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Circuits, electronics and noise
Sallen-Key quality factor with equal components
Quality factor of a Sallen–Key low-pass section with equal resistors and equal capacitors, where K is the gain of the non-inverting stage between 1 and 3.
- Ref
- ELN-37
- Rev
- v1
- Field
- Circuits, electronics and noise
- Result
- Q
For a Sallen–Key section with equal resistors R and equal capacitors C, the amplifier gain K between 1 and 3 sets the damping: Q = 1/(3 − K). K = 1.586 gives the Butterworth value of 0.707, and K = 3 gives Q = ∞, an oscillator. Gains near 3 make the filter very sensitive to the amplifier gain and to component tolerance.
Quality factor of a Sallen–Key low-pass section with equal resistors and equal capacitors, where K is the gain of the non-inverting stage between 1 and 3. As K approaches 3 the damping goes to zero.
Calculator
- Damping ratio, 1/(2Q) = (3 − K)/20.75
- Gain for Butterworth Q = 0.707, 3 − √21.586 V/V
- Passband gain, K1.5 V/V
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.
- A gain of 1.5 gives Q = 0.6667, close to the flat Butterworth response.
- Q rises quickly near K = 3: at K = 2.9 it is 10, and a 1 % change in K changes Q by about 29 %.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| K Non-inverting stage gain | 1 to 2.9 V/V | 0.5 | 10 | 9.5 | +10% | +11.1 % |
Sweep of Non-inverting stage gain
| Non-inverting stage gain (V/V) | Quality factor |
|---|---|
| 1 | 0.5 |
| 1.238 | 0.5674 |
| 1.475 | 0.6557 |
| 1.712 | 0.7767 |
| 1.95 | 0.9524 |
| 2.188 | 1.231 |
| 2.425 | 1.739 |
| 2.662 | 2.963 |
| 2.9 | 10 |
Sweep one variable
Variables and how each one contributes
Quality factor
What the equation gives you. Everything else feeds it.
Non-inverting stage gain
The gain of the non-inverting amplifier stage, 1 ≤ K < 3. A larger K raises Q, and sharply so as K nears 3. Raising it by 10 % (to 1.65) moves the result by +11.1 %, from 0.6667 to 0.7407.
Worked example with the values above
- Distance from instability: 3 − K = 3 − 1.5 = 1.5.
- Take the inverse: 1 / 1.5.
- Q = 0.66667.
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
- Using a gain K of 3 or above, where the filter oscillates.
- Using this formula when the resistors or the capacitors are not equal.
- Forgetting that Q is very sensitive to K close to 3.
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