In the RC Phase shift oscillator using op-amp, what is the value of the op-amp feedback resistance (Rf), if R1 = 10K Ohm?

1. Break Down the Problem

To find the feedback resistance $R_f$ in an RC Phase Shift Oscillator using an operational amplifier (op-amp), we need to understand the relationship between resistances in the circuit. The configuration typically consists of resistors and capacitors that determine the oscillation frequency, and there’s a feedback loop involving $R_f$ and $R_1$.

2. Relevant Concepts

The formula for the gain in an RC Phase Shift Oscillator can be derived from the principle of positive and negative feedback in op-amps. The gain $A$ required for sustained oscillation is determined by the phase shift condition and the feedback path:

$A = \frac{V_{out}}{V_{in}} = 1 + \frac{R_f}{R_1}$

For an oscillator to function, it requires a gain $A$ that is greater than or equal to 3. This gives us the relationship:

$1 + \frac{R_f}{R_1} \geq 3$

3. Analysis and Detail

To find $R_f$:

1. Start with the inequality for $A$: $1 + \frac{R_f}{R_1} \geq 3$
2. Rearranging gives: $\frac{R_f}{R_1} \geq 2$
3. Multiply both sides by $R_1$: $R_f \geq 2 \times R_1$
4. Substituting $R_1 = 10 \text{K} \, \Omega$: $R_f \geq 2 \times 10 \text{K} \, \Omega = 20 \text{K} \, \Omega$

4. Verify and Summarize

We derived that $R_f$ must be at least 20K Ohms to satisfy the gain condition for oscillation. This means that any feedback resistor value equal to or greater than this will ensure operation.

Final Answer

The feedback resistance $R_f$ must be at least $20K \, \Omega$ for the RC Phase Shift Oscillator to function correctly.