Sallen Key High Pass Filter: Circuit Theory and Component Selection

A Sallen Key high pass filter is an active second-order RC circuit topology that uses an operational amplifier to pass high-frequency signals while attenuating low-frequency noise at a steep 40dB/decade roll-off rate. Unlike a simple passive RC filter, it changes a real circuit by providing a high input impedance that prevents loading down your signal source, while simultaneously buffering the output to drive low-impedance loads like ADCs or audio amplifiers. People most commonly confuse it with the Multiple Feedback (MFB) high pass topology; while both are second-order active filters, Sallen-Key uses the op-amp as a non-inverting buffer (or gain stage), whereas MFB uses it in an inverting configuration, making MFB better for high-Q narrow bandpass designs but Sallen-Key vastly superior for standard Butterworth or Bessel high-pass alignments due to lower sensitivity to op-amp Gain-Bandwidth Product (GBW) limitations.

The Core Topology and Component Roles

The standard Sallen Key high pass filter relies on two capacitors and two resistors arranged in a specific RC network feeding the non-inverting input of an op-amp. Think of a passive first-order RC filter as a single speed bump that slows down low-frequency traffic; a second-order Sallen-Key filter is a speed bump followed immediately by a physical roadblock—the attenuation compounds, dropping the signal amplitude much faster past the cutoff frequency.

The op-amp in this configuration does not just amplify; it isolates. The high input impedance of the non-inverting pin ensures that the RC network's Q-factor isn't degraded by whatever load you connect to the output. This is critical when feeding a 10kΩ ADC input or a long shielded cable.

Worked Numeric Example: 1 kHz Butterworth Design

Let's design a 2nd-order Butterworth high pass filter with a cutoff frequency ($f_c$) of 1 kHz. We will use the standard equal-component design approach where $C_1 = C_2$ and $R_1 = R_2$, which requires a gain of $K = 1.586$.

Step 1: Select the Capacitors

Capacitor values dictate the physical size and parasitic behavior of the filter. For audio and general sensor work in the 100 Hz to 10 kHz range, 10 nF is the sweet spot. It avoids the microphonic effects of large ceramic caps and the leakage issues of electrolytics.

• C1 = 10 nF (C0G/NP0 ceramic, 50V)
• C2 = 10 nF (C0G/NP0 ceramic, 50V)

Step 2: Calculate the Resistors

Using the standard cutoff frequency formula $R = \frac{1}{2 \pi f_c C}$:

$R = \frac{1}{2 \pi \times 1000 \times 10 \times 10^{-9}} = 15,915 \Omega$

Instead of chaining series resistors, we select the closest standard 1% E96 value: 15.8 kΩ.

• R1 = 15.8 kΩ (1% metal film)
• R2 = 15.8 kΩ (1% metal film)

Step 3: Set the Gain for Butterworth Response

To achieve the required $K = 1.586$, we use the non-inverting gain formula $K = 1 + \frac{R_f}{R_g}$.

$\frac{R_f}{R_g} = 0.586$

Choose a standard 1% value for $R_g$: 10.0 kΩ.
Calculate $R_f$: $10,000 \times 0.586 = 5,860 \Omega$.
The closest 1% E96 value is 5.90 kΩ.

• Rg = 10.0 kΩ (1% metal film)
• Rf = 5.90 kΩ (1% metal film)

For a deeper dive into the transfer functions and pole-zero placements for this topology, refer to the Texas Instruments Application Report SLOA049 on active filter design.

Where You Meet This in Practice

You will rarely see a Sallen Key high pass filter used for power electronics; it is strictly a signal-conditioning topology. Here is where it earns its keep on the bench:

• Piezo Vibration Sensors: Piezo elements generate massive DC drift and low-frequency thermal noise. A 5 Hz Sallen Key high pass filter strips the baseline wander while passing the 100 Hz+ mechanical impact spikes to the microcontroller's ADC.
• Audio Crossovers (Tweeter Protection): In active speaker crossovers, a 2 kHz or 3 kHz 2nd-order high pass filter prevents low-frequency excursion from destroying a fragile tweeter dome, while the 40dB/decade roll-off ensures the midrange driver handles the fundamental frequencies.
• AC-Coupling Sensor Interfaces: When measuring small AC signals (like current transformer outputs) superimposed on a noisy DC bias, a Sallen Key high pass filter removes the DC offset cleanly without the phase-shift anomalies introduced by simple capacitor-resistor AC coupling.

Decision Tree: Choosing Your Filter Topology and Op-Amp

Not every filtering problem requires a Sallen Key. Use this decision matrix to lock in your topology and select the exact silicon for the job.

Op-Amp Selection and Board-Level Gotchas

Selecting the right op-amp for a Sallen Key high pass filter requires checking two critical datasheet parameters that frequently brick amateur designs: Gain-Bandwidth Product (GBW) and Slew Rate.

The GBW Rule of Thumb

The op-amp's open-loop gain rolls off with frequency. If your op-amp's GBW is too close to your filter's cutoff frequency, the actual filter response will peak or ring instead of following the smooth Butterworth curve. The Analog Devices Active Filter Design Guide mandates that the op-amp GBW must be at least 100 times the product of the filter's cutoff frequency and its Q-factor.

Example: For our 1 kHz filter with Q=0.707, the minimum GBW is $100 \times 1000 \times 0.707 = 70.7 \text{ kHz}$. A standard LM358 (GBW ~1 MHz) easily passes this test. However, if you shift that same design to a 50 kHz ultrasonic high pass filter, you need a GBW of at least 3.5 MHz, pushing you toward parts like the TLV3501 or OPA350 series.

Parasitic Capacitance and PCB Layout

At the workbench, a Sallen Key high pass filter built on a solderless breadboard will often exhibit a mysterious high-frequency roll-off above 100 kHz. This is due to the ~2pF to 5pF of parasitic capacitance between breadboard rows interacting with the high-impedance feedback nodes. When moving to a PCB, keep the traces from the op-amp's inverting input and the Rf/Rg junction as short as physically possible, and avoid routing ground planes directly under the non-inverting input pin to minimize stray capacitance to ground.

Frequently Asked Questions