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سجل التغييرات الكاملالمحرّك 1.26.4

Active Filter: Sallen-Key / MFB

الإشارة والتوقيت

Which parts give this filter, and is the op-amp fast enough?

هذه الصفحة غير مترجمة بالكامل بعد. الأجزاء غير المترجمة معروضة بالإنجليزية.

المدخلات

Check mode tests one second-order stage.
The −3 dB point relative to the DC gain (the high-frequency gain for the high-pass), for all four responses.
Hz
10 kHz
Scales every resistor; a larger C₁ lowers them. Sallen-Key low-pass: the capacitor to ground. MFB: the feedback capacitor. High-pass and Sallen-Key with gain: both capacitors.
F
1 nF
Hz
95 kHz
dB
The datasheet GBW (unity-gain bandwidth) at your supply.
Hz
1 MHz
Part series
Each resistor is the nearest value in it.
C₂ is rounded up to it.
Tolerances

النتائج

Parts
R₁ 7.87kΩ, R₂ 14.7 kΩ, C₁ 1 nF, C₂ 2.2 nF
ناجح

المعيار: −3 dB corner within 3 % of your fc, failing past 5 % (design margin): 9.9725 kHz as built, −0.28 % from 10 kHz, with E96 resistors and E12 C₂
أظهر خطوات الحساب
  1. Smallest C₂C₂ ≥ C₁ × 4b / a² = 1 nF × 4 × 1 / 1.4142² = 2 nF, rounded up in E12 to 2.2 nF
  2. ResistorsR₁,₂ = (a C₂ ∓ √(a² C₂² − 4b C₁ C₂)) / (4π f_c C₁ C₂) = 7.861 kΩ and 14.65 kΩ, nearest E96 7.87 kΩ and 14.7 kΩ
  3. Natural frequency, as builtf₀ = 1 / (2π √(R₁R₂C₁C₂)) = 1 / (2π √(7.87 kΩ × 14.7 kΩ × 1 nF × 2.2 nF)) = 1 / (2π × 15.95 µs) = 9.976 kHz
  4. Q, as builtQ = √(R₁R₂C₁C₂) / (C₁(R₁ + R₂)) = 15.95 µs / (1 nF × (7.87 kΩ + 14.7 kΩ)) = 0.7068
  5. Corner, as built−3 dB corner = 9.9725 kHz; error = 9.9725 kHz / 10 kHz − 1 = −0.28 %
Q error, worst stage
−0.0376%
GBW / needed
1.003
Attenuation at fstop
39.15dB
كل النتائج (10)
Corner frequency, as built
9.9725kHz
f₀ as built
9.9761kHz
f₀ target
10kHz
Q as built
0.70685
Q target
0.70711
GBW needed
997.2kHz
Values before rounding
R₁ 7.86101kΩ, R₂ 14.6467 kΩ
C₂ minimum
2nF
Pass-band gain
1V/V
f₀ spread (RSS)
Butterworth, as built

إلى أي حد نعرف هذا الرقم

من أين تأتي الطريقة
  • T. Kugelstadt

T. Kugelstadt, Active Filter Design Techniques, in Op Amps for Everyone, Texas Instruments SLOD006, also published as SLOA088 المعادلة

يُقاس مقابل: J. Karki, Analysis of the Sallen-Key Architecture, Texas Instruments SLOA024

مدى دقتها
Exact arithmetic on the ideal-op-amp transfer functions of SLOA088, with the parts as rounded to their series.

The op-amp's finite gain-bandwidth is checked separately. Its output impedance and input capacitance are left out, so a Sallen-Key stage's attenuation is trusted only below a tenth of GBW / NG. The capacitors' tolerance usually dominates where the corner lands on the bench.

عملياً يهيمن على عدم اليقين مدخل واحد: C₁. ضيّقه يضِق معه الجواب.

يُقاس مقابل

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Circuit

المبدأ

An active filter is a cascade of second-order stages, Sallen-Key or multiple-feedback (MFB), each an op-amp with two capacitors and two or three resistors around it (two more resistors set a Sallen-Key stage's gain). The response, Butterworth, Bessel or Chebyshev, fixes each stage's coefficients a and b, and from them its Q and its natural frequency. This page works the parts out from TI's coefficient tables, rounds them to preferred values, and checks the filter as built: the corner, each stage's Q, the attenuation you need, and whether the op-amp is fast enough.

Cascade
A(s)=A0∏i(1+ais+bis2)A(s) = \frac{A_0}{\prod_i \left(1 + a_i s + b_i s^{2}\right)}
From SLOA088, with s normalised to ωc = 2π fc. The tables put fc at −3 dB relative to A₀ for all four responses, Chebyshev included.
Stage target
Qi=biaif0,i=fcbiQ_i = \frac{\sqrt{b_i}}{a_i} \qquad f_{0,i} = \frac{f_c}{\sqrt{b_i}}
From the stage's coefficients. For a high-pass stage, f₀ = fc √bi.
Unity-gain Sallen-Key low-pass
R1,2=aC2∓a2C22−4bC1C24πfcC1C2C2≥C1 4ba2R_{1,2} = \frac{a C_2 \mp \sqrt{a^{2} C_2^{2} - 4 b C_1 C_2}}{4\pi f_c C_1 C_2} \qquad C_2 \geq C_1\,\frac{4b}{a^{2}}
C₁ to ground, C₂ in the feedback path. C₂ is rounded up so the square root stays real.
Equal-part Sallen-Key with gain
R=b2πfcCA0=3−1Q=1+R4R3R = \frac{\sqrt{b}}{2\pi f_c C} \qquad A_0 = 3 - \frac{1}{Q} = 1 + \frac{R_4}{R_3}
With R₁ = R₂ = R and C₁ = C₂ = C, Q sets the gain, so it is solved, not chosen. This page takes R₃ = R.
Unity-gain Sallen-Key high-pass
R1=1πfcCaR2=a4πfcCbR_1 = \frac{1}{\pi f_c C a} \qquad R_2 = \frac{a}{4\pi f_c C b}
C₁ = C₂ = C, R₁ to ground and R₂ in the feedback path.
MFB low-pass
R2=aC2−a2C22−4bC1C2(1+∣A0∣)4πfcC1C2R1=R2∣A0∣R3=b4π2fc2C1C2R2R_2 = \frac{a C_2 - \sqrt{a^{2} C_2^{2} - 4 b C_1 C_2 (1 + |A_0|)}}{4\pi f_c C_1 C_2} \qquad R_1 = \frac{R_2}{|A_0|} \qquad R_3 = \frac{b}{4\pi^{2} f_c^{2} C_1 C_2 R_2}
C₁ in the feedback path and C₂ to ground, with C₂ ≥ C₁·4b(1 + |A₀|)/a². A fourth-order MFB here splits the gain equally between its two stages.
As built
f0=12πR1R2C1C2Q=R1R2C1C2C1(R1+R2)+(1−A0)R1C2f_0 = \frac{1}{2\pi\sqrt{R_1 R_2 C_1 C_2}} \qquad Q = \frac{\sqrt{R_1 R_2 C_1 C_2}}{C_1 (R_1 + R_2) + (1 - A_0) R_1 C_2}
The Sallen-Key low-pass from its rounded parts. The high-pass and the MFB have their own forms. The corner is found from the product of the stages.
Op-amp gain-bandwidth
fT=100 G fCi    (Q≤1)fT=100 G fcaQ2−0.5Q2−0.25    (Q>1)f_T = 100\, G\, f_{Ci} \;\; (Q \leq 1) \qquad f_T = 100\, G\, \frac{f_c}{a} \sqrt{\frac{Q^{2} - 0.5}{Q^{2} - 0.25}} \;\; (Q > 1)
The open-loop gain 40 dB above the stage's peak gain, for at most 1 % gain error. The page evaluates the second form as 100 G Q f₀ √(…) with the parts as built. For a high-pass stage it applies the same rule at the stage's own corner, or at its peak when Q > 1.
  • a,ba, ba stage's coefficients, from the SLOA088 tables
  • fcf_ccorner frequency, −3 dB relative to the pass-band gain
  • f0f_0a stage's natural frequency
  • QQa stage's quality factor
  • A0A_0pass-band gain: at DC for a low-pass, at high frequency for the high-pass
  • GGa stage's pass-band gain, as a magnitude
  • fCif_{Ci}a stage's own −3 dB corner
  • fTf_Tthe op-amp unity-gain bandwidth a stage needs
  • R1,R2,R3,R4,C1,C2R_1, R_2, R_3, R_4, C_1, C_2the parts, named as in the figure
  • Vin,VoutV_{in}, V_{out}the filter's input and output
تفاصيل إضافية

Choosing the topology

The unity-gain Sallen-Key is the usual choice for low Q, below about 3, and an accurate gain of 1. The multiple-feedback (MFB) stage suits high Q and high gain, and each stage inverts. The equal-part Sallen-Key is simple to build, but Q fixes its gain: 1.586 for Butterworth.

Parts

  • Resistors from 1 kΩ to 100 kΩ. Lower values draw too much current from the op-amp output; higher ones add resistor noise.
  • Capacitors from 1 nF, so the parasitic capacitance stays small beside them. C0G (NP0) parts hold their value over temperature and voltage.
  • Capacitors come in coarser steps than resistors, so the capacitors are fixed first and the resistors computed from them, as SLOA088 does.

The op-amp

SLOA088 sizes the op-amp so that its open-loop gain stays 40 dB above each stage's peak gain. A high-pass on a real op-amp is a band-pass: the op-amp's own roll-off ends its pass band. A Sallen-Key low-pass leaks far above its corner: the input reaches the output through R₁ and C₂, held down only by the op-amp's output impedance, which rises as the loop gain falls (SLOA024). So this page trusts its attenuation only below a tenth of GBW / NG, a design margin.

Check the filter as built. Rounding to preferred values moves each stage's f₀ and Q, so the corner and Q checks use the rounded parts. The parts' tolerances add the f₀ spread on top.
References: T. Kugelstadt, "Active Filter Design Techniques", in Op Amps for Everyone, Texas Instruments SLOD006, also published as SLOA088. Consult it for higher orders and other responses. J. Karki, "Analysis of the Sallen-Key Architecture", Texas Instruments SLOA024, for the low-pass leak. Related: RC / RL Filter & Time Constant, Op-Amp Gain & Bandwidth, ADC Resolution & Noise Budget.

إصدار المحرك ⁨1.26.4⁩