# Bandgap references

The last post kind of violated Chekhov's Law.. I started off talking about bandgap references, went off into details, then never came back to them.

I showed the schematic, just didn't give it the correct name:

This, ladies and gentlemen, is a truly pathetic bandgap reference. Seriously.. the first primitive ancestor to the modern bandgap reference was more sophisticated.

It does use the same basic principle though.

### So what is a bandgap reference?

It's a circuit whose output voltage remains stable when the temperature changes.

All components behave differently as the temperature changes, but semiconductors are especially vulnerable. All the equations for current and voltage depend strongly on the thermal voltage Vt, and sending current through any semiconductor generates heat. Thermal effects also depend on voltage and current, so they don't happen uniformly across a whole circuit. As you can guess, that plays hell with a circuit's long-term stability.

Back around 1964, a scientist working for Fairchild Semiconductor named David Hilbiber used strings of diode-connected transistors to see if the thermal effects would cancel out. They did, at least over a certain ratio of currents through the strings and over a 5C range. Even with those limits though, the circuit worked better than anything else available at the time.

The circuit above does the same thing with the LED and the diode. The diode and LED have different bandgap voltages, but they both respond to changes in temperature the same way.. the ratio of 'change in LED current' to 'change in diode current' will be nearly constant.

If you choose the right current and resistor values, the constant will be 1.. any change in the voltage across the LED1 will be cancelled by an equal change across D1, and the voltage across R2 will stay fixed.

If you were able to control the doping of the LED and diode, you could design for values that would work on a large scale. For the home experimenter who gets parts already packaged, you'd have to test each pair separately.

### Nobody wants to do that.

Around 1970, Bob Widlar picked up the problem.

Widlar was one of the great circuit designers of the early silicon era. He was a major contributor to the first mass-produced silicon op amps and voltage regulators, and several basic current and voltage control circuits carry his name.

Instead of using different materials (which would be difficult for an IC), he took advantage of the fact that raising the forward voltage of a diode by 60mV lets ten times as much current flow through.. at least at room temperature. Changing the temperature changes the relationship between forward voltage and current, but it does so in a predictable way.

A simplified version of his circuit looks like this:

The dark bar on Q10 means it's larger than the others. In this case, ten times larger.

### An aside about IC design:

The basic component of integrated circuit design is the transistor. It's the smallest, most versatile, and most predictable thing you can make with doped silicon.

In particular, IC designers spend a lot of time deciding how big to make their transistors. The amount of current you get from a transistor, given a certain bias voltage, is directly proportional to the transistor's size.. twice as big means twice as much current.

There's really nothing mysterious about that.. making one transistor twice as big is exactly the same as using two transistors of the original size. That's often how large-area transistors are made in fact: you put several copies of the transistor side by side and wire the collectors, emitters, and bases together.

You can do the same thing with discrete transistors if you want to:

.. it's a handy way to sneak around the maximum current limits of smaller transistors.

The nice thing about size ratios is that they're independent of voltage or current. A 10:1 size ratio means you always get a 10:1 current ratio for the same bias voltage, no matter what the exact voltage and currents happen to be.

The entire modern electronics industry relies on that fact.. the absolute accuracy of devices you can only see with an electron a microscope tends to suck.

Tolerances from one side of a chip to the other are often +/-20% or so. Variation from one batch of chips to the next is even wider. Transistors that are physically close to each other (within a few microns) tend to match pretty well though.. the things that change don't have room to change much.

That means you can get well-controlled ratios from a process whose overall accuracy is moderate at best. IC fabrication facilities ('fabs') define the quality of their processes in those terms, in fact: component ratios are guaranteed to be accurate to +/-0.5% (or whatever) as long as you obey certain design rules.

The upshot is that IC designers use transistor scaling the way discrete circuit designers use resistors (and a good thing too.. IC resistors make IC transistors look like finely-tuned precision instruments).

### Back to the bandgap:

Q10 is ten times larger than Q1, so ideally the current through Q10 would be ten times as large as the current through Q1.

R2 and R3 don't let that happen though. They're both the same size, so R3 throttles Q10 down to the same current as Q1. That means each Q1-sized piece of Q10 sees 1/10th as much current as Q1, a property called 'current density'. In this case, Q10's current density is 1/10th that of Q1.

The base-emitter junction in a transistor is just a diode, and hopefully you know by now what happens when one diode sees ten times as much current as another: you get a 60mV difference in voltages.

That 60mV shows up across R4, and since R4's voltage is controlled by two diodes (the base-emitter junctions of Q1 and Q10), it will stay remarkably stable.

The rest of the circuit is arranged to make that stability contagious.

R4 controls the current through R3. R3's resistance is ten times that of R4, so the voltage across R3 will be a stable 600mV. The stability of R3's voltage makes it look like a constant current source to the base of Q3, so Q3's base-emitter voltage will be stable.

Connecting Q3's collector to the top of R3 means we add the stable voltage across R3 to the stable voltage across Q3's base-emitter junction. Those are both about 600mV, so the output voltage will be stable somewhere near 1.2v.

Q3 also forms strong a negative feedback loop around Q1 and Q10. Any change in the voltage at the top of Q10 will produce a change in the current through R3. That will produce a change in the current flowing into the base of Q3, which will produce a much larger change in the current flowing through Q3's collector. If the voltage through R3 rises, Q3 will pull it down again. If the voltage across R3 falls, Q3 will force it to rise.

The average transistor's current gain (the ratio between the base-emitter current and the collector-emitter current) is somewhere between 50 and 100. That means Q3 pushes back against change 50-100 times harder than the change itself, so the output voltage remains stable.

### Thermal effects:

The only thing that can change all that is a change in the thermal voltage. If the transistors start to conduct differently as the temperature changes, it can move the output voltage around.

That's where this circuit becomes absolutely beautiful.

If you use low-value resistors for R1, R2, and R3, the output voltage will drift downward as the temperature rises.. its 'temperature coefficient' (usually called 'tempco') is negative.

If you use high-value resistors, the tempco is positive.

If you use resistors somewhere in the middle (around 25k for R3), the tempco will be close to zero.

That gives you a stable voltage reference whose value is more or less independent of temperature.

And that, gentlemen and ladies, is the reference point we use for almost all modern circuits. Modern designs are more complicated and deliver even better performance (a lot of smart people followed the path once Widlar showed the way), but the general strategy is the same.