First-Order Active Filters

Core Idea

First-order active filters combine op-amps with RC networks to achieve frequency-selective behavior with passband gain — something passive filters cannot provide. An active low-pass filter places a capacitor in the feedback path of an inverting amplifier (or uses a non-inverting topology with an RC network at the input), producing a transfer function with a flat passband gain of -R_f/R_in and a -20 dB/decade roll-off above the cutoff frequency f_c = 1/(2*pi*R_f*C). The active high-pass filter places the capacitor in the input path, passing high frequencies with gain while attenuating frequencies below f_c. Unlike passive filters, active filters can provide gain greater than unity in the passband, have low output impedance (driven by the op-amp output), and do not suffer from loading effects when cascaded. The cutoff frequency and passband gain are independently adjustable through separate component choices. However, active filters are limited by the op-amp's gain-bandwidth product, supply voltage, and power consumption — constraints absent in passive designs.

How It's Best Learned

Start from the inverting amplifier and replace either R_in or R_f with an impedance (R + 1/jwC or R || 1/jwC). Derive the transfer function, identify the cutoff frequency and passband gain, then sketch the Bode magnitude and phase plots. Compare directly to the equivalent passive RC filter to see the gain advantage and the independence of gain and cutoff frequency settings.

Common Misconceptions

• Assuming active filters are always superior to passive filters — at very high frequencies (above the op-amp's useful bandwidth), passive filters outperform active ones, and passive filters require no power supply.
• Setting the passband gain arbitrarily high without considering the gain-bandwidth product — a first-order active filter with 40 dB passband gain and a 10 kHz cutoff requires an op-amp GBW of at least 1 MHz, and the roll-off will deviate from ideal well before that.
• Believing first-order active filters have sharper roll-off than first-order passive filters — both roll off at -20 dB/decade; the active version adds gain and buffering but does not change the filter order or roll-off rate.

Explainer

From your study of passive filters, you know that an RC low-pass filter attenuates signals above a cutoff frequency f_c = 1/(2πRC) at a rate of -20 dB/decade, while passing lower frequencies. The limitation is fundamental: a passive filter can only attenuate — it cannot amplify. Its passband gain is at most 0 dB (unity), and when you cascade two passive RC stages to improve roll-off, the second stage loads the first, shifting the cutoff frequency in a way that's hard to predict without careful analysis. From your study of op-amps, you know the op-amp has near-infinite input impedance and near-zero output impedance — properties that directly solve both of these problems.

An active low-pass filter combines an RC network with an op-amp amplifier. In the simplest inverting topology, you replace the feedback resistor R_f of an inverting amplifier with a parallel combination of R_f and a capacitor C. At DC and low frequencies, the capacitor is an open circuit, so the gain is simply -R_f/R_in — the ordinary inverting amplifier gain. At high frequencies, the capacitor's impedance Z_C = 1/(jωC) drops, progressively shorting out R_f and reducing the gain. The frequency where this transition occurs is the cutoff f_c = 1/(2πR_f C). The transfer function is H(f) = -(R_f/R_in) × 1/(1 + j f/f_c), which combines a flat passband gain of -R_f/R_in with the familiar first-order rolloff. Crucially, the gain magnitude R_f/R_in and the cutoff frequency 1/(2πR_f C) are controlled by separate components: you can independently set gain by choosing R_in and set cutoff by choosing C, then pick R_f to satisfy both.

The active high-pass filter reverses the placement: a capacitor C in series with the input resistor R_in. At low frequencies, C has high impedance and blocks the signal; at high frequencies, C acts like a short circuit and the circuit behaves like a standard inverting amplifier. The cutoff frequency is again f_c = 1/(2πR_in C), and the passband gain above f_c is -R_f/R_in. The Bode plot is the mirror image of the low-pass case: flat gain above f_c, -20 dB/decade rolloff below it.

The op-amp's output impedance advantage becomes clear when you cascade stages. Because the op-amp drives the next stage from a near-zero output impedance, the second stage has no effect on the first stage's transfer function — each stage is perfectly buffered. This lets you build a second-order active filter simply by cascading two first-order stages, with predictable independent cutoff frequencies. The one constraint to respect: op-amps have a finite gain-bandwidth product (GBW). An op-amp with GBW = 1 MHz used at a passband gain of 100 (40 dB) has only 10 kHz of usable bandwidth before the op-amp's own rolloff interferes. Always verify that your desired passband gain × cutoff frequency is well within the op-amp's GBW.