Who is Paul Brokaw?

Voltage reference circuits, you know, those mythical chips which produce a fixed voltage seemingly independent of supply fluctuations, temperature changes and circuit loading!?!!

I'm sure you have, however, have you ever paused to think about where these circuits came from or even the humans behind these truly groundbreaking circuits? Stay tuned, we are now going to explore both of these facets by taking the practical circuit known as the Brokaw bandgap reference as a case study via a deep dive.

Introducing the Brokaw bandgap reference

Theoretical circuit of the brokaw bandgap reference

There you have it, feast your eyes on this truly elegant circuit!

We provide a supply voltage, denoted V+ in the schematic above, connect up a few resistors, transistors and an op-amp and we have an apparently constant voltage output!!

Now for the million dollar question I hear you all asking in unison, "how does it work?"

Let's take it from first principles by labelling all of the voltages and currents and see where it takes us! Time to fire up our old friend LTSpice:

Annotated currents and voltages

Let's now try to unpack the dynamic behaviour from a high level without diving into mathematics.

First of all recall NPN transistors have a negative temperature coefficient i.e. higher temp means higher β (current gain)! Also take note that Q2 has a base emitter area that is 8 times larger than Q1, which means it requires a smaller base–emitter voltage for the same current (So if Vbe1=Vbe2 we expect Q2 to have roughly 8 times the current flowing through it from collector to emitter).

Note also that the output is fed back to the positive and negative terminals, i.e. we have negative feedback which means the inverting terminal voltage will track the non-inverting terminal (virtual connection between them exists, Vp=Vn).

So to summarise. The base emitter voltages will be equal and have a negative temperature coefficient while the voltage difference of Vbe1 and Vbe2 has a positive temperature coefficient and Q2 has 8 times the area of Q1.

Due to negative feedback the currents flowing through R3 and R4 will be equal, i.e. collector currents equal, Q2 base emitter voltage will be lower than Q1 by a magnitude of kT/q*ln(8). This voltage is generated across R2 and so defines the current I2 as kT/q*ln(N)/R2.

The output voltage is therefore VBE(Q1)+2*kT/q*ln(8)*R1/R2.

The first term has a negative temperature coefficient and the second term has a positive temperature coefficient (proportional to T). By careful selection of N, R1 and R2, these temperature coefficients can cancel out, i.e. the output voltage is independent of temperature.

Paul Brokaw invented this brilliant circuit (Brokaw, P., "A simple three-terminal IC bandgap reference", IEEE Journal of Solid-State Circuits, vol. 9, pp. 388–393, December 1974.) and holds over 100 patents. I highly recommend reading this interview with him to better understand Paul, as well as this recent video:

The wisdom I personally take from Paul is to do more with less.