{"id":394927,"date":"2024-06-29T11:54:41","date_gmt":"2024-06-29T11:54:41","guid":{"rendered":"http:\/\/savepearlharbor.com\/?p=394927"},"modified":"-0001-11-30T00:00:00","modified_gmt":"-0001-11-29T21:00:00","slug":"","status":"publish","type":"post","link":"https:\/\/savepearlharbor.com\/?p=394927","title":{"rendered":"4th Order Low-pass Filter with 1 Op Amp<\/span>"},"content":{"rendered":"
<\/div>\n
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\"Amateur<\/p>\n

The idea to build a 4th order low-pass filter looks simple: add one more feedback loop. But there are pitfalls, as always.
<\/a> <\/p>\n

Basic equations for fourth order low-pass filters<\/h2>\n

The transfer function of a 4th order low-pass filter is:<\/p>\n

                              A H(s) = \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014        (1 + s\/(Q\u2081 \u03c9\u2081) + s\u00b2\/\u03c9\u2081\u00b2)(1 + s\/(Q \u03c9\u2082) + s\u00b2\/\u03c9\u2082\u00b2) <\/pre>\n
  where:
  A \u2013 DC gain;
  \u03c9\u2081 \u2013 radial frequency of the first stage, K1\u00a0\u00d7\u00a0\u03c9;
  \u03c9\u2082 \u2013 radial frequency of the second stage, K2\u00a0\u00d7\u00a0\u03c9;
  Q\u2081 \u2013 quality factor of the first stage;
  Q\u2082 \u2013 quality factor of the second stage;
  \u03c9 \u2013 pass-band radial frequency of the filter;
  s \u2013 complex frequency.<\/p>\n
  Open brackets to get another form of the transfer function:<\/p>\n
                           A H(s) = \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014        1 + ps1 s + ps2 s^2 + ps3 s^3 + ps4 s^4  ps1 = 1 \/ (Q\u2081 \u03c9\u2081) + 1 \/ (Q\u2082 \u03c9\u2082) ps2 = 1 \/ (Q\u2081 \u03c9\u2081 Q\u2082 \u03c9\u2082) + 1\/\u03c9\u2081\u00b2 + 1\/\u03c9\u2082\u00b2 ps3 = 1 \/ (Q\u2081 \u03c9\u2081 \u03c9\u2082\u00b2) + 1 \/ (\u03c9\u2081\u00b2 Q\u2082 \u03c9\u2082) ps4 = 1 \/ (\u03c9\u2081\u00b2 \u03c9\u2082\u00b2) <\/pre>\n
  Knowing \u22123\u00a0dB frequency and filter type, we can get K1, Q1, K2 and Q2 from tables in books [1] or compute them.<\/p>\n
Circuits<\/h2>\n
  Let\u2019s start by adding one more feedback loop to well-known Multiple Feedback (MFB) and Sallen-Key (SK) second order low-pass filter topologies.  <\/p>\n
\n\n\n\n
\"Second  <\/td>\n<\/tr>\n
Second Order Topologies<\/td>\n<\/tr>\n<\/table>\n<\/div>\n
  <\/p>\n
\n\n\n\n
\"Fourth  <\/td>\n<\/tr>\n
Fourth Order Topologies<\/td>\n<\/tr>\n<\/table>\n<\/div>\n
  Write equations and try to solve them.<\/p>\n
  Surprise: there are no solutions with positive component values. Varying gain, it turns out that there are solutions only for the Sallen-Key topology when gain is more than 3.<\/p>\n
  The equations for MFB look like:<\/p>\n
 psN = a + b + c + d <\/pre>\n
  The equations for SK look like:<\/p>\n
 psN = a + b + c \u2212 d <\/pre>\n
  So, possible, we need to find a way to add members with the negative sign to the equations.<\/p>\n
4-th Order One Op Amp Sallen-Key Low-pass Filter<\/h2>\n
  For SK, a resistor divider helps. Now the equations have solutions with gains around 1.  <\/p>\n
\n\n\n\n
\"Fourth  <\/td>\n<\/tr>\n
Fourth Order Sallen-Key Low-pass Filter<\/td>\n<\/tr>\n<\/table>\n<\/div>\n
  <\/p>\n
Design Equations<\/h2>\n
  The transfer function is:<\/p>\n
                           A H(s) = \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014        1 + ps1 s + ps2 s^2 + ps3 s^3 + ps4 s^4 <\/pre>\n
  Assuming an ideal Op Amp, the factors are:<\/p>\n
       R6 (C1 R1 (R2 + R3) + C2 (R1 + R2) (R3 + R4) + C4 (R4 R5 + (R1 + R2 + R3) (R4 + mR5)))       \u2212 R4 R7 (C1 R1 + C3 (R1 + R2 + R3)) ps1 = \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014                                       (R1 + R2 + R3 + R4) R6 <\/pre>\n
  <\/p>\n
       R6 (R1 C1 (R2 C2 (R3 + R4) + C4 ((R2 + R3 + R4) R5 + (R2 + R3) R4))       + C4 (C2 (R1 + R2) ((R3 + R4) R5 + R3 R4) + R4 R5 C3 (R1 + R2 + R3)))       \u2212 C3 R4 R7 (R1 C1 (R2 + R3) + R3 C2 (R1 + R2)) ps2 = \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014                            (R1 + R2 + R3 + R4) R6 <\/pre>\n
  <\/p>\n
       C4 R6 (R1 R2 C1 C2 (R4 R5 + R3 (R5 + R4)) + R4 R5 C3 (C1 R1 (R2 + R3) + R3 C2 (R1 + R2)))       \u2212 R1 R2 R3 R4 R7 C1 C2 C3 ps3 = \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014                                        (R1 + R2 + R3 + R4) R6 <\/pre>\n
  <\/p>\n
       R1 R2 R3 R4 R5 R6 C1 C2 C3 C4 ps4 = \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014          (R1 + R2 + R3 + R4) R6 <\/pre>\n
  the DC gain is:<\/p>\n
     R4 (R7 \/ R6 + 1) A = \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014     R1 + R2 + R3 + R4 <\/pre>\n
  But there is another problem: the circuit is very sensitive to component tolerances.<\/p>\n
  A fourth order Butterworth filter circuit is shown with a pass-band of 1\u00a0kHz and a gain of 2 with computed and the closest standard component values.  <\/p>\n
\n\n\n\n
\"4  <\/td>\n<\/tr>\n
4 Pole Sallen-Key Low-pass Filter, Simulation<\/td>\n<\/tr>\n<\/table>\n<\/div>\n
  Thus, this 4th pole SK topology is useless in practice. Drop it.<\/p>\n
4-th Order One Op Amp Multiple Feedback Low-pass Filter<\/h2>\n
  For MFB topology, members with negative signs can be added with positive feedback.  <\/p>\n
\n\n\n\n
\"Fourth  <\/td>\n<\/tr>\n
4-th Order Multiple Feedback Low-pass Filter<\/td>\n<\/tr>\n<\/table>\n<\/div>\n
  <\/p>\n
Design Equations<\/h2>\n
  The transfer function is:<\/p>\n
                           A H(s) = \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014        1 + ps1 s + ps2 s^2 + ps3 s^3 + ps4 s^4 <\/pre>\n
  Assuming an ideal Op Amp, the factors are:<\/p>\n
       R7 (R1 C1 (R2 + R3) + C2 (R1 + R2) (R3 + R4) + C4 (R4 R5 + (R4 + R5) (R1 + R2 + R3)))       \u2212 R4 R6 (C1 R1 + C3 (R1 + R2 + R3)) ps1 = \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014                                  (R1 + R2 + R3) R7 \u2212 R4 R6 <\/pre>\n
  <\/p>\n
       R7 (R1 R2 C1 C2 (R3 + R4) + C4 (R1 C1 (R4 R5 + (R2 + R3) (R4 + R5))       + C2 (R1 + R2) (R4 R5 + R3 (R4 + R5)) + R4 R5 C3 (R1 + R2 + R3)))       \u2212 C3 R4 R6 (C1 R1 (R2 + R3) + C2 R3 (R1 + R2)) ps2 = \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014                            (R1 + R2 + R3) R7 \u2212 R4 R6 <\/pre>\n
  <\/p>\n
       R7 C4 (R1 R2 C1 C2 (R4 R5 + R3 (R4 + R5)) + R4 R5 C3 (R1 C1 (R2 + R3) + R3 C2 (R1 + R2)))       \u2212 R1 R2 R3 R4 R6 C1 C2 C3 ps3 = \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014                                    (R1 + R2 + R3) R7 \u2212 R4 R6 <\/pre>\n
  <\/p>\n
       R1 R2 R3 R4 R5 R7 C1 C2 C3 C4 ps4 = \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014         (R1 + R2 + R3) R7 \u2212 R4 R6 <\/pre>\n
  the DC gain is:<\/p>\n
             R3 (R6 + R7) A = \u2212 \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014       (R1 + R2 + R3) R7 \u2212 R4 R6 <\/pre>\n
Example<\/h2>\n
  Let\u2019s compute a unity gain fourth order Butterworth filter with 150\u00a0kHz pass-band.<\/p>\n
  Set R = R2 = R3 = R4 = 1 kOhm (E96)<\/nobr> to optimize our bill of materials.<\/p>\n
  The DC gain now is:<\/p>\n
             R3 (R6 + R7)              R (R6 \/ R7 + 1) A = \u2212 \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014 = \u2212 \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014       (R1 + R2 + R3) R7 \u2212 R4 R6     R1 + R (2 \u2212 R6 \/ R7) <\/pre>\n
  Thus, R6 \/ R7 = 2<\/nobr> helps to simplify the equation.
  Choose R6 = 1.18 kOhm (E96)<\/nobr>, R7 = 590 Ohm (E96)<\/nobr>.<\/p>\n
      (A (2 \u2212 R6 \/ R7) \u2212 1 \u2212 R6 \/ R7) R   (\u22121 \u00d7 (2 \u2212 2) \u2212 1 \u2212 2) \u00d7 1 kOhm R1 = \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014 = \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014 = 3000 \u2248 3.01 kOhm (E96)                       A                                 \u22121 <\/pre>\n
  For a 4th order Butterworth filter K1 = K2 = 1<\/nobr>, so \u03c91 and \u03c92 are equal to its radial pass-band frequency, and Q1 = 0.5412<\/nobr>, Q2 = 1.3066<\/nobr>.<\/p>\n
          1         1                  1                           1 ps1 = \u2014\u2014\u2014\u2014\u2014\u2014\u2014 + \u2014\u2014\u2014\u2014\u2014\u2014\u2014 = \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014 + \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014 \u2248 2.773e-6       Q\u2081 K1 \u03c9   Q\u2082 K2 \u03c9   0.5412 \u00d7 1 \u00d7 2\u03c0 \u00d7 150 kHz   1.3066 \u00d7 1 \u00d7 2\u03c0 \u00d7 150 kHz               1             1        1                              1                                      1                   1 ps2 = \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014 + \u2014\u2014\u2014\u2014\u2014\u2014\u2014 + \u2014\u2014\u2014\u2014\u2014\u2014 = \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014 + \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014 + \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014 \u2248 3.834e-12       K1 \u03c9 Q\u2081  K2 \u03c9 Q\u2082  (K1 \u03c9)\u00b2   (K2 \u03c9)\u00b2  1 \u00d7 2\u03c0 \u00d7 150 kHz \u00d7 0.5412 \u00d7 1 \u00d7 2\u03c0 \u00d7 150 kHz \u00d7 1.3066   (1\u00d7 2\u03c0 \u00d7150 kHz)\u00b2   (1 \u00d7 2\u03c0 \u00d7 150 kHz)\u00b2               1                 1                              1                                                1 ps3 = \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014 + \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014 = \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014 + \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014 \u2248 3.121e-18       Q\u2082 K2 \u03c9 (K1 \u03c9)\u00b2   Q\u2081 K1 \u03c9 (K2 \u03c9)\u00b2   1.3066 \u00d7 1 \u00d7 2\u03c0 \u00d7 150 kHz \u00d7 (1\u00d72 \u03c0\u00d7150 kHz)\u00b2   0.5412 \u00d7 1 \u00d7 2\u03c0 \u00d7 150 kHz \u00d7 (1 \u00d7 2\u03c0 \u00d7 150 kHz)\u00b2             1                           1 ps4 = \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014 = \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014 \u2248 1.267e-24       (K1 \u03c9 K2 \u03c9)\u00b2   (1 \u00d7 2\u03c0 \u00d7 150 kHz \u00d7 1 \u00d7 2\u03c0 \u00d7 150 kHz)\u00b2 <\/pre>\n
  There are 4 equations and 5 unknown values: C1, C2, C3, C4, R5.
  We can set an R5 value and solve the equations numerically to find C1, C2, C3, C4.<\/p>\n
  Let\u2019s set R4 = 154 Ohm (E96)<\/nobr>.
  The solution is:<\/p>\n
 C1 \u2248 1.341 nF \u2248 1.3 nF (E24) C2 \u2248 1.286 nF \u2248 1.3 nF (E24) C3 \u2248 1.782 nF \u2248 1.8 nF (E24) C4 \u2248 2.677 nF \u2248 2.7 nF (E24) <\/pre>\n
  Simulation confirms that the solution is correct and that sensitivity to component tolerances is acceptable.  <\/p>\n
\n\n\n\n
\"Fourth  <\/td>\n<\/tr>\n
4-th Order Multiple Feedback Low-pass Filter, Frequency response<\/td>\n<\/tr>\n<\/table>\n<\/div>\n
  Magnitude of the input impedance is defined by the R1 value.  <\/p>\n
\n\n\n\n
\"Fourth  <\/td>\n<\/tr>\n
4-th Order Multiple Feedback Low-pass Filter, Input impedance<\/td>\n<\/tr>\n<\/table>\n<\/div>\n
  Step response confirms that the circuit is stable. An Op Amp model with unity gain frequency of 8\u00a0MHz and DC gain of 120\u00a0dB is used. Parasitic capacitances at inputs are added to do it more realistic.  <\/p>\n
\n\n\n\n
\"Fourth  <\/td>\n<\/tr>\n
4-th Order Multiple Feedback Butterworth Low-pass Filter, Step response<\/td>\n<\/tr>\n<\/table>\n<\/div>\n
  <\/p>\n
Conclusion<\/h2>\n
  A 4th order low-pass filter can be designed using only one Op Amp. All filters parameters are linked, so sensitivity to component tolerances should be verified.<\/p>\n
  Sensitivity of the 4th order Sallen-Key topology to component tolerances is too high, so this topology cannot be recommended to use.<\/p>\n
  If relatively high output impedance is acceptable, an RC network can be added at the output to get a 5th order low-pass filter.<\/p>\n
  Unfortunately, I could not find a solution with Fully Differential Amplifier.<\/p>\n
References<\/h2>\n
  <\/p>\n
    \n
  1. Analog Devices. \u201cLinear Circuit Design Handbook\u201d. Chapter 8, \u201cAnalog Filters\u201d.<\/a><\/li>\n
  2. Nuhertz Technologies, Active Filter Module.<\/a><\/li>\n
  3. \u00abidealCircuit\u00bb, a simulator.<\/a><\/li>\n
  4. \u00abFilter Designer\u00bb, a multistage analog active filter design tool for Android<\/a><\/li>\n
  5. \u00abCircuit Calculator\u00bb, an electronics circuit design tool for Android.<\/a><\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n
    <\/div>\n

     \u0441\u0441\u044b\u043b\u043a\u0430 \u043d\u0430 \u043e\u0440\u0438\u0433\u0438\u043d\u0430\u043b \u0441\u0442\u0430\u0442\u044c\u0438      https:\/\/habr.com\/ru\/articles\/559714\/<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"
    <\/div>\n
    \n
    \n
    \n
    \"Amateur<\/p>\n
      The idea to build a 4th order low-pass filter looks simple: add one more feedback loop. But there are pitfalls, as always.  <\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[],"tags":[],"class_list":["post-394927","post","type-post","status-publish","format-standard","hentry"],"_links":{"self":[{"href":"https:\/\/savepearlharbor.com\/index.php?rest_route=\/wp\/v2\/posts\/394927","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/savepearlharbor.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/savepearlharbor.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/savepearlharbor.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/savepearlharbor.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=394927"}],"version-history":[{"count":0,"href":"https:\/\/savepearlharbor.com\/index.php?rest_route=\/wp\/v2\/posts\/394927\/revisions"}],"wp:attachment":[{"href":"https:\/\/savepearlharbor.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=394927"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/savepearlharbor.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=394927"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/savepearlharbor.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=394927"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}