Oscillator Derivations – Analog Electronics

Oscillators are circuits that generate periodic signals without external input. They use positive feedback to sustain oscillations.

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1. Barkhausen Criterion Derivation

Consider an amplifier with gain A and feedback factor β.

Output voltage = A × input

Feedback signal:

Vf = β Vo

For sustained oscillations:

Input = Feedback signal

Therefore

Vi = βVo

But

Vo = A Vi

Substitute Vi

Vo = A (βVo)

Simplify

Aβ = 1

Thus the two conditions are:

• Loop gain |Aβ| = 1
• Total phase shift = 360°

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2. RC Phase Shift Oscillator Derivation

The RC phase shift oscillator uses three RC sections to produce 180° phase shift.

Each RC section produces:

60° phase shift

Total phase shift:

3 × 60° = 180°

Amplifier (CE stage) provides:

180°

Total loop phase shift:

360°

Solving the RC network gives frequency:

f = 1 / (2πRC√6)

Minimum amplifier gain required:

A ≥ 29

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3. Wien Bridge Oscillator Derivation

The Wien bridge oscillator uses a bridge network of resistors and capacitors.

At resonance condition:

R1 = R2 = R C1 = C2 = C

The frequency of oscillation becomes:

f = 1 / (2πRC)

For sustained oscillations:

Loop gain Aβ = 1

For Wien bridge:

Amplifier gain A = 3

Thus oscillations occur when amplifier gain equals 3.

 

GATE Electrical Engineering

Analog Electronics – Complete Formula Sheet

This page provides important formulas for quick revision from the entire Analog Electronics syllabus.

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1. Diode Equations

Diode Current Equation

I = Is (e^(V/ηVT) − 1)

Thermal Voltage

VT = kT/q ≈ 25 mV

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2. Rectifier Formulas

Half Wave Rectifier

Vdc = Vm / π

Efficiency = 40.6%

Ripple Factor = 1.21

Full Wave Rectifier

Vdc = 2Vm / π

Efficiency = 81.2%

Ripple Factor = 0.482

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3. BJT Transistor Formulas

Ic = β Ib

β = α / (1 − α)

α = β / (1 + β)

gm = Ic / VT

re = VT / Ic

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4. CE Amplifier Gain

Av = −Rc / re

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5. h-Parameter Gain

Av = − (hfe RL / hie)

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6. Differential Amplifier

Vd = V1 − V2

Vc = (V1 + V2)/2

CMRR = Ad / Ac

CMRR(dB) = 20 log (Ad/Ac)

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7. Op-Amp Gains

Inverting Amplifier

Av = −Rf / Rin

Non-Inverting Amplifier

Av = 1 + (Rf / R1)

Voltage Follower

Av = 1

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8. Integrator

Vo = − (1/RC) ∫ Vin dt

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9. Differentiator

Vo = − RC (dVin/dt)

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10. Filters

fc = 1 / (2πRC)

First Order Slope = −20 dB/decade

Second Order Slope = −40 dB/decade

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11. Oscillators

RC Phase Shift Oscillator

f = 1 / (2πRC√6)

Wien Bridge Oscillator

f = 1 / (2πRC)

Barkhausen Criterion

Loop Gain = 1 Phase Shift = 360°

 

GATE Electrical – Analog Electronics

Practice Problems – Page 19

This section contains mixed problems from Diodes, Transistors, Operational Amplifiers, Filters and Oscillators.

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Problem 181

Question:

What is the cut-in voltage of a silicon diode?

Answer:

≈ 0.7 V

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Problem 182

Question:

What is the cut-in voltage of a germanium diode?

Answer:

≈ 0.3 V

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Problem 183

Question:

If β = 100 and base current is 40 μA, find collector current.

Solution:

Ic = β × Ib

Ic = 100 × 40 μA = 4 mA

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Problem 184

Question:

What is the ripple factor of a full wave rectifier?

Answer:

0.482

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Problem 185

Question:

What is the efficiency of a full wave rectifier?

Answer:

81.2%

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Problem 186

Question:

If op-amp gain = 20 and input voltage = 0.2 V, find output voltage.

Solution:

Vo = Av × Vin

Vo = 20 × 0.2 = 4 V

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Problem 187

Question:

What is the slope of first order filter?

Answer:

−20 dB/decade

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Problem 188

Question:

What is the slope of second order filter?

Answer:

−40 dB/decade

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Problem 189

Question:

Which oscillator uses positive feedback?

Answer:

All oscillator circuits use positive feedback.

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Problem 190

Question:

What is the ideal CMRR of an op-amp?

Answer:

Infinite

 

GATE Electrical – Analog Electronics

Practice Problems – Page 18

This section contains problems based on Oscillators and feedback principles.

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Problem 171

Question:

What is an oscillator?

Answer:

An oscillator is an electronic circuit that generates periodic signals without any external input signal.

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Problem 172

Question:

What are the two essential conditions for oscillation?

Answer:

• Loop gain must be equal to 1
• Total phase shift must be 360° or 0°

This is known as Barkhausen Criterion.

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Problem 173

Question:

Write the frequency formula of an RC phase shift oscillator.

Answer:

f = 1 / (2πRC√6)

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Problem 174

Question:

What is the frequency formula of Wien bridge oscillator?

Answer:

f = 1 / (2πRC)

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Problem 175

Question:

Which oscillator is commonly used for audio frequency generation?

Answer:

Wien Bridge Oscillator

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Problem 176

Question:

Why is positive feedback required in oscillators?

Answer:

To sustain continuous oscillations.

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Problem 177

Question:

If R = 10 kΩ and C = 0.01 μF, find frequency of Wien bridge oscillator.

Solution:

f = 1 / (2πRC)

RC = 10,000 × 0.01 × 10⁻⁶

f ≈ 1591 Hz

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Problem 178

Question:

Which oscillator uses three RC sections?

Answer:

RC Phase Shift Oscillator

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Problem 179

Question:

What type of feedback is used in oscillators?

Answer:

Positive feedback

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Problem 180

Question:

Which device is commonly used as amplifier in RC oscillator?

Answer:

Transistor or Operational Amplifier.

 

GATE Electrical – Analog Electronics

Practice Problems – Page 17

This section contains problems based on Active Filters using Operational Amplifiers.

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Problem 161

Question:

What is a low pass filter?

Answer:

A low pass filter allows low frequency signals to pass and attenuates high frequency signals.

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Problem 162

Question:

What is a high pass filter?

Answer:

A high pass filter allows high frequency signals to pass and blocks low frequency signals.

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Problem 163

Question:

Write the cut-off frequency formula of an RC filter.

Answer:

fc = 1 / (2πRC)

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Problem 164

Question:

If R = 10 kΩ and C = 0.01 μF, find the cut-off frequency.

Solution:

fc = 1 / (2πRC)

RC = 10,000 × 0.01 × 10⁻⁶

fc ≈ 1591 Hz

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Problem 165

Question:

What is a band pass filter?

Answer:

A band pass filter allows signals within a certain frequency range to pass.

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Problem 166

Question:

What is a band stop filter?

Answer:

It blocks a specific band of frequencies while allowing others to pass.

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Problem 167

Question:

Why are active filters preferred over passive filters?

Answer:

Because they provide amplification and better frequency control.

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Problem 168

Question:

Which component in a filter determines cut-off frequency?

Answer:

Resistor and Capacitor

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Problem 169

Question:

What is the slope of a first order filter?

Answer:

−20 dB/decade

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Problem 170

Question:

What is the slope of a second order filter?

Answer:

−40 dB/decade

 

GATE Electrical – Analog Electronics

Practice Problems – Page 16

This section contains problems based on Operational Amplifier Comparator and Schmitt Trigger circuits.

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Problem 151

Question:

What is a comparator circuit?

Answer:

A comparator compares two voltages and produces a high or low output depending on which input voltage is larger.

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Problem 152

Question:

What happens when Vin > Vref in a comparator?

Answer:

Output becomes positive saturation.

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Problem 153

Question:

What happens when Vin < Vref?

Answer:

Output becomes negative saturation.

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Problem 154

Question:

What is a Schmitt trigger?

Answer:

A Schmitt trigger is a comparator circuit with positive feedback that provides hysteresis.

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Problem 155

Question:

What is hysteresis in Schmitt trigger?

Answer:

It is the difference between upper and lower threshold voltages.

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Problem 156

Question:

Write the formula for hysteresis voltage.

Answer:

VH = VUT − VLT

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Problem 157

Question:

Why is Schmitt trigger used?

Answer:

To eliminate noise and produce clean digital signals.

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Problem 158

Question:

What is the output of a comparator if input is a sine wave?

Answer:

Square wave

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Problem 159

Question:

Which op-amp application converts analog signals to digital signals?

Answer:

Comparator

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Problem 160

Question:

What is the advantage of positive feedback in Schmitt trigger?

Answer:

It creates hysteresis which improves noise immunity.

 

GATE Electrical – Analog Electronics

Practice Problems – Page 15

This section contains problems based on Operational Amplifier Integrator and Differentiator circuits.

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Problem 141

Question:

What is an integrator circuit?

Answer:

An integrator is an op-amp circuit whose output voltage is proportional to the integral of the input voltage.

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Problem 142

Question:

Write the output voltage equation of an op-amp integrator.

Answer:

Vo = − (1/RC) ∫ Vin dt

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Problem 143

Question:

If R = 10 kΩ and C = 0.1 μF, find RC.

Solution:

RC = 10,000 × 0.1 × 10⁻⁶

RC = 0.001 sec

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Problem 144

Question:

What is the output waveform of an integrator if input is a square wave?

Answer:

Triangular waveform

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Problem 145

Question:

Define differentiator circuit.

Answer:

A differentiator produces an output proportional to the rate of change of input voltage.

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Problem 146

Question:

Write the output equation of an op-amp differentiator.

Answer:

Vo = − RC (dVin/dt)

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Problem 147

Question:

If input is a triangular waveform, what is the output of a differentiator?

Answer:

Square wave

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Problem 148

Question:

Why is practical differentiator used instead of ideal differentiator?

Answer:

To avoid excessive noise amplification and instability.

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Problem 149

Question:

Where are integrator circuits commonly used?

Answer:

Analog computers, signal processing and waveform generation.

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Problem 150

Question:

Where are differentiator circuits used?

Answer:

Edge detection and wave shaping circuits.