Active Filters & Sinusoidal Oscillators

An active filter uses an amplifier to realize a transfer function that passive RC sections alone cannot provide with the same gain, buffering, or pole quality. An oscillator uses feedback to make a chosen natural response sustain itself. Both are pole-placement problems, but one is designed to remain stable and the other to sit deliberately on the stability boundary before amplitude limiting establishes a cycle.

Read the transfer function #

A normalized second-order low-pass section is

$$H(s)=\frac{K\omega_0^2}{s^2+\frac{\omega_0}{Q}s+\omega_0^2}.$$

$\omega_0$ sets natural frequency, $Q$ sets damping and peaking, and $K$ sets passband gain. Low-pass, high-pass, band-pass, and notch responses differ by numerator zeros. Higher orders are normally implemented as cascaded first- and second-order sections so component sensitivity and op-amp requirements remain local.

flowchart TB
  SPEC["passband + stopband + phase/step needs"] --> APPROX["Butterworth / Bessel / Chebyshev / elliptic"]
  APPROX --> POLES["pole-zero set"]
  POLES --> SOS["cascade biquad sections"]
  SOS --> REAL["choose topology, values, op amp"]
  REAL --> VERIFY["tolerance + noise + GBW + slew + load"]

Butterworth maximizes passband flatness; Bessel preserves transient shape and group-delay smoothness; Chebyshev and elliptic responses buy sharper transitions with ripple and greater sensitivity. The “best” approximation depends on what error matters.

The amplifier is part of the filter #

Ideal equations assume infinite op-amp gain, bandwidth, and output drive. Finite gain-bandwidth shifts poles and $Q$; slew rate distorts large signals; input/output limits clip; amplifier noise and resistor thermal noise set floor; source/load impedance alters component ratios. High-$Q$ and high-frequency sections usually demand far more op-amp bandwidth than a simple “ten times cutoff” rule.

Component tolerance moves both frequency and $Q$. Put the most sensitive/highest-$Q$ section last when overload permits, scale impedances to avoid excessive noise or drive current, and simulate a tolerance distribution rather than only nominal values.

Oscillation condition and startup #

For loop gain $L(j\omega)=A\beta$, sustained sinusoidal oscillation requires phase $0°$ modulo $360°$ and magnitude one at $\omega_0$. Exact unity gives no startup from a zero state. Real oscillators begin with noise because small-signal loop gain is slightly above one; an amplitude-control nonlinearity then reduces effective gain to one.

A Wien bridge provides zero phase at its selected frequency; an RC phase-shift network supplies the necessary phase with an inverting amplifier; LC and crystal networks provide much higher $Q$ where appropriate.

Verification #

For a filter, verify magnitude, phase/group delay, step response, noise, maximum internal node swing, and stability with source/load. For an oscillator, verify startup over process and temperature, steady amplitude, spectral purity, frequency pulling by load/supply, and recovery after disturbance. The schematic's nominal $RC$ product is only the first estimate.