Log amplifier

A log amplifier (logarithmic converter) is one for which the output voltage V_out is K times the natural log of the input voltage V_in. This can be expressed as,

$V_\mathrm{out} = K \ln\frac{V_\mathrm{in}}{V_\mathrm{ref}}$

where V_ref is the normalization constant in volts and K is the scale factor.

Basic op-amp diode circuit

Basic op-amp diode log converter

The relationship between the input voltage $v_{\text{in}}$ and the output voltage $v_{\text{out}}$ is given by:

$v_{\text{out}} = -V_{\text{T}} \ln \left( \frac{v_{\text{in}}}{I_{\text{S}} \, R} \right)$

where $I_{\text{S}}$ is the saturation current and $V_{\text{T}}$ is the thermal voltage.

Transdiode configuration

A transdiode configuration with a BJT connected in the negative feedback loop.

A necessary condition for successful operation of a log amplifier is that the input voltage, V_in is always positive. This may be ensured by using a rectifier and filter to condition the input signal before applying to the log amp input. As V_in is positive, V_out is obliged to be negative (since the op amp is in the inverting configuration) and is large enough to forward bias the emitter-base junction of the BJT keeping it in the active mode of operation. Now,

$V_\mathrm{BE} = -V_\mathrm{out}\,\!$

$I_\mathrm C = I_\mathrm{SO}(e^{V_\mathrm{BE} / V_\mathrm T} - 1) \approx I_\mathrm {SO} e^{V_\mathrm{BE} /V_\mathrm T}$

$\Rightarrow V_\mathrm{BE} = V_\mathrm T \ln \frac{I_\mathrm C}{I_\mathrm{SO}}$

where $I_\mathrm{SO}\,$ is the saturation current of the emitter-base diode and $V_\mathrm T\,$ is the thermal voltage. Due to the virtual ground at the op amp differential input,

$I_\mathrm C = \frac{V_\mathrm{in}}{R_1}$, and

$V_\mathrm{out} = -V_\mathrm T \ln \frac{V_\mathrm{in}}{I_\mathrm{SO} R_1}$

The output voltage is expressed as the natural log of the input voltage. Both the saturation current $I_\mathrm{SO}\,$ and the thermal voltage $V_\mathrm T\,$ are temperature dependent, hence, temperature compensating circuits may be required.

See also

Diode
Logarithmic output

External links

Integrated DC logarithmic amplifiers from Maxim's AN 36211
Analog electronics with Op Amps by A. J. Peyton, V. Walsh