Answer 4—The circuit is a logarithmic converter. The output voltage represents the logarithm of the input voltage (or current). Carefully constructed, this circuit can perform well over five decades or more of input (nanoamps to milliamps).

The collector current of a transistor is an exponential function of the base-emitter voltage, as defined by the Ebers-Moll equation:

I_S is the reverse saturation current of the base-collector junction, which is a function of transistor geometry, temperature, and other factors. V_T is kT/q, about 25.3 mV at room temperature. q is the electron charge (1.6E – 19 C). k is Boltzmann’s constant (1.38E – 23 J/K). T is absolute temperature (K).

Solving for V_BE yields:

This is the output of the first op-amp, which changes about 60 mV per decade of input current, but both V_T and I_S have a strong dependence on temperature. The second transistor, which has the same construction and is held at the same temperature as the first, is fed a constant reference level of collector current. This offsets the converter so that the output voltage is zero when the input current equals the reference current. The output voltage varies with the logarithm of the ratio I_IN/I_REF, with proportionality constant set by the gain of the second op-amp. A gain of 16 gives an output of –1 V per decade.

Note that the V_T factor in the V_BE equation has a dependency on temperature, which means that for input currents other than the reference current, there will be a scale factor error. The usual approach to addressing this is to incorporate a thermistor into the feedback network of the second op-amp so that its gain varies in such a way as to compensate for the varying scale factor.

Contributor: David Tweed