Multiple-Feedback Band-Pass Design: Q, Gain, Tolerance, and Validation

Start With a Measurable Response

Specify the desired center frequency, lower and upper half-power frequencies, center gain magnitude and polarity, passband input amplitude, source impedance, load, allowable error, supply rails, and expected temperature. For the ideal second-order band-pass, Q and bandwidth are linked; selecting all three as independently adjustable specifications without checking consistency creates an impossible brief. The Filter Q Factor & Bandwidth Calculator is useful for translating between response targets. This guide then addresses how one particular active stage realizes them.

Keep the Exact MFB Convention Together

The Multiple-Feedback Band-Pass Filter Calculator adopts one inverting R1, R2, R3, C1, C2 convention. Its R1 is the input-resistor term in the numerator; R2 and R3 participate in denominator damping; C1 and C2 have distinct positions in the feedback network. Other application notes can reuse those labels for different nodes. Before transferring values, redraw or inspect the actual netlist and compare its complete transfer function, not just a familiar-looking frequency formula.

These equations reproduce the project calculator's declared ideal transfer function. They are not a universal MFB pinout. The op amp's non-inverting input is at the circuit signal reference; with a single supply, that reference and bias path must be implemented deliberately.

Work an Existing Stage Before Solving a New One

Take R1 = R2 = 10 kΩ, R3 = 100 kΩ, and C1 = C2 = 10 nF in the adopted convention. The denominator constant is 2×10⁷ s⁻², so w0 ≈ 4472.136 rad/s and f0 ≈ 711.763 Hz. The s coefficient is 2000 s⁻¹, giving Q ≈ 2.236 and BW ≈ 318.310 Hz. Center gain is −5 V/V, or about 13.98 dB in magnitude. The ideal half-power points are about 570.185 Hz and 888.495 Hz; their difference and geometric mean recover BW and f0. A 0.5 V peak input would demand about 2.5 V peak at the center before considering supply, common-mode, slew, or loading limits.

Move From Nominal Values to Buildable Values

Choose available resistor and capacitor series values, then rerun all five actual values together. A substitution that preserves f0 can still change Q or gain. Keep resistor thermal noise and bias-current error reasonable; very large resistors are susceptible to leakage and board contamination, while very small resistors load the source and op amp. Capacitor dielectric, bias dependence, tolerance, and temperature drift matter. Do not average tolerances into one generic Q error: sweep independent component corners and correlated matched-pair cases, or use a distribution-based simulation when production yield matters.

Check the Op-Amp and Measurement Boundary

Ideal R-C coefficients do not include finite open-loop gain, phase shift, noise gain, input bias, common-mode limits, output swing, slew, or capacitive-load stability. At higher f0 and Q, finite loop gain can alter the apparent center and bandwidth. Signed signal gain is not the same thing as op-amp noise gain; do not select GBW using 5 V/V from the example alone. Use the chosen device model and the Active Filter and Op-Amp Dynamic Limits Guide for device checks.

Sweep a small sinusoid across both skirts, recording gain magnitude and phase with the real source and load. Find the measured center and the two points 3 dB below its measured gain. Then repeat at maximum signal amplitude and supply/temperature corners. A probe or analyzer with finite input impedance can itself alter a high-impedance node; specify measurement loading in the validation record. Use SPICE for corner response and hardware measurements for final sign-off.

Common Design Errors

• Copying an MFB equation from a differently labeled schematic.
• Treating f0, fL, fH, and the peak as interchangeable without checking response shape.
• Forgetting the output is inverted.
• Using signed negative gain inside log10 instead of its magnitude.
• Changing one resistor without recomputing Q and gain.
• Ignoring finite op-amp loop gain at high Q or frequency.
• Sizing GBW solely from signed signal gain.
• Ignoring input and output headroom at passband peak.
• Using nominal capacitors despite bias or temperature sensitivity.
• Calling a nominal result a worst-case guarantee.
• Measuring bandwidth from an arbitrary 0 dB threshold instead of the stage peak.
• Letting a scope probe or unexpected load change the network.

Primary Engineering Sources

See Analog Devices AN-649 for active-filter realization tradeoffs and TI's multiple-feedback band-pass design resource for another practical implementation. Their schematic labels and equations must be checked against the adopted ECParts convention before reuse.