While operational amplifiers are foundational to linear circuit design, their high gain and differential inputs also make them ideal for a wide range of non-linear applications. These circuits exploit the properties of other non-linear components, like diodes, to achieve complex signal processing tasks. This section explores circuits that perform logarithmic operations, voltage comparisons, and signal rectification, demonstrating the op-amp’s remarkable versatility.

4.1 Logarithmic and Anti-Logarithmic Amplifiers

Logarithmic Amplifier

A logarithmic amplifier, or log amp, produces an output voltage that is proportional to the logarithm of the input voltage. This is achieved by placing a forward-biased diode, a component with an exponential current-voltage relationship, in the feedback path of an inverting op-amp configuration.

The analysis combines the nodal equation at the inverting terminal with the fundamental diode current equation. The current through the feedback path (I_f) is equal to the input current (V_i/R_1). I_{f}=\frac{V_i}{R_1} The diode current is also given by: I_{f}=I_{s} e^{(\frac{V_f}{nV_T})} In the feedback loop, the voltage across the diode (V_f) is equal to -V_0. Substituting and equating the expressions for I_f yields: \frac{V_i}{R_1}=I_{s}e^{\left(\frac{-V_0}{nV_T}\right)} By taking the natural logarithm of both sides and solving for V_0, we arrive at the final output voltage equation: V_{0}=-{nV_T}In\left(\frac{V_i}{R_1I_s}\right) Here, I_s is the diode’s saturation current, V_T is its thermal equivalent voltage, and n is an ideality factor.

Anti-Logarithmic Amplifier

An anti-logarithmic amplifier, or anti-log amp, performs the inverse operation, producing an output voltage that is proportional to the anti-logarithm (exponential) of the input voltage. The circuit configuration is the reverse of the log amp, with the diode serving as the input element and the resistor in the feedback path.

The derivation follows a similar path. The feedback current I_f is equal to -V_0/R_f, and the input diode current is an exponential function of the input voltage V_i. V_{0}=-R_{f}I_{f} I_{f}=I_{s} e^{\left(\frac{V_i}{nV_T}\right)} Substituting the expression for I_f into the output equation gives the final relationship: V_{0}=-R_{f}{I_{s} e^{\left(\frac{V_i}{nV_T}\right)}}

4.2 Comparators

A comparator leverages the op-amp’s full open-loop gain to resolve minute differences between two input voltages, driving the output to one of its supply rails (+V_{sat} or -V_{sat}) to produce a binary representation of the input polarity. It is a fundamental non-linear application that forms the basis of many signal-processing and data-conversion circuits.

Inverting Comparator

In an inverting comparator, the input signal (V_i) is applied to the inverting terminal, and a reference voltage (V_{ref}) is applied to the non-inverting terminal.

• When V_i > V_{ref}, the output voltage swings to negative saturation, V_0 = -V_{sat}.
• When V_i < V_{ref}, the output voltage swings to positive saturation, V_0 = +V_{sat}.

For example, an inverting zero crossing detector uses V_{ref} = 0V. For a sinusoidal input, the output is -V_{sat} during the positive half-cycle and +V_{sat} during the negative half-cycle.

Non-Inverting Comparator

In a non-inverting comparator, the input signal (V_i) is applied to the non-inverting terminal, and the reference voltage (V_{ref}) is applied to the inverting terminal.

• When V_i > V_{ref}, the output voltage swings to positive saturation, V_0 = +V_{sat}.
• When V_i < V_{ref}, the output voltage swings to negative saturation, V_0 = -V_{sat}.

A non-inverting zero crossing detector (V_{ref} = 0V) with a sinusoidal input produces an output of +V_{sat} during the positive half-cycle and -V_{sat} during the negative half-cycle.

4.3 Rectifiers

A rectifier is a circuit designed to convert an AC signal into a pulsating DC signal by passing only one polarity of the waveform. Op-amp based rectifiers offer precision and the ability to rectify very small signals, overcoming the forward voltage drop limitations of simple diode rectifiers.

Half-Wave Rectifier

An op-amp half-wave rectifier provides a rectified output by utilizing diodes in its feedback loop to dynamically alter the circuit configuration based on the input signal’s polarity.

• During the positive half-cycle of the input: The op-amp’s output swings negative, forward-biasing D1. This creates a low-impedance feedback path directly from the op-amp’s output to its inverting input, effectively putting the op-amp in a voltage follower configuration with a -0.7V output. However, diode D2 is reverse-biased, isolating this behavior from the circuit’s main output node, which remains at zero volts.
• During the negative half-cycle of the input: The op-amp’s output swings positive, reverse-biasing D1 and forward-biasing D2. The circuit now functions as a standard inverting amplifier with gain determined by R_f and R_1. The output is an inverted, amplified version of the input: V_0=-\left(\frac{R_f}{R_1}\right)V_i The result is an output that only contains positive-going half-cycles corresponding to the negative input cycles.

Full-Wave Rectifier

An op-amp full-wave rectifier, often called a precision rectifier, produces a positive output for both positive and negative halves of the input signal. It is typically constructed as a two-stage circuit.

• During the positive half-cycle:
  • The first stage op-amp acts as an inverting amplifier, producing a negative output. Diode D1 is forward-biased, and D2 is reverse-biased.
  • This negative signal is fed through resistor R4 to the inverting input of the second op-amp. The second stage also functions as an inverting amplifier, inverting the negative signal to produce a final positive output. The output voltage is: V_{0}=\left(\frac{R_2R_5}{R_1R_4}\right)V_{i}
• During the negative half-cycle:
  • The first stage op-amp’s output swings positive. Diode D1 is reverse-biased, and D2 is forward-biased.
  • This positive signal is fed directly to the non-inverting input of the second op-amp. The second stage is now configured as a non-inverting amplifier, resulting in a final positive output. The output voltage is: V_{0}=-\left(\frac{R_3}{R_1}\right)\left(1+\frac{R_5}{R_4}\right)V_{i} By carefully selecting resistor values, the circuit can be designed for unity gain in both half-cycles, producing a precise, full-wave rectified output.

These non-linear circuits demonstrate the op-amp’s ability to manipulate signals in complex ways. Building on this, the next section will explore circuits designed specifically to alter or “shape” waveforms.