Subscribe Recommend		Low-Sensitivity Sallen-Key Filter Design with the HP-41C/CV/CX Programmable Calculator This program, which I originally wrote for the HP-67 calculator, addresses the problem of designing second order single op-amp low- and high-pass filters using the Sallen-Key topology. The Sallen-Key filter topology has the advantage of using a minimum of components. The simplest Sallen-Key filters use only two resistors and two capacitors. Additional resistors may be added for input attenuation (low-pass only) and gain adjustment. The following schematics illustrate these generalized Sallen-Key circuits: Generalized Sallen-Key low-pass filter Generalized Sallen-Key high-pass filter The equations governing the low-pass filter are as follows, where f is the filter's cut-off frequency, Q is its "quality", and H is its gain at the cut-off frequency. The corresponding equations for the high-pass filter are: The mathematically inclined will notice that in each case, there are three equations in either nine or ten variables. Thus there is no single "right" solution. At least six or seven of the variables have to be decided arbitrarily (f, Q, H, and three or four component values), at which point the remaining variables (component values) can be solved for. An article by Texas Instruments suggests a number of simplifications to help one choose component values, but this just adds the complication of which simplification to choose. I recently came across a pair of application notes by National Semiconductor which gives a procedure for designing Sallen-Key filters to minimize the effect of component value tolerances on the performance of the filter. A side-effect of this procedure is to reduce the number of inputs to five: f, Q, H, RF, and R, where the latter is simply an indication of the magnitude of resistor values desired for R1, R2, and R3. The procedure then dictates how all the other values are chosen, even adjusting for available "real-world" values part way through the solution. The program presented here implements this procedure, with some minor changes: Instead of asking for a desired resistor magnitude, R, the program asks for a capacitor magnitude, C, since in my experience, the capacitors drive the design. The formula given for internal gain variable K in the procedure (please refer to the application note) seems to have been derived empirically, and has a jump at Q = 1.1. To make the formula simpler to implement, I modified it slightly to: The graphs of the original (pink) and revised (blue) formulae show the difference. Testing has shown that the resulting solution is generally at most one real-world capacitor value increment different (with corresponding changes in resistor values of course). Also, if H > K, then K is set equal to H, since otherwise it will not be possible to achieve the desired gain. For the low-pass circuit, the desired gain, H, is achieved by a combination of input attenuation, α (controlled by R1 and R3), and the internal gain, K, of the filter (controlled by RF and RG). The division of this gain between the two stages depends on H and Q and is chosen to minimize the sensitivity of the circuit to component tolerances. For example, for H = 1 and Q = 2, the attenuator gain is α = 0.629 and the internal gain is K = 1.59, for a net gain of H = αK = 1. The high-pass circuit has no attentuation stage, so H must be at least 1, and higher for Q > 0.917. If too low a value is entered for H, it is increased as necessary to make the circuit solvable. For either circuit, to achieve a result with minimum component sensitivity without regard to gain, set H = 0. The program will automatically choose the optimal value for K (and thus H). The resulting gain will be output when determining the performance. Using the Program First type in the program and save it, or read it from a previously recorded magnetic card. The card should be labelled as follows: LOW-SENSITIVITY SALLEN-KEY FILTER DESIGN f Q H C RF LP→RGC1C2R1R2R3   HP→RGC1C2R1R2   →f,Q,H Filter Design from Specifications Example: Design a 500Hz low-pass unity-gain filter with a Q of 2, using capacitors in the 10nF range, and a 47kΩ resistor for RF: Description Keystrokes Display Use engineering notation      ENG 2   0.00  00  Enter f  500       a   500.  00  Enter Q  2       b   2.00  00  Enter H  1       c   1.00  00  Enter C  10 EEx CHS 9       d   10.0 -09  Enter RF  47 EEx 3       e   47.0  03  Compute RG  A   79.5  03  Enter real-world RG and compute C1  82 EEx 3   R/S   31.6 -09  Enter real-world C1 and compute C2  33 EEx CHS 9   R/S   3.16 -09  Enter real-world C2 and compute R1  3.3 EEx CHS 9   R/S   15.4  03  Enter real-world R1 and compute R2  15 EEx 3   R/S   95.9  03  Enter real-world R2 and compute R3  100 EEx 3   R/S   26.1  03  Enter real-world R3  27 EEx 3   R/S   27.0  03  Notes If a resistor is to be omitted (open circuit), this program displays a value of zero for the resistance. This is different than some of my other programs, which display a "large" value representing infinity. When the value for RG is displayed as zero, meaning it can be omitted, the value of RF will not matter any more, and RF can be replaced by a direct connection. Filter Performance from Chosen Components During the calculation of the solution above, we've entered real-world values in response to each computed value. The real-world values of C1 and C2 are used when computing the values of R1, R2, and R3. However, the real-world values of each of those resistors does not affect the computed value of the remaining ones. Thus, the final filter may not perform exactly as specified. To find out how it does perform, follow these steps: Description Keystrokes Display Compute resulting f  E   491.  00  Compute resulting Q  R/S   1.86  00  Compute resulting H  R/S   1.01  00  A High-Pass Example Using the parameters already entered for the low-pass filter above, determine the components for a high-pass filter: Description Keystrokes Display Compute RG for high-pass filter  C   79.5  03  Enter real-world RG and compute C1  82 EEx 3   R/S   31.4 -09  Enter real-world C1 and compute C2  33 EEx CHS 9   R/S   3.18 -09  Enter real-world C2 and compute R1  3.3 EEx CHS 9   R/S   9.59  03  Enter real-world R1 and compute R2  10 EEx 3   R/S   97.0  03  Enter real-world R2  100 EEx 3   R/S   0.00  00  Now determine the predicted actual performance: Description Keystrokes Display Compute resulting f  E   482.  00  Compute resulting Q  R/S   1.96  00  Compute resulting H  R/S   1.57  00  Notice that H is higher than the specified unity gain. This is because the filter is not possible to construct with unity gain when Q = 2. The smallest possible gain is H = 1.59 (which due to real-world components, has become Q = 1.89 and H = 1.57). To achieve H = 1, you will need either a pre-attenuator with low output impedance, or a post-attenuator with high input impedance. Program Listing Line Instruction Comments 01♦   LBL "SK"    02♦   LBL a  Enter and store f 03  STO 11    04  RTN    05♦   LBL b  Enter and store Q 06  STO 12    07  RTN    08♦   LBL c  Enter and store H (gain) 09  STO 13    10  RTN    11♦   LBL d  Enter and store C (capacitor scale) 12  STO 00    13  RTN    14♦   LBL e  Enter and store RF 15  STO 04    16  RTN    17♦   LBL A  Low-pass filter: RG,C1,C2,R1,R2,R3 18  CF 00    19  GTO 00    20♦   LBL C  High-pass filter: RG,C1,C2,R1,R2 21  SF 00    22♦   LBL 00  Forward solution 23  RCL 12    24  2.2    25  ×    26  .9    27  −    28  RCL 12    29  .2    30  +    31  ÷  (2.2Q-0.9)/(Q+0.2) 32  1    33  x≤y?    34  x↔y    35  STO 15  K = max(1,(2.2Q-0.9)/(Q+0.2)) 36  RCL 13    37  x>y?    38  STO 15  K = max(H,1,(2.2Q-0.9)/(Q+0.2)) 39  RCL 15    40  ÷  H/K 41  x=0?    42  1  Use α = 1 if H/K = 0 (because H was 0) 43  FS? 00    44  1  Always use α = 1 for a high-pass filter 45  STO 14  α = H/K (always 1 for a high-pass filter or H = 0) 46  RCL 04    47  RCL 15    48  1    49  −    50  x≠0?    51  ÷  RG = RF/(K-1) if K ≠ 1, or zero if K = 1 52  R/S  Display RG and let user change it 53  STO 05    54  .1  Initialize n to √0.1; 55  √x    56  XEQ 07  n(1+√(1+4Q2(1+n2)(K-1)))/(2Q(1+n2)) 57  RCL 08  √0.1 was stored here by subroutine 7 58  x≤y?    59  x↔y  max(n, n(1+√(1+4Q2(1+n2)(K-1)))/(2Q(1+n2))) 60  FS? 00  High-pass filter? 61  XEQ 06  2nQ/(1+√(1+4Q2(K-1-n2))) 62  STO 08    63  RCL 00    64  RCL 08    65  ÷  C1 = C/n 66  R/S  Display C1 and let user change it 67  STO 06    68  RCL 08    69  RCL 00    70  ×  C2 = nC 71  R/S  Display C2 and let user change it 72  STO 07    73  RCL 06    74  ×    75  √x  √(C1C2) 76  XEQ 04  Multiply by 2πf and take reciprocal 77  STO 09  N = 1/(2πf√(C1C2)) 78  RCL 07    79  RCL 06    80  ÷    81  √x  n = √(C2/C1) 82  XEQ 03  2nQ/(1+√(1+4Q2(K-1-n2))) or n(1+√(1+4Q2(1+n2)(K-1)))/(2Q(1+n2)) 83  STO 08    84  RCL 09    85  ×    86  STO 03  (R1||R3) = nN 87  RCL 14    88  ÷  R1 = (R1||R3)/α 89  R/S  Display R1 and let user change it 90  STO 01    91  RCL 09    92  RCL 08    93  ÷  R2 = N/n 94  R/S  Display R2 and let user change it 95  STO 02    96  RCL 03    97  1    98  RCL 14    99  −  ( 1-α (R1||R3) ) 100  x≠0?    101  ÷  R3 = (R1||R3)/(1-α) if α ≠ 1, or zero otherwise 102  R/S  Display R3 and let user change it 103  STO 03    104  RTN    105♦   LBL 04  Multiply by 2πf and take reciprocal 106  RCL 11  f 107  ×    108♦   LBL 01  Multiply by 2π and take reciprocal 109  2    110  ×    111  π    112  ×    113  1/x    114  RTN    115♦   LBL 03  Compute either 2xQ/(1+√(1+4Q2(K-1-x2))) or x(1+√(1+4Q2(1+x2)(K-1)))/(2Q(1+x2)) 116  FS? 00  High-pass filter? 117  GTO 07    118♦   LBL 06  Subroutine to compute 2xQ/(1+√(1+4Q2(K-1-x2))) 119  STO 08  Save x for later use 120  RCL 12    121  2    122  ×  ( 2Q x ) 123  ×  ( 2xQ ) 124  LASTx  ( 2Q 2xQ ) 125  x2  ( 4Q2 2xQ ) 126  RCL 15    127  1    128  −    129  RCL 08    130  x2    131  −  ( K-1-x2 4Q2 2xQ ) 132  ×  ( 4Q2(K-1-x2) 2xQ ) 133  XEQ 09  ( 1+√(1+4Q2(K-1-x2)) 2xQ ) 134  ÷    135  RTN    136♦   LBL 07  Subroutine to compute x(1+√(1+4Q2(1+x2)(K-1)))/(2Q(1+x2)) 137  STO 08  Save x in register 8 for later use both by this subroutine and the caller 138  XEQ 08  ( 2Q 1+x2 ) 139  x2    140  ×  ( 4Q2(1+x2) ) 141  RCL 15    142  1    143  −  ( K-1 4Q2(1+x2) ) 144  ×  ( 4Q2(1+x2)(K-1) ) 145  XEQ 09  ( 1+√(1+4Q2(1+x2)(K-1)) ) 146  RCL 08    147  ×  ( x(1+√(1+4Q2(1+x2)(K-1))) ) 148  RCL 08    149  XEQ 08  ( 2Q 1+x2 x(1+√(1+4Q2(1+x2)(K-1))) ) 150  ×  ( 2Q(1+x2) x(1+√(1+4Q2(1+x2)(K-1))) ) 151  ÷    152  RTN    153♦   LBL 08  Subroutine to populate stack with 2Q 1+x2 154  x2    155  1    156  +    157  RCL 12    158  2    159  ×    160  RTN    161♦   LBL 09  Subroutine to compute 1+√(1+x) 162  1    163  +    164  √x    165  1    166  +    167  RTN    168♦   LBL E  Compute actual f, Q, and Gain 169  XEQ 02  R1 or (R1||R3) 170  RCL 06    171  ×    172  STO 08  Save R1C1 for use in calculating Q 173  RCL 02    174  RCL 07    175  ×    176  STO 09  Save R2C2 for use in calculating Q 177  ×    178  √x    179  XEQ 01  Multiply by 2π and take reciprocal 180  STO 10  Save actual f for use in calculating Q 181  R/S  Display actual f 182  RCL 09  ( R2C2 ) 183  RCL 08  ( R1C1 R2C2 ) 184  FS? 00  High-pass filter? 185  x↔y  ( R2C2 R1C1 ) 186  1    187  RCL 15    188  −    189  ×    190  +    191  XEQ 02  R1 or (R1||R3) 192  RCL 07    193  ×    194  +    195  RCL 10  Recall actual frequency 196  ×    197  XEQ 01  Multiply by 2π and take reciprocal 198  R/S  Display actual Q 199  RCL 04    200  RCL 05    201  x≠0?    202  ÷    203  1    204  +    205  XEQ 02  R1 or (R1||R3) 206  ×    207  RCL 01    208  ÷    209  RTN  Return actual Gain 210♦   LBL 02  Return either R1 or (R1||R3) 211  RCL 01    212  FS? 00  High-pass filter? 213  RTN  Return just R1 214  1/x    215  RCL 03    216  x≠0?  R3 exists? (0 means not) 217  1/x    218  +    219  1/x    220  RTN    Registers and Flags Register Use  00  C - capacitor scale  01,02,03  R1, R2, R3 - filter resistors  04,05  RF, RG - feedback resistors  06,07  C1, C2 - capacitors  08  n - variable used during computation  09  N - variable used during computation  11  f - cutoff frequency  12  Q - filter quality  13  H - overall gain  14  α - input attenuator gain  15  K - internal gain  10  Temporary register Flag Meaning  00  High-pass filter Revision History 2008-Dec-21 — Initial release. References 1. Analysis of the Sallen-Key Architecture (Rev.B), Texas Instruments Application Report SLOA024B, James Karki, 2002 2. Low-Sensitivity, Lowpass Filter Design, National Semiconductor Application Note OA-27, Kumen Blake, 1996 3. Low-Sensitivity, Highpass Filter Design, National Semiconductor Application Note OA-29, Kumen Blake, 1996 Other HP Calculator Programs I've written programs for many of the HP calculators calculators in my collection. You may be interested in some of these: HP 35s Curve Fitting Matrix Multi-Tool HP-41C/CV/CX Op-Amp Gain and Offset Stage Design Op-Amp Oscillator Design HP-67 Low-Sensitivity Sallen-Key Filter Design Op-Amp Gain and Offset Stage Design Op-Amp Oscillator Design Resistor Networks of 1 or 2 Nodes     Buy Stefan a coffee! If you've found this article useful, consider leaving a donation to help support stefanv.com   Last updated Sunday December 21, 2008. E-mail Stefan   Disclaimer: Although every effort has been made to ensure accuracy and reliability, the information on this web page is presented without warranty of any kind, and Stefan Vorkoetter assumes no liability for direct or consequential damages caused by its use. It is up to you, the reader, to determine the suitability of, and assume responsibility for, the use of this information. Copyright: All materials on this web site, including the text, images, and HTML mark-up, are Copyright © 2009 by Stefan Vorkoetter unless otherwise noted. All rights reserved. Unauthorized duplication prohibited. You may link to this site or pages within it, but you may not link directly to images on this site, and you may not copy any material from this site to another web site or other publication without express written permission. You may make copies for your own personal use.