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

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 \(10-50~\mu V/^{\circ}C\))

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 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

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

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

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

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

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 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 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

• 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

• 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 (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

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

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

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 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

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

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