Active Filters : Single-stage, Sallen Key, Butterworth, VCOs & Timers - GATE Quick Notes

Active Filters Fundamentals

Active vs Passive Filters

Active Filters:

  • Use active components (Op-Amps, transistors)

  • Provide gain (amplification)

  • No inductors required

  • Input/output impedance can be controlled

  • Require power supply

Advantages over Passive:

  • High input impedance, low output impedance

  • Gain adjustment possible

  • No loading effects

  • Better isolation between stages

Filter Classification

By Frequency Response:

  • Low Pass Filter (LPF): \(H(j\omega) = \dfrac{A}{1 + j\omega/\omega_c}\)

  • High Pass Filter (HPF): \(H(j\omega) = \dfrac{A \cdot j\omega/\omega_c}{1 + j\omega/\omega_c}\)

  • Band Pass Filter (BPF): Passes specific frequency band

  • Band Stop Filter (BSF): Rejects specific frequency band

By Order:

  • First Order: \(20~\mathrm{dB/decade}\) roll-off

  • Second Order: \(40~\mathrm{dB/decade}\) roll-off

  • Higher Order: \(n \times 20~\mathrm{dB/decade}\) roll-off

Frequency Response Parameters

Key Parameters:

  • Passband: Frequency range where \(|H(j\omega)| \geq -3dB\)

  • Stopband: Frequency range where signal is significantly attenuated

  • Transition band: Region between passband and stopband

  • Passband ripple: Maximum variation in passband gain

  • Stopband attenuation: Minimum attenuation in stopband

Bode Plot Characteristics:

  • Magnitude plot: \(20\log|H(j\omega)|\) vs \(\log\omega\)

  • Phase plot: \(\angle H(j\omega)\) vs \(\log\omega\)

  • Corner frequency: Where response is 3dB down

  • Asymptotic approximations for analysis

Single-Stage Active Filters

First Order Low Pass Filter

Non-inverting Configuration:

  • Gain: \(A_v = 1 + \dfrac{R_2}{R_1}\)

  • Cut-off frequency: \(f_c = \dfrac{1}{2\pi RC}\)

  • Transfer function: \(H(j\omega) = \dfrac{A_v}{1 + j\omega RC}\)

  • Roll-off: 20 dB/decade beyond \(f_c\)

Inverting Configuration:

  • Gain: \(A_v = -\dfrac{R_2}{R_1}\)

  • Same cut-off frequency formula

  • \(180^{\circ}\) phase shift

First Order High Pass Filter

Non-inverting Configuration:

  • Gain: \(A_v = 1 + \dfrac{R_2}{R_1}\)

  • Cut-off frequency: \(f_c = \dfrac{1}{2\pi RC}\)

  • Transfer function: \(H(j\omega) = \dfrac{A_v \cdot j\omega RC}{1 + j\omega RC}\)

  • Roll-off: 20 dB/decade below \(f_c\)

Key Points:

  • Capacitor and resistor positions are swapped vs LPF

  • Phase leads by \(90^{\circ}\) at very high frequencies

  • Gain approaches \(A_v\) at high frequencies

Band Pass Filter

Cascade of HPF and LPF:

  • \(f_{c1} < f_{c2}\) (HPF cut-off < LPF cut-off)

  • Bandwidth: \(BW = f_{c2} - f_{c1}\)

  • Center frequency: \(f_0 = \sqrt{f_{c1} \cdot f_{c2}}\)

  • Quality factor: \(Q = \dfrac{f_0}{BW}\)

Single Op-Amp BPF:

  • Multiple feedback topology

  • Gain: \(A_v = -\dfrac{R_2}{2R_1}\)

  • \(f_0 = \dfrac{1}{2\pi RC}\)

  • \(Q = \dfrac{1}{3 - A_v}\)

All-Pass Filter

Characteristics:

  • Constant magnitude response: \(|H(j\omega)| = A\) (constant)

  • Variable phase response: \(\angle H(j\omega) = f(\omega)\)

  • Used for phase correction and delay equalization

First Order All-Pass:

  • Transfer function: \(H(s) = \dfrac{s - \omega_0}{s + \omega_0}\)

  • Phase shift: \(\phi(\omega) = -2\tan^{-1}(\omega/\omega_0)\)

  • Phase shift at \(\omega_0\): \(-90^{\circ}\)

  • Group delay: \(\tau_g = \dfrac{2\omega_0}{\omega_0^2 + \omega^2}\)

Second Order All-Pass:

  • Higher phase shift range: \(0^{\circ}\) to \(-180^{\circ}\)

  • Used in multi-stage phase shifters

Integrator and Differentiator

Op-Amp Integrator:

  • Transfer function: \(H(s) = -\dfrac{1}{sRC}\)

  • Magnitude: \(|H(j\omega)| = \dfrac{1}{\omega RC}\)

  • Phase: \(\angle H(j\omega) = -90^{\circ}\) (constant)

  • Slope: \(-20\) dB/decade

  • DC gain: Infinite (requires reset mechanism)

Op-Amp Differentiator:

  • Transfer function: \(H(s) = -sRC\)

  • Magnitude: \(|H(j\omega)| = \omega RC\)

  • Phase: \(\angle H(j\omega) = +90^{\circ}\) (constant)

  • Slope: \(+20\) dB/decade

  • Prone to noise amplification at high frequencies

Practical Considerations:

  • Integrator needs feedback resistor for DC stability

  • Differentiator needs series resistor for stability

Sallen Key Topology

Sallen Key - Introduction

Characteristics:

  • Voltage Controlled Voltage Source (VCVS) topology

  • Non-inverting configuration

  • Unity gain buffer or non-inverting amplifier

  • Two-pole (second-order) response

  • Popular for realizing standard filter responses

General Form:

  • Uses 2 resistors, 2 capacitors, 1 op-amp

  • Feedback from output to input through components

  • Gain: \(K = 1 + \dfrac{R_4}{R_3}\) (if gain \(> 1\))

Sallen Key Low Pass Filter

Equal Component Design:

  • \(R_1 = R_2 = R\), \(C_1 = C_2 = C\)

  • Cut-off frequency: \(f_c = \dfrac{1}{2\pi RC}\)

  • Damping factor: \(\zeta = \dfrac{3-K}{2}\)

  • For critical damping: \(K = 1\) (unity gain)

Transfer Function:

\[H(s) = \dfrac{K}{1 + sRC(3-K) + s^2R^2C^2}\]

Design Constraints:

  • For stability: \(K < 3\)

  • For Butterworth response: \(K = 1.586\)

Sallen Key High Pass Filter

Equal Component Design:

  • \(R_1 = R_2 = R\), \(C_1 = C_2 = C\)

  • Cut-off frequency: \(f_c = \dfrac{1}{2\pi RC}\)

  • Same damping factor: \(\zeta = \dfrac{3-K}{2}\)

Transfer Function:

\[H(s) = \dfrac{K \cdot s^2R^2C^2}{1 + sRC(3-K) + s^2R^2C^2}\]

Key Difference:

  • Resistors and capacitors are interchanged

  • High frequency gain = K

  • Roll-off below cut-off: 40 dB/decade

Sallen Key Band Pass Filter

Configuration:

  • More complex than LPF/HPF

  • Uses different component arrangement

  • Can be realized using multiple feedback

Parameters:

  • Center frequency: \(f_0 = \dfrac{1}{2\pi\sqrt{R_1R_2C_1C_2}}\)

  • Quality factor: \(Q = \dfrac{1}{2\zeta}\)

  • Gain at \(f_0\): \(A_0 = \dfrac{K}{2\zeta}\)

Design Considerations:

  • Higher Q requires more precision

  • Stability issues with high Q

  • Component tolerance effects

Butterworth Filters

Butterworth Response

Characteristics:

  • Maximally flat response in passband

  • No ripple in passband

  • Monotonic roll-off in stopband

  • Also called "maximally flat" filter

Transfer Function (nth order):

\[|H(j\omega)|^2 = \dfrac{1}{1 + (\omega/\omega_c)^{2n}}\]

Properties:

  • At \(\omega_c\): \(|H(j\omega_c)| = \dfrac{1}{\sqrt{2}} = -3dB\)

  • Roll-off: \(n \times 20\) dB/decade

  • Phase response: \(n \times 90^{\circ}\) at \(\omega \gg \omega_c\)

Butterworth Polynomial

Normalized Polynomials:

  • \(B_1(s) = s + 1\)

  • \(B_2(s) = s^2 + 1.414s + 1\)

  • \(B_3(s) = (s + 1)(s^2 + s + 1)\)

  • \(B_4(s) = (s^2 + 0.765s + 1)(s^2 + 1.848s + 1)\)

Pole Locations:

  • Poles lie on unit circle in s-plane

  • Equally spaced by \(\pi/n\) radians

  • Left half-plane poles for stable filter

Damping Factors:

  • 2nd order: \(\zeta = 0.707\)

  • 4th order: \(\zeta_1 = 0.383\), \(\zeta_2 = 0.924\)

Butterworth Filter Design

Order Determination:

\[n \geq \dfrac{\log(10^{A_s/10} - 1) - \log(10^{A_p/10} - 1)}{2\log(\omega_s/\omega_p)}\]

Where:

  • \(A_p\): Passband attenuation (dB)

  • \(A_s\): Stopband attenuation (dB)

  • \(\omega_p\): Passband edge frequency

  • \(\omega_s\): Stopband edge frequency

Frequency Scaling:

  • Replace \(s\) with \(s/\omega_c\)

  • Scale all component values accordingly

Higher Order Butterworth Realization

Cascade Approach:

  • Decompose into 1st and 2nd order sections

  • Each section implemented using Sallen Key

  • Odd orders: One 1st order + multiple 2nd order

  • Even orders: Only 2nd order sections

Design Steps:

  1. Determine required order

  2. Factor the polynomial

  3. Design each section with appropriate Q

  4. Cascade the sections

Practical Considerations:

  • Component tolerance effects

  • Op-amp bandwidth limitations

  • Noise considerations

Chebyshev Filters

Type I Chebyshev:

  • Ripple in passband, monotonic in stopband

  • Steeper roll-off than Butterworth

  • Passband ripple: \(\epsilon\) parameter

  • \(|H(j\omega)|^2 = \dfrac{1}{1 + \epsilon^2 T_n^2(\omega/\omega_c)}\)

  • \(T_n(x)\): Chebyshev polynomial of first kind

Type II Chebyshev (Inverse):

  • Monotonic in passband, ripple in stopband

  • Finite transmission zeros

  • Better phase response than Type I

Comparison with Butterworth:

  • Chebyshev: Faster roll-off, phase distortion

  • Butterworth: Slower roll-off, better phase response

  • Trade-off between magnitude and phase performance

Elliptic (Cauer) Filters

Characteristics:

  • Ripple in both passband and stopband

  • Steepest roll-off for given order

  • Minimum order for given specifications

  • Complex design procedure

Comparison Summary:

Filter Type Passband Stopband Roll-off
Butterworth Flat Monotonic Moderate
Chebyshev I Ripple Monotonic Steep
Chebyshev II Monotonic Ripple Steep
Elliptic Ripple Ripple Steepest

Selection Criteria:

  • Butterworth: General purpose, good phase response

  • Chebyshev: When steep roll-off is critical

  • Elliptic: Minimum order requirement

Bessel Filters

Characteristics:

  • Maximally flat group delay in passband

  • Linear phase response

  • Constant time delay for all frequencies

  • Slower roll-off than Butterworth

Transfer Function:

  • Based on Bessel polynomials

  • Denominator: \(B_n(s) = \sum_{k=0}^{n} \dfrac{(2n-k)!}{2^{n-k}k!(n-k)!}s^k\)

  • Poles lie on circle in s-plane (different from Butterworth)

Applications:

  • Pulse shaping circuits

  • Data communication systems

  • Any application requiring minimal phase distortion

Group Delay:

  • Definition: \(\tau_g = -\dfrac{d\phi}{d\omega}\)

  • Bessel filters have constant group delay

  • Important for preserving waveform shape

Filter Transformations

Frequency Transformations:

  • Low-pass to High-pass: \(s \rightarrow \dfrac{\omega_c^2}{s}\)

  • Low-pass to Band-pass: \(s \rightarrow \dfrac{s^2 + \omega_0^2}{s \cdot BW}\)

  • Low-pass to Band-stop: \(s \rightarrow \dfrac{s \cdot BW}{s^2 + \omega_0^2}\)

Impedance Scaling:

  • Multiply all resistors by \(K_R\)

  • Divide all capacitors by \(K_R\)

  • Inductors remain unchanged

Frequency Scaling:

  • Divide all reactive components by \(K_f\)

  • Resistors remain unchanged

  • New frequency = \(K_f \times\) old frequency

Multiple Feedback Topology

Characteristics:

  • Inverting configuration

  • Single op-amp implementation

  • Good for band-pass filters

  • Multiple feedback paths

MFB Band-Pass Filter:

  • Gain: \(A_v = -\dfrac{R_2}{2R_1}\)

  • Center frequency: \(f_0 = \dfrac{1}{2\pi\sqrt{R_1R_2C_1C_2}}\)

  • Quality factor: \(Q = \dfrac{\pi f_0 R_2 C_2}{1}\)

  • Bandwidth: \(BW = \dfrac{1}{\pi R_2 C_2}\)

Advantages:

  • Single op-amp design

  • Independent gain and Q adjustment

  • Lower component count

State Variable Filters

Configuration:

  • Uses 3 op-amps

  • Simultaneous LP, HP, BP outputs

  • Universal filter topology

  • Also called KHN (Kerwin-Huelsman-Newcomb) filter

Transfer Functions:

  • Low-pass: \(H_{LP}(s) = \dfrac{\omega_0^2}{s^2 + \dfrac{\omega_0}{Q}s + \omega_0^2}\)

  • High-pass: \(H_{HP}(s) = \dfrac{s^2}{s^2 + \dfrac{\omega_0}{Q}s + \omega_0^2}\)

  • Band-pass: \(H_{BP}(s) = \dfrac{\dfrac{\omega_0}{Q}s}{s^2 + \dfrac{\omega_0}{Q}s + \omega_0^2}\)

Advantages:

  • Independent control of \(\omega_0\) and Q

  • Multiple outputs available

  • Good stability

  • Easy to tune

Switched Capacitor Filters

Principle:

  • Capacitor switched between two nodes

  • Equivalent resistance: \(R_{eq} = \dfrac{1}{f_s C}\)

  • \(f_s\): Switching frequency

  • CMOS technology compatible

Advantages:

  • No external resistors required

  • High integration density

  • Programmable frequency response

  • Good matching in IC process

Disadvantages:

  • Requires clock signal

  • Limited frequency range

  • Aliasing considerations

  • Clock feedthrough

Applications:

  • Audio processing

  • Telecommunications

  • Anti-aliasing filters

Gyrator-Based Filters

Gyrator Concept:

  • Converts capacitor to inductor behavior

  • Impedance: \(Z_L = \dfrac{s R_1 R_2 C}{1}\)

  • Simulated inductance: \(L = R_1 R_2 C\)

  • No actual inductor required

Implementation:

  • Two op-amps and few passive components

  • Realizes LC ladder filters

  • Good for low-frequency applications

Applications:

  • Low-frequency filters

  • Precision filters

  • Replacement for bulky inductors

Limitations:

  • Requires matched components

  • Sensitive to op-amp characteristics

  • Limited frequency range

Sensitivity Analysis

Definition:

  • Sensitivity: \(S_x^y = \dfrac{x}{y} \dfrac{\partial y}{\partial x}\)

  • Measures how output parameter y changes with component x

  • Important for practical filter design

Component Sensitivity:

  • Frequency sensitivity: \(S_R^{\omega_0} = S_C^{\omega_0} = -\dfrac{1}{2}\)

  • Gain sensitivity varies with topology

  • Q sensitivity is critical for high-Q filters

Design Guidelines:

  • Low sensitivity to component variations

  • Use precision components for critical parameters

  • Consider temperature effects

  • Active filters generally have lower sensitivity than passive

Practical Impact:

  • Component tolerance effects

  • Aging and temperature drift

  • Manufacturing variations

Filter Specifications and Testing

Key Specifications:

  • Passband gain and ripple

  • Stopband attenuation

  • Transition bandwidth

  • Group delay variation

  • Input/output impedances

Testing Methods:

  • Frequency response measurement

  • Step response analysis

  • Impulse response testing

  • Distortion measurements

  • Noise figure measurements

Common Problems:

  • Oscillation due to instability

  • Frequency response deviation

  • Excessive noise

  • Distortion at high levels

  • Temperature drift

Design Verification:

  • Simulation vs measurement

  • Component tolerance analysis

  • Temperature testing

  • Long-term stability

Voltage Controlled Oscillators (VCOs)

VCO Fundamentals

Definition:

  • Oscillator whose frequency is controlled by input voltage

  • Linear relationship: \(f_{out} = f_0 + K_v \cdot V_{in}\)

  • \(K_v\): VCO sensitivity (Hz/V)

  • \(f_0\): Free-running frequency

Applications:

  • Frequency modulation (FM)

  • Phase-locked loops (PLLs)

  • Function generators

  • Frequency synthesizers

Key Parameters:

  • Frequency range

  • Linearity

  • Temperature stability

  • Phase noise

VCO Topologies

RC Oscillators:

  • Wien bridge oscillator

  • Phase shift oscillator

  • Voltage-controlled through varactor diodes

LC Oscillators:

  • Hartley oscillator

  • Colpitts oscillator

  • Higher frequency operation

Integrated VCOs:

  • 555 timer as VCO

  • Dedicated VCO ICs

  • Voltage-to-frequency converters

Wien Bridge VCO

Basic Operation:

  • Frequency: \(f = \dfrac{1}{2\pi RC}\)

  • Voltage control through FET resistance

  • Good frequency stability

Control Methods:

  • Varactor diode in parallel with C

  • FET as voltage-controlled resistor

  • Dual-gate MOSFET control

Design Considerations:

  • Gain must be exactly 3 for oscillation

  • Amplitude stabilization required

  • Temperature compensation

VCO Applications in PLLs

Phase-Locked Loop:

  • VCO is controlled element

  • Phase detector compares input and VCO

  • Loop filter smooths control voltage

  • Negative feedback system

PLL Equation:

\[K_v = \dfrac{\Delta f}{\Delta V_c}\]

Loop Parameters:

  • \(K_d\): Phase detector gain (V/rad)

  • \(K_v\): VCO gain (rad/s/V)

  • \(F(s)\): Loop filter transfer function

Timers

555 Timer - Introduction

Internal Structure:

  • Two comparators

  • SR flip-flop

  • Discharge transistor

  • Voltage divider (three \(5~\mathrm{k \Omega}\) resistors)

Reference Voltages:

  • Upper threshold: \(V_{CC}/3 \times 2 = 2V_{CC}/3\)

  • Lower threshold: \(V_{CC}/3\)

Pin Configuration:

  • Pin 1: Ground, Pin 2: Trigger, Pin 3: Output

  • Pin 4: Reset, Pin 5: Control, Pin 6: Threshold

  • Pin 7: Discharge, Pin 8: \(V_{CC}\)

555 Timer - Monostable Mode

Operation:

  • Triggered by negative edge on trigger pin

  • Output goes HIGH for predetermined time

  • Time period: \(T = 1.1 \times R \times C\)

  • Returns to LOW state after timeout

Timing Sequence:

  1. Trigger pulse (LOW) starts timing

  2. Output goes HIGH, discharge transistor OFF

  3. Capacitor charges through R

  4. When \(V_C = 2V_{CC}/3\), output goes LOW

  5. Discharge transistor ON, capacitor discharges

Applications:

  • Pulse width modulation

  • Time delays

  • Missing pulse detection

555 Timer - Astable Mode

Operation:

  • Continuous oscillation

  • No external trigger required

  • Square wave output

Timing Equations:

  • HIGH time: \(t_H = 0.693(R_A + R_B)C\)

  • LOW time: \(t_L = 0.693 R_B C\)

  • Period: \(T = t_H + t_L = 0.693(R_A + 2R_B)C\)

  • Frequency: \(f = \dfrac{1.44}{(R_A + 2R_B)C}\)

  • Duty cycle: \(D = \dfrac{R_A + R_B}{R_A + 2R_B}\)

Design Notes:

  • Minimum duty cycle \(\approx 50\%\)

  • For 50% duty cycle: \(R_A \ll R_B\)

555 Timer - VCO Configuration

Voltage Controlled Oscillator:

  • Control voltage applied to pin 5

  • Changes threshold voltages

  • Frequency varies with control voltage

Modified Equations:

  • Upper threshold: \(V_{control}\)

  • Lower threshold: \(V_{control}/2\)

  • Frequency: \(f = \dfrac{1.44}{(R_A + 2R_B)C} \times \dfrac{V_{CC}}{V_{control}}\)

Characteristics:

  • Linear frequency vs voltage relationship

  • Frequency range: few Hz to several MHz

  • Simple implementation

Other Timer ICs

556 Dual Timer:

  • Two independent 555 timers

  • Separate power and ground pins

  • Can be cascaded or used independently

LM555 vs NE555:

  • CMOS version: Lower power consumption

  • Better temperature stability

  • Higher input impedance

Precision Timers:

  • ICM7555: CMOS version of 555

  • LTC1799: Resistor-set oscillator

  • XR-2206: Function generator IC

Additional GATE Calculation Examples

Filter Order Calculation:

  • Given: \(A_p = 1dB\) at \(f_p = 1kHz\), \(A_s = 40dB\) at \(f_s = 10kHz\)

  • Butterworth: \(n = \dfrac{\log\sqrt{10^4 - 1} - \log\sqrt{10^{0.1} - 1}}{2\log(10)} = 2.9\)

  • Choose \(n = 3\) (next higher integer)

Component Scaling:

  • Normalized design: \(R = 1\Omega\), \(C = 1F\), \(f_c = 1Hz\)

  • For \(f_c = 1kHz\): \(K_f = 1000\)

  • For \(R = 10k\Omega\): \(K_R = 10^4\)

  • Final: \(R = 10k\Omega\), \(C = \dfrac{1}{K_f \times K_R} = 100nF\)

Q Calculation:

  • BPF with \(f_0 = 1kHz\), \(f_1 = 900Hz\), \(f_2 = 1100Hz\)

  • \(BW = f_2 - f_1 = 200Hz\)

  • \(Q = \dfrac{f_0}{BW} = \dfrac{1000}{200} = 5\)

Advanced Timer Applications

555 Timer as Schmitt Trigger:

  • Upper threshold: \(\dfrac{2V_{CC}}{3}\)

  • Lower threshold: \(\dfrac{V_{CC}}{3}\)

  • Hysteresis: \(\dfrac{V_{CC}}{3}\)

  • Applications: Noise immunity, waveform shaping

Precision Timing:

  • Crystal oscillators for accurate timing

  • Temperature compensation techniques

  • Voltage regulation for stability

Timer Cascading:

  • Frequency division

  • Sequential timing

  • Complex waveform generation

Power Considerations:

  • CMOS vs bipolar power consumption

  • Supply voltage effects on timing

  • Load current limitations

GATE Specific Points

Important Formulas for GATE

Filter Cutoff Frequencies:

  • Single RC: \(f_c = \dfrac{1}{2\pi RC}\)

  • Sallen Key: \(f_c = \dfrac{1}{2\pi RC}\) (equal components)

  • Butterworth: Same, but with specific Q values

Quality Factor:

  • \(Q = \dfrac{f_0}{BW}\)

  • For Butterworth 2nd order: \(Q = 0.707\)

  • Higher Q \(\to\) sharper response

555 Timer:

  • Monostable: \(T = 1.1RC\)

  • Astable: \(f = \dfrac{1.44}{(R_A + 2R_B)C}\)

Common GATE Questions

Filter Analysis:

  • Given circuit, find cutoff frequency

  • Calculate gain at specific frequency

  • Determine filter order and type

  • Find component values for given specs

Timer Calculations:

  • Time period calculations

  • Duty cycle problems

  • Frequency calculations

  • Component selection

Op-Amp Considerations:

  • Gain-bandwidth product effects

  • Slew rate limitations

  • Input offset voltage effects

Design Guidelines

Active Filter Design:

  • Use standard component values

  • Consider op-amp bandwidth limitations

  • Account for component tolerances

  • Ensure stability (\(K < 3\) for Sallen Key)

Practical Considerations:

  • Power supply requirements

  • Temperature effects

  • Noise considerations

  • Loading effects

GATE Exam Tips:

  • Memorize key formulas

  • Practice frequency response plots

  • Understand pole-zero concepts

  • Know standard filter responses

GATE Problem-Solving Strategy

Filter Analysis Steps:

  1. Identify filter type and order

  2. Write transfer function

  3. Find poles and zeros

  4. Calculate key parameters (\(f_c\), Q, gain)

  5. Verify stability conditions

Common Mistakes to Avoid:

  • Confusing magnitude and phase responses

  • Incorrect pole-zero identification

  • Wrong stability analysis

  • Unit conversion errors

  • Ignoring op-amp limitations

Quick Check Methods:

  • Dimensional analysis

  • Limiting case verification

  • Symmetry properties

  • Physical realizability

Summary

Key Takeaways:

  • Active filters provide gain and good isolation

  • Sallen Key topology is popular for 2nd order filters

  • Butterworth filters have maximally flat response

  • VCOs are essential for frequency synthesis

  • 555 timer is versatile for timing applications

For GATE Success:

  • Practice numerical problems

  • Understand circuit analysis methods

  • Know applications of each topology

  • Remember key formulas and relationships