Bandgap voltage references are widely used in electronics to generate stable voltages that do not change with temperature. This is surprisingly difficult in electronics since many semiconductor parameters are highly temperature dependent. The solution is ingenious and comes down to adding a voltage with a positive temperature coefficient to another voltage with a similar negative coefficient to get a resultant voltage with (near) zero coefficient.

The idea was first mooted by David Hilbiber in 1967, but first used commercially by Bob Widlar in 1971. Bob Widlar was an erratic electronics genius who developed the first commercially successful op-amps, comparators and three terminal voltage regulators among many others. Part of his success was to recognise and work with the strengths and weaknesses of the IC production process. He recognised that it was very hard to manufacture components of precise absolute value in silicon but was relatively easy to make components with precisely matched values.

**Figure 1** shows the Widlar bandgap reference. The output voltage is the sum of Q3’s base-emitter voltage and voltage across R2. The base-emitter voltage will be around 0.65V at room temperature with a temperature coefficient (tempco) of -2.01mVK^{-1}. But how do we get a voltage with a positive one? The clever answer is to exploit the fact that, while a base-emitter voltage has a negative tempco, the *difference* between the base-emitter voltages of two identical transistors with different current densities will have a positive tempco.

That’s where identical transistors Q1 and Q2 come in. Q1 has ten times more collector current than Q2 because of the relative values of R1 and R2. Both transistors’ bases are at the same potential, so the voltage across R3 will be the difference between the two transistors’ base-emitter voltages. The Ebers-Moll model allows us to calculate the voltage across R3 as follows:

Note that the terms for reverse saturation current have cancelled out a and we are left with just the thermal voltage V_{T} multiplied by a constant. In our case with a 10:1 current ratio, so the voltage across R2 will be 23.0 V_{T}. We can now work out the reference voltage and its tempco.

The thermal voltage V_{T} is about 26.0mV at room temperature, with a temperature coefficient of 8 6.2 µVK^{-1}. The output voltage is therefore 0.65V + 23.0 × 26.0mV = 1.25V. The all-important temperature coefficient of the output voltage will be -2.01 mVK^{-1} + 23.0 × 86.2 µVK^{-1} = 27.4 µVK^{-1}. That’s about 10ppm per degree which is not bad.

As with any circuit there are some limitations. The output voltage shows some dependence on the source current and this circuit can’t supply arbitrary voltages. A better version was devised by Paul Brokaw in 1974.

**Figure 2** shows a simplified version of the Brokaw bandgap reference. An op amp drives the bases of transistors Q1 and Q2 such that their collector currents are the identical (since their emitter resistors are of equal value). This time the current density difference is created by having Q2’s physical area larger than Q1’s by a factor of N. Q2 will therefore have a lower current density and therefore a lower V_{BE} for the same collector current.

As for Widlar’s circuit the voltage is developed across R1 is the difference in base-emitter voltages between Q1 and Q2. This time the relationship is:

This voltage also has a positive tempco due to the thermal voltage V_{T}. The voltage across R2 must therefore be:

… since the collector currents are equal. The reference voltage will be:

Choosing the right resistor values and area ratio allows us to cancel the temperature dependency as before.

A small tweak, as shown in **Figure 3**, allows us to achieve an arbitrary output voltage higher than the 2.5V imposed on the earlier circuits by the need to balance the temperature effects. This is just a small introduction to bandgap voltage references. As usual there are a whole lot more variations and improvements out there.

**References**

“Bob Widlar.” In Wikipedia, November 28, 2020. https://en.wikipedia.org/w/index.php?title=Bob_Widlar&oldid=991059969.

“Paul Brokaw.” In Wikipedia, November 26, 2020. https://en.wikipedia.org/w/index.php?title=Paul_Brokaw&oldid=990831978.

Horowitz, Paul, and Winfield Hill. The Art of Electronics. Third edition, 11th printing, with Corrections. Cambridge New York, NY: Cambridge University Press, 2017.

“Chapter 14: Voltage References [Analog Devices Wiki].” Accessed March 17, 2021. https://wiki.analog.com/university/courses/electronics/text/chapter-14.

Brokaw, Paul. “How to Make a Bandgap Voltage Reference in One Easy Lesson.” A. Paul Brokaw and Integrated Device Technology, 2011. https://www.renesas.com/sg/en/document/whp/how-make-bandgap-voltage-reference-one-easy-lesson-paul-brokaw.

Sponsor this ArticleAndrew Levido (andrew.levido@gmail.com) earned a bachelor’s degree in Electrical Engineering in Sydney, Australia, in 1986. He worked for several years in R&D for power electronics and telecommunication companies before moving into management roles. Andrew has maintained a hands-on interest in electronics, particularly embedded systems, power electronics, and control theory in his free time. Over the years he has written a number of articles for various electronics publications and occasionally provides consulting services as time allows.