A Comparison of Tolerance Analysis Methods by Steven M. Sandler AEi Systems, LLC. We have seen many methods of calculating the worst case tolerance limits for electronic circuits. The intent of this paper is to demonstrate several different methods, and determine the results, and the corresponding confidence factors for each method. The calculation methods addressed, and a brief description of each method is shown below: 1. Extreme Value Analysis - Each component is varied in the direction of the sensitivities to obtain the absolute worst case values of the circuit performance. 2. EVA Sensitivity Analysis - The parameter sensitivities are computed by evaluating the derivative of the output with respect to each component. The algebraic sum of the individual component tolerances (ie temperature, radiation, initial..) is multiplied by the sensitivity to determine the voltage variance for the component. The voltage variances of each part are summed algebraically to obtain the worst case circuit performance. 3. RSS Sensitivity 1 - The parameter sensitivities are computed at the nominal values. sum of each individual component tolerance (ie temperature, radiation, initial..) is multiplied by the sensitivity to determine the voltage variance for the component. The square root of the sum of the squares of each voltage variance is defined as the worst case circuit performance 4. RSS Sensitivity 2 - The parameter sensitivities are computed at the nominal values. The square root of the sum of the squares of each individual component tolerance (ie temperature, radiation, initial..) is multiplied by the sensitivity to determine the voltage variance for the component. The square root of the sum of the squares of each voltage variance is defined as the worst case circuit performance. 5. Monte Carlo - The component tolerances are algebraically added and entered into a SPICE simulator. The simulator randomly selects component values within the specified tolerance range, following a 12 point gaussian distribution. The results of the simulation include the population standard deviation, the population mean and normally, the 3 sigma limits for the worst case circuit performance. Copyright © 1998 – AEi Systems, LLC All Rights Reserved Proprietary and Confidential Page 1 A simple circuit was selected to apply each of these methods to The confidence level of each approach is defined later in this article in order to compare the results from each method. The circuit selected for this example is an LM117 linear regulator circuit. The schematic of the circuit is shown in figure 1. LM117 OUT IN ADJUST LOAD INPUT 124 374 Figure 1 - Simple Evaluation Circuit For this simple case the following symmetrical tolerances are defined for each part: Component Tolerances Part R1 R2 LM117 Rout LM117 Ref LM117 Iadj Load Current 124 374 .00625 1.25 55 uA 0.75A Copyright © 1998 – AEi Systems, LLC All Rights Reserved initial 1.00% 1.00% 100.00% 4.00% 7.00E-05 100.00% temp 0.19 0.19 0 2 0 0 Proprietary and Confidential age 0.05 0.05 0 2.64 0 0 Total 1.24% 1.24% 100.00% 8.64% 7.00E-05 100.00% Page 2 Extreme Value Analysis nominal voltage and sensitivity calculations R2 374 V out dIadj dR1 dR2 R1 Vref 124 I o. R o . 1 1.25 R2 R1 R2 R2 Vref I o. R o . 2 R1 Vref I o. R o R1 I o. 1 R2 dIo R o. 1 R2 1 Iadj Iadj. R2 6 55. 10 Ro .00625 Io 0.75 V out = 5.022 dIadj = 374 dRo dVref Vref Iadj R1 R1 R2 R1 Copyright © 1998 – AEi Systems, LLC All Rights Reserved dR1 = 0.03 dR2 = 0.01 dRo = 3.012 dIo = 0.025 dVref = 4.016 Proprietary and Confidential Page 3 Extreme Value Worst Case Maximum and Minimum Voltages Maximum Voltage R2 R1 V out 374. 1 124. 1 1.24 100 1.24 100 1.25. 1 Vref Ro .00625. 1 I o. R o . 1 Vref R2 R1 8.64 100 Iadj 100 100 Io ( 55 70 ) . 10 0.75. 1 6 100 100 Iadj. R2 Extreme Value Minimum output voltage V out = 5.604 Minimum Voltage R2 R1 Vout 374. 1 124. 1 1.24 100 1.24 Vref 100 Vref Ro I o. R o . 1 1.25. 1 .00625. 1 R2 R1 8.64 100 Iadj 100 100 Io ( 55 70 ) . 10 0.75. 1 6 100 100 Iadj. R2 Extreme Value Minimum output voltage Vout = 4.423 Copyright © 1998 – AEi Systems, LLC All Rights Reserved Proprietary and Confidential Page 4 EVA Sensitivity Analysis Part R1 R2 ROUT VREF IADJ ILOAD Vnominal Value Sensitivity 1.24E+02 -3.03E-02 3.74E+02 1.01E-02 6.25E-03 -3.05E+00 1.25E+00 4.02E+00 5.50E-05 3.74E+02 7.50E-01 -2.51E-02 Relative -3.76E-02 3.78E-02 -1.91E-04 5.02E-02 2.06E-04 -1.88E-04 initial 1.00% 1.00% 100.00% 4.00% 7.00E-05 100.00% 5.023 temp 0.19 0.19 0 2 0 0 EVA Tol age 0.05 0.05 0 2.64 0 0 Tol Abs Value 1.24% 0.047 1.24% 0.047 100.00% 0.019 8.64% 0.434 7.00E-05 0.026 100.00% 0.019 0.591 Volts RSS1 Sensitivity Analysis Part R1 R2 ROUT VREF IADJ ILOAD Vnominal Value 1.24E+02 3.74E+02 6.25E-03 1.25E+00 5.50E-05 7.50E-01 Sensitivity -3.03E-02 1.01E-02 -3.05E+00 4.02E+00 3.74E+02 -2.51E-02 Relative -3.76E-02 3.78E-02 -1.91E-04 5.02E-02 2.06E-04 -1.88E-04 initial 1.00% 1.00% 100.00% 4.00% 7.00E-05 100.00% temp 0.19 0.19 0 2 0 0 age 0.05 0.05 0 2.64 0 0 Tol Abs Value 1.24% 0.047 1.24% 0.047 100.00% 0.019 8.64% 0.434 7.00E-05 0.026 100.00% 0.019 5.023 RSS Tol 0.440 Volts RSS2 Sensitivity Analysis Part R1 R2 ROUT VREF IADJ ILOAD Vnominal Value 1.24E+02 3.74E+02 6.25E-03 1.25E+00 5.50E-05 7.50E-01 Sensitivity -3.03E-02 1.01E-02 -3.05E+00 4.02E+00 3.74E+02 -2.51E-02 Relative -3.76E-02 3.78E-02 -1.91E-04 5.02E-02 2.06E-04 -1.88E-04 5.023 Copyright © 1998 – AEi Systems, LLC All Rights Reserved initial 1.00% 1.00% 100.00% 4.00% 7.00E-05 100.00% temp 0.19 0.19 0 2 0 0 RSS2 Tol Proprietary and Confidential age RSS Tol RSS Row 0.05 1.02% 3.83E-02 0.05 1.02% 3.85E-02 0 100.00% 1.91E-02 2.64 5.19% 2.61E-01 0 7.00E-05 2.62E-02 0 100.00% 1.88E-02 0.268973 Volts Page 5 SPICE Monte Carlo Analysis The SPICE circuit for this circuit is shown in figure 2. 1 ROUT 6.2M V(5) VOUT 5.03 VREF 1.25 5 5.02 R1 124 6 3 3.78 3.78 IADJ 55U ILOAD .75 R2 374 E1 10000 Figure 2 Spice Schematic Spice Monte Carlo Netlist F:\TEMP\lm117 .OP .TRAN 1U 100U .PRINT TRAN V(5) *ALIAS V(5)=VOUT VREF 1 3 1.25 TOL=8.64% RTOP 5 6 124 TOL=1.24% RBOT 6 0 374 TOL=1.24% IADJ 0 6 55U TOL=70U ROUT 1 5 6.25M TOL=6.25M ILOAD 5 0 .75 TOL=.75 E1 3 0 3 6 10000 .END Copyright © 1998 – AEi Systems, LLC All Rights Reserved Proprietary and Confidential Page 6 Monte Carlo Results The results of 100 cases, performed in a Monte Carlo SPICE simulation, are shown below. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 5.1924 5.0099 5.107 5.1488 5.0898 4.9969 5.0128 5.3198 4.8813 5.099 5.1541 5.3301 4.915 5.1745 5.1916 4.9378 4.9538 4.7793 5.0798 5.2465 4.7666 5.056 5.056 5.2808 4.9699 4.9986 4.9667 4.9621 5.1424 5.0145 4.9698 5.0635 4.7392 4.6917 5.0959 5.0351 4.8466 4.987 5.207 Copyright © 1998 – AEi Systems, LLC All Rights Reserved 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 5.1717 5.1344 4.9889 5.0089 4.8358 4.8783 4.9972 5.0934 4.8432 5.1633 5.0917 4.7834 4.9136 4.8822 4.9711 5.1044 5.0693 5.0622 5.3137 4.8554 5.1819 4.9533 5.1175 4.9699 5.0142 5.2482 5.2886 5.0802 4.9757 5.0127 4.9546 5.0153 5.1374 5.1161 5.0907 4.9393 5.1218 4.9501 5.1876 Proprietary and Confidential 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 Mean Pop Stdev 5.1839 5.0727 5.1752 4.8133 4.9281 5.0251 4.8478 4.8168 4.9089 5.1033 5.0903 4.9508 5.1078 4.7521 5.1305 4.8451 5.2143 4.9566 5.3247 4.758 5.1846 4.6512 5.032088 0.14441 Page 7 Monte Carlo Histogram Results 40.00 Number of Number per Cell 30.00 20.00 10.000 x 4.579 < 1.000 x 5.397 < 0 > > 1 0 4.434 4.582 4.731 4.880 5.028 5.177 5.325 5.474 5.623 .5 Sigma Cells ∆ x = 817.3M ∆ y = -1.000 Copyright © 1998 – AEi Systems, LLC All Rights Reserved Proprietary and Confidential Page 8 Statistical Evaluation The results of the Monte Carlo Analysis yield a population mean and a population standard deviation. In order to determine the Extreme Value Worst Case circuit performance we need to select a confidence level. Selecting a confidence level of 0.99998, meaning that we have 99.998 percent confidence that any device will remain within these limits, we can define the number of standard deviations from the mean. EXCEL was used to compute the confidence results. The resulting number of standard deviations for a confidence level of 0.99998 is 4.265. The resulting minimum and maximum values can be computed as V min = mean − (4.265 * σ ) V max = mean + (4.265 * σ ) The results of the Monte Carlo analysis provided a population mean of 5.032 volts with a population standard deviation of 0.14441 volts. This results in a maximum of 5.648 volts and a minimum of 4.416 volts. Comparative Results The table below shows the mean, minimum, and maximum output voltages, the effective number of standard deviations from the mean and the respective confidence level. Method Extreme Value EVA Sensitivity RSS1 RSS2 Monte Carlo Mean 5.013 5.023 5.023 5.023 5.032 Minimum 4.423 4.432 4.583 4.754 4.416 Maximum 5.604 5.614 5.463 5.292 5.648 # of STDEV Confidence 4.414* 4.092 3.048 1.863 4.265 100% 99.996 99.770% 93.750% 99.998 * 4.414 yields a confidence of 99.999% Conclusions Different circuits will result in different tolerances. This paper merely demonstrates the relative performance of each of the methods. It does show that Monte Carlo may be a Copyright © 1998 – AEi Systems, LLC All Rights Reserved Proprietary and Confidential Page 9 reasonable method of determining the Extreme Value performance, if it is combined with a confidence level for the result. Copyright © 1998 – AEi Systems, LLC All Rights Reserved Proprietary and Confidential Page 10