Design Description

The Butterworth Sallen-Key low-pass filter is a second-order active filter. V_ref provides a DC offset to accommodate for single-supply applications. A Sallen-Key filter is usually preferred when small Q factor is desired, noise rejection is prioritized, and when a non-inverting gain of the filter stage is required. The Butterworth topology provides a maximally flat gain in the pass band.

Design Notes

1. Select an op amp with sufficient input common-mode range and output voltage swing.
2. Add V_ref to bias the input signal to meet the input common-mode range and output voltage swing.
3. Select the capacitor values first since standard capacitor values are more coarsely subdivided than the resistor values. Use high-precision, low-drift capacitor values to avoid errors in f_c.
4. To minimize the amount of slew-induced distortion, select an op amp with sufficient slew rate (SR).

Design Steps

The first step is to find component values for the normalized cutoff frequency of 1 radian/second. In the second step the cutoff frequency is scaled to the desired cutoff frequency with scaled component values.

The transfer function for second order Sallen-Key low-pass filter is given by:

Here,

1. Set normalized values of R₁ and R₂ (R_1n and R_2n) and calculate normalized values of C₁ and C₂ (C_1n and C_2n) by setting w_c to 1 radian/sec (or f_c = 1 / (2 × π) Hz). For the second-order Butterworth filter, (see the Butterworth Filter Table in the Active Low-Pass Filter Design Application Report).
   ${\text{ω}}_{\text{c}}=\text{1}\frac{\text{radian}}{\text{second}}\text{→}{a}_0=\text{1,}{\text{ a}}_1=\sqrt{2}{\text{, let R}}_{1n}={\text{R}}_{2n}=\text{1, then}{\text{ C}}_{\text{1n}}\times {\text{C}}_{\text{2n}}={\text{1 or C}}_{\text{2n}}=\frac{1}{C_{\mathrm{1n}}},{\text{ a}}_{\text{1}}\text{=}\frac{2}{C_{1n}}=\sqrt{2}$
   ${\text{∴C}}_{1n}=\sqrt{2}{\text{= 1.414 F, C}}_{\mathrm{2n}}=\frac{1}{C_{1n}}\text{= 0.707 F}$
2. Scale the component values and cutoff frequency. The resistor values are very small and capacitors values are unrealistic, hence these have to be scaled. The cutoff frequency is scaled from 1 radian/sec to w₀. If m is assumed to be the scaling factor, increase the resistors by m times, then the capacitor values have to decrease by 1/m times to keep the same cutoff frequency of 1 radian/sec. If the cutoff frequency is scaled to be w₀, then the capacitor values have to be decreased by 1 / w_o. The component values for the design goals are calculated in steps 3 and 4.
   Equation 1. $R_1=R_{1n}\times m, R_2=R_{2n}\times m$
   Equation 2. $C_1=\frac{C_{1n}}{m\times {\omega }_0}=\frac{1.414}{m\times {\omega }_0}F$
   Equation 3. $C_2=\frac{C_{2n}}{m\times {\omega }_0}=\frac{0.707}{m\times {\omega }_0}F$
3. Set R1 and R2 values:
   $\text{m = 10000}$
   Equation 4. $R_1=\left(R_{1n}\times m\right)=10k\Omega$
   Equation 5. $R_2=\left(R_{2n}\times m\right)=10k\Omega$
4. Calculate C₁ and C₂ based on m and w₀.
   ${\text{Given ω}}_0{\text{= 2 × π × f}}_c{\text{, where f}}_c\text{= 10kHz and m = 10000 = 10 k}$
   ${\text{C}}_1\text{=}\frac{\text{1.414}}{{\text{m × ω}}_0} \text{F =}\frac{\text{1.414}}{\text{10 k × 2 × π × 10kHz}}\text{= 2.25nF ≈ 2.2nF (Standard Value)}$
   $C_2=\frac{\text{0.707}}{{\text{m × ω}}_0}\text{ F =}\frac{\text{0.707}}{\text{10 k × 2 × π × 10kHz}}\text{= 1.125nF ≈ 1.1nF (Standard Value)}$
5. Calculate the minimum required GBW and SR for f_c.
   ${\text{GBW = 100 × Gain × f}}_{\text{c}}\text{ = 100 × 1 × 10kHz = 1MHz}$
   ${\text{SR = 2 × π × f}}_c{\text{ × V}}_{\mathrm{ipeak}}\text{ = 2 × π × 10kHz × 2.45V = 0.154}\frac{V}{\mathrm{\mu s}}$

   The TLV9062 device has a GBW of 10MHz and SR of 6.5V/µs, so the requirements are met.

Design Simulations

AC Simulation Results

Transient Simulation Results

The following image shows the filter output in response to 5-Vpp, 1-kHz input signal (gain = 1V / V).

The following image shows the filter output in response to 5-Vpp, 100-kHz input signal (gain = 0.01 V/V).

Design References

1. See Analog Engineer's Circuit Cookbooks for TI's comprehensive circuit library.
2. SPICE Simulation File SBOC598.
3. TI Precision Labs.
4. Active Low-Pass Filter Design Application Report

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