Low Pass Filter Calculator

Circuit Topology

Calculation Target

Target Cutoff Frequency (fc)

Cutoff Frequency Unit

Resistance (R)

Resistance Unit

Capacitance (C)

Capacitance Unit

Inductance (L)

Inductance Unit

Test Signal Frequency (Optional)

Test Frequency Unit

Frequency Response & Time Domain Analysis

Cutoff Frequency (-3 dB fc) —

Time Domain & Angular Metrics —

Signal Attenuation & Phase Shift —

Filter Dynamics, Roll-Off & Damping —

Calculations assume ideal passive components without parasitic winding capacitance, core saturation, or source/load impedance loading. For 1st-order filters, roll-off is -20 dB/decade (-6 dB/octave) with a -45° phase shift at the half-power cutoff frequency. For 2nd-order filters, roll-off is -40 dB/decade (-12 dB/octave).

What Is a Low Pass Filter Calculator?

A Low Pass Filter Calculator determines the cutoff frequency and related characteristics of passive resistor-capacitor (RC), resistor-inductor (RL), and resistor-inductor-capacitor (RLC) filters. For first-order RC and RL circuits, it can also calculate the resistance, capacitance, or inductance needed to achieve a chosen cutoff frequency. Results include signal gain, phase shift, and time-domain measurements.

A low-pass filter allows lower-frequency signals to pass while reducing the amplitude of higher-frequency signals. Its cutoff frequency marks the point where the voltage gain falls to approximately 70.71% of the reference passband gain, corresponding to about -3.01 decibels (dB).

The calculator supports three circuit topologies: passive first-order RC, passive first-order RL, and passive second-order RLC. Each uses different equations to determine cutoff frequency and filter behavior. The calculations assume ideal passive components and do not include the effects of source and load impedance.

How the Low Pass Filter Formulas Work

The calculator uses different mathematical formulas depending on the selected circuit topology and calculation target. Resistance is converted to ohms, capacitance to farads, inductance to henries, and frequency to hertz before calculation.

1. RC Low Pass Filter Formula

A first-order RC low-pass filter uses a resistor and capacitor. Its cutoff frequency depends on the product of resistance and capacitance.

f_c=\frac{1}{2\pi RC}

Where:

f_c = Cutoff frequency in hertz (Hz)
R = Resistance in ohms (Ω)
C = Capacitance in farads (F)
π = Pi, approximately 3.14159

The calculator can also rearrange this equation to determine the resistance or capacitance required for a target cutoff frequency.

R=\frac{1}{2\pi f_c C}

C=\frac{1}{2\pi f_c R}

The RC time constant is:

\tau=RC

The time constant, represented by τ, describes the circuit’s characteristic response time. Increasing either resistance or capacitance increases the time constant and reduces the cutoff frequency.

2. RL Low Pass Filter Formula

A first-order RL low-pass filter uses a resistor and inductor. The calculator determines its cutoff frequency using:

f_c=\frac{R}{2\pi L}

Where L is inductance in henries (H), and R and f_c have the same meanings as in the RC equations.

For RL filter design, the calculator can solve for the required resistance or inductance:

R=2\pi f_c L

L=\frac{R}{2\pi f_c}

The RL time constant is:

\tau=\frac{L}{R}

Increasing resistance raises the RL cutoff frequency when inductance remains constant. Increasing inductance lowers the cutoff frequency when resistance stays the same.

3. RLC Low Pass Filter Formula

For the second-order RLC topology, the calculator uses resistance, capacitance, and inductance to determine the natural frequency, quality factor, damping ratio, and cutoff frequency.

First, it calculates the natural frequency:

f_0=\frac{1}{2\pi\sqrt{LC}}

It then calculates the quality factor (Q) and damping ratio (ζ):

Q=\frac{1}{R}\sqrt{\frac{L}{C}}

\zeta=\frac{1}{2Q}

The RLC cutoff frequency is determined using:

f_c=f_0\sqrt{1-2\zeta^2+\sqrt{(1-2\zeta^2)^2+1}}

This formula accounts for damping when locating the -3 dB cutoff relative to the filter’s unity DC gain. The natural frequency and cutoff frequency are not necessarily equal.

For this RLC topology, the calculator also uses the following time-constant expression:

\tau=\frac{2L}{R}

4. Time-Domain and Frequency-Response Calculations

For all three topologies, the calculator determines angular cutoff frequency and a displayed 10%–90% rise-time figure:

\omega_c=2\pi f_c

t_r=\ln(9)\times\tau

Angular cutoff frequency (ω_c) is measured in radians per second. The rise-time calculation uses ln(9), approximately 2.197224577, multiplied by the calculated time constant.

The calculator also evaluates how much a signal is reduced at a selected test frequency. For first-order RC and RL filters, the voltage gain magnitude is:

|H(f)|=\frac{1}{\sqrt{1+(f/f_c)^2}}

For the second-order RLC filter, it uses:

|H(f)|=\frac{1}{\sqrt{[1-(f/f_0)^2]^2+[2\zeta(f/f_0)]^2}}

The gain is converted into decibels and a percentage of input voltage:

G_{dB}=20\log_{10}(|H(f)|)

\text{Voltage Transfer (\%)}=100|H(f)|

For RC and RL filters, the phase shift in degrees is:

\phi=-\tan^{-1}\left(\frac{f}{f_c}\right)\frac{180}{\pi}

For RLC filters, the calculator uses the two-argument inverse tangent to determine the correct phase angle:

\phi=-\operatorname{atan2}\left(2\zeta\frac{f}{f_0},\,1-\left(\frac{f}{f_0}\right)^2\right)\frac{180}{\pi}

If no valid positive test frequency is entered, the calculator evaluates gain and phase shift at the calculated cutoff frequency.

Worked Example: RC Low Pass Filter

Suppose you have an RC circuit with the following component values:

Resistance (R): 1 kΩ = 1,000 Ω
Capacitance (C): 100 nF = 0.0000001 F

Substitute these values into the cutoff frequency formula:

f_c=\frac{1}{2\pi(1000)(100\times10^{-9})}

f_c\approx1591.549\ \mathrm{Hz}

The calculator displays a cutoff frequency of 1.592 kHz.

The time constant and rise time are:

\tau=1000\times100\times10^{-9}=0.0001\ \mathrm{s}

t_r=2.197224577\times0.0001\approx0.000219722\ \mathrm{s}

The time constant is displayed as 100.000 µs, and the rise time as 219.722 µs. The angular cutoff is 10,000 rad/s.

With the test frequency left blank, the calculator evaluates the signal at cutoff and displays approximately -3.01 dB gain, 70.71% voltage transfer (0.7071), and -45.00° phase shift.

How to Use the Low Pass Filter Calculator

The available calculation targets change according to the selected circuit topology.

Select Circuit Topology: Choose Passive 1st-Order RC Filter, Passive 1st-Order RL Filter, or Passive 2nd-Order RLC Filter.
Choose Calculation Target: For RC filters, calculate cutoff frequency, required resistance, or required capacitance. For RL filters, calculate cutoff frequency, required resistance, or required inductance. RLC mode calculates cutoff frequency from all three component values.
Enter the Required Values: Enter resistance, capacitance, inductance, or target cutoff frequency according to the selected calculation.
Select Units: Choose the units displayed alongside each required input.
Enter Test Signal Frequency (Optional): Provide a positive frequency to evaluate gain and phase shift at a specific signal frequency. Leave the field blank to evaluate at the cutoff frequency.
Click Calculate: View the calculated cutoff frequency or required component value, time-domain measurements, signal response, and filter dynamics.
Click Reset if Needed: Clear entered values, restore the default RC topology and unit selections, and hide the previous results.

The calculator accepts resistance in Ω, kΩ, or MΩ; capacitance in pF, nF, µF, or mF; and inductance in µH, mH, or H. Cutoff and test frequencies can be entered in Hz, kHz, or MHz. All required numerical values must be positive.

The primary result represents either the cutoff frequency or the required component value, depending on the selected target. The remaining results describe the frequency response and dynamic characteristics of the calculated circuit.

Understanding Your Filter Results

The calculator presents several measurements to help you interpret the expected behavior of an ideal low-pass filter.

Cutoff Frequency and Signal Attenuation

The cutoff frequency indicates where the filter’s voltage gain reaches its -3 dB level. Signal attenuation describes the reduction in output voltage compared with input voltage.

The calculator displays gain in decibels, voltage transfer as both a percentage and a decimal ratio, and phase shift in degrees. Gain in decibels and phase shift are displayed to two decimal places, while the decimal voltage-transfer ratio uses four decimal places.

For RC and RL filters, the calculated first-order response has a nominal high-frequency roll-off of -20 dB per decade, approximately -6 dB per octave. For the second-order RLC filter, the listed roll-off is -40 dB per decade, approximately -12 dB per octave.

Time Constant and Rise Time

The time constant describes a characteristic time scale for circuit response. The calculator also displays a 10%–90% rise-time value derived from the time constant.

For first-order RC and RL circuits, this relationship describes the standard exponential step response. For RLC circuits, actual rise time may differ because second-order step response also depends on damping and possible oscillations. The calculator still applies its stated rise-time expression in RLC mode.

RLC Quality Factor and Damping Ratio

In RLC mode, the calculator displays the natural frequency, quality factor (Q), damping ratio (ζ), and damping classification.

Damping Ratio	Calculator Classification	General Behavior
ζ < 1	Underdamped	May exhibit oscillatory behavior.
ζ = 1	Critically Damped	Boundary between underdamped and overdamped response.
ζ > 1	Overdamped	Non-oscillatory response with stronger damping.

The calculator’s underdamped label also states that a resonant peak is present. However, an underdamped response does not always produce a peak above unity gain. For this second-order transfer function, a gain peak above unity requires ζ to be less than 1/√2.

Important Calculation Assumptions

The calculator assumes ideal passive circuit components. Its equations do not account for parasitic winding capacitance, inductor core saturation, or changes caused by connected source and load impedances.

Real components also have tolerances and non-ideal behavior that can affect measured cutoff frequency and signal response. Use the calculated values as theoretical circuit estimates and verify actual performance when designing or testing physical circuits.

Frequently Asked Questions

What is the cutoff frequency of a low-pass filter?

The cutoff frequency is the point where a low-pass filter’s voltage gain falls to approximately 70.71% of its reference passband gain. This corresponds to about -3.01 dB. For RC and RL filters, cutoff frequency depends on their component values. The calculator uses a damping-dependent formula for its second-order RLC topology.

How do I calculate the cutoff frequency of an RC filter?

Divide 1 by 2π multiplied by resistance in ohms and capacitance in farads. For example, a 1 kΩ resistor and 100 nF capacitor produce a cutoff frequency of approximately 1,591.55 Hz. The calculator performs the necessary unit conversions before applying this formula.

What is the difference between RC, RL, and RLC low-pass filters?

RC filters use resistance and capacitance, while RL filters use resistance and inductance. Both are first-order filters in this calculator. The RLC option combines resistance, inductance, and capacitance to model a second-order response. RLC behavior also depends on damping and quality factor, which can influence its frequency response.

Can I calculate resistance or capacitance for a target cutoff frequency?

Yes. Select the RC topology and choose either required resistance or required capacitance as the calculation target. Enter the target cutoff frequency and the other component value. For RL circuits, you can similarly calculate required resistance or inductance. The RLC mode only calculates cutoff frequency from the entered component values.

Why is the phase shift -45 degrees at the cutoff frequency?

For the first-order RC and RL models, phase shift is the negative inverse tangent of the signal frequency divided by cutoff frequency. At cutoff, this ratio equals 1, and the inverse tangent is 45 degrees. Therefore, the calculator displays -45.00°. RLC phase shift follows a different equation and is not necessarily -45 degrees at cutoff.

What happens if I leave the test frequency blank?

The calculator automatically evaluates signal gain, voltage transfer, and phase shift at the calculated cutoff frequency. You can instead enter a positive test frequency to examine the expected response at another frequency. If the entered test frequency is invalid or nonpositive, the calculation also falls back to the cutoff frequency.

How accurate are the low-pass filter calculations?

The calculator follows its programmed equations for ideal passive RC, RL, and RLC circuits. Actual circuit behavior may differ because of component tolerances, parasitic effects, loading, and other physical conditions. The calculated frequency response is useful for theoretical analysis and initial circuit design but does not guarantee measured performance.

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