Active Filter Designer

Method, formulas and references

Cascaded sections

An nth-order filter is a cascade of second-order sections and, for odd n, one first-order section. Each low-pass section has the transfer function A0 / (1 + a s + b s²) with s normalised to the corner, s = jf/fc; a first-order section has b = 0. The high-pass equivalent replaces s by 1/s. The coefficients set the pole positions; the pole quality of a section is Q = √b / a. As in TI's design method the sections are ordered by rising Q, with the first-order section first, so that a high-Q section does not saturate on a signal it has already amplified.

Coefficients

The coefficients are computed from the poles, then checked against TI SLOA088 tables 16-4 to 16-9 for orders 1 to 8 (all agree to the four printed decimals except three misprints noted below):

• Butterworth: poles on the unit circle at angles (2k − 1)π/2n; a = 2 sin((2k − 1)π/2n), b = 1.
• Chebyshev (0.5, 1, 2, 3 dB ripple): ε = √(10^(r/10) − 1), poles −sinh(μ) sin θk ± j cosh(μ) cos θk with μ = asinh(1/ε)/n. These are normalised to the ripple band edge; they are rescaled so the gain at fc is 3.01 dB below the DC gain. For odd n that frequency is cosh(acosh(1/ε)/n) times the band edge; for even n, where DC sits at a ripple trough, cosh(acosh(√(1 + 2ε²)/ε)/n).
• Bessel: roots of the reverse Bessel polynomial θn(s), found numerically and rescaled so the gain at fc is −3.01 dB.

So fc is the −3.01 dB frequency for every response, as in the TI tables; for a Chebyshev filter it lies above the edge of the ripple band, which is shown separately. The even-order Chebyshev filter has its DC gain at a ripple trough, so its passband peaks are the ripple value above the DC gain.

Section circuits and design equations (TI SLOA088 sections 16.3, 16.4)

The first-order sections are an RC network buffered by a voltage follower: R1 = a / (2π fc C1) for the low-pass and R1 = 1 / (2π fc a C1) for the high-pass.

Unity-gain Sallen-Key low-pass (figure 16-16), C1 to ground, C2 in the feedback path. Given C1, C2 must satisfy the first condition for real resistors:

Multiple-feedback low-pass (figure 16-19) with passband gain A0 = −R2/R1:

Unity-gain Sallen-Key high-pass (figure 16-28) with C1 = C2 = C:

Multiple-feedback high-pass (figure 16-29) with C1 = C3 = C and passband gain A∞ = −C/C2:

TI prints b = (2C + C2) / (ωc R1 C C2) for this circuit, which repeats the expression for a. Nodal analysis of figure 16-29 gives H(s) = −(C1/C2) / (1 + (C1 + C2 + C3)/(s C2 C3 R1) + 1/(s² C2 C3 R1 R2)), so b = 1 / (ωc² R1 R2 C C2); TI's resistor equations (4) agree with this corrected b.

Choosing components

C1 is either the entered capacitor or, from the resistance level R0, the standard value nearest 1 / (2π f0 R0 √m), where f0 is the section's natural frequency and m the capacitor ratio C2/C1 the section uses (the minimum of (1) or (2) for the low-pass circuits, 1/|A∞| for the MFB high-pass, otherwise 1). The two resistors that set f0 then have a geometric mean near R0. C2 of the low-pass sections is the next standard value at or above the minimum of (1) or (2), and C2 of the MFB high-pass is the standard value nearest C/|A∞|, which sets the realised gain; for that circuit the automatic choice takes, among the series values within √10 of the target C1, the one whose C1/C2 comes closest to |A∞|. The resistors are then calculated exactly for these capacitors and rounded to the nearest value of the resistor series. The solid curves, f0 and Q actual use the rounded values; nothing else is adjusted. TI recommends resistors from 1 kΩ to 100 kΩ and capacitors from 1 nF upward, preferably C0G (NP0) ceramic or film.

Response

Each section is evaluated exactly as gain · s^m / (d0 + d1 s + d2 s²) from its component values. The magnitude and phase are the products and sums over the sections (an inverting MFB section adds −180°), and the group delay is the sum of d1 (d0 + d2 ω²) / ((d0 − d2 ω²)² + d1² ω²) over the sections. The corner with standard values is the highest frequency (lowest for a high-pass) where the gain is 3.01 dB below the passband gain. The op amp is ideal in these curves.

Op amp requirements (TI SLOA088 section 16.8.4)

TI asks for open-loop gain 40 dB above the peak gain of each section, which gives these minimum unity-gain bandwidths for a low-pass section of gain A:

k = fci/fc is the ratio of the section's own −3.01 dB frequency to fc (the k column of the TI tables). For a first-order section the tool uses its own corner fc/a in place of fc. A section with Q = 1 exactly uses the Q < 1 rule. The slew rate for full-power output at fc is SR = π · Vpp · fc. TI gives no corresponding rule for high-pass filters.

Sensitivity

TI gives the approximate sensitivity of Q and fc of a Sallen-Key or MFB section to each R and C as ±0.5 %/% (section 16.8.2). The errors add up across the sections, and TI shows an 8th-order Butterworth turning into a Chebyshev-like response with 0.35 dB of peaking through capacitor tolerance alone. Compare the dashed and solid curves, and use tighter tolerances for high-Q and high-order filters.

Differences from the TI document

• Table 16-6 (0.5 dB), n = 3: a2 is printed 0.0640; the poles and the printed Q = 1.71 give 0.6402.
• Table 16-7 (1 dB), n = 7: b4 is printed 1.0432; the poles and the printed k = 1.520 give 1.0423.
• Table 16-8 (2 dB), n = 8: Q4 is printed 18.39; the printed a4 and b4 give 18.70.
• Example 16-2 prints R1 = 1.26 kΩ and R2 = 1.30 kΩ. TI's own equation with C1 = 22 nF, C2 = 150 nF gives 1.235 kΩ and 1.333 kΩ (same sum); only these give back a = 1.065, b = 1.9305.

References

1. TI SLOA088, Active Filter Design Techniques (T. Kugelstadt), excerpted from Op Amps for Everyone, SLOD006
   Sections 16.2–16.4 (transfer functions, figures 16-12 to 16-29, design equations, examples 16-1 to 16-4), 16.8.2–16.8.4 (sensitivity, component values, op amp selection), 16.9 (tables 16-4 to 16-9).
2. IEC 60063, Preferred number series for resistors and capacitors (E6, E12, E24, E96).