Active Filter Gain⚠ unverified

Understanding the Parameters

• $R_f$ and $R_{in}$ — only their ratio sets the gain. Replacing them with frequency‑dependent impedances $Z_f$, $Z_{in}$ turns the ratio into a filter transfer function (the basis of active low/high/band‑pass stages).

• $A$ — negative means a $180^\circ$ phase inversion; its magnitude $|A| = R_f/R_{in}$ is the gain.

Derivation (Approaching a Proof)

Model the op‑amp as ideal: infinite open‑loop gain and no input current. Negative feedback then forces the two inputs to the same voltage; with the non‑inverting input grounded, the inverting input sits at a virtual ground ($V_- = 0$). All the input current therefore flows on through the feedback resistor (none enters the op‑amp):

$$I = \frac{V_{in} - 0}{R_{in}} = \frac{0 - V_{out}}{R_f}.$$

Equating and solving,

$$\frac{V_{in}}{R_{in}} = -\frac{V_{out}}{R_f} \quad\Rightarrow\quad A = \frac{V_{out}}{V_{in}} = -\frac{R_f}{R_{in}}.$$

The result depends only on the resistor ratio — not on the op‑amp's exact gain — which is precisely why feedback makes the stage accurate and stable.

History

The inverting‑amplifier relation dates to the vacuum‑tube operational amplifiers of 1940s analog computers and became ubiquitous with the integrated op‑amp (Fairchild μA709/741, 1960s). Active‑RC filters built on it let designers realise sharp, tunable responses without bulky inductors.

Related Concepts: Sallen Key Lowpass, Ohm's Law solve for current, Impedance Capacitor

Notes: Registry calculator active-filter-gain (unverified; $R_f$, $R_{in}$ mislabelled dimensionless — should be ohms). Ideal‑op‑amp result; real gain rolls off at high frequency (finite gain–bandwidth).