Would it surprise you that the likelihood of two people out of any twenty-four having the same birthday is 50/50? At my last gig three of us out of a group of maybe 30 had the same birthdate! There's even a name for this coincidence: The Birthday Paradox. If you've ever cherry-picked analog components looking for ones that precisely matched, you've depended upon a similar phenomenon. As it turns out, the mathematics behind the Birthday Paradox shows cherry picking doesn't work when it comes to designing a profit-making product — or does it?
I've been working for a while now on a variant of classical cherry picking that works great when solving a challenge like ultrasound analog front-end design where the future is lower power at lower cost with literally hundreds and even thousands of precision matched channels.
I put together a chart, Figure 1, which quantifies why old-school cherry picking doesn't work and why this relatively new variant does. I designed the chart contemplating capacitor matching for high precision, low power, data converters, but to keep it simple let me use amplifier-offset voltage as the objective.
Let's say you bought a boatload of LM324 quad opamps real cheap, but you only want those with input offset voltages equal to the typical 3mV specification or better. So the question is, “How many amplifiers will you need to test before you find an amplifier with the typical 3mV offset?” The answer can be found looking at the chart.
Think of the X-axis on Figure 1 as the maximum number of times you'd need to pick until you found that 3mV amplifier. Since you aren't looking for any reduction relative to the typical, find the Y-axis intercept where the Cherry Picking curve intersects one since a 3mV objective / 3mV typical = 1. Then follow the dashed line down to the X-axis intercept. Here you find that you may need to pick up to 15 times before you find your first one. Now it won't be that bad all the time. The curves represent a 3-ppm failure rate on picking which is about 6-sigma. More likely you'd only need to look at one or two which is why you can usually just buy the next higher grade for about that same effective cost ratio. Essentially, the manufacturer cherry picks for you: for example, LM124s versus LM324s. But classical cherry picking falls apart for even the manufacturer when one needs to be lucky hundreds or thousands of times on a thousand-channel, ultrasound, analog front-end.
To get around cherry picking, analog IC designers use the Brute Force curve. Looking back at the chart one sees that the 1.0 Y-axis intercept for Brute Force is at the right most edge of the drawing crossing over the series of blue dots at about 26. What this tells a designer is they need to increase the total area of the input transistors by 26. So what they do is lay down 26 typical transistors in two composite groups of 13. In the context of The Birthday Paradox, this is choosing 13 from a group of 26, once. The key word here is once . And then they crank up the operating current to keep the bandwidth constant for the accompanying higher capacitances.
Now consider if the design enabled choosing 13 from 26 in all possible ways after the IC was manufactured. Well, tracing 26 down to the X-intercept shows that there are literally millions of ways to cherry pick from a pool of 26 transistors. And perhaps even more astounding is that at least one of the combinations would be a factor of 100,000 better as indicated on the Cherry Picking curve! This is a ridiculous example of course, but done to make a very important point.
Nothing in this variant of cherry picking has anything to do with size unlike with Brute Force where larger always has limits and always translates to more power loss. Like The Birthday Paradox, it is only combinations that drive the result. So not only can this variant greatly improve matching, but also lower power. As the chart indicates, the blue dot intercept is at about 8 typical transistors, so 35 possible combinations guarantee with 6-sigma confidence that 3mV becomes 1.5mV with 65 percent less power. These same benefits can work for matched capacitors in data converters.
Consider driving the huge capacitance at the frontend of an ADC. Manufacturers will tell you that it is large to minimize KTC noise and sell you an external amplifier to drive the large capacitance consuming even more power. In a low power SAR, this is a dominant point of energy loss. But, ultrasound designers use other tricks to minimize noise rather than Brute Force. They take advantage of the phenomenon whereby noise adds as the square root and signals add linearly and hence build parallel signal processing systems.
Why not do the same with an ADC and put down two ADCs in parallel? Maybe it's time to rethink the old methods of Brute Force, lasers, auto-zero switching clocks, and programming trim code ROMs. Cherry picking a static combination is a digital problem: cheap and low power without sacrificing performance. How far can this be carried? Well, how small can one make a capacitor on an IC? Start there.