The Barkhausen criterion is a fundamental principle that establishes the conditions necessary for an electronic circuit to generate and sustain oscillations. It is a critical concept in the design and analysis of various types of oscillators, which are circuits that produce repetitive, time-varying signals without an external input signal.

Barkhausen Criterion Explained

For an oscillator to produce sustained oscillations, two primary conditions must be met simultaneously around the feedback loop. These conditions relate to the magnitude and phase of the loop gain, which is the product of the amplifier gain and the feedback network's transfer function.

Conditions for Sustained Oscillation

The Barkhausen criterion specifies two essential conditions for a circuit to operate as a self-sustaining oscillator:

• Magnitude Condition: The magnitude of the loop gain ($A\beta$) must be equal to unity (1).
• Phase Condition: The total phase shift around the closed loop must be zero degrees ($0^{\circ}$) or an integer multiple of $360^{\circ}$ ($n \times 360^{\circ}$, where $n = 0, 1, 2, \ldots$).

Understanding Loop Gain ($A\beta$)

The term $A\beta$ represents the loop gain, where:

• $A$ is the voltage gain of the active amplifying device (e.g., a transistor or op-amp) in the oscillator circuit.
• $\beta$ (beta) is the feedback factor, which is the fraction of the output voltage that is fed back to the input of the amplifier. It represents the voltage transfer function of the feedback network.

The product $A\beta$ quantifies the overall gain experienced by a signal as it completes one full cycle through the amplifier and the feedback network.

Magnitude Requirement for Oscillation

For oscillations to be sustained, the signal fed back to the input must have an amplitude equal to the original signal. This ensures that the oscillations do not die out or grow indefinitely, leading to stable output. Mathematically, this condition is expressed as:

\[|A\beta| = 1\]

• If $|A\beta| < 1$, the amplitude of the signal will decrease with each cycle, and the oscillations will eventually decay to zero.
• If $|A\beta| > 1$, the amplitude of the signal will increase with each cycle. This would cause the amplifier to saturate and distort the waveform, or it might lead to instability unless some amplitude limiting mechanism is in place.

Phase Requirement for Oscillation

The second crucial aspect of the Barkhausen criterion involves the phase shift. For a signal to reinforce itself (positive feedback), it must arrive back at the amplifier input in phase with the original signal. This means the total phase shift introduced by the amplifier and the feedback network combined must be $0^{\circ}$ or a multiple of $360^{\circ}$. Mathematically:

\[\angle A\beta = n \times 360^{\circ} \quad \text{or} \quad n \times 2\pi \text{ radians}\]

where $n$ is an integer ($0, 1, 2, \ldots$). If the phase shift is not $0^{\circ}$ (or a multiple of $360^{\circ}$), the feedback will be negative, leading to cancellation of the signal and preventing sustained oscillations.

Barkhausen Criterion Combined

When both the magnitude and phase conditions are simultaneously satisfied, the circuit will generate continuous, stable oscillations at a specific frequency. The most concise way to express the Barkhausen criterion, encompassing both the magnitude and phase requirements, is:

\[A\beta = 1\]

This equation, representing a complex number equality, implies that the magnitude of $A\beta$ is 1 and its phase angle is $0^{\circ}$ (or $360^{\circ}$), which are the precise conditions for the self-starting and sustained operation of an oscillator.