Oscillators and Feedback Amplifiers for GATE Exam: Operational Amplifiers, Characteristics, and Applications

Operational Amplifiers - Basics

Operational Amplifier (Op-Amp) Introduction

  • High-gain differential amplifier with very high input impedance

  • Ideal Op-Amp characteristics:

    • Infinite open-loop gain (\(A_0 = \infty\))

    • Infinite input impedance (\(Z_i = \infty\))

    • Zero output impedance (\(Z_0 = 0\))

    • Infinite bandwidth

    • Zero offset voltage

  • Two inputs: Inverting (-) and Non-inverting (+)

  • Output: \(V_0 = A_0(V_+ - V_-)\)

Op-Amp Parameters - Real vs Ideal

Real Op-Amp:

  • Open-loop gain: \(10^4\) to \(10^6\)

  • Input impedance: \(10^6\) to \(10^{12}\) \(\Omega\)

  • Output impedance: 10 to 100 \(\Omega\)

  • Bandwidth: Limited

  • Offset voltage: 1-10 mV

  • Slew rate: 0.1 to 100 V/\(\mu\)s

Key Parameters:

  • CMRR: 80-120 dB

  • Input bias current: 10 pA to 1 \(\mu\)A

  • Input offset current

  • Power supply rejection ratio (PSRR)

  • Gain-bandwidth product (GBW)

  • Temperature drift

Op-Amp Non-Idealities and Compensation

  • Input Offset Voltage (\(V_{OS}\)):

    • Causes DC error at output

    • Compensation: External potentiometer or chopper stabilization

  • Input Bias Current (\(I_B\)):

    • \(I_B = \dfrac{I_{B+} + I_{B-}}{2}\)

    • Compensation: Equal source resistances

  • Input Offset Current (\(I_{OS}\)):

    • \(I_{OS} = |I_{B+} - I_{B-}|\)

    • Creates additional DC error

  • Temperature Drift: \(\Delta V_{OS}/\Delta T\) (typically \(10-50~\mu V/^{\circ}C\))

Op-Amp Configurations

Inverting Amplifier

  • Gain: \(A_v = -\dfrac{R_f}{R_i}\)

  • Input impedance: \(Z_i = R_i\)

  • Output impedance: \(Z_o \approx 0\)

  • Virtual short concept: \(V_+ = V_- = 0\) (for ideal op-amp)

  • Bandwidth: \(BW = \dfrac{GBW}{|A_v|}\)

  • DC Error Analysis:

    • Due to \(V_{OS}\): \(V_{o,error} = V_{OS}(1 + \dfrac{R_f}{R_i})\)

    • Due to \(I_B\): \(V_{o,error} = I_B \times R_f\)

Non-Inverting Amplifier

  • Gain: \(A_v = 1 + \dfrac{R_f}{R_i}\)

  • Input impedance: Very high (ideally infinite)

  • Output impedance: \(Z_o \approx 0\)

  • Minimum gain = 1 (unity gain buffer)

  • Better input characteristics than inverting amplifier

  • DC Error Analysis:

    • Due to \(V_{OS}\): \(V_{o,error} = V_{OS}(1 + \dfrac{R_f}{R_i})\)

    • Compensation resistor: \(R_c = R_i || R_f\)

Other Op-Amp Configurations

  • Voltage Follower (Buffer):

    • Gain = 1, very high input impedance

    • Used for impedance matching

  • Summing Amplifier:

    • \(V_o = -R_f(\dfrac{V_1}{R_1} + \dfrac{V_2}{R_2} + \dfrac{V_3}{R_3})\)

  • Difference Amplifier:

    • \(V_o = \dfrac{R_f}{R_i}(V_2 - V_1)\) (for balanced resistor ratios)

  • Instrumentation Amplifier:

    • High CMRR, high input impedance

    • Used in precision measurements

Integrator and Differentiator

  • Integrator:

    • \(V_o = -\dfrac{1}{RC}\int V_{in} dt\)

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

    • DC gain = \(\infty\), requires bias resistor

    • Applications: Analog computers, waveform generation

  • Differentiator:

    • \(V_o = -RC\dfrac{dV_{in}}{dt}\)

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

    • Susceptible to noise, requires input resistor

    • Applications: Edge detection, wave shaping

Feedback Amplifiers

Feedback Amplifier Fundamentals

  • Closed-loop gain: \(A_f = \dfrac{A}{1 + A\beta}\)

  • Loop gain: \(A\beta\) (determines stability)

  • Positive feedback: \(A\beta > 0\) (can cause oscillation)

  • Negative feedback: \(A\beta < 0\) (stabilizes amplifier)

  • Feedback factor: \(\beta = \dfrac{V_f}{V_o}\)

  • Desensitivity factor: \(D = 1 + A\beta\)

Effects of Negative Feedback

Advantages:

  • Stabilizes gain: \(\dfrac{dA_f}{dA} = \dfrac{1}{(1+A\beta)^2}\)

  • Reduces distortion by factor \((1+A\beta)\)

  • Increases bandwidth by factor \((1+A\beta)\)

  • Improves input/output impedance

  • Reduces noise

Disadvantages:

  • Reduces overall gain

  • Requires additional components

  • Can cause instability if not properly designed

  • Increased complexity

Four Types of Feedback Amplifiers

  1. Voltage Series (Series-Shunt):

    • \(Z_{if} = Z_i(1 + A\beta)\), \(Z_{of} = \dfrac{Z_o}{1 + A\beta}\)

    • Example: Non-inverting op-amp

  2. Current Series (Series-Series):

    • \(Z_{if} = Z_i(1 + A\beta)\), \(Z_{of} = Z_o(1 + A\beta)\)

  3. Voltage Shunt (Shunt-Shunt):

    • \(Z_{if} = \dfrac{Z_i}{1 + A\beta}\), \(Z_{of} = \dfrac{Z_o}{1 + A\beta}\)

    • Example: Inverting op-amp

  4. Current Shunt (Shunt-Series):

    • \(Z_{if} = \dfrac{Z_i}{1 + A\beta}\), \(Z_{of} = Z_o(1 + A\beta)\)

Feedback Amplifier Analysis Method

  • Step 1: Identify feedback topology

  • Step 2: Find open-loop parameters \(A\), \(Z_i\), \(Z_o\)

  • Step 3: Calculate feedback factor \(\beta\)

  • Step 4: Determine loop gain \(A\beta\)

  • Step 5: Calculate closed-loop parameters

  • Loading effects:

    • Input loading: \(A' = A \times \dfrac{Z_i}{Z_i + Z_s}\)

    • Output loading: \(A'' = A' \times \dfrac{Z_L}{Z_o + Z_L}\)

Oscillators

Oscillator Fundamentals

  • Oscillator: Circuit that generates periodic waveforms without external input

  • Barkhausen Criteria for Oscillation:

    • Loop gain \(|A\beta| = 1\)

    • Phase shift around loop = \(0^{\circ}\) or \(360^{\circ}\)

  • Oscillation condition: \(A\beta = 1 \angle 0^{\circ}\)

  • Startup condition: \(|A\beta| > 1\) initially

  • Steady state: \(|A\beta| = 1\) (amplitude limiting)

  • Frequency of oscillation: Determined by phase shift network

Types of Oscillators

Sinusoidal Oscillators:

  • RC Oscillators

  • LC Oscillators

  • Crystal Oscillators

  • Negative resistance oscillators

Non-Sinusoidal Oscillators:

  • Relaxation oscillators

  • Astable multivibrators

  • Monostable multivibrators

  • Bistable multivibrators

RC Oscillators

  • Phase Shift Oscillator:

    • Uses RC network providing \(180^{\circ}\) phase shift

    • Op-amp provides additional \(180^{\circ}\) shift

    • Frequency: \(f = \dfrac{1}{2\pi RC\sqrt{6}}\)

    • Gain required: \(A = 29\) minimum

  • Wien Bridge Oscillator:

    • Uses Wien bridge network

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

    • Gain required: \(A = 3\) minimum

    • Better frequency stability

    • Amplitude stabilization using AGC

LC Oscillators

  • Hartley Oscillator:

    • Uses tapped inductor

    • Frequency: \(f = \dfrac{1}{2\pi\sqrt{L_T C}}\)

    • \(L_T = L_1 + L_2 + 2M\) (where M is mutual inductance)

    • Good frequency stability

  • Colpitts Oscillator:

    • Uses capacitive voltage divider

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

    • \(C_T = \dfrac{C_1 C_2}{C_1 + C_2}\)

    • Better frequency stability than Hartley

  • Clapp Oscillator:

    • Modified Colpitts with series capacitor

    • Frequency: \(f = \dfrac{1}{2\pi\sqrt{LC_3}}\) (approximately)

    • Excellent frequency stability

Crystal Oscillators

  • Piezoelectric effect: Mechanical stress produces electrical charge

  • Equivalent circuit: Series RLC with parallel capacitance

  • Two resonant frequencies:

    • Series resonance: \(f_s = \dfrac{1}{2\pi\sqrt{LC}}\)

    • Parallel resonance: \(f_p = \dfrac{1}{2\pi\sqrt{L(C + C_p)}}\)

  • Advantages:

    • Very high Q factor (10,000 to 100,000)

    • Excellent frequency stability (\(\pm 10\) ppm)

    • Low temperature coefficient

  • Applications: Clock generators, frequency standards

Negative Resistance Oscillators

  • Principle: Use devices with negative resistance characteristics

  • Oscillation condition: \(R_{neg} + R_{pos} = 0\)

  • Tunnel Diode Oscillator:

    • Exhibits negative resistance in certain region

    • Very high frequency operation possible

    • Low power consumption

  • Gunn Diode Oscillator:

    • Microwave frequency oscillator

    • Based on Gunn effect in GaAs

    • Frequency: 1-100 GHz

  • Applications: Microwave oscillators, high-frequency sources

Multivibrators

Multivibrator Types

  • Astable Multivibrator:

    • No stable state, continuously oscillates

    • Generates square/rectangular waves

    • Period: \(T = 0.693(R_1 + R_2)C_1 + 0.693R_2C_2\)

    • Duty cycle: \(D = \dfrac{t_{ON}}{T}\)

  • Monostable Multivibrator:

    • One stable state, triggered to quasi-stable state

    • Pulse width: \(t_p = 0.693RC\)

    • Used for pulse generation and timing

  • Bistable Multivibrator (Flip-Flop):

    • Two stable states

    • Triggered between states

    • Memory element

555 Timer IC

  • Internal components:

    • Two comparators, SR latch, discharge transistor

    • Voltage divider providing reference voltages (\(V_{CC}/3\), \(2V_{CC}/3\))

  • Astable Mode:

    • Frequency: \(f = \dfrac{1.44}{(R_1 + 2R_2)C}\)

    • Duty cycle: \(D = \dfrac{R_1 + R_2}{R_1 + 2R_2}\)

    • \(t_{HIGH} = 0.693(R_1 + R_2)C\)

    • \(t_{LOW} = 0.693R_2C\)

  • Monostable Mode:

    • Pulse width: \(t_p = 1.1RC\)

    • Triggered by negative edge on trigger pin

  • Applications: Timing circuits, PWM generation, frequency division

Waveform Generators

Function Generator Circuits

  • Triangular Wave Generator:

    • Integrator + Schmitt trigger combination

    • Frequency: \(f = \dfrac{1}{4RC}\) (for symmetric output)

    • Slope: \(\dfrac{dV}{dt} = \dfrac{V_{sat}}{RC}\)

  • Sawtooth Wave Generator:

    • Uses constant current source for linear ramp

    • Reset transistor for flyback

    • Applications: CRO timebase, ADC ramp

  • Square Wave from Triangular:

    • Use comparator with reference

    • Adjustable duty cycle possible

Voltage-Controlled Oscillator (VCO)

  • Principle: Output frequency varies with input voltage

  • Implementation methods:

    • Varactor diode in LC tank

    • Current-controlled integrator

    • Voltage-controlled delay line

  • Key parameters:

    • Sensitivity: \(K_o = \dfrac{df}{dV}\) (Hz/V)

    • Linear range

    • Output frequency range

  • Applications:

    • Phase-locked loops (PLL)

    • Frequency modulation

    • Frequency synthesis

Frequency Response and Stability

Frequency Response of Op-Amps

  • Open-loop gain: Decreases at -20 dB/decade after unity gain frequency

  • Gain-Bandwidth Product (GBW): Constant for given op-amp

  • Unity gain frequency: \(f_t = A_0 \times f_0\)

  • Closed-loop bandwidth: \(BW = \dfrac{GBW}{|A_f|}\)

  • Slew rate: Maximum rate of change of output voltage

    • \(SR = \dfrac{dV_o}{dt}_{max}\)

    • Causes distortion in high-frequency signals

    • Full power bandwidth: \(f_{FP} = \dfrac{SR}{2\pi V_m}\)

Stability Analysis

  • Nyquist Stability Criterion:

    • System stable if Nyquist plot doesn’t encircle (-1, 0)

    • Critical for feedback amplifier design

  • Phase Margin: \(PM = 180^{\circ} - \phi\) at gain crossover frequency

  • Gain Margin: \(GM = -|A\beta|\) at phase crossover frequency

  • Stability criteria:

    • \(PM > 45^{\circ}\) for good stability

    • \(GM > 6\) dB for good stability

  • Compensation techniques:

    • Dominant pole compensation

    • Lead/lag compensation

    • Miller compensation

Active Filters

Active Filter Advantages

  • Advantages over passive filters:

    • No loading effects

    • Gain possible

    • No inductors required

    • Better isolation

    • Easy to tune

  • First-order filter responses:

    • Low-pass: \(H(s) = \dfrac{K}{1 + sRC}\), \(f_c = \dfrac{1}{2\pi RC}\)

    • High-pass: \(H(s) = \dfrac{sRC}{1 + sRC}\), \(f_c = \dfrac{1}{2\pi RC}\)

    • Roll-off: 20 dB/decade

Second-Order Active Filters

  • Sallen-Key Low-Pass:

    • \(H(s) = \dfrac{K}{s^2 + s\omega_0/Q + \omega_0^2}\)

    • \(\omega_0 = \dfrac{1}{\sqrt{R_1 R_2 C_1 C_2}}\)

    • \(Q = \dfrac{\sqrt{R_1 R_2 C_1 C_2}}{R_1 C_2 + R_2 C_2(1-K)}\)

  • Multiple Feedback (MFB) Low-Pass:

    • Uses inverting op-amp configuration

    • Better for high-Q applications

  • Band-Pass Filter:

    • \(H(s) = \dfrac{K s\omega_0/Q}{s^2 + s\omega_0/Q + \omega_0^2}\)

    • Gain at center frequency: \(K\)

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

Filter Design Considerations

  • Filter approximations:

    • Butterworth: Maximally flat passband

    • Chebyshev: Equiripple in passband

    • Elliptic: Equiripple in both bands

  • Quality factor (Q):

    • \(Q = \dfrac{f_0}{BW}\) for band-pass

    • Higher Q = sharper cutoff

    • Stability concerns for high Q

  • Sensitivity:

    • \(S_x^F = \dfrac{x}{F}\dfrac{\partial F}{\partial x}\)

    • Lower sensitivity = better performance

Comparators and Schmitt Triggers

Comparators

  • Function: Compares two analog voltages, outputs digital signal

  • Op-amp as comparator:

    • Open-loop configuration

    • Output saturates to \(\pm V_{sat}\)

    • Very sensitive to noise

    • Slow switching due to saturation

  • Dedicated comparator ICs:

    • Faster switching than op-amps

    • Better suited for digital interface

    • Examples: LM311, LM393

    • Open-collector outputs

  • Window comparator: Detects if input is within specified range

Schmitt Trigger

  • Purpose: Provides hysteresis to reduce noise sensitivity

  • Inverting Schmitt Trigger:

    • \(V_{TH} = +\dfrac{R_1}{R_1 + R_2}V_{sat}\) (Upper threshold)

    • \(V_{TL} = -\dfrac{R_1}{R_1 + R_2}V_{sat}\) (Lower threshold)

    • Hysteresis: \(V_H = V_{TH} - V_{TL} = \dfrac{2R_1 V_{sat}}{R_1 + R_2}\)

  • Non-inverting Schmitt Trigger:

    • Different threshold calculations

    • Opposite switching behavior

  • Applications: Noise immunity, waveform squaring, oscillators

Precision Rectifiers and Peak Detectors

Precision Rectifiers

  • Need: Eliminate forward voltage drop of diodes

  • Half-wave precision rectifier:

    • Uses op-amp to overcome diode drop

    • Output follows input for positive half-cycle

    • Zero output for negative half-cycle

  • Full-wave precision rectifier:

    • Rectifies both half-cycles

    • Two op-amp design for better performance

    • Output: \(V_o = |V_i|\)

  • Applications:

    • AC-DC conversion

    • RMS-to-DC conversion

    • Signal processing

Peak Detectors

  • Function: Captures and holds peak value of input signal

  • Basic peak detector:

    • Op-amp + diode + capacitor

    • Charges capacitor to peak input voltage

    • Holds peak value when input decreases

  • Precision peak detector:

    • Uses op-amp to eliminate diode drop

    • Better accuracy for small signals

    • Faster response time

  • Reset mechanism:

    • MOSFET switch to discharge capacitor

    • Allows detection of new peaks

  • Applications:

    • Peak voltage measurement

    • Envelope detection

    • Signal analysis

Sample-and-Hold Circuits

Sample-and-Hold (S/H) Circuits

  • Purpose: Capture and hold analog signal value at specific time

  • Basic S/H circuit:

    • Analog switch + hold capacitor + buffer amplifier

    • Sample mode: Switch closed, capacitor charges to input

    • Hold mode: Switch open, capacitor holds voltage

  • Key parameters:

    • Acquisition time: Time to charge capacitor to final value

    • Droop rate: Rate of voltage decay during hold

    • Feedthrough: Coupling from input to output during hold

    • Aperture time: Delay between command and actual switching

  • Applications:

    • ADC front-end

    • Data acquisition systems

    • Signal processing

Analog-to-Digital Converters

ADC Fundamentals

  • Quantization: Converting continuous analog signal to discrete digital values

  • Resolution: Number of bits determines quantization levels

    • n-bit ADC has \(2^n\) levels

    • LSB = \(\dfrac{V_{ref}}{2^n}\)

  • Quantization error: \(\pm \dfrac{LSB}{2}\)

  • Key specifications:

    • Resolution (bits)

    • Conversion time

    • Sampling rate

    • Accuracy (INL, DNL)

    • Signal-to-noise ratio (SNR)

ADC Types

  • Flash ADC:

    • Parallel comparison with \(2^n - 1\) comparators

    • Fastest conversion time

    • High power consumption

    • Limited to low resolution (6-8 bits)

  • Successive Approximation (SAR) ADC:

    • Binary search algorithm

    • Conversion time: n clock cycles

    • Moderate speed and power

    • Most common type (8-18 bits)

  • Dual-slope ADC:

    • Integrating type converter

    • High accuracy, noise rejection

    • Slow conversion time

    • Used in digital multimeters

ADC Errors and Specifications

  • Integral Non-Linearity (INL):

    • Deviation from ideal transfer function

    • Measured in LSB units

  • Differential Non-Linearity (DNL):

    • Variation in code width from ideal 1 LSB

    • DNL > 1 LSB causes missing codes

  • Offset Error: DC shift in transfer function

  • Gain Error: Slope error in transfer function

  • Aperture Jitter: Timing uncertainty in sampling

  • Effective Number of Bits (ENOB):

    • \(ENOB = \dfrac{SNR - 1.76}{6.02}\)

    • Accounts for all noise sources

Digital-to-Analog Converters

DAC Fundamentals

  • Function: Convert digital code to analog voltage/current

  • Transfer function: \(V_o = V_{ref} \times \dfrac{D}{2^n}\)

  • Key specifications:

    • Resolution (bits)

    • Settling time

    • Accuracy (INL, DNL)

    • Monotonicity

    • Glitch energy

  • DAC types:

    • R-2R ladder

    • Weighted resistor

    • Current steering

    • Delta-sigma

R-2R Ladder DAC

  • Advantages:

    • Only two resistor values required

    • Good accuracy with matched resistors

    • Easy to implement in IC form

  • Operation:

    • Binary switches connect to \(V_{ref}\) or ground

    • Current division by factor of 2 at each node

    • MSB contributes most current

  • Output current: \(I_o = \dfrac{V_{ref}}{2R} \times \dfrac{D}{2^n}\)

  • Disadvantages:

    • Switch resistance affects accuracy

    • Glitches during code transitions

Phase-Locked Loop (PLL)

PLL Fundamentals

  • Function: Synchronize output frequency to input frequency

  • Basic PLL blocks:

    • Phase Detector (PD)

    • Loop Filter (LF)

    • Voltage-Controlled Oscillator (VCO)

  • Operation modes:

    • Free-running: No input signal

    • Acquisition: Trying to lock to input

    • Locked: Output frequency = Input frequency

  • Lock range: Frequency range where PLL can maintain lock

  • Capture range: Frequency range where PLL can acquire lock

PLL Analysis

  • Loop gain: \(K = K_d \times K_v \times F(s)\)

  • Closed-loop transfer function:

    • \(H(s) = \dfrac{K \times F(s)}{s + K \times F(s)}\)

  • First-order PLL: \(F(s) = 1\)

    • \(H(s) = \dfrac{K}{s + K}\)

    • Always stable

    • Type 1 system (zero steady-state error for step input)

  • Second-order PLL: \(F(s) = \dfrac{s + a}{s}\)

    • \(H(s) = \dfrac{K(s + a)}{s^2 + Ks + Ka}\)

    • Type 2 system (zero steady-state error for ramp input)

    • Stability depends on damping factor

PLL Applications

  • Frequency synthesis:

    • Divide-by-N counter in feedback

    • Output frequency = N × Input frequency

    • Fractional-N synthesis for fine resolution

  • Clock recovery:

    • Extract clock from data stream

    • Used in digital communication

  • Frequency modulation/demodulation:

    • VCO output follows input frequency variations

    • Control voltage represents demodulated signal

  • Motor speed control:

    • Maintain constant motor speed

    • Feedback from tachometer

Power Amplifiers

Power Amplifier Classes

  • Class A:

    • Conducts for full cycle (\(360^{\circ}\))

    • Low distortion, low efficiency ( 25%)

    • High heat dissipation

  • Class B:

    • Conducts for half cycle (\(180^{\circ}\))

    • Push-pull configuration required

    • Efficiency  78.5%, crossover distortion

  • Class AB:

    • Conducts for more than \(180^{\circ}\) but less than \(360^{\circ}\)

    • Compromise between Class A and B

    • Efficiency  50-70%, reduced crossover distortion

  • Class C:

    • Conducts for less than \(180^{\circ}\)

    • High efficiency ( 80%), high distortion

    • Used in RF applications with tuned loads

Power Amplifier Design Considerations

  • Thermal management:

    • Heat sinks for power dissipation

    • Thermal resistance calculations

    • Temperature-dependent biasing

  • Protection circuits:

    • Short-circuit protection

    • Over-temperature protection

    • Current limiting

  • Efficiency: \(\eta = \dfrac{P_{out}}{P_{in}} \times 100\%\)

  • Total Harmonic Distortion (THD):

    • \(THD = \dfrac{\sqrt{V_2^2 + V_3^2 + V_4^2 + ...}}{V_1} \times 100\%\)

    • Measure of signal quality

GATE Problems and Solutions

Typical GATE Problems - Op-Amps

  • Problem 1: Find gain of inverting amplifier with \(R_f = 100k\Omega\), \(R_i = 10k\Omega\)

  • Solution: \(A_v = -\dfrac{R_f}{R_i} = -\dfrac{100k}{10k} = -10\)

  • Problem 2: Calculate bandwidth of non-inverting amplifier with gain = 20 and GBW = \(1 ~\mathrm{MHz}\)

  • Solution: \(BW = \dfrac{GBW}{|A_v|} = \dfrac{1~\mathrm{MHz}}{20} = 50kHz\)

  • Problem 3: Find oscillation frequency of Wien bridge oscillator with \(R = 10~\mathrm{k\Omega}\), \(C = 100~\mathrm{nF}\)

  • Solution: \(f = \dfrac{1}{2\pi RC} = \dfrac{1}{2\pi \times 10k \times 100n} = 159.2 Hz\)

Typical GATE Problems - Oscillators

  • Problem 1: Phase shift oscillator with \(R = 1~\mathrm{k\Omega}\), \(C = 1~\mathring{\mu F}\)

  • Solution: \(f = \dfrac{1}{2\pi RC\sqrt{6}} = \dfrac{1}{2\pi \times 1k \times 1\mu \times \sqrt{6}} = 65.0 Hz\)

  • Problem 2: Find minimum gain for oscillation in phase shift oscillator

  • Solution: Minimum gain = 29

  • Problem 3: Colpitts oscillator with \(L = 100~\mu H\), \(C_1 = C_2 = 100~\mathrm{pF}\)

  • Solution: \(C_T = \dfrac{C_1 C_2}{C_1 + C_2} = 50~\mathrm{pF}\)

  • \(f = \dfrac{1}{2\pi\sqrt{LC_T}} = \dfrac{1}{2\pi\sqrt{100\mu \times 50p}} = 7.12 ~\mathrm{MHz}\)

Typical GATE Problems - Feedback

  • Problem 1: Feedback amplifier with \(A = 1000\), \(\beta = 0.01\)

  • Solution: \(A_f = \dfrac{A}{1 + A\beta} = \dfrac{1000}{1 + 1000 \times 0.01} = \dfrac{1000}{11} = 90.9\)

  • Problem 2: Find desensitivity factor for above amplifier

  • Solution: \(D = 1 + A\beta = 1 + 10 = 11\)

  • Problem 3: If open-loop gain changes by \(20\%\), find change in closed-loop gain

  • Solution: \(\dfrac{dA_f}{A_f} = \dfrac{1}{1 + A\beta} \times \dfrac{dA}{A} = \dfrac{1}{11} \times 0.2 = 1.8\%\)

Important Formulas Summary

Key Formulas - Op-Amps and Feedback

  • Inverting amplifier: \(A_v = -\dfrac{R_f}{R_i}\)

  • Non-inverting amplifier: \(A_v = 1 + \dfrac{R_f}{R_i}\)

  • Feedback amplifier: \(A_f = \dfrac{A}{1 + A\beta}\)

  • Bandwidth with feedback: \(BW_f = BW \times (1 + A\beta)\)

  • Integrator: \(V_o = -\dfrac{1}{RC}\int V_{in} dt\)

  • Differentiator: \(V_o = -RC\dfrac{dV_{in}}{dt}\)

  • Voltage follower: \(A_v = 1\)

  • Summing amplifier: \(V_o = -R_f\left(\dfrac{V_1}{R_1} + \dfrac{V_2}{R_2} + \dfrac{V_3}{R_3}\right)\)

Key Formulas - Oscillators

  • Wien bridge: \(f = \dfrac{1}{2\pi RC}\), Gain = 3

  • Phase shift: \(f = \dfrac{1}{2\pi RC\sqrt{6}}\), Gain = 29

  • Hartley: \(f = \dfrac{1}{2\pi\sqrt{L_T C}}\)

  • Colpitts: \(f = \dfrac{1}{2\pi\sqrt{LC_T}}\), \(C_T = \dfrac{C_1 C_2}{C_1 + C_2}\)

  • 555 Timer (Astable): \(f = \dfrac{1.44}{(R_1 + 2R_2)C}\)

  • 555 Timer (Monostable): \(t_p = 1.1RC\)

  • Crystal oscillator: \(f_s = \dfrac{1}{2\pi\sqrt{LC}}\)

  • Barkhausen criterion: \(|A\beta| = 1\), \(\angle A\beta = 0^{\circ}\)

Key Formulas - Filters and Converters

  • First-order filter: \(f_c = \dfrac{1}{2\pi RC}\)

  • Second-order filter: \(Q = \dfrac{f_0}{BW}\)

  • Schmitt trigger hysteresis: \(V_H = \dfrac{2R_1 V_{sat}}{R_1 + R_2}\)

  • ADC resolution: LSB = \(\dfrac{V_{ref}}{2^n}\)

  • DAC output: \(V_o = V_{ref} \times \dfrac{D}{2^n}\)

  • PLL (first-order): \(H(s) = \dfrac{K}{s + K}\)

  • Power amplifier efficiency: \(\eta = \dfrac{P_{out}}{P_{in}} \times 100\%\)

  • ENOB: \(ENOB = \dfrac{SNR - 1.76}{6.02}\)