Library / Circuits, electronics and noise / ELN-36
Circuits, electronics and noise
Sallen-Key low-pass natural frequency
Pole frequency of a standard second-order Sallen–Key low-pass section.
- Ref
- ELN-36
- Rev
- v1
- Field
- Circuits, electronics and noise
- Result
- f0 [Hz]
A Sallen–Key low-pass section has a pole (natural) frequency f₀ = 1/(2π√(R₁R₂C₁C₂)), the geometric mean of the two RC corner frequencies. The −3 dB frequency equals f₀ only when Q = 0.707 (Butterworth); for a higher Q the response peaks near f₀, and for a lower Q it rolls off earlier. The quality factor is set by the ratio of the components and the amplifier gain.
Pole frequency of a standard second-order Sallen–Key low-pass section. It equals the −3 dB frequency only when the quality factor is the Butterworth value of 0.707.
Calculator
- Angular frequency, ω₀25,000 rad/s
- Time constant, √(R₁R₂C₁C₂)4×10−5 s
- Equal-component frequency using the geometric means, R = √(R₁R₂), C = √(C₁C₂)3,979 Hz
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.
- The pole frequency is 3,979 Hz (3.979 kHz).
- Doubling any one component lowers f₀ by a factor of √2, and doubling both resistors halves it.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (Hz) | Local step | Change in result |
|---|---|---|---|---|---|---|
| C2 Capacitor 2 | 1e-11 to 1e-6 F | 251.6 | 79,580 | 79,330 | +10% | −4.65 % |
| R2 Resistor 2 | 100 to 1,000,000 Ω | 795.8 | 79,580 | 78,780 | +10% | −4.65 % |
| C1 Capacitor 1 | 1e-11 to 1e-6 F | 125.8 | 39,790 | 39,660 | +10% | −4.65 % |
| R1 Resistor 1 | 100 to 1,000,000 Ω | 397.9 | 39,790 | 39,390 | +10% | −4.65 % |
Sweep of Resistor 1
| Resistor 1 (Ω) | Pole frequency (Hz) |
|---|---|
| 100 | 39,790 |
| 316.2 | 22,370 |
| 1,000 | 12,580 |
| 3,162 | 7,076 |
| 10,000 | 3,979 |
| 31,620 | 2,237 |
| 100,000 | 1,258 |
| 316,200 | 707.6 |
| 1,000,000 | 397.9 |
Sweep one variable
Variables and how each one contributes
Pole frequency
What the equation gives you. Everything else feeds it.
Resistor 1
The first resistor, in Ω. A larger R₁ lowers the pole frequency, with a square-root dependence. Raising it by 10 % (to 11,000) moves the result by −4.65 %, from 3,979 to 3,794 Hz.
Resistor 2
The second resistor, in Ω. A larger R₂ lowers the pole frequency, with a square-root dependence. Raising it by 10 % (to 44,000) moves the result by −4.65 %, from 3,979 to 3,794 Hz.
Capacitor 1
The first capacitor, in F. A larger C₁ lowers the pole frequency, with a square-root dependence. Raising it by 10 % (to 1.1×10−9) moves the result by −4.65 %, from 3,979 to 3,794 Hz.
Capacitor 2
The second capacitor, in F. A larger C₂ lowers the pole frequency, with a square-root dependence. Raising it by 10 % (to 4.4×10−9) moves the result by −4.65 %, from 3,979 to 3,794 Hz.
Worked example with the values above
- Product: R₁R₂C₁C₂ = 1.6×10−9 s².
- Square root: 4×10−5 s.
- f₀ = 1 / (2π × 4×10−5) = 3,978.9 Hz.
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