Introduction to Logarithmic and Anti-Logarithmic Amplifiers

Logarithmic and anti-logarithmic amplifiers are specialized circuits that exploit the nonlinear characteristics of semiconductor devices to produce outputs that are logarithmic or exponential functions of the input. When built around operational amplifiers (op amps), these circuits offer precision, stability, and ease of integration into larger systems. Their ability to compress or expand signal dynamics makes them indispensable in audio processing, instrumentation, analog computation, and communication systems.

A logarithmic amplifier (log amp) outputs a voltage proportional to the logarithm of the input current or voltage. An anti-logarithmic amplifier (exp amp or antilog amp) does the opposite: its output is proportional to the exponential of the input. By pairing these two functions, engineers can create companding systems that first compress a signal’s dynamic range for transmission or storage and later expand it back to its original form. This technique is especially valuable in environments where bandwidth or signal-to-noise ratio is limited.

This article provides a detailed guide to designing, implementing, and applying logarithmic and anti-logarithmic amplifiers using op amps. It covers core principles, practical circuit configurations, temperature compensation methods, and real-world applications, with a particular focus on dynamic range compression. By the end, you will have a solid foundation for building these circuits in your own projects.

Fundamentals of Logarithmic and Anti-Logarithmic Amplifiers

At their heart, log and antilog amplifiers rely on the exponential current‑voltage relationship of a PN junction. For a silicon diode or a bipolar junction transistor (BJT) operated in the forward active region, the collector current is approximately IC = IS × exp(VBE / VT), where IS is the saturation current and VT is the thermal voltage (≈26 mV at 300 K). By placing such a nonlinear element in the feedback loop of an op amp, the circuit forces a logarithmic or exponential relationship between input and output.

Why Op Amps?

Operational amplifiers provide the high gain and feedback that make it possible to precisely control the voltage across the nonlinear element. In a log amp, the op amp maintains a virtual ground at its inverting input, so the input current is forced through the diode or transistor. The op amp’s output then adjusts to keep the junction voltage at a level that satisfies the exponential equation. The result is a highly accurate logarithmic conversion over several decades of input current.

Similarly, in an antilog amp, the op amp forces the input voltage to appear across the base‑emitter junction of a BJT, causing a collector current that is an exponential function of that voltage. The op amp converts this current into a voltage output. The high open‑loop gain of the op amp minimizes errors, while its low input bias current and offset voltage allow operation over wide dynamic ranges.

Core Design Principles

Using the Diode Exponential Characteristic

The simplest log amp uses a diode in the feedback path. When the input current flows through the diode, the voltage across the diode is proportional to the logarithm of the current. However, a diode’s behavior deviates from the ideal exponential law due to series resistance and high‑level injection effects. For better accuracy over a wider dynamic range, designers replace the diode with a bipolar transistor connected as a diode (base and collector shorted). The transistor exhibits a more ideal exponential characteristic over up to six or more decades.

Feedback Loop Configuration

The basic inverting amplifier topology is used for both log and antilog circuits. In the log amp, the nonlinear element is placed in the feedback path from output to inverting input, while the input signal is applied through a resistor to the same inverting node. The op amp’s non‑inverting input is connected to ground. For the antilog amp, the nonlinear element is placed in the input path, and the feedback network uses a linear resistor. The op amp still forces a virtual ground, but now the input voltage controls the current through the nonlinear device, which is then mirrored by the feedback resistor to produce the output voltage.

Temperature sensitivity is a major challenge. The thermal voltage VT and saturation current IS both vary with temperature, causing significant drift unless compensated. Modern log amp designs often incorporate matched transistor pairs and temperature‑sensitive resistors to cancel these effects.

Designing a Logarithmic Amplifier

Basic Circuit with Diode

A simple log amp can be built with a single op amp, a resistor, and a diode. The input voltage is applied through resistor Rin to the inverting input. The diode is connected with its anode at the op amp output and its cathode at the inverting input (for positive input voltages). The output voltage is then Vout ≈ –VT × ln(Vin / (IS × Rin)). This circuit works over about two to three decades, limited by the diode’s non‑idealities.

Temperature Compensation Techniques

To achieve stable operation over temperature, three techniques are commonly used:

  • Matched transistor pair: A second transistor, thermally coupled to the first, provides a reference against which temperature‑dependent terms cancel. This is the foundation of many integrated log amps.
  • PTAT current source: A proportional‑to‑absolute‑temperature (PTAT) current can be injected into the compensation network to offset the VT variation.
  • Thermistor or diode temperature sensing: A temperature sensor (e.g., a silicon diode) adjusts the gain of a subsequent stage to correct for the VT coefficient.

Using Transistors for Better Performance

Replacing the diode with a bipolar transistor (base and collector shorted) dramatically improves dynamic range. The transistor’s collector current follows the ideal exponential law over five to seven decades. A typical circuit uses an NPN transistor (e.g., 2N2222 or BC547) with its base and collector connected together. The input resistor is chosen so that the transistor operates in its forward active region. With proper layout and a low‑offset op amp like the OP07 or OP27, a dynamic range exceeding 100 dB is achievable.

Precision log amp ICs such as the Analog Devices AD8306 or the Texas Instruments LOG112 internally implement these principles with matched transistors and on‑chip temperature compensation. They simplify design while offering excellent accuracy and bandwidth.

Designing an Anti-Logarithmic Amplifier

Circuit Configuration

The antilog amplifier is essentially the log amp turned inside out. The nonlinear element (diode or transistor) is placed in the input path, and a linear feedback resistor sets the gain. For a bipolar transistor, the input voltage is applied to the base through a high‑value resistor, or directly via an op amp buffer to avoid loading. The emitter is connected to the inverting input of the op amp, and the collector is connected to the op amp output. The output voltage becomes Vout = –Rf × IS × exp(Vin / VT) (for positive inputs).

Precision requires careful matching of the transistor pair if a compensation transistor is used. As with the log amp, temperature compensation is critical. Often a log amp and antilog amp are paired together in a feedback loop to create a linear gain block whose overall transfer function is immune to temperature variations.

Relationship to Log Amp

The antilog amp is the exact reciprocal function of the log amp. When cascaded—first a log amp, then an antilog amp—the overall output is linear, provided the two circuits are matched. This property is the basis for analog multipliers and dividers. In dynamic range compression, a controlled amount of log amplification followed by antilog expansion (or the reverse) allows flexible shaping of the signal envelope.

Practical Considerations

Component Selection

Choose op amps with low input bias current (FET‑input types like TL072, OPA134, or AD8629) to avoid errors when handling nanoamp‑level input currents. Low offset voltage (under 1 mV) and low drift are essential for DC accuracy. For the nonlinear element, use a high‑gain, matched transistor pair such as the LM394, MAT‑02, or the transistor arrays in the SSM2212. These integrated pairs provide excellent thermal tracking.

Bandwidth and Stability

Log and antilog amplifiers are inherently nonlinear, which complicates frequency compensation. The small‑signal bandwidth depends on the instantaneous gain, which varies with the signal level. At very low input currents, the gain of the transistor in the feedback loop can be extremely high, leading to instability. Adding a small compensating capacitor (10–100 pF) across the feedback transistor often tames oscillations. For wideband applications, consider using a log amp IC with internal compensation.

Noise and Offset

Because log amps operate over extremely wide dynamic ranges, noise at low input levels can be problematic. Use low‑noise resistors (metal film) and bypass power supplies carefully. Offset voltage at the op amp input appears as a current error at the inverting node, which can shift the log output significantly at low currents. Offset nulling or autozero techniques (chopper‑stabilized op amps) are recommended for precision measurements.

For antilog amps, noise on the input voltage is exponentially amplified, so low‑noise design is even more critical. Keep signal paths short and use guard rings around high‑impedance nodes.

Applications in Dynamic Range Compression

Audio Compression Circuits

In professional audio, dynamic range compression reduces the volume of loud peaks while boosting quiet passages. A log amp converts the audio envelope into a logarithmic control voltage. This voltage then adjusts the gain of a voltage‑controlled amplifier (VCA), which compresses the signal. The compression ratio (e.g., 2:1, 4:1) is set by scaling the log‑domain control voltage. Many classic analog compressors, such as the Urei 1176, use a log/antilog approach with a special VCA (e.g., the THAT 2180 series).

Modern implementations use a log amp to measure the RMS or peak level of the audio, then feed a scaled antilog stage to control the VCA. The time constants (attack and release) are applied to the control voltage in the log domain, which gives better tracking of the ear’s psychoacoustic response.

Log/Antilog Pair for Compander Systems

Compander (compressor/expander) systems like the dbx or Dolby noise reduction encode audio by compressing the dynamic range before recording or transmission, then expand it during playback. The encoder uses a log amp to compress, while the decoder uses a matched antilog amp to expand. Perfect matching between the two ensures no net amplitude distortion. The AN‑112 application note from Analog Devices provides detailed circuit examples of such companders.

Wireless communication systems also employ log/antilog pairs to linearize power amplifiers or to implement automatic gain control (AGC) loops that maintain a constant signal level across wide fading ranges.

Other Applications

Measurement and Instrumentation

Log amps are ideal for measuring signals that span many orders of magnitude, such as optical power in fiber‑optic receivers, acoustic intensity, or radioactive decay rates. A photodiode can provide a current proportional to light intensity; the log amp converts this to a voltage proportional to dBm. Instruments like spectrum analyzers, network analyzers, and multi‑meter RMS converters use log amps internally. For example, the Texas Instruments application note on logarithmic amplifiers describes a circuit that achieves 80 dB dynamic range with a single op amp.

Analog Computation

Before the dominance of digital signal processors, analog computers used log and antilog amplifiers to perform multiplication, division, and exponentiation. By adding or subtracting log‑domain voltages, multiplication becomes addition, and division becomes subtraction. An antilog stage then recovers the linear result. This technique is still used in some high‑speed analog multipliers, such as the ADI AD734 or the Burr‑Brown OPA580.

Log/antilog blocks also form the basis of function generators that produce sine, cosine, or arctangent outputs from polynomial approximations.

Advanced Variations

Multidecade Log Amps

To achieve nine or more decades of operation, designers use multiple cascaded log amp stages, each handling a portion of the dynamic range. The outputs are summed to produce a piecewise‑linear approximation of the logarithmic function. This “successive detection” architecture is common in RF power detectors and wideband receivers. The AD8307, for example, uses six cascaded logarithmic amplifier stages to cover 80 dB.

Precision Log Amps with Matched Transistors

Integrated solutions such as the LOG101 from Texas Instruments include an on‑chip matched transistor pair, temperature compensation, and a precision reference. These devices provide logarithmic accuracy of better than 0.1% over 10 decades of input current. They also offer a logarithmic slope pin that allows the user to set the volts‑per‑decade scaling. External resistors determine the scale factor. These ICs are the easiest way to implement high‑performance log amps without discrete design complexity.

For educational projects, a discrete circuit using a cheap dual transistor (e.g., BC847BS) and a rail‑to‑rail op amp can still achieve 60 dB of usable range with careful temperature compensation. Many hobbyist audio compressor designs follow this approach.

Conclusion

Logarithmic and anti-logarithmic amplifiers built around operational amplifiers provide a robust and flexible method for manipulating signal dynamics. By exploiting the exponential characteristics of semiconductor junctions, these circuits can compress or expand signals across many decades of amplitude. Their applications range from essential audio compression and companding systems to precision instrumentation and analog computation.

Successful design requires careful attention to temperature compensation, component selection, and stabilization. Whether you choose a discrete transistor pair or an integrated log amp IC, the principles remain the same. With a solid understanding of feedback topology and the exponential law, you can tailor log and antilog amplifiers to meet the needs of any nonlinear signal processing task.

For further reading, consult the Analog Devices logarithmic amplifier tutorial and the Wikipedia article on logarithmic amplifiers for additional theory and historical context.