Working principle of Phase shift oscillator

Phase shift oscillator

The circuit diagram for R-C phase shift oscillator is shown in figure. It consists of C-E amplifier and phase shift network. The phase shift network consists of 3 R-C section which are identical. The resistor \(R_B,R_C\) and \(R_E\) are used for stabilization of circuit providing DC-biasing path. The bypass capacitor \(C_E\) allows the access ac signal to ground by maintaining the temperature variation.

Working principle:

The circuit oscillation initiates due to random variation in base current or inherent (own) noise in transistors. The random or chance variation of base current sets the circuit into oscillation. This variation of base current is due to

1. Noise inherent in a transistor or
2. Minor variation in voltage of the dc source

Due to variation is supply voltage \(V_{CC}\). This oscillation in collection region is amplified by amplifier. Hence the phase shift f \(180^\circ\) occurs at collector regions. Also each R-C section produce the phase shift of \(60^\circ\). Therefore, producing the overall phase shift of R-C network as \(180^\circ\). Hence, total phase sift of \(360^\circ\) occurs. So that positive feedback is given to the base of amplifier circuit. Which continues the production of sine wave.

AC- equivalent circuit:

To draw the ac-equivalent circuit of phase shift network, Take

$$R_1=R_2=R_3=R$$

$$C_1=C_2=C_3=C$$

Now from the ac equivalent circuit,

The voltage droop across \(R_2\),i.e.

$$V_2=i_3 x_C+V_3$$

$$v_2=\frac{v_3}{j \omega C R}+v_3\dotsm(1)$$Also,$$v_1=i_2 X_C+V_2$$

$$=\biggl(\frac{V_2}{R}+i_3\biggr)X_C+V_2$$Where,\(i_2=\frac{X_2}{i_2}+i_3\)

$$=\biggl(\frac{V_2}{R}+i_3\biggr).\frac{1}{j\omega C}+\biggl(\frac{V_3}{j \omega C R}+V_3\biggr)$$

$$=\biggl(\frac{V_2}{R}+\frac{V_3}{R}\biggr)\frac{1}{j\omega C}+\biggl(\frac{v_3}{j \omega C R}+V_3\biggr)$$

$$V_1=\frac{V_3}{j^2\omega^2 C^2 R^2}+\frac{2V_3}{j \omega C R}+\frac{V_3}{j \omega C R}+V_3$$

$$V_1=V_3+\frac{3V_3}{j \omega C R}+\frac{V_3}{j^2 \omega^2 C^2 R^2}\dotsm(2)$$Now, the output voltage \(V_circ\) is given by,

$$V_\circ=i_1 X_c+V_1$$

$$=\biggr(\frac{V_1}{R}+i_2\biggr)\frac{1}{j\omega C}+V_1$$

$$=\frac{1}{j \omega C}\biggl[\biggl(\frac{V_2}{R}+i_3\biggr)+\frac{V_1}{R}\biggr]$$

$$V_\circ=\biggl[\biggl(\frac{2 V_3}{R}+\frac{V_3}{j \omega C R^2}\biggr)+\biggl(\frac{3V_3}{j \omega C R^2}+\frac{V_3}{R}\biggr)\biggr]+\biggl(\frac{3V_3}{j \omega C R}+\frac{3V_3}{j\omega C R}+\frac{V_3}{j^2 \omega^2 C^2 R^2+V_3}\biggr)$$

$$V_\circ=V_3+\frac{6V_3}{j \omega C R}+\frac{5V_3}{j^2 \omega^2 C^2 R^2}+\frac{V_3}{j^3\omega^3 C^3 R^3}$$Now the feedback ratio is given by,

$$\beta’=\frac{v_f}{V_\circ}=\frac{V_3}{v_\circ}$$

$$=\frac{1}{1+\frac{6}{j\omega C R}-\frac{5}{\omega^2 C^2 R^2}-\frac{1}{j \omega^3 C^3 R^3}}$$

$$=\frac{1}{\biggl(1-\frac{5}{\omega^2 C^2 R^2}\biggr)+j\biggl(\frac{1}{\omega^3 C^3 R^3}-\frac{6}{\omega C R}\biggr)}$$

Oscillation frequency:

Since, the feedback ratio should be real. So comparing imaginary part of with zero,i.e.

$$\frac{1}{\omega^3 C^3 R^3}-\frac{3}{\omega R}=0$$

$$\frac{1}{\omega^2 C^2 R^2}=6$$

$$\omega^2=\frac{1}{6C^2R^2}$$

$$\omega=\frac{1}{2\pi \sqrt6 RC}$$This is required expression of frequency of phase shift oscillator.

For positive, \(\beta\)’should be real this means imaginary part of \(\beta\)’ is zero i.e.

$$\beta’=\frac{1}{\biggl(1-\frac{5}{\omega^2 C^2 R^2}\biggr)+0}$$

$$\beta’=\frac{1}{1-5\times 6}$$Where,\(\frac{1}{\omega^2 C^2 R^2}=6\)

$$\beta’=\frac{-1}{29}$$This negative sign is due to phase inversion by phase shift network.

$$|\beta’|=\frac{1}{29}$$Since for oscillator we should have,

$$A\beta’=1$$

$$A=\frac{1}{\beta’}=29$$This gain of oscillator is equal to the reciprocal value of feedback ratio which is 29.

Advantages:

• Doesn’t require any bulky and expensive inductors.
• Well suited for frequencies below 10Hz.
• Pure sine wave output is possible as positive feedback occurs for only one frequency.
• It provides good frequency stability
• Circuit is simpler than other oscillator
• It doesn’t need any negative feedback and stabilization arrangements.
• Having wide operating frequency range.

Disadvantages:

• Not suited to variable frequency usage.
• High distortion level I.e. nearly 5% in output is produced.
• High \(\beta\)-transistor should be used to overcome losses in the RC network.
• The output is small due to smaller feedback.
• It is very difficult for the circuit tos start the oscillation
• The frequency stability is not so good.
• It requires high \(V_{CC}\) for large feedback.

Application of phase shift oscillator:

Phase shift oscillator is used for generating signals over the a wide frequency range. The frequency may be varied from a few Hz to 200 Hz by employing one set of resistor with three capacitors ganged together to vary over range in the 1:10 ratio. Similarly the frequency ranges of 200 Hz to 2KHz,2 KHz to 20 KHz and 20 KHz to 200 KHz can be obtained by using other sets of resistors.

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. Prin