---
title: "Barkhausen criteria; Wien bridge frequency"
subject: "Basic Electronics Engineering"
chapter: 4
key: "semester_2/basic_electronics_engineering:barkhausen-wien-derive-frequency"
source: https://www.bctengineer.com/q/semester_2/basic_electronics_engineering/barkhausen-wien-derive-frequency
site: BCT Engineering
---

# Barkhausen criteria; Wien bridge frequency

Basic Electronics Engineering · Chapter 4: The Operational Amplifier and Oscillator

- Asked in: 2075 Baisakh, 2+2+2 marks
- Video: [BARKHAUSEN CRITERIA, WEIN BRIDGE, AND PHASE SHIFT OSCILLATOR](https://www.youtube.com/watch?v=YpKGCwkt5Xk) — Prof. Barapate's Tutorials
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_2/basic_electronics_engineering/barkhausen-wien-derive-frequency
- Whole chapter: https://www.bctengineer.com/md/semester_2/basic_electronics_engineering/chapter-4

## Question

State Barkhausen criteria for oscillation. Draw Wien bridge oscillator circuit to generate sine wave and derive the frequency of the generated sine wave.

## Short answer

Barkhausen criteria: |Aβ|=1 and loop phase shift 0°. The Wien bridge oscillator uses a non-inverting amplifier with gain A=3 (Rf=2R1) and a lead-lag RC network; setting imaginary part of feedback factor to zero gives ω=1/(RC), so oscillation frequency f=1/(2πRC).

## Answer

#### Barkhausen criteria

1. Loop gain magnitude |Aβ| = 1.
2. Total loop phase shift = 0° (or 360°).

#### Wien bridge oscillator

```text
           +------[ Rf ]-------+
           |                   |
 GND--[R1]-+        |\         |
           +--------|- \       |
                    |   >------+----o Vo
           +--------|+ /       |
           |        |/         |
           |                   |
           +--[ R ]--||--------+
           |         C     (Zs: series R and C)
         +-+-+
         |   |
        [R] === C   (Zp: R parallel C)
         |   |
         +-+-+
           |
          GND
```

The amplifier is a non-inverting op-amp with gain A = 1 + Rf/R1. The positive feedback is taken through a lead-lag network: a series R and C from the output to the non-inverting input, and a parallel R and C from that input to ground. Rf and R1 form the negative feedback and fix the gain.

#### Derivation of frequency

Series arm: Zs = R + 1/(jωC)
Parallel arm: Zp = R/(1 + jωRC)

Feedback factor:
β = Zp/(Zs + Zp) = 1/(1 + Zs/Zp)

```text
β = 1 / [ (1 + jωRC)(R + 1/jωC)/R + 1 ]
  = 1 / [ 1 + (1 + jωRC)(1 + 1/jωRC) ]
  = 1 / [ 3 + j(ωRC − 1/ωRC) ]
```

For a 0° phase shift, the imaginary part must be zero:

ωRC = 1/(ωRC), so ω² = 1/(RC)², i.e. ω = 1/(RC)

**f = 1/(2πRC)**

At this frequency β = 1/3. For |Aβ| = 1 we need A = 3, i.e. 1 + Rf/R1 = 3, so Rf = 2R1 (in practice slightly more than 2R1, so that oscillation can start).

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Which is the phase condition of Barkhausen criteria?
   A. Total loop phase shift is 90°
   B. Total loop phase shift is 180° exactly
   C. Total loop phase shift is 0° (or 360°)
   D. Phase shift is irrelevant if gain is 1
   Answer: C. Positive feedback with zero net phase shift is needed for sustained oscillation. (Key point: Barkhausen criteria)

2. What is β of the Wien network at the oscillation frequency?
   A. β = 1/3
   B. β = 1
   C. β = 1/2
   D. β = 3
   Answer: A. Putting the imaginary part zero leaves β = 1/(3 + 0) = 1/3. (Key point: Feedback factor)

3. How is the oscillation frequency found?
   A. By making the real part of β zero
   B. By setting Rf = R1
   C. By making the imaginary part of β zero: ωRC = 1/(ωRC)
   D. By setting ω = 0
   Answer: C. Zero phase needs the imaginary term to vanish, giving f = 1/(2πRC). (Key point: Condition for frequency)

4. What gain A and resistor ratio are needed for |Aβ| = 1?
   A. A = 3, so Rf = 2R1
   B. A = 1/3, so Rf = R1/2
   C. A = 3, so Rf = 3R1
   D. A = 2, so Rf = R1
   Answer: A. A = 1 + Rf/R1 = 3 gives Rf = 2R1 (slightly more in practice to start oscillation). (Key point: Gain required)

## Related in Basic Electronics Engineering

- [Wien bridge RC oscillator, Barkhausen criteria](https://www.bctengineer.com/question.md?key=semester_2%2Fbasic_electronics_engineering%3Awien-bridge-barkhausen) — Basic Electronics Engineering, chapter 4
- [Wien bridge oscillator circuit and operation](https://www.bctengineer.com/question.md?key=semester_2%2Fbasic_electronics_engineering%3Awien-bridge-draw-explain) — Basic Electronics Engineering, chapter 4, asked 4×
- [Barkhausen criteria and square wave generator](https://www.bctengineer.com/question.md?key=semester_2%2Fbasic_electronics_engineering%3Abarkhausen-square-wave-generator) — Basic Electronics Engineering, chapter 4

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