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Sun May 11 15:37:25 EDT 2014

## Two coupled NPN 2N3904 transistors + resistor divider

```Assumption:

Using two thermally coupled BJTs at the same V_BE, the current ratio
will be constant, but not equal to one.

To calibrate, two things need to be done:
- Constant component offsets
- Temperature.

To "set the stage", build two reference rails to set max and min
references to be used in the DAC.

For a 3.3V supply e.g. from a zener from 5V, a ratio of 12-1-2 seems
to be the right range.  With standard component values this will be:

120k   10k   22k  -> 152k
1      0.21  0.14
3.3V   0.69V 0.47V

Let's simplify that to
100k   10k   22k  -> 132k

Note that using a zener is maybe not the best approach.  These are
specified at 20mA which is quite a lot, and they're not very precise
either.  Best to use a linear regulator and decouple at the 32k tap.

Another one: how to limit the current at the high point? Probably best
to leave about a 10x current error margin or 60mV.  Note that is only
2% error in the resistors.

Maybe best to stick to transistor biasing?

So this is a bit dangerous.  Can be mediated by not driving the
transistor in full scale during calibration.

Probably not a problem since the circuit is supposed to run with a
digital control loop, but it feels very wrong to amplify errors like
that..

The essential thing to keep in mind is that V_be = kT/q log(I/I_s).
Knowing that T can vary about 10% in typical applications, the
resulting current difference is quite large.

Relative difference I_A/I_B = exp( (V_BE / V_TA) (T_B - T_A) / T_b) ).

Or exp( 25 * DT/T )

Scaling of V_BE is just proportional to T.

Alternative to resistor divisors is zener biasing.  I still need to
get a good idea of zener characteristics.

```
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