# Explanation of Barkhausen Criterion

# Explanation of Barkhausen Criterion

Barkhausen criterion is that in order to produce continuous undamped oscillations at the output of an

amplifier, the positive feedback should be such that :

m_{v} A_{v} = 1

Once this condition is set in the positive feedback amplifier, continuous undamped oscillations

can be obtained at the output immediately after connecting the necessary power supplies.

#### (i) Mathematical explanation.

The voltage gain of a positive feedback amplifier is give by;

If m_{v} A_{v} = 1, then A_{vf} → ∞.

We know that we cannot achieve infinite gain in an amplifier. So what does this result infer in

physical terms ?

(Explanation of Barkhausen Criterion) It means that a vanishing small input voltage would give rise to finite (i.e., a definite

amount of) output voltage even when the input signal is zero. Thus once the circuit receives the input

trigger, it would become an oscillator, generating oscillations with no external signal source.

#### (ii) Graphical Explanation

Let us discuss the condition mν Aν = 1graphically. Suppose the

voltage gain of the amplifier without positive feedback is 100. In order to produce continuous

undamped oscillations, m_{v} A_{v} = 1 or m_{v} × 100 = 1 or m_{v} = 0.01. This is illustrate in Fig-1.

Since the condition mνAν = 1 is met in the circuit shown in Fig-1, it will produce sustained oscillations.

Suppose the initial triggering voltage is 0.1V peak.

Starting with this value, circuit (A_{ν} = 100 ;

m_{ν} = 0.01) will progress as follows.

The same thing will repeat for 3rd, 4th cycles and so on. Note that during each cycle, V_{f}=0.1Vpk and V_{out} = 10 V_{pk}. Clearly, the oscillator is producing continuous undamped oscillations.

##### Note. The relation m_{ν} A_{ν} = 1 holds good for true ideal circuits.

However, practical circuits need an m_{ν} A_{ν} product that is slightly greater than 1. This is to compensate for power loss (e.g., in resistors) in the circuit.

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