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First-Order Low-Pass Filter Design

Calculate cutoff frequency and ideal passband gain for passive RC or RL and active inverting or non-inverting first-order low-pass filters, with component inverses.

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How to use this calculator

  1. Enter the known values in the units shown. Results update as you type.
  2. Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
  3. Use the worked example to check the method. Reset restores the starting fields.

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Formula and method

A 1000 Ω and 1 μF RC stage has a cutoff frequency of approximately 159.155 Hz. At cutoff the magnitude is 1/√2 of its passband magnitude.

RC: fc = 1/(2πRC). RL: fc = R/(2πL). Active gain: −Rf/Rin (inverting), 1 + Rf/Rg (non-inverting).

Worked example

Enter these known values and leave the other values blank.

Filter type
RC (passive)
Resistance (R)
1000 Ω
Capacitance (C)
1 μF
Cutoff frequency (Fc)
0.15915 kHz

Assumptions and limitations

  • All four modes are ideal first-order low-pass circuits with positive finite components and positive cutoff frequency. The passive output is unloaded; connecting a load changes the transfer function.
  • The RC time constant is in seconds when resistance is in ohms and capacitance in farads. R/L has dimensions s⁻¹. Divide angular cutoff frequency by 2π to obtain hertz.
  • For the inverting low-pass topology, cutoff uses the feedback resistor Rf and feedback capacitor C. Its signed passband gain is −Rf/Rin. The non-inverting mode uses the input RC network and gain 1+Rf/Rg.
  • An ideal op-amp is assumed: unlimited gain bandwidth, no slew-rate limit, zero bias/offset currents and no output clipping. Check real supply voltage, stability, loading, component tolerance and manufacturer bandwidth limits separately.
  • A first-order cutoff is the frequency at which amplitude falls to 1/√2 of the applicable passband amplitude, approximately −3.0103 dB relative to that passband. It does not imply a unity-gain filter.
  • Each component, cutoff and gain can be edited where the inverse has a unique physical value. The non-inverting gain must exceed one with both feedback resistors positive; unity is a limiting case outside the positive-resistor inverse.
  • Controls accept decimal/scientific-notation numbers and bounded exact finite-decimal arithmetic. Expressions that lose input digits or exceed the proof budget are rejected. Nonzero entered values below 10⁻¹⁰⁰⁰⁰ in equation base units are outside the supported working range. Arithmetic results use 64 significant digits; tiny representable values use scientific notation, and an overflowing or zero-losing unit conversion is withheld with an error.

Common questions

Which resistor determines an active cutoff?

In the inverting low-pass circuit, the parallel feedback Rf and C set the cutoff. Rf/Rin separately sets gain magnitude.

What is the cutoff of 1 kΩ and 1 μF?

The time constant is 0.001 seconds, so fc = 1/(2π × 0.001) ≈ 159.155 Hz.

Can a passive filter provide voltage gain above one?

These unloaded first-order passive RC and RL topologies have unity asymptotic passband gain. Active op-amp modes can have larger passband gain.

References

The calculation equations, inverse formulas, units, and input rules were imported from this source. Bookify provides the interface and equation solver.

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