Start from the governing relation
\[H(s)=\frac{\omega_0^2}{s^2+\frac{\omega_0}{Q}s+\omega_0^2}\]
This is the smallest equation that preserves the physics needed for the formula.
Choose the model state or domain.
FORMULA / 006
Active filter design: Sallen-Key topology
Interactive derivation module for Active filter design: Sallen-Key topology.
\[H(s)=\frac{\omega_0^2}{s^2+\frac{\omega_0}{Q}s+\omega_0^2}\]
Connects the formula to engineering interpretation, limits, and computation.
Problem Definition
Problem statement and variable definition
A sensor front-end needs to remove high-frequency noise without adding visible ringing after a step.
Core variable legend
| Symbol | Quantity | Unit | Meaning |
|---|---|---|---|
| $\omega_0$ | Natural frequency | rad/s | Center of the second-order denominator. |
| $Q$ | Quality factor | 無 | Inverse damping measure. |
| $f_c$ | Cutoff frequency | Hz | Frequency where response begins to roll off. |
Analytical Derivation
Model assumptions and analytical derivation
Model simplifications
| Item | Assumption |
|---|---|
| Assumption 1 | Linear time-invariant or locally linear behavior is assumed when using transfer functions. |
| Assumption 2 | Parameters are treated as constant over the analysis interval. |
| Assumption 3 | Numerical plots are educational approximations, not full circuit or field solvers. |
Apply engineering assumptions
\[H(s)=\frac{\omega_0^2}{s^2+\frac{\omega_0}{Q}s+\omega_0^2}\]
The model is simplified so the dominant mechanism is visible.
Drop second-order parasitics unless they are the topic.
Rearrange to the working formula
\[Q=\frac{1}{\sqrt{2}}\text{ gives Butterworth flatness}\]
The final form is the one used for design, simulation, or measurement review.
Algebraic rearrangement.
parameter $\to0$dominant term remainsThe formula reduces to its simplest physical behavior.
parameter $\to\infty$parasitics and implementation limits dominateThe closed-form equation becomes a warning rather than a full implementation model.
Parametric Verification
Parametric simulation and numerical verification
Use the sliders to change the physical parameters and watch the formula expose its behavior. The metric panel keeps sample count and compute latency visible so the visualization remains an engineering instrument, not only a picture.
INTERACTIVE MODULE
Second-order low-pass Bode plot
Engineering Application
Engineering application and implementation notes
Design rules
- Check units before trusting numeric output.
- Sweep the key parameter through the critical region, not only the nominal value.
- Validate the simplified formula against measurement or simulation before committing hardware.
Core algorithm
python
import numpy as np
def second_order_mag(f, fc, q):
w = f / fc
return 1 / np.sqrt((1-w*w)**2 + (w/q)**2)