Log amplifier

The basic opamp diode log amplifier shown in the diagram utilizes the diode's exponential current-voltage relationship for the opamp's negative feedback path, with the diode's anode virtually grounded and its cathode connected to the opamp's output V_\text{out}, used as the circuit output. The Shockley diode equation gives the current–voltage relationship for the ideal semiconductor diode in the diagram to be: : I_\text{D} = I_\text{S} \left(e^{\frac{-V_\text{out}}{V_\text{T}}} - 1\right), where I_\text{D} flows from the diode's anode to its cathode, I_\text{S} is the diode's reverse saturation current and V_\text{T} is the thermal voltage (approximately 26 mV at room temperature). When -V_\text{out} \gg V_\text{T}, the diode's current is approximately proportional to an exponential function: : I_\text{D} \simeq I_\text{S} e^{\frac{-V_\text{out}}{V_\text{T}}}. Rearranging this equation gives the output voltage V_\text{out} to be approximately: :V_{\text{out}} = -V_\text{T} \ln \left(\frac{I_\text{D}}{I_\text{S}} \right) \, . An input voltage can easily be scaled and converted into the diode's current I_\text{D} using Ohm's law by sending the input voltage through a resistance R to the virtual ground, so the output voltage V_\text{out} will be approximately: :V_{\text{out}} = -V_\text{T} \ln \left(\frac{V_\text{in}}{I_\text{S} \, R} \right) \, . A necessary condition for successful operation of this log amplifier is that V_\text{in} is always positive. This may be ensured by using a rectifier and filter to condition the input signal before applying it to the log amplifier's input. V_\text{out} will then be negative (since the op amp is in the inverting configuration) and is negative enough to forward bias the diode. Drawbacks The diode's saturation current I_\text{S} doubles for every ten kelvin rise in temperature and varies significantly due to process variation. And because thermal voltage V_\text{T}{=}\tfrac{kT}{q}, the output voltage is also proportional to its kelvin temperature. Hence, it is very difficult to set the reference voltage for the circuit. Additionally, the bulk resistance R_{\text{B}} of a real diode limits accuracy at high currents due to an added I_{\text{D}} R_{\text{B}} voltage term. And, diffusion currents in surface inversion layers and generation-recombination effects in space-charge regions cause a scale factor m at low currents that varies (between 1 and 4) with current. With inputs near 0 volts, log amps have a linear V_\text{in} to V_\text{out} law. But this non-logarithmic behavior itself is often lost in this device noise, which limits the dynamic range to 40-60 dB, but the dynamic range can be increased to over 120 dB by replacing the diode with a transistor in a "transdiode" configuration. To address inaccuracies for small inputs the size of V_\text{T} or smaller and the question of how to handle negative inputs, one solution uses a symmetric function such as the inverse hyperbolic sine, whose graph approximates \ln(2x) for large positive and negative \ln(\text{-}2x) for large negative inputs, but which linearly goes through 0 for small inputs. This function may be implemented with a combination of N and P diodes (sold many years ago in a temperature compensated module) to make what is called a "true log" amp or "baseband log" amp (which may instead use a multistage amplifier architecture, as described in ). == Transdiode configuration ==

connected in the negative feedback loop. While the floating diode in the earlier basic opamp implementation causes the output voltage to depend on the opamp's input offset current, the grounded-base or "transdiode" configuration shown in the diagram does not possess this problem. Negative feedback causes the opamp to output enough voltage on the base-emitter junction of the bipolar junction transistor (BJT) to ensure that all available input current is drawn through the collector of the BJT, so the output voltage is then referenced relative to the true ground of the transistor's base rather than the virtual ground. While the circuit in the diagram uses an npn transistor and produces a negative V_\text{out} and sinks input current, a pnp will instead result in positive V_\text{out} and a current-sourcing input. With a positive V_\text{in} large enough to make V_\text{out} negative enough to forward bias the emitter-base junction of the BJT (to keep it in the active mode of operation), then: :\begin{align} V_\text{BE} &= -V_\text{out} \\ I_\text{C} &= I_\text{S}\left(e^\frac{V_\text{BE}}{V_\text{T}} - 1\right) \approx I_\text{S} e^\frac{V_\text{BE}}{V_\text{T}} \\ \Rightarrow V_\text{BE} &= V_\text{T} \ln\left(\frac{I_\text{C}}{I_\text{S}}\right) \end{align} where I_\text{S} is the saturation current of the emitter-base diode and V_\text{T} is the thermal voltage. Due to the virtual ground at the opamp's inverting input, :I_\text{C} = \frac{V_\text{in}}{R}, and :V_\text{out} = -V_\text{T} \ln \left(\frac{V_\text{in}}{I_\text{S} R}\right) The output voltage is expressed as the natural log of the input voltage. Both the saturation current I_\text{S} and the thermal voltage V_\text{T} are temperature dependent, hence, temperature compensation may be required. Temperature compensation Because temperature compensation is generally needed, it is often built into log amplifier ICs. Some analog computation chips that follow log operations by an antilog may conveniently compensate the log circuit's temperature variation by a similar variation in the antilog circuit. Texas Instruments application note AN-311 describes another temperature-compensated circuit which only uses two opamps instead of three and maintains 1% log conformity. It also uses a matched BJT configured with the second opamp to compensate for the first BJT's V_\text{BE} temperature dependence by cancelling I_\text{S} out from \Delta V_\text{BE}, the difference between the first BJT's V_\text{BE} minus the second BJT's V_\text{BE}. The second BJT's collector is fed a constant current from a temperature-compensated Zener diode voltage reference and its emitter is tied to the emitter of the first BJT, which also connects through a resistor the output of the second opamp. The second BJT's V_\text{BE} is fixed by its constant collector current. The second BJT's base voltage relative to ground is \Delta V_\text{BE}, so it will lack any I_\text{S} component. This \Delta V_\text{BE} is outputted through the midpoint of a temperature-compensated voltage divider (where one resistor has a much higher temperature coefficient) to counteract V_\text{T}'s temperature dependence. This circuit can also be inverted to form an exponentiator. == Multistage log amp architectures ==

While the previous circuits utilized the p–n junction's exponential current–voltage relationship for computing the log function, the following approaches instead approximate the log function by cascading multiple simpler amplifiers. Basic multistage log amp A basic multistage log amp works by cascading a series of linear amplifiers, each with gain of  dB, and then summing the result. For small signals such that the final amplifier doesn't saturate, the total gain will be  dB. However, as the input signal level increases, the final amplifier will limit and thus make a fixed contribution to the sum, so that the gain will drop to  dB. As the signal increases, the second to last amplifier will limit, and so on, until the first limits. The resulting curve is a piecewise linear function approximation of the log function. == See also ==