Now that we have a calibrated programmable resistance decade, we can try to make the programmable decade resistor more accuracte - for higher resistance values. Actually, the idea behind that is trivial.
The idea
I'd like to start with an example in which we set a resistance value of Rset = 100.000 kΩ. (Let's not worry about temp drift, contact resistance etc.) In my case all resistors have a specified tolerance of 0.1. So we'd expect a resistance somewhere in the range of
Let's assume we estimated a resistance of Rset = 99.952 kΩ based on the calibration values. Now it's pretty obvious that we can use the lower two decades to add some additional resistance - 40 Ω from the second decade and 8 Ω from the first decade should do the trick. And with 0.1 % resistors we know that we'll achieve a value very, very close to 48 Ω. The actual setpoint that I will call the hardware setpoint has now become (Rhwset = 100.048 kΩ).
But care must be taken: In a more generalized sense the lower decades deviate from the theoretical value as well due to their tolerance. Had I chosen an estimated resistance Rest > 100.000 kΩ, it would have been not only a little more paperwork, but we would have ended up with a hardware setpoint below 100.000 kΩ, e. g. Rhwset = 99,910 Ω. Now it's easy to see that not only the highest decade might introduce a relevant error, but also multiple lower decades: Decade 4 (i. e. the fifth decade) could deviate by 90 Ω, decade 3 by 9 Ω and decade 2 by 0.9 Ω.
A simple, but unpractical approach
A simple approach for calculating the optimal hardware setpoint would be to use brute force and set up a look-up table that keeps the hardware setpoint in memory for any given setpoint. Certainly, such a look-up table can be accessed very quickly. But even on a computer a completely non-optimized approach for generating the look-up table can be rather time consuming. Whereas memory usage isn't a concern in that case, with microcontrollers this would be a completly different story: Good luck finding a (cheap) microcontroller with >4 MByte of Flash memory (1M datapoints with 4 Byte each if we are talking 6 decades). Also, the look-up table would have to be re-calculated whenever the programmable decade resistor is calibrated and this would take really long on a microcontroller.
In many cases we'd get away with such a lazy approach, but not this time. We need an algorithm that uses the current calibration values as an input and is able to narrow down the solution space in a way that allows for calculation in real time: Branch and bound.
Branch and bound
The obvious idea is to start the selection process at the highest decade. In our previous example we wanted to achieve Rset = 100.000 kΩ. With a tolerance of the resistors of 0.1% we can be sure that decade 5 has to be either a "1" (100.000 kΩ) or a "0" (0 Ω). If option "1" results in a resistance value greater than the setpoint, then we don't have to try to add any more resistance - this already would be the best we can do with this option. If it is below, then we apply the algorithm again, but now limited to the 5 lower decades and with the remaining Ohms as the new target. After that we repeat for option "0" and ultimately figure out who the winner was.
A few notes:
- The algorithm described can be implemented fairly easily with recursion. A good thing: The recursion depth is limited by the (fixed) number of decades, so that shouldn't become a problem if done properly
- It's the resistors' tolerances that decides which options have to be checked
- Eliminate as many of the options as required to reach the performance goals while factoring in the constraints imposed by the tolerances of the resistors
- For obvious reasons the accuracy of the calibration directly impacts the results of this approach
- The algorithm will achieve benefits for larger resistance values only
As for the calibration procedure, I'll share the results of this optimization in a future log.
Next steps
After many log entries about specific design decisions regarding the decade resistor itself it's time to discuss the system design.
Discussions
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