Why three RC networks are needed for RC phase shift oscillator? Describe the construction of RC phase shift oscillator and explain its working. Also derive the expression for frequency of oscillation.

Why Three RC Networks are Needed for RC Phase Shift Oscillator?

An RC phase shift oscillator is designed to produce a sine wave output. It uses RC networks, each of which provides a phase shift. To achieve oscillation, the total phase shift around the loop must be 360 degrees or 0 degrees (which are effectively the same in terms of phase). A single RC network provides a phase shift of up to 90 degrees. Therefore, to achieve the necessary 360-degree phase shift, we need multiple RC networks.

Here's why three RC networks are specifically needed:

1. Phase Shift Requirement: Each RC network can provide a maximum of 90 degrees phase shift. To get 360 degrees, we need four times 90 degrees, which means at least four networks. However, the amplifier itself can provide an additional phase shift of 180 degrees. Hence, only three RC networks are needed to contribute the remaining 180 degrees.

2. Stability and Selectivity: Three RC networks provide a good balance between stability and frequency selectivity. More networks would increase selectivity but decrease stability, while fewer networks would do the opposite.

3. Feedback Fraction: The amount of feedback needs to be just right to sustain oscillations without causing distortion or dying out. Three networks provide a feedback fraction that is suitable for many practical applications.

Construction of RC Phase Shift Oscillator

The RC phase shift oscillator consists of the following components:

1. Amplifier: A transistor or operational amplifier (op-amp) is used to amplify the signal. It also provides the necessary gain and the additional 180 degrees of phase shift.

2. RC Networks: Three RC networks are connected in series, and each network consists of a resistor (R) and a capacitor (C).

3. Feedback Path: The output of the last RC network is fed back into the amplifier's input.

Working of RC Phase Shift Oscillator

Here's a step-by-step explanation of how the RC phase shift oscillator works:

1. Initial Signal: The oscillator starts with some noise or an initial disturbance in the circuit.

2. Amplification: The amplifier boosts the signal and inverts its phase by 180 degrees.

3. Phase Shift Through RC Networks: The signal then passes through the three RC networks. Each network shifts the phase of the signal by approximately 60 degrees, adding up to 180 degrees.

4. Total Phase Shift: Combining the phase shift from the amplifier and the RC networks, the total phase shift becomes 360 degrees, which is necessary for sustained oscillations.

5. Feedback for Oscillation: The output of the last RC network is fed back to the input of the amplifier. If the loop gain (product of amplifier gain and feedback fraction) is equal to or greater than one, the oscillations will sustain.

Expression for Frequency of Oscillation

The frequency of oscillation for an RC phase shift oscillator can be derived using the properties of the RC networks. The formula for the frequency of oscillation (f) is:

[ f = \frac{1}{2\pi RC\sqrt{6}} ]

Here's the derivation:

1. Reactance of Capacitor: The reactance of a capacitor at a frequency ( f ) is given by ( X_C = \frac{1}{2\pi fC} ).

2. Phase Shift Condition: For each RC network, the phase shift is 60 degrees when the resistance equals the reactance (( R = X_C )).

3. Substitute ( X_C ) with ( R ): ( R = \frac{1}{2\pi fC} ).

4. Solve for Frequency (f): ( f = \frac{1}{2\pi RC} ).

5. Total Phase Shift: Since we need a total phase shift of 180 degrees from the three RC networks, we must consider the phase shift condition for the network as a whole. The factor of (\sqrt{6}) comes from the condition for sustained oscillations in the feedback loop, which involves complex calculations beyond the scope of this explanation.

Example

Let's consider an RC phase shift oscillator with each resistor ( R = 10 k\Omega ) and each capacitor ( C = 100 pF ).

Using the formula for frequency of oscillation:

[ f = \frac{1}{2\pi RC\sqrt{6}} ]

[ f = \frac{1}{2\pi \times 10 \times 10^3 \times 100 \times 10^{-12} \times \sqrt{6}} ]

[ f \approx 258 Hz ]

So, the oscillator would produce a sine wave with a frequency of approximately 258 Hz.