Question

A phase shift oscillator requires minimum gain of

• A. \(1 \)
• B. \(10 \)
• C. \(29 \)
• D. \(100 \)

Hint:

For a standard 3-stage RC Phase Shift Oscillator: - Minimum Gain required: \( A \ge 29 \) - Frequency of oscillation: \( f_0 = \frac{1}{2\pi RC\sqrt{6}} \) Always remember that if the RC network uses a different arrangement (e.g., buffered stages or FET variations), this minimum gain requirement changes, but for a standard unbuffered RC ladder network, it is always 29.

Answer 1

The Correct Option is C

Concept: An RC phase shift oscillator uses an amplifier network combined with a feedback loop consisting of three identical RC sections. Each RC network section introduces a specific phase shift, and together the three sections create a total phase shift of exactly $180^\circ$ at the frequency of oscillation. To achieve sustained oscillations, the system must meet the Barkhausen criterion:
• The total loop phase shift must be $0^\circ$ or $360^\circ$. Because the three RC stages contribute $180^\circ$, the amplifier must be an inverting configuration (providing another $180^\circ$).
• The magnitude of the loop gain must satisfy: \[ |\beta A| \ge 1 \] where $A$ represents the voltage gain of the active amplifier and $\beta$ represents the feedback factor of the passive RC network.

Step 1: Determining the feedback attenuation factor \(\beta\).
For a standard three-stage ladder RC network where all resistors have value $R$ and all capacitors have value $C$, the feedback transfer function can be derived using nodal analysis or mesh equations. The attenuation factor $\beta$ at the exact frequency where the phase shift is $180^\circ$ is found to be: \[ \beta = \frac{1}{29} \] This means the output signal passing backward through the three RC network pairs is attenuated by a factor of 29.

Step 2: Calculating the minimum amplifier gain \(A\).
To maintain continuous, non-decaying sinusoidal oscillations, the total gain around the closed loop must be at least equal to unity: \[ |A| \cdot |\beta| \ge 1 \] Substituting the network attenuation factor $\beta = \frac{1}{29}$ into this condition: \[ |A| \cdot \left(\frac{1}{29}\right) \ge 1 \] Multiplying both sides by 29 gives: \[ |A| \ge 29 \] Therefore, the active amplifier configuration (such as a BJT common emitter or an op-amp inverting amplifier) must be set up to deliver a minimum voltage gain magnitude of 29 to compensate for the loss inside the feedback path. Hence, the correct choice is option (3).