Log amplifier

A log amplifier, also known as logarithmic amplifier or logarithm amplifier or log amp, is an amplifier whose output voltage $V_{\text{out}}$ is approximately proportional to the logarithm of the applied input voltage $V_{\text{in}}$:

$V_{\text{out}}\approx K\cdot \ln \left({\frac {V_{\text{in}}}{V_{\text{ref}}}}\right)\,,$

where $V_{\text{ref}}$ is a normalization constant in volts, $K$ is a scale factor, and $\ln$ is the natural logarithm.

Log amplifier circuits designed with operational amplifiers (opamps) use the exponential current–voltage relationship of a p–n junction (either from a diode or bipolar junction transistor) as negative feedback to compute the logarithm. Multistage log amplifiers instead cascade multiple simple amplifiers to approximate the logarithm's curve. Temperature-compensated log amplifiers may include more than one opamp and use closely-matched circuit elements to cancel out temperature dependencies. Integrated circuit (IC) log amplifiers have better bandwidth and noise performance and require fewer components and printed circuit board area than circuits built from discrete components.

Log amplifier applications include:

• Performing mathematical operations like multiplication (sometimes called mixing), division, and exponentiation. This ability is analogous to the operation of a slide rule and is used for:
  • Analog computers
  • Audio synthesis
  • Measurement instruments (e.g. power as multiplication of current and voltage)
• Decibel (dB) calculation
• True RMS conversion
• Extending the dynamic range of other circuits for:
  • Automatic gain control of transmit power in radio frequency circuits
  • Scaling a large dynamic range sensor (e.g. from a photodiode) into a linear voltage scale for an analog-to-digital converter with limited resolution

The log amplifier's operation can be inverted by an exponentiator, such as an opamp configured for exponential output. An exponentiator may be needed at the end of a series of analog computation stages done in a logarithmic scale in order to return the voltage scale back to a linear output scale. Additionally, signals that are first companded by a logarithmic amplifier may eventually need to be expanded out by an exponentiator to restore to their original scale.

Basic opamp diode circuit

Basic opamp diode log converter

The basic opamp diode log converter 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. Similarly the emitter saturation current varies significantly from one transistor to another and also with temperature. Hence, it is very difficult to set the reference voltage for the circuit.

Additionally, the bulk resistivity $R_{\text{B}}$ of silicon in a real diode limits accuracy at high currents due to the addition of a $I_{\text{D}}\cdot R_{\text{B}}$ term to the diode's voltage. Additionally, 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.

Transdiode configuration

A transdiode configuration with a BJT 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:

${\begin{aligned}V_{\text{BE}}&=-V_{\text{out}}\\I_{\text{C}}&=I_{\text{S}}\left(e^{\frac {V_{\text{BE}}}{V_{\text{T}}}}-1\right)\approx I_{\text{S}}e^{\frac {V_{\text{BE}}}{V_{\text{T}}}}\\\Rightarrow V_{\text{BE}}&=V_{\text{T}}\ln \left({\frac {I_{\text{C}}}{I_{\text{S}}}}\right)\end{aligned}}$

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. Alternatively, for analog computers that follow log operations by an antilog, then temperature variation in the log operation is unimportant, since it will be compensated by a similar variation in the antilog circuit in thermal equilibrium (e.g. on the same chip).

One method to remove $I_{\text{S}}\,$temperature dependence is to copy the basic uncompensated BJT-based log amplifier, but use a constant current source instead of resistor $R$ for this copy, and then follow both log amplifiers by a difference amplifier. The BJTs should be matched and in thermal equilibrium, so that the difference amplifier subtracts the second BJT's junction voltage to cancel out $I_{\text{S}}\,$in the difference amplifier's output. The constant current source can also be used to set the desired x-axis intercept and allows users to make ratiometric measurements that are relative to a desired reference. Using a resistive temperature detector (e.g. a thermistor) in the difference amplifier's gain-setting resistors can minimize the remaining $V_{\text{T}}\,$dependence. This architecture can be very accurate; for instance, the LOG200 chip released in 2024 achieves 160 dB dynamic range with under 0.2% log conformity error.

Texas Instruments application note AN-311 describes another temperature-compensated circuit which only uses two opamps instead of three. 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 N linear amplifiers, each with gain of A dB, and then summing the result. For small signals such that the final amplifier doesn't saturate, the total gain will be N·A 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 (N-1)·A 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.

True log amp

If each limiting amplifier clips "softly" and are cascaded without summing, the approximation can be within 0.1 dB and is sometimes called a "true log amp". The response of both this true log amp and the basic multistage log amp are not truly logarithmic, because they are symmetric about zero (while the mathematical logarithm function is indeterminate for negative inputs) and are linear for small inputs. But, such a symmetrical transfer function is fine for capacitively coupled AC inputs, such as from radar receivers. The term "logarithmic converter" may better describe such functionality than "logarithmic amplifier".

Successive detection log amp

The successive detection log amp architecture is a variant of this which uses full or half wave detectors from the output of each amplifier stage, all connected to the log amplifier's output node.

See also

• Diode
• Operational amplifier applications § Logarithmic output

External links

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