4.1 Comparators

A comparator is a fundamental non-linear circuit that compares two input voltages and produces a digital-level output indicating which of the two is larger. The output swings to one of two saturation levels, either positive saturation (+V_{sat}) or negative saturation (-V_{sat}), representing a binary outcome.

Inverting Comparator

In an inverting comparator configuration, a reference voltage (V_{ref}) is applied to the non-inverting terminal, and the input signal (V_i) is applied to the inverting terminal. The circuit’s operation is straightforward:

• If the input voltage is greater than the reference voltage (V_{i} > V_{ref}), the output will swing to negative saturation, -V_{sat}.
• If the input voltage is less than the reference voltage (V_{i} < V_{ref}), the output will swing to positive saturation, +V_{sat}.

When the reference voltage is set to zero volts (V_{ref}=0V), the circuit is known as an inverting zero crossing detector. The output transitions from one saturation level to the other precisely when the input signal crosses zero.

Non-Inverting Comparator

In a non-inverting comparator, the configuration is reversed: the reference voltage is applied to the inverting terminal, and the input signal is applied to the non-inverting terminal. Its operation is complementary to the inverting type:

• If the input voltage is greater than the reference voltage (V_{i} > V_{ref}), the output will swing to positive saturation, +V_{sat}.
• If the input voltage is less than the reference voltage (V_{i} < V_{ref}), the output will swing to negative saturation, -V_{sat}.

Similarly, when the reference voltage is zero, this circuit functions as a non-inverting zero crossing detector, with the output changing state as the input signal passes through zero volts.

From comparison logic, we move to another class of non-linear circuits: logarithmic amplifiers.

4.2 Logarithmic and Anti-Logarithmic Amplifiers

Logarithmic and anti-logarithmic amplifiers are non-linear circuits that perform these specific mathematical operations. They achieve this by utilizing the inherent non-linear voltage-current characteristic of a semiconductor diode, typically placed in the feedback path of an Op-Amp.

Logarithmic Amplifier

A logarithmic amplifier produces an output voltage that is proportional to the natural logarithm of its input voltage. The analysis begins with the diode current equation and Kirchhoff’s Voltage Law (KVL). The current through the feedback diode, I_f, is equal to the input current, \frac{V_i}{R_1}. This current is also described by the diode equation:

I_{f}=I_{s} e^{(\frac{V_f}{nV_T})}

where I_s is the saturation current, V_f is the forward voltage across the diode, and nV_T represents thermal constants. Since the inverting terminal is at virtual ground, the KVL equation for the feedback loop gives V_f = -V_0. Substituting these relationships yields:

\frac{V_i}{R_1}=I_{s}e^{\left(\frac{-V_0}{nV_T}\right)}

Solving for V_0 by taking the natural logarithm of both sides results in the final expression:

V_{0}=-{nV_T}In\left(\frac{V_i}{R_1I_s}\right)

This shows that the output voltage is proportional to the natural log of the input voltage. The negative sign indicates phase inversion.

Anti-Logarithmic Amplifier

An anti-logarithmic amplifier performs the inverse operation, producing an output voltage proportional to the exponential (anti-logarithm) of its input voltage. In this circuit, the diode is placed in the input path and the resistor is in the feedback path. The feedback current I_f is equal to -\frac{V_0}{R_f}. The KVL equation at the input shows that the voltage across the diode is simply the input voltage, V_f = V_i. The diode current is therefore:

I_{f}=I_{s} e^{\left(\frac{V_i}{nV_T}\right)}

Equating the two expressions for I_f and solving for V_0 gives the final equation:

V_{0}=-R_{f}{I_{s} e^{\left(\frac{V_i}{nV_T}\right)}}

This demonstrates that the output voltage is proportional to the anti-natural logarithm (exponential) of the input voltage, again with a phase inversion.

These circuits form the basis for more complex signal processing, including signal generation and shaping.