Unit 5: Fundamentals of filters and operational amplifier

I. Orientation

Filters selectively pass or suppress frequency components, while an operational amplifier is a high-gain differential voltage amplifier used with feedback to perform amplification, filtering, comparison, and mathematical operations.

• Frequency convention: Frequency is measured in hertz (Hz), and angular frequency is ω = 2πf rad/s.
• Transfer function: A circuit is characterized by H(jω) = Vout/Vin, where Vin and Vout are input and output phasor voltages.
• Cutoff convention: At cutoff frequency fc, voltage gain falls to 1/√2 ≈ 0.707 of its passband value, corresponding to −3 dB.
• Reactive components:
  • Capacitive reactance is XC = 1/(2πfC).
  • Inductive reactance is XL = 2πfL.
• Op-amp convention: The output depends on the differential input V+ − V−, where V+ is non-inverting and V− is inverting input voltage.
• Feedback assumption: Most linear op-amp circuits use negative feedback to obtain stable, resistor-controlled gain.

II. Frequency-Selective Networks

A. Band-pass filter

A band-pass filter passes frequencies between a lower cutoff fL and an upper cutoff fH while attenuating frequencies outside this interval.

• Passband: The useful frequency range is fL < f < fH.
• Bandwidth:

BW = fH - fL

Here, BW is bandwidth in Hz, while fH and fL are upper and lower cutoff frequencies.

• Centre frequency:

f0 = √(fL fH)
Q = f0 / BW

Here, f0 is centre frequency and Q is quality factor; a larger Q indicates a narrower, more selective passband.

• Construction: A simple passive band-pass response can be obtained by cascading high-pass and low-pass sections, provided their loading interaction is controlled.
• Applications: Radio channel selection, audio equalization, sensor-signal extraction, and noise rejection.

B. Low-pass filter

A low-pass filter passes low-frequency signals and attenuates frequencies above its cutoff frequency.

• RC circuit: A resistor in series and capacitor to ground, with output across the capacitor, forms a first-order low-pass filter.
• Cutoff frequency:

fc = 1 / (2πRC)
H(jω) = 1 / (1 + jωRC)

Here, R is resistance in ohms, C is capacitance in farads, j = √(-1), and ω is angular frequency.

• Response: Above fc, a first-order filter decreases at approximately 20 dB/decade.
• Phase: The output phase approaches −90° at very high frequencies.
• Application: It smooths rectifier outputs, removes high-frequency noise, and reconstructs slowly varying signals.

C. High-pass filter

A high-pass filter passes frequencies above its cutoff and suppresses DC and low-frequency components.

• RC circuit: A series capacitor followed by a resistor to ground, with output across the resistor, gives a first-order high-pass response.
• Transfer relation:

fc = 1 / (2πRC)
H(jω) = jωRC / (1 + jωRC)

Here, all symbols have their usual RC-filter meanings.

• Response: Below fc, gain rises at approximately 20 dB/decade; at high frequency, gain approaches unity.
• DC blocking: At f = 0, the capacitor behaves as an open circuit, so the output is zero.
• Applications: AC coupling, removal of sensor drift, and separation of rapid changes from slowly varying signals.

III. Operational-Amplifier Foundations

A. Operational amplifier abstraction

The op-amp abstraction treats the device as a differential voltage-controlled voltage source without requiring analysis of its internal transistor stages.

• Open-loop equation:

Vout = AOL(V+ - V-)

Here, AOL is open-loop voltage gain and Vout is limited by the supply rails.

• Ideal model: AOL → ∞, input resistance Rin → ∞, and output resistance Rout → 0.
• Input terminals: A positive change at V+ drives the output positive, while a positive change at V− drives it negative.
• Linear use: Negative feedback keeps the differential input extremely small and prevents uncontrolled saturation.

B. Device properties of the operational amplifier

A physical op-amp approximates the ideal abstraction but has finite electrical and dynamic limits.

• Input offset voltage: A small differential voltage, often in microvolts or millivolts, may be required to make Vout = 0.
• Input bias current: Small currents flow into or out of both input terminals and create errors across source resistances.
• Gain-bandwidth product: Closed-loop bandwidth generally decreases as closed-loop gain increases.
• Slew rate: Maximum output slope is specified in V/µs; exceeding it distorts fast, large-amplitude signals.
• Output limits: Output voltage and current cannot exceed values allowed by the supply rails and output stage.

C. Properties of operational amplifier

The principal op-amp properties determine its accuracy, speed, stability, and compatibility with external circuits.

• Common-mode rejection ratio: CMRR measures rejection of voltage common to both inputs; larger CMRR improves differential measurement.
• Power-supply rejection ratio: PSRR describes immunity to changes or noise in supply voltage.
• Input and output impedance: High input impedance minimizes source loading, while low output impedance supports the load.
• Frequency response: Internal compensation usually causes open-loop gain to decrease as frequency rises.
• Supply operation: Devices may use dual supplies such as ±15 V or a single supply such as 0–5 V.

D. Simple op-amp circuits

Simple op-amp circuits use feedback and impedance networks to control gain and signal behavior.

• Open-loop operation: Without negative feedback, even a small differential voltage normally drives the output toward positive or negative saturation.
• Closed-loop operation: Feeding part of Vout to V− establishes negative feedback and predictable gain.
• Voltage follower: Connecting Vout directly to V− and applying the signal to V+ gives unity gain.
• Buffering: A follower isolates a high-resistance source from a low-resistance load.

E. Virtual ground concept

A virtual ground is a feedback-maintained node whose voltage is approximately zero even though it is not physically connected to ground.

• Input equality: For an ideal op-amp in linear negative-feedback operation, V+ ≈ V−.
• Ground condition: If V+ = 0 V, feedback makes V− ≈ 0 V.
• No current path: The virtual-ground node cannot supply arbitrary current because ideal input current is zero.
• Circuit consequence: Current entering the inverting node through one component must leave through another component, enabling nodal analysis.

IV. Basic Amplifier and Arithmetic Circuits

A. Inverting op-amp

An inverting amplifier produces an amplified output with a 180° phase reversal.

• Gain equation:

Vout / Vin = -Rf / Rin

Here, Rf is feedback resistance and Rin is input resistance.

• Current relation: With the inverting node at virtual ground, input current is Vin/Rin and ideally flows through Rf.
• Impedance: The source sees approximately Rin.
• Example: If Rin = 10 kΩ, Rf = 50 kΩ, and Vin = 0.2 V, then Vout = −1.0 V.

B. Non-inverting op-amp

A non-inverting amplifier produces an output with the same polarity as its input.

• Gain equation:

Vout / Vin = 1 + Rf/Rg

Here, Rf connects output to the inverting input, and Rg connects the inverting input to ground.

• Minimum gain: The closed-loop gain cannot be less than unity with this standard resistor arrangement.
• Input impedance: The signal is applied directly to V+, giving very high input impedance.
• Follower limit: Direct feedback with no divider gives Vout = Vin.

C. Op-amp as an adder

An op-amp adder, or summing amplifier, combines several input voltages into one weighted output.

• Inverting summer:

Vout = -Rf(V1/R1 + V2/R2 + ... + Vn/Rn)

Here, V1…Vn are input voltages, R1…Rn their input resistors, and Rf is feedback resistance.

• Equal weighting: If every input resistor equals Rf, then Vout = −(V1 + V2 + ... + Vn).
• Weighted addition: Different resistor ratios implement scaling, as required in audio mixers and digital-to-analog converters.

D. Op-amp as a subtractor

An op-amp subtractor amplifies the difference between two input voltages while rejecting their common component.

• Difference equation:

Vout = (R2/R1)(V2 - V1)

This relation holds when the two resistor pairs have matched ratios; V2 is the non-inverting-side signal and V1 is the inverting-side signal.

• Unity subtraction: When R2 = R1, the output is V2 − V1.
• Accuracy: Resistor-ratio matching determines subtraction accuracy and common-mode rejection.
• Applications: Bridge sensors, balanced signal receivers, and measurement of voltage differences.

V. Active Frequency Filters

A. Active filter

An active filter combines resistors and capacitors with an amplifying device, normally an op-amp, to shape frequency response.

• Advantages: It can provide voltage gain, buffering, high input impedance, and low output impedance.
• Component choice: Low-frequency active filters avoid bulky inductors by using only R, C, and an op-amp.
• Order: Each first-order section contributes approximately 20 dB/decade; a second-order section contributes 40 dB/decade.
• Limitation: Performance is constrained by op-amp bandwidth, slew rate, noise, and output range.

B. Active high-pass filter

An active high-pass filter combines an RC high-pass network with op-amp gain and isolation.

• Cutoff:

fc = 1/(2πRC)
AF = 1 + Rf/Rg

Here, AF is non-inverting passband gain; the remaining symbols denote filter and feedback components.

• Operation: Low frequencies are blocked by the input capacitor, while frequencies well above fc receive approximately gain AF.
• Design requirement: The op-amp gain-bandwidth product must substantially exceed the highest operating frequency multiplied by the required gain.

C. Active band-pass filter

An active band-pass filter passes a selected frequency band while providing buffering or gain.

• Realization: Cascading active high-pass and low-pass sections gives approximate cutoff frequencies fL and fH.
• Centre and bandwidth:

f0 = √(fL fH)
BW = fH - fL

Here, f0 is the geometric centre frequency and BW is the passband width.

• Selectivity: Increasing Q = f0/BW narrows the passed frequency range.
• Applications: Biomedical signal conditioning, communication receivers, and tone detection.

D. Active band-stop filter

An active band-stop, or notch, filter rejects a chosen frequency range while passing frequencies below and above it.

• Response: Attenuation is greatest near the notch frequency f0.
• Realization: Low-pass and high-pass paths may be summed, or a twin-T RC network may be buffered by an op-amp.
• Notch condition: In a balanced twin-T design, component ratios establish deep cancellation at approximately f0 = 1/(2πRC).
• Application: A notch near 50 Hz or 60 Hz suppresses mains interference in instrumentation.

VI. Time-Domain RC Operations

A. Op-amp RC circuits

Op-amp RC circuits use frequency-dependent capacitor impedance inside input or feedback paths.

• General principle: Replacing a resistor with a capacitor makes closed-loop gain dependent on frequency.
• Capacitor relation:

iC = C(dvC/dt)

Here, iC is capacitor current, C is capacitance, vC is capacitor voltage, and t is time.

• Uses: RC feedback supports integration, differentiation, filtering, waveform shaping, and timing.
• Practical design: Added resistors limit DC gain and high-frequency noise amplification.

B. Op-amp integrator

An op-amp integrator produces an output proportional to the time integral of the input.

• Ideal equation:

Vout(t) = -(1/RC) ∫ Vin(t) dt + Vout(0)

Here, R is input resistance, C is feedback capacitance, and Vout(0) is initial output voltage.

• Waveform action: A constant input produces a linear ramp; a square wave produces an approximately triangular output.
• Practical circuit: A resistor placed parallel to the feedback capacitor limits low-frequency and DC gain, reducing saturation caused by offsets.
• Applications: Analog computation, ramp generation, and low-pass signal processing.

C. Op-amp differentiator

An op-amp differentiator produces an output proportional to the input signal’s rate of change.

• Ideal equation:

Vout(t) = -RC[dVin(t)/dt]

Here, R is feedback resistance, C is input capacitance, and t is time.

• Waveform action: Rapid transitions produce large output pulses, while a constant input ideally produces zero output.
• Practical limitation: Gain increases with frequency, strongly amplifying noise and risking instability.
• Improvement: Series input resistance and parallel feedback capacitance restrict operation to a useful frequency band.

VII. Comparison and Control

A. Op-amp as a comparator

An op-amp comparator determines which of two input voltages is greater and drives its output toward a supply limit.

• Decision rule:
  1. If V+ > V−, then Vout approaches positive saturation.
  2. If V+ < V−, then Vout approaches negative saturation or ground in a single-supply circuit.
• Reference comparison: Applying a fixed Vref to one input allows detection of whether a sensor voltage is above or below a threshold.
• Hysteresis: Positive feedback creates separate switching thresholds and prevents noisy signals from causing repeated output transitions.
• Limitation: General-purpose op-amps may recover slowly from saturation; dedicated comparators are preferred for high-speed switching.

B. Application of op-amp comparator in anti-lock braking systems

In a simplified anti-lock braking system, comparator stages can identify excessive wheel deceleration or wheel-speed differences associated with impending lock.

• Sensing: Wheel-speed sensors produce voltages or pulses related to wheel rotational speed.
• Conditioning: Filters suppress electrical noise, and op-amp circuits amplify or convert the sensor signal into a usable voltage.
• Comparison: A comparator tests the conditioned wheel signal against a reference representing acceptable wheel slip or deceleration.
• Control action: When the threshold is crossed, the electronic controller commands hydraulic valves to reduce brake pressure; pressure is then restored when traction recovers.
• Closed-loop cycle: Measurement, comparison, pressure reduction, and reapplication repeat rapidly to maintain wheel rotation and steering control.
• Practical architecture: Modern ABS primarily uses digital controllers and dedicated interface circuits, but the op-amp comparator illustrates the essential threshold-detection principle.