Nano Ammeter

One of "self-explanatory" circuits in AoE2 is a nano-ammeter circuit; no explanation is given.  I doubt it is really self-explanatory to a beginner.

This circuit appears to originate from PMI's OP-41 datasheet.  (PMI is Precision Monolithics, Inc which later became a part of Analog Devices.)  OP-41 is a JFET input opamp with very low input bias (5pA @ 25C) in the inverting configuration as a logarithmic amplifier.  (OP-41 also tauts the excellent CMRR of over 100dB for a FET input opamp.)  Q1 and Q2 are matching transistors.  Their current ratio is the exponential of the difference in Vbe. If the difference in Vbe is proportional to the temperature, the current ratio is constant.  Since the Q1 base is grounded, the difference in Vbe is just the Q2 base voltage.  The resistor divider provides the Q2 base voltage that is proportionally to the temperature, as the bandgap reference subtracts the diode connected Q3 to produce a voltage is approximately proportional to Vt (although there is more complex 2nd order temperature dependency).   How good is the approximation?   We can run a Spice simulation stepping the temperature.  The Q3 current is about 65uA at -55C and 115uA at 125C and approximately linear  I (uA) ~ 0.28 * T (K) . 

The trimmer adjusts the full scale value of the current meter. 

Also the compensation network requires some explanation, which is deferred to a separate discussion.