The Barkhausen criterion for oscillation is Aß = 1.

Where A-> gain of amplifier

ß-> transfer ratio.

When Aß = 1, the feedback signal will be equal to the input signal. At this condition, the circuit will continue to provide output, even if the external signal is disconnected. This is because the amplifier cannot distinguish between external signal and signal from the feedback circuit. Thus, the output signal is continuously obtained.

An oscillation is a type of feedback amplifier in which a part of the output is fed back to the input via a feedback circuit.

The condition to achieve oscillations are

1. |Aß| = 1
2. ∠Aß = 0^o
3. ∠Aß = multiples of 2π

All the conditions should be simultaneously satisfied to achieve oscillations.

If |Aß| becomes less than unity, the feedback signal goes on reducing in each feedback cycle and oscillation will die down eventually.

When |Aß| is kept slightly greater than unity the signal, however, cannot go on increasing and gets limited due to the non-linearity of the device (that is transistor enters into saturation). Thus, it is the non-linearity of the transistor because sustained oscillation can be achieved.

The Barkhausen condition |Aß| = 1 is usually difficult to maintain in the circuit as the value of A and ß vary due to temperature variations, aging of components, change of supply voltage, etc.

Schottky noise is the noise signal always present at the input of the transistor due to variation in the carrier concentration.

Barkhausen criterion for oscillation is given as

Aß = 1

= > A = 1/ ß

= 1/0.7072 = 1.414.

The feedback signal of an oscillator is given as the product of the externally applied signal & the loop gain of the system.

= > V_f = Aß ×V_i.

There are two requirements for oscillation

1. The magnitude of Aß = 1
2. The total phase shift of Aß = 0^o or 360^o.

The RC feedback network provides a 180^o phase shift and the amplifier used in the RC phase shift oscillator provides a 180^o phase shift (op-amp is used in the inverting mode) to obtain a total phase shift of 360^o.

The RC stage forms the feedback network of the oscillator. It consists of three identical RC stages, each of 60^o phase shifts to provide a total phase shift of 180^o.

The frequency of oscillation of RC phase shift oscillator is,

f_o = 1/(2πRC√6)

= 1/(2×3.14×√6×3.7µF×35Ω)

= > f_o = 1/ 1.9921×10^-3  = 502Hz.

For a sustained oscillation in the RC phase shift oscillator, the gain of the inverting op-amp should be at least 29. Therefore, the gain is kept greater than 29 to ensure the variation in circuit parameter will not make |Aß|<1, otherwise, oscillation will die out.

Applying KVL equation to the circuit, we get

= > I₁(R+(1/SC.)-I₂ = V_o -> Equ1

= > -I₁R+ I₂(2R+(1/SC.)- I₃R = 0 -> Equ2

= > 0- I₂R+ I₃(2R+(1/SC. = 6 -> Equ3

V_f  = I₃× 2R,

Solving Equ 1, 2, and 3 for I₃

= > I₃ = ( V_oR²S³C³)/ (1+ 5SRC+6S²C²R²+S³R³C³)

= > V_f  = I₃× 2R

= ( V_oR³S³C³) / (1+ 5SRC+6S²C²R²+S³R³C³).

Op-amp741 is generally used for low frequencies < 1 kHz.

From phase shift network, we obtain

ß = 1/(1-5 α²)+j α(6- α²) -> Equ1

For Aß = 1, ß should be real and the imaginary terms must be zero

α(6- α²) = 0

= > α = √6

Now substituting α² = 6 in Equ 1, we get

ß = -1/29 (Negative sign indicates that the feedback network produces a phase shift of 180^o).

For phase shift oscillator the value of Rf is

R = 1/(2πC√6×f_o)

= > R = 1/7.153×10^-4

= 1398 = 13.9 kΩ.