Discover the Barkhausen Criterion for oscillator and how it enables sustained oscillations in electronic circuits. Learn the conditions, importance, and applications of Barkhausen’s criterion for oscillator in this detailed guide.
Oscillators play a vital role in electronic systems—from generating clock signals to enabling wireless communication. But what ensures that these oscillations are continuous and stable? The answer lies in a fundamental principle known as the Barkhausen Criterion for Oscillator. In this article, we’ll break down what is Barkhausen Criterion for oscillator, its importance, the conditions it sets for sustained oscillations, and how it’s applied in designing oscillators.
What is Barkhausen Criterion?
The Barkhausen criterion is a set of conditions that must be met in an electronic feedback circuit to produce continuous (sustained) oscillations without any external input signal. It is named after German physicist Heinrich Barkhausen, who formulated it in the early 20th century.
In simple terms, the Barkhausen criterion for oscillator ensures that the loop gain and phase conditions of a feedback circuit are just right to maintain oscillations indefinitely.
Barkhausen’s Criterion for Oscillation: The Two Conditions
For sustained oscillations to occur, a feedback system must satisfy two essential conditions, known collectively as the Barkhausen criterion:
- Loop Gain = 1 (or unity):
The product of the amplifier gain (A) and feedback network gain (β) must equal 1:
Aβ = 1 - Total Phase Shift = 0° or 360°:
The total phase shift around the loop must be a multiple of 360° (or 0°), which means the feedback signal is in phase with the original input signal.
These two conditions are also called the Barkhausen criteria of oscillator, and they ensure that the signal reinforces itself in each cycle, instead of fading away or growing uncontrollably.
Understanding the Feedback Loop
To understand how the Barkhausen criterion for oscillator works, let’s look at a typical positive feedback loop:
- The amplifier amplifies the input signal.
- The feedback network sends part of the output back to the input.
- If the gain and phase conditions are met, the circuit starts to oscillate at a particular frequency.
When the loop gain is exactly 1, and the feedback signal is in phase, the system maintains stable oscillations without increasing or decreasing in amplitude.
In practical circuits, the loop gain doesn’t remain constant—it changes over time. To initiate and sustain oscillations, the system must allow this time-varying behavior in the loop gain. For example, in a basic second-order oscillator circuit, the loop gain typically starts off greater than one, causing the oscillation amplitude to increase. As the active components begin to saturate or the gain is dynamically controlled, the circuit enters a nonlinear operating region. This nonlinear behavior stabilizes the oscillation amplitude. Once this steady state is reached, the average loop gain ∣Aβ∣| settles at exactly 1, satisfying the Barkhausen criterion for sustained oscillations.
Mathematical Representation of Barkhausen Criterion
For Positive Feedback System
To understand Barkhausen’s criterion mathematically, consider a system arranged with positive feedback, as illustrated in Figure 1.
To determine the transfer function, we can rearrange the equation given above as follows:
Where,
- A = Amplifier gain
- β = Feedback network gain
- Aβ = Loop gain
In an oscillator, no external input is required (or it is set to zero), making it an autonomous circuit—one that generates periodic waveforms inherently from within. Even when the input is zero, the circuit produces a finite, non-zero output. This behavior implies the presence of infinite gain. The transfer function serves as a fundamental descriptor of the system, helping us understand the specific conditions under which this infinite gain occurs.
To determine the condition for infinite gain in the circuit:
We get,
From the above equation, we can deduce that the conditions for oscillation are:
- The magnitude of the loop gain must be equal to one:
∣Aβ∣=1 - The phase angle of the loop gain must be an integer multiple of 2π:
∠Aβ=2Nπ
where N∈{…,−2,−1,0,1,2,…}
These two conditions together define the Barkhausen criterion for oscillator circuits.
If |Aβ| < 1, oscillations die out.
If |Aβ| > 1, oscillations grow uncontrollably.
Only when |Aβ| = 1, the oscillations are sustained.
For Negative Feedback System
Let’s consider a system arranged with negative feedback, as illustrated in Figure 2. By examining the relationship between the amplifier’s input and the output voltage (Vout), we can express the system as follows:
To derive Barkhausen’s criterion, we can rearrange the above equation to express the transfer function as follows:
The condition for infinite gain:
From the above equation, we can conclude that the conditions for sustained oscillation are:
- The magnitude of the loop gain must be unity:
∣Aβ∣=1 - The phase angle of the loop gain must be an odd multiple of π:
∠Aβ=(2N+1)π
where N∈{…,−2,−1,0,1,2,…}
Practical Applications of Barkhausen Criteria
The Barkhausen criterion for oscillator is used in designing different types of oscillators:
- RC Phase Shift Oscillator
- Wien Bridge Oscillator
- Hartley Oscillator
- Colpitts Oscillator
- Crystal Oscillator
In each case, designers ensure that the gain and phase shift conditions match the Barkhausen criteria at the desired frequency of oscillation.
Why is Barkhausen Criterion Important?
Understanding what is Barkhausen criterion helps electronics engineers:
- Design stable and reliable oscillator circuits.
- Prevent circuits from producing noise or distortion.
- Fine-tune frequency in communication systems.
- Ensure self-starting oscillation conditions in feedback loops.
It’s not just about starting oscillations—it’s about maintaining them stably over time.
Summary Table
| Condition | Requirement |
|---|---|
| Loop Gain | Aβ = 1 |
| Phase Shift | 0° or 360° (in phase feedback) |
| Result | Sustained Oscillations |
Conclusion
The Barkhausen criterion for oscillator forms the foundation for building stable, self-sustaining oscillators in electronic circuits. By ensuring the correct gain and phase shift around a feedback loop, designers can achieve continuous oscillation without distortion or signal loss. Whether you’re working with RF, audio, or digital systems, mastering the Barkhausen’s criterion for oscillation is key to success.
So next time you’re designing or analyzing an oscillator, remember the golden rule: gain must be unity, and phase must be zero—that’s the Barkhausen criterion.
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