--- title: "Barkhausen criteria and relaxation oscillator" subject: "Fundamental of Electrical and Electronics Engineering" chapter: 6 key: "semester_1/fundamental_of_electrical_and_electronics_engineering:barkhausen-relaxation-oscillator" source: https://www.bctengineer.com/q/semester_1/fundamental_of_electrical_and_electronics_engineering/barkhausen-relaxation-oscillator site: BCT Engineering --- # Barkhausen criteria and relaxation oscillator Fundamental of Electrical and Electronics Engineering · Chapter 6: Operational Amplifier and Oscillator - Asked in: 2081 Baishakh (new course), 1+3+1 marks - Video: [Oscillators & Barkhausen Criterion - Basic Introduction](https://www.youtube.com/watch?v=skDVcnVwK3A) — The Organic Chemistry Tutor - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_1/fundamental_of_electrical_and_electronics_engineering/barkhausen-relaxation-oscillator - Whole chapter: https://www.bctengineer.com/md/semester_1/fundamental_of_electrical_and_electronics_engineering/chapter-6 ## Question State Barkhausen criteria. Draw and explain circuit diagram of Relaxation oscillator (square wave generator). Write its frequency of oscillation. ## Short answer Barkhausen: |Aβ| = 1 and loop phase = 0° or 360°. A relaxation oscillator uses a Schmitt trigger (R1‑R2) and an RC timing network. The capacitor charges/discharges between ±βV_sat, producing a square wave. Frequency f = 1/[2RC ln[(1+β)/(1−β)]]; with R2 ≈ 1.16 R1, f ≈ 1/(2RC). ## Answer #### Barkhausen criteria For sustained oscillations in a feedback circuit with amplifier gain A and feedback factor β: 1. The loop gain magnitude must be unity: **|Aβ| = 1**. 2. The total phase shift around the loop must be **0° or 360°** (positive feedback). (In practice |Aβ| is made slightly greater than 1 to start oscillations.) #### Relaxation oscillator (op-amp square wave generator) A relaxation oscillator produces a non-sinusoidal output by repeatedly charging and discharging a capacitor. The op-amp works as a comparator (Schmitt trigger) with positive feedback through R1–R2 and an RC timing circuit on the inverting input. ```text R +---/\/\/\-----+ | | | |\ | Vc +--+----|-\ | | | >------+---o Vo (±Vsat) C +---|+/ | | | |/ | GND +----/\/\/\---+ | R2 R1 | GND ``` #### Operation - Positive feedback gives V+ = β Vo, where **β = R1/(R1 + R2)**. The output is always at +Vsat or −Vsat. - Suppose Vo = +Vsat. Then V+ = +β Vsat. C charges through R towards +Vsat. - When Vc just exceeds +β Vsat, V− > V+ and the output switches to −Vsat. Now V+ = −β Vsat. - C now discharges and charges in the negative direction through R towards −Vsat. - When Vc falls below −β Vsat, the output switches back to +Vsat. The cycle repeats. The output is a **square wave** (±Vsat) and the capacitor voltage is an exponential triangle-like wave between ±β Vsat. ```text Vo +Vsat |‾‾‾‾| |‾‾‾‾| 0 +----+----+----+---- t -Vsat |____| |___ Vc +βVsat /\ /\ -βVsat / \__/ / \__/ ``` #### Frequency of oscillation ```text T = 2RC ln[(1 + β)/(1 − β)] f = 1/T = 1/(2RC ln[(1 + β)/(1 − β)]) ``` If R2 = 1.16 R1 (β ≈ 0.462), ln[(1 + β)/(1 − β)] ≈ 1, so **f ≈ 1/(2RC)**. If R1 = R2 (β = 0.5), T = 2RC ln 3 ≈ 2.2RC. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Which pair states the Barkhausen criteria for sustained oscillations? A. |Aβ| = 1 and total loop phase shift 0° or 360° B. |Aβ| = 0 and phase shift 90° C. |Aβ| < 1 and phase shift 180° D. |Aβ| = 1 and phase shift 180° only Answer: A. Unity loop gain magnitude and 0° or 360° loop phase (positive feedback) are required. (Key point: Barkhausen criteria) 2. In the relaxation oscillator, how does the op-amp work? A. As a linear inverting amplifier B. As a comparator (Schmitt trigger) with positive feedback, switching between +Vsat and -Vsat C. As a voltage follower D. As an integrator only Answer: B. Positive feedback through R1-R2 makes the output saturate at either +Vsat or -Vsat. (Key point: Role of op-amp) 3. The output switches from +Vsat to -Vsat when the capacitor voltage Vc: A. reaches zero B. just exceeds +βVsat, where β = R1/(R1+R2) C. reaches +Vsat D. falls below -βVsat Answer: B. When Vc exceeds V+ = βVsat, V- > V+ and the output flips. (Key point: Switching threshold) 4. What is the frequency of oscillation? A. f = 1/(2RC ln[(1+β)/(1-β)]) B. f = 1/(2πRC) C. f = 2RC ln[(1+β)/(1-β)] D. f = 1/(RC ln 3 only) Answer: A. T = 2RC ln[(1+β)/(1-β)], and f = 1/T. 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