In electronic systems, it is often necessary to either generate specific periodic signals or to selectively pass or reject certain frequencies from a complex signal. Operational amplifiers play a critical role in both of these domains. They are the core active components in oscillators, which create new periodic waveforms, and in active filters, which shape the frequency content of existing signals. This section provides the design principles for these essential building blocks.

6.1 Active Filters

Active filters are frequency-selective circuits that contain an active element, such as an op-amp, in addition to passive resistors and capacitors. The inclusion of an op-amp allows for voltage gain, buffering to prevent loading effects, and the ability to create complex filter responses without the need for inductors. The four main types are:

1. Active Low Pass Filter: This filter passes low-frequency components and attenuates high-frequency components. A common implementation consists of a passive RC low-pass filter stage connected to the input of a non-inverting op-amp amplifier, which provides gain and buffering.
2. Active High Pass Filter: This filter passes high-frequency components while rejecting low-frequency ones. Its construction is similar, with a passive RC high-pass filter feeding the input of a non-inverting amplifier.
3. Active Band Pass Filter: This filter is designed to pass a specific band of frequencies, rejecting frequencies both below and above this band. It can be constructed by cascading an active high-pass filter and an active low-pass filter. The high-pass filter sets the lower cutoff frequency, and the low-pass filter sets the upper cutoff frequency.
4. Active Band Stop Filter: Also known as a band-reject or notch filter, this circuit rejects a specific band of frequencies while passing all others. It can be implemented using a summing amplifier that combines the outputs of an active low-pass filter and an active high-pass filter, where the cutoff frequency of the low-pass filter is set lower than that of the high-pass filter.

6.2 Sinusoidal Oscillators

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

1. The magnitude of the total loop gain must be greater than or equal to unity (|A_v\beta| \ge 1).
2. The total phase shift around the feedback loop must be exactly 0^\circ or an integer multiple of 360^\circ.

RC Phase Shift Oscillator

This oscillator uses an inverting amplifier and a feedback network composed of three cascaded RC sections.

• Phase Shift: The inverting amplifier provides a 180° phase shift. The three-section RC network is designed to provide an additional 180° phase shift at a specific frequency. This satisfies the 360^\circ phase shift requirement.
• Frequency of Oscillation: The frequency at which the RC network provides a 180° shift is given by: f=\frac{1}{2\Pi RC\sqrt[]{6}}
• Condition for Oscillation: To satisfy the unity gain criterion, the gain of the inverting amplifier must compensate for the attenuation of the RC network. The condition is: \frac{R_f}{R_1}\geq29

Wien Bridge Oscillator

This oscillator is known for producing a stable, low-distortion sine wave. It uses a non-inverting amplifier and a lead-lag feedback network.

• Phase Shift: The non-inverting amplifier provides a 0° phase shift. Therefore, the feedback network must also provide a 0° phase shift at the desired frequency of oscillation.
• Frequency of Oscillation: The frequency at which the Wien bridge network exhibits a 0° phase shift is: f=\frac{1}{2\Pi RC}
• Condition for Oscillation: At the oscillation frequency, the feedback network has an attenuation factor of 3. The non-inverting amplifier’s gain must be at least 3 to sustain oscillations: 1+\frac{R_f}{R_1}\geq3 \quad \text{or} \quad R_{f}\geq2R_{1}

6.3 Waveform Generators

In addition to sine waves, op-amps can generate other standard waveforms, such as square and triangular waves.

Square Wave Generator

An op-amp square wave generator, also known as an astable multivibrator, uses both positive feedback to create hysteresis and negative feedback to set the timing. The circuit’s operation relies on the charging and discharging of a capacitor through the negative feedback resistor. The op-amp output, driven by the positive feedback, switches between the positive (+V_{sat}) and negative (-V_{sat}) saturation levels. When the output is at +V_{sat}, the capacitor charges until its voltage exceeds the threshold set by the positive feedback network, causing the op-amp output to flip to -V_{sat}. The capacitor then discharges until its voltage drops below the negative threshold, causing the output to flip back to +V_{sat}, and the cycle repeats.

Triangular Wave Generator

A triangular wave can be generated by integrating a square wave. This is accomplished by cascading a square wave generator and an integrator circuit. The output of the square wave generator serves as the input to the integrator. As the square wave alternates between its positive and negative constant voltage levels, the integrator output ramps linearly up and down, producing a clean triangular waveform.

These signal generation and filtering circuits are crucial in signal processing. The final section of this handbook will explore how op-amps are used to bridge the gap between the analog and digital worlds through data conversion.