Barkhausen Criterion for Oscillation

Before starting with the Barkhausen Criterion for Oscillation, we will see what oscillations are, and What the oscillator principle is. So let’s start with the Oscillator Principle and then we will discover the Barkhausen Criterion for Oscillation.

Oscillator Principle

An oscillator is basically an “amplifier” which does not have any AC input but it operates on the principle of positive feedback to generate an AC signal at its output.

Thus it is clear that an amplifier can work as an oscillator if positive feedback is made to exist. However positive feedback does not always guarantee oscillations.

An amplifier will work as an oscillator if and only if it satisfies a set of conditions called the Barkhausen Criterion for Oscillation

So We will see the Barkhausen Criterion for Oscillation in detail.

Barkhausen Criterion for Oscillation

The Barkhausen criteria should be satisfied by an amplifier with positive feedback to ensure sustained oscillations.

For an oscillator circuit, there is no input signal V_s. Hence the feedback signal V_f itself should be sufficient to maintain the oscillations.

Refer to the below figure to understand the Barkhausen Criterion for Oscillation.

• From the figure, the expression for output voltage V₀ is,

V₀ = A V_i

• But V_i is the sum of V_s and V_f.

∴ V_i = V_s + V_f

Note that we need to add V_s and V_f because in positive feedback, V_s and V_f will be in phase with each other and hence it will get added.

• The expression for feedback voltage is,

V_f = β V₀

• Substitute the value of V₀, we get,

V_f = β A V_i

• From the equation of input voltage V_i, we get,

V_i = V_s + A β V_i

∴ (1 – A β) V_i = V_s

• For an oscillator, the input voltage V_s is absent i.e. V_s = 0, and the feedback signal V_f is supposed to maintain oscillations. Therefore substitute V_s = 0 into the above equation to get,

(1 – A β) V_i = 0 or A β = 1

This condition must be satisfied in order to obtain sustained oscillations. Along with this condition, the condition for the positive feedback states that the phase shift between V_s and V_f must be zero, and should also be satisfied.

With an inverting amplifier introducing an 180⁰ phase shift between Vi and Vo, the feedback network must introduce another 180⁰ phase shift to ensure that V_i and V_f are in phase.

These two conditions which are required to be satisfied to operate the circuit as an oscillator are called as the “Barkhausen Criterion for Oscillation”.

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Statement for Barkhausen Criterion for Oscillation

Barkhausen Criterion states that:

1. An oscillator will operate at the frequency for which the total phase shift is introduced, as the signal proceeds from the input terminals, through the amplifier and feedback network, and back again to the input precisely 0⁰ or 360⁰ or an integral multiple of 360⁰.
2. At the oscillator frequency, the magnitude of the product of open loop gain of the amplifier A and the feedback factor β is equal to or greater than unity.

\mathbf{\therefore |A\beta| \geq 1}

The product Aβ is called the open loop gain.

These conditions are diagrammatically illustrated in the below figures.

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