RC Low-Pass Calculator
Find the cutoff of a first-order RC low-pass and the gain, attenuation, and phase at the frequency you type.
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Result
About this tool
This is the series-resistor, shunt-capacitor low-pass, one pole, ideal parts. The form is the same shape as the high-pass page: resistance, capacitance, and a test frequency, each multiplied by its unit. The defaults are again 10 kilohms, 100 nanofarads, and 1000 hertz, which the formula sees as 10000 ohms, 1e-7 farads, and 1000 hertz. If any of the three is not strictly positive, the report stops. Cutoff and time constant do not depend on which filter mode you are in. Cutoff is still 1 / (2πRC) = 159.1549430919 Hz, and τ = RC = 1 ms. Reactance at the test frequency is still 1 / (2πfC) = 1.591549430919 kΩ at 1000 Hz.
What changes is the shape of the gain and the sign of the phase. Low-pass gain is 1 / sqrt(1 + (f/fc)²). At 1000 Hz, f/fc is about 6.283, so the gain prints as 0.1572x. Attenuation is 20 log10 of that gain, which prints as -16.07 dB. Phase is −atan(f/fc) converted from radians to degrees, which prints as -80.96°. At the cutoff, f/fc is 1, the gain prints as 0.7071x, and the phase is -45.00°. Far below the cutoff the gain is near 1 and the phase is near 0°. Far above it the gain keeps falling by about 20 dB for each decade of frequency, which is all a single real pole can do.
The same defaults on the high-pass page print gain 0.9876x and phase 9.04°, because that page uses (f/fc) / sqrt(1 + (f/fc)²) and 90° minus the arctangent. The two pages share the cutoff helper and the formatting. They do not model a load across the capacitor, a series inductor, or the output impedance of a real driver. A test frequency of twice the cutoff gives a ratio of 2, a gain of 1/sqrt(5) = 0.4472x, and a phase of −atan(2) = -63.43°. That is a useful check if you want a number that is not the default 1000 Hz. Gain is a voltage ratio to four decimal places, not a power ratio. Attenuation is that ratio in decibels to two decimal places. The engineering formatter prints the cutoff, the reactance, and the time constant, so a cutoff above 1000 Hz is shown in kilohertz and a time constant of 0.001 seconds is shown as 1 ms. The page will not search the E12 or E24 tables for a nearby resistor. Use the preferred-value RC designer when you need a stock part close to a target cutoff.
Example. 10 kΩ and 100 nF, tested at 1000 Hz, report a cutoff of 159.1549430919 Hz, gain 0.1572x, attenuation -16.07 dB, and phase -80.96°. At twice the cutoff the gain is 0.4472x and the phase is -63.43°.
Limit. The transfer function is one series resistor into a shunt capacitor. A non-positive resistance, capacitance, or test frequency stops the report. There is no load and no inductor.
What do the default 10 kΩ and 100 nF produce?
Cutoff is 1/(2πRC) = 159.1549430919 Hz and the time constant is 1 ms. At the default test frequency of 1000 Hz the low-pass gain is 0.1572x, the attenuation is -16.07 dB, and the phase is -80.96°. Reactance of the 100 nF capacitor at 1000 Hz is 1.591549430919 kΩ.
What is the output at the cutoff frequency?
Set the test frequency equal to the cutoff. The gain prints as 0.7071x and the phase as -45.00°. That gain is 1/sqrt(2). Twenty times its log10 is about −3 dB. The cutoff number itself does not change when you edit only the test frequency.
How is this different from the high-pass page?
Both pages use cutoff 1/(2πRC). The low-pass gain is 1/sqrt(1+(f/fc)²) and the phase is −atan(f/fc). The high-pass gain is (f/fc) over that same square root, and its phase is 90° minus the arctangent. With the defaults, high-pass gain is 0.9876x and phase is 9.04°.
Will it account for the next stage loading the capacitor?
No. The capacitor is assumed to be unloaded, and the resistor is assumed to be driven from a zero-ohm source. Tolerance, ESR, and a second pole are not in the formula. Enter only positive R, C, and test frequency.