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Barkhausen criteria and square wave generator

Short answer

Barkhausen criteria: |Aβ|=1 and loop phase shift of 0° or 360°. The square wave generator is an astable multivibrator with an op-amp comparator, positive feedback (R1,R2), and an RC timing network; the capacitor voltage cycles between ±βVsat, causing output switching and yielding a square wave of period T=2RC ln[(1+β)/(1-β)].

State Barkhausen criteria. Draw the circuit diagram of square wave generator and explain how it works.

Barkhausen criteria

For sustained oscillations in a feedback amplifier with gain A and feedback factor β:

  1. The magnitude of the loop gain must be unity: |Aβ| = 1.
  2. The total phase shift around the loop must be 0° or 360° (positive feedback): ∠Aβ = 0° or 360°.

Square wave generator (astable multivibrator)

          +----------[ R ]-----------+
          |                          |
          |         |\               |
          +---------|- \             |
          |         |   >------------+----o Vo
         === C   +--|+ /             |
          |      |  |/               |
         GND     +-------[ R2 ]------+
                 |
               [ R1 ]
                 |
                GND

The op-amp works as a comparator. R2 and R1 give positive feedback (β = R1/(R1 + R2)). R and C form the timing network on the inverting side.

How it works

  1. When Vo = +Vsat, the non-inverting input is at +βVsat. C charges through R towards +Vsat.
  2. When Vc just exceeds +βVsat, the comparator output switches to −Vsat.
  3. The reference becomes −βVsat, and C discharges through R and then charges towards −Vsat.
  4. When Vc falls below −βVsat, the output switches back to +Vsat.
  5. The cycle repeats continuously.

The output is a square wave of amplitude ±Vsat. Time period T = 2RC ln[(1 + β)/(1 − β)]; for R1 = R2, f ≈ 1/(2.2 RC). The duty cycle is 50% because the charge and discharge paths are the same.

✓ Answer checked by Claude Opus 5.5 (high effort)

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