Unit 6: Active Filters and Wave Shaping Circuits - Subjective Questions

Active Filter: An active filter is a frequency-selective circuit that uses active components (such as op-amps or transistors) along with passive components (resistors and capacitors) to pass or reject certain bands of frequencies.

Advantages:

• Gain and frequency adjustment: Op-amps provide gain, so the signal is not attenuated as in passive filters.
• No loading effect: High input impedance and low output impedance prevent loading of the source and load.
• Cost effective: No bulky and expensive inductors are required.
• Small size and light weight: Absence of inductors makes them compact.
• Easy tuning: Cut-off frequency can be easily varied by changing R and C.

Disadvantages:

• Limited bandwidth: Performance depends on the bandwidth of the op-amp.
• Requires power supply: Active devices need a DC power source.
• Not suitable for high frequency/high power: Op-amps cannot handle very high frequencies or large power levels.

Circuit: A first order low pass Butterworth filter consists of a single RC network connected to the non-inverting input of an op-amp, with and providing the pass band gain.

Derivation:

The voltage at the non-inverting terminal is obtained by the RC voltage divider:

Since the op-amp is in non-inverting configuration with gain :

Therefore the transfer function is:

where the high cut-off frequency is:

Magnitude of gain:

Behaviour:

• When : gain (pass band)
• When : gain (–3 dB point)
• When : gain decreases at –20 dB/decade.

Circuit: A first order high pass filter is obtained by interchanging the positions of R and C in the low pass filter. The capacitor is placed in series with the signal and the resistor to ground, feeding the non-inverting terminal of the op-amp.

Working:

• At low frequencies, the reactance of the capacitor is very high, so the signal is blocked (attenuated).
• At high frequencies, becomes small, allowing the signal to pass to the op-amp.

Transfer function:

where the low cut-off frequency is:

and is the pass band gain.

Frequency Response:

• Below : gain increases at +20 dB/decade.
• At : gain is (–3 dB point).
• Above : gain is constant at (pass band).

Band Pass Filter: A band pass filter passes a band of frequencies between a lower cut-off frequency and a higher cut-off frequency , while rejecting frequencies outside this band.

Wide Band Pass Filter Construction:

• Formed by cascading a high pass filter and a low pass filter.
• The high pass filter sets the lower cut-off frequency .
• The low pass filter sets the higher cut-off frequency .
• Condition: .

Working:

• Frequencies below are blocked by the high pass section.
• Frequencies above are blocked by the low pass section.
• Only the band between and passes through.

Bandwidth:

Centre Frequency:

Quality Factor:

A wide band pass filter has , while a narrow band pass filter has .

Band Reject Filter: Also called a band stop or notch filter, it rejects (attenuates) a particular band of frequencies while passing all other frequencies above and below that band.

Types:

• Wide Band Reject Filter: Constructed by combining a low pass filter, a high pass filter, and a summing amplifier.
• Narrow Band Reject (Notch) Filter: Uses a Twin-T network (two T-shaped RC networks) with an op-amp.

Working of Wide Band Reject Filter:

• The low pass filter cut-off frequency must be lower than the high pass filter cut-off frequency .
• The outputs of both are summed. Frequencies in the stop band are attenuated by both filters.

Notch Frequency:

Frequency Response: The gain is high in the pass bands and drops sharply to a minimum at the notch frequency .

Applications:

• Elimination of 50/60 Hz power line hum in instrumentation.
• Removing unwanted frequencies in biomedical instruments (e.g., ECG).
• Communication systems to reject a specific interfering frequency.

All Pass Filter: An all pass filter passes all frequencies equally in magnitude (constant gain), but introduces a predictable phase shift for different frequencies. It is also called a delay equalizer or phase corrector.

Purpose: Used to introduce a desired amount of phase shift without altering the amplitude of the signal.

Transfer Function:

Magnitude:

Phase Shift:

Explanation:

• The phase angle varies from to as frequency increases from to .
• By interchanging R and C, the phase shift can be made positive (leading).

Applications: Phase compensation, signal delay, and correcting phase distortion in communication systems.

Square Wave Generator: An op-amp square wave generator (astable multivibrator) produces a square wave output without any external triggering. It uses positive feedback (through resistors and ) and a negative feedback RC timing network.

Working:

• The output swings between and .
• A fraction of the output is fed back to the non-inverting terminal, setting the threshold voltages .
• The capacitor C charges through R toward the output voltage. When the capacitor voltage reaches the threshold, the output switches state.
• This continuous charging and discharging produces a square wave.

Waveforms:

• Output: square wave between and .
• Capacitor voltage: exponential triangular-like (charging/discharging) waveform.

Derivation of Time Period:

The capacitor charges/discharges between and . Solving the RC charging equation gives:

Substituting :

Frequency of Oscillation:

If , then and .

Triangular Wave Generator: A triangular wave is generated by integrating a square wave. Hence the circuit consists of a comparator (square wave generator) followed by an integrator.

Block Diagram:

Comparator (Square Wave) → Integrator → Triangular Output

Working:

• The first op-amp acts as a comparator/square wave generator, producing a square wave.
• The second op-amp acts as an integrator. When a constant DC (square wave level) is applied to the integrator, its output ramps linearly.
• A positive square wave input produces a negative-going ramp, and a negative input produces a positive-going ramp, resulting in a triangular waveform.
• Feedback from the integrator output back to the comparator determines the switching, keeping oscillation continuous.

Peak Amplitude of Triangular Wave:

Frequency of Oscillation:

Note: The triangular wave has equal rise and fall times (symmetrical), unlike a sawtooth wave.

Key Point: A sawtooth wave is obtained from a triangular wave generator by making the charging and discharging times of the integrating capacitor unequal, usually by adding a diode and a potentiometer (variable resistor) in the integrator feedback path.

Voltage Controlled Oscillator (VCO): A VCO is an oscillator whose output frequency is controlled by an external DC input (control) voltage. As the control voltage increases, the output frequency increases (or decreases) linearly.

Principle of Operation:

• The control voltage determines the charging/discharging current of a capacitor.
• A higher control voltage charges the capacitor faster, increasing the switching rate and hence the output frequency.
• The IC NE566 is a popular VCO that produces simultaneous square and triangular wave outputs.

Output Frequency of NE566 VCO:

where is the control voltage, and , are the timing components.

Characteristics:

• Output frequency is linearly proportional to control voltage.
• Also called a voltage-to-frequency converter.

Applications:

• Frequency modulation (FM) and frequency shift keying (FSK).
• Phase Locked Loops (PLL).
• Function generators and tone generators.
• Frequency synthesizers and signal generators.

555 Timer IC: The 555 is an 8-pin timer IC widely used for timing, pulse generation, and oscillator applications. It contains two comparators, an SR flip-flop, a discharge transistor, and a voltage divider of three resistors (hence the name '555').

Pin Configuration (8-pin DIP):

• Pin 1 – Ground (GND): Connected to the negative supply/ground.
• Pin 2 – Trigger: Activates the timing cycle. When voltage falls below , output goes HIGH.
• Pin 3 – Output: The output terminal; swings between 0 and near .
• Pin 4 – Reset: Active LOW; resets the timer (output LOW) when grounded. Usually tied to .
• Pin 5 – Control Voltage: Allows external control of the threshold voltage. Usually connected to ground through a capacitor.
• Pin 6 – Threshold: When voltage exceeds , output goes LOW.
• Pin 7 – Discharge: Connected to the discharge transistor; discharges the external timing capacitor.
• Pin 8 – : Positive supply voltage (typically +5V to +18V).

Internal Reference Levels:

• Upper comparator triggers at .
• Lower comparator triggers at .

The 555 Timer operates in three main modes:

1. Monostable Mode (One-Shot):

• Has one stable state and one quasi-stable state.
• Produces a single output pulse of fixed width when triggered.
• The pulse width (time delay) is:

• Applications: Timers, pulse generation, missing pulse detection, switch debouncing.

2. Astable Mode (Free-Running):

• Has no stable state; it continuously oscillates between HIGH and LOW.
• Acts as an oscillator/square wave generator.
• Time periods:

• Frequency of oscillation:

• Duty cycle:

• Applications: Clock generation, LED flashers, tone generation.

3. Bistable Mode (Flip-Flop):

• Has two stable states; requires external triggers to change state.
• Acts as a flip-flop / latch (no timing capacitor needed).
• Applications: Latches, toggle switches, memory elements.

Astable Multivibrator using 555: In astable mode, the 555 timer runs freely, generating a continuous square wave without external triggering.

Circuit Connections:

• Two resistors and and a capacitor form the timing network.
• is connected between and pin 7 (Discharge).
• is connected between pin 7 and pins 6 & 2 (Threshold & Trigger joined together).
• Capacitor from pins 6/2 to ground.

Working:

• Charging: Capacitor charges through toward . Output is HIGH.
• When reaches , the discharge transistor turns ON.
• Discharging: Capacitor discharges through only. Output is LOW.
• When falls to , the cycle repeats.

Derivation:

Charging time (output HIGH), capacitor charges from to through :

Discharging time (output LOW) through :

Total Time Period:

Frequency:

Duty Cycle:

Note: Duty cycle is always greater than 50% in this basic configuration.

Monostable Multivibrator: The 555 in monostable mode produces a single output pulse of fixed duration each time it is triggered. It has one stable state (output LOW).

Circuit Connections:

• A resistor connected from to pins 6 & 7.
• A capacitor from pins 6/7 to ground.
• A negative trigger pulse applied at pin 2.

Working:

• In the stable state, the output is LOW and the discharge transistor keeps the capacitor discharged.
• When a negative trigger (< ) is applied to pin 2, the output goes HIGH.
• The capacitor now charges through R toward .
• When reaches , the output returns to LOW and the capacitor discharges rapidly.
• The output remains LOW until the next trigger.

Waveforms:

• Trigger: brief negative pulse.
• Output: rectangular pulse of width .
• Capacitor voltage: exponential rise up to then sharp fall.

Derivation of Pulse Width:

The capacitor charges from to :

Setting :

Solving:

This is the fixed output pulse width.

Summary:

• Butterworth: Best trade-off — flat pass band with reasonable roll-off. Most commonly used.
• Chebyshev: Best when a sharp transition is required and pass band ripple is acceptable.
• Bessel: Best when preserving the waveform shape (linear phase) is critical.

Given:

• ,

Step 1: Cut-off Frequency

Step 2: Pass Band Gain

Results:

• Cut-off frequency
• Pass band gain (or )

Second Order Low Pass Filter: A second order filter uses two RC networks with a single op-amp (Sallen-Key configuration), giving a steeper roll-off than the first order filter.

Differences from First Order Filter:

Transfer Function (magnitude):

Cut-off Frequency:

For equal components and :

Advantages of Second Order Filter:

• Steeper roll-off (–40 dB/decade) provides better selectivity.
• Closer to an ideal filter response.
• Better rejection of unwanted frequencies beyond cut-off.

Note: To maintain Butterworth (maximally flat) response, the pass band gain must be set to .

Narrow Band Pass Filter: A narrow band pass filter has a high quality factor () and passes a narrow band of frequencies centered around a resonant frequency . It uses a multiple feedback topology with a single op-amp.

Circuit:

• Uses one op-amp in the inverting configuration.
• Two feedback paths through capacitors and resistors provide the band pass response.
• Typically uses two capacitors and three resistors (, , ).

Working:

• At the center frequency , the circuit provides maximum gain.
• Above and below , the gain falls off sharply.

Key Expressions:

Center Frequency:

Quality Factor:

Gain at Center Frequency:

Advantage: One of the useful features is that can be changed without altering the bandwidth or gain by simply adjusting the resistor .

Applications: Used in communication receivers, tone detectors, and biomedical signal processing where selecting a specific frequency is required.

Recent Trends in Electronics:

• Switched Capacitor Filters: Analog filters implemented using capacitors and switches (clocks) instead of resistors. They allow programmable cut-off frequencies controlled by clock frequency and are ideal for IC integration.

• Digital Signal Processing (DSP): Many filtering functions are now performed digitally using DSP processors and FIR/IIR digital filters, offering flexibility, programmability, and no component drift.

• System on Chip (SoC): Integration of analog, digital, and RF circuits on a single chip, reducing size and cost.

• MEMS (Micro Electro-Mechanical Systems): Miniaturized sensors and RF filters used in mobile phones, IoT devices, and biomedical applications.

• Software Defined Radio (SDR): Filtering, modulation, and demodulation implemented in software, replacing fixed analog hardware.

• Low Power / Low Voltage Design: Growing demand for battery-operated and IoT devices drives ultra-low-power analog circuit design.

• CMOS Technology Scaling: Continued miniaturization enabling faster, denser, and cheaper mixed-signal circuits.

• AI/ML in Signal Processing: Machine learning algorithms used for adaptive filtering, noise cancellation, and pattern recognition.

• Programmable Analog (FPAA): Field Programmable Analog Arrays allow reconfigurable analog signal processing similar to FPGAs in the digital domain.

These trends reflect the movement toward integration, programmability, miniaturization, and low power consumption in modern electronics.

Given:

Step 1:

Step 2:

Step 3: Total Time Period

Step 4: Frequency

Step 5: Duty Cycle

Results:

• Duty cycle