Sallen-Key Filter Explained

 

Sallen-Key Filter

A Sallen-Key filter is a popular type of active analog filter made using:

• An operational amplifier (op-amp)
• Resistors
• Capacitors

It is widely used to build:

• Low-pass filters
• High-pass filters
• Band-pass filters

The circuit was developed by R. P. Sallen and E. L. Key.

Circuit Diagram

Transfer Function Derivation

Using Laplace-domain impedances:

Z_C1 = 1/(sC₁)
Z_C2 = 1/(sC₂)

Applying Kirchhoff's Current Law at the intermediate nodes and assuming an ideal unity-gain buffer:

H(s)=V_out(s)/V_in(s)

H(s) = 1 / [ s²R₁R₂C₁C₂ + s(C₁R₁ + C₂R₁ + C₂R₂) + 1 ]

Standard Second-Order Form

H(s) = ω₀² / ( s² + (ω₀/Q)s + ω₀² )

Comparing coefficients gives:

ω₀ = 1 / √(R₁R₂C₁C₂)

f₀ = 1 / (2π√(R₁R₂C₁C₂))

Q = √(R₁R₂C₁C₂) / (C₁R₁ + C₂R₁ + C₂R₂)

Equal Component Design

R₁ = R₂ = R

C₁ = C₂ = C

Then:

f_c = 1/(2πRC)

Q = 1/3

A Butterworth response requires a non-unity gain Sallen-Key stage. For Butterworth operation:

K ≈ 1.586

Q ≈ 0.707

Design Example

Choose:

R₁ = R₂ = 10 kΩ

C₁ = C₂ = 10 nF

Then:

f_c = 1/(2π × 10k × 10nF) ≈ 1.59 kHz

Main Idea

A Sallen-Key filter combines:

• An RC network (frequency selection)
• With an op-amp buffer/amplifier

This gives:

• Better filtering
• Stable response
• Higher gain
• Less loading effect

Most Common Type: Low-Pass Sallen-Key Filter

It allows:

• Low frequencies to pass
• High frequencies to be blocked

Cutoff Frequency

The cutoff frequency is:

fc = 1 / (2π √(R1 × R2 × C1 × C2))

Where:

• R1, R2 = resistors
• C1, C2 = capacitors
• fc = cutoff frequency

Types of Sallen-Key Filters

Type	Function
Low-pass	Pass low frequencies
High-pass	Pass high frequencies
Band-pass	Pass middle frequencies
Band-stop	Reject certain frequencies

Advantages

• Simple design
• Uses few components
• Good frequency stability
• Easy to tune
• Works well with op-amps

Disadvantages

• Not ideal for very high frequencies
• Op-amp bandwidth limits performance
• Sensitive to component tolerances

Applications

Used in:

• Audio systems
• Equalizers
• Sensor circuits
• ADC anti-aliasing filters
• Communication systems

Example applications:

• Removing noise from audio signals
• Smoothing sensor outputs

Why It’s Popular

Compared to passive RC filters:

• Gives sharper response
• Can provide gain
• Prevents signal loss due to loading

Example

Suppose:

• R1 = R2 = 10kΩ
• C1 = C2 = 0.01µF

Then:

fc ≈ 1.59 kHz

This means frequencies above about 1.59 kHz start getting attenuated.

Try Interactive Online Simulator (web-based)

1.  1. Sallen-Key Filter Simulation (2nd order LPF)
2.  2. Interactive RLC Simulator (Cut-off, Impedance, Phase, etc.) (first order)

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