A filter is designed using opamps, resistors, and capacitors as shown below. Opamps are ideal with infinite gain and infinite bandwidth. If \( \frac{V_o(s)}{V_i(s)} \) is an all-pass transfer function, the value of resistor R2 is _________ kΩ.
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In opamp all-pass filters, the resistor ratios must create a mirror pole-zero structure to keep magnitude constant and alter only phase.
An all-pass filter has a transfer function of the form:
\( H(s) = \frac{a - sRC}{a + sRC} \) which must have a magnitude equal to 1 for all frequencies. This requires that resistor ratios around the opamp stages satisfy symmetry conditions.
Step 2: Observing the Circuit Structure
The given circuit contains multiple opamp stages with \( R \)-\( C \)-\( R \) symmetry. The 5 kΩ resistor in the feedback path indicates that the input resistor to the same opamp must also match appropriate scaling for all-pass behavior.
Step 3: Condition for All-Pass Realization
In standard opamp-based all-pass filters, the resistor connected in the feedback path must match the resistor connected in the forward path (or maintain a known ratio). Here, the required resistor \( R_2 \) must match the 10 kΩ resistor feeding the final opamp to ensure correct pole-zero reflection.
Step 4: Conclusion
To satisfy the all-pass condition and symmetry of the transfer function, \( R_2 \) must be 10 kΩ.
