Barkhausen Criterion

Every oscillator is just an amplifier with positive feedback, but not every amplifier with feedback will oscillate. The Barkhausen criterion defines the exact conditions under which a feedback amplifier sustains steady oscillations. Understanding it explains why your Wien bridge oscillator starts up, stabilizes, and holds a clean sine wave.

Core Concept

An oscillator is a feedback system with no external input. The output feeds back into the input through a frequency-selective network. For the output to sustain itself at one particular frequency, the signal must arrive back at the input with the same amplitude and phase it left with. This is the physical meaning of the Barkhausen criterion.

The loop gain is the product of the amplifier gain A and the feedback fraction β. If |Aβ| is greater than 1, oscillations grow and clip. If |Aβ| is less than 1, oscillations decay and die. Sustained clean oscillations need |Aβ| = 1 exactly. The phase condition requires the total phase shift around the entire loop to be 0° or a multiple of 360°.

Real oscillator circuits use automatic gain control to maintain |Aβ| = 1. The Wien bridge oscillator uses a thermistor or a JFET as a nonlinear element to reduce gain if the amplitude grows. Without this, the output clips into a square wave. The LM741 was historically used, but the TL071 is better because of its lower distortion and higher slew rate.

Key Equations

Barkhausen magnitude condition:

|A(jω) × β(jω)| = 1 where A is the amplifier voltage gain (dimensionless) and β is the feedback fraction (dimensionless).

Barkhausen phase condition:

∠A(jω) + ∠β(jω) = 0° or ±360° meaning the total phase shift around the loop is exactly zero (or a full multiple of 360°).

Closed-loop gain of a feedback amplifier (for reference):

Af = A / (1 - Aβ) At the oscillation condition Aβ = 1, the denominator is zero, so gain is theoretically infinite. This represents sustained oscillation with no input.

Given:
  Wien bridge oscillator using TL071
  Feedback network β = 1/3 at the resonant frequency f0
  Required amplifier gain A to satisfy Barkhausen magnitude condition

Why this formula:
  Barkhausen: |A × β| = 1, so A = 1/β

Formula:
  A = 1 / β

Substitution:
  A = 1 / (1/3)

Calculation:
  A = 3

The non-inverting op-amp gain is set by:
  A = 1 + Rf/R1 = 3
  Rf/R1 = 2
  Choose R1 = 10 kΩ, Rf = 20 kΩ

Phase check:
  Wien network phase shift at f0 = 0°
  Non-inverting amplifier phase shift = 0°
  Total loop phase = 0° (satisfies phase condition)

Final Answer:
  Gain A = 3, Rf = 20 kΩ, R1 = 10 kΩ
  Both Barkhausen conditions satisfied at f0.

Exam Tip: The most common GATE trap is confusing the two Barkhausen conditions. Magnitude condition alone is not enough. Students often write |Aβ| = 1 and forget the phase condition must also be 0° or 360°. Another trap: Barkhausen gives necessary conditions for oscillation, not sufficient ones. A circuit satisfying Barkhausen may still not oscillate if it cannot self-start. Startup requires |Aβ| slightly greater than 1 initially, then gain compression brings it back to 1.

Key Properties

• Barkhausen criterion applies to linear feedback oscillators only. It does not apply to relaxation oscillators like the 555 timer.
• The phase condition is the most restrictive. The feedback network must provide exactly the right phase shift at f0.
• A Wien bridge feedback network gives β = 1/3 and 0° phase at f0 = 1/(2πRC). The amplifier must provide A = 3.
• If |Aβ| > 1, the oscillator self-starts but the output clips at the supply rail unless automatic gain control is applied.
• The Barkhausen criterion predicts frequency and amplitude conditions but does not predict startup transient behavior.
• In practice, |Aβ| is set slightly above 1 during startup. Nonlinear gain reduction in the amplifier (via diode, thermistor, or JFET) brings it down to 1 in steady state.

Quick Revision

• Barkhausen magnitude: |Aβ| = 1 at the oscillation frequency.
• Barkhausen phase: total loop phase = 0° or 360° at the oscillation frequency.
• Both conditions must be met simultaneously. One alone is not sufficient.
• For startup, |Aβ| > 1. Nonlinear gain compression brings it to 1 in steady state.
• Wien bridge: β = 1/3, phase = 0° at f0. Requires amplifier gain A = 3.
• Barkhausen applies to sinusoidal (linear) oscillators, not relaxation oscillators.
• Loop gain = Aβ. At oscillation, closed-loop gain denominator (1 - Aβ) = 0.
• Exam trap: Writing |Aβ| = 1 as the only condition. Phase condition ∠(Aβ) = 0° is equally required and is separately tested in GATE.