Explain the Barkhausen criterion for the oscillation. How is it used in oscillators?

Barkhausen Criterion for Oscillation

The Barkhausen criterion is a fundamental principle in electronics that provides the conditions necessary for a circuit to produce sustained oscillations. Oscillations are periodic variations in voltage or current and are essential in various applications such as radio transmitters, clocks, and signal generators.

The criterion states that for a circuit to oscillate, it must satisfy two main conditions:

1. The loop gain must be equal to unity (1).
2. The phase shift around the loop must be zero or an integer multiple of 2π radians.

Let's break down these conditions and understand their implications:

1. Loop Gain Must Be Equal to Unity

The loop gain of an oscillator circuit is the product of the gains of all the components within the feedback loop. For sustained oscillations, the magnitude of the loop gain (|Aβ|) must be exactly 1. If the gain is less than 1, the oscillations will die out over time. If the gain is greater than 1, the oscillations will grow until they are limited by some non-linear effects in the circuit.

The loop gain is given by the formula:

[ |Aβ| = |A| \cdot |β| ]

Where:

• ( |A| ) is the magnitude of the amplifier gain.
• ( |β| ) is the magnitude of the feedback factor.

2. Phase Shift Around the Loop Must Be Zero or an Integer Multiple of 2π Radians

The total phase shift around the feedback loop must be such that the signal returns to the input in phase with the original signal. This ensures constructive interference, which is necessary for sustained oscillations. The phase shift is the sum of the phase shifts introduced by each component in the loop.

The phase shift condition can be expressed as:

[ \angle Aβ = \angle A + \angle β = 0° \text{ or } 360° \text{ or } ±n \cdot 360° ]

Where:

• ( \angle A ) is the phase angle of the amplifier.
• ( \angle β ) is the phase angle of the feedback network.
• ( n ) is an integer.

How the Barkhausen Criterion is Used in Oscillators

Oscillator circuits are designed to meet the Barkhausen criterion to ensure that they can start and maintain oscillations. Here's how the criterion is applied in practice:

Step-by-Step Approach

1. Designing the Amplifier: An amplifier is designed with a gain greater than necessary to compensate for any losses in the circuit. This is often done with a transistor or an operational amplifier.

2. Incorporating the Feedback Network: A feedback network is added to the circuit to feed a portion of the output signal back to the input. This network is designed to provide the necessary phase shift to meet the Barkhausen criterion.

3. Adjusting the Gain: The gain of the amplifier is adjusted (often through a variable resistor or a feedback network) so that the loop gain is as close to unity as possible without exceeding it.

4. Ensuring Phase Shift: The phase shift introduced by the feedback network is adjusted to ensure that the total phase shift around the loop is 0° or ±n \cdot 360°.

5. Testing and Tuning: The oscillator is tested to ensure that it starts oscillating and that the oscillations are stable. Fine-tuning may be necessary to ensure that the Barkhausen criterion is precisely met.

Example: LC Oscillator

An LC oscillator is a common type of oscillator that uses an inductor (L) and a capacitor (C) to determine the frequency of oscillation. The feedback network typically consists of these two components forming a tank circuit.

The frequency of oscillation (fo) is given by the formula:

[ f_o = \frac{1}{2\pi\sqrt{LC}} ]

The amplifier provides the necessary gain, and the LC tank circuit provides the phase shift. By designing the LC network to have a phase shift of 0° or 360° at the desired frequency of oscillation, the Barkhausen criterion is satisfied, and the circuit will oscillate at the frequency ( f_o ).

In summary, the Barkhausen criterion is a theoretical guideline for designing and analyzing oscillators. It ensures that the circuit will produce stable and sustained oscillations by carefully balancing the gain and phase shift within the feedback loop.