic" voltage near 1.22V to 1.25V, which corresponds to the theoretical bandgap energy of silicon extrapolated to absolute zero. The result is a rock-solid voltage reference that remains stable regardless of whether the surrounding electronic environment heats up or cools down.