5.1 Rectifiers

A rectifier is an electronic circuit designed to convert an Alternating Current (AC) signal, where the charge flow periodically reverses direction, into a pulsated Direct Current (DC) signal, where the charge flows in only one direction. Op-Amps can be used to create high-precision rectifiers.

Half-wave Rectifier

A half-wave rectifier produces an output for only one half-cycle of the AC input, while producing zero output for the other half-cycle. In the op-amp-based inverting configuration:

• During the positive half-cycle of the input, the op-amp’s output attempts to go negative. This forward-biases diode D1, clamping the op-amp output at -0.7V, and reverse-biases diode D2. Consequently, the final circuit output voltage is zero.
• During the negative half-cycle of the input, the op-amp’s output goes positive. This reverse-biases D1 and forward-biases D2, allowing the circuit to act as a standard inverting amplifier. The output voltage is given by: V_0=-\left(\frac{R_f}{R_1}\right)V_1

The result is a series of positive-going half-cycles at the output corresponding to the negative half-cycles of the input.

Full-wave Rectifier

A full-wave rectifier produces a positive-going output for both the positive and negative half-cycles of the AC input. A common implementation uses a two-op-amp structure.

• During the positive half-cycle of the input, the first op-amp acts as an inverting amplifier, while the second is also configured as an inverting amplifier that takes its input from the first. The combined effect results in a positive output given by: V_{0}=\left(\frac{R_2R_5}{R_1R_4}\right)V_{i}
• During the negative half-cycle of the input, a different signal path through the diodes is activated. The first op-amp still inverts, but the second op-amp now acts as a non-inverting amplifier. The resulting output is also positive, given by: V_{0}=-\left(\frac{R_3}{R_1}\right)\left(1+\frac{R_5}{R_4}\right)V_{i}

To achieve a precise unity-gain full-wave rectifier, the component values must be selected to satisfy the conditions for both half-cycles. For instance, setting R_{1}=2R_{3}=R_{4}=R_{5}=R and R_2 = R will ensure a gain of one during both the positive and negative portions of the input waveform.

From rectification, we now turn to other forms of wave shaping.

5.2 Clippers and Clampers

Wave shaping circuits are designed to alter the form of an applied input waveform. Their two primary functions are to attenuate parts of the wave and to alter its DC level. The two main categories of these circuits are clippers and clampers.

Clipper

A clipper is a circuit that removes, or “clips,” a part of the input signal that is either above or below a specified reference value.

• Positive Clipper: This circuit clips the positive portion of the input signal. When the input voltage (V_i) is less than the reference voltage (V_{ref}), the circuit behaves as a voltage follower, and the output matches the input. However, when V_i exceeds V_{ref}, the feedback diode turns off, and the output is clipped, remaining at the level of V_{ref}.
• Negative Clipper: Conversely, this circuit clips the negative portion of the input. It acts as a voltage follower when V_i is greater than V_{ref}. When V_i drops below V_{ref}, the feedback diode turns off, and the output is clipped at the V_{ref} level.

Clamper

A clamper is a circuit that shifts the entire DC level of an input signal up or down without changing its peak-to-peak amplitude or shape.

• Positive Clamper: This circuit shifts the input waveform vertically upward. The amount of the shift is determined by a positive DC reference voltage, effectively adding a positive DC offset to the entire signal.
• Negative Clamper: This circuit shifts the input waveform vertically downward. The amount of the shift is determined by a negative DC reference voltage, which adds a negative DC offset to the signal.

Beyond shaping existing signals, Op-Amps can also be used to generate new waveforms, as seen in oscillators.

5.3 Sinusoidal Oscillators

An oscillator is an electronic circuit that converts energy from a DC source into a periodic AC output signal without any external AC input. For a circuit to produce sustained sinusoidal oscillations, it must satisfy the Barkhausen criteria:

1. The magnitude of the loop gain must be at least unity (A_v\beta \ge 1).
2. The total phase shift around the feedback loop must be 0° or an integer multiple of 360°.

RC Phase Shift Oscillator

The RC phase shift oscillator uses an inverting amplifier and a feedback network consisting of three cascaded RC sections. The inverting amplifier provides a 180° phase shift. The three-stage RC network is designed to provide the remaining 180° of phase shift at a specific frequency.

• Output Frequency: The frequency of oscillation is given by the formula: f=\frac{1}{2\Pi RC\sqrt[]{6}}
• Condition for Oscillation: To satisfy the loop gain criterion, the gain of the inverting amplifier must be at least -29, meaning the feedback resistor must satisfy the condition: R_{f}\geq29R_{1}

Wien Bridge Oscillator

The Wien bridge oscillator utilizes a non-inverting amplifier, which provides 0° of phase shift. Therefore, the feedback network must also provide 0° of phase shift at the desired frequency of oscillation. This is achieved using a lead-lag RC network configured as a Wien bridge.

• Output Frequency: The frequency of oscillation is given by: f=\frac{1}{2\Pi RC}
• Condition for Oscillation: To sustain oscillations, the gain of the non-inverting amplifier must be at least 3. This imposes the following condition on the feedback resistors: R_{f}\geq2R_{1}

We now shift our focus from generating sinusoidal waveforms to creating non-sinusoidal ones.

5.4 Waveform Generators

Waveform generators are circuits specifically designed to produce standard, non-sinusoidal waves such as square waves and triangular waves. These are fundamental signals in digital electronics and testing.

Square Wave Generator

An op-amp-based square wave generator, also known as an astable multivibrator, utilizes both positive and negative feedback. The operation is based on the charging and discharging of a capacitor. The output of the op-amp is fed back to the non-inverting input through a resistive divider (positive feedback), which sets the switching thresholds. The output is also fed back to the inverting input through an RC network (negative feedback). The capacitor charges and discharges through the feedback resistor, and when its voltage crosses the thresholds set by the positive feedback, the op-amp’s output rapidly switches between its positive (+V_{sat}) and negative (-V_{sat}) saturation levels, creating a continuous square wave.

Triangular Wave Generator

A triangular wave generator is typically constructed by cascading a square wave generator with an integrator circuit. This elegant configuration works because the mathematical integration of a square wave results in a triangular wave. The square wave generator produces a constant positive or negative output, and the integrator, in response, generates a linearly increasing or decreasing voltage ramp. When the square wave switches polarity, the direction of the ramp reverses, resulting in a continuous triangular waveform at the output of the integrator.

The principles of signal generation and shaping lead naturally to active filters, which modify signals based on their frequency content.