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

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 - 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:

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:

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 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):

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:

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:

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

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:

Loop Parameters:

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

• \(K_v\): VCO gain (rad/s/V)

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

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

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