# 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

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:

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.