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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.