Design Description

The multiple-feedback (MFB) high-pass (HP) filter is a 2nd-order active filter. V_ref provides a DC offset to accommodate for single-supply applications. This HP 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).
5. For HP filters, the maximum frequency is set by the gain bandwidth (GBW) of the op amp. Therefore, be sure to select an op amp with sufficient GBW.

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 2nd-order MFB high pass filter is given by:

1. Set normalized values of C₁, C₂, and C₃ (C_1n, C_2n, and C_3n) and calculate normalized values of R₁ and R₂ (R_1n and R_2n) by setting w_c to 1radian/sec (or f_c = 1 / (2 × π)Hz). For a 2nd-order Butterworth filter, (see the Butterworth Filter Table in the Active Low-Pass Filter Design Application Report).
   ${\text{ω}}_{\text{c}}\text{= 1}\frac{\text{radian}}{\text{second}}\text{→} a_0\text{= 1,}{\text{ a}}_1\text{=}\sqrt{2}{\text{, let C}}_{\mathrm{1n}}{\text{= C}}_{\mathrm{2n}}{\text{= C}}_{\mathrm{3n}}\text{= 1 F}$
   $\text{Then}{\text{ R}}_{\text{1n}}{\text{× R}}_{\text{2n}}{\text{= 1 or R}}_{\text{2n}}=\frac{1}{R_{\mathrm{1n}}},{\text{ a}}_{\text{1}}\text{=}\frac{3}{R_{\mathrm{2n}}}\text{=}\sqrt{2}$
   $\therefore R_{2n}=2.1213, R_{1n}=\frac{1}{R_{2n}}=0.4714$
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 we assume m 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 we scale the cutoff frequency to be w₀ then the capacitor values have to be decreased by 1/w₀. The component values for the design goals are calculated in step 3 and 4.
   $R_1=R_{1n}\times m=(0.4714\times m), R_2=R_{2n}\times m=(2.1213\times m)$
   $C_1=\frac{C_{1n}}{m\times {\omega }_0}=\frac{1}{m\times {\omega }_0}F$
   $C_2=\frac{C_{2n}}{m\times {\omega }_0}=\frac{1}{m\times {\omega }_0}F$
   $C_3=\frac{C_{3n}}{m\times {\omega }_0}=\frac{1}{m\times {\omega }_0}F$
3. Set C₁, C₂, and C₃ to 1nF and calculate m.
   ${\text{Given ω}}_0{\text{=2 × π × f}}_c{\text{, where f}}_c\text{= 1kHz,}$
   ${\text{C}}_1=C_2=C_3=\frac{1}{m\times {\omega }_0} F=\frac{1}{m\times 2\times \pi \times \mathrm{1kHz}}$
   $\text{So, }m\text{= 159155}$
4. Calculate R₁ and R₂ based on m.
   $R_1{\text{= R}}_{\mathrm{1n}}\times m\text{= 0.4714 × 159155 ≈ 75kΩ (Standard Value)}$
   $R_2{\text{= R}}_{\mathrm{2n}}\times m\text{= 2.1213 × 159155 ≈ 336kΩ (Standard Value)}$
5. Calculate minimum required GBW and SR for f_max. Be sure to use the noise gain for GBW calculations. Do not use the signal gain of –1V/V.
   ${\text{GBW = 100 × Noise Gain × f}}_{\text{max}}\text{ = 100 × 2 × 10kHz = 2MHz}$
   ${\text{SR = 2 × π × f}}_{\max }{\text{ × V}}_{\mathrm{iMax}}\text{ = 2 × π × 10kHz × 2.45V = 0.154}\frac{V}{\mathrm{\mu s}}$

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

Design Simulations

AC Simulation Results

Transient Simulation Results

Design References

1. See Analog Engineer's Circuit Cookbooks for TI's comprehensive circuit library.
2. SPICE Simulation File: SBOC599.
3. TI Precision Labs.
4. Active Low-Pass Filter Design Application Report

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