Subscribe Recommend		Op-Amp Gain and Offset Design with the HP-41C/CV/CX Programmable Calculator This is a program I originally wrote for the HP-67 calculator, and then ported to the HP-41C series. This program is for designing offset-and-gain stages using a single operational amplifier. Such stages are often necessary to convert an input signal covering one range of voltages (e.g., 0.1V to 0.2V from a sensor) to an output signal covering a different range (e.g., 1.0 to 4.0V into an A/D converter). Mathematically, such a stage performs a linear transformation on the input voltage, VOUT = m VIN + b where m is the slope and b is the intercept or offset. Given an input voltage range, VIL to VIH, and an output voltage range, VOL to VOH, the slope and offset are given by: There are four main cases to consider, with different circuits for each, that are addressed by this program: Positive slope and offset (m > 0 and b > 0) Positive slope and negative offset (m > 0 and b < 0) Negative slope and positive offset (m < 0 and b > 0) Negative slope and offset (m < 0 and b < 0) Positive Slope and Offset A positive slope and offset stage is implemented by the following circuit: Positive gain, positive offset amplifier The designer must select values for R1 and RF, and then appropriate values for R2 and RG can be calculated using the following formulae: After determining theoretically ideal values for R2 and RG, real-world values can be chosen and the following formula applied to VIL and VIH to see the resulting values of VOL and VOH respectively: This case is also used to handle the special case where b = 0 (postive gain with no offset). In such a case, the formula for R2 would result in dividing by zero, which means R2 is infinite. In other words, R2 and VREF are not needed. The value of R1 won't matter then either, and can be replaced by a direct connection. The circuit reduces to: Positive gain, zero offset amplifier Positive Slope and Negative Offset The following circuit implements a positive slope and negative offset stage: Positive gain, negative offset amplifier After choosing values for R1 and RF, values for R2 and RG can be be calculated using these formulae: The following formula can then be used to determine the effect of real-world values of R2 and RG on the transformation: Negative Slope and Positive Offset A negative slope and postive offset stage is implemented by the following circuit: Negative gain, positive offset amplifier Given values for R1 and RF, values for R2 and RG can be be calculated as follows: The following formula can then be used to determine the effect of substituting real-world values of R2 and RG: This case is also used to handle the special case where b = 0 (negative gain with no offset). As in case 1, the formula for R2 would involve division by zero, so R2 and VREF are not needed. The non-inverting input of the op-amp can be connected directly to ground, giving the following circuit: Negative gain, zero offset amplifier Negative Slope and Offset The following circuit implements a negative slope and offset stage: Negative gain, zero offset amplifier This circuit has no R1, so it is only necessary to choose a value for RF, after which R2 and RG are given by: The effect of using real-world values of R2 and RG can then be tested using this formula: Limitations Theoretically, the formulae presented here work perfectly well for gains between -1 and 1 (i.e. |m| < 1). However, many real-world op-amps are unstable in such cases. Instead, it will usually be necessary to design for a higher gain (|m| ≥ 1), with an attenuator on the input side. The design procedures for this are described on Texas Instruments' Op Amp Gain and Offset Page. 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: OP-AMP GAIN AND OFFSET STAGE DESIGN VREF VIL,VIH VOL,VOH     →m,b →CASE R1,RF→R2,RG R2,RG→VOL,VOH   Forward Solution: Finding R2 and RG Consider the following example. A sensor has an output ranging from 0.5V to 0.7V, and we want to interface it to an A/D converter that is expecting an input between 1V and 4V. There is no reference voltage available other than the well regulated 5V supply voltage of the circuit. Use a 10kΩ resistor for R1, and 100kΩ for RF. Follow these steps to solve the problem: Description Keystrokes Display Use engineering notation      ENG 2   0.00  00  Enter VREF  5       a   5.00  00  Enter VIL and VIH  0.5 ENTER 0.7       b   500. -03  Enter VOL and VOH  1 ENTER 4       c   1.00  00  Compute slope m  A   15.0  00  Compute offset b  R/S  -6.50  00  Determine case number  B   2.00  00  Enter R1 and RF, compute R2  10 EEx 3 ENTER 100 EEx 3   C   1.02  03  Compute RG  R/S   6.21  03  Notes It is not necessary to compute the slope and offset (by pressing A) before determining the case number. Likewise, it isn't necessary to determine the case number (by pressing B) before computing resistor values (although you'll want to know the case number in order to know which circuit to build). The program keeps track of which information is up to date, and will (re)compute anything that it needs that hasn't already been computed. Reverse Solution: Finding the Effect of R2 and RG on VOL and VOH The closest available 5% resistor values for R2 and RG are 1kΩ and 6.2kΩ respectively. What effect does using these have on the solution? Follow these steps to find out: Description Keystrokes Display Enter R2 and RG, compute VOL  1 EEx 3 ENTER 6.2 EEx 3   D   1.13  00  Compute VOH  R/S   4.15  00  This is within the A/D's input range at the lower bound, but outside the range at the upper bound. What happens if we use the next available value for RG, 6.8kΩ, instead? Description Keystrokes Display Enter R2 and RG, compute VOL  1 EEx 3 ENTER 6.8 EEx 3   D   1.09  00  Compute VOH  R/S   3.88  00  This is almost centered within the desired output range, and covers 93% of it. The only remaining concern is how component tolerances might affect the solution. This can be analyzed by trying different combinations of R1, R2, RF, and RG representing resistors that are maximally out of tolerance (±5%) in each direction. For example, to test the case where R1 and RF are 5% low and R2 and RG are 5% high, follow these steps: Description Keystrokes Display Enter low R1 and RF, ignore computed R2  0.95 EEx 3 ENTER 95 EEx 3   C   920.  00  Enter high R2 and RG, compute VOL  1.05 EEx 3 ENTER 7.14 EEx 3   D   808. -03  Compute VOH  R/S   3.48  00  The results show that in this case, the lower limit of the output is out of range, meaning either a redesign is necessary, or tighter tolerance components are needed. Cases where b = 0 In cases where the offset, b, is zero, the program will instead use b = 10-9 as the offset. This will avoid any division-by-zero errors. The program will then use case 1 (if m > 0) or case 3 (if m < 0) to compute the solution. In both cases, the computed value for R2 will be very large, typically around 109 times the value specified for R1. This indicates that R2 and VREF can be omitted, and that R1 can be replaced by a direct connection. Program Listing Line Instruction Comments 01♦   LBL "GOFF"    02♦   LBL a  Enter available reference voltage 03  STO 09    04  RTN    05♦   LBL b  Enter VIL and VIH 06  STO 06  Store VIH 07  x↔y    08  STO 05  Store VIL 09  CF 00  Must recompute m and b 10  RTN    11♦   LBL c  Enter VOL and VOH 12  STO 08  Store VOH 13  x↔y    14  STO 07  Store VOL 15  CF 00  Must recompute m and b 16  RTN    17♦   LBL A  Compute transfer function slope (m) and intercept (b) 18  FS? 00  Are m and b already up to date? 19  GTO 05    20  RCL 08  Compute and store m = (VOH - VOL) ÷ (VIH - VIL) 21  RCL 07    22  −    23  RCL 06    24  RCL 05    25  −    26  ÷    27  STO 20    28  RCL 05  Compute and store b = VOH - m VIL 29  ×    30  RCL 07    31  −    32  CHS    33  x≠0?    34  GTO 07    35  1 E-9  Make sure b is never exactly 0; 36♦   LBL 07    37  STO 21    38  SF 00  Stored m and b are now up to date 39  CF 01  Must recompute case number now 40♦   LBL 05  Display computed or already up-to-date m and b 41  RCL 20    42  RTN  Return, leaving m on stack 43  RCL 21  Display b if user pressed R/S after return 44  RTN    45♦   LBL B  Compute case number and store in I 46  XEQ A  Make sure m and b are up to date 47  FS? 01  Is case number already up to date 48  GTO 08    49  x<0?    50  GTO 06  Handle cases where m < 0 51  1    52  STO 25    53  RCL 21    54  x<0?    55  ISZ 25  Case 2: m positive and b negative 56  GTO 08  Case 1: m positive and b positive 57♦   LBL 06  Cases where m is negative 58  3    59  STO 25    60  RCL 21    61  x<0?    62  ISZ 25  Case 4: m negative and b negative 63♦   LBL 08  Case 3: m negative and b positive 64  SF 01  Case number is now up to date (or was already up to date) 65  RCL 25  Display case number and return 66  RTN    67♦   LBL C  Compute solution using appropriate case 68  STO 03  Store RF 69  x↔y    70  STO 01  Store R1 71  XEQ B  Get case number 72  GTO IND 25  Branch to appropriate case 73♦   LBL 01  Positive m and positive b 74  RCL 01  Compute and store R2 = VREF R1 m ÷ b 75  RCL 09    76  ×    77  RCL 20    78  ×    79  RCL 21    80  ÷    81  STO 02    82  R/S  Display R2; 9.99e99 means "open circuit" 83  RCL 09  Compute and store RG = VREF RF ÷ (VREF (m - 1) + b) 84  RCL 03    85  ×    86  XEQ 09  Compute VREF (m - 1) + b 87  GTO 05  Divide, store RG, and return, leaving RG on stack 88♦   LBL 02  Positive m and negative b 89  RCL 01  Compute and store R2 = -R1 b ÷ (VREF (m - 1) + b) 90  RCL 21    91  CHS    92  ×    93  XEQ 09  Compute VREF (m - 1) + b, leaving VREF (m - 1) in register 0 94  ÷    95  STO 02    96  R/S  Display R2 97  RCL 01  Compute and store RG = (R1 b + VREF RF) ÷ (VREF (m - 1)) 98  RCL 21    99  ×    100  RCL 09    101  RCL 03    102  ×    103  +    104  RCL 00  Subroutine 9 left VREF (m - 1) in register 0 for us 105  GTO 05  Divide, store RG, and return, leaving RG on stack 106♦   LBL 03  Negative m and positive b 107  RCL 01  Compute and store R2 = R1 (VREF (m - 1) + b) ÷ -b 108  XEQ 09  Compute VREF (m - 1) + b 109  ×    110  RCL 21    111  CHS    112  ÷    113  STO 02    114  R/S  Display R2 115  RCL 03  Compute and store RG = -RF ÷ m 116  CHS    117  RCL 20    118  GTO 05  Divide, store RG, and return, leaving RG on stack 119♦   LBL 04  Negative m and negative b 120  RCL 09  Compute and store R2 = VREF RF ÷ -b 121  RCL 03    122  ×    123  RCL 21    124  CHS    125  ÷    126  STO 02    127  R/S  Display R2 128  RCL 03  Compute and store RG = RF ÷ m 129  RCL 20    130  CHS    131♦   LBL 05  Common code to compute y ÷ x and store in RG 132  ÷    133  STO 04    134  RTN  Return, leaving RG on stack 135♦   LBL 09  Compute VREF (m - 1) + b, leaving VREF (m - 1) in register 0 136  RCL 20    137  1    138  −    139  RCL 09    140  ×    141  STO 00  Save VREF (m - 1) in register 0 for later 142  RCL 21    143  +    144  RTN    145♦   LBL D  Compute VOL,VOH using supplied R2,RG 146  STO 04  Store RG 147  x↔y    148  STO 02  Store R2 149  XEQ B  Get case number 150  RCL 05    151  XEQ IND 25  Compute VOL from VIL 152  R/S  Display computed VOL 153  RCL 06    154  GTO IND 25  Compute VOH from VIH 155♦   LBL 01  Compute VOUT for positive m and positive b 156  RCL 02    157  ×    158♦   LBL 00  Entry point to compute (x + VREF R1) (1 + RF ÷ RG) ÷ (R1 + R2) 159  RCL 09    160  RCL 01    161  ×    162  +    163  RCL 03    164  RCL 04    165  ÷    166  1    167  +    168  ×    169  RCL 01    170  RCL 02    171  +    172  ÷    173  RTN    174♦   LBL 02  Compute VOUT for positive m and negative b 175  RCL 01    176  1/x    177  RCL 02    178  1/x    179  +    180  1/x    181  RCL 04    182  +    183  STO 00  Store RG + R1 R2 ÷ (R1 + R2) for later 184  RCL 03    185  +    186  ×    187  RCL 00    188  ÷    189  RCL 09    190  RCL 02    191  ×    192  RCL 03    193  ×    194  RCL 01    195  RCL 02    196  +    197  RCL 00    198  ×    199  ÷    200  −    201  RTN    202♦   LBL 03  Compute VOUT for negative m and positive b 203  0    204  XEQ 00  Compute (VREF R1) (1 + RF ÷ RG) ÷ (R1 + R2) 205  GTO 09  Subtract VIN RF ÷ RG 206♦   LBL 04  Compute VOUT for negative m and negative b 207  RCL 09    208  RCL 03    209  ×    210  RCL 02    211  ÷    212  CHS    213♦   LBL 09  Entry point to compute x - y RF ÷ RG 214  x↔y    215  RCL 03    216  ×    217  RCL 04    218  ÷    219  −    220  RTN    Two interesting aspects of this program are its use of indirect addressing for branching to the forward and reverse solution subroutines for each of the four cases, and its use of repeated labels. The forward solution for each of the four slope/offset cases is implemented by a sequence of instructions starting with the label corresponding to the case number (1 to 4). When the user presses C, the case number is computed if necessary, and a GTO (i) instruction then branches to the appropriate case. Similarly, the reverse solution for each case is also labeled according to the case number. When the user presses D, VIL is recalled to the stack, after which a GSB (i) instruction causes VOL to be calculated. Then VIH is recalled, and GTO (i) is used to calculate VOH and then return. So, there are two each of LBL 1 through LBL 4. This works because the HP-41C/CV/CX (and other vintage HP calculators) search forwards from the current step for the matching label. Thus the GTO (i) in step 73 will branch to the appropriate forward solution, and the GSB (i) in step 152 and GTO (i) in step 155 will branch to the appropriate reverse solution. This program also makes use of many small subroutines to compute sub-expressions common to multiple solutions. Since there were not enough labels for all the subroutines needed, the same labels were used more than once. Had this not been done, the same sequence of steps would have been repeated several times, and the program would not have fit into the calculator's 224 step memory. Registers and Flags Register Use  00  Temporary register  01  R1 (Ohms)  02  R2 (Ohms)  03  RF (Ohms)  04  RG (Ohms)  05,06  VIL and VIH, input voltage range  07,08  VOL and VOH, output voltage range  09  VREF  11  m, slope of transfer function  12  b, intercept of transfer function  10  Case number (1,2,3,4) Flag Meaning  00  m and b are up to date  01  Case number is up to date Revision History 2009-May-26 — Initial release. 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 Low-Sensitivity Sallen-Key Filter 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 Tuesday May 26, 2009. 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. 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