Multiple-Feedback Band-Pass Filter Calculator
This calculator designs and analyzes ideal second-order multiple-feedback active band-pass filters. It is built around a documented inverting MFB topology so R1, R2, R3, C1, C2, center gain, phase, and feasibility constraints stay aligned between the calculator, formula section, and tests.
It is not a passive RLC replacement and it is not a generic active-filter synthesizer. Use it when the design intent is a specific op-amp MFB band-pass stage with center frequency, Q, bandwidth, center gain, and frequency response.
Engineering tool
Multiple-Feedback Band-Pass Filter Calculator
Design and analyze ideal second-order inverting MFB active band-pass filters with center frequency, Q, bandwidth, gain, half-power frequencies, phase, and sweep response.
Input resistor in the adopted MFB numerator convention.
Additional input/feedback damping resistor in the denominator constant term.
Feedback damping resistor that sets the denominator s coefficient.
Input-side feedback capacitor in the adopted MFB convention.
Output feedback capacitor used in the numerator and denominator convention.
Result console
- Center frequency
- 1.125395kHz
- Q factor
- 1.06066017
- Bandwidth
- 1.061033kHz
- Lower -3 dB frequency
- 713.654945Hz
- Upper -3 dB frequency
- 1.774688kHz
- Fractional bandwidth
- 0.9428%
- Center gain magnitude
- 1.5
- Center gain
- -1.5V/V
- Center gain
- 3.52182518dB
- Polarity
- Inverting
- Selectivity
- Low Q: broad bandwidth and lower frequency selectivity.
MFB band-pass stages are topology- and labeling-dependent; keep schematic labels aligned with the documented convention.
Solved or entered values have a very wide component spread; check practicality, noise, leakage, and tolerance.
Signal gain is not the same as op-amp noise gain. Use datasheet guidance or simulation for stability and GBW margin.
Formula reference
MFB Band-Pass Topology and Formulas
Adopted topology: Inverting multiple-feedback active band-pass filter. The op-amp is used in an inverting multiple-feedback configuration with the non-inverting input at signal ground or reference.
H(s) = -(s/(R1C2)) / (s² + s(C1+C2)/(C1C2R3) + (1/(C1C2R3))(1/R1+1/R2))ω0 = sqrt((1/(C1C2R3))(1/R1 + 1/R2))f0 = ω0 / (2π)Q = ω0 / ((C1 + C2)/(C1C2R3))H(jω0) = -C1R3 / (R1(C1 + C2))Center gain magnitude = |H(jω0)|Gain dB = 20log10(|H|)BW = f0 / QFBW = BW / f0 = 1 / QfL = (sqrt(BW² + 4f0²) - BW) / 2fH = (sqrt(BW² + 4f0²) + BW) / 2f0 = sqrt(fL × fH)Design feasibility: target gain magnitude < Q²(C1 + C2)/C2Variable definitions
- R1 is the input resistor associated with the band-pass numerator term.
- R2 is the additional input/feedback damping resistor used in the denominator constant term.
- R3 is the feedback damping resistor that sets the denominator s coefficient with C1 and C2.
- C1 is the input-side feedback capacitor in the adopted MFB denominator convention.
- C2 is the output feedback capacitor used in the numerator and denominator convention.
- Center gain is reported as both signed gain and magnitude; dB uses 20log10(|gain|).
- f0
- center frequency
- Q
- quality factor
- BW
- bandwidth
- fL and fH
- lower and upper half-power frequencies
MFB Band-Pass Formula Audit
| Adopted Topology | Inverting multiple-feedback active band-pass filter |
|---|---|
| R1 Position | R1 is the input resistor associated with the band-pass numerator term. |
| R2 Position | R2 is the additional input/feedback damping resistor used in the denominator constant term. |
| R3 Position | R3 is the feedback damping resistor that sets the denominator s coefficient with C1 and C2. |
| C1 Position | C1 is the input-side feedback capacitor in the adopted MFB denominator convention. |
| C2 Position | C2 is the output feedback capacitor used in the numerator and denominator convention. |
| Op-Amp Input Configuration | The op-amp is used in an inverting multiple-feedback configuration with the non-inverting input at signal ground or reference. |
| Gain Sign Convention | Center gain is reported as both signed gain and magnitude; dB uses 20log10(|gain|). |
| Low-Frequency Limit | |H| approaches 0 as frequency approaches DC. |
| High-Frequency Limit | |H| approaches 0 as frequency becomes much larger than f0. |
| Phase Convention | Displayed in the -180° to +180° range using atan2 of the complex transfer function. |
Worked Examples
Independent component set
Known: R1 = 10 kΩ, R2 = 20 kΩ, R3 = 30 kΩ, C1 = C2 = 10 nF
f0, Q, BW, and center gain are computed from the documented denominator coefficients.
Bandwidth relation
Known: Known f0 and Q
BW = f0 / Q using the shared FIL-001 utility.
Half-power spread
Known: Known fL and fH
fH - fL equals BW.
Geometric center
Known: Known fL and fH
fL × fH equals f0² for the standard band-pass definition.
Center response
Known: Frequency = f0
Gain magnitude equals the calculated center gain magnitude.
Lower half-power
Known: Frequency = fL
Relative gain is approximately -3.0103 dB.
Upper half-power
Known: Frequency = fH
Relative gain is approximately -3.0103 dB.
Low-frequency limit
Known: f << f0
Gain tends toward zero because the numerator is proportional to s.
High-frequency limit
Known: f >> f0
Gain tends toward zero because the denominator is dominated by s².
Q = 10
Known: f0 = 100 kHz, Q = 10
FBW = 0.1 and BW = 10 kHz.
Exact cutoffs
Known: f0 = 100 kHz, Q = 10
The exact fL/fH pair preserves the geometric-center invariant.
Target design
Known: f0, Q, gain, C1, C2 selected
The solver returns R1, R2, R3 and round-trips through the analyzer.
Equal-capacitor design
Known: C1 = C2 = C
Equal-C design uses the same general topology constraint.
Unit equivalence
Known: 100 kHz and 0.1 MHz
Both represent the same analysis frequency.
Log sweep
Known: 50 points
The calculator returns finite ordered frequency response rows.
Invalid target
Known: Gain too high for selected Q and capacitor ratio
The design is rejected instead of returning a negative resistor.
Inverting topology
Known: At f0
Polarity is displayed as Inverting and dB is based on gain magnitude.
Unequal capacitors
Known: C1 ≠ C2
Target design supports unequal selected C1 and C2 values when feasible.
Multiple-Feedback Filter
MFB band-pass filters are active second-order op-amp filters with topology-dependent equations.
Center Frequency
f0 comes from the denominator coefficients, not from a generic 1/(2πRC) shortcut.
Quality Factor
Q sets fractional bandwidth and affects component sensitivity, ringing, and settling.
Bandwidth
BW = f0/Q and the exact half-power points preserve fL × fH = f0².
Center Gain
The adopted topology is inverting, so signed center gain is negative while magnitude is positive.
Voltage dB
Use 20log10 of gain magnitude. Do not place signed gain directly inside log10.
Op-Amp GBW
Real MFB filters can require more bandwidth margin than simple center signal gain times f0.
Noise Gain
Signal gain and op-amp noise gain are not the same; verify stability and bandwidth with datasheet guidance.
Component Tolerance
Capacitor tolerance and resistor ratios can shift f0, Q, and center gain.
Validation
Critical designs should be checked with SPICE, tolerance analysis, and bench measurement.
Common Mistakes
Support reference
FAQ
What is a multiple-feedback band-pass filter?
A multiple-feedback band-pass filter is an active second-order op-amp filter topology that uses multiple feedback paths to set center frequency, Q, bandwidth, and midband gain.
How does an MFB band-pass filter work?
The adopted topology creates a second-order band-pass transfer function with a zero at DC, rolloff at high frequency, and an inverting center-frequency gain.
How do I calculate its center frequency?
The calculator derives center frequency from the denominator coefficient: f0 = sqrt((1/(C1C2R3))(1/R1 + 1/R2)) / (2π).
How do I calculate Q?
Q is calculated from the denominator as Q = ω0 / ((C1+C2)/(C1C2R3)). This is then cross-checked with Q = f0 / BW.
How do I calculate bandwidth?
Bandwidth uses the standard second-order band-pass relation BW = f0 / Q and is reused from the shared Filter Q Factor & Bandwidth utilities.
How do I calculate the lower and upper -3 dB frequencies?
The exact half-power frequencies are fL = (sqrt(BW² + 4f0²) - BW)/2 and fH = (sqrt(BW² + 4f0²) + BW)/2, not f0 ± BW/2.
Why is an MFB filter usually inverting?
The adopted MFB topology drives the inverting op-amp input and reports center gain as signed negative gain plus positive magnitude. Voltage gain dB uses the magnitude.
How is center gain calculated?
For the adopted convention, H(jω0) = -C1R3/(R1(C1+C2)). The calculator reports signed center gain, magnitude, dB, and inverting polarity.
What is fractional bandwidth?
Fractional bandwidth is BW/f0. For the standard band-pass definition used here, FBW = 1/Q.
Why do MFB formulas vary between schematics?
MFB filters are sensitive to schematic convention, resistor numbering, capacitor numbering, and gain definition. Always match R1, R2, R3, C1, and C2 to the documented topology.
How does Q affect component sensitivity?
Higher Q gives narrower bandwidth but usually increases component tolerance sensitivity, op-amp bandwidth demands, settling time, and ringing risk.
How much op-amp bandwidth is required?
This calculator uses an ideal op-amp model. Real designs need gain-bandwidth comfortably above the center frequency and noise-gain requirements, especially for high-Q filters.
What is the difference between signal gain and noise gain?
Signal gain is the band-pass transfer gain from input to output. Op-amp noise gain is a separate stability and bandwidth consideration and may be larger than the center signal gain.
What is the difference between an MFB and RLC band-pass filter?
An MFB filter is an active op-amp topology. An RLC band-pass filter is a passive second-order network. They can share f0/Q/BW terminology but have different components, loading, and practical limits.
Planned Engineering Guides
Planned guide
Multiple-Feedback Band-Pass Filters Explained
Planned guide
MFB Filter Design
Planned guide
Band-Pass Q and Bandwidth
Planned guide
Active Band-Pass Filter Topologies
Planned guide
Op-Amp Noise Gain in Active Filters
Planned Filter Calculators
Notch Filter Calculator
Butterworth Filter Calculator
Multiple-Feedback Low-Pass Filter Calculator
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Engineering Disclaimer
This calculator uses an ideal multiple-feedback band-pass model. It does not model real op-amp GBW, slew rate, output swing, input bias current, voltage noise, output current, source impedance, loading, capacitor non-ideal behavior, or PCB parasitics. Validate critical designs with op-amp datasheets, SPICE, tolerance analysis, and bench measurement.