Design Description

The multiple-feedback (MFB) low-pass filter (LP filter) is a second-order active filter. V_ref provides a DC offset to accommodate for single-supply applications. This LP filter inverts the signal (Gain = –1V/V) for frequencies in the pass band. An MFB filter is preferable when the gain is high or when the Q-factor is large (for example, 3 or greater).

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 in design 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 a second-order MFB low-pass filter is given by:

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 fc = 1 / (2 × π) Hz). For a 2nd-order Butterworth filter, (see the Butterworth Filter Table in the Active Low-Pass Filter Design Application Report).
   ${\text{ω}}_{\text{c}}\text{= 1} \frac{\text{radian}}{\text{second}}\text{→} a_0\text{= 1,}{\text{ a}}_1\text{=}\sqrt{2}{\text{, let R}}_{1n}{\text{= R}}_{2n}{\text{= R}}_{3n}\text{= 1}$
   $\text{Then}{\text{ C}}_{\text{1n}}{\text{× C}}_{\text{2n}}{\text{= 1 or C}}_{\text{2n}}=\frac{1}{C_{\mathrm{1n}}},{\text{ a}}_{\text{1}}\text{=}\frac{3}{C_{1n}}\text{=}\sqrt{2}$
   $\therefore C_{1n}=\frac{3}{\sqrt{2}}=2.1213 \text{F}, C_{2n}=\frac{1}{C_{1n}}=0.4714 \text{F}$
2. Scale the component values and cutoff frequency. The resistor values are very small and capacitors values are unrealistic, hence these must be scaled. The cutoff frequency is scaled from 1 radian/second 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/second. 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.
   $R_1=R_{1n}\times m, R_2=R_{2n}\times m, R_3=R_{3n}\times m$
   $C_1=\frac{C_{1n}}{m\times {\omega }_0}=\frac{2.1213}{m\times {\omega }_0}F$
   $C_2=\frac{C_{2n}}{m\times {\omega }_0}=\frac{0.4714}{m\times {\omega }_0}F$
3. Set R₁, R₂, and R₃ to 10kΩ.
   $R_1=R_{1n}\times m=10k\Omega , R_2=R_{2n}\times m=10k\Omega , R_3=R_{3n}\times m=10k\Omega$
   $\text{Therefore,} m=10000$
4. Calculate C₁ and C₂ based on m and w₀.
   $C_1=\frac{2.1213}{m\times {\omega }_0} F=\frac{2.1213}{\mathrm{10k}\times 2\times \pi \times \mathrm{10kHz}}=\mathrm{3.376nF}\approx \mathrm{3.3nF}\text{ (Standard Value)}$
   $C_2=\frac{0.4714}{m\times {\omega }_0} F=\frac{0.4714}{\mathrm{10k}\times 2\times \pi \times \mathrm{10kHz}}=\mathrm{0.75nF}\approx \mathrm{0.75nF}\text{ (Standard Value)}$
5. Calculate the minimum required GBW and SR for f_c. Be sure to use the noise gain for GBW calculations. Do not use the signal gain of –1V/V.
   ${\text{GBW = 100 × Noise Gain × f}}_{\text{c}}\text{ = 100 × 2 × 10kHz = 2MHz}$
   ${\text{SR = 2 × π × f}}_c{\text{ × V}}_{\mathrm{iMax}}\text{ = 2 × π × 10kHz × 2.45V = 0.154}\frac{V}{\mathrm{\mu s}}$

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

Design Simulations

AC Simulation Results

Transient Simulation Results

The following image shows the filter output in response to a 5-V_pp, 100-Hz input signal (gain = –1V/V).

The following image shows the filter output in response to a 5-V_pp, 10-kHz input signal (gain = –0.01V/V).

Design References

1. See Analog Engineer's Circuit Cookbooks for TI's comprehensive circuit library.
2. SPICE Simulation File SBOC597
3. TI Precision Labs.
4. Active Low-Pass Filter Design Application Report

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