Barkhousen Criterion and explanation of negative feedback amplifier:

Barkhousen Criterion and explanation of negatige feedback amplifier:

Oscillator

The oscillator are the electronic device or circuit which generates the periodic ac output signal without input signal. Thus oscillator are basically ‘+’ve feedback amplifier without input signal. The difference between amplifier and oscillator are studied by below figure

The oscillator acts as frequency generator and are of following two types:-

• Sinusoidal oscillator

These generate the sine wave. There may be Hartley, Culpits, R-C phase shift oscillator.

• Relaxation oscillator

These generators are non sinusoidal wave and generates square wave, rectangular wave, triangular wave, saw tooth wave etc. Multivibrators are used to generate the various periodic wave.

Barkhousen criterion for sinusoidal oscillation:

As we know that, the close loop voltage gain of positive feedback amplifier is as

$$A’=\frac{A}{1-A\beta}$$ Where

A=open loop voltage

\(\beta\)=feedback ratio

A\( \beta\)=feedback ratio

\(v_{\circ}’\)=total output voltage

\(v_{in}\)=input voltage

$$\frac{v_{\circ}’}{v_{in}}=\frac{A}{1_A\beta}$$

$$v_{\circ}’(1-A\beta)=Av_{in}$$Since oscillation has no input signal i.e.\(v_{in}=0\)

$$\Rightarrow v_{circ}’(1-A \beta)=0$$

$$\Rightarrow v_{circ}’\neq 0$$

$$\Rightarrow (1-A\beta)=0$$

$$A\beta=1$$Thus for the sustained oscillation any oscillator circuit must have,

1. The positive feedback which is given to the input so that total phase change should be zero degree or integral multiple of \(360^\circ\)(2\pi).
2. The feedback factor (\(A\beta\)) should be unity.

Thus the oscillation gain should be equal to reciprocal of feedback ratio.

These are necessary Barkhousen criterion for sustained oscillator.

Explanation of advantages of negative feedback amplifier:

The numerous advantages of negative feedback outweigh its only disadvantage of reduced gain.

Among the advantages are:

1. Higher fidelity, i.e., more linear operation
2. Higher stabilized gain,
3. Increased bandwidth, i.e., improved frequency response,
4. Less amplitude distortion,
5. Less harmonic distortion
6. Less frequency distortion
7. less phase distortion
8. reduced noise
9. input and output impedances can modified as desired
10. band with improvement

Let us discuss briefly about some advantages such as

• Gain stability

As we know that, the closed loop voltage gain of negative feedback amplifier is as,

$$A’=\frac{A}{1+\beta A}$$ Where ,A=open loop gain ( gain without feedback), \(\beta\)=feedback ratio

Taking log on both,

$$log A’=log A- log(1+A /beta)$$

$$\frac{dA’}{A’}=\frac{dA}{A}-\frac{d(1+\beta A}{(1+\beta A)}$$

$$ \frac{dA’}{A’}=\frac{dA}{A}-\frac{(0+\beta dA)}{(1+\beta A)}$$

$$\frac{dA’}{A’}=\frac{dA}{A}-\frac{\beta dA}{(1+a \beta)}$$

$$\frac{dA’}{A’}=\frac{dA}{A}\biggl[1-\frac{\beta A}{(1+A \beta)}\biggr]$$

$$\frac{dA’}{A’}=\frac{dA}{A} \frac{1}{(1+A \beta)}$$Since,A(\beta\)>>1

$$\frac{dA’}{A’}=\frac{dA}{A} \biggl(\frac{1}{A\bata}\biggr)$$

$$ \Rightarrow \frac{dA’}{A’}<\frac{dA}{A}$$

This shows that the percentage change in gain with negative feedback is always less than that without feedback. Hence, gain stability increase with negative feedback.

• Decrease in distortion

The distortion appears an output of amplifier when it is driven beyond the linear region. This distortion produced can be reduced with application of negative feedback on the amplifier.

Let D and \(D_1\) be output distortion voltage without and with negative feedback for an amplifier circuit.

Suppose,\(D’=xD\) where ,x=multiplier

Now the fraction of output distortion voltage which is given to input is given by \(\beta D’\)=\(\beta\) xD

Let A be the amplifier gain after amplification the feedback voltage becomes A(\(\beta\)xD). This feedback voltae (A\(\beta\)xD) is always in opposite phase to that of input distortion voltage. Thus, total output distortion voltage can be written as,

$$D’=D-A(\beta xD)$$

$$xD=D-D(A\beta x)$$

$$x=1-A\beta x$$

$$x=\frac{1}{1+A \beta}$$

$$D’=xD$$

$$\Rightarrow D’=\frac{D}{1+A \beta}$$

$$\Rightarrow D’<D$$

This shows that the distortion produced on output of amplifier is reduced by the factor \((1+A\beta\)) by the application of negative feedback.

3. Increased bandwidth

The band width of electronic circuit without feedback is defined as the frequency difference between upper cutoff frequency and lower cutoff frequency i.e.

Bandwidth (B.W.)=\(f_2-f_1\)

When the negative feedback is given to the amplifier circuit it reduce their gain of amplifier but increases its bandwidth. The lower cutoff frequency is reduced by the factor (1+A \(/beta\)) with negative feedback i.e

$${f_1}’=\biggl(\frac{1}{1+A \beta}\biggr) \times f_1$$

$${f_2}’=(1+A \beta) \times f_2$$Thus the band width with negative feedback increases. So that \({f_2}’-{f_1}’<f_2-f_1\).

Hence the bandwidth increases showing better frequency response. So that the negative feedback reduce the distortion appear in phase. However, the product of gain and band width is always constant.i.e

(B.W.) \(\times\)A=(B.W.)\(\times\)A’

\(\Rightarrow (f_2-f_1)A=({f_2}’-{f_1}’)A’\)

References:

(1)Theraja, B.L. Basic Electronics. N.p.: S.Chand, n.d. Print.

(2)C.L.Arora. Refresher Course in Physics. Vol. II and III. N.p.: S.Chand, 2006. Print.

(3)Malvino. Electronic Principles. N.p.: Tata McGraw-Hill, n.d. Print.

(4)N.Nelkon and P.Parker. Advanced Level Physics. 5th ed. N.p.: Arnold Heinemann, n.d. Print.

(5)Priti Bhakta Adhikari,Diya Nidhi Chaatkuli, Ishowr Prasad Koirala. A Textbook of Physics (2nd Year). N.p.: Sukunda Pustak Bhawan, 2070. Print.