Unit 5: Fundamentals of filters and operational amplifier

ECE131 — Basic Electrical And Electronics Engineering 7 min read

Filters shape a signal's frequency content and op-amps provide the active gain that makes high-performance filtering, arithmetic, and decision circuits possible. This unit moves from passive RC filters through the ideal op-amp model to active filters and comparator applications.

I. Governing Concepts and Conventions

A filter passes a chosen band of frequencies and attenuates the rest; an operational amplifier is a very high-gain differential voltage amplifier used as the building block of active circuits.

Cutoff frequency: frequency where output power falls to half (−3 dB), given by fc = 1 / (2πRC) for a single RC stage; symbols: R resistance (Ω), C capacitance (F).
Reactance: capacitor reactance Xc = 1 / (2πfC) falls as frequency f rises, which is why RC networks discriminate between frequencies.
Decibel gain: A(dB) = 20·log10(Vout/Vin); roll-off of a first-order filter is 20 dB/decade.
Op-amp terminals: inverting input (−), non-inverting input (+), output; output Vout = A·(V+ − V−) where open-loop gain A is very large.
Feedback convention: negative feedback (output fed to −) stabilises gain; positive or no feedback gives switching behaviour.

II. Passive Filter Examples

RC networks that filter without amplification

A passive filter uses only resistors and capacitors; output amplitude never exceeds input.

A. low-pass filter

Passes frequencies below fc and attenuates those above.

Topology: series R, then C to ground; output taken across C.
Behaviour: at low f, Xc is large so most voltage appears across C (pass); at high f, Xc is small so C shorts the signal.
Cutoff: fc = 1/(2πRC); e.g. R = 1.6 kΩ, C = 0.1 µF gives fc ≈ 1 kHz.

B. high-pass filter

Passes frequencies above fc and attenuates those below.

Topology: series C, then R to ground; output across R.
Behaviour: high f → small Xc → signal passes; low f → large Xc → signal blocked.
Use: DC blocking and coupling between stages.

C. band-pass filter

Passes a band between a lower and upper cutoff.

Construction: a high-pass stage (sets lower cutoff fL) cascaded with a low-pass stage (sets upper cutoff fH).
Bandwidth: BW = fH − fL; centre frequency f0 = √(fL·fH).
Quality factor: Q = f0 / BW; higher Q means a narrower, more selective band.

III. Operational Amplifier Abstraction

Modelling the op-amp as an ideal block

The abstraction treats the op-amp as a near-perfect differential amplifier so circuit behaviour depends on external components, not the device.

A. device properties of the operational amplifier

Real-device parameters set the limits of the abstraction.

Open-loop gain A: typically 10⁵–10⁶ for a 741-type device.
Input offset voltage: small DC (~mV) needed at inputs to null the output.
Slew rate: maximum dVout/dt (e.g. 0.5 V/µs for a 741), limiting large-signal speed.
Bandwidth: gain falls with frequency; gain–bandwidth product is roughly constant.

B. properties of operational amplifier

The idealised properties used for analysis.

Infinite input impedance: input currents I+ = I− ≈ 0.
Zero output impedance: output voltage independent of load.
Infinite open-loop gain: forces V+ ≈ V− under negative feedback.
Infinite bandwidth, zero offset: assumed ideal for first-pass design.

IV. Simple Op Amp Circuits

Negative-feedback configurations

With negative feedback the two input terminals are driven to nearly equal voltage, which is the key to analysing these circuits.

A. virtual ground concept

Explains why the inverting node sits at 0 V without being physically grounded.

Principle: with V+ grounded and infinite gain, feedback forces V− ≈ V+ = 0, so the − node is a virtual ground.
Consequence: input current flows entirely through the feedback resistor since I− = 0.
Anchor: Vin/R1 = −Vout/Rf because both currents meet at the 0 V node.

B. inverting and non-inverting op-amp

Two gain configurations distinguished by where the signal enters.

Inverting: input through R1 to the − node (virtual ground), feedback Rf. Gain Vout/Vin = −Rf/R1; output is phase-inverted. Input impedance ≈ R1.
Non-inverting: input applied to +, feedback divider R1–Rf from output to −. Gain Vout/Vin = 1 + Rf/R1; output in phase, gain always ≥ 1. Input impedance ≈ infinite.

V. Active Filters

Op-amp filters with gain and buffering

A. active filter

An RC filter combined with an op-amp to add gain and isolate stages.

Advantage: no bulky inductors, provides gain >1, and the low output impedance prevents loading of the next stage.
Order: cascading RC stages increases roll-off (second-order = 40 dB/decade).

B. active high pass filter

Passes high frequencies with gain.

Topology: input capacitor C and resistor R on the non-inverting input, op-amp set for gain 1 + Rf/R1.
Cutoff: fc = 1/(2πRC); below fc, Xc blocks the signal; above it, gain is flat.

C. active band pass filter

Passes a band with gain.

Construction: active high-pass followed by active low-pass, or a single op-amp with both Cs.
Design: set fL from the high-pass RC, fH from the low-pass RC; f0 = √(fL·fH).

D. active band stop filter

Rejects a band and passes all else (notch filter).

Construction: sum of a low-pass and a high-pass path whose outputs are added, leaving a gap between fL and fH.
Use: removing a specific interferer, e.g. 50 Hz mains hum.
Selectivity: notch depth and width set by Q.

VI. Op-amp as an Adder and Subtractor

Analog arithmetic from the virtual-ground rule

A. op-amp as an adder and subtractor

Performs weighted addition and difference of voltages.

Adder (summing amplifier): inputs V1, V2, V3 through R1, R2, R3 into the virtual ground, feedback Rf.
TEXT
```
   Vout = −Rf·(V1/R1 + V2/R2 + V3/R3)
```
With all resistors equal to Rf: Vout = −(V1 + V2 + V3).
Subtractor (difference amplifier): one signal on −, one on +, matched resistors.
TEXT
```
   Vout = (Rf/R1)·(V2 − V1)
```
Output tracks the difference, rejecting any common-mode voltage.

VII. Op-amp RC Circuits

Time-domain operations using a feedback capacitor

These circuits replace a resistor with a capacitor so the op-amp performs calculus on the input.

A. op-amp integrator

Output is proportional to the time integral of the input.

Topology: inverting amplifier with input R and feedback capacitor C.
Relation:
TEXT
```
  Vout = −(1/RC) ∫ Vin dt
```
symbols: RC the time constant (s).
Response: a step input gives a ramp output; a square wave becomes a triangular wave.
Note: a large resistor across C is added to stop DC drift saturating the output.

B. op-amp differentiator

Output is proportional to the rate of change of the input.

Topology: inverting amplifier with input capacitor C and feedback resistor R.
Relation:
TEXT
```
  Vout = −RC·(dVin/dt)
```
Response: a ramp gives a constant output; a triangular wave becomes a square wave.
Note: amplifies high-frequency noise, so a small series resistor limits gain.

VIII. Op-amp as a Comparator and its Application in Anti-lock Braking Systems

Open-loop switching for decision-making

Without feedback the huge gain drives the output to one supply rail or the other, turning the op-amp into a two-state decision element.

A. op-amp as a comparator

Compares two voltages and outputs a logic-level decision.

Operation: if V+ > V−, output saturates to +Vsat; if V+ < V−, output goes to −Vsat.
Reference: one input is held at a reference Vref; the other receives the signal.
Hysteresis: adding positive feedback (Schmitt trigger) gives two thresholds, preventing chatter near the switching point.

B. application in anti-lock braking systems

Comparators detect imminent wheel lock and trigger brake modulation.

Sensing: a wheel-speed sensor voltage is compared with a reference representing expected vehicle speed.
Decision: if a wheel decelerates too fast (signal crosses Vref), the comparator output switches state.
Action: that switch signals the ABS controller to release and reapply brake pressure rapidly, keeping the wheel rolling.
Benefit: maintains tyre–road grip and steering control, shortening stopping distance on slippery surfaces.