RC and RLC Filter Calculator
RC and RLC Filter Calculator
Cut-off or resonant frequency, Q and bandwidth of an RC low-pass or high-pass filter, a series RLC band-pass or a second-order RLC low-pass, with the gain and phase at any frequency and a Bode plot.
Passive filter response
RC low-pass, 10 kΩ and 10 nF, tested at 10 kHz
Transfer functions
- ω
- 2π × frequency
- gain (dB)
- 20·log10|H|; −3 dB is half power, |H| = 0.7071
- phase
- the angle of H: how far the output leads (+) or lags (−) the input
- Q
- quality factor of the series RLC; ζ = 1/(2Q)
Worked example
RC low-pass, 10 kΩ and 10 nF, tested at 10 kHz
fc = 1 ÷ (2π × 10,000 × 10 nF) = 1.592 kHz; RC = 100 µs
At 10 kHz, ωRC = 2π × 10,000 × 100 µs = 6.283
Gain = −10·log10(1 + 6.283²) = -16.07 dB (× 0.1572)
Phase = −atan(6.283) = -81.0°
Reading a passive filter’s response
RC filters. A resistor and a capacitor make the simplest filter. Take the output across the capacitor and it passes low frequencies: the capacitor’s reactance 1/(2πfC) is large at low frequency and small at high. Take it across the resistor and it passes high frequencies. Either way the corner is at fc = 1/(2πRC), where the output is 3 dB down (0.707 of the input) and the phase is 45° off; well past it the gain falls by 20 dB per decade. 10 kΩ and 10 nF give 1.592 kHz, and at 10 kHz the output is down 16.1 dB.
Series RLC. Adding an inductor gives a second-order circuit that resonates at f0 = 1/(2π√(LC)), where the inductor’s and capacitor’s reactances cancel. Taken across the resistor, the output is a band-pass: 0 dB and 0° at f0, falling on both sides, with a −3 dB bandwidth of f0/Q. Taken across the capacitor, it is a low-pass that falls at 40 dB per decade. Q = √(L/C)/R measures how sharp the resonance is: at Q = 0.707 the low-pass is maximally flat (Butterworth); above it, the response peaks near f0 — at Q = 10 the capacitor sees ten times the input voltage at resonance — and a step input rings.
The Bode plot. The chart shows gain in decibels and phase in degrees on one scale, against frequency spaced logarithmically: each step is a tenth of a decade, over four decades around the corner, and the axis labels are the real frequencies. The table lists every point.
In a real circuit. These results assume a perfect source and no load. The driving circuit’s output resistance adds to R, and anything connected to the output draws current and shifts the response; buffer the filter with an op-amp if that matters, or include the load in your values. Component tolerance moves the corner too: ±5% parts can shift fc by about ±10%. For the step response of the same RC pair use the RC and RL time constant calculator.
Frequently asked questions
How do you calculate the cut-off frequency of an RC filter?
fc = 1 ÷ (2π × R × C). 10 kΩ with 10 nF gives 1.592 kHz; the same formula applies to the high-pass.
What is the resonant frequency and Q of a series RLC circuit?
f0 = 1 ÷ (2π√(LC)) and Q = √(L/C) ÷ R. 10 mH, 10 nF and 100 Ω resonate at 15.92 kHz with Q = 10, a bandwidth of 1.592 kHz.
What does −3 dB mean?
Half the power, or 0.7071 of the voltage. The cut-off frequency of a filter is conventionally where its gain has fallen by 3 dB.
How fast does an RC filter roll off?
20 dB per decade (6 dB per octave) well beyond the cut-off; a second-order RLC filter falls at 40 dB per decade.
Related calculators
References
- Horowitz P, Hill W. The Art of Electronics, 3rd ed. Cambridge University Press, 2015. Ch. 1 (RC circuits, the time constant, RC low-pass and high-pass filters).
- Alexander CK, Sadiku MNO. Fundamentals of Electric Circuits, 7th ed. McGraw-Hill, 2021. Ch. 7 (first-order RC and RL circuits), Ch. 14 (frequency response, series resonance, Q and bandwidth).