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

An electrical filter is a frequency-selective circuit that allows signals within a desired frequency range to pass while attenuating signals outside that range.

Classification according to frequency response:

• Low-pass filter: Passes frequencies below a cutoff frequency and attenuates frequencies above it.
• High-pass filter: Passes frequencies above and attenuates frequencies below it.
• Band-pass filter: Passes frequencies between a lower cutoff frequency and an upper cutoff frequency .
• Band-stop filter: Rejects frequencies between and while passing frequencies below and above this range.

The cutoff frequency of a first-order RC filter is

Filters are used for noise removal, signal separation, frequency selection, waveform shaping, and communication-system channel selection.

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

The capacitor impedance is

Using the voltage-divider rule,

Therefore,

Its magnitude is

At the cutoff frequency, the output magnitude is times its passband value. Hence,

• For , the capacitor offers high reactance and .
• For , the capacitor offers low reactance and the output is attenuated.
• Beyond cutoff, the response falls at approximately dB/decade.

A passive RC high-pass filter contains a capacitor in series with the input and a resistor connected from the output to ground. The output is measured across the resistor.

Using the voltage-divider rule,

Thus,

The magnitude response is

The cutoff frequency is

Frequency response:

• At very low frequencies, the capacitor behaves nearly as an open circuit, so .
• At the cutoff frequency, the output is times its high-frequency value.
• At high frequencies, the capacitor behaves nearly as a short circuit, so .
• Below cutoff, the gain increases at approximately dB/decade.

A band-pass filter passes signals within a selected frequency band and attenuates signals below and above that band. It may be formed by combining high-pass and low-pass sections such that .

The important parameters are:

• Lower cutoff frequency:
• Upper cutoff frequency:
• Bandwidth:

• Center or resonant frequency:

• Quality factor:

A high value of indicates a narrow and highly selective passband, while a low value indicates a wide passband. At and , the output power is half its maximum value and the voltage gain is times the maximum gain.

An operational amplifier abstraction represents the op-amp as a high-gain differential voltage amplifier having two input terminals and one output terminal. The non-inverting input is marked and the inverting input is marked .

The open-loop relation is

where is the open-loop voltage gain.

Ideal op-amp assumptions:

• Infinite open-loop gain:
• Infinite input resistance:
• Zero output resistance:
• Infinite bandwidth
• Infinite slew rate
• Infinite common-mode rejection ratio
• Infinite power-supply rejection ratio
• Zero input offset voltage
• Zero input bias and offset currents

For a negatively fed-back ideal op-amp operating in its linear region, the two useful rules are and .

Ideal and practical op-amps differ as follows:

• Open-loop gain: Ideal gain is infinite; practical gain is very high but finite, typically to at low frequency.
• Input resistance: Ideal input resistance is infinite; practical input resistance is finite but high.
• Output resistance: Ideal output resistance is zero; practical output resistance is small but nonzero.
• Bandwidth: An ideal op-amp has infinite bandwidth; a practical op-amp has a limited gain-bandwidth product.
• Slew rate: An ideal op-amp responds instantaneously; a practical op-amp has a finite maximum rate of output change.
• Input offset: The ideal value is zero; a practical device has input offset voltage and bias currents.
• CMRR: The ideal common-mode rejection ratio is infinite; the practical value is finite.
• Output swing: An ideal output can reach any required value; a practical output is limited by its supply voltages.
• Noise and drift: These are absent in an ideal op-amp but present in practical devices.

These non-ideal properties limit the accuracy, speed, frequency range, and output capability of real op-amp circuits.

A virtual ground is a point that remains approximately at ground potential even though it is not physically connected to ground.

Consider an ideal op-amp with its non-inverting terminal grounded and negative feedback applied. Since the open-loop gain is extremely high, linear operation requires

Because , it follows that

Therefore, the inverting input becomes a virtual-ground node.

Important points:

• It has approximately zero voltage but is not directly connected to ground.
• It cannot act as a general-purpose current source or current sink.
• Ideally, no current enters the op-amp input.
• Currents entering the node through external components must leave through other external paths, usually the feedback network.

The concept is valid only when negative feedback is present, the op-amp operates in its linear region, and the output is not saturated.

In an inverting amplifier, the input signal is connected to the inverting terminal through , the non-inverting terminal is grounded, and a feedback resistor connects the output to the inverting terminal.

For an ideal op-amp with negative feedback,

The input current is

Since no current enters the op-amp, the same current flows through :

Equating the currents,

Therefore,

The negative sign shows that the output is out of phase with the input. The input resistance seen by the source is approximately , and the gain is determined accurately by the resistor ratio.

In a non-inverting amplifier, the input is applied to the non-inverting terminal. The inverting terminal is connected to a feedback divider consisting of from output to the inverting terminal and from the inverting terminal to ground.

For an ideal op-amp,

The feedback-divider voltage is

Therefore,

Hence, the closed-loop gain is

Characteristics:

• The output is in phase with the input.
• The minimum closed-loop gain is unity.
• It has very high input resistance.
• Its gain is controlled by the feedback-resistor ratio.
• When and is omitted, the circuit becomes a voltage follower with .

An op-amp adder produces an output proportional to the algebraic sum of two or more input voltages. In an inverting summing amplifier, input voltages , , and are applied to the inverting terminal through , , and . The non-inverting terminal is grounded.

By the virtual-ground principle, the input currents are

The total current flows through the feedback resistor . Therefore,

If all input resistors are equal to ,

For ,

The circuit is used in audio mixing, digital-to-analog conversion, signal weighting, and analog computation.

An op-amp subtractor, also called a difference amplifier, produces an output proportional to the difference between two input voltages.

Let be applied to the inverting input through , with feedback resistor . Let be applied to the non-inverting input through a divider formed by and .

For correct subtraction, the resistor ratios must satisfy

Under this matched condition, the output is

If all four resistors are equal,

Important features:

• It amplifies the differential component of the inputs.
• Common signals ideally cancel.
• Accurate resistor matching is required for good common-mode rejection.
• It is used in sensor interfaces, bridge circuits, measurement systems, and signal conditioning.

An ideal op-amp integrator is an inverting circuit with an input resistor and a feedback capacitor . The non-inverting terminal is grounded.

The input current is

Since no current enters the op-amp, this current flows through the feedback capacitor. For the virtual-ground node,

Therefore,

Integrating with respect to time gives

Its transfer function is

A constant input produces a ramp output, while a square-wave input produces an approximately triangular output. A practical integrator includes a resistor in parallel with the feedback capacitor to limit low-frequency and DC gain.

An ideal differentiator has an input capacitor and a feedback resistor . Its non-inverting terminal is grounded.

The current through the input capacitor is

Since the op-amp input draws no current, this current flows through the feedback resistor. Thus,

Therefore,

The transfer function is

A ramp input produces a constant output, and rapid input changes produce large output pulses.

An ideal differentiator strongly amplifies high-frequency noise and may become unstable. A practical differentiator adds a resistor in series with the input capacitor and a small capacitor in parallel with the feedback resistor. These components limit the high-frequency gain, improve stability, and reduce noise sensitivity.

Op-amp RC circuits combine resistors, capacitors, and operational amplifiers to obtain frequency-dependent gain or time-domain operations. Capacitor impedance varies with frequency according to

Therefore, changing frequency changes the feedback or input impedance and hence the closed-loop gain.

Main op-amp RC circuits include:

• Active low-pass filter: Attenuates high-frequency signals.
• Active high-pass filter: Attenuates low-frequency and DC signals.
• Active band-pass filter: Passes a selected frequency band.
• Active band-stop filter: Rejects a selected frequency band.
• Integrator: Produces an output proportional to the time integral of the input.
• Differentiator: Produces an output proportional to the rate of change of the input.

Op-amp RC circuits can provide gain and buffering in addition to frequency selection. Their performance is limited by practical properties such as bandwidth, slew rate, output swing, and component tolerances.

An active filter uses an active device such as an op-amp along with resistors and capacitors. It can provide amplification, buffering, and frequency selection without requiring inductors.

A first-order active high-pass filter is formed by connecting an RC high-pass network to the non-inverting input of an op-amp. The op-amp is commonly configured as a non-inverting amplifier.

The RC section has the transfer function

The non-inverting passband gain is

Therefore, the complete transfer function is

The cutoff frequency is

Below , the gain rises at dB/decade. Above , the gain approaches . Active filters offer high input impedance, low output impedance, and controllable passband gain.

An active band-pass filter can be constructed by cascading an active high-pass filter with an active low-pass filter and including op-amp amplification or buffering.

• The high-pass section determines the lower cutoff frequency:

• The low-pass section determines the upper cutoff frequency:

For proper band-pass operation,

The bandwidth and center frequency are

Operation:

• Frequencies below are attenuated by the high-pass section.
• Frequencies between and are passed and may be amplified.
• Frequencies above are attenuated by the low-pass section.

Active band-pass filters are used in communication receivers, audio equalizers, biomedical instruments, and frequency-selective measurement circuits.

An active band-stop filter, also called a band-reject filter, attenuates a selected range of frequencies between and while passing frequencies outside this range. It can be constructed by combining low-pass and high-pass signal paths with a summing amplifier. A narrow band-stop filter is often called a notch filter.

Its important parameters are

Difference from a band-pass filter:

• A band-pass filter passes frequencies from to and rejects frequencies outside that range.
• A band-stop filter rejects frequencies from to and passes frequencies outside that range.
• The band-pass response is maximum near , whereas the band-stop response is minimum near .

Band-stop filters are used to remove power-line interference, suppress unwanted tones, eliminate communication interference, and reject mechanical resonance frequencies.

An op-amp comparator operates in open-loop mode and compares an input voltage with a reference voltage. Because the open-loop gain is very high, even a small differential input drives the output toward one of its saturation levels.

The output behavior is approximately

Non-inverting comparator:

• The signal is applied to and the reference to .
• The output is high when .
• The output is low when .

Inverting comparator:

• The signal is applied to and the reference to .
• The output is low when .
• The output is high when .

A comparator is used in zero-crossing detectors, level detectors, alarm circuits, waveform generators, and sensor interfaces. Positive feedback may be added to create hysteresis and prevent rapid output switching in noisy conditions.

An anti-lock braking system (ABS) prevents the wheels of a vehicle from locking during heavy braking. Wheel-speed sensors generate electrical signals whose frequency or amplitude represents wheel speed.

Role of op-amp comparators:

• Sensor signals are first amplified and filtered to reduce noise.
• Comparators convert the conditioned analog sensor signals into clean digital pulses.
• A reference level is applied to one comparator input, while the wheel-sensor signal is applied to the other.
• The comparator switches whenever the sensor signal crosses the reference level.
• The resulting pulse frequency allows the controller to calculate wheel speed.
• The controller compares the speeds and deceleration rates of different wheels.
• If one wheel decelerates too rapidly or approaches lock, the hydraulic modulator temporarily reduces brake pressure at that wheel.
• Pressure is then restored and adjusted repeatedly to maintain traction.

Hysteresis is often included in the comparator to prevent false transitions caused by electrical noise, vibration, or small sensor-signal variations.

Passive filters use only resistors, capacitors, and inductors, whereas active filters use an active device such as an op-amp together with resistors and capacitors.

Comparison:

• Gain: Passive filters cannot provide power gain; active filters can amplify the passband signal.
• Components: Passive filters may require bulky inductors; op-amp active filters commonly use only resistors and capacitors.
• Input impedance: Passive-filter impedance depends directly on the network; active filters can provide high input impedance.
• Output impedance: Passive output impedance may be significant; active filters can provide low output impedance.
• Loading: Passive stages may load each other; op-amp buffering reduces loading between active-filter stages.
• Frequency range: Op-amp active filters are well suited to low and moderate frequencies but are limited by op-amp bandwidth and slew rate. Passive filters can operate at very high frequencies.
• Power supply: Passive filters require no external power, while active filters require a DC supply.
• Applications: Active filters are common in audio, instrumentation, and control systems. Passive filters are common in power circuits, radio-frequency circuits, and simple signal networks.