Unit 5: Fundamentals of filters and operational amplifier - Subjective Questions

An electrical filter is a circuit that allows signals of certain frequencies to pass through while attenuating (blocking) signals of other frequencies. Filters are widely used in signal processing, communication, and audio systems.

Four basic types of filters:

• Low-Pass Filter (LPF): Allows frequencies below a cutoff frequency to pass and attenuates higher frequencies.
• High-Pass Filter (HPF): Allows frequencies above a cutoff frequency to pass and attenuates lower frequencies.
• Band-Pass Filter (BPF): Allows a specific band of frequencies between a lower cutoff and an upper cutoff to pass.
• Band-Stop Filter (BSF)/Notch Filter: Rejects a specific band of frequencies while passing all others.

The cutoff frequency is defined as the point where output power drops to half ( dB point), given by for simple RC filters.

A passive RC low-pass filter consists of a resistor in series with the input and a capacitor connected across the output.

Working:

• At low frequencies, the capacitor's reactance is very high, so most of the input voltage appears across the capacitor (output). Signal passes.
• At high frequencies, becomes very small, so the capacitor acts like a short circuit and the output voltage drops. Signal is attenuated.

Transfer Function Derivation:

Using the voltage divider rule with capacitor impedance :

Magnitude:

Cutoff frequency occurs when :

At , the output is of input (the dB point).

A passive RC high-pass filter consists of a capacitor in series with the input and a resistor connected across the output.

Working:

• At low frequencies, the capacitor's reactance is high, blocking the signal, so output is low.
• At high frequencies, is low (capacitor acts as a short), allowing the signal to pass to the output.

Transfer Function:

Cutoff frequency:

Applications:

• AC coupling / DC blocking in amplifiers
• Removing low-frequency noise/hum
• Audio systems (treble boost, tweeters in crossover networks)
• Edge detection in signal processing

A band-pass filter (BPF) passes a specific range of frequencies between a lower cutoff and an upper cutoff , while attenuating frequencies outside this range.

Formation:
A band-pass filter can be formed by cascading a high-pass filter and a low-pass filter:

• The high-pass filter sets the lower cutoff (blocks frequencies below ).
• The low-pass filter sets the upper cutoff (blocks frequencies above ).
• For proper operation, .

Key parameters:

• Bandwidth:
• Center frequency:
• Quality factor:

Frequency Response:

• Below : attenuated
• Between and : passed (passband)
• Above : attenuated

Applications: Radio/TV tuning, audio equalizers, communication receivers.

An operational amplifier (op-amp) is a high-gain, direct-coupled differential amplifier that amplifies the voltage difference between its two input terminals. It is one of the most versatile building blocks in analog electronics.

Symbol: A triangle with two inputs on the left and one output on the right.

Terminals:

• Inverting input (): Signal applied here produces an output that is out of phase (inverted).
• Non-inverting input (): Signal applied here produces an in-phase output.
• Output (): The amplified difference signal, given ideally by where is the open-loop gain.
• Power supply terminals ( and ): Provide the DC bias; output cannot exceed these supply rails.
• Offset null terminals: Used to nullify the output offset voltage.

The op-amp is typically used with negative feedback to control gain precisely.

Properties of an Ideal Op-Amp:

• Infinite open-loop gain:
• Infinite input impedance: (draws no input current)
• Zero output impedance: (acts as ideal voltage source)
• Infinite bandwidth: amplifies all frequencies equally
• Zero offset voltage: output is zero when inputs are equal
• Infinite CMRR (Common Mode Rejection Ratio): rejects common signals completely
• Infinite slew rate: output responds instantly to input changes

Comparison with Practical Op-Amp (e.g., 741):

The virtual ground concept applies to an op-amp operating with negative feedback where the non-inverting input is grounded.

Explanation:
For an ideal op-amp with negative feedback, two golden rules apply:

1. The input terminals draw no current (infinite input impedance).
2. The op-amp adjusts its output to make the voltage difference between inputs zero ().

If the non-inverting input () is connected to ground ( V), then the inverting input () is also forced to V by the feedback action.

Thus, the inverting node behaves like a ground — it is at zero potential — but no actual physical connection to ground exists. This is called a virtual ground.

Importance:

• Simplifies circuit analysis (node voltage known = 0 V).
• Allows easy derivation of gain for inverting amplifier, adder, integrator, differentiator.
• Ensures input current flows entirely through the feedback resistor.

Example (Inverting amplifier): Since , input current flows through , giving .

Inverting Amplifier Circuit:
The input signal is applied through resistor to the inverting input. A feedback resistor connects the output to the inverting input. The non-inverting input is grounded.

Derivation (using virtual ground):

Since (grounded) and , the inverting node is at virtual ground, so .

The current through the input resistor:

Since the op-amp draws no input current, this entire current flows through :

Applying KCL at the inverting node:

Output voltage:

Closed-loop gain:

The negative sign indicates a phase inversion between input and output.

Non-Inverting Amplifier Circuit:
The input signal is applied directly to the non-inverting input (). The feedback network consists of (from output to inverting input) and (from inverting input to ground).

Derivation:

Since and (ideal op-amp), we have .

The voltage at the inverting node due to the voltage divider:

Setting :

Output voltage:

Closed-loop gain:

Differences from Inverting Amplifier:

An op-amp adder (summing amplifier) produces an output proportional to the sum of multiple input voltages.

Circuit:
Several inputs are applied through resistors respectively to the inverting input. A common feedback resistor connects output to the inverting node. The non-inverting input is grounded.

Derivation (virtual ground, ):

Currents through each input resistor:

Applying KCL at the virtual ground node (no current enters op-amp):

Therefore:

Special case — equal resistors ():

This gives the direct (inverted) sum. Used in audio mixers and digital-to-analog converters.

An op-amp subtractor produces an output proportional to the difference between two input voltages.

Circuit:

• applied to inverting input through , with feedback resistor .
• applied to non-inverting input through , with to ground.

Derivation using superposition:

Voltage at non-inverting input (voltage divider):

Since , applying KCL at inverting node:

Solving:

Special case — all resistors equal ():

Thus the circuit gives a scaled or direct difference of inputs. Used in instrumentation amplifiers and differential signal processing.

An op-amp integrator produces an output that is the time integral of the input signal.

Circuit:
Input resistor in series with the inverting input, and a feedback capacitor between output and inverting input. Non-inverting input grounded.

Derivation (virtual ground, ):

Input current:

The capacitor current (charging current):

Since (no current into op-amp):

Rearranging and integrating:

where is the initial capacitor voltage.

Observations:

• A square wave input produces a triangular wave output.
• is the time constant.

Applications: Waveform generation, analog computers, ramp generators.

An op-amp differentiator produces an output proportional to the rate of change (derivative) of the input signal.

Circuit:
Input capacitor in series with the inverting input and a feedback resistor between output and inverting input. Non-inverting input grounded.

Derivation (virtual ground, ):

Current through capacitor:

Current through feedback resistor:

Since :

Output voltage:

Observations:

• A triangular wave input produces a square wave output.

Limitations:

• High-frequency noise amplification: gain increases with frequency, making it unstable and noisy.
• Prone to oscillations/instability.
• Practical differentiators add a series resistor with and a capacitor across to limit high-frequency gain.

An active filter is a filter circuit that uses active components (op-amps or transistors) along with resistors and capacitors to perform filtering, while also providing gain.

Comparison of Active and Passive Filters:

Advantages of active filters: No inductors, provide gain, good isolation, easy tuning, compact.

Disadvantages: Require power supply, limited by op-amp bandwidth, generate noise.

An active low-pass filter combines an RC low-pass network with an op-amp to pass low frequencies and attenuate high frequencies, while also providing gain.

Circuit (first-order non-inverting):

• An RC low-pass network ( and ) feeds the non-inverting input.
• The op-amp is configured as a non-inverting amplifier with feedback resistors and .

Working:

• At low frequencies: capacitor reactance is high, signal passes to the amplifier → amplified output.
• At high frequencies: capacitor shorts the signal to ground → output attenuated.

Transfer Function:

where pass-band gain:

Cutoff frequency:

• Below : gain (passband).
• Above : gain rolls off at dB/decade (first order).

Applications: Audio processing, noise removal, anti-aliasing filters.

An active high-pass filter uses an op-amp with an RC high-pass network to pass high frequencies while attenuating low frequencies, and provides gain in the passband.

Circuit (first-order non-inverting):

• A capacitor in series with a resistor (to ground) forms the high-pass network feeding the non-inverting input.
• The op-amp with and provides the non-inverting gain.

Operation:

• At low frequencies: capacitor reactance is high, blocking the signal → output attenuated.
• At high frequencies: is low, signal passes to the amplifier → amplified output.

Transfer Function:

Pass-band gain:

Cutoff frequency:

• Below : gain rises at dB/decade.
• Above : gain is constant at .

Applications: Removing DC offset, audio treble boost, high-frequency signal selection.

An active band-pass filter (BPF) passes a specific band of frequencies between a lower cutoff and upper cutoff while attenuating frequencies outside this band.

Construction:
A common approach cascades an active high-pass filter and an active low-pass filter:

• The high-pass stage sets the lower cutoff frequency .
• The low-pass stage sets the upper cutoff frequency (with ).
• Op-amps also provide gain and isolation between stages.

Working:

• Below : blocked by high-pass stage.
• Between and : passed (passband).
• Above : blocked by low-pass stage.

Key parameters:

• Lower cutoff:
• Upper cutoff:
• Bandwidth:
• Center frequency:
• Quality factor:

Applications: Radio receivers, audio equalizers, biomedical signal (ECG/EEG) processing.

An active band-stop filter (also called a band-reject or notch filter) attenuates a specific band of frequencies while passing all frequencies outside this band.

Construction:
It can be realized by combining a low-pass filter and a high-pass filter whose outputs are summed using an op-amp adder:

• The low-pass filter passes frequencies below .
• The high-pass filter passes frequencies above .
• A summing amplifier adds both outputs, so the band between and is rejected.

Working:

• Below : passed by LPF.
• Between and : rejected (stopband).
• Above : passed by HPF.

Key parameters:

• Center (notch) frequency:
• Bandwidth of rejection:
• Quality factor: (higher = sharper notch)

Applications:

• Removing 50/60 Hz power line hum from signals
• Eliminating interference in audio and communication systems
• Removing specific noise frequencies in biomedical (ECG) instruments

An op-amp comparator compares two input voltages and produces a digital (saturated) output indicating which input is larger. It is operated in open-loop (no feedback) mode, so the very high gain drives the output to saturation.

Operation:

• Output: when
• Output: when

The output switches sharply as the input crosses the reference voltage .

Non-Inverting Comparator:

• Input signal applied to non-inverting () terminal; to inverting terminal.
• When → output = .
• When → output = .

Inverting Comparator:

• Input signal applied to inverting () terminal; to non-inverting terminal.
• When → output = .
• When → output = .

Applications: Zero-crossing detectors, level detectors, analog-to-digital converters, Schmitt triggers.

In an Anti-Lock Braking System (ABS), an op-amp comparator plays a key role in detecting wheel lock-up and preventing skidding during braking.

Role of the Comparator:
The comparator continuously compares two signals:

• A voltage proportional to the actual wheel speed (from a wheel-speed sensor).
• A reference voltage proportional to the expected vehicle speed/deceleration threshold.

Working Principle:

• During normal braking, wheel speed decreases smoothly, and the sensor voltage stays close to the reference.
• If a wheel is about to lock up (decelerating too fast), its speed voltage drops below the reference voltage.
• The comparator detects this condition: , switching its output state (e.g., to or logic HIGH signal to the controller).
• This output signals the ABS control unit to momentarily release and reapply brake pressure (pulsing the brakes) to that wheel.
• This prevents the wheel from locking, maintaining traction and steering control.

Benefits:

• Prevents skidding and loss of steering control.
• Reduces stopping distance on slippery surfaces.
• Enhances vehicle safety.

The comparator thus acts as a fast decision-making element, converting the analog speed comparison into a digital control signal for the braking actuator.