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Circuits, electronics and noise
Phase lag of a first-order RC low-pass filter
Phase of the output across the capacitor relative to the input for a series R and shunt C driven from an ideal source and with negligible load, in degrees.
- Ref
- ELN-40
- Rev
- v1
- Field
- Circuits, electronics and noise
- Result
- φ [°]
A series resistor R and a shunt capacitor C, driven from an ideal source and unloaded, make a low-pass filter with output phase φ = −arctan(2πfRC) relative to the input. The output lags by 45° at the cut-off frequency f = 1/(2πRC) and tends to 90° far above. The result is in degrees. The lag is also a time delay of φ/(360 f).
Phase of the output across the capacitor relative to the input for a series R and shunt C driven from an ideal source and with negligible load, in degrees. It is −45° at the cutoff frequency and approaches −90° well above it.
Calculator
- Cut-off frequency, 1/(2πRC)159.2 Hz
- Gain magnitude, 1/√(1 + (2πfRC)²)0.1572 V/V
- Time delay of the phase lag2.249×10−4 s
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- At 1,000 Hz the output lags the input by 80.96°, with a corner at 159.2 Hz.
- A lag of 45° at the corner, 5.7° one decade below it and 84.3° one decade above it is the general pattern for any RC low-pass filter.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (°) | Local step | Change in result |
|---|---|---|---|---|---|---|
| f Frequency | 1 to 1,000,000 Hz | −89.99 | −0.36 | 89.63 | +10% | −0.81 ° |
| C Shunt capacitance | 1e-9 to 0.001 F | −89.99 | −0.36 | 89.63 | +10% | −0.81 ° |
| R Series resistance | 100 to 1,000,000 Ω | −89.99 | −32.14 | 57.85 | +10% | −0.81 ° |
Sweep of Frequency
| Frequency (Hz) | Phase of the output (°) |
|---|---|
| 1 | −0.36 |
| 5.623 | −2.024 |
| 31.62 | −11.24 |
| 177.8 | −48.17 |
| 1,000 | −80.96 |
| 5,623 | −88.38 |
| 31,620 | −89.71 |
| 177,800 | −89.95 |
| 1,000,000 | −89.99 |
Sweep one variable
Variables and how each one contributes
Phase of the output
What the equation gives you. Everything else feeds it.
Frequency
The signal frequency, in Hz. A higher frequency increases the lag towards −90°. Raising it by 10 % (to 1,100) moves the result by +1 %, from −80.96 to −81.77 °.
Series resistance
The series resistance, in Ω. A larger R lowers the corner frequency, so the lag at a given frequency increases. Raising it by 10 % (to 1,100) moves the result by +1 %, from −80.96 to −81.77 °.
Shunt capacitance
The shunt capacitance, in F. A larger C lowers the corner frequency, so the lag at a given frequency increases. Raising it by 10 % (to 1.1×10−6) moves the result by +1 %, from −80.96 to −81.77 °.
Worked example with the values above
- Product: 2πfRC = 2π × 1,000 × 1,000 × 1×10−6 = 6.2832.
- Arctangent: arctan(6.2832) = 80.957°.
- φ = −80.957° (the output lags).
Assumptions and limits
- Ideal amplifiers with enough bandwidth, and ideal resistors and capacitors.
- A source with low output impedance and a load that is much larger than the filter’s output impedance.
Common mistakes
- Treating f₀ as the −3 dB frequency when Q is not the Butterworth value.
- Using angular frequency where frequency in hertz is needed, a factor of 2π.
- Rounding the filter order down: the ceiling is needed.
Statistics and uncertainty
Component tolerance moves the cut-off frequency and the quality factor: 1 % parts give about 1 % in f₀ but can move Q a lot when the design sits near instability. Real amplifier gain-bandwidth limits the filter at high frequency.
Where it is used
- Anti-alias and noise-reduction filters before an ADC.
- Choosing order, cut-off and quality factor of analogue filters from a specification.
Maths you need
- Transfer function: output over input as a function of frequency
- Decibel: 20 log₁₀ of a voltage ratio
- Quality factor: Q, how sharply a resonance peaks